Handbook of Thin Film Deposition || Optical Thin Films

Handbook of Thin Film Deposition. DOI: http://dx.doi.org/© 2012 Elsevier Inc. All rights reserved.2012

10.1016/B978-1-4377-7873-1.00009-7

Optical Thin FilmsAngus MacleodThin Film Center Inc., Tucson, AZ

9

9.1 Introduction

A typical optical system might consist of a source of illumination and a receiver separated by a series of optical components designed to manipulate the light in vari-ous ways in order that the function of the optical system should be achieved. Most frequently, the components consist of a body of suitable optical material with sur-faces worked to present a desired shape and character. The surfaces are generally smooth, with properties described as specular, so that the light is directed according to the laws of reflection and refraction. The shape of the surface and the properties of the incident and emergent materials are chosen to assure the desired direction of the light that is transmitted through the surface and reflected by it. But the direction of the light is only part of the desired function. The amount and character of the light transmitted or reflected is equally important, and the properties of the primi-tive surface in these respects are rarely satisfactory. Modification of these properties, without changing the power of the surface, is the primary role of an optical coat-ing [1]. There is virtually no modern optical system that could function correctly without optical coatings. In the case of lenses, for example, the coating, known as an antireflection coating, operates to reduce the reflectance and increase the transmit-tance. This not only improves the strength of the desired signal but also reduces the stray light that might otherwise find its way to the receiver and corrupt the signal. A component intended as a mirror will obtain its necessary high reflectance from an applied optical coating. A special mirror coating, known as a cold mirror, is designed to reflect the visible region and to transmit the infrared, thus reducing the thermal content of an image. The inverse is accomplished by a hot mirror. Thermal insula-tion of buildings is partially assured by such coatings applied to their glazing. Lasers could never have functioned without the special laser mirror coatings for their reso-nators. The multiplexing and demultiplexing of optical telecommunication channels is achieved by special thin-film coatings that transmit one channel and reflect all others. Forgery of banknotes and other documents is inhibited by patches of iridescent color created by thin-film coatings.

The coatings consist of assemblies of thin films of various materials, and their optical properties are determined partly by interference in the light reflected back and forth between the interfaces and partly by the optical properties of the materials. In the normal way, they will consist of anything from one single layer to several tens of

http://dx.doi.org/10.1016/B978-1-4377-7873-1.00009-7

Handbook of Thin Film Deposition272

layers, but coatings of several thousand layers are not unknown. The thicknesses of the individual layers are of the same order as the wavelength of the relevant light and so the coatings, even those with large numbers of layers, are normally sufficiently thin that, except in exceptional cases, the direction of the light is not perturbed.

The principal reason for the application of an optical coating is the modification of the surface optical properties. However, many optical materials lack satisfactory environmental resistance and the applied coating, therefore, is expected to afford an element of environmental protection in addition to its optical effects. Environmental properties, such as abrasion resistance, and resistance to humidity and similar agents, even sometimes to carbonated soft drinks, are of considerable importance. Adhesion is, of course, of primary importance. Significant aspects of optical coating perform-ance, therefore, are not limited to optical, but range over mechanical, environmental, and chemical. An unfortunate, but inevitable, feature of an optical coating is that its application is usually the final operation in the manufacture of an optical component, in the production of which considerable effort may already have been invested, and so the reliability of the deposition of what is frequently a structure of great complex-ity is of major importance.

9.2 Nature of Light

Light is a propagating electromagnetic wave [2,3]. In other words, it has both an electric field and a magnetic field associated with it, both of which are necessary for the propagation of energy. Except when extremely high power density is involved, a special field of study completely out of the scope of this chapter, the interactions between the light and the media through which it propagates are entirely linear. In such cases, we can decompose the wave into a spectrum of plane harmonic waves, each of which can be considered separately. The resultant system response is sim-ply the sum of the individual responses. This decomposition is such a normal proce-dure in optics that we readily refer to spectral output of sources, spectral sensitivity of receivers, and spectral characteristics of coatings, without having to justify the use of such terms. Because the interactions are linear, the frequencies are constant. We make our fundamental spectral components as simple as possible and so we limit our discussion to isotropic materials and choose as component the plane, harmonic, and linearly polarized wave. In such a wave, the electric field, magnetic field, and direction of propagation are mutually perpendicular and, in that order, form a right-handed set. Both electric and magnetic fields have a sine or cosine profile and as long as we limit ourselves to linear combinations of such waves, where we add elec-tric and magnetic fields but do not multiply them, we can use the complex form of the wave:

E t zE exp i ( )ω κ ϕ

where we are assuming propagation along the positive direction of the z-axis. ω is the angular frequency (invariant), κ is the wavenumber, and ϕ is an arbitrary phase.

(9.1)

Optical Thin Films 273

The expression contains a sign convention in the order in which we have placed the t-term and the z-term. This is the normal convention in optical coatings. A similar expression applies to the magnetic field H. In a linearly polarized wave, the direction of E, and of H, is constant.ω is given by 2π/τ, where τ is the period of the wave. κ, similarly, is given by

2π/λ, where λ is the wavelength, and herein lies a problem. The wave velocity is constant in free space but varies in every other medium. Since ω is invariant then so is τ, and, since the velocity of the wave is given by λ/τ, clearly, λ cannot be constant. Unfortunately, λ is the parameter we normally use to characterize the wave and it would be unthinkable to have to qualify any statement of λ by an identification of the particular medium supporting the propagation. The convention adopted (except by a few spectroscopists) is to use λ to refer to the wavelength the wave would have in free space. Refractive index, n, is the ratio of the velocity of the wave in free space to the velocity in a medium and the actual wavelength is then λ/n and κ, therefore, 2πn/λ. Then the normal way of expressing the wave becomes

E i tnz

E exp2

ωπλ

ϕ

The quantity nz is then known as the optical thickness.Electric and magnetic fields can interact with charged particles, the electric field

exerting a force on a charged particle even when stationary, but the magnetic field requiring movement of the charged particle before exerting any force. At optical fre-quencies, even including the infrared, the cycle time is so short that the velocity of the particle never reaches a sufficiently high value for magnetic effects to be other than negligible, and so it is the electric field that is responsible for the interaction between the light and any material. At shorter wavelengths, the visible and ultravio-let, the interactions with the electrons of the material dominate. In the infrared, the heavier molecular units also become important. We shall return to this in Section 9.4. Here, we note that terms like amplitude and phase are assumed to refer to the electric field of the wave.

Some media possess loss and a distributed loss causes an exponential decay of a propagating wave. We can accommodate such an exponential decay by changing n into (n ik) where k is known as the extinction coefficient and (n ik) as the com-plex refractive index. n and k are also known as the optical constants. Then the wave expression becomes

E i tn ik z

E

E

exp2

exp2

ωπλ

ϕ

π

[ ]

kkzi t

nz

λω

πλ

ϕ

exp2

with a similar expression for H.

(9.2)

(9.3)


The power carried by the wave per unit area is given by the product of the electric and magnetic fields, the Poynting expression. This brings two problems. The first is that the product of the fields, involving the square of either a sine or cosine, fluctu-ates at twice the wave frequency, and it is the mean rate of power propagation that interests us. The second is that the product is a nonlinear operation that implies that real rather than complex waves should be used. These problems are neatly solved in what is known as the complex Poynting expression.

I EH1

2( *)Re

This quantity gives the mean directly. It is known as the irradiance and is measured in W/m2. Unfortunately, the international symbol for irradiance is E and so in an attempt to avoid confusion we are using in Eq. (9.4) the nonstandard symbol I (i.e., strictly, the international symbol for intensity, a measure of the output from a point source).

We do need to be able to calculate H and not just because it is required for irradi-ance. It also takes part in the boundary conditions at interfaces. For a plane harmonic wave, H is proportional to E with the constant of proportionality a material constant known as the characteristic admittance, y.

H yE

At optical frequencies (greater than roughly 100 cm1 or with wavelengths less than around 100 μm), there is the already-mentioned absence of direct magnetic effects, and, therefore, the relative permeability is unity. This leads to a simple rela-tionship between characteristic admittance and refractive index.

y n ik( )Y

where Y is the characteristic admittance of free space, i.e., 1/377 siemens. Since, for most purposes, y is used in the form of a ratio, the actual units are less important and y can be expressed in units of Y, or free space units.

y n ik( ) free space units

The numerical equivalence of y and (n ik) is a welcome simplification, but it can lead to the erroneous conclusion that y and (n ik) are physically equivalent.

