The Airborne Microparticle || Inelastic Light Scattering

8 Inelastic Light Scattering

Rayleigh's light scattering studies of sulphur hydrosols provided the impetus forRaman's work on this system and ultimately to his discovery of the Raman effect.The circle is completed with the realization that the enhancement of Ramanscattering (SERS) which occurs when molecules are adsorbed upon metalhydrosols such as those studied by Faraday is caused by the very optical effectthat is responsible for the colors ofthese sols.

M. Kerker (1989)

8.1Introduction

If light interacts with matter without changing its frequency, the process is calledelastic scattering because the photons change only their direction and not theirenergy . The scattered light has the same frequency as the incident light. Rayleighscattering is one particular elastic scattering process. The key assumption inRayleigh's theory is that the scattering particles are small enough compared to thewavelength of the incident light to consider the electric field independent of spacewithin the particles, as was pointed out in Sect. 3.5. In most cases light scatteringby molecules can be considered to be Rayleigh scattering.

Scattering processes in which the interaction of the incident photons with thescattering particles (atoms, molecules) causes a change of direction of the photonsand a change in energy are called inelastic (scattering) processes. Ramanscattering, where the energy change is caused by interaction with the vibrationaland rotational movement of the scattering molecule, is such an inelastic scatteringprocess. We will not consider Raman processes related to transitions within theelectronic energy levels. Changes in the wavelength caused by interaction withelastic waves in the scattering medium constitute yet another inelastic scatteringprocess. We will discuss Raman scattering in considerable detail here because thisprocess is an important tool for the chemical characterization of microparticles.Although not everyone would consider fluorescence to be a scattering processbecause energy absorption is involved, we include fluorescence in ourconsiderations.

The emission, absorption, and scattering of light are due to the interaction oflight waves with bound electrons. These are electrons that cannot move freely inspace because they are bound by electric forces to the atomic nuclei. For an exactdescription of the interaction of light with matter quantum theory must be used,

E. J. Davis et al., The Airborne Microparticle© Springer-Verlag Berlin Heidelberg New York 2002

494 8 Inelastic LightScattering

but in a number of cases the microscopic interaction process can be condensedinto a material property, such as the dielectric constant, and the process can betreated by methods of continuum theory. This was the approach used quitesuccessfully in Chap. 3 to describe elastic scattering. We have shown that theelectric field of an electromagnetic wave induces a separation of charge in thescattering medium , so the originally electrically neutral particle becomespolarized. This polarization depends on the applied field and the dielectricconstant . In general, the dielectric constant associated with polarization is asecond rank tensor and depends on the amplitude, direction, and frequency of theincident field . It can also vary with time and position. However, a number ofmaterials can be considered to be linear, isotropic, and homogeneous, in whichcase the dielectric constant is assumed to be independent of the amplitude of thefield and does not depend on direction and position.

Continuum theory is not adequate to describe all details of inelastic scatteringprocesses, so we must turn to a molecular viewpoint to describe inelasticscattering. We first apply classical methods and then sketch the quantummechanical treatment.

The effect of an electric field on a molecule is very similar to that known fromcontinuum theory . The Coulomb forces of the electromagnetic wave act on eachmolecule, inducing a molecular dipole moment. The proportionality constantrelating the induced molecular dipole moment and the electric field is called the(molecular) polarizability and is usually designated by (J. Analogous to themacroscopic material property , the dielectric constant, the polarizability depends,in general, on the amplitude, direction, and frequency of the incident wave. It canalso vary with time and position. This is no surprise , because the microscopicquantity polarizability and the macroscopic quantity dielectric constant are closelyrelated. In a number of cases, the dependence on amplitude, direction , andfrequency can be neglected . We consider this case first, concentrating on theinvestigation of the effect of a time-dependent polarizability.

The time dependence of the polarizability is a result of the thermal motion ofthe nuclei . These nuclei oscillate and rotate about the center of gravity of themolecule, and the motion modulates the scattered light, resulting in the appearanceof new spectral lines not present in the incident beam. Such processes constituteinelastic scattering becau se the energy of the scattered photons is not preserved.We will see that classical electrodynamics cannot explain these processessatisfactorily. Consequently, the interaction of light with atoms and molecule s canbe treated robustly only by means of quantum mechanics. For this reason, weinclude an outline of the quantum mechanical treatment in this chapter, but arigorous quantum mechanical analysi s of scattering processes is beyond the scopeof this book. We will restrict our presentation to some basic concepts and presentsome quantum mechanical results without detailed derivation. We have no need toconfront the reader with the powerful but complicated mathematical apparatusnecessary for quantum mechanical calculations.

Raman scattering by microparticles differs in at least one essential aspect fromRaman scattering by bulk material. The induced dipole moment depends on thelocal field of the incident wave and the local field depends, in turn, on properties

8.2 Raman Scattering: Classical Description 495

of the particle such as size, shape and refractive index. The wave emitted by theinduced dipole moment is also affected by the particle. As a result, the Ramanscattered light depends in a rather complicated way on the size, shape, and opticalproperties of the scattering particle. Finally, spherical and ellipsoidal particles withlow absorptivity are optical resonators, and these resonators can have very highquality factors discussed in Sect. 3.11. On resonance, the transmitted field withinthe particle can rise dramatically over the non-resonant case. This is especiallyimportant for the excitation of nonlinear processes.

8.2Raman Scattering: Classical Description

In this section we show that modulation of the polarizability caused by thermalmotion results in the appearance of frequency-shifted lines. In the classicaltreatment of the scattering process, the source of light scattering is the oscillatingmolecular dipole moment induced by the incident radiation. The dipole oscillateswith the frequency of the incident radiation and emits radiation with the samefrequency , as considered in detail in Chap. 3.

The molecules are also subject to rotational and vibrational motions, whichcause periodic changes in the polarizability. The associated dipole radiation ismodulated, and frequency-shifted bands appear in the scattered light. Theseadditional bands are called Raman lines in honor of Chandrasekar V. Raman, whowas the first to observe them. The Raman lines are shifted to higher as well as tolower frequencies. Raman lines with lower frequencies than the incident radiationare called Stokes Raman lines", and those with higher frequencies are called antiStokes Raman lines. The scattered light flux is proportional to the fourth power ofthe scattered frequency as in elastic scattering. The induced dipole moment ofnon-spherical molecules depends on their orientation. If these molecules havearbitrary orientation , the scattered radiation is at least partially depolarized , whichis analogous to elastic scattering.

There are a number of good textbooks on Raman scattering (e.g., Herzberg1989,1991 ; Anderson 1971, 1973; Brandmliller and Moser 1962; Long 1977), anda more recent book edited by Schrader (1995) covers infrared spectroscopy as wellas Raman spectroscopy . We restrict ourselves to a presentation of the physicalprocesses only in sufficient detail to understand the most important properties ofRaman scattering by microparticles. The classical treatment is much easier tocomprehend than the formal quantum mechanical description, and many of theproperties of Raman scattering can be explained at least qualitatively by classicalmethods. Thus, we first present the electrodynamic analysis of elastic and inelasticscattering processes.

13 The designation band is used in place of line if Raman scattering extends over a broaderspectral range.


8.2.1Stokes and anti-Stokes Raman Scattering

The polari zability, a, which is a function of the coordinates, Xb of the nuclei,represents the response of a specific molecule to an external electrical field, E.This field induces a molecular dipole moment, p. If the molecules are notrotationally symmetric, a is a second rank tensor, and we can write the induceddipole moment as the vector product,

(8.1)14

Rotational and vibrational thermal motions change the internuclear distances,thereby changing the polarizability. Other phenomena also alter the polarizability.These include electronic motion, intermolecular colli sions, photon interaction,etc., but we will exclude them from our consideration, recognizing that they canexplain the appearance of additional frequency-shifted lines in the scatteringspectrum.

The effects of internal molecular motion on the polarizability caused byvibrations and rotations of the nuclei are weak , so we can represent thedependence of the polarizability on the coordin ates of the internuclear distancesby a Taylor series expansion about the equilibrium position. Thus, for therepresentative element <Xxxwe write

(8.2)

(8.3)

Here the vector (XbYk,Zk) represents the displacement of the kth-nucleus from itsequilibrium position. Quantities evalu ated at the equilibrium position are denotedby the subscript "0". The displacement is a function of time.

It is well known from classical dynamics that the motion of N centers of masscoupled by elastic forces can be represented as the superposition of harmonicoscillations in 3N - 6 distinct directions (3N - 5 directions for linear molecules) .These directions are called the normal coord inates, qb of the system. Expressingthe polarizability tensor in terms of the normal coordin ates yields

a xx=a~x +L(oaXX Jq~COS«(J)kt+~\)+ ...k Oqk

The frequencies of the molecular oscillations (rotations) are designated by ~ andthe amplitude by qkO, the index k refers to a particular eigenmode. The phase, 8b isarbitrary and varies from molecule to molecule because the molecular oscillationsare not generally in-phase with the incident radiation. We assume that the

14 Also other formulations are in use, such as P=XII)EoE, where XiI) is the first order(dielectric) permittivity tensor also called (optical) susceptibility. Some authors include Eo' theelectric permittivity of free space, in XII). In the older literature 41tEoa is used in place a .Inthis case the polarizability has the dimension of volume, and is also called polarizabilityvolume (Schrader and Moore 1997).

8.2Raman Scattering: Classical Description 497

wavelength of the incident light is much larger than the molecular dimensions.The electric field of the incident wave can be considered independent of thecoordinates of the electrons and nuclei of the scattering molecule. Furthermore,we assume that the frequency of the incident light is sufficiently far from anyeigenfrequency of the system to neglect resonance effects . Only smalldisplacements from the equilibrium position of the nuclei result from theseconditions. Furthermore, we will consider only the motion of nuclei and neglectelectron motion.

The electric field, E, in Eq. (8.1) oscillates with the frequency ~ of theincident wave and has the Cartesian coordinates

Ex = Eo ex cose-0ot, Ey = Eo eycosui-t, E, = Eo ez cosO)ot. (8.4)

Here Eo is the amplitude of the incident wave, and e =(ex, ey, ez) is a unit vector inthe direction of the electric field.

Introducing Eqs. (8.3) and (8.4) into Eq. (8.1), we obtain the x-component ofthe dipole moment,

roo 0]Px = ~1t£oEolaxx ex +aXY, ey+axz ez cosO)o~

elastic - scattering

+EOf[[aa~~ 1e, fa:~ 1e,+[a~~ 1e,]q: (85)

x~ to, [("'0 +.",,)t +0, l+,os [("'0 -.",,)t - 0, l).1 anti-Stokes Stokes

The components py and pz of the induced dipole moment can be calculatedaccordingly. Equation (8.5) shows that the incident wave induces a dipole momentthat has one component oscillating with the same frequency as the incident wave,the elastic scattering component. It also has components involving the sum anddifference frequencies ~ ± CDJc. As stated above, the spectral lines that havefrequencies ~ - CDJc are called Stokes-Raman-lines, and spectral lines shifted tohigher frequencies , ~ + CDJc, are called anti-Stokes-Raman-lines. The amplitude ofthe elastic scattering component is proportional to the equilibrium polarizabilityand the amplitude of the incident wave. The amplitudes of the frequency-shiftedcomponents are also proportional to the amplitude of the incident wave butproportional to the derivatives of the polarizabillty with respect to the direction ofthe normal coordinates.

8.2.2Intensity of Raman Scattering

The elastically scattered light is in-phase with the incident radiation, but theRaman scattered light is not. In the normal situation a large number of moleculesparticipate in the scattering process. Due to the phase relation in elastic scattering ,

(8.6)


interference effects between the radiation from the individual molecules can playan important role, and the total scattered radiation is not simply the algebraic sumof the individual contributions. This is not so for Raman scattering, for the Ramaneffect involves energy transfer between the incident light field and the moleculebecause of the interaction of the incident field with the internal motion of thescattering molecule. This results in the scattering of frequency-shifted photonswith random phases.

It is convenient to treat the induced dipole moment as split between a dipolemoment oscillating with the incident frequency and dipole moments oscillatingwith frequencies tOo ± ~, that is,

Ptotal = Eo[P~ cos (ffiot)+ tP~ COS(ffio± ffik±Ok)}

o - 0 I ( aa: ) 0Po = <xo·e, and Pk =- -;-- ·eqk'2 aqk 0

where Poo and Pko are the amplitudes of the induced dipole moments. The totalemissive power follows from Eq. (3.55). If we introduce Eq. (8.6) into Eq. (3.55)and express the amplitude Eo by the power density Sine, which is identical to thePoynting vector given by Eq. (3.39) , we get

<Plotal = _1_2Sine[k~la:l .:L (ko±k, r(~a: ) q~ 2]. (8.7)61t£ 4 k oqk 0

The first term on the right side of this equation represents elastic scattering, thesecond term Raman scattering. The intensity of Raman scattering is linearlyproportional to the incident power, so this type of scattering is called linearRaman scattering. It is also proportional to the fourth power of the Ramanfrequency and is a function of the derivative of the polarizability at theequilibrium position. If the polarizability changes symmetrically with theequilibrium position, (aalaqk)O = 0 , and no Raman scattering is possible. FromEq. (8.6), we see that the phase of the Raman scattered light varies randomly frommolecule to molecule. Consequently, linear Raman scattering is an incoherentscattering process. In this classical treatment of Raman scattering the frequencyshift is predicted correctly, but the amplitudes of the Stokes and anti-Stokes linesare equal, which is in contradiction to observation. The quantum mechanicaltreatment yields the correct results.

8.2.3Selection Rules

As indicated above, Raman lines are only possible if the polarizability a: changeswith the coordinate q and if the gradient of the polarizability is non-zero at theequilibrium position. The calculation of the dependence of the polarizability onthe coordinates of the nuclei is a complicated quantum mechanical problem

8.2 RamanScattering: Classical Description 499

Modeof

-o-Qo -000~ A A ?oVibration

Quantitative

+--f f-+-t-tVariationofpolarlzabilitywith nonnalcoordinate

Ramanactivity Yes No No Yes Yes Yes

Infraredactivity No Yes Yes Yes Yes Yes

Fig. 8.1. Relationship between oscillation modes and selection rules

beyond the scope of this monograph, so we will restrict our treatment to anintuitive discussion of the problem (see Long 1977).

Let us consider first a diatomic molecule. As the internuclear separationincreases, the binding forces decrease until the molecule finally breaks apart.Hence, we expect the polarizability to increase with increasing molecularseparation from the equilibr ium position because the binding forces decrease . Theeffect of a change in polarizability with decreasing internuclear separation is notas obvious . Quantum mechanical calculations show that in the neighborhood ofthe equilibrium position the polarizability decreases with decreasing internucleardistance. During one cycle of a molecular oscillation, the polarizability has itsminimum at the turning point where there is a minimum intermolecular separationand has its maximum at the second turning point where the separation of the twonuclei has a maximum. In this case the gradient of the polarizability at theequilibrium distance is not zero. As a result, the diatomic molecule is Ramanactive in agreement with observation and quantum mechanical calculations.

For triatomic molecules such as CO2, a linear molecule , certain modes ofoscillation are not Raman-active. Figure 8.1 shows the relation between theRaman oscillation modes and the variation of the polarizability and theconsequences for scattering. Relations are depicted for linear molecules like CO2and nonlinear molecules such as water. In the case of water the stretching modesand bending modes shown are Raman-active, but for CO2 only the symmetricstretching mode shown is Raman-active .

8.2.4State of POlarization of Scattered Light

Let us consider a non-spherical molecule. The polarizability is a tensor, and, ingeneral , the direction of the induced dipole moment is different from the directionof the incident electric field. We define a coordinate system x', y', z' fixed with the

500 8 Inelastic Light Scattering

molecules and orientated parallel to the principal axis of the polarizability tensor.In this system the polarizability is a trace tensor with components b., b2, and b.,For simplicity, let us assume that the molecules have rotational symmetry and letthe x'-axis be the symmetry axis. In this case polarizability components b2 and b,are identical.

Let us investigate the state of polarization for a standard situation in which theincident light is linearly polarized, and the scattering angle is 90°. Assume that theincident beam propagates in the z-direction of a space-fixed rectangular coordinatesystem, and the electric vector of the incident field points in the x-direction. Weare interested in light scattered in the y-direction.

Now let x',y',z' be Cartesian coordinates defining the orientation of a moleculethat has a rotationally symmetric polarizability. In this case a rotation about the x'axis would give nothing new, and the angles 8 and are sufficient to describe theorientation of the molecule with respect to the space-fixed coordinate systemx,y,Z. The coordinate systems and angles 8 and are shown in Fig. 8.2. Thespace-fixed system is transformed into the molecule-fixed system by a rotationaround the x-axis through angle and around the y'-axis by angle 8.

The analysis of the scattering process involves projecting the electric field ofthe incident wave on the molecule-fixed coordinate system , calculating theinduced dipole moment , and projecting it back on the space-fixed system. Thisprocedure gives

(8.8)

and

(8.9)

The y-component is irrelevant here because we are interested In the state ofpolarization of light scattered into the y-direction.

In gases and most liquids , the orientation of the molecules is random. To takeinto account the statistical orientation of the molecules we must average over all

~x x'

z

y"'::::::.::::;r

z·

Fig. 8.2. Orientation of the space-fixed coordinate system x, y, z relative to the molecule-fixedsystem x', s: z'

8.2 Raman Scattering: Classical Description 501

possible orientations. No interference effects need to be considered. In linearRaman scattering the scattered light is incoherent, and in the case of elasticscattering the large number of randomly orientated molecules typicall y containedin the scattering volume causes interference effects to be averaged out. As we areprimarily interested in the radiation flux density of the scattered light (the lightemitted by the dipole), we calculate the orientation-averaged square of the induceddipole moment, obtaining

and

2 l! l!

fdcpfPx2 sin SdS

p~ = ° 2l! 0 l! = /5 (3b; +8b~ +4blb3)E~ ,fdcpfsin SdS° 0

(8.10)

2 l! l!

fdcpfP;sin SdS2 0 0

Pz = 2n l!

fdcpfsin SdSo 0

(8.11)

Here the bars indicate spatial averaging.The degree of depolarization or depolarization ratio is defined as the ratio of

the averages given by Eqs. (8.1l) and (8.10), that is,

(8.12)

The degree of polarization can also be written in terms of two rotationalinvariants of the polarizability tensor, the mean polarizability, ex, and theanisotropy, -;. These quantities are defined in terms of the principal axes as

(8.13)

or, in general, as

and

(8.14)

With these definitions the depolarization ratio for 90° elastic scattering of aplane polarized incident wave becomes

(8.15)


The orientation-averaged dipole components for incident light whose E-fieldoscillate s in the yz-plane can be calculated in a manner similar to that for the xzplane. Once this is determined, the depolarization ratio for unpolarized incidentlight is calculated to be

(8.16)

The depolarization ratio for Raman scattering differs only in the elements of thepolarizability tensor. These elements can be different for different eigenmodes.Therefore, the total intensity is simply found by calculating the intensity for anarbitrary orientation of the dipole and averaging over all possible orientations. Theresult of this procedure is that the scattered light always has non-zero componentsof the electri c field in any two directions perpendicular to each other and to thedirection of the observer. The scattered light is at least partiall y depolarized.

The anisotropic part of the polarizability and its derivative are usually muchsmaller than the average isotropic part. The intensity of the scattered light asfunction of the scattering geometry is, therefore, often not very different from thedipole radiation. The scattered radiation perpendicular to the oscillation plane ofthe incident wave is usually much larger than for any other scattering geometry inthe case of vibrational Raman scattering. In rotational Raman scattering, thescattered light is highly depolarized.

8.3Quantum Mechanical Description

By the beginning of the 20th Centu ry, it became abundantly clear that classicalmechani cs fails to describe a number of experimental findings, especially in thefield of the interaction of radiation with matter. New concepts presented byEinstein, Schrodinger, de Broglie , Heisenberg, and others caused shock waves inthe scientific community whose reverberations are still felt. Most of those ideasare now part of the extremel y successful concept called quantum mechanic s. Nocontradictions between quantum mechanical predictions and experimentalfinding s have been found up to now, but the interpretation of quantum mechanicsis still controversial in some respects. The importance of this theory in explainingthe behavior of the microcosm as well as its often puzzling results and its impacton our picture of nature is reflected in the large number of books on quantummechani cs that have been published. A comprehensive list can be found in thetreatise by Morri son (1990). Books dealing with the basic ideas that areinformative for lay readers include those by Hoffmann (1963), Wolf (1981) andGribbin (1984). In this chapter we do not intend to add significantly to the largeamount of material on quantum mechanics in the literature, but the spectroscopyof the microparticle cannot be understood without the use of some quantummechanical results. Our intention is to provide the reader with enough material tograsp the essential ideas and methods used in quantum mechanics and their impacton experiments involving microparticle analysis by spectroscopic method s.

8.3 Quantum Mechanical Description 503

8.3.1Review of Some Basic Relations

In the first half of the 20th Century Heisenberg put forth the revolutionary ideathat in the world of atoms and molecules well defined entities such as position,r =(x,y,z), momentum, p =(Px,Py,Pz), or energy , E, can only be determined withina certain degree of uncertainty, expressed mathematically by the HeisenbergUncertainty Principle ,

and

!i.E!i.t »»/2,

(8.17)

(8.18)

where Ii =hl2n and h is Planck 's constant. The energy-time inequality given byEq. (8.18) is conceptually different from Eq. (8.17) because in quantum mechanicstime is not an observable but a parameter (e.g., see Morrison 1990). The rules ofquantum mechanics come into play if Planck's constant is considered to be acharacteristic quantity of the problem. As a consequence of the uncertainty, it isnot meaningful in the quantum world to define such quantitie s as position andmomentum.

A wave function , also called a state function or state vector, is now introducedto describe the state of the quantum mechanical system. Let 'P(r,t) be this functionthat, in general, depends on position and time. The wave function is a complexfunction, and 'P(r,t) and its first derivatives must be finite, continuous, and singlevalued throughout all space. In addition, it must be square-integrable.

Without dwelling on the mathematical properties of this function , we proceedto consider its use and physical interpretation. Since the posit ion of a quantum orquantum mechanical entity such as an electron , atom, or photon must satisfy theHeisenberg uncertainty principle, we must deal with probabilities. Thus , we definethe probability density , p(r,t), to be the probability of finding a quantum in thevolume dr centered on the point r at time t given by

per , t) = 'P*(r, t) · 'P(r, t) = 1'P(r, 012, (8.19)

in which 'P*denotes the complex conjugate of 'P .Since the quantum (electron , proton, photon, etc....) must be somewhere in

space, the wave function is normali zed, that is, it satisfies the relation

+00 +00fper, t)dr = f'P *(r, t) 'P(r, t)dr = I . (8.20)

An alternate notation, the Dirac notation corresponding to Eq. (8.20), is givenby

1'P12

= ('P *I'P ) =f'P *(r , t)'P(r, t)dr. (8.21)

From classical mechanics, such as considered in Chapt. 6, we are familiar withthe change of the state of a physical system in space and time. This change isusually described quantitatively by an equation of conservation such as the energy


equation. In quantum mechanics the evolution of the state function in space andtime is given by the time-dependent Schriidinger equation,

h2h a

- -V'2'P(r, t)+ V(r, t)'P(r, t)= ih- 'P(r, r).2m at (8.22)

Here V(r,t) is the potential energy operator.Equation (8.22) can be considered to be the quantum mechani cal equivalent of

the Newtonian equation of motion . It bears the name of the extraordinary Austrianphysicist Erwin Schrodinger, who published this equation in 1926. His portraitformerl y decorated the Austrian 1000Schill ing note.

A commonly used representation of this equation is

h aH 'P (r, t) = ih-'P(r, t) ,at

where the Hamilton operator, iI is

(8.23)

(8.24)h h2

0 h ( )H=--V'"+Vr,t .2m

If the potential energy operator does not depend explicitly on time , V(r,t) =VCr), we can write the state function as the product,

'P(r,t)=e- i(Enlli)t 'Pn(r) , (8.25)

in which 'P n(r) is independent of time and satisfies the time-independentSchriidinge r equation,

(8.26)

Mathematically, this equation is an eigenvalue equation of the operator iIwhose eigenvalues are En and whose eigenf unctions are 'P n(r). Usually theequation has a large number (sometimes infinit y) of eigenvalues andeigenfunction s.

An important property of the time-independent state functi ons is that they arenormalized and orthogonal (orthonormal), which means that these function ssatisfy the orthogonality conditi on

+=f'P; (r )'Pe(r )d3r

== (k If)= Ske' (8.27)

where SkI is the Kronecker delta functi on.An important question is still unanswered. How can we calculate quantities that

can be validated by experiment using quantum mechanics? Again , becausequantum mechani cs deal s with probabilities, we anticipate that quantummechanics will only provide us with a prediction of the probability that anobse rvable has a specific value that can be measured in an experiment. The recipe ,found in any textbook on quantum mechanics, involves determ ining an operator,

8.3 auantum Mechanical Description 505

6 , from which we can extract the quantity or information that ought to bemeasurable. The resulting expectation is given by

(8.28)

Because physical quantities are represented by real numbers, the expectationmust be real. In this case, the following relation holds

The last expression in Eq. (8.29) symbolizes that the operator can act either to theleft on the function, 'P ' , called the bra-vector, or to the right on 'P, called the ketvector.

The last very useful and important formali sm we want to point out is that anystate function, 'P(r.t), can be expre ssed as a superpo sition of eigenstates,

(8.30)n

where In) are the eigenfunctions of the corresponding time-independentSchrodinger equation

(8.31 )

A useful technique for representing any state function in terms of theorthogonal eigenfunction, 'Pn, is to use the projection operator, P, written as

n

Eq. (8.30) can be written in terms of this operator as

I'P(r, t)) =LI n)(n I'P(r,O)) e- i (En I h)t .

n

8.3.2Quantum States of Atoms and Molecules

(8.32)

(8.33)

As mentioned in the introduction, atoms and molecules consist of one or severalatomic nuclei surrounded by electrons that are bound to a region around the nucleiby Coulomb forces" . To calculate the possible states and energies, one must solvethe Schrodinger equation formulated for the particular problem. This can be atremendous task, which is beyond the scope of this introduction. For the relativelysimple case of a hydrogen atom, the time-independent Schrodinger equation reads

IS We ignore here more subtle effects such as relativistic motion, magnetic interaction or nuclearor electron spin.


(8.34)

in which mel and mnll are the masses of the electron and nucleus, respectively.The operator 'Ve

2acts only on the coordinates of the electron, feb and v: acts only

on the coordinates of the nucleus, f nll . The solution of this problem is described indetail in many textbooks on quantum mechanics. The state function is found to bea product of a radial function, Rne(r), and an angular dependent function , Y em(8,<j»,

where

and

13 ]1 /2

R (r)= _ [2Z) (n - f -I)! e-p/2 €LU+1(r),n€ 2 r( fi) ,] 3 P n+€na o n~ n + 1: •

(8.35)

(8.36)

(8.37)li 2 m m 2Z

ao = --2 ' ~ = en, p = -- r .ue me +mn na.,

Here Z is the nuclear charge number, Lqj represents the associated Laguerre

polynomials, and Yme<8,<j» are spherical harmonic functions . Finally, the energyeigenvalues are found to be

E =_ Z2[£)n n2 2ao '

n = 1,2,.. .. (8.38)

The two most important results are that the energy of bound electrons can haveonly certain discrete values , and the state function of the electron can becharacterized unambiguously by a set of four integers called quantum numbersdesignated by the letters n, f , m and s. The integers nand f specify the radialfunction Rne, and the integer f , and m specify Y em. The last integer, s, the spinquantum number, is not a result of the simple theory outlined above .

The Pauli exclusion principle demands that in a multi-electron atom eachelectron must differ in at least one quantum number. This rule is the buildingblock of the periodic system of the chemical elements. Some of the possiblequantum states of one electron and their designations are listed in Table 8.1.


Table 8.1. Some possiblequantumstates of electrons in an atom

Shell K L

Principalquantum n I 2number

Azimuthalquantum e 0 0 I

numberDesignation s s p

Magnetic quantum mt 0 0 -I 0 +1number

Spin quantum number ms n n n ri n

Shell M

Principal quantum n 3

number

Azimuthal quantum e 0 I 2numberDesignation s p d

Magnetic quantum mt 0 -I 0 +1 -2 -I 0 +1 +2number

Spin quantumnumber ms ri u n u n n n ri ti

The arrow i indicates m, =+1/2, and the arrow J. indicates m, =-1/2.

The energy of each electron is uniquely defined by the set of the quantumnumbers assigned to that electro n. The energies of the variou s states may or maynot be diffe rent for each electron. If electrons with different quantum numbershave the same energy , the state is said to be degenerated. Not only is the energy ofone individual electron quantized, but the same holds for the electron cloud. As anexample of the discrete energy structure of an atom, Fig. 8.3 shows an energylevel diagram (Grotrian-diagram) for a potassium atom.

Additional kinetic energy states are possible for molecules due to the rotationaland vibrational motion of the nuclei. The energy states of vibrations can becharacterized by the vibrational quantum number, v, the rotational states by therotational quantum number, J. Figure 8.4 provides a graphical representation ofsome possible energy states of a molecule to illustrate the rotational andvibrational states of molecules. The letters A and B designate electronic states. Inmost cases , the actual energy difference between different electronic states ismuch larger as shown in Fig. 8A.

In a freely orientable system such as the molecules in an ideal gas, therotational energy is independent of the orientation of the rotational axis . Therotational energy is degenerated. If a preferential direction exists in the system, therotational energy depends on the orientation of the molec ule and the dege neracy ofthe rotational energy is removed.


o 25112 2P3/2 2P112 20512 203/2 2F712,512 0

-!!!I!!!!!!!!!!!!===----------

';"

E'"u

~20...-): a:

Cl wa: 20 mw :Ez :::;)w z

3 w><t3=

30

4

4.32

Fig . 8.3. Energy level diagram of a potassium atom sowing radiative transitions with theassociated wave lengths

__J" v '

__J' •

--- - - --== = '!!.!' !!!!!!!!~ ,0--3__.1'=--10=6

--.1'=~O-5

-.,B

- v"-- - -__J" _

--'0====:__3__J~===

~O==== 5_ _ J " I)

-10-'

-------.._iiiO- - - - - - - -

A

--,~_--___:-------O

Fig. 8.4. Vibrational rotational states of a molecule


8.3.3Light Scattering

With this background, we can proceed to the origin al goal of this chapter. Usinginformation based on the classical treatment of scattering and the quantum mechanical recipes sketched above, we know the way. First, we need to calculate theexpectation of the dipole moment of the quantum system (atom, molecule) in thefield of the incident light wave . Next, we use Eq. (3.55) to calculate the averagepower emitted by one atom or molecule. For this purpo se, we replace the clas sicaldipole moment by its quantum mechanical equivalent, the expectation of thedipole operator. To do so we require the state funct ion of the combined system, ofthe molecule and of the light wave . To make things easier we treat the light fieldclassically; that is, we do not quantize the light field . This is usually justified because the dimensions of the molecule are much smaller than the wavelength of theelectromagnetic radiation. In addition, we apply the well-justified assumption thatthe light field adds only a small disturbance to the energy of the atom . Consequently, we can apply perturbation theory, writing the Hamiltonian as the sum ofthe unperturbed operator and a small perturbation, that is,

H = Ho- HI . (8.39)

The first term on the right, iIo is the Hamiltonian of the unperturbed system,and the second term represents the small perturbation caused by the light field".Furthermore, we assume that the state function can be expressed as the sum of thestate function of the unperturbed system 1\fI)o and an additional term caused by theperturbation 1\fI)I ' Thus, for a specific state k we write

\fIk(r,t)=\fI~O)(r,t)+\fI~I )(r,t)=lk,t)o +lk,t)I ' (8.40)

For convenience we will use the Dirac representation. Thus, the Schrodingerequation for the kth perturbed state, using Eqs. (8.39) and (8.40), is

( H o - HI - iii :t ) I\fIk (r, t)) = ( Ho - HI - iii :t ) (I k, t)o + Ik, t)]) = 0

or (8.41)

( H o - iii :t ) 1k, t) 0 + ( H o - iii :t ) 1k, t) I = HI (I k, t)0 + 1k, t)] ).

A corresponding equation can be formulated for the complex conjugate (c.c.)quantities.

The first term on the left side of Eq. (8.41) vanishes, and Ik,t)o is aneigenfunction of the unperturbed Schrodinger equation. Neglecting theperturbation term in the state function on the right side, we obtain

16We choose a negative sign for the disturbance because the electric field of the incident waveacts only on the negatively-charged electrons.


(8.42)

We represent the electromagnetic wave interacting with the quantum systemconventionally by

(8.43)

To avoid any confusion between the amplitude of the electromagnetic wave andthe energy of the quantum system, we have used A in place of Eo in Eq. (8.43) .

The Coulomb force of this wave contributes to the potential energy of thesystem. This additional energy is linearly proportional to the space coordinate, andwe can express this contribution as

HI = eXE= M · Ae- iliwot + M· A *eiliwot , (8.44)

where xis the position operator and M the dipole operator. The state function ofthe unperturbed system can be factored into time- and space-dependent parts,

Ik,t)o =e-iEk tllilk)o' (8.45)

We now assume that we can do the same for the perturbation term of the statefunction and write

Ik,t)\ =e-i(Ek+IiWo)t llil+,k)\ +e-i(Ek-IiWo)tllil_,k)[. (8.46)

Here we have split the perturbation into two parts, one that depends only onpositive frequencies, the other on negative frequencies . Introducing Eq. (8.44),(8.45) and (8.46) into (8.42), there results

[fIo- (Ek

-liwo)]. e-i(Ek-liwo)l 11i1_,k)t

+ [fIo- (Ek + hwo)]. e- i(Ek+liwo )tllil+ , k)[ (8.47)

= A .Me- i(Ek+liwo)t llilk)o + A *M e- i(Ek-IiWo)tllilk)o'

The equation must be satisfied for positive and negative frequenciesindependently. Consequently, we obtain

(Ho-(Ek +hwo))I+ ,k)\ =A .Mlk)o' (8.48)

and

(8.49)

Expansion of the perturbation term of the state function into an eigenfunction ofthe unperturbed state yields

(8.50)

(8.51)


We calculate the expan sion coefficients by exploiting the orthogonality of theeigenfunctions Ik)o and get

( I ) - A ·Mkro r +,k 1 - ( ) 'E, - Ek + liwo

(8.52)

( 1_ ) - A ·Mkr

or ,k,- ( ).E, - Ek -liwo

(8.53)

(8.54)

(8.55)

The quantity Mkr is called the transition moment, in which the first subscriptindicates the initial state and the second subscript the final state. Using Eqs. (8.52)and (8.53) , Eq. (8.46) becomes

Ik,t), = Llr) [ AM kr . e-i(Ek +liwo )t / li

E,- (Ek + liwo)r 0

+ A*Mkr -i(Ek-IiWO) t/ li]

E, - (Ek -liwo) e .

Writing Er-Ek =Ii~n the denominator of the right hand side reduces to Ii(w,.k ±Wo) , and the equation becomes

I jA.M A*·M 1Ik, t)l = - Llr)o kr e-i(Ek+h~) l lh + kr e-i (Ek-hWO )tlh .

Ii r Wkr - Wo Wkr +Wo

At this point we have all we need to calculate the expectation of the dipolemoment. First, we calculate the expectation of the kth pure state to be

This equation can be written in the more compact form,

(8.56)

where (8.57)

The first term, which is time-independent, represents the permanent dipolemoment of state k. The second, and the third terms describe oscillations at thefrequency of the incident radiation and correspond to coherent elastic scattering.


The expectation for mixed states is

(8.58)

(8.59)

The expectation of the dipole moment, Mkn, is also called the transtttonmoment for the transition k -7 n. To elucidate the significance of the three terms inEq. (8.59) , we use the correspondence between the radiation field of a classicaldipole and the radiation field associated with the transition from state n to a lowerstate k (En> Ek). The following correspondence between classical and quantummechanical approaches holds (Corney 1988)

(8.60)

It follows that

Pclassic = 2(kIMln) = 2Mnk· (8.61)

The first term on the right site of Eq. (8.59) is proportional to the amplitude ofspontaneous emission, the second term is proportional to the emission of radiationwith frequency (COo ± Wod > 0 17

• The third term represents a special case ofstimulated emission not considered here" , It is no surprise that the second termrepresents spontaneous Raman scattering. If the energy of the initial state, n, ishigher than that of the final state, k, radiation with frequency (00 + Wok) is emitted.This is anti-Stokes Raman scattering. In the opposite case, radiation withfrequency (00 - Wok) is emitted, which is Stokes Raman scattering.

From (8.61) and Eq. (3.55) we find that the expectation of the power emittedper molecule from state n to state k by spontaneous emission is

17 The sign depends whether Ek > Eo or vice versa. In any case Wo » IWimI because we treatedthe effect of the light field as a small perturbation .

18 The exponent of the third term is always positive, independent of the sign of Wok' becausewo» Iwk.l ·

8.3 Quantum Mechanical Description

004

<l> ffiolecule = __Ok_1M 12

spont 3 3 nk'1tEC

and the emission by Raman scatte ring is

{oo +oo )4<l> molecuJe = ~ ok 1P 1

2

Raman 31tEC 3 ok '

513

(8.62)

(8.63)

The power emitt ed from volume V by spontaneous emission is proportional tothe number of molecules in the initial state n. If we designate the number densityof molecules by N, and the occupation number that is the fraction of molecules inthe exci ted state with energy Enby n. , we get

(8.64)

Here we have assumed that En> Ek• For a scattering process where the molecule isinitially in the energy state En. the second term of Eq. (8.59) yields the anti-StokesRaman emissive power

(8.65)

For the opposite case, that is, for a transition from a lower energy state to a higherstate, E, ~ En' the Stokes Raman result is

(8.66)

In the classical treatment, we found no differen ce between the intensity of antiStokes and Stokes Raman scattering, but the quantum mechani cal calculationsshow that Raman scattering depend s on the occupation number of the initial state,which differs for anti-Stokes and Stokes Raman scattering. At low temperatures,the number of molecul es in higher energy states is much smaller than the numberin the ground state, so we can expect the Stokes Raman emission to be muchlarger than the anti-Stokes emis sion.

8.3.4Placzek's Polarizability Theory

The intensity of Raman scattering can be calculated, at least in principle , as shownin the forgoing section. However, the calculation of all transition moments Mkr

can be a formidable, and often impossible, task because all intermediate quantumstates must be known. An approximate method , which avoids the necessity ofcalculating all these transition moments, was proposed by Placzek (1934). We willfollow his treatment here.


First, we relate the concept of polarizability, used successfully in the classicaltreatment, to quantum mechanical calculations, defining a quantum mechanicalpolarizability tensor, U kn- or transition polarizabllity. This tensor relates theexpectation, Pkn. of the induced dipole moment to the amplitude A, of the electricfield of the incident radiation,

(8.67)

To show the relation between the expectation of the transition moments, Mkrcalculated in Sect. 8.3.3 and Ukn. we choose a coordinate system with thecoordinates in the directions of the three unit vectors a, ~,y. The amplitude vectorin this coordinate system is

(8.68)

The product A·Mkr becomes

A·M kr = Au (Mu)kr + A~(M~)kr+ Ay(Myk" (8.69)

A comparison of Eqs. (8.59) and (8.67), taking into account Eqs. (8.68) and(8.69), shows that the matrix element of the polarizability tensor is

(8.70)

Due to the large mass difference between an electron and a nucleus, theelectronic motion is much faster than the motion of the nuclei. The couplingbetween electronic and nuclear motions is weak, therefore, and the wave functioncan be factorized to give

(8.71)

The first term, the wave function of the electronic motion, [etq.r) depends on thecoordinates r of the electrons at fixed nuclear positions, q. The function ischaracterized by a set of quantum numbers represented here by a single letter, e.The second term, Ive(q), is a function of the motion of the atomic nuclei and ischaracterized by the quantum number v. This function depend s only on thecoordinates of the nuclei and is slightly different for different electronic states.This function can, in tum, be factorized into vibrational and rotational modes. Thefirst depend on the internuclear distance , the latter on the orientation. Vibrationalstates are conventionally identified by the letter, v, rotational states by, J.