9.3 Surfaces and Films

The behavior of a plane harmonic wave at a surface is determined by the bound-ary conditions that are the continuity of the total tangential components of E and H. At a simple interface, experience tells us that some of the incident light is reflected

(9.4)

(9.5)

(9.6)

(9.7)


and some is transmitted. In order for a quantitative examination of the relative amounts of light involved, we need to introduce a convention for the positive direc-tions of the fields. Since the electric field is responsible for material interactions, we first fix the positive directions of the tangential components of electric field in the incident, reflected, and transmitted waves to be all parallel. Then the positive direc-tion of the tangential magnetic field of the reflected wave will be opposite to that of the incident and transmitted waves. Should the incident waves arrive at the sur-face obliquely then the fields must be resolved. Fortunately, at oblique incidence, the wave directions are coplanar, the plane being known as the plane of incidence, but a moment’s consideration shows that the refraction of the transmitted wave will generally cause a problem because the tangential fields of the various waves will generally not be parallel implying a shift in polarization. The problem is avoided by introducing the eigenmodes of polarization, two orthogonal modes where the tan-gential vectors are parallel and polarization, therefore, unperturbed. These are known as p-polarization, where the electric vector is in the plane of incidence for the three waves, and s-polarization where it is perpendicular to the plane of incidence. Any arbitrary polarization is then expressed as a combination of these two modes. The sign convention for oblique incidence and positive directions of the electric field components is shown in Figure 9.1. This convention is usually known by the name Abelès [4,5]. For the normal incidence convention, we can imagine the angle of inci-dence shrinking to zero when there is, strictly, no longer any plane of incidence and the two polarization modes are equivalent.

We can imagine that we measure the power density of the various beams by plac-ing a receiver in each of them in turn. Reflectance and transmittance are then defined as the appropriate ratios of the receiver outputs. In the normal way, the measurement involves limited narrow beams, the total power of which can be measured by the receiver, and in the absence of any loss the sum of reflectance and transmittance will be unity. Here, however, we are dealing with infinite plane waves, and, in order that the reflectance and transmittance should add to unity should there be no source of loss, the receiver must always be arranged to be parallel to the interface, otherwise a

Transmitted

Incident Reflected

z

x

sp

s s

p

p

ϑ 1

ϑ 0ϑ 0

Figure 9.1 The sign convention for the positive directions of electric field for oblique incidence. p and s indicate the two planes of polarization.


different area of the interface will be involved in the refracted, or transmitted, wave. The irradiances involved with infinite waves must be, therefore, those normal to the interface that are calculated using the electric and magnetic field components parallel to the interface, i.e., the tangential components. It is sensible and convenient then to define the amplitude reflection and transmission coefficients in terms of the tangen-tial components of the electric field amplitudes. Note that this is a departure from the Fresnel coefficients that use the full electric field amplitudes. The necessary relation-ship between the tangential components of electric and magnetic fields of the waves, which is through y at normal incidence, is then through what is known as the tilted admittance [1] and written as η. Application of the boundary conditions for a simple interface between two media characterized by y0 and n0 (incident medium) and y1 and n1 (emergent medium), where these parameters may be complex, yields the fol-lowing relationships:

n n0 0 1 1sin sin Snell’s Lawϑ ϑ ( )

The amplitude reflection and transmission coefficients, ρ and τ, respectively, are then

ρη ηη η

τη

η η0 1

0 1

0

0 1

and2

where

η ϑ η ϑps cos and /cos y y

both being given simply by y at normal incidence.The reflectance and transmittance are then given by:

R T ρρηη

ττ** ( )and 1

0

Re⋅

where Eq. (9.11), but not Eq. (9.10), requires that η0 be real, i.e., no absorption in the incident medium. This condition is not as restrictive as it may seem at first sight. It is impossible to measure reflectance in an incident medium that has even slight absorption and inside any coating we deal solely with amplitudes so that no restric-tion applies.

The derivation of the necessary angles may appear rather involved and confus-ing when the refractive indices are complex. We can assume that the ultimate inci-dent medium is free from absorption, and so n0 and ϑ0 in the incident medium, and n0 sin ϑ0 will be real. All possible n sin ϑ values will be referred back to the incident medium because all the interfaces are parallel. We have no need for ϑ itself but rather cos ϑ. Thus, we replace Eq. (9.10) by:

η ϑ η ηs2 2

02 2

p02

ssin 2 and /n k n ink y

(9.8)

(9.9)

(9.10)

(9.11)

(9.12)


where the fourth quadrant solution for the square root is the correct one and where we have used Eq. (9.7) to replace y.

To include thin films in our calculations, we additionally need to derive an expres-sion for the phase change suffered by a wave on one traversal of a film. This is known as the phase thickness of the layer, is usually denoted by δ, and is a parameter of fundamental importance. At normal incidence, it is easy to see from Eq. (9.2) that δ is given by:

δπλ

2 nd

where d is the physical and nd the optical thickness of the layer. For an absorbing film, we simply use the complex index in Eq. (9.13) to give

δπλ

2 ( )n ik d

It is often useful to refer the layer parameters to a reference wavelength that we will write as λ0. Equation (9.14) then becomes

δπλ

λλ

2

0

0( )n ik d⋅

The quantity (λ0/λ) is dimensionless, and δ is also proportional to it, and so it is a useful parameter that is usually given the symbol g.

At oblique incidence, it is a little more difficult. From Figure 9.2, the tilted phase thickness is

δπ ϑλ

2 cosnd

that is, effectively thinner as the tilt increases. If the layer is absorbing, then n becomes (n ik) a very easy replacement in Eq. (9.13) but not as easy in Eq. (9.16)

(9.13)

(9.14)

(9.15)

(9.16)

z-axis

Planes ofconstant phase

Direction oflight ray

A

B

ndcosϑ

nd

ϑFilmboundaries

Figure 9.2 A and B are the points where we assess the phase of the obliquely propagating wave. The gap between the relevant planes of constant phase is nd cos ϑ and the phase thickness, therefore, 2πnd cos ϑ/λ.


where we adopt the technique of Eq. (9.12) to give

δπλ

ϑ2

sin 2 fourth quadrant root2 202 2

0d

n k n ink− ( )

Note that this result is polarization insensitive.The most used technique for the calculation of the optical properties of a thin-film

coating transfers the total tangential fields from the emergent surface of the assem-bly, usually the input surface of the substrate assumed semi-infinite, to the surface at the side of incidence. Since the processes are linear, it is usual to normalize the exit fields so that the electric field is unity and the magnetic field therefore ηsub, that is, the admittance (tilted if appropriate) of the substrate since there is no returning wave. Then, the transfer operation is expressed for each layer in terms of a 2 2 transfer matrix known as the characteristic matrix. The complete expression for a multilayer coating is

B

C

i

i

jj

j

j j jj

∏

cossin

sin cos

δδ

η

η δ δ

1

subη

where B and C represent the normalized tangential electric and magnetic fields, respec-tively, at the front of the coating. Reflectance and transmittance are then given by:

RB C

B C

B C

B CT

ηη

ηη

η0

0

0

0

0and4

*(Re ηη

η ηsub

0 0

)

( )( )*B C B C

9.4 Optical Materials

We have already explained that although electric and magnetic fields can interact with charged particles, because of the very high frequencies and the fact that mag-netic fields can interact only with moving particles, it is the electric field that is responsible for the interaction between the light and the material supporting propa-gation. An atom can be considered to consist primarily of two parts, a very heavy positively charged part consisting of the nucleus and the tightly bound electrons and a light part consisting of the valence electron bound to the positive part. The bond between the electron and the positive part can be thought of as a kind of spring that allows relative movement against the force of the bond. The atoms may be assem-bled into molecules, but the important feature is that the valence electrons remain bound to very heavy positively charged masses.

Now let us consider the influence of a harmonic plane wave on a single atom or molecule. In the steady state, the separation of the positive and negative charges will

(9.17)

(9.18)

(9.19)


oscillate at the frequency of the wave and the resulting oscillating dipole will scat-ter the light in its usual doughnut-shaped distribution. If more molecules are added, as long as the separation between them is very large compared with the wavelength of light, the scattering of each will be independent of all the others and the scattered light will be stronger by the number of molecules involved. This type of scattering is known as incoherent.