Generally, electronic, vibrational, and rotational states are coupled. This makesthe summation in Eq. (8.70) often impossible , or at least difficult, to evaluate.Placzek showed how different modes of vibrational and rotational Ramanscattering in the electronic ground state can be decoupled based on the followingrestrictions:

• The degeneration of the electronic ground state is low.• The frequency, % , of the incident radiation is much larger than the rotational

and vibrational frequencies.


• The difference between electronic frequencies and the incident frequency ismuch larger than the rotational and vibrational frequencies .

Under these conditions the transition frequencies, ~r and CUrn, in thedenominator of Eq. (8.70) can be replaced by a mean frequency, w Oe' ,that isindependent of the nuclear motion. With this approximation and considering onlytransitions that begin and end in the electronic ground state Eq. (8.70) may bewritten as

(8.72)

(8.74)

where (a°aj3)kn is an element of the transition polarizability tensor of the electronicground state. The superscript, 0, indicates this. The rotational and vibrationalquantum numbers of the initial state are represented by k, that of the final state byn. With the aid of Eq. (8.71) and taking into account the completeness of thefunctions IVeJ ' the summation can be carried out to obtain

This tensor element can be considered as the expectation of the transitionpolarizability operator,cia, which depends on the nuclear coordinates.

The dependence is weak, so we can expand the operator as a Taylor seriesanalogous to the classical treatment,

A ° A a aaoI aao

a = a +L-;-qk +-L-;-:;-qkqj +...k oqk 2 kj oq koqj

The derivatives are evaluated at the equilibrium position. The expectation of thepolarizability operator is calculated in the usual manner, and the result is

(8.75)

We neglect the weak dependence of the vibrational state function on the electronicmotion and, similarly, the dependence of the rotational state function on theelectronic and vibrational motion . The quantum states of the initial state areindicated by the subscript, n, and that of the final state by, k. Introducing Eq.(8.71) into (8.75) yields

(8.76)


The first term on the right side of Eq. (8.76), which is associated with elasticscattering, is zero except when k = n. The second term is responsible for Ramanscattering19. We will examine this in somewhat more detail in the next section.The terms <eo!d aU/dqleo> are called componentsof the tensor of the polarizabilitychange.

The attractive aspect of the polarizability theory is that a ° and its derivativesare independentof the vibrational and rotational states. This allows the calculationof the relative intensities of rotational vibrational Raman transitions withoutknowledge of the polarizability operator and its derivatives. This drasticallysimplifies the evaluation of Eq. (8.74) for rotational and vibrational Ramanscattering.

8.3.4.1Vibrational Raman Scattering

The calculation of the expectationof the positionoperator is a standard problem inquantum mechanics. The result is

o

(v~ + l)n2f.llil

~~~

(8.77)

in which l.l is the reduced mass and 00 = (Ev+,-Ev)/n is the frequency of thevibration transition. The total power of Raman scattering follows from Eq. (8.65)through (8.67) by introducing for the transition polarizability the second term ofEq. (8.76). We omit the differentiation of different vibrational modes labeled by iand, using Eq. (8.77), we obtain the emissive power for Stokes and anti-Stokesscattering, respectively, in the forms

_ _ (ooo-ookn)41121( Idaol )12

(v k +l). n<PSlokes - <P~v =+ ' - VNnk 3 A e -- e ,3nc E dq 2f.llil

( ) 4 1 :l A ° 1

2\000 +OOkn 2 oa Vk·n

<Panli-SIOkes=<P~V=-l=VNnk 3 IAI (el-Ie) - .3nc E dq 2f.llil

(8.78)

(8.79)

The occupation numbers, nb are given by the Boltzmann distribution function.According to this distribution the fraction of molecules, n, , in a specific energystate, En , is given by:

19 To avoid confusion between the index for the mode with the quantum number of the initialstate, we now identify the mode by the superscript i.


(8.80)

where N is the total number of molecules, gk is the degree of degeneracy, kB is theBoltzmann constant, and T is the temperature .

The vibrational energy is given by (Herzberg 1989)

E, = nwe(v +1/2) -nuVe(v +1/2)2 +.... (8.81)

The quantities We and x, are molecular constants, we being the angular frequencyof a vibrational transition. The subscript e indicates that the quantities depend alsoon the electronic state of the molecule s, but this dependence is usually weak, asmentioned before.

If we neglect the higher order terms in Eq. (8.81) and write the vibrationalenergy as E, =nwe(v+ 112), Eq. (8.80) becomes with En=Ev

e - hUle(V+1I2)/ kBT e - flUleV/kBT

n = =v ~ () (l_e-hUle/ kBT)LJe-hUle v+ll2/ kBT

v= l

(8.82)

The total intensity of the vibrational Raman scattering follows by introducingEq. (8.82) into Eq. (8.78) or Eq. (8.79) and summing over all vibrational levels. Ifwe finally express the amplitude A of the incident wave by the power density, Sinousing Eqs. (3.39) and (3.40), we obtain

IEI 2= 2S inc/ sc, (8.83)

The total scattering cross section, c, for Raman scattering becomes

<P kn (wo±wkn ! I( lalio l )12

!( II )1 2

cr=--=2VN 2 4 I eo -a- eo v. q v. nk •Sinc 31tE C n q

(8.84)

If we consider only transitions with I~vl = ±1, use Eq. (8.78), and neglecthigher order terms in Eq. (8.81) as before, the Raman scattering cross section forStokes scattering becomes

<P 8V=+1 = 2NV (wo - ~~r I(eoIalio leo)12

_n_Sinc 31tE c aq 2/-M

hCUle(vl+l l2 h BT

xI(vk+l) e ( } .Vk Ie- hcUle vj +1I2 kBT

j

(8.85)

In this case the vibrational frequencies are all the same, that is, ~ = Wv. Thesummation in Eq. (8.85) must be carried out over all quantum numbers, k, for agiven vibrational mode, i (e.g., all bending modes of a triatomic molecule). Thedegeneracy is assumed to be unity. For Stokes transitions the summation is


and the intensity for Stokes Raman scattering is,

lI> ~V=+ l ((00- (O Yr I( IdaoI )1

2

Ii 2NV---0;:-= 31tc2C4 eo dq eo 211W (l_e -hw'kBT)

For anti-Stokes transitions the summation yields

1~v n -...,....---._£." k k - ( hwlk T ) 'Yk \e B-1

and the corresponding intensity is

(8.86)

(8.87)

(8.88)

(8.89)

(8.91)

lI> ~v=-l ((00 +(OY rI( IdaO I )12

n 2NV~= 31tc2c4 eo-aq-eo 211W(e-hwlkBT_l)

The differential scattering cros s section can be calculated in a similar manner. Thestarting point in this case is Eq. (3.70).

For randomly oriented molecules the average over the squares of thecomponents of the tensor of the polarizability change, I<eold ao/dqleo>1

2, has to be

calculated in much the same way as in Sect. 8.24. The scattered intensity must beindependent of the coordinate system and depends only on the two tensorinvariants, the mean polarizability and anisotropy, defined by Eqs. (8.13) and(8.14), respectively.

For Stokes lines with the electric vector orientated perpendicular to the plane ofobservation (defined by the incident direction and the direction of the scatteredlight) , and for a scattering angle of 900

, we get

(de )SIOkeS= 1dll> = ((00 -(0) Ii NV (a~ +~Y~). (8.90)dn 1- Dine dn 81t2c2C4 211W (1- e-hWl kBT) 45

We have added to the invariants of the Raman polarizability tensor (tensor ofpolarizability change) the subscript R to indicate that the components of thistensor are actually the derivatives (dao~/aq). For anti-Stokes lines thecorresponding scattering cross section is

(de )anli-Slokes ((00 + (0)4 Ii NV (-Z 4 2)

dn 1- = 81tZcZC4 211W(e-T'WlkBT -1) U R + 45 YR .

The mean polarizability and the anisotropy are calculated with the matrix elementsof the tensor of the polarizability change.


8.3.4.2Rotational Raman scattering

Rotational Raman spectra or the rotational structure of rotational vibrational bandscan be resolved only in the gas phase. In the liquid phase, intermolecularinteractions mask the rotational bands completely, and line broadening overlapsthe rotational structure overlaps. These intermolecular interactions are responsiblein most cases for the shape of the Raman bands and for a shift in frequencycompared to gas phase Raman bands.

To calculate the intensity of the rotational lines, the corresponding coefficientsin the Raman polarizability tensor need to be evaluated and averaged over allpossible orientations by a procedure similar to that described in Sect. 8.2.4 . Thecomponents of the tensor of the polarizability change can be factored into a termthat depends only on the vibrational structure and one that depends only on therotational quantum number. We skip details of the derivation of invariants of thecorresponding tensor and refer to the literature (e.g., Long 1977).

One of the results of this analysis is that the intensity of the rotational lines isproportional to YRbJJ• The coefficients, bJJ, depend on the change of the rotationalquantum number. For linear or rotationally symmetric molecules these coefficientsare

b = 3(J+l)(J+2) b = J(J+l)1+21 2(2J+l)(2J+3)' II (2J-l)(2J+3)'

b = 3J(J+l) .HI 2(2J + 1)(2J -I)

(8.92)

(8.94)

The coefficient, bJJ, is only relevant to vibrational rotational transitions(scattering processes with simultaneous change in the rotational and vibrationalquantum number). The Boltzmann distribution for freely rotating molecules with2J+ 1 possible orientations is

g, (2J + l)e-ER(e,v,J)/kBTn - (8.93)1- LgJ(2J + l)e-ER(e,v.J)/kBT

I

where g, is the degeneracy of the nuclear spin. The rotational energy, ER, ofsimple molecules can be expressed as follows

ER =hc[B)(J + 1)- DY(J + If + "J, B, =Be -a..{v + 1/2),

D, = De +~.{v + 1/2).

The dependence of the expansion coefficients Be' De, o, and ~e, on theelectronic state is expressed by the subscript e. The dependence on v is weakbecause o, « Be and ~e « De. Sometimes the amplitudes of rotational lines witheven and odd quantum numbers have, despite their temperature dependence, afixed amplitude ratio . This ratio depends on the nuclear spin statistics and is two,for example, for the rotational spectrum of nitrogen.


The most important difference between the classical and the quantummechanical treatment of Raman scattering is that the quantum mechanical analysispredicts a dependence of the Raman lines on the occupation numbers of the initialstates. These numbers are temperature dependent, and consequently, the Ramanline intensity depends on temperature too. In effect, this dependence is used todetermine the local temperature in gases by Raman scattering, a method widelyapplied in combustion research, and Rassat and Davis (1994) showed thatStokes/anti-Stokes measurements can be used to determine the temperature ofheated microparticles. Their approach is discussed below.

8.3.5Properties of Raman Scattering

Raman scattering can be applied very successfully without understanding thedetails of the theoretical analyses , but it is desirable to know the main propertiesof this scattering process. This section therefore summarizes the properties ofRaman scattering. Its purpose is to serve as a guide for the experimentalist tooutline Raman scattering experiments.

8.3.5.1The Rule of the Intermediate States

In fluorescence" the transition probabilities for absorption or emission of radiationare proportional to the square of the transition momentMnb Eq. (8.64). In Ramanscattering, the transition probabilities depend on products of the transitionmoment, Mnr·Mrk" from Eqs. (8.58) and (8.59). Consequently, it is possible thattransition moments of different states can reinforce or weaken each other. We seefrom Eq. (8.72) that the matrix elements (aaphn are only different from zero if atleast one of the intermediate states r combines with both the initial state n and thefinal state k. The transition moments for both states, Mnr and Mrk must be differentfrom zero. However, even in this case the transition probability can be zerobecause the two transition moments can interfere destructively.

Although intermediate states are involved in Raman scattering, this does notmean at all that it is a two-step process, that is, transition from energy stateEn~ E, and then from E, ~ Ek. Such a two-step process would depend on I Mnr 1

2

and I M rk 12 but not on the product (Mnr·Mrk). The energy of the intermediate states

can lie above or below the final state. The theory of Raman scattering shows thatin a molecular system perturbed by the field of an electromagnetic wave,transitions between energy states are induced without absorption or emission ofthe incident radiation. In terms of quantum mechanics, one speaks of theannihilation of the photon of the incident radiation and the creation of a newphoton with a different frequency .

A popular graphic representation of Raman scattering is shown in Fig. 8.5. Theenergy of an incident photon does not correspond to any energy difference of the

20 Fluorescence is simply spontaneous emission where the upper level was populated byabsorption of light.


K=4 }K=3K=2 V= 1K=1K=O

K=4 }K=3K=2 V=OK=lK=O

ntl- Stokesine

okesne

brationalamanffeet

-- -Rotational

ViR

Raman EEffeet

'-- .--- I-- - r-r- I-Stokes St

vLine Li

/

V VAnti- A

LStokes /)-ine

,if

E

Fig. 8.5. Schematic representation of Raman scattering by virtual transitions in theenergy level diagram

eigenstates of the molecules. The absorption is therefore only virtual. Themolecule can not really dwe ll in a virtual state, which is indicated by the brokenlines in the figure . The molecule falls back into an eigenstate that might have ahigher energy than the initial level , and a photon with a lower freque ncy isemitted. This is Stokes Raman scattering. If the final level is lower than the init iallevel, a line appears in the spectrum of the scattered light, which is shifted tohigher frequencies. This is anti -Stokes Raman scattering .

The quantum mechanical treatment has shown that the interpretation of theRaman effect as a virtual absorption and emission process is not quite correct. Noabsorption of the incident radiation takes place, and the molecule is, in effec tnever in any kind of virtual state . Raman scattering is, as the name tells, ascattering and not an absorption-emission process. For this reason Ramanscattering is much less sensitive to the environment of the molecule thanfluorescence, where an absorbed photon, for example, can transfer its energy bycollision to other molecules, thereby quenching the fluorescence. This is alsoreflected by the speed of the two processes. The life time for spontaneousemission is about 10.8 s, whereas Raman scattering is a picosecond process.

8.3.5.2Coherence Properties

Each eige nvalue solution contains an arbitral)' phase factor , <p, because, if \f' r is asolution of Schrodinger's wave equa tion, \f' r e'" is also a solution. If we introduce


this phase factor, we see that the phase factor of the transition moment Mkr isexp[i«Jlk-<j)r)] ' If the initial and final states are the same, the phase factors cancel,and the scattered light is coherent. This is the case for elastic scattering. Forinelastic scattering, the phase factor is not zero. The phase factor variesstatistically from molecule to molecule, and the scattered light that is the result ofscattering on a great number of molecules is incoherent.

8.3.5.3Frequency Dependence

The theory of Raman scattering is nothing else than the analysis of the effect ofthe time dependence of the dipole moment induced by the incidentelectromagnetic field. It is no surprise, therefore , that the characteristics of Ramanscattering are closely related to the characteri stics of dipole radiation that dependon the fourth power of frequency . This is correct for elastic scattering but not quitefor inelastic scattering , as we can see by examination of Eq. (8.65) or (8.66). Wesee from Eq. (8.59) that the expectation of the transition moment and the transitionpolarizability, respectively, depend also on frequency. Only if the frequency of theincident radiation roo is much smaller than the transition frequencies, <llnb theeffect of roo on the transition polarizability is negligible . If the incident frequencyis close to one of the transition frequencies roo ::::: Wrb then the term containing thistransition frequency will dominate the right side of Eq. (8.59), and the frequencydependence of the inelastic scattering is (roo+~n)4/(Wrk-roo)2 . This is the case ofresonance Raman scattering .

8.3.5.4Selection rules

As already mentioned, a transition is only possible if at least one intermediate energy level, r, of the molecule combines in absorption and emission with the statesnand k, the initial and final states of the Raman process. This is called the thirdcommon-level-rule. It is important to mention that the transition moment forspontaneous or stimulated emission or absorption related to a direct transitionn ~ k does not appear at all in the scattering formula. The appearance of Ramanlines in the scattering spectrum, shifted by <llnk relative to the frequency of theincident wave, is independent of the absorption or emission at this frequency.

The selection rules can be found by the same symmetry considerations as forspontaneous or stimulated emission and are, in effect, independent of theintermediate states. Whether or not a transition is optically allowed is closelyrelated to the symmetry properties of the system. We will not explore this topicbut refer the reader to the literature (Placzek 1934; Brandmtiller and Moser 1962;Long 1977) and the references cited therein. Table 8.2 lists some of the moreimportant rules, but this list should not be considered exhaustive. For example ,Raman lines with L1v =±2 etc. are frequently observed . They are called overtonesand have been used to determine the gas temperature in combustion processes .


Table 8.2. Selection rules

Diatomic molecules

Symmetric topmolecules

8.3.5.5Frequency Shift

/::"v =0, LV= +2; /::"v =±I , LV=0, ±2

/::"v =0, LV= 0, +1, +2; /::"v =±1, /::,.K= 0"

In the foregoing paragraphs the relations between the Raman spectrum and thevibrational and rotational energy levels were outlined. One of the main fields ofapplication of the Raman effect is the revelation of molecular structure. Togetherwith infrared spectroscopy, Raman spectroscopy plays a crucial role in thedetermination of molecular structure. The other classical domain of Ramanspectroscopy is the identification of molecules, that is, the analysis of the chemicalcomposition of a sample. Similar to the infrared spectrum, the Raman spectrumrepresents a 'fingerprint ' of the scattering molecule. The Raman spectra of manychemical substances can be found in the literature (e.g., Schrader and Meier1974,1975,1976; Brame and Graselli 1976,1977 ; Graselli et a1. 1981; Freeman1974; Ross 1972; Dolish et a1. 1974).

Raman spectra are compri sed of relatively few lines if the molecules arecomposed of only a few atoms (simple molecules) . The spectra becomeincreasingly complicated as the size of the molecule and number of atomic speciesin it increases. The identification of the different chemical components in anunknown sample is only possible as long as the characteristic lines of thesecomponents can be identified in the spectrum. Raman scattering has been usedvery successfully in the last 25 years for the determination of the compo sition ofgaseous media, even in such a hostile environment as combustion processes.

The application of this technique for the identification of molecularcomposition of gases at elevated densities and for liquids or solids is not asstraightforward as for gases at moderate densities. The reason is the increasingimportance of intermolecular forces, which affect the line shift, band shapes, orboth . In effect, the Raman spectrum can be used to investigate intermolecularforces , as outlined by Srivastrava and Zaidi (1979). Obviously, the frequency shiftdepends strongly on the chemical substances under investigation. Not onlypressure, but also phase changes can affect the Raman spectrum. The frequencyshift due to phase changes depends on the vibrational mode of the particularRaman line and is usually accompanied by intensity changes. An example of thephase shifts for liquid and gaseous states is shown in Fig. 8.6. The frequency shiftprovides a discrimination between Raman scattering from the liquid phase andRaman scattering from the gas phase, and the simultaneous determination of liquidphase and gas phase concentrations is possible.

211n polyatomic molecules, J is the rotational quantum number of the total angular momentumandK is thequantum number associated with rotation about themolecular symmetry axis.


~

:::J

.i

...J«zoCi5z«::a:«a:

1200 1300 1400 1500 1600 1700 1800

RAMAN SHIFT, 1/cm

Fig. 8.6. Effect of phase change on Raman bands in ethanol : a) CO, (g), (b) pure H20 , and (c)CO, (aq) and H,G vertically shifted by 10%, from Vehring et al. (1995)'

8.3.5.6Intensity

The intensities of Raman lines depend on the number of molecules of a specificchemical component in the scattering volume and on temperature, as indicated byEqs. (8.84) trough (8.91). The pronounced dependence of the intensity of therotational Raman lines on temperature has found widespread application in fluiddynamics and combustion research for the local determination of gas temperatureas indicated by the survey of Ledermann and Sacks (1984) . In liquids, therotational structure cannot be resolved, and the effect of temperature, ifobservable, is mainly on the shape of the Raman band.

The intensity of Raman scattering is proportional to the local electric field,which may be different from the external incident field . Whereas in gases atmoderate pressure, the deviation of the local field from the external field is usuallynegligible, the situation is different in liquids or solids. Due to the close neighborhood of the surrounding molecules, the local electromagnetic field differs from theexternal incident field . For scattering on bulk material the local field can usuallybe expressed as a relatively simple function of the refractive index and the energyflux density of the incident radiation (Kaiser et al. 1992). However, for scatteringfrom microparticles the local internal field can differ dramatically from the

• Reproduced with permission from Vehring R, Moritz H, Niekamp D, Schweiger G, andHeinrich P Linear Raman Sprctroscopy on Droplet Chains: A New Experimental Method forthe Analysis of Fast Transport Processes and Reactions on Microparticles Appl Spectrosc49:1215. © (1995) Society of Applied Spectroscopy


external field, as we have seen in Chap. 3. This important point is examined inSect. 8.6.

If the refractive index depends on the concentration (composition), the localfield and the intensities of the Raman lines depend on concentration too. Anothersource for deviation from a linear dependence on the incident intensity isintermolecular forces . This effect depends on the chemical components in thesample, and the effect is usually weak for nonpolar liquids . A quantitativeprediction of this effect is difficult because it depends on the distribution and typeof molecular neighbors, on the interaction strength between dissimilar neighborsand on the electronic properties (transition frequencies). Complexes are oftenformed, which affect the band shape and line frequencies as well as the intensity,especially in aqueous solution . We will return to this point. The solvent effect isusually different for different Raman lines of the same solute . Several researchgroups have investigated this effect. Fini et al. (1968) investigated the solventeffect for some hydrocarbons, and other examples are given by Schweiger(l990c). These effects are usually not large and can often be evaluatedquantitatively by calibration measurements on bulk material under well-definedconditions.

8.3.5.7Bandwidth and BandShape

Intermolecular forces often not only affect intensities but also the bandwidth.Band intensities can be enhanced or reduced due to the presence of other chemicalcomponents. In the gaseous state at moderate pressures the effect is negligible inmost cases. In the liquid state, however, where the intermolecular distances areshort, the bandwidths and shapes are usually affected. The Raman band of CO2,for example, is broader in the liquid state, and the opposite holds for S02.

Not only the width but also the shape can change by a transition from thegaseous to the liquid state. This is particularly pronounced in water, as can be seenin Fig. 8.7. In addition, Quillon and Le Duff (1973) and also Tanabe (1984) haveshown that the line width can depend on the solvent.Not only intermolecular forces affect the intensity , shape and width of Ramanbands, but temperature also does . Sometimes the interplay of intermolecular forcesand thermal motion can cause that the shape of Raman lines to depend sensitivelyon temperature. This is the case for water. The formation of hydrogen bonds" issensitive to temperature, as shown in Fig. 8.7.

An alternate approach to particle temperature measurement was explored byRassat and Davis (1994). They used Stokes/anti-Stokes Raman measurements todetermine the temperature of electrodynamically levitated microparticles. Figure8.8 shows Stokes and anti-Stokes peaks for a 52 urn particle of Ca(N03h that washeated from two sides using a split infrared laser beam (6.2 W, A =10.6 urn). An

22There are also other interpretations for the temperature dependence as discussed by Walrafenet aI. (I 986a,b) and Zhelyaskov et aI. (1988) .


I(J(XXITre:

~:.5

::::: 20.~ ~l«XXI 35 // /~7..

50

fI • //

65 I' .///6CXXI 1/://

v:: 0 I . I.-Z ;, : ,/1/

~ 4000 1 I

ii/Ill

2000 #j~/~

OL......~..........~.........L~~..L-.o.-~ ..........~~'-'-~..........~.........L~..........

3000 3100 3200 3300 3400 3500 3600 3700 3800

RAMAN-SI-IIFr. em'

Fig. 8.7. The effect of temperature on the OH stretching band of water

600

400

>- 200I-enzwI- a~w>i= 16:5wa: 12

8

4

a

I-- (a)

I--

I-- Stokes

I--

... !~............ anti-Stokes

(b)

I--

anti -Stokes:

~

-, . ', ~ ~.,r

. ..',- , J r · , o'. .:.;.':.,;;~'>:~;';;: :~';~:~;;.''.. .; ~ :j'

"....

~~:"'-\ :.- ....

1200 1100 1000 900 800 700RAMAN SHIFT. em ·1

600

Fig. 8.8. (a) Stokes and anti-Stokes Raman spectra for a 52 11m Ca(N0 3)2 microparticle heatedwith an infrared laser. The N-O symmetric vibrational stretch mode at 1065 em" and the in-plane

bending vibration mode at 740 em" are observed. (Rassat and Davis 1994)'

Reproduced with permission from Rassat SO, Davis EJ (1994) Temperature measurement ofsingle levitated microparticles using Stokes/anti-Stokes Raman intensity ratios, Appl Spectrosc48:1503. © (1994) Society of Applied Spectroscopy

(8.95)


argon-ion laser (0.5 W, A = 488 nm) was the illumination source for Ramanscattering. The peaks can be identified as the N-O symmetric stretch mode at 1065cm-' and the in-plane bendin g vibration mode at 740 crn'. The peaks near 1065cm' were used to calculate the temperature. Figure 8.8a shows a very weakintensity for the anti-Stokes Raman scattering compared with the Stokes peak. Theanti-Stokes spectrum is magnified in Fig. 8.8b to show that it is readily identified.

Rassat and Davis fitted the relatively narrow band near 1065 em" and used theStokes/anti-Stokes intensity ratio to determine that the temperature was 450 K. Byvarying the infrared laser power they were able to explore the temperature as afunction of the laser power.

The quantitative evaluation of Raman spectra is usually quite straightforwardfor gases, but can be complicated for liquids. For application of this technique toaerosol analy sis, this is not a disadvantage, becau se the different effects can bedetermined by Raman scattering on bulk materials .

8.3.5.8Polarization of Scattered Light

The polarization of the Raman lines depends on the transition polarizability tensorand whether or not the molecules are freely orientable. In this case, the rotationallines are usually highly depolarized compared with the vibrational lines. In theliquid state, the intermolecular forces complicate the situation. The interactionprocesses can affect the symmetry properties of the molecules and change thepolarizabilit y, but it can also hinder the free rotation of the molecules. Bothinteraction processes influence the degree of polarization of the scattered light.Quantitatively, the polarization can be calculated from the transition polarizab ilitytensor in much the same way as for elastic scattering. The only difference is thatthe elements of the tensor for Raman scattering have to be used. Qualitatively, theRaman bands of liquids are usually more or less polarized parallel to the incidentradiation.