Let us now assemble the molecules into a solid. The spacing in the solid is very small compared with the wavelength of light. In one class of materials known as die-lectrics, the valence electrons remain essentially attached to their positively charged units or possibly shared with a neighboring unit, and so they form oscillating dipoles as before, but now the spacing between the dipoles is so small compared with the wavelength that the phases of the emission from the various dipoles are related. Now all the scattered light interferes destructively except that part that is radiated in the same direction as the primary beam. This is known as coherent scattering. The beam that traverses the material is then an inseparable combination of primary and scat-tered light. It takes time for the energy to be fed into the dipoles and time for it to reemerge as scatter, and so there is a phase lag in the coherently scattered beam. The resultant, primary plus scattered, then appears to move more slowly through the material leading to a refractive index greater than unity.

The spacing between molecules in a dielectric does not vary greatly from one material to another. Thus, the refractive index depends much more on the binding of the electrons than on the molecular spacing: the weaker the binding, the greater the dipole moment and the greater the refractive index. The delivery of energy from a light beam to an electron is in discrete units known as photons and the photon energy is proportional to frequency. In the linear regime, the dipoles receive zero or one pho-ton. Multiple photons belong to the nonlinear regime. As the light frequency increases, so does the photon energy and eventually it reaches a level sufficient to rupture the bond. The electron is now free and no longer part of the dipole and so does not radiate back the received energy, which is lost to absorption. Roughly, this onset of absorption occurs at lower frequencies, the weaker the binding of the electrons and, hence, the higher the refractive index. Thus, at the high frequencies of the ultraviolet, correspond-ing to high photon energy, there are no materials with reasonably high indices. In the visible region, the situation is a little better with materials like titanium dioxide. At the low frequencies of the infrared, there are many materials with very high indices of refraction. At very long wavelengths, the frequency can reach the natural vibrational frequency of the molecular units. There can be mechanical coupling into vibrations of the lattice of the material and, again, energy delivered from the light is not returned and consequently absorption becomes high and the materials lose their transparency. Clearly, the heavier the molecules, the lower the frequency and the longer the wave-length at which this occurs. This is why so many materials that exhibit excellent far infrared transparency are heavy metal compounds of unfortunately high toxicity.

There is another class of materials where the spacing between the atomic and molecular units is slightly less than in dielectrics. These materials tend, therefore, to have somewhat higher densities but, more important from our point of view, the properties of the valence electrons are so considerably perturbed that they become


free to move through the material and no longer form part of a dipole. The free elec-trons absorb the light, the penetration depth of which is then very small, and they also allow the conduction of electricity. This class of materials is known as the metals.

There is yet another class of materials where the binding of the electrons is weak and the refractive index high. Thermal fluctuations are enough to shake some of the electrons free and so there is some electrical conduction, but it is weak compared with that with the metals and, unlike metals, it disappears at low temperatures. These mate-rials are known as semiconductors. There are usually insufficient free electrons at room temperature to cause serious absorption, and so the transparency at lower frequencies, or longer wavelengths, is not much affected. The properties, therefore, are similar to those of the dielectrics and in thin-film work, semiconductors are treated as though a special case of dielectrics. They are particularly important in the infrared where they are transparent. Germanium, for example, has a refractive index of around 4.0 and is quite transparent from just below 2 μm to longer wavelengths. Lead telluride at longer wavelengths from roughly 3.7 μm has a refractive index of around 5.7.

The two classes of materials that we deal with, therefore, are dielectrics and metals.

9.5 Metals and Dielectrics in Coatings

Both metals and dielectrics (and semiconductors) are characterized by a refractive index, n, and an extinction coefficient, k.

In dielectrics, n is large, greater than unity, while k in the region of transparency is vanishingly small. The refractive index, n, varies only slightly with wavelength, fall-ing to a very small degree as the wavelength increases, so that to a first approxima-tion it can be considered substantially constant.

In metals, the primary interaction is with the free electrons, and this interaction increases as frequency falls and wavelength increases. The extinction coefficient, k, is large and becomes even more so with increasing wavelength. Indeed, k is roughly proportional to wavelength and this tendency continues out into the far infrared. At very high frequencies, beyond what is known as the plasma frequency and usually in the ultraviolet, the interaction with the electrons becomes vanishingly small and the optical properties of the metal become like a dielectric and close to those of free space (Figures 9.3 and 9.4).

Multilayer coatings tend to belong to two different types, those made up purely of dielectric (including semiconductor) materials, known as all-dielectric, and those made up of a mixture of metals and dielectrics, known as metal-dielectric. These two classifications lend themselves to two different types of performance.

The characteristic matrix of a thin film, Eq. (9.18), depends on two variables, the characteristic admittance, y (n ik) free space units, and the phase thickness δ 2π(n ik)d/λ. Here, we are referring to the normal incidence performance. For die-lectric materials, we can neglect k so that the variables become y n and δ 2πnd/λ. Since n is substantially constant with wavelength, y remains constant, while δ shrinks with wavelength. This means that as the wavelength increases any interference


structure based on dielectric materials will tend to weaken. Dielectric systems, there-fore, will be more suitable for performance characteristics that reflect shorter wave-lengths and transmit longer. Metals on the other hand are dominated by the extinction coefficient, k. If we neglect n, then admittance, y ik, and δ 2πkd/λ. Since k is proportional to λ, y increases with wavelength while δ remains constant. Thus, the

0

2

4

6

8

10

12

14

16

0 2000 4000 6000 8000 10,000 12,000

Refractive index

Wavelength (nm)

0

10

20

30

40

50

60

70

80

90

Extinction coefficient

Refractiveindex

Extinctioncoefficient

Figure 9.4 The optical constants, n and k, of silver. Note the linear dependence of the extinction coefficient k on wavelength. At the very shortwave end, the refractive index, n, rises to around unity while k remains very small, typical behavior beyond the plasma frequency.

0

1

2

3

10,0001000100

Refractive index

Wavelength (nm)

0

1

2

3

Extinction coefficient

Refractiveindex

Extinctioncoefficient

Figure 9.3 The optical constants, n and k, of SiO2 over its region of transparency.


metal becomes stronger in its effect with increasing wavelength. By adding dielec-tric systems to either side of a metal film, its reflectance can be reduced and transmit-tance enhanced. This is an interference effect and weakens with increasing wavelength where the strengthening metal will reflect more and more strongly. Thus, a metal-dielectric assembly will lend itself to a characteristic where transmittance is high at shorter wavelengths and reflectance high at longer, the opposite of the case with the all-dielectric structure. We can expect considerable difficulties if we attempt character-istics that are completely in opposition to the natural trends of the materials.

9.6 Admittance Transformer

The behavior of a plane harmonic wave at a surface is determined by the bound-ary conditions that are the continuity of the total tangential components of E and H. Since they are continuous through the surface, their ratio can be considered as a property of the surface. Since it must be an admittance, it is known as the surface admittance and usually written as Y to distinguish it from y, a material property. The B and C in Eq. (9.18) are normalized total tangential electric and magnetic fields, respectively, and so we can write at any interface

Y C B /

Should the value of Y be known at the front interface, then the amplitude reflec-tion coefficient of the system will be given by:

ρy Y

y Y

y B C

y B C0

0

0

0

which is consistent with Eq. (9.19). Of course, in order to be able to calculate trans-mittance, we need to retain the values of B and C separately, but if we confine our-selves to reflectance, then a knowledge of Y is sufficient. For accurate calculations, we make use of the computer. It is completely unproductive to try to derive accurate performance parameters in any other way. The computer, however, has no under-standing of the results it derives. For understanding, a knowledge of Y and the way in which it varies is very helpful. A particularly useful way of viewing the action of a thin film is as an admittance transformer.

For a single film, Eq. (9.18) becomes

B

C

i

y

iyY

cossin

sin cos

1

exit

δδ

δ δ

where we have assumed normal incidence and an admittance at the rear surface of Yexit. We look first for simple cases, and to avoid the complications of a complex δ, we limit our search to dielectric materials. We see immediately that values for δ

(9.20)

(9.21)

(9.22)


of 0°, 90°, 180°, 270°, and so on will yield quite simple terms in the characteristic matrix. A value of zero implies a film of zero thickness. The matrix is then the unity matrix and there is no change in the surface admittance. A film of 90° is the first really interesting case. Here, the optical thickness is one-quarter of the wavelength and it is readily shown that

YC

B

y

Y

2

exit

This is known as the quarterwave rule, and its simplicity is one of the reasons why quarterwaves are so important in optical coatings.