8.3.6Resonance Raman Scattering

One important restriction of the polarizability theory is that the frequency of theincident radiation must be far from all eigenfrequencies. There are often situationswhere the incident radiation corresponds to or is at least close to an absorptionfrequency. In this case, the use of a mean frequency is not justified . Suppose thatthe incident frequen cy, COo, is equal or nearly equal to one of the intermediatefrequencies, 0lJ<r- In the development of Eq. (8.70), damping was ignored and oneelement of the polarizability tensor would become infinity. In reality, damping isalways present, but this term would nevertheless become dominant. Introducing adampin g constant, I', and assuming OlJ<r - W,e Eq. (8.70) reads:

(ao ) =.!.I [(M ~(q,))oJM u(q ,))eJ kna~ kn h r 00; -00

0-if


The intermediate states have all the same electronic state, e, but differentvibrational , rotational states, r.

The dipole transition moment operator is only a weak function of the nuclearcoordinates, q, and can be expanded about the equilibrium position, qo. For aparticular mode, the expansion reads

, ()_ ' ( ) [dMoe(q)]MOe q - MOe qo + dq qo q + ..... (8.96)

We introduce the Taylor expansion given by Eq. (8.96) into Eq. (8.95), use Eq.(8.71), designate particular vibrational states by Vk or vr and get

(8.97)

The electronic transition moment, Mtl'" is much larger than its derivatives. Thepolarizability is, therefore, dominated by the first term in Eq. (8.97) as long as thevibrational overlap integrals (Frank-Condon factor), (vkivr) and (VrIVk)' do notvanish. Although the first term ofEq. (8.97) looks quite similar to Eq. (8.73), thereare some important differences. In Eq. (8.73) the summation includes allelectronic states, in contrast to Eq. (8.97), where only one intermediate electronicstate is included . In the non-resonance case , the differences between the incidentfrequency and the frequencies of the intermediate states are large, and dampingcan be ignored. In resonance Raman scattering the incident frequency is not muchdifferent from the intermediate frequencies, (0/, so the denominator is small, andthe polarizability can be orders of magnitude larger than for non-resonant linearRaman scattering. Enhancements as much as 106 have been observed.

There are a number of textbooks which contain chapters on the theory ofresonance Raman scattering (e.g., Lee and Albrecht 1985; Behringer 1974; Martinand Falicov 1975; Rousseau et al. 1979). Overviews of experimental application sof resonance Raman spectroscopy in various fields can be found in the articles ofSpiro and Loehr (1975), Kiefer (1977), Stockburger et al. (1986) , and Harada andTakeuchi (1986).

The practical application of resonance Raman Scattering for microparticleanalysis is hindered by the lack of suitable light sources . Examples of theapplication of this technique are given in Chap. 9.

8.4 Absorption and Emission of Radiation 529

8.4Absorption and Emission of Radiation

Light can interact with matter in a number of ways. Some are readily observablein everyday life. Everybody has seen the dispersion of light by scattering fromwater droplets in fog or clouds. It is common experience that light can passthrough some materials seemingly without any attenuation. Some materials seemto be completely impenetrable by light; others weaken the intensity withoutchanging the color , and some selectively weaken some colors but not others.These absorption processes are invest igated in this section as well as the oppositeprocess, the emission of light.

In this section, we consider two processes of light emission, luminescence andthermal emission . Luminescence compri ses the two processes, fluorescence andphosphorescence, illustrated in Fig. 8.9. Fluorescence is a relatively fast process .After light is absorbed, it decays very quickly by internal conversion to a lowerenergy state. This internal conversion is a non-radiative proces s by collisionalvibrational relaxation . If the multiplicity" is changed, such a non-radiating processis called intersystem crossing. In this case, the electron has to flip its spin. Oncethe electron is in the triplet state, it may persist in this state for quite a while . Toreach the singlet ground state by radiation, it has to change its spin orientation.

Spin change is an improbable process, and the lifetime is much longer than fortransition s within the singlet or triplet states. In this so-called phosphorescenceprocess the time between absorption and reemission is delayed . We will restrictour discussion to fluorescence .

SINGLET-STATES TRIPLET-STATESSPIN

ORIENTATION

'I:~INTERNAL CONVERSION10''' ,

~~

SPINORIENTATION

~ii'

Fig. 8.9. Jablon sky diagram , showing radiative (straight vertical lines) and non-radiative (wavylines) transitions in a typical molecule, with two electrons in the highest orbit

23The multiplicity depends on the orientation of the electron spins and is three for two electronswith parallel spin and one for two electrons with anti-parallel spins.


The other process treated in this section is thermal radiation or, more precisely,black body radiation. This emission process is characterized by thermalequilibrium between emission and absorption processes . Thermal emission, that is,the emission of light by a hot body is certainly the most frequent process of lightemission .

8.4.1The Electron Oscillator Model

The starting point of our analysis is again the polarizability. In Sect. 8.2.1 weanalyzed the effect of thermal oscillations on the polarizability. The frequencies ofthese oscillations are very different from the frequency of the incident light. Wewill now treat the molecule as a classical oscillator that is damped.

Assuming that the electric field points in the z- direction , the equation ofmotion reads,

•. . 2 eE'z + yz + Ole Z = - - ,

m(8.98)

(8.99)

where m is the mass of the electron, y is the damping constant, and We is theeigenfrequency (resonance frequency) . The local electric field, E' , acts on theelectron with charge, e. Multiplying both sides by -e, we obtain the equation ofmotion for the dipole moment , p =-ez,

a2 a e2

--R+y-E. +Ol2p = - E' .at 2 at e m

If the electric field oscillates with the frequency ~, the steady state solution ofthis equation is

(8.100)

The energy of the oscillating dipole is taken from the applied oscillating field.The total energy, Etot, stored in the system is given by

(8.101)

(8.102)

where Po is the amplitude of the dipole moment. If the external field is switchedoff, the oscillator continuously loses energy by emission of radiation. The fractionof this energy lost per cycle follows from Eq. (3.55) and is

(2n liE lot J_p~ Ol3

- -;;;-~ - 6ec 3 •

We eliminate the dipole amplitude using Eq. (8.101) to obtain

8.4Absorption andEmission of Radiation

~E lot = e2w

2E

~t 61tcmc3 101 ·

531

(8.103)

The relative energy loss per cycle is extremely small; it is of the order 10-8

;

therefore, it is justifiable to write Eq. (8.102) as a differential equation with thesolution

where

61tcmc3

't= 2 2 •eW

(8.104)

(8.105)

(8.106)

This result is accurate to within an order of magnitude of the quantum mechanicalcalculations.

8.4.2Line Shape

In the previous section we considered the emission process for one particularfrequency, but the light emitted by physical light sources always extends oversome frequency range. In this section we will discuss the most import antprocesses that cause spreading of the emitted light over a finite spectral range.

8.4.2.1Natural LineShape

We have shown that the emISSIOn of light can be represented within theframework of classical electrod ynamics as emission by an electrical dipole.Because of this emission of radiation , the energy must be reduced, and itsamplitude must decay. Furthermore, the dipole is subject to damped oscillation s.To include the effect of damping we write the electric field in the form,

E - E -yt12- e e cos wet ,

in which y is a damping coefficient. From the classical electrodynamics treated inChap. 3 we know that the power of electromagnetic radiation is proportional toE·E*. We conclude from Eq. (8.104) that (yl2) is simply the inverse of therelaxation time, r , given by Eq. (8. 105).

Fourier transformation of Eq. (8.106) yields

(8.107)


Integration gives

(8.108)

We are now able to calculate the spectral distribution of the radiant power, pew),as

(8.109)

Usually , y« ~ and the term (y!2)24 in the square bracket can be neglected. Inthat case the line width is small, and we can set 00;::: We. Thus, the equation for thespectral distribution reduces to

(8.110)

The spectral distribution of electromagnetic radiation is usually expressed by theline shape function, g((0), which satisfies the normalization condition

(8.111)

From Eq. (8.110) the line shape function follows

(8.112)

The line shape described by Eq. (8.112) and shown in Fig. 3.4 is called theLorentzian, or natural line shape. The full width of the distribution at halfmaximum, called FWHM is LlWi/2' which equals y or l/t, where "'C is the lifetime ofthe radiating dipole". We have seen that the damping of the oscillator causes abroadening of the emitted line. Because every spontaneous emission process froma microscopic oscillator (atom, molecule) is unavoidably connected with thisbroadening mechanism, it is called the natural broadening. This is a process thataffects all radiating dipoles in the same way and is, therefore, a homogeneousbroadening process .

24 The lifetime is l/y and not 2/y because it is related to the energy relaxation and not therelaxation of the electric field amplitude.


8.4.2.2Doppler Broadening

Some broadening processes act individually on each oscillator (atom, molecule).Such processes are called inhomogeneous broadening processes. An importantexample is Doppler broadening , which depends on the velocity of the oscillatorrelative to the observer. Atoms and molecule s in thermal motion cover a widerange of velocities, and, consequently, the Doppler broadening is aninhomogeneous proces s. If We is the frequency of molecules observed at rest, agroup of molecules moving away from an observer with velocity vz' will have thefrequency given by

(8.113)

The line shape function of the radiation emitted by a group of molecule s movingwith velocity Vz is given by

(8.114)

For a system in thermodynamic equilibrium, the probability that a group ofmolecules with mass m has velocities between Vz and Vz + dvz is given by theMaxwell-Boltzmann distribution function. The fraction of molecule s havingvelocities in that range is (see Chap. 6)

[ )

1/ 2 ( 2 Jdn m mv,-; = 2nkBT exp - 2kBT dv z .

(8.115)

Multiplying Eq. (8.114) by the fraction given by Eq. (8.115) and integrating overall molecular velocities, one obtains the line-shape function called the Voigtfun ction, which has been numerically integrated and tabulated

[ 2)mv z1/2 ~ LlWhomog exp - - -

g(W) = [_m_) f [ 2kBT rv, . (s .1l 6)

2nk T ( )2 [ )2B -~ 2n W _ We +We vcZ + LlW;mog

If the natural line width Ll~omog is small compared to the Doppler broadening, itcan be approximated by a delta function and the integral of Eq. (8.116) can besolved to get the Doppler (line) profile and


( ]1/2 {( ]2 ]I mc' W-We mc'

g(w)= - -- ex - - - --We 21tk BT We 2k BT .

The FWHM (full width at half maximum) line width is given by

_(8kBT ln 2) 112~WD - 2 we·

mc

(8.117)

(8.118)

Except at temperatures far below room temperature, the Doppler width for gases isat least an order of magnitude larger than the natural line width.

8.4.2.3Collision Broadening

Colli sions represent an important mechanism for broadening the line width. If amolecule co llides with another molecule during the emis sion process, the collisioncauses a statistical phase jump in the oscillating dipole. Thi s dephasing scramblesthe phase of the great number of molecular oscillators and increases the effectivedecay of the macro scopic dipole moment. The short distance between twomolecules during colli sion can cause a shift in the resonance frequency, and thisprocess also contributes to line broadening.

8.4.3Depolarization

The treatment of the polarization of the emitted light is completely analogous tothe corresponding section of Raman scattering. The only difference is that theelements of the polarization matri x are now the elements of the dipole transitionmatrix and are, therefore, different from the corresponding elements in Ramanscattering. Generally, the orientation of the dipole moment in gases and liquids iscompletely random, and the light is completely depolarized, that is, natural light.Thi s is not the case for stimulated processes, for stimulated light has the samepolarization as the stimulating light.

8.4.4Einstein Coefficients

In the classical treatment, the emISSIOn of radiation is due to electric dipolemoments in thermal motion undergoing damped oscillations. Thi s description is inquite good agreement with observed line shapes, but it doe s not explain why therad iation emitted from atom s or simple molecules is concentrated in small spectralranges, the spectral lines. Niel s Bohr, who postulated that atom s and moleculeshave certain energy states, where the electrons can dwell without emission ofradiation, gave the explanation. The se stationary states are called eigenstates. Inthese states the atom or molecule has no time-dependent dipol e moment and emitsno radiation. Thi s cannot be explained by cla ssical electrodynamics and is one of


the famous Bohr's postulates. Straight lines in Figs. 8.3, 8.4, and 8.10 symbolizethese states .

In his analysis of the photoelectric effect, Einstein discovered that radiation wasabsorbed in discrete energy packets. With that discovery it quickly became clearthat radiation is also emitted in discrete quantities now called photons. Thecorpuscular theory of light championed by Newton experienced a renaissance.Each photon carries a discrete quantity of energy, the photon energy, Ep, which isequal to the energy difference between the initial state, the energy state of themolecule before the emission, and the final state, the energy state after theemission , and is related to the frequency by

(8.119)

where ~} is the frequency of the radiation, and. Here E2 and E( are the energies ofthe excited and the lower states, respectively . Obviously, E2 must be larger thanE l . In Einstein's concept, three types of radiative transitions are possible, and theseare illustrated in Fig. 8.10. Transition from higher energy levels can occurvirtually without any external stimulation". These transitions are calledspontaneous transitions. Transitions from the lower level E} to the higher level E2

induced by the incident radiation are called absorption. The reverse process iscalled stimulated emission

E

E2 1-- - - ---.- - - ...,-- - - - - - - -.- - - - -

ABSORPTION

SPONTANEOUSEMISSION

STIMULATEDEMISSION

E11-- ..........---------J.--------.........---

Fig. 8.10. Emission and absorption transitions in a two level system

26 Today one knows that these transitions are closely related with the so-called vacuumfluctuations .


8.4.4.1Spontaneous Emission

It is reasonable to assume that the number of emission processes per unit time,ZezI. is related to the number of molecules or atoms that can, in principle, radiate.These are the electrons in the excited state with energy Ez. As before, wedesignate the volume of interest by V, the number of molecules (or atoms) per unitvolume by N, and the fraction of molecules in the energy level E, by nz. The totalnumber of transitions per unit time in volume V is given by:

(8.120)

(8.121)

The quantity AzI. the Einstein coefficient for spontaneous emission, is the numberof transitions from Ez to E, per molecule per unit time. The lifetime, t2] , of theexcited state", the state with energy Ez, is related to the Einstein coefficient forspontaneous emission by

t ZI =1/ Azl.

The power emitted by volume V is

<1>;1 = Z;I!iwZI = VNnzAzl!iw. (8.122)

Setting <l>ez1 = <l>spont , where <l>spont is the quantum mechanical expectation of thepower emitted per molecule for a transition from n = 2 to k = 1 given by Eq.(8.64), we can solve for AZI to obtain a relation between the Einstein coefficientand the quantum mechanical transition moment,

W3

I IzA z_l_M21 - 31tEc 3!i ZI ·

(8.123)

The numerical value of AZI can be calculated, in principle, using quantummechanics, but exact calculations are available only for atoms with a moderatenumber of electrons.

8.4.4.2Absorption and Stimulated Emission

Light is absorbed as well as emitted, and the number of absorption processes, Z'IZ,depends on the power density, p(W12), per unit bandwidth of the incident radiationin the frequency range from WIZ to W12+dw. In addition, the absorption ratedepends on the number of molecules in the initial energy state Elo Einsteindetermined that the rate of absorption processes is

z, =p(wlz)vNnIBIZ ' (8.124)

27 If transitions from Ez to other levels with energies E, are also possible, the lifetime of theexcited state is reduced and is 1I"tz =Lj 1I"t,;, where t ,; is the lifetime of a transition from E, toall lower states E;.

8.4Absorption and Emission of Radiation 537

where B12, the Einstein coefficient for absorption, is a material property. Itdetermines how "easily" the incident radiation stimulates an energy change withinthe absorbing material by a transition from the lower energy level to a higherenergy level. The Einstein coefficient for absorption quantifies the energy transferfrom the radiation field to the material.

If the incident radiation field stimulates upward transitions, that is, transitionsfrom lower energy levels to higher energy levels, it is reasonable to assume thatthe radiation field can also stimulate the reverse process, downward transitions.The transition rate for this stimulated emission process is given by

Z~l = P (W21)VNn 2Bz1' (8.125)

where B21 is the material property called the Einstein coefficient for stimulatedemission.

8.4.5Black Body Radiation

Einstein calculated the energy density of black body radiation by assuming that inthermal equilibrium the emission processes must be balanced by the absorptionprocesses, and hence,

(8.126)

(8.128)

(8.127)

(8.129)

(8.130)

Using Eqs. (8.120) , (8.124), and (8.125) and equating the frequencies for upwardor downward transitions (W12 =Ulzl =to), we obtain

VNnzAZl + VNnzp(w)gzBzl = VNn1P(w)B12nl ,

which may be rearranged to give

nz p(w)B12

~= A21+P(w)B z1

The occupation numbers, nl and n2, are given by Boltzmann statistics, that is

~ = he-(Ez-E1)/k BT = he-IiOl /k BT,

n l g, g\

where g228 is the degeneracy of energy state, E2, and kB is the Boltzmann constant,and T is the temperature.

Introducing Eq. (8.129) into Eq. (8.128), yields the power density,

g A e- IiOl / k BT

( )_ 2 21

P W - -IiOl /kBT 'g,B 12 - g 2B21e

This equation should be compared with the radiation density of a black bodyderived by Planck,

28 The degeneracy gi is the number of different quantum states with the same energy E, see alsoSect. 8.3.4.


(8.131)

Based on Planck's equation, peW) -7 00 as T -7 00. To get the same result from Eq.(8.130) we must take

g lB12 = g2B21' (8.132)

Comparing the low temperature limits of Eqs. (8.130) and (8.131) , we find

Thus, the Einstein coefficients are related.To elucidate the significance of these results it is convenient to write Eq.

(8.131) in the form

(8.134)

The term identified as I on the right hand side is the spectral mode density of theelectromagnetic field in free space. The term II represents the number of photonswith energy liw per mode . We will discuss this topic in Sect. 8.4.7.

Comparing Eq. (8.134) with Eq. (8.133), one recognizes that the Einsteincoefficient Al2 for spontaneous emission corresponds to the Einstein coefficientB12 for a transition with emission into a particular mode multiplied by the spectralmode density of free space. One can speculate, therefore, that the mode densityaffects the number of spontaneous transitions. The mode density in a dielectricmesoscopic resonator is different from the mode density in free space .Consequently, the rate of spontaneous transitions is different than the rate ofspontaneous transitions in free space.

8.4.6Amplification of Light

In this section we consider the change in the total radiative power, dW, of a lightbeam travelling in the z-direction as it passes through volume element dV =dAdz,where dA is the cross-sectional area of the beam. The change in the radiativepower is

dW =Suo.z +dz)dA dQ-S(w,z)dA dQ =oS(w,z) dV dQ ,oz

(8.135)

where we have expanded the energy flux density, Suo.z), in a Taylor series. Thelight emitted and/or absorbed per unit time per space angle dQ by differentialvolume dV is


dW =oS(w,z) dz dA dQ =hi»[_1dZ;1+dZ~ 1 - dZ~Z] dQ . (8.136)oz 41t

Writing Eqs. (8.120), (8.124) and (8.125) for the volume dV, we can express Eq.(8.136) as

oS(w, z) _ dQ[ () ()]-l........:......!.dVdQ - nw- NnzA zl +P\W2l NnzBzl -p\w12 Nn]BlZ dV .(8.137)oz 41t

Let us introduce the relations,

S(oi) =.2:.... p(oi), Nj =Nn., and P(WI Z) =P(Wzl) =p(w),41t

where v is the velocity of light, into Eq. (8.137) to get

(8.138)

(8.139)

Since we are only interested in the coherent interaction, we ignore the first termon the right side of Eq. (8.139). The Einstein coefficients for stimulated emissionand absorption can be expressed in terms of AZI using Eqs (8.132) and (8.133).The Einstein coefficients given by Eq. (8.133) apply to black body radiation,which has a broad band of frequencies . The energy density, p(w), is a slowlyvarying function of to, For transitions in atomic or molecular systems, thespectrum of the emitted light is described by the line shape function discussed inSect. 8.4.2. The Einstein coefficients must be modified for this situation bymultiplication by the line shape function . Replacing c by v in Eq. (8.133) andmultiplying by the line shape function , we get

The solution of this differential equation is

S(w, z) = S(w,O)e -K Z ,

where

(8.140)

(8.141)

(8.142)

in which gz and g, are the degrees of degeneracy of the upper and lower levels ofthe transition and should not be confused with the line shape function, g(w).

The absorption coefficient, K, depends on the lifetime, 'tZl, of the excited stateand the difference in the occupation numbers, LlNz1. If the energy levels areoccupied thermally, this difference is always negative, the absorption coefficient ispositive, and energy is transferred from the light field to the medium. It is alsopossible to generate inversion, which means that LlNz] is positive. In this case,energy is transferred from the material to the electromagnetic field, that is, light is


amplified by stimulated emISSIOn of radiation . This is the basis of LightAmplification by Stimulated Emission of Radiation, that is, the LASER.

8.4.7Resonant Cavities

In the classical description of the interaction of radiation with a spherical dielectricmicroparticle discussed in Chap. 3, we have seen that such particles can undergooptical resonance. If resonances are excited, the strength of the radiation fieldswithin the particle can exceed the non-resonant case by orders of magnitude.These properties of microcavities can also be treated quantum mechanically, andwe shall sketch the quantum mechanical approach here. The reader with aninterest in more details is referred to the literature of quantum electrodynamics(e.g., Heitler 1954; Feynman 1962; Power 1964 or Loudon 1983).

Now the Einstein coefficient for stimulated emission can be calculated usingthe rules of quantum mechanics by introducing Eq. (8.123) into Eq. (8.133). Theresult is

(8.143)

This quantity depends only on molecular properties and universal constants . Theconcept of spontaneous emission is somehow unsatisfactory because it postulatesthe existence of a process that takes place "without reason" . It is more reasonableto assume that the spontaneous emission is proportional to some kind of "radiationdensity", Po. To get more information about Po , we compare the emission rates forspontaneous and stimulated emission. If we express B21 in Eq. (8.125) in terms ofA21 using Eq. (8.133), we see that the emission rate for stimulated emissionreduces to the emission rate for spontaneous emission given by Eq. (8.120) whenPo is given by

(8.144)

In effect, spontaneous emission can be interpreted as induced by vacuumfluctuations and radiation reaction, which both contribute to the energy density(see MiIIoni 1993). The energy density given by Eq. (8.144) is the mode densityof free space multiplied by the photon energy. We would not be surprised if themode density and transition rates in a cavity differ from those in free space. Thestudy of quantum mechanical effects in cavities has led to an offspring of quantummechanics called cavity quantum electrodynamics.

The following treatment outlines the presentation given by Ching et ai. (1996)in which they define a microcavity as a mesoscopic and open system. The gaugefor a radiating microcavity is the wavelength or size parameter x. For an atomic ormolecular system x « 1, and the situation is straightforward and well known. Ifthe system is macroscopic (x » 1), geometrical optics applies, and a photon canbe regarded as a point in space. The intermediate or mesoscopic case is more

8.5 Nonlinear Processes 541

complicated. Furthermore, energy leaks from the resonator, so it must beconsidered an open system.

For a complete quantum mechanical treatment, the electromagnetic radiationfield must also be quantized, and the state function include s the state of theradiation field . The transition rate remains proportional to the transition momentand may be written as (Ching et al. 1996)

Ze oc 1(1, y(s)IM. AI2,0)12

dn 8(00 - Q). (8.145)

Here 12,0) denotes the atom in state 2 with energy Ez, and the field is in thevacuum state. Similarly, 11 ,y(s) is the state function of the atom in state 1 withenergy Eh and the field in state s. Here M is the dipole operator, and A is thefield operator as before . The frequency Q = (E, - Et)lh is defined by the initial andfinal states of the atom, and dn sums over the photon states s with frequency to.The transition rate can be factorized into atomic and field dependent parts to give

(8.146)

It can be shown that the second part is nothing else than the mode density of freespace. Thus, Eq. (8.146) takes the familiar form

r:= ~1(IIMI2)12 Po(oo),3Eh

(8.147)

where we restored the numerical factor , and Po(oo) is the mode density given byEq. (8.144) . The density of states depends on the environment. In a mesoscopiccavity the electric field depends on posit ion and so does the field operator. In thiscase, Eq. (8.147) reads

(8.149)

(8.148)

with

r:=~I(IIMI2)12 p(oo,r),3Eh

l(y(s)IE!O)12

=p(ffi,r)doo .

The effect of the cavity on the mode density is a redistribution, which meansthe total number of modes remains unchanged . The mode density, however,depends on position and can sharply peak at the location of the resonant modes ofthe cavity . As a result, the emission rate within the mode volume can be greatlyenhanced. This dramatically reduces the threshold for stimulated processes such asstimulated Raman scattering, stimulated Brillouin scattering , or lasing.

8.5Nonlinear Processes

The sources of light scattering, spontaneous or stimulated, are inhomogeneities inthe optical properties of the material. If these inhomogeneities are independent ofthe incident light field, linear scattering takes place. Examples are elasticscattering and linear Raman scattering. In macroscopically homogeneous bulk


material, elastic light scattering is caused by thermal density fluctuation s. Thisprocess is also called Rayleigh scattering. The perturbation of the index ofrefraction caused by thermal vibrations and rotation s of the molecules causeslinear Raman scattering. Perturbations in the index of refraction can also becaused by sound waves, for sound waves modulate the local density therebyaltering the index of refraction. Light scattering caused by thermally inducedsound waves is called (linear) Brillouin scattering. These acoustic perturbationsmove through the medium with the velocity of sound and with statisticallydistributed amplitudes and frequencies. Light scattered by these perturbations is,therefore, Doppler-shifted.

In the preceding sections, we investigated scattering processes in which theperturbations in the index of refraction are independent of the incident lightwaves . In this section we investigate scattering processes by perturbations inducedby the light field. From a microscopic point of view, the coupling of the light fieldto the index of refraction takes place through the polarizability, while in themacroscopic sense, the coupling is via the polarization. As long as thepolarizability is independent of the applied electric field, the scattering processdepend s linearly on the incident field and cannot affect the material properties.The assumption of a polarizability that is independent of the incidentelectromagnetic field is not quite correct, for the polarizability is a weak functionof the applied electric field . Because of this weak dependence, it is reasonable torepresent the induced dipole moment as a power series of the form

p =aE+~~EE+~yEEE+ ..., (8.150)2 6

in which ~ is the hyperpolarizability, and y the second hyperpolarizability. Theparameters a , ~, y, ... are tensors of rank 2, 3, 4, ... In most textbooks, nonlineareffects are treated in the framework of continuum theory in which the polarization,P, (the dipole moment per unit volume) is used to describe nonlinear materialproperties. The polarization can also be represented as a power series in theelectric field

(8.151)

(8.153)

where the quantities X (2) and X C} ) are known as the second- and third-ordernonlinear optical susceptibilities". The relative ratio of these three susceptibilitiesis approximately I :5x1O-8:3x 10-15

. Alternately, the polarizability is oftenrepresented as the sum of the linear polarizability pO ) and the nonlinearpolarizability, p NL,

p (l) = EoX(I)E, and p NL = Eo(X(2)EE+ X(3)EEE+....) . (8.152)

The effect of the nonlinear susceptibility becomes clear if we consider theinteraction of a nonlinear material with an electromagnetic wave composed of twofrequencies, WI and (j}z, described by

E() E -iWll E - iW21t = Ole + 02e +C.C.

29 Different notations can be found in the literature. Boyd (1992) uses P = X(l)E + X(2):EE+X(3):EEE+..., Yariv (1989) prefers P = €oXE+2d:EE+4X(3):EEE+...

8.5NonlinearProcesses 543

The second order nonlinear polarization is found by introducing Eq. (8.153) into(8.151) and taking the second term only, yielding

p(2) = coX(2)E(t )E(t) = cox (2 ) [E~ le-2iw)1 + E~2e-2iw21+ 2Eo1Eo2e-i(w) +(2)1

(8.154)+ 2EolE~2e -i(Wl -(2)1 + c.c] + 2cOX(2)[EolE~1 + E02E~2].

The nonlinear polarization contains terms representing oscillations at frequencies200, and 2Wz. Thus, the nonlinear response causes f requency doubling . The thirdterm corresponds to oscillations with the sum-frequency, (00, + Wz), and the fourthwith the difference-frequency, (WI - Wz). This effect is called frequency mixing.The last term is time-independent and is known as optical rectification. Higherorder terms in the nonlinear polarizability cause frequency tripling and sum anddifference generation of three waves, and so on.

There are a number of good textbooks that deal with nonlinear optics , includingBlombergen (1965), Reintjes (1984), Shen (1984), Hopf and Stegeman (1986),Schubert and Wilhelmi (1986) , Delone (1988) , Agrawal (1989) , Butcher andCotter (1990), Boyd (1992), Yariv (1991), and Mukamel (1995). Armstrong'sarticle in the book edited by Chang and Campillo (1996) deals specifically withnonlinear optical effects in microcylinders and microdroplets.

8.5.1The Nonlinear WaveEquation

In Sect. 3.2 we have shown that the Maxwell equations can be transformed intowave equations for the electric and magnetic fields , Eqs. (3.17) and (3.18) . Theseequations hold for non-conducting materials with constant properties. We nowpresent a slightly different solution that is especially useful for the analysis of thepropagation of optical waves in nonlinear materials.

Introducing Eq. (3.7) into (3.6), we have

aE apVXH =Eo- +- (8.155)at at

This result together with the curl of Eq. (3.14) yields

a2E a2pVxVxE=-llco at 2 -Il at 2 • (8.156)

We can split the polarization into its linear and nonlinear parts, expressing thelinear part by Eq. (3.10), to get

(8.157)

Evaluating the left side, and writing co(1 +X(l ») =c(w), we obtain


(8.158)

(8.159)

(8.160)

For a dielectric material with spatially uniform properties, it follows from Eq.(3.5) that the first term on the left side of Eq. (3.158) is zero. We assume that theelectric field can be represented as a plane wave and get

_ V2E = {J2E

o +2ik JEo _ k2E ]ei(k.r-Oll )Jr2 Jr 0

. J2pNL= /lE{<o)<o2E e,(k.r-w t) - /l-- .

o J t 2

In the slowly-varying-amplitude approximation (SVAPA) , it is assumed that thetemporal change of any relative field amplitude, F, is small compared to thefrequency ; and the spatial change is small compared to the magnitude of the wavevector, that is,

I~II~~I« <OF' and I~II~:I« k F•

As a consequence of SVAPA, V2 E « kJE/Jr. Recalling that I-IE<o2 = k2, thereresults

(8.161)

8.5.2Stimulated Raman Scattering

Raman scattering is caused by thermal motion of the nuclei that produces amodulation of the polarizability. In Sect. 8.2.1 we assumed that these motions arenot affected by the incident radiation, an assumption that is well-justified foroptical radiation from classical sources because the intramolecular fields arestrong. However, the electromagnetic field in a focused laser beam can be so largethat its effect on the motion of the nuclei can no longer be neglected. Themolecule is subjected to forced oscillations. The equation of motion describing themolecular vibration is formally the same as Eq. (8.98), but we are now interestedin the forces acting only on the vibrational degree of freedom . This force is foundconveniently in terms of the work, W, done by the electric field on the dipolemoment, which is

W =~(p{z , t) . E{z, t)) =~ a{q, t). (E2 (z, t)) , (8.162)

where the angular brackets denote a time average over an optical period . Thus, theoptical field exerts a force on the vibrational degrees of freedom given by

8.5Nonlinear Processes 545

We know from Sect. 8.2.1 that the interaction of the incident radiation with asystem oscillating freely with a frequency filk, causes the generation of sidebandsoscillating with frequencies % ± filk.

For forced oscillations the phase is not random, and we can assume that theradiation field, E, is composed of the laser radiation, with amplitude, Eo, andfrequency, %, and a Stokes field with amplitude E, and frequency, Ws,propagating in the z-direction, that is,

E(z, t)= Ao(z)ei(koZ-uJot)+ As(z)ei(ksz-wst)+ c.c. (8.164)

Introducing this field into Eq. (8.163), we obtain the force acting on the moleculeswith beat frequency, Q = %-Ws, as

(8.165)

(8.166)

Here we define K = ko - ks. We can now write the equation of motion given byEq. (8.98) in terms of normal coordinates, qb designate the eigenfrequencies byWI<, and replace the Coulomb force, F = - eE, on the right hand side of Eq. (8.98)by Eq. (8.165) to get

•. . 2 - 1 [ au J [A A * i(Kz-Qt) ]qk +yq k + (J)kqk -- -- 0 se +c.c. .m aqk 0

Here WI< is the eigenfrequency of the kth vibrational mode. We look for a solutionof the form

qk = q~e i (KZ-QI ) + C.C. ,

and introduce this ansatz into Eq. (8.166) to get

Hence, we find that the amplitude is given by

(8.167)

(8.168)

(8.169)q~(Q)= 1 [ au JAoA; .m(J)~ -Q2-iyQ) aqk 0

If we introduce Eq. (8.167) into Eq. (8.3), using the amplitude given by Eq.(8.169), and assume a scalar polarizability, U= o; we obtain

( )- 0 [ I [ auJA A * i(Kz-QI ) ; (8.170)U,(J)k - U + ( 2 2 . ) - 0 se +c.c. .m\(J)k-Q -lyQ aqk 0


The nonlinear part of the polarizability, pNL , is found by multiplication of thesecond term of Eq. (8.170) with the incident field given in Eq. (8.164) . This yields

(8.171)

The nonlinear dipole moment contains several different frequency components.We extract the Stokes dipole moment for one particular mode, the kth, oscillatingwith frequency Ws and get

p NL(Z,t)= ~ 1 ) [au J2IAlAsei(kSZ-WSI)+c.c.. (8.172)mW~-Q2+iyQ aqk 0

We assume that the nonlinear polarization, p NL, is simply the product of theinduced molecular dipole , p'", and the number of molecules per volume, N. Thus

(8.173)

(8.174)

The amplitude of the nonlinear polarization is therefore

pNL(Z)= N 1 [ au J

2

1Aol2Ase iksz.

m (w~ _Q 2 +iyQ) aqk 0

The Raman susceptibility (second-order hyperpolarizability), XR, follows from

pNL (z) = 6XR (ws~Al AseiksZ.

Comparing Eq. (8.175) and (8.174) gives

(8.175)

(8.176)

Near resonance, Wo - Ws '" ~, so the Raman susceptibility can be approximated by

(8.177)

The Stokes-wave, the second term in Eq. (8.164) , satisfies the nonlinear waveequation. Using Eq. (8.175) in Eq. (8.161), the amplitude of the Stokes wavesatisfies the equation

8.6 Particle Specific Effects 547

(8.178)dAs . ~~ 6 IA 12A A--=)-- XR 0 S =-KR S'dz 2ks

and KR is obtained by using Eq. (8.177) in (8.178), recognizing that ks = NsWslcand separating the real and imaginary parts, to give

(8. 179)

If the frequency difference, Wo - fils , corresponds to the eigenfrequency, (1\ , theimaginary part of Eq. (8.179) vanishes. Solving Eq. (8.178), we see that theamplitude of the Stoke s wave grows exponentially with z. The complexabsorption coefficient, KR, depends only on the modulus of the complex amplitudeof the laser field . Consequently, stimulated Stokes amplification is a process forwhich the phase-matching condition is automatically satisfied, and Stokes Ramanampli fication is said to be a pure gain process.

8.6Particle Specific Effects

In the preceding sections of this chapter we have seen, that for all inelasticscattering proce sses, knowledge of the local electromagnetic field is necessary topred ict the scattering characteristics . In scattering processes on bulk material, thelocal field is usually a plane wave or the field of a Gaussian laser beam, and thecalculation of the local field is straightforward. However, the electromagneticfield within a microparticle is a complicated function of the size parameter and theindex of refraction. Let us now consider methods for calcul ating the inela sticfields inside and outside of microparticles, starting with exact methods based onelectrodynamic theor y. Then, we outline approximate methods using geometricaloptic s.

As we have seen in Sect. 3.6, the electric and magnetic field s within amicroparticle can deviate appreciably from the incident field s. The incident field,often the field of a plane wave, is deformed by the sudden change of the refractiveindex at the surface of the particle . This change acts on the incident and scatteredfield as well. Consequently, not only the transmitted field is affected by theboundary of the particle but also the radiation fields emitted by the induceddipoles are affected. The particle acts as a micro-lens on the incident and theemitted radiation.

The foc using effect of the incident radiation is illustrated in the left part of Fig .8.11. The ray paths were construct ed by geometrical optics. The figure shows thatrays refle cted from the back of the particle are concentrated to a focal spot that is


Fig. 8.11. Focusing effect of a particle on the incident radiation , the rays are truncated after thesecond reflection (left figure) and projection effect of a particle for the emitted radiation, the rays

are truncated after the first refraction or total reflection (right figure)

called hot spot . A second hot spot, not shown in the figure, is formed by internalreflection on the front site of the particle. This second hot spot is located close tothe front side and is much weaker than the first. The effect of the particle on thedipole radiation is shown in the right part of Fig. 8.11. The particle causesrefraction of the emitted light in a preferential direction . This projection effectdepends on the position of the dipole within the particle.

8.6.1Multipole expansion

We have shown in Sect. 3.6 how an exact solution for the transmitted andelastically scattered fields can be found from the Maxwell equations by expandingall fields involved into series of multipole radiation. The same procedure will befollowed here except that in the analysis of inelastic scattering the multipoleexpansion must be made for the incident radiation with frequency, %, and for theinelastic scattered or emitted radiation with frequency, COs.

To distinguish the transmitted field of the incident radiation from the fieldwithin the particle that oscillates with shifted frequency, COs, , we designate thelatter as the inelastic source function. This function is composed of twocontributions, the inelastic dipole field and the boundary field. The boundary fieldis generated by the partial reflection of the dipole radiation on the boundary of theparticle . Finally, we have to calculate the external inelastic scattered field.

Theoretical analysis of Raman and fluorescent scattering processes, which areof particular interest, was developed by Kerker and his coworkers (Chew et al.1976a,b) . Numerical results were presented for incoherent scattering (Chew et al.1976a,b) as well as coherent scattering (Chew et al. 1978). Reviews of thetheoretical analyses of elastic and inelastic scattering have been published by


McNulty et al. (1980) and Schweiger (1990c), while Davis (1992) and Widmannet al. (1998) reviewed Raman scattering by microparticles.

In the model outlined below, we assume that the transmitted field, oscillatingwith frequency Wo, induces dipoles moments that radiate at a different frequency,o)s. These dipoles may be induced by linear or stimulated processes. We knowalready from Sect. 3.6 how to calculate the transmitted field. The procedure is:

• Calculate the expansion coefficients, (XE(n,m) and (XM(n,m) for the incidentwave Einc(r,Wo), Binc(r,Wo).

• Expand the transmitted fields, Et(r,Wo), Bt(r,Wo), in a series of multipoles,using spherical Bessel functions as the radial functions.

• Expand the scattered fields, EscaCr,Wo), Bsca(r,Wo) , into a series of multipolesusing spherical Hankel functions as the radial functions .

• Calculate the expansion coefficients, cn(n,m) and dn(n,m), of the transmittedfields and the expansion coefficients, an(n,m) and bn(n,m), of the scatteredfields from the boundary conditions on the surface of the particle.

For a linearly polarized incident wave, the results of this procedure are given byEqs. (3.104) through (3.115) .

We restrict the following discussion to linear processes, and we neglectmultiple scattering processes. This means that we neglect the interaction ofradiation emitted from an induced dipole with other molecules in the scatteringparticle. With these simplifications, the procedure for calculating the inelasticallyscattered light is:

• Calculate the amplitude and orientation of the dipole moments, p, induced bythe transmitted field. The exact procedure depends on the type of scattering,e.g., for Raman scattering the induced dipole moment is given by the secondterm in Eq. (8.6) or by Eq. (8.67).

• Expand the induced dipole field of one particular dipole moment, p(r',Ws),located at position r' within the particle and oscillating with frequency, Ws'into a series of multipoles with expansion coefficients, aE(n,m) and aM(n,m).

• Represent the boundary field for this dipole as a series of multipoles withexpansion coefficients, bE(n,m) and bM(n,m).

• Represent the inelastic scattered field outside of the particle for this dipole as aseries of multipoles with expansion coefficients, cE(m,n) and cM(m,n).

• Calculate the expansion coefficients bE(m,n), bM(m,n), cE(n,m) and cM(m,n)from the boundary conditions on the surface of the particle .

• Repeat these steps for as many dipoles as necessary to represent the inelasticsource function to the accuracy needed.

• Calculate the incoherent sum of the scattered field of all dipoles for linearRaman scattering and fluorescence, or the coherent sum for coherent Ramanscattering or lasing.

(8.180)


8.6.1.1The Dipole Field

The representation of a dipole field by muitipole radiation is a standard problem inelectrodynamics, and we refer to the literature for a detailed treatment of thisproblem. Nevertheless, we sketch the main steps in the following. A verypowerful method for solving electrodynamic problems is the definition of a scalarpotential cjl(r,t) and a vector potential A(r,t), which satisfy the Maxwell equations.The field quantities, E and B, are related to these potentials by

B(r,w)= VxA and E = -Vcjl- aA .at

These relations are introduced into the Maxwell equations. General solutions forthe potentials are

cjl(r,t) =-l-f [p(r',t')] dY' and A(r,t)=~f [J(r',t')] dY' , (8.181)4n£ Ir - r1 4n Ir - r1

where the integrals have to be evalu ated at the retarded time t = t'-I r - r' IIv, inwhich v is the velocity of the electromagnetic wave. If the current density is thatof a dipole located at r ' and oscillating with frequenc y to, the vector potential isgiven by

ikl r -r'l

A ( ) . 11 I ') - jto t edip r .t = -\-w'p\r e -1--1'

4n r - r '

We omit the time-dependent part, as we did before.The multi pole expan sion of the last term for r > r' is (Rose 1955)

ikl r-r] m=+n

Ir-r1

= i4nk~jJk lsr')h ~l ) (kr)m~n Y:.m(a' ,cjl') · Yn,Ja,cjl)

= L([h ~n(kr)X n ,m(e,cjl)] . UJkr')X: .m(a',cjl')]n,m

+:2V[h ~I)(kr)Yn.Ja,cjl)] . VuJkr')Y:,Ja',cjl')]

+:2vx[h ~l )(kr)xn,Ja,cjl)] . V' x UJkr')x:.Je',cjl')]}

(8.182)

(8.183)

Introducing Eq. (8.183) into Eq. (8.182) and calculating the magnetic field from(8.180), we obtain

B(r, w)= urokL(p ,Un(kr')X: ,m(a',cjl')]V X k n (krix.; (a, cjl)]n.m (8.184)

+n- v: X Un(kr')X: .m(a', cjl' )]k 1) (kr rx.; (e, cjl)]).

(8.185)


Comparing this result for the magnetic induction with Eq. (3.96), we obtain theexpansion coefficients for the dipole radiationas

AE(n, m)= /lWck(p.v:« Gn (kr')X~ ,m (eA')])

AM (n,m] = i/lW2k(p. Un (kr')X ~,m (e', <1>')]).

Let k1s be the wave vector of the dipole radiation within the particle, where theindex, I, refers to the interior of the particle as before, and S refers to the scatteredradiation. Using the notation of Druger et al. (1987) and Lange and Schweiger(1994) and setting the functions fn(kr) and gn(kr) in Eqs. (3.95) and (3.96) equal tothe spherical Hankel function, hn(l)(k1Sr'), the expansioncoefficients for the dipolefield become"

where

AE= aE(n,m)= p(r',wo,ws)' VE(n,m~

AM = aM (n,m)= p(r',wo ,ws) · VM(n,m]

VE(n,m)=c2kl~ ~l V'XUn(klSr')X~,m(e',<1>')],1

VM(n,m)= iC2kl~ ~12 jn(klSr')X~ ,m (e' , <1>').1

(8.186)

(8.187)

The boundary conditions that apply to the inelastic scattered fields are [see Eq.(3.103)]

nxEsca =nx(Eb +EdiP ) ' nxHSca =nx(H b +H diP ) (8.188)

where the subscripts sea, dip and b refer to the scattered field, the dipole field andthe boundary field, respectively.

Again, Eqs. (3.95) and (3.96) apply for representing the fields. For theboundary fields, the expansion coefficients are designated bdn,m) and bM(n,m),and the radial functions are the spherical Bessel functions, jn(klsr). Similarly, forthe scattered field, we write the expansion coefficients as cE(n,m) and cM(n,m)and choose spherical Hankel functions, hn(l)(k2Sr), as the radial functions.

Introducing the boundaryfields, the dipole fields, and the scattered fields in Eq.(8.188) and using the orthogonalityof the vector spherical harmonics, we obtain aset of equations from which the expansion coefficients can be calculated. Theequations are

jn(k1sa)bM(n,m)+ h~l)(klSa)aM(n,m)= h~I)(k2sa)cM(n,m]

~2jn (k1sa)bE(n,m)+~2h~1)(klSa)aE (n,m)= ~lh~I)(k2Sa)cE[n.rn]

Ni",: (k1sa)bE(n,m)+ Ni~: (k1Sa)a E(n,m)= N~~: (k2Sa)cE(n, m]

~2"': (k1sa)bM(n,m)+ ~2S: (kISa)aM(n, m)= ~lS: (k2Sa)cM(n,m]

(8.189)

30 We remind the reader that we are using the SI system here in contrast to the references cited .


where \lfn(x) = xjn(x), and ~n(x) = xh/)(x) are Ricatti-Bessel functions.The boundary field is given by Eqs. (3.95) and (3.96) with the following

specifications

fn(kr) = gn(kr) = jn(kISr'), (8.190)

(8.191)

bE(n,m)= ~~ (klSa)~Jk2Sa)-(Nl~2/N2~J~n,(klSa)~~ (k2Sa) aE

(n,m}, (8.192)(Nl~2 /N2~1)\Ifn(klSa)~n (k2Sa)- \Ifn(klSa)~n (k2Sa)

bM

[n,m)= (Nl~2/N2~1 )~~ (klSa)~n (k2Sa)- ~n,(klSa)~~ (k2Sa) aM (n,m}, (8.193)\Ifn(klSa)~n (k2Sa)- (Nl~2 / N2~ 1)\Ifn(klSa )~n (k2Sa)

where aE(n,m) and aB(n,m) are given by Eq. (8.186). For non-magnetic particlesthe ratio NI~i N2~1 is m, where m is simply the relative index of refraction.

Finally, the inelastic scattered fields are obtained from Eqs. (3.95) and (3.96),using

fn(kr) = gn(kr) = h~I)(k2Sr)

AE(n,m)= CE (n,m)= fE(n)aE(n,m},

AM (n,m)= CM (n,m)= fM(n)aM (n,m)

(8.194)

(8.195)

(8.196)

(8.197)

Computation time is still a point of concern in inelastic scattering calculations.The contribution of a large number of dipoles has to be calculated to map thecomplicated field distribution of the transmitted field. The contribution of eachdipole has to be determined by a separate multipole expansion. The number ofexpansion terms is approximately of the order of the size parameter. As a result,multipole expansion techniques become very time-consuming for large particles.This led to the development of alternative computational techniques such as raytracing described in the next section.

8.6.2Ray Tracing

We have seen in Sect. 3.10 that geometrical optics provides us with a powerfultool for calculating the transmitted field and the elastically scattered field.Calculating the elastic scattering by spherical or rotationally symmetric particles


using the exact methods considered in Chap. 3 is no longer a problem of longcomputation times with the modem computer. With PCs, the transmitted andscattered fields can be calculated in minutes, even for large particles . Ray-tracinghas an important advantage in computation time for inelastic scattering. However,the main advantage of ray-tracing is that it can also be applied to complicatedshapes, as we have pointed out in Sect. 3.10.1.