A halfwave layer can be looked on either as a δ of 180° or as a double application of the quarterwave rule. The result is simply

YC

BY exit

In other words, the halfwave layer does not alter the admittance. For this reason, a halfwave layer is sometimes called an absentee layer. Of course, halfwave layers are halfwaves only at one wavelength and so, although they have no effect on performance at that wavelength, they do affect performance elsewhere. This makes them particularly useful in many flattening and broadening roles, particularly in antireflection coatings.

9.7 Applications to Coatings

The simplest application of the quarterwave rule is in the single-layer antireflec-tion coating. Here, we imagine an incident medium of admittance y0, a substrate of admittance ysub, and a quarterwave film of admittance yf. We assume all these materi-als to be dielectric. The amplitude reflection coefficient, ρ, of the structure is

ρy y y

y y y0 f

2sub

0 f2

sub

/

/

−

and since this is completely real, the reflectance is just the square of ρ.

Ry y y

y y yρ2 0 f

2sub

0 f2

sub

/

/

Clearly R will be zero if, and only if,

y y yf 0 sub=

This is the well-known condition for a perfect antireflection coating at one wavelength.

(9.23)

(9.24)

(9.25)

(9.26)

(9.27)


Crown glass has a refractive index and optical admittance of around 1.52 in the visible region, although there is some dispersion. The reflectance of each surface in air, assuming a refractive index and optical admittance of 1.00, is

R1.00 1.52

1.00 1.524.26

2

%

From Eq. (9.27) the perfect antireflection coating would have an admittance of (1.00 1.52) which is 1.233. This is a very low value and there is no completely solid material with such a low admittance in the visible region. A very porous film could be created to have such a value, but it would be quite weak from the point of view of abrasion resistance. Antireflection coatings by their nature are always on the outside of a component. In many cases, they must withstand environmental attacks of all kinds and therefore must be tough and resistant. The material that has been found to be most satisfactory in this application is magnesium fluoride, which has a refractive index of around 1.38 in the visible. The quarterwave rule tells us that the minimum reflectance for a single film of this material is

R 1.00 1.38 /1.52

1.00 1.38 /1.521.26

2

2

2−

+

%

Using Eqs (9.18) and (9.19) we calculate, see Figure 9.5, the performance of the coating in terms of g, i.e., λ0/λ.

(9.28)

(9.29)

0

1

2

3

4

5

0 1 2 3 4

Reflectance (%)

g (dimensionless)

Figure 9.5 The reflectance in air, with admittance unity, of a layer of admittance 1.38 over a substrate of admittance 1.52 as a function of g.


We would like to arrange the thickness of this film so that it gives good perform-ance over the visible region, i.e., 400–700 nm. The criterion we will use to deter-mine the correct thickness is that the reflectances at 400 and 700 nm should be equal. We will specify the thickness as a quarterwave and then find the reference wave-length at which the optical thickness is our quarterwave. The fringe is completely symmetrical about g 1, and so our design criterion will be satisfied if 1Δg cor-responds to λ0/700 and if 1Δg corresponds to λ0/400. This gives two simultaneous equations that can be solved for Δg and λ0. The values we get are Δg 0.273 and λ0 509.1 nm, which we will round off as 510 nm. The performance of a film equiv-alent to quarterwave of magnesium fluoride at 510 nm is shown in Figure 9.6. The increase in reflectance at both the shortwave (blue) and longwave (red) ends of the visible region implies that the residual reflected color will be magenta. This distinc-tive color is normally used to control the deposition of the coating.

Since we do not have the necessary material to achieve zero reflectance with a glass substrate and air as the incident medium, we need more adjustable parameters. If we introduce two quarterwaves, then it is readily shown from the quarterwave rule that the relationship that needs to be satisfied is

y y y y1 2 0 sub/ /

In the visible region, we have materials sufficiently rugged for antireflection coatings that range in refractive index from the 1.38 of magnesium fluoride up to around 2.4 or 2.45 for titanium dioxide. The refractive index of a thin film does depend on deposition conditions and, with certain processes, the index of titanium

(9.30)

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Figure 9.6 The performance over the visible region of the film of Figure 9.5 with thickness one quarterwave at 510 nm.


dioxide may be still higher. If the substrate index is 1.52 and the incident medium index is 1.00, then there is an infinite number of combinations of indices within the range 1.38–2.45 that will satisfy Eq. (9.30). The problem, then, is to decide which combination should be used. It is fairly easy to see instinctively, but more difficult to prove, that the broadest antireflection coating of this type would be achieved by a combination of materials such that the amplitude reflection coefficient at each interface is of the same order. Examination of the range of available indices shows that the strongest reflection invariably occurs at the front surface and all we can do is to reduce that as far as possible by choosing the lowest refractive index for the outermost layer. This implies 1.38 for the outermost layer and, therefore, 1.70 for the innermost. The calculated performance is shown in Figure 9.7. The value, 1.70, does not correspond to any of the more common coating materials and a close, but not exact, match is the 1.65, or slightly less, of aluminum oxide. Figure 9.8 shows a comparison of the coating of Figure 9.7 with a calculation using typical real optical constants. The minimum reflectance is now no longer zero, but the curve is a little flatter.

It was found at an early stage [6] that a three-layer coating consisting of the design of Figures 9.7 and 9.8 with a halfwave layer of high index inserted between them gave a much wider performance (Figure 9.9). This idea of the halfwave layer as a performance broadening device in an antireflection coating is much used in antire-flection coating design [1].

Achieving zero (i.e., theoretically zero) reflectance with two quarterwaves requires that the characteristic admittances should be related through Eq. (9.30). The number of suitable materials at our disposal is limited and this implies difficulties in satis-fying Eq. (9.30) exactly. It is possible, however, by departing from the quarterwave

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Figure 9.7 The reflectance, as a function of g, of a two-layer antireflection coating consisting of two quarterwaves. The outer one, next to the air incident medium, is of admittance 1.38 and the inner one, next to the substrate of admittance 1.52, is of admittance 1.70.


condition to achieve zero reflectance with any pair of characteristic admittances as long as the higher admittance is greater than or equal to that given by Eq. (9.30). The usual design involves a thin layer of high-admittance material next to the sub-strate that can be thought of as raising the surface admittance until it reaches a level that can be exactly matched to the incident medium by the low-admittance material.

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Perfect

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Figure 9.8 The broken line shows the performance of the coating of Figure 9.7 transferred to a wavelength scale using λ0 510 nm. The full line shows the calculated performance with layers and substrate with practical dispersive optical properties corresponding to magnesium fluoride, aluminum oxide, and borosilicate crown glass.

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Figure 9.9 The coating of Figure 9.8 labeled real materials with a halfwave layer of tantalum pentoxide added between the two existing layers.


The admittance presented by the high-admittance layer is complex and the thickness of the following low-admittance layer consequently has to be somewhat greater than a quarterwave in order to make a perfect match. Figure 9.10 shows an example.

The caption of Figure 9.10 shows an example of a typical formula that is used to indicate the design of an optical coating. A quarterwave is denoted as a capital letter that is chosen to indicate the particular material. There are no strict rules, but, usu-ally, if there are only two materials in a coating, the capital letter L will denote the low-admittance material and H the high admittance. When more than two materials are employed, A might indicate aluminum oxide and T, tantalum oxide, and so on. A qualifying multiplier, m, say, indicates that the thickness should be m quarter-waves, as in 1.5 L meaning 1.5 quarterwaves of low-admittance material, i.e., three-eighths of a full wave. A set of layers in brackets and raised to a power, q, implies q repeats of the system in brackets. A substrate and incident medium can be added. There is no strict rule about the order. Those who deposit coatings usually prefer the substrate to be first, while designers normally prefer the incident medium to be first.

A flattening halfwave can be added to the design of Figure 9.10 in much the same way as in the quarter–quarter coating. The best place is after the outer quarterwave of low admittance. The performance can then be optimized by a gentle process of computer refinement to give the performance shown in Figure 9.11. From the optical point of view, magnesium fluoride, because of its lower index, has superior perform-ance to that of silica, but it suffers from rather high tensile stress and a somewhat poorer environmental resistance. Silica, SiO2, is preferred and the performance shown in Figure 9.11 is typical of high-performance antireflection coatings for the visible region.

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Figure 9.10 The performance over the visible region of a two-layer coating consisting of a thin layer of Ta2O5 next to the glass substrate followed by a thicker layer of SiO2. The design is Air | 1.2914L 0.3422H | Glass.


High-reflectance coatings normally consist of thin layers of a suitable metal (Figure 9.12). Most metals have reasonably high reflectance due to their free elec-trons. Silver has the highest reflectance in the visible and infrared but suffers from poor environmental resistance. Aluminum has good ultraviolet performance and reflects well into the infrared and so is the most common coating for general applica-tions. For those situations where the mirror coating will be exposed to an aggressive

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Figure 9.11 The refined four-layer antireflection coating with design: Air | L 2.0883H 0.3771L 0.2285H | Glass where L is a quarterwave of SiO2 and H of Ta2O5. The reference wavelength, λ0, is 510 nm.

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AlAg

Au

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Figure 9.12 Reflectance of various metals commonly used as front-surface reflector coatings.


environment, rhodium, or chromium, is commonly used. Gold has poor visible per-formance but high performance, almost as good as that for silver, in the infrared. It possesses the advantage of resistance to corrosion and so is frequently used for infrared applications. Aluminum and silver are normally protected by dielectric over-coats. Except for abrasion, gold has good environmental resistance and so is often unprotected unless abrasion is a problem. Rhodium and chromium are tough and resistant and are usually unprotected. The addition of a thin dielectric layer to the metal will reduce reflectance, a high-admittance material giving much greater reduc-tion than a low admittance one. For aluminum, a single layer of SiO2 is normal. If luminous reflectance is important, then a halfwave of SiO2 (λ0 510 nm) is best (Figure 9.13). For performance over extended spectral regions, and where reflec-tors in series are concerned, the best strategy might be to mix coating thicknesses. SiO2 does not stick quite as well to silver, and a thin layer of Al2O3 next to the silver greatly improves the adhesion. Dielectrics do not stick well to gold. SiO is some-times used; but in many cases where the metal is not subject to handling, no protec-tion is often better than a somewhat unreliable overcoat of dielectric. Chromium and rhodium are sufficiently rugged not to require protection.

Metals in reflection always have losses. Even silver with its highest reflectance loses several percent by absorption. There are many applications where higher reflectance and lower loss are required. Provided the increased reflectance is required over a limited spectral interval only, it is possible to satisfy it with a sim-ple dielectric system of quarterwaves of alternate high and low admittance. The most efficient designs will terminate on either end with a high-admittance material. The design can be written as Incident medium | (HL)q H | Substrate. A typical perform-ance is shown in Figure 9.14.

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Halfwave SiO2

Quarterwave SiO2

Figure 9.13 The performance of a front-surface aluminized mirror protected with a quarterwave and a halfwave of SiO2 (λ0 510 nm). The halfwave clearly gives best performance over the visible region.


The width of the high-reflectance zone in Figure 9.14 is clearly limited. In fact, it is a function of the ratio of the high admittance to the low. The higher this ratio, the broader the regions of high reflectance, although the gap between the various orders always remains very wide. Figure 9.15 shows the performance of a typical reflector centered on 1200 nm.

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Figure 9.14 The reflectance of a typical quarterwave stack as a function of g. This particular coating consists of 25 quarterwave layers of alternate 1.38 and 2.35 admittances.

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Figure 9.15 The performance versus wavelength of the design of Figure 9.14 with materials Ta2O5 as high admittance and SiO2 as low. Reference wavelength λ0 is 1200 nm and the third-order high-reflectance peak can be seen at the 400 nm end of the scale.


Such reflectors can have very low losses since they are constructed entirely from dielectric materials. This implies that the transmittance is the complement of reflect-ance. In the transmission mode, the coating can form the basis for a longwave pass or a shortwave pass filter or even what is usually known as a notch filter, i.e., a fil-ter that rejects a region surrounded by pass regions. The deficiencies in the simple quarterwave stack are the fringes in the pass regions, better known as ripple, and the limited range of the rejection regions. The limited rejection range can be tackled by adding additional filter, sometimes colored glass or other materials with a suitable pattern of absorption, or perhaps additional thin-film components. Such additional filters are usually known as blocking filters and the unwanted regions of transmis-sion as sidebands. The ripple is essentially due to a mismatch between the basic structure and the surrounding media and can usually be dealt with by adding several layers and using computer refinement to transform them into matching structures [1]. Figure 9.16 is typical of such filters.

The quarterwave stack can also be used to enhance the reflectance of a front-surface metallic mirror coating. The metal has a phase shift that is reasonably close to 180°, and this corresponds to the phase shift of exactly 180° that occurs at the reference wavelength for a quarterwave stack ending with a high-admittance layer. The addition of several, usually just two, low–high admittance pairs can therefore enhance the reflectance just as though they were the outer part of a conventional quarterwave stack. The enhancement is, unfortunately, confined to a region corre-sponding to that of the quarterwave stack and so is suitable primarily for a specifica-tion involving luminous reflectance. The most important factor is to remember that a low-admittance layer must be next to the metal (Figure 9.17).

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Figure 9.16 A longwave pass filter based on a quarterwave stack such as that in Figure 9.14 with additional matching layers on either side and backed up with a colored glass absorption filter to remove the unwanted shortwave transmission. Blocking on the shortwave side is now complete.


Two similar quarterwave stacks, one deposited over the other, have an interesting property. A typical structure is

Air |HLHLHLHLHLH HLHLHLHLHLH| Glass

The two high-admittance layers in the center form a halfwave, and this is effectively surrounded by two reflectors making the structure that of a thin-film version of the Fabry–Perot etalon [7]. This has the characteristic of a narrow band of high transmit-tance surrounded by two regions of low transmittance, in other words, a narrowband filter. Increasing the number of layers in the quarterwave stacks narrows the pass-band. A similar effect can be achieved by adding additional halfwave thicknesses to the central layer, known as the cavity layer, or, sometimes, spacer layer by analogy with the Fabry–Perot structure. The effect of the additional layers is weaker than that of the additional layers in the stacks, and so the two together form a type of fine-coarse control of the passband width. Unfortunately, the shape of the passband is rather triangular (Figure 9.18).

A much improved passband shape results when two or more of these single-cavity structures, as they are called, are coupled together into a multiple-cavity filter. For correct performance, each single-cavity structure must be separated from the adja-cent one by a single quarterwave of alternate material, known as a coupling layer. Without these coupling layers, the characteristic shows spurious unwanted peaks in the rejection regions (Figure 9.19).

Metal layers can be used to replace some of the quarterwave structures in the multiple-cavity filters. Allowance has to be made for the reflectance phase of the metal layers in the thicknesses of the adjacent cavities. The big advantage of such structures is the high reflectance out into the infrared so that they require additional

(9.31)

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Figure 9.17 Typical performance of an enhanced metal reflector consisting of an aluminum film followed by two pairs of quarterwaves of SiO2 and Ta2O5.


blocking filters on the shortwave side of the passband. There are special techniques for assuring maximum transmittance from the metal, such structures usually being referred to as induced transmission filters [1,8].

Metals are used in a number of important coatings from decorative to coatings for thermal control of buildings. The performance of dielectric layer over a metal can be

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Figure 9.18 Typical performance of a filter based on the structure given in Eq. (9.31) and using Ta2O5 and SiO2 as materials on a glass substrate. The shape shows clearly the narrow peak but broad base.

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Figure 9.19 Typical performance of a multiple-cavity filter. This particular filter has three cavities of the structure given in Eq. (9.31) together with a simple two-layer V-coat matching structure outermost to reduce ripple. A more complex matching structure would be required to remove the remaining residual ripple.


reasonably well explained as interference between the ray reflected at the outer sur-face of the dielectric and that by the interface between the dielectric and the metal. The thickness of the dielectric can be adjusted so that these rays are out of phase and, therefore, interfere destructively. If their amplitudes are similar, then an antire-flection effect is the result. A thick metal will usually have a larger reflectance than the outer surface of a dielectric, even with a high-admittance material and some way of balancing the reflectances is necessary. In decorative coatings where transmis-sion is not required, the balancing is usually done by adding an additional thin metal layer, often chromium, to the outer dielectric surface.

For thermal control, the transmittance of the coating is important and the balancing of the interfering beams is usually made by way of thinning the metal. A typical struc-ture is a dielectric layer on either side of a thin metal, usually silver. This gives the correct optical properties and good transmittance over the visible region but increas-ing reflectance into the infrared. This prevents the entry of heat into the building in the summer and the loss of heat in the winter. Unfortunately, silver is not environmen-tally resistant and oxidizes readily unless protected by additional very thin diffusion barriers, and so the structure of the practical coatings is slightly more complicated.