Velesco and Schweiger (1999) showed that even for relatively small particles(x =30) ray tracing requires two orders of magnitude less CPU time than thedipole model. They used a reversed ray tracing technique , which is illustrated inFig. 8.12. The conventional method used in analytical ray tracing (ART) to findthe scattered radiance is as follows : The rays emitted from a representativenumber of points in the particle are traced from the point source to the far field.An appropriate number of reflections on the surface must be taken into account.Then the contribution of the point sources that contribute to the emission in agiven angular range must be summed. It is not known a priori which of the raysemitted by the point sources propagate in a specific direction after refraction andone or more reflections. Consequently , a numerical procedure has to beimplemented to find all of the rays contributing to scattering in a particulardirection .

In reversed ray tracing (RRT), rays are traced that impinge on the particle fromthe scattering direction. All those point sources in the particle contribute to thescattering in the direction given by the reversed ray that are located on the path of

analyticalray

tracing

L.

pointsources

o

reversedray

tracing

y

Fig. 8.12. Illustrationof the concept of reversed ray tracing compared with analytical ray tracing(Velesco and Schweiger 1999)'

• Reproduced with permission from Velesco N, Schweiger G (1999) Geometrical optics calculation of inelastic scatteringon large particles, Appl Opt 38:1049. © (1999) Optical Society ofAmerica


the reversed rays through the particle. Their relative contribution to the scatteredradiance is identical to the transmitted field generated by a plane wave incident onthe particle from the direction of interest. However, this is only correct if thepolarization of the scattered light is perpendicular to the scattering plane . If this isnot the case, slight modifications are necessary , but the concept of RRT is stillapplicable. Some results of this method are given in the next section.

8.6.3Scattering Cross Section

For practical applications, the differential scattering cross section is of majorinterest. In the following sections, we present exact methods to calculate thedifferential and total scattering cross sections using the multipole expansiontechnique. This technique is very computer time-consuming and has been usedonly for particles with small x, so we also present some results based on raytracing methods. Although ray tracing is much faster than the multipole expansionmethod, its application is questionable for small x.

Kerker and Druger (1979) published some numerical results using themultipole expansion method, and geometrical optics was used successfully forparticles as large as x = 500 by Velesco and Schweiger (1999). Zhang andAlexander (I 992a,b) applied a hybrid technique to calculate the angulardependence for light emission of particles with size-parameters x =102_103

• Theycalculated the transmitted field using Mie-theory and used geometrical optics forthe inelastically scattered light.

The scattering cross section of an individual molecule located within amicroparticle is affected in a threefold way by the particle. Firstly, the curvature ofthe particle surface deform s the incident field so that the transmitted field differsappreciably from the incident field. We call this the focu sing effect. Secondl y, theamplitude of the induced dipole moment is affected by the partial reflection of theemitted light from the surface . This reflection causes an enhancement of the localfield depending on its radial position . This effect may also be investigated bymethods of cavity quantum mechanics mentioned in Sect. 8.4.5. Thirdly, thescattered field, the field outside of the particle, usually differs considerably fromthe dipole field in bulk material due to the refraction and diffraction of theoutgoing waves on the surface of the particle. We called this the projection effect.Quantitatively, we take this into account by defining a geometry function . Thisfunction modifies the dipole emis sion, depending on the position and orientationof the dipole within the particle.

The quantitative calculation of the transmitted field was treated in detail inSect. 3.6. Here we can concentrate on the effect of the particle on the emission ofradiation from a molecule located at an arbitrary position within the particle. Wedistinguish between scattering from a single molecule within the particle and theinelastic scattering of all active molecules within the particle . The corre spondingscattering cross section of the former we call the molecular particle scatteringcross section (MPS), and for the latter , the inelastic particle scattering crosssection (IPS). As common practice, we also differentiate between differential and

(8.198)


total scattering cross sections. Molecular cross sections will be designated by oand cross sections related to the particle as a whole by C, as before . Thesescattering cross sections are the ratios of scattered to incident radiation calculatedusing the corresponding quantities outside of the particle in the medium identifiedby the subscript, 2.

8.6.3.1Differential Scattering Cross Section

With the results of the foregoing section, we can calculate the power emitted intoa solid-angle element dQ from a dipole arbitrarily located in a microparticle. Weuse Eqs. (3.38) and (3.39) to express the power in terms of the magnetic induction ,B. Knowing that in the farfield the electric and magnetic field is perpendicular tothe scattering direction, and using the asymptotic form of the spherical Hankelfunction given by Eq. (3.123), we get

d = .!.r2 _C_IB 12 = 1

dQ 2 N21!2 sea 2N21!2ck;2

LI(- itt [cE (n, m)Xn ,m (8,<jl)+ N2cM (n, m)roxx.; (8, <jl)W ,n,m

where we have used Eqs. (8.194), (8.195). The index, S, refers to the frequency ofthe inelastically scattered light.

Using Eqs. (8.186) and (8.187) and the identity,

(8.199)

(8.200)

we can find, after some algebra, the intensity, d<jl/dQ, emitted in the space angledQ by an oscillating dipole located within a sphere. The size-parameters Xo and Xsare calculated for frequency , %, of the incident wave and COs of the scatteredwave, respectively. The result is

d N~II! S t ( ('\ )-=61tdiP-3--gQ xO,x S ,Nl' N2 , P O r /oro ,dQ NS21! S2

where dip, is the radiation emitted by an oscillating electric dipole in bulkmaterial, and dip is given by Eq. (3.55) using the optical properties of theparticle. The subscript, S, relates to the frequency of the scattered light. Thedirection of the induced dipole moment is Po, and scattering direction is given bythe unit vector roo We define the differentialgeometryfunction, gQ,as


n.m

The functions Yn,m are the spherical harmonics given by Eq, (3,99), and r'o is aunit vector pointing in the direction of r', the position vector of the dipole , The

180o

~ 10 1--1---;1';-....-::==---+--1-----1z~

>0::

~liD0::«

aC>

102 1-- - - +-- - - +-- - -+--1

10-11--- - - +----.--r-++-- - - -1

60 120

SCATTERING ANGLE, ~

Fig. 8.13. Angular dependence of the different ial geometry function gQ for a dipole withorientation perpendicular, L , and parallel , IH, to the scattering plane, The dipole is located atz =O.Ola, x =y =0 (dashed curves), and at z =0,07a, x =y =0; x =5, m =1.5, respectively . 1st

direction is always perpendicular to the z-axis (Kerker et al. 1978)'

* Reproduced with permission from Kerker M, McNulty PJ, Sculley M, Chew H, Cooke DD(1978) Raman and fluorescent scattering by molecules embedded in small particles: numericalresults for incoherent optical processes, J Opt Soc Am 68:1679. © (1978) Optical Society ofAmerica


1 HHO+-----+-------;

(j)

!::z::>>I«II:!::mII:«a

Ol

10 + 2 I-----------I-----+-;'--~

10 -1 1--4f---------i--------1

. -a 0 +a

POSITION ALONG THE Z AXIS

Fig . 8.14. Differentia l geometry function , gil for back-scattering, as a funct ion of position of the

dipole moment on the z-axis; x =5, As =1.5"'0' m =1.5. The incident radiation propagate s in thepositive z-direction and is polarized parallel to the scattering plane. The dotted curve gives the

power of the transmitted field along the z-axis, and the dashed curve shows the geometry

function for a dipole moment with constant amplitude (Kerker et al. 1978)'

orientation and magnitude of the dipole follow from the direction and magnitudeof the transmitted field, and for Raman scattering, from the tensor of thepolarizability change ; see Eq. (8.6).

Kerker et al. (1978) investigated Raman and fluorescent scattering by singlemolecules embedded in spheres with radii up to a size parameter of x = 5. Some of' their results were used to plot the angular dependence of the differential geometryfunction shown in Fig. 8.13. It is no surprise that the geometry function is nearlyidentical to the angular distribution of dipole radiation in bulk material if thedipole is located close to the center of the sphere, but the conventional dipolecharacteristic is altered appreciably if the dipole is shifted from the center. Thedependence of the geometry function on the position along the z-axis for back-

, Reproduced with permission from Kerker M, McNulty PJ, Sculley M, Chew H, Cooke DD(1978) Raman and fluorescent scattering by molecules embedded in small particles : numericalresults for incoherent optical processes, J Opt Soc Am 68:1679. © (1978) Optical Society ofAmerica


scattering is shown as the full line in Fig. 8.14. It was assumed that the induceddipole moment has the same direction as the transmitted electric field. This figurewas also redrawn from Kerker's data. The dotted line shows the focusing effect ofthe incident radiation, and the dashed line the projection effect. The particle causesincreasing focusing of light as the emitting dipole is shifted along the positive zaxis. The solid line shows the combined effect.

We assume a scalar polarizability", a, and get from Eqs. (8.1) and (3.121)

where Eine is the amplitude of the incident field, and EtO is the normalized transmitted field . Dividing Eq. (8.198) by the power density of the incident radiation,and using Eq. (8.201) and (8.202) we obtain the differential MPS given by

(8.205)

The same result follows from (8.200) after division by the incident irradiance, Sine'

The differential MPS is a relatively complicated function of the scatteringgeometry. The contribution to inelastic scattering from molecules located atdifferent positions in the particle can vary appreciably, as shown in Figs. 8.13 and8.14. This is important if the Raman-active or fluorescent molecules are nothomogeneously distributed within the particle. This space-dependent contributionto inelastic scattering has to be taken into account if Raman or fluorescencescattering characteristics are evaluated for evaporating multicomponent particlesthat have concentration gradients. We give examples in the next chapter.

In most cases, we are interested in the intensity of light scattered inelasticallyfrom all molecules in the particle rather than from a single molecule . We assumethat the spatial distribution of these molecules is described by the number densityfunction nA(r'), where A stands for a specific type of molecule. The differentialinelastic particle scattering cross section follows by multiplying the differentialMPS with the number density of the molecules and integrating over the particlevolume. This yields

raC Ajan part

f ra(JAjmOlnA(r')dV' .particle an

part

(8.206)

There is no analytical solution of this equation. For a numerical solution theparticle is divided into a number of sub-volumes, and the contribution of each sub-

3' We remember , in case of Raman scattering a is the gradient of the molecular polarizability . Influorescence a has to be replaced by a corresponding quantity relating the fluorescenceamplitude to the incident field.


2.5 ...---r----,---,---,----.-----,

2.0....:~-

V5Z 1.5wI-Z0 m= 1.50

~ 1.0w

§v>

0.5x =2.25

00 30 60 90 120 150 180

SCATTERING ANGLE, ep

Fig. 8.15. Perpendicular component, L; of light scattered inelastically for a microparticlehomogeneously filled with Raman active molecules for different size parameters. The incident

radiation is polarized perpendicular to the scattering plane (Kerker and Druger 1979)'

volume is represented by one dipole moment. The amplitude of this dipolemoment is proportional to the local transmitted field averaged over the subvolume. Usually, the size of the sub-volumes must be chosen to be small becausethe transmitted field can vary appreciably with position. The total number of subvolumes can become quite large for bigger particles and increases roughly withthe third power of the particle radius. This makes the application of this methodextremely computer time-consuming for larger particles ,'

Numerical calculations of Raman scattered light taking into account thepolarization were published by Kerker et al. (1978) and Kerker and Druger (1979)for incoherent optical processes, and Chew et al. (1978) calculated the angulardependence for coherent scattering on particles with size parameters x =1, 2, 3.An example of the angular dependence for particles uniformly filled with Ramanactive molecules is shown in Fig. 8.15. Vibrational Raman scattering is oftenhighly polarized in the same direction as that of the incident light, and the verticalcomponent shown in Fig. 8.15 has nearly the same shape as the differential MPS.

The angular dependence of the differential IPS for particles with sizeparameters up to x =435 was calculated by a modified ray-tracing technique byVelesco and Schweiger 1999. Example s are shown in Figs. 8.16 and 8.17. Themaximum of the angular distribution in the back-scattering direction is typical for

* Reproduced with permission from Kerker M, Druger SD (1979) Raman and fluorescentscattering by molecules embedded in spheres with radii up to several multiples of the wavelength, Appl Opt 18:1176. © (1979) Optical Society of America


Pxv

....

40

Reversed Ray Trac ing MethodClassical Dipole Solution

-.

20

1.6

104

1.2........0'-' 1.0~a..I1l 0.80''1' 0.6a.

004

0.2

0.0a 60 80 100 120 140 160 180

e (degree)

Fig. 8.16. Angular distribution of inelastically scattered radiance for a particle with sizeparameter, x =30 based on the incident radiation, and. x, =27 b~sed on the emitted radiation,

Velesvo and Schweiger (1999)

Raman scattering proces ses. This is in sharp contrast to elastic scattering wherethe scattering maxima are in the forward direction .

For elastic light scattering, methods have been developed for analyticallyintegrating the differential scattering cross section over a given space angle, Q(Chylek 1973; Wiscombe and Chylek 1977; Chu and Robinson 1977; Pendleton1982; Son et al. 1986). Using such methods , Pendleton and Hill (1997) showedthat the geometry function can be integrated over a circular aperture . The effect ofposition , orientation, and frequency of a single fluorescing molecule in a spherewas investigated by Hill et at. (1996) . Hill and his associates also studied theeffect of the illumination geometry (Hill et at. 1997b). Kerker and Druger (1979)published numerical solutions for particles up to a size parameter of 20.

Reciprocity methods have proved to be very useful to treat inelastic scatteringproblem s. The basic idea is that the farfield generated by a source inside amicroparticle is the same as the field generated by a plane wave at the position ofthe source. This concept is well known in geometri cal optics . Following Hill et at.(l997a), the mathematical formulation of this concept reads:

• Reproduced with permission from Velesco N, Schweiger G (1999) Geometrical opticscalculation of inelastic scattering, Appl Opt 38:1050. © (1999) Optical Society of America


where the 3x3 matrix, usually labelled G(ra,rb), is the dyadic Green's function(Chew 1995; Tai 1994). The Green's function of Eq. (8.207) obeys the reciprocityrelation,

(8.208)

where T indicates the transposed matrix. Given a solution for which a plane waveat r, generates an electromagnetic field (source) within a particle at r a, we canreadily calculate the farfield at r, generated by a radiating source located at r a

from the transposed matrix (Hill et al. 1996; 1997a,b).Ray-tracing is an attractive alternative to the exact methods of electrodynamics.

For larger particles , x > 10, the results of ray-tracing are quite accurate, andusually much simpler and faster than the exact methods . They are especiallyadvantageous for complicated geometries such as crystals or particles withinclusions .

A comparison of the angular distribution of inelastically scattered lightcalculated by RRT with the classical multipole technique is shown in Fig. 8.16.The incident light is a plane wave propagating in the z-direction and the scatteringplane is the yz-plane . Here Pxv is the inelastically scattered radiance for anincident wave polarized in x-direction . The scattered light is polarizedperpendicular to the scattering plane. Also shown is the inelastically scatteredlight, Py H, polarized parallel to the scattering plane if the incident wave is

4020o

/,/,II

/3 --,,_. InputTE~OutputTEt, t :

I ,- '.. - ,. Input TE~ I

, -+- Output TEt, J i0' \ Non resononceli\ , i

'1'2\ ,Ii/a. - \ ,Ii ,'/-

I ~I : /

~ /.' ,, \ /I. . ..... /

' - ' .. /-./ .-¥'" ..... ,'" / ,."' .............. ... , ..... ' ........, 11./ ....-....... __-e_--::~~:.-- .........._...._-.._.-~:.• ~

'~="'-:.f'':=f.:~'"

60 80 100 120 140 160 180

o (degree)

Fig. 8.17. Same as Fig 8.16. for the non-resonant case, for the input resonance TE508

24, emission

size parameter x, =435, and a double resonance, (input TEs08' 4 - output TE44/ \ Velesco andSchweiger (I 999}'

• Reproduced with perrmssion from Velesco N, Schweiger G (1999) Geometrical opticscalculation of inelastic scattering, Appl Opt 38:1051. © (1999) Optical Society of America

(8.209)


polarized in the y-direction. In Fig 8.17 the angular distribution is shown for alarge particle (x :::: 500) for the non-resonant case, for the case of the excitation ofan input, and for double resonance.

The dependence of the intensity of inelastically scattered light on themicroparticle size was investigated experimentally by Schweiger (1991). Hemeasured the intensity of Raman scattered light as function of the particle size inthe size range 60 ~ x ~ 120, and Vehring et aI. (1995) made similar measurementsin the size range 110 ~ x ~ 240. They found that the intensity is approximatelyproportional to the particle volume.

Only relative concentrations are usually of interest, and the explicit calculationof the differential MPS may be avoided under certain conditions. The followinganalysis is based on a concept presented by Schweiger (1987). The numberdensity, nA, of molecules of species A can be expressed as:

nA(r')= ~: PA(r')= ~: pcA(r'),

where No is Avogadro's number, MAis the molecular weight of species A, PA itsmass density, CA its mass fraction, and P is the mean particle density. IntroducingEqs. (8.209) and (8.205) into Eq. (8.206) and using the total scattering crosssection" for dipole radiation , O"dip, that is, the total emitted radiation given byEq.(3.55) divided by the local irradiance, yields the differential IPS in thefollowing form:

We define a morphology function , PQ , as follows

Pu(ko,ks,a, NS1' NSZ,cA (r')) =~ fgu(S,<j>,r')cA(r')dV',VC A particle

(8.211)

that depends on the concentration profile of the corresponding chemicalcomponent, the size of the particle, the wave vectors of the incident and scatteredwaves, and the indices of refraction . Assuming that the particle is illuminated by aplane wave with power density Do, the radiance, </lA, scattered into the space angleQ by molecules of type A, is given by

(8.212)

32 Note, the molecular scattering cross section is calculated with the transmitted incident field incontrast to the IPS where the scattered radiation is normalized by the external incidentradiation.


The ratio of light scattered inelastically on a particle containing molecules ofspecies A and B, respecti vely, is

(8.213)

(8.214)

in which m« and mB are the relative indices of refraction for the Raman frequenc yWA and CDs of molecul ar species A and B, respectively. In the gas phase NA2 == NB2and CA2 == CB2 and the second bracket can be set equal to I. All quantitie s in thefirst square bracket on the right side of Eq. (8.2 13) can be determined fromexperiments on bulk material. Usually Eq. (8.213) is used to determine the ratiocA/cB from measurements of <l>Aand <l>Bassuming Pn(CA) '" Pn(CB)' However, thisassumption is only justified as long as concentration gradients in the particle aresmall and WA '" CDs .

8.6.3.2Total Scattering Cross Section

Let us first calculate the total scattering cross section of a single molecule, theMPS. The molecule is locate at r. Next, we calculate the total inelastic scatteringcross section, IPS, of a particle with a distribution, c(r '), of Raman-activemolecules within the particle. The overall radiation emitted by a dipole located atr' follow s from Eq. (8.200) by integrat ing over the full space angle 41t. Thi sintegration can be carried out analytically and gives (Chew 1987, 1988a; Langeand Schweiger 1996; Holler et al. 1998)

!hmol 6!h N~,lls , f dQ 6 '" N~llls i G( ')'i' pan= 1t'i'dip-3-- go = 1tO/bulk- 3- - xo,xs, m,r,Ns2 1lS2 411 NS21lS2

The geometry function is

G(xo, xS' ms' r') = L (2n + l) f[n(n + I)Ijn (k IS~')1 2 (Po . r~)2n 1 klSr

+ I (k1srJn (k,ISr')) 2(PoX r~)2] IfE (n) 12

2 klSr

+~m~UJk lsr')12(Po xr~) IfM(n)1

2}.

(8.215)

Note that the geometry function depends only on the position and orientation ofthe dipole relative to the radius vector. The function G is sometimes also callednormalized rate or relative rate. A dipole orientated parallel to the radius vector,r ", couples only to TM-modes. A dipole orientated parallel to the surface of theparticle couples to TE-modes and TM-modes. This can be verified by experiment.

(8.216)


The scattering cross section from a particle with a distribution of Raman activemolecules follows by multiplying the MPS given by Eq.(8.214) with themolecular number density nA(r'), integration over the particle volume, anddividing by the power density of the incident radiation . Alternatively, we find theIPS simply by integrating Eq. (8.210) over the full space angle. For a scalarpolarizability this yields

C part = ,.,4",2 4 NS2 1l~11l 2 Nop Y- PA UJSu- ms CA '

N2 Ils2 M A

where

p(c A)= JpQ (ko,ks ,a,ms'c A(r'))dQ =~ J G(x o, xs,ms,r')c A(r')dY'. (8.217)41t cA Y V

The total power scattered by the particle and the total scattering cross sectionhave been investigated for various cases. Numerical results for certaindistributions of molecules are available (Kerker and Druger 1979; Lange andSchweiger 1996). Hartmann et al. (1997) calculated the total scattering crosssection for linear Raman scattering as function of time for slowly evaporatinghomogeneous droplets . The numerical results show good agreement withexperimental data.

For the special case of uniformly distributed molecules, cA(r') = const, themorphology function Pn reduces to the volume integration of the geometryfunction G(r') given in Eq. (8.215). This integral can be solved analytically (Chew1988a). However, the solution is quite complicated, so we present only thelimiting case of particles that are small compared with the wavelength (Rayleighlimit). Adapting Chew's results to the present analysis, the geometry function inthe Rayleigh limit reads

(8.218)

For materials with negligible frequency dependence of E, Eq. (8.218) reduces to

(8.219)

It is interesting to note that molecules of a dielectric sphere embedded in amedium with lower E than the particle (for m > 1) emit less radiation than thesame numbers of molecules in bulk material.

8.6.4Morphology Dependent Resonance Effects

A spherical cavity represents a case of three-dimensional enclosure. If the radiusof this cavity cannot be considered large compared to the wavelength, the mode


density of such a cavity differs appreciably from free space, and the emission ofradiation can be alternatively enhanced or inhibited. The effect of theredistribution of the mode density in the cavity on the spontaneous emission ofradiation was examined in Sect. 8.4.7. In this section we discuss the effect ofmesoscopic cavities on fluorescence, Raman transitions or stimulated processes(Chew 1987, 1988b; Ching et al. 1987a,b; Lange and Schweiger 1994; Brorsonand Skovgaard 1996; Barnes et al. 1996a; Lin and Campill o 1994; Campillo et al.1996).

If the frequency emitted by a molecule located in the cavity corresponds to aneigenmode, the emission rate is enhanced . The lifetime of the excited state isreduced accordingly, as shown in Fig. 8.18 (Barnes et al. I996b). Thefluorescence decay rate increases as the particle radius decreases. In bulk materialthe decay is a simple exponential. For smaller particle sizes the decay consists of afast and a slow component. The slow component shows the same time behavior asbulk material. This is considered to be caused by molecule s in the core of theparticle , which are practically unaffected by resonances, as pointed out by Arnold(1997). The narrow resonance modes are close to the surface, and only molecule sin this region can couple into narrow resonance modes.' A closer inspection would

aJc:::s(/)fZ:Joo~c:::oz

o 5

TIME (ns)

---*- 4 urn (10 '6 M)

--- 7 urn (10 -6M)

--+- 9 urn (10 ' 6M)

- - BULK (10 -6M)INSTRUMENTRESPONSE

10 15

Fig. 8.18. Fluorescence decay curves for R6G dye in glycerol droplets of varying diameter(Barnes et al. 1996b)'

• Reproduced with permission from Barnes MD, Kung CY, Whitten WB, Ramsey JM (1996)Molecular fluorescence in a microcavity: solvation dynamics and single molecules detection,Advanced Series in Appl Phys, Vol. 3. © (1996) World Scientific Publishing