Then there are coatings that intentionally use the variation of properties with angle of incidence. These include polarizers, polarizing beam splitters, and phase retarders. There are also coatings designed for oblique incidence that attempt to reduce the polar-ization effects. The word attempt is used deliberately because it is virtually impossible to remove all the consequences of tilting. Color separation filters are an example.

A more recent development in optical coatings is the use of their transient response in reflecting and shaping ultrafast pulses. It is impossible to cover all of these in this chapter.

9.8 Coating Manufacture

There is an enormous number of different processes for the production of thin-film coatings [1]. Virtually all have been used at one time or another for the construc-tion of optical coatings. However, at the present time, the principal techniques can be lumped together under the heading of physical vapor deposition and we shall there-fore concentrate on these.

In physical vapor deposition, the material to be deposited is in the vapor phase and condenses in the form of a thin solid film on the substrates that are held at tem-peratures well below the freezing temperature of the material. To avoid disturbances to the vapor, such as turbulence, interaction with the atmosphere, and so on, that would compromise the process, it is normally conducted under vacuum. The various techniques differ primarily in the method used to produce the required vapor.

The simplest, and the traditional method, called thermal evaporation, is to heat the material in a crucible until it melts and then boils. The crucible is usually situ-ated in the base of the vacuum chamber so that the vapor rises and the molten mate-rial is contained. The substrates to be coated are then held in the upper part of the chamber facing downward toward the crucibles or sources. Uniformity of thickness is very important in optical coatings and the deposition follows roughly the same


laws as illumination, i.e., an inverse square law together with the cosines of the emission and incidence angles. Generally, the variation over even quite small station-ary substrates is too high, and so it is normal to move the substrates during deposi-tion to even out the thickness. The movement may be as simple as rotation about the center of the machine above offset sources or it may involve double rotation sys-tems termed planetary because of the resemblance to planets rotating about the sun. Here, the substrates are held in small fixtures that are rotated rapidly about their axes while turning more slowly about the machine center (Figure 9.20). Masks to trim the uniformity are also common.

At a fairly early stage in the development of thermal evaporation, it was found that layers were much tougher and more compact if substrates were heated during deposi-tion [9] and this is now a standard practice. The temperature is usually around 300°C, provided the substrates are not disturbed by such an elevated temperature. Heating is almost invariably by radiation. Radiant heaters are mounted in the machine either behind or in front of the substrates, care being taken not to obscure the stream of evaporant. With simple rotation, the heaters can readily be mounted behind, but, with the more complicated planetary motion, front-mounted heaters are more common. It is a matter of some debate which heating arrangement is to be preferred and there seems no clear answer. Temperature measurement is difficult particularly because the substrates are moving. Often, a thermocouple is simply mounted somewhere in the

Electron beamsource

Boat source

Radiant heaters

To pumpsSubstrate

Planetarywork holder

Light guide

Light guide

Monitor chip

Shutter

Figure 9.20 Sketch of a typical arrangement of a machine for the deposition of optical coatings by thermal evaporation. Such a machine is box shaped with a door that opens and so is usually referred to as a box coater. The entire planetary work holder rotates about the center of the machine, while the individual substrate carriers spin about their centers to give enhanced layer thickness uniformity.


machine and tests are carried out to roughly calibrate it. But the calibration is dif-ficult and changes with the type of substrate. The ultimate technique is an infrared temperature sensing device that looks at the rotating substrates through a suitable port. Here, the coating itself changes its emittance as it is deposited, so that again calibration is difficult. At present, there seems no perfect solution and temperature is one of the coating parameters that is still rather poorly controlled.

The crucibles in which the material is heated are frequently constructed from refractory metals like tungsten or molybdenum and heated electrically by passing a current though them. The crucibles, therefore, are usually rather long with flat lands at either end for connection to electrodes. They have the appearance of a boat, like a punt, and so are usually referred to as boats.

Boats are cheap and easy to install, but there is often some reaction with the con-tents. They frequently have to be heated to very high temperatures when they warp and cause changes in the evaporant plume with consequences for uniformity. A more stable source involves a water-cooled crucible normally of copper and perhaps including a disposable refractory metal or carbon lining. The material in the crucible is then heated by directing a beam of energetic electrons onto it. A beam current of several amps, at a voltage drop of several kilovolts, represents an enormous power directed into a quite small volume and the temperatures reached are high enough to melt and vaporize any material. Reaction with the crucible is inhibited by the water cooling. The source of electrons is usually a refractory metal filament that would be corroded by the evaporant in the same way as the boat source if the filament were not buried within the structure of the source and the electron beam bent through at least 180° by the field of two magnets arranged along the side of the device. This magnetic field also serves to focus the beam and, by varying it through the means of subsidiary small electromagnets, tracks it across the surface. The electron beam source is so versatile, stable, and, nowadays, reliable; in spite of its greater initial cost, it has become the standard source in thermal evaporation.

Heated sources cannot be immediately turned off even if the power can immedi-ately be cut. Shutters, therefore, are provided so that the deposition can be instantly terminated by interposing the shutter between source and substrate. This is impor-tant in the control of thickness. Many techniques can be, and are, used for deposi-tion control. Control of refractive index is rather more difficult than thickness and so refractive index is generally controlled in open-loop mode simply by using the correct material and making sure that the evaporation conditions, temperature, and rate of deposition are exactly similar each time. Thickness control can be as simple as measuring time, provided the process is sufficiently stable, but more frequently some measure of thickness during deposition is obtained so that the control can be essentially closed loop. This thickness measurement can be made optically on the actual component, or one of the components, being manufactured, or on a separate test substrate, or it can be a measure of deposited mass by specially constructed microbalances. These microbalances operate through a quartz crystal vibrating in a shear mode, similar in construction to those crystals used to establish the frequency of radio transmitters. Addition of mass to the crystal causes its vibrational frequency to vary, and this can be measured electrically through the piezoelectric effect.


Vapor deposition almost invariably leads to a columnar structure in the films. In the case of thermal evaporation, this structure is often very pronounced. The major problem with such columnar structures is the packing of the columns, which, if not tight, defines pore-shaped voids in between the columns. These pores represent parts of the film that consist of a different material. This can be, perhaps, air but more often the pores are filled or partially filled with liquid water through a proc-ess of capillary condensation of atmospheric moisture. The refractive index of the film is therefore not simply that of the original film material but that of a compos-ite material consisting of film material, air, and water. Variation of the water content leads to an instability of refractive index. Loose packing of the columnar structure also implies a reduction in resistance to abrasion, an intrinsic tensile stress, and an increased temperature coefficient of optical changes.

Thermal evaporation has the immense advantage of being a suitable deposition technique for virtually any material, but it carries the price of the aforementioned defi-ciencies. In recent years, attention has therefore been more focused on what are collec-tively known as the energetic processes [10–12]. In an energetic process, momentum is transferred to the growing film [13], sometimes through deliberate bombardment by energetic particles and sometimes by the addition of energy to the condensing thin-film material. This additional momentum disrupts the loose columnar structure and increases the packing of the material. The columns may remain but much more closely packed, or the material may even be forced into an amorphous form where the columnar structure has more or less disappeared altogether. This is the case with many common coating materials including SiO2, Ta2O5, Nb2O5, and TiO2. The amorphous form is popular in optical coatings because of the absence of grain boundaries that can cause scattering losses and the smoother interfaces between layers. The momen-tum transfer also forces material deeper into the coating or through the interface with the substrate, and this contributes to an enhanced adhesion. Unfortunately, not all materials benefit from such treatment. The fluorides, especially, tend to lose fluorine under bombardment. This is particularly unfortunate because the fluorides are impor-tant materials for the ultraviolet having regions of transparency well beyond those of oxides. Bombardment of the fluorides has to be kept very light, and so the beneficial effects are not nearly as great as they are in the case of the oxides.

The simplest energetic process involves the addition of a broad-beam ion source to the thermal evaporation chamber. A beam of energetic positive ions, up to several hundred electron volts in energy, with enough electrons added to assure the continued neutralization of the growing film, is directed at the film. Momentum is transferred to the particles of film, forcing them deeper into the film and compacting it. This technique carries the advantage that the bombarding ion beam is separately control-lable from the other process parameters. Thus, even the fluorides can benefit to some extent from this process provided their bombardment is very light so that the loss of fluorine is minimized. A second advantage is that thermal evaporation machines can readily be modified to apply this process by the addition of an ion source.

Sputtering is an old process that has gained much ground in recent years. At first it was primarily employed in large area coatings but more recently has been adopted


throughout the optical thin-film field. A plasma is electrically excited in the gas atmosphere of the machine and positive ions from this plasma are attracted to a cathode. Momentum is transferred to the material of the cathode resulting in colli-sion cascades that eventually expel molecules from the cathode with considerable energy. The expelled cathode material then coats the anode of the process and the high energy of the arriving material compacts the growing film in the same way as the beam of bombarding ions in ion-assisted deposition. There are some problems. If the discharge is DC, maintained by the anode and cathode, then the two elec-trodes should be metallic in order that the process be maintained. Dielectric anode and cathodes have very small capacitance and immediately charge up and prevent further action. The time constant is so short that it takes radio frequency plasmas to discharge the electrodes sufficiently fast to permit the process to continue. The grounding and matching problems in RF processes make DC processes preferred and conducting cathodes can be used in reactive deposition where the metallic layers that are deposited continuously react with oxygen or nitrogen that is introduced into the atmosphere of the machine. Then the anode of the process is either the structure of the machine or, sometimes, a suitable conducting element. This works well for a wide range of oxides and nitrides, but there are considerable difficulties with fluo-rides and other compound materials.

There are two serious problems. The first is that the plasma tends to be situated in between cathode and substrates and passage of the sputtered material through the plasma causes a reduction in energy due to scattering. This has been mitigated by introducing a magnetic field that crosses the electric field causing the electrons in the plasma to move in cycloids, greatly increasing their paths and improving their ioni-zation efficiency so that the gas pressure can be reduced. Such processes are simi-lar to that in the magnetron oscillator, and so the term magnetron is applied to this form of sputtering, although no magnetron oscillator is involved. The magnetic field is derived from permanent magnets mounted behind the cathode. The ideal crossed fields are limited to a track around the area of cathode that results in an erosion pat-tern reminiscent of a map of a racetrack, and so it is referred to as the racetrack. The second problem is more serious. The reacting gas also reacts with the cathode, and although the resulting dielectric material is rapidly scrubbed out of the racetrack, it does introduce some hysteresis into the control characteristic of the source, which has to be dealt with. The greater problem is that it builds up in the other regions that are subject to reduced bombardment. There it forms capacitors that are charged by the arrival of positive ions. Since the insulating films are quite thin, the capacitances are large and the stored energy is high. The capacitors tend to break down result-ing in powerful arcs that can cause ejection of melted cathode material and, even, in severe cases, destruction of the cathode itself. Fortunately, the time constant is long and so periodic reversal of the polarity of the power can successfully discharge the capacitors. Suitable power supplies that accomplish this are commercially avail-able. A recent development has been the introduction of what is usually called mid-frequency sputtering. This uses two similar magnetron sources that are attached to opposite poles of a power supply operating at a frequency of around 40 kHz. Thus, in


alternate half cycles, one of the sources acts as cathode and the other as anode, while their roles are reversed in the other half cycles. This keeps the capacitors discharged and brings another advantage in that the process removes the disappearing anode problem. In reactive deposition, the structure of the machine and, if present, the addi-tional anode element are also coated with the insulating film so that the electrical properties of the discharge are adversely altered. Now with mid-frequency sputter-ing, the anode is the other source and the disappearing anode problem vanishes.

In ion-beam sputtering, the sputtering ions are derived from a broad-beam ion source so that the generating discharge is isolated from the deposition process. Since electrons can be added to the beam to deal with the charging problems, dielectric materials can readily be sputtered. A second ion source is often added directly to bombard the growing film and control its stress levels in what is termed dual ion-beam sputtering. Although it is rather slower than other forms of sputtering and has smaller areas of uniform deposition, it produces films of exceptionally high quality.

Chemical vapor deposition involves a reaction between precursors to produce the final film material. The precursors in the vapor phase are moved into the reaction chamber by a carrier gas. In the traditional form of the process, the reac-tion is activated by the high temperature of the substrate. The more usual form of the process today uses a cooler substrate and the reaction is induced by a plasma discharge. A very high rate of deposition is the normal result. If this is permitted to be continuous, then the accommodation of the deposited material to the substrate is likely to be inhibited by the rapid rate of arrival of further material, causing a rather poorly packed and hence environmentally weak film. This problem is prevented by pulsing the process. Radio or microwave frequencies are used to induce the plasma; the process is known by various names, often pulsed plasma-induced chemical vapor deposition with an inevitably large number of acronyms. Uniformity depends very much on the flow pattern through the reactor, and so the process tends to be used pri-marily for long runs involving similar products.

Atomic layer deposition is a similar process where instead of feeding both precur-sors into the reaction chamber simultaneously, they are pulsed in alternate cycles. In each half cycle, a thin layer of precursor, roughly one molecule thick, is deposited on the substrate. In the second half cycle, the other precursor interacts with the first to produce a thin layer of the compound product on the substrate. The uniformity is vir-tually perfect and does not depend on the surface shape. Pinholes vanish. The nature of the process is that it is rather slow, but it can be arranged to be completely auto-matic and optical filter coatings as complex as dense wavelength multiplexing and demultiplexing filters have been successfully produced. The process has not yet pen-etrated the general optical coating field and a current impediment to progress, which will doubtless be overcome, is the absence of a good, reliable, and straightforward technique for the deposition of silicon dioxide, the preferred low-index material.

There are many other processes from liquid to powder that can be, and are, used in the construction of optical coatings. Those mentioned above are the major ones at the present time, but the character of the field is such that the number of viable processes is constantly increasing.


9.9 Control

The term generally used for the measurement and control of layer parameters in the deposition of an optical coating is monitoring. Although monitoring in its usual sense means observation, in the thin-film field, it has always been used in the sense of measurement and control. Normally, the control of refractive index is completely open loop. The parameters of the deposition process, material, deposition power, substrate temperature, and so on, are controlled sufficiently tightly to assure a con-sistent refractive index for the deposited material. The correct film thickness is the usual objective of the monitoring process. There are many different techniques that can be, and have been, used for the monitoring of the layer thicknesses in a thin-film coating. Nowadays, the three most often employed are simple timing, mass measure-ment by quartz crystal microbalance, and optical monitoring.

Timing is simple and depends on very stable deposition rates. This is achieved in some of the latest sputtering machines.

Quartz crystal microbalances are specially constructed for thin-film thickness monitoring and are derived from the crystal tuned oscillator used in transmitters and receivers. Here, the quartz crystal is exposed to the vapor stream. The addition of mass to the face of the crystal changes its vibrational frequency and, hence, by cou-pling through the piezoelectric effect, the frequency of the associated oscillator. This frequency change is then a measure of the deposited mass.

Optical monitoring involves the measurement of some aspect of optical perform-ance from which the thickness of the deposited film can be deduced. Often a single wavelength is used for the measurement of either transmittance or reflectance of the deposited film. The signal, similar to that shown in Figure 9.5, oscillates with each extremum indicating the addition of a further quarterwave. The film may be part of the coating being manufactured or may be deposited on a separate test glass usually referred to as a monitoring chip. Special holders that have the ability to change the chips are commonly employed. Recently, more attention has been paid to broadband monitoring systems where the spectral performance over a range of wavelengths is measured and analyzed.

Each of these techniques has strengths and weaknesses and the optimum depends very much on the coating being manufactured.

9.10 Production Tolerances

Assigning tolerances to layer thicknesses and optical properties is a complex and involved task (Figure 9.21). Errors in one layer tend to affect the performance of others, and this interaction is a function not only of the design but also of the particu-lar process used for monitoring the parameter values. It has been found that the most useful tolerancing approaches involve a computer model of the actual process that can be run many times with random errors drawn from an infinite population with


statistical properties similar to those in the actual equipment to be used for the depo-sition. Such models are known by the term Monte Carlo after the famous European casino. Thus, the question of tolerances is effectively replaced by the question: in this equipment with these properties and methods is it possible to manufacture this filter with reasonable yield?

Such models also provide a test of the production plan or run sheet as it is some-times called. Often a plan that fails to deliver an acceptable yield can be corrected by altering the sequence of chip changes, readily accomplished by the modeling process and avoiding a usually expensive series of actual test runs.