have to take into account the excitation of the fluorescence by the transmittedfield, which depends strongly on position, as shown in Fig. 3.9.

The coupling to an eigenmode depends not only on the location of themolecule but also on the orientation of the transition moment (Barnes et al. 1996b;Arnold et al. 1997). The spectral location of the eigenmodes is identical to themaxima of the expansion coefficients of the electrodynamic analysis. They can befound from Eqs. (8.202), (8.203) or (8.215). Eversole et al. (1992) developed analgorithm for identification of the eigenmodes. The radial component of amolecular dipole moment couples only into the electric TM-modes, as indicated

WAVELENGTH (nm)572 574 576 578 580 582 584

60

0.5

2.5

~ :, {'

$ 55A)

I-Z 50::::>

aia: 45s>- 40I-UiZUJI-~ 30

B) TE ~9

TE ~3

f-i_ _ -_-+_-+- _........+-l 01.5

C )

;;'..~

0.5

572 574 576 578 580 582 584

WAVELENGTH (nm)

Fig. 8.19. Comparison of experimental emission spectrum (A) and theoretical emission ratespectra for tangentially (B) and perpendicularly (C) to the interface orientated molecules Thetransition moment of the surfactant molecules couples favorably into the TE modes. Theemission is recorded from a dilute layer of DiI(3) surfactant on a glycerol particle of radius

a =7.3656l!m (Holler at al. 1998)'

, Reproduced with permission from Holler S, Doddard NL, Arnold S (1998) Spontaneousemission spectra from microdroplets, J Chem Phys 108:6564. © (1998) American Institute ofPhysics'


by Eq. (8.215) . The tangential component couples to both mode types. Theresonance frequencie s, or eigenmode s, are given by the poles (for complexarguments and maxima for real arguments) of the expansion coefficients, fE• Theresonances of the TE-modes and TM-modes, respectively, are located at the polesof the magnetic expansion coefficients, fM, and the electric expansion coefficients,

aE·Holler et al. (1998) investigated the fluorescent emission of oriented surfactant

molecules on a levitated microdroplet. A comparison of the measured fluorescence with calculations of the transition rates using Eq. (8.214) is shown inFig.8.l9. The transition moment of the surfactant molecules couples favorablyinto the TE-modes. From these results, Holler at al. (1998) concluded that thesemolecules have their transition moments preferentially orientated tangentially tothe surface of the glycerol droplet.

Inelastic emission of radiation from microparticles can show two types ofresonances. Resonances in the inelastic radiation field, given by the poles of the fE

and fM expansion coefficients, are called output resonances. The appearance ofresonances in the inelastic scattered spectrum is nearly unavoidable because thefluorescence or Raman bands are usually broad enough that at least one frequencyin the spectrum fulfills the resonance condition . Resonances in the spectrum ofinelastic scattered light were reported for a number of experiments. The firstobservations of resonance peaks in a fluorescing microparticle were reported byBenner et al. (1980), Owen et al. (1982a,b), and Hill et al. (1984). Resonance s inRaman-scattered light from microparticles have been observed by Thurn and

(a)

~Ci5zWIZ

W>~u:J (b)0:::

16250 17250 18250

WAVE NUMBER (em")

Fig. 8.20. Fluorescence spectra from R6G-doped ethanol droplets . Output resonances are shownfor (a) simultaneou s excitation of an input resonance and for (b) a non-resonant case. The spectra

were recorded from droplets with a size difference of only - 4nm (Eversole et al. 1995)'

* Reproduced with permission from Eversole 10, Lin HB, Campillo AJ (1995) Input/outputresonance correlation in laser-induced emission from microdropl ets , J Opt Soc Am B 12:288.© (1995) Optical Society of America


Kiefer (1984, 1985), Lettieri and Preston (1985), Schweiger (1990a,b), and others.A glimpse at Eqs. (8.215) and (8.204) shows that resonances also appear if theexpansion coefficients Cn and d, have poles. These resonances are called inputresonances . Double resonances, the simultaneous excitation of input and outputresonances can be observed in fluorescence (Eversole et al. 1995) as well as inRaman scattering (Schweiger 1990a,b; Schaschek et al. 1993; Popp et al. 1997;Kaiser et al. 1996). Hartmann et al. (1997) calculated the spectrum of evaporatingtetraethylene glycol droplets in the spectral range of the C-H Raman stretchingmode showing output and double resonances . They found excellent agreementwith experiment.

The enhancement of an output resonance by the excitation of an inputresonance depends on the overlap of the mode volume of the input mode with thatof the output modes. The mode volume is that part of the microparticle where theelectromagnetic field within the particle is enhanced due to the excitation of theresonant modes. Figure 8.20, reproduced from Eversole et aI. (1995), shows thatthe energy from the input mode is preferentially transferred to the output modesdue to the overlap of the mode volumes, whereas the non-resonant background isnot much enhanced.

The interplay of output and input resonances can be very well observed forevaporating particles if, for example , the Raman spectra are plotted as gray scale

5

2

o

28002900

--- WAVENUMBER I em _1

o....:':::O"~I:WIj..Qll;.,3000

100

300

t

Fig. 8.21. Contour map of Raman spectra recorded at subsequent time steps on evaporatingglycol droplets (Popp et al. 1997)'

• Observability of morpholog y-dependent output resonances in the Raman spectra of opticallylevitated microdroplets, Popp J, Trunk M, Lankers M, Hartmann 1, Schaschek K, Kiefer W.1997 © John Wiley & Sons Limited . Reproduced with permission

8.6Particle Specific Effects 569

lines as function of time. Figure 8.21 shows a contour map of Raman spectrarecorded at subsequent time steps for evaporating glycol droplets (Popp et al.1997). Each horizontal line represents a Raman spectrum recorded at the timeindicated on the vertical axis. Pixels of equal intensities of the spectra recorded atsubsequent times are connected by contour lines. One can identify two types oflines in the spectrum: a number of inclined lines running from bottom left to topright, and a few horizontal lines. The first are output resonances, the second inputresonances . As the particle shrinks, the wavelength of a resonant output modemust by reduced to fulfil the resonance condition. The wave number shift of theoutput resonances decreases with decreasing particle size. This causes the tiltedlines. At some specific sizes the condition for an input resonance is fulfilled, andthe spectrum is enhanced over the whole range causing horizontal lines.

8.6.5Stimulated Processes

Practically all nonlinear processes known for bulk material can also be observedin micro-resonators. Most of the nonlinear processes studied with droplets haveinvolved moving droplets, but some work has been done using a levitated droplet.The surface tension causes the formation of nearly ideal spheres with very highresonator qualities. Nonlinear optics associated with droplets was reviewed by Hilland Chang (1995). Many aspects of nonlinear optical effects in microparticles arealso treated in the book on optical processes in microcavities by Chang andCampillo (1996). Some of these effects are useful for microparticle analyses, andtheir applications are treated in the next section.

The excitation of stimulated Brillouin scattering (SBS) or stimulated Ramanscattering (SRS) in microdroplets is relatively easy to achieve due to the lowthresholds of these nonlinear effects. A number of investigators reported theobservation of SBS (Zhang and Chang 1989; Wirth et al. 1992; Leung and Young1991) or SRS (Biswas et aI1989a,b; Xie et al. 1993; Chen et al. 1991; Kwok andChang 1993; Zhang et al. 1988; Lin et al. 1992). Second harmonic generation inoptically-trapped nonlinear particles with pulsed laser excitation was observed byMalmqvist and Hertz (1995). Third-order sum frequency generation was reportedby Leach et al. (1990), and Hartings et al. (1997) generated second harmonicsfrom surfactants on pendant droplets. Since laser emission from individualdroplets was first observed (Tzeng et al. 1984; Qian et al. 1986; Lin et al. 1986;Biswas et al. 1989b), investigation of lasing from microspheres has foundincreasing interest. Various aspects of lasing from microspheres that have beeninvestigated were reviewed by Hill and Chang (1995). Stimulated processes arealways excited at resonance modes.

The theory of stimulated processes initially developed for bulk material wasextended to spherical particles by Chitanvis and Cantrell (1989) using geometricaloptics. A more refined treatment of SBS was presented by Cantrell (1991a,b,c).He solved the nonlinear wave equations in spherical coordinates with theappropriate boundary conditions . He used a series expansion of Debye'selectromagnetic potential because this is a more powerful technique for the


treatment of nonlinear processes than the E and H field representation (Nisbet1955). He applied the conventional spherical harmonics for the angular dependentpart, but a Fourier-Dini expansion and the conventional Fourier-Bessel functionswere used to represent the radial partial-wave eigenfunctions for the acoustic andelectromagnetic fields (Dini 1880). Cantrell also derived the boundary conditionsfor the acoustic pressure, including surface-tension effects . He showed that anSBS wave must be resonant to satisfy the electromagnetic boundary conditions. Asolution of SBS in microspheres with special emphasis on double resonances wasdeveloped by Leung and Young (1991).

The theoretical analysis of stimulated processes in microparticles can be treatedby following the heuristic approach of Serpenguzel et al. (1992). The mainfeatures of stimulated Raman scattering and lasing can be elucidated that way.Applications of nonlinear effects to microparticle analyses will be disussed inChap. 9.

8.6.5.1Coupled Partial Wave Theory

The excitation of stimulated processes differs in several ways from that for bulkmaterial. The internal-input intensity, which is proportional to IEolZ, is usuallyconcentrated in two small regions, the hot spots, caused by the focusing of theincident beam depicted in Fig.8.20b. The stimulated waves always propagate aslow-order MDR-modes and are confined to regions close to the surface. Finally,the spontaneous Raman (Brillouin) cross sections can be larger than in bulkmaterial due to the redistribution of the mode density in the microparticle (cavityquantum effects). The feedback caused by the total reflection of the nonlinearwaves can excite several orders of stimulated waves. For example, the input beamstimulates a wave propagating with the Raman shifted frequency WI = % - ~,

where % is the frequency of the incident wave as before and ~ is aneigenfrequency of the molecule, say, for a vibrational mode. The stimulatedRaman beam can be strong enough to stimulate a second-order Raman beam. Thissecond order SR-beam, oscillating with frequency 002, is again shifted by ~ butwith respect to the frequency of the first-order SR line. Thus,002 = WI - Wv = % - 2Wy, and so on.

The gain and coupling between higher order SR-modes can be calculated fromthe theory sketched in Sect. 8.5.2 by including higher order waves in Eq. (8.164).The electromagnetic field now contains higher order SR waves and reads

E(z, t) = Eo (z, t)+ E IS (z, t)+ Ezs (z, t)+ ...+ C.C. , (8.219)

where

Ej(z, r) = A j (z)ei(kjH!ljt}. (8.220)

We use the second Raman susceptibility given by Eq. (8.176) and get, for thenonlinear polarization,


(8.221)

(8.223)

p NL (Wo,W1S'W2S ,W3S , .. . ) =6X(wls - Wo)EoE;s + 6X(w2S - Wo)EoE;s

+6X(w3S - Wo)EoE;s + 6X(w2S - WS1)EIsE;s

+6X(w3S - W1S)EIsE;s + 6X(w3S - W2S )E2SE ;S

+ ...c.c)x(Eo + E 1S +E2S+ ... +c.c) .

Extracting the terms involving exp( -WIS), we get

p NL (z)e - iwlS = 6X(wo- wlS )EoE~Els + 6X(w2S - wlS )EISE;sE2S (8.222)+6X(w2S - w3S )EoE;sE3S+ 6X(wls - Wo)EoE~sE2S + ... + C.c.

Using Eq. (8.176), we have

X(w 3S - w2S )=X(W 2S - W1S)=X(w1S - wo)=X(- wk ) ,

( )_ ( )_. N(aajaq k)~X- wk - -X\wk - 1 •

6mw k Y

Finally, we introduce Eqs. (8.220) and (8.223) into Eq. (8.222) to obtain from Eq.(8.161)

(8.224)

where

(8.225)

and N1s is the refractive index at the frequency of the first-order Stokes wave. Thenonlinear polarization for oscillations at frequencies ~s, ~s, etc. can be foundaccordingly.

Low order resonances propagate near the surface, so Serpenglizel et al. simplyassumed that the stimulated wave circulates at a distance r =± a, where a is theradius of the particle. The incremental length dz of an optical cell transforms intoa-do in the spherical resonator. Including absorption losses and cavity-radiationleakage losses, the coupled nonlinear wave equations for the first- order StokesSRS amplitude reads :

(8.226)

where Kjs is the coefficient for radiation losses, and Cjs is the absorptioncoefficient.

(8.227)


The amplitude equation for the second-order SRS is found correspondingly andreads

jf.nal

xexp(-i~k 2sa<!» - L~ Ul2S AoA(j'_2)sAj'sexp(idkj'sa<!» .

j' ; 3 2 UlIS

The amplification, gR, is related to KR given in Eq. (8.179) by 2KR =- gR 1 Ao12

•

The electric field amplitudes of the internal , the first-, second-, third-, and jthorder Stoke SRS are Ao, A1S, Azs, A3S, and Ajs. The corresponding wave vectorsand angular frequencie s are kjs and Uljs, respectively. The integers j and j'designate the order of the SRS, and jfinal is the highest order Stokes SRS. Here ~kjs

=(kjs + ko - klS - k(j _ I)S) is the wave vector mismatch among the jth - orderStokes SRS (kjs), the input wave (ko), the first-order Stokes SRS (kIS) ' and the(j - l)th - order Stokes SRS. Similarly , ~kj's =(kjs + ko- kzs - k(j' - Z)s) is the wavevector mismatch among the internal wave vectors of the j'th - order Stokes SRS(kj,s), the input wave (ko), the second-order Stokes SRS (kzs), and the (j' - 2)th order Stokes SRS.

The first term on the right-hand side of Eq. (8.226) takes into account thedepletion caused by generating the second order Stokes wave with irradianceproportional to 1 Ezs 1

2 and losses by radiative leakage from the droplet andabsorption. The second term describes the depletion by the frequency mismatchwith the second order Stoke wave, and the third term takes into account thecorresponding process by higher order Stokes than the second-order SRS. Thecorresponding interpretation holds for the amplification of the second-orderStokes SRS given in Eq. (8.227). Equations (8.226) and (8.227) can be extendedto include SBS (Serpengiizel et al. 1992).

8.6.5.2Stimulated Raman and Brillouin Scattering

The threshold for SRS (stimulated Raman scattering) is relatively low, and theobservation of SRS was reported by a number of investigators (Snow et al. 1985;Qian and Chang 1986b; Chang et al. 1986; Zhang et al. 1988; Pinnick et al. 1988;Hsieh et al. 1988; Biswas et al. 1989a; Pinnick et al. 1989; Zhang et al. 1990; Qianet al. 1990; Lin et al. 1990a,b,c; Chen et al. 1991; Srivastrava and Jarzembski1991; Kwok and Chang 1992, 1993; Knight et al. 1992; Lin et al. 1992;Armstrong et al. 1993). Suppression of SRS can be achieved by artificiallylowering the cavity Q value, for example, by adding nanometer-sized particles(Xie et al. 1993).

Snow et al. (1985) were the first to report the excitation of SRS inmicrodroplets. A modified Berglund Liu generator was used to generate a chain ofnearly identical water, heavy water, and ethanol droplets in the diameter range40 - 80 urn. The droplets were illuminated by a frequency-doubled Nd:YAG laser


(A = 532 nm, 20 mJ/pulse). A number of discrete peaks were observed in thespectral range of the spontaneous Raman scattering of H20 and 0 20, as shown inFig. 8.22. This multiorder Stokes emission is observed if the Raman gain profilespans several MOR's, which is typical for the asymmetric stretching mode ofwater. At resonance, the large internal field of the first Stokes emission, whichhas achieved SRS threshold, can act as a pump and provides significant gain forsubsequent SRS processes (Qian and Chang 1986b). Qian and Chang (l986a,b)observed SRS with droplets of various liquids such as CS2 and CCI4•

Biswas et al. (l989b) and Lin et al. (l990a) identified the resonance mode andorder number of SRS in ethanol droplets. The simultaneous excitation of an input

a) H20, BULK

SPONTANEOUS

\

3500 3400 3300 3200

b) 020

SPONTANEOUS

" BULK,-STIMULATED-, DROPLET

I

2600 2500 2400 2300

RAMAN SHIFT (crn-t)

Fig. 8.22. Spontaneous Raman scattering from bulk and SRS from a single droplet a) H,o andb) D,o. The spectral position of the discrete peaks of SRS corresponds to the spectral position of

MDR's (Snow et al. 1985)'

, Reproduced with permission from Snow JB, Qian SX, Chang RK (1985) Stimulated Ramanscattering from individual water and ethanol droplets at morphology-dependent resonances,Opt Lett 10:38. © (1985) Optical Society of America


resonance and SRS could also be observed (Biswas et al. 1989a; Lin et al. 1990b).The angular scattering characteristics of SRS differ from that of elastic scattering.Angular scattering from SRS is more isotropic , and its intensity is two to threeorders of magnitude less than elastic scattering (Pinnick et al. 1988).

The fine structure in the angular distribution was examined by Chen et al.(1991). They concluded from their findings that the mode number of the SRSdepends on the illumination. Focused center illumination of the droplet favors theexcitation of larger mode orders. Edge illumination favors coupling to SRS withsmaller order numbers because the mode volume of low order resonances isconfined closer to the surface. If several SRS lines are excited simultaneously, theangular distribution is smoothed because the different SRS lines are excited atdifferent MDR's and do not have the same angular distribution. Aker et al. (1996)proposed to excite different resonance orders by selecting the illuminationgeometry appropriately. Because the mode volume of different orders occupiesdifferent regions in the particle, they speculated that a simple tailoring of theillumination geometry allows the generation of stimulated Raman signals fromdifferent regions within the particle . Thus, stimulated Raman scattering could beused to image spatial variations in chemical composition and/or molecularstructure .

The investigation of the temporal behaviour of SRS and stimulated Brillouinscattering (SBS) indicates that in microdroplets the first order SRS is pumped bystimulated Brillouin scattering rather than by the internal intensity of the incidentlaser if a single mode laser is used (Zhang et al 1990). This can be understoodfrom the lower threshold of SBS.

The excitation of SRS depends on the droplet size, the concentration of theRaman-active species, and the laser power density . Lin and Campillo (1994)observed the nonlinear growth of SRS in a CS2 droplet with a diameter of - 4.211m at a laser power density of 30 W em" , In the same paper, they showed that thesize dependence of the threshold follows a strict d

4-law and investigated the

threshold laser power for a number of fluids. In contrast , Vehring (1998)investigated the SRS threshold on water droplets and found nearly no dependenceon the particle size.

Stimulated Raman scattering builds up from the spontaneous Raman signal,which is commonly referred to as noise. The Raman gain has to be high enough toexceed the leakage and other losses within the droplet. This is a severe limitationon the lower concentration limit for the detection of a chemical species. Thelowest detectable concentration can be significantly reduced by injection seeding.Conventionally, seeding is done by using two lasers. The second laser operates atthe Stokes wavelength. This technique, called external seeding, was applied tomicrodroplets by Pasternack et al. (1996) and by Fields et al. (1996). Pasternak etal. dissolved ammonium sulfate in water and generated droplets with a diameter of- 50 11m in a chain of microdroplets. Two laser beams with different frequencieswere used. The frequency difference corresponds to the Raman shift of the speciesof interest.

The enhancement of SRS by seeding is shown in Fig.8.23. A similar effect canbe achieved by internal seeding. In this case the second wave overlapping with the


SRSIN =0.2M AQUEOUS (NH.lzSO. DROPLETS

::i~

~(ijzWIZ

532 + 612 nm

532 nm only

645 650 655 660 665

WAVELENGTH (nm)

Fig. 8.23. Effect of seeding on the excitation of stimulated Raman scattering. The lower curvewas recorded using a single laser (frequency-doubled Nd:YAG, A = 532 nm). The upper curvewas obtained under similar conditions, except that a dye laser operating at 612 nm was also

incident upon the droplets (Pasternack et al. 1996)'

Raman band is generated within the droplet. Kwok and Chang (1992, 1993)seeded microdroplets with R6G. The laser which also generated Raman bands,excited the fluorescence. These Raman bands were in the same spectral range asthe dye fluorescence.

8.6.5.3Lasing

The illumination of dye-doped microparticles results in a fluorescence spectrumquite distinct from the spectrum of the bulk solution under similar conditions (seeFig.8.20). The peaks in the fluorescence spectrum of the droplet are, as discussedabove , caused by MDR's. If the microparticle is exposed to a light wave whosewavelength corresponds to a resonance frequency, light couples preferentially inthis low loss resonant mode, and the transmitted electromagnetic energy risesquite dramatically in the volume occupied by this mode. If the incident intensity issufficiently high, the light wave propagating in the resonant mode can beamplified by stimulated emission of radiation. Lasing can be achieved relativelyeasily in spherical non-absorbing microparticles due to the high Q (Lin et al.1986). The first laser emission from liquid droplets was reported by Tzeng et al.(1984) , and Baer (1987) demonstrated that lasing can be stimulated in solid

, Reproduced with permission from Pasternack L, Fleming JW, Owrutzky JC (1996) Opticallyseeded stimulated Raman scattering of aqueous sulfate microdroplets, J Opt Soc Am B13:1513. © (1996) Optical Society of America


spheres", Fig.8.24 shows lasing from a polystyrene sphere with a diameter of41 um.

This particle was doped with Nile Red and expo sed to a pulsed dye laser(Kuwata-Gonokami et al. 1992). The figure shows that lasing occur ssimultaneously at many different wavelengths; it takes place at several modeorders at once. The intensity of the emitted light rises sharply once the thresholdfor stimulated emission is exceeded. Eversole et al. (1992) showed that the modeand order of lasing can be identified by matching observed fixed angle elasticscattering as function of droplet size to calculated intensities from Lorenz-Mietheory and, independently, by matching observed resonance peaks to computedcavity-mode positions .

A variety of techniques were investigated to control the wavelength of the laseremission . Chen et aI. (1993) demon strated that the wavelength of a non-sphericalparticle varies along the entire rim of the particle. The wavelength of the laseremission can be shifted by controlling the cavity Q value. This can be done byincreasing the concentration of an absorbing liquid in the micro-re sonator(Mazumder et aI. 1995) or by adding scattering particles (Taniguchi and

~wzWf~Zow(/)

~w

PUMPPOWER

"w~20W~0

4W W ~o

570 580 590 600 610 620 630

WAVELENGTH (nm)

Fig. 8.24. Pumping power dependence of laser emission from a 41 urn dye-doped polystyrenesphere (Kuwata-Gonokami et al. 1992)'

33The Nd:YAG sphere used by Baer was about 5 mm in diameter and could actually not becalled a microsphere .

• Reproduced with permission from Kuwata-Gonokami M, Takeda K, Yasuda H, Ema K (1992)Laser emission from dye-doped polystyrene microsphere, J Appl Phys 31:100. © (1992)American Institute of Physics


Tomisawa 1994). Reduction of the cavity Q results in a blue shift of the laseremission. Depending on the concentration, size, and optical properties of theseeding particles, the lasing intensity can be increased (Taniguchi et al. 1996) orsuppressed (Armstrong et al. 1993).

The fabrication of micro-lasers using spherically or rotationally symmetricparticles is attractive because of the very high quality of these micro-resonators.Different materials have been used successfully to fabricate lasing microparticles(Armstrong et al. 1992; Shibata et al. 1997; Nagai et al. 1997; Taniguchi et al.1993, 1996; Taniguchi and Tomisawa 1994).

In resonance, the amplitude is enhanced not just within the particle but alsoclose to its surface. Borchers et al. (2001) have shown that in resonance thisevanescent field can be strong enough to excite lasing in a second microparticle incontact with the resonant particle .

Nonlinear effects in microparticles have not attracted much interest in theaerosol community mainly because these effects depend sensitively on the shapeand surface properties of the particles. These properties are often not knownprecisely enough to make a quantitative interpretation of nonlinear scatteringposs ible. However, nonlinear effects in microparticles have potential applicationsin photonics and in the field of microsensors.


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