9.11 Optical Instruments. Modeling Their Optical Behavior

The model of the optical properties of an optical coating is exceptional in its validity. It can readily be extended to model the properties of an instrument or system containing coatings. However, there are some pitfalls that must be avoided. The normal calcula-tion of optical coating performance is with reference to an infinite, linearly polarized, monochromatic plane wave. This simple type of illumination never exists in practice. Instead there are cones of illumination, wide spectral regions, different polarizations, scattered light, and other departures from perfection, especially in the coatings them-selves. Fortunately, these effects are linear and, with the exception of scattered light that needs special treatment, can be expressed as a sum of ideal components.

0

1

2

3

4

5

700600500400

Reflectance (%)

Wavelength (nm)

Figure 9.21 A simple tolerancing exercise that models the production of a filter as if all layers were controlled on separate monitoring chips. This shows 20 plots of the four-layer antireflection coating from Figure 9.11 with independent random thickness errors of 1% standard deviation. A good rule of thumb is that most monitoring systems will readily achieve thicknesses within 2% standard deviation of the correct values. 1% requires greater care and attention.


The concept of coherence [3] is important in this summation. When two ideal beams of light of identical frequency, but with phase difference ϕ, are combined, the irradiance of the sum can be written as:

I I I I I1 2 1 22 cosϕ

The cosine term is known as the interference term and it can be either positive or negative, depending on ϕ. Now let the two ideal light rays each be replaced by a bundle of rays. The interference term that results will be the sum of a large number of like terms, each with a different ϕ. If the interference term that results is essen-tially undiminished, then we term this combination of rays coherent. Should the interference term disappear then the term is incoherent. Partially coherent is the term we employ when the interference term is diminished but not eliminated. The two most important cases are the coherent case and the incoherent case. Although the incoherent case is simply the coherent case integrated over a sufficiently wide range of parameters to eliminate the interference effects, it is usual to calculate those com-ponents that clearly support incoherent beam combination by simply summing the irradiances.

A substrate is normally quite thick in optical terms, i.e., it is many wavelengths in thickness. Small changes in thickness together with small differences in angle of incidence over the aperture of the substrate translate into quite significant shifts in ϕ that will usually wash out the interference term completely. Substrates, therefore, can usually be considered to involve incoherent beam combination. With no absorption and two surfaces of transmittances T1 and T2, the incoherent combination gives a net transmittance of

T T T 1 2

given that all reflected beams are deflected out of the aperture of the receiver, or

TT T

1

1/ 1/ 11 2( ) ( )

if all reflected beams are collected by the receiver. In either case, the transmittance cannot be higher than either T1 or T2 but when the transmittances are low, Eq. (9.34) gives a much higher net transmittance than does Eq. (9.33). This is of considerable importance when filters are being placed in series to secure a wide region of low transmittance. A sufficiently wide illuminating cone can also assure the incoherent case in the substrate.

Cones can also cause problems with coatings. The change in angle of incidence through the cone will alter the optical properties of a coating, with the movement of the performance toward shorter wavelength, with perhaps some distortion and some polarization effects. They are calculable but complicated. The polarization effects can be particularly serious. A severe degradation of performance occurs in polarizing

(9.32)

(9.33)

(9.34)


coatings that are necessarily tilted with respect to the incident light. The effect in this case has its origin in the fact that s- and p-polarization directions are defined by the local plane of incidence that varies throughout the illuminating cone. For col-limated light, incredibly high degrees of polarization can be obtained, but these can-not be sustained in a cone of illumination. Let E denote the extinction ratio that is defined by the polarization leakage. E will then also represent the ratio below which no improvement is possible even with the most advanced polarizer design. Then it can be shown [14] that an approximate value for E is

E Ω2

24 tan ϕ

where Ω is the semiangle of the cone expressed in radians (Ω/57.3 if Ω is given in degree) and ϕ the angle of incidence of the cone axis on the polarizer coating. The expression is good for sermiangles up to around 8°. Note that both these angles are measured in whatever medium is the incident one for the coating, usually glass in a cube beam splitter. As an example we can take a 3° cone at 45° incidence to a cube polarizer coating, giving a limiting performance of 0.07%.

Stray light can be an insidious problem in a system that uses thin-film opti-cal filters. The problem is that optical coatings most frequently reject a good por-tion of unwanted light by reflection rather than absorption. The reflected light does not disappear but rather is redirected, and, if the design of the system does not pre-vent it, may return toward the component that rejected it, either at a different angle of incidence and meeting a higher transmittance than before or escaping around it. Effective baffles to prevent this behavior are important.

Scattering is a still more difficult topic [15–19]. Surfaces and bulk material suf-fer from imperfections of one kind or another. These imperfections are local dis-turbances of what is otherwise a uniform property, either the homogeneity of the material or the smoothness of the surface. Such defects are spaced much wider apart than the atoms or molecules of the intrinsic material and so scatter the light inco-herently rather than coherently, rather like isolated atoms or molecules. A problem is that it is impossible exactly to specify the form either of the solid parts or of the surfaces of the films, and we take refuge in statistical parameters and they depend on the scale with which we measure the imperfections. Scattering theory, therefore, is approximate, unlike the exact nature of the normal interference calculations for thin films. As far as the transmittance and reflectance of the coating are concerned, scat-tering acts much like normal absorption. It is locally proportional to the square of the electric field amplitude. However, the particular theory of scattering that should be used to determine the scattered light distribution depends very much on the scale of the imperfections. Apart from the performance reduction caused by the loss in scat-tering, the scattered light, like that lost by reflection in the previous paragraph, does go somewhere else. Some is trapped in waveguide modes in the coating, or substrate, but the rest can cause difficulties such as coupling between the elements of an array receiver, especially when an array filter is used over it to define the passbands of

(9.35)


the elements. Scattered light tends to emerge from an optical coating along the easy directions, i.e., those directions where the transmittance is high. This can help to give some idea of the variation of scattering with wavelength and angle of incidence.

There are various definitions used in connection with the distribution of scattered light. The light is measured as a power distribution, and so the units will be power per unit solid angle. However, just like the specular reflectance and transmittance, the effect is linear and so we will compare the scattered light with that incident. The incident light will generally illuminate a small spot and the total power incident on that spot can be written Pinc. The receiver measures a power of Ps and can be con-sidered to be of an area that will subtend a solid angle of ΔΩ at a direction given by angle ϑ with respect to the normal. The bidirectional scattering distribution function (BSDF) is based on what is sometimes called the AΩ product (which can be detected in the denominator of the first version in Eq. (9.36)). It is given by:

BSDFcos

/

coss

inc

s

inc

P

P

P

P( )

( )

ϑ ϑ∆Ω∆Ω

the second definition being the common one.Angle-resolved scattering (ARS) sometimes called the cosine-corrected BSDF

simply omits the cosine in Eq. (9.36) to give

ARS/s

inc

( )P

P

∆Ω

and is often preferred because it is a little closer to what is actually measured.Total integrated scatter (TIS) is a measure of the total scattering into the appropri-

ate hemisphere, invariably the reflected one. The incident beam is usually directed normally onto the surface and the specularly reflected beam is excluded from the measurement. If the integrated scatter is normalized with respect to the specularly reflected light for a completely smooth surface, rather than the incident light, the TIS becomes purely a surface property related to the surface roughness.

TISd 4

s

reflected

P

P∆Ω

Ω∫

πσλ

2

where σ is the rms surface roughness.Figure 9.22 shows a scatter distribution calculated for a multiple-cavity filter sim-

ilar to that of Figure 9.19 where the various interfaces are assumed to have similar, but uncorrelated, roughness.

To understand the variation in the scattered light, we can very roughly consider it as depending on two primary factors, the ease of entry of the incident light, which in turn determines the magnitude of the electric field through the coating, and the ease of emergence of the scattered light. It can be thought of as very roughly varying in

(9.36)

(9.37)

(9.38)


the same sense as the product of these two factors. Figure 9.23 shows the transmit-tance of the filter on a log scale with the angle of incidence corresponding to that in the glass substrate. The filter was assumed surrounded by glass for this calculation. Since normal incidence was assumed for the incident light in the scattering calcu-lations, the ease of entry as the wavelength is permitted to vary, is represented by

960 980 1000 1020 1040

Wavelength (nm)

0

10

20

30

40

50

60

70

80S

catte

r an

gle

(°)

Log (transmitted ARS)

–2 .. 0

–3 .. –2

–4 .. –3

–5 .. –4

–6 .. –5

–7 .. –6

–8 .. –7

–9 .. –8

–10 .. –9

–11 .. –10

–12 .. –11

Figure 9.22 Typical calculation of the variation of angle-resolved scattering from a multiple-cavity narrowband filter of the type of Figure 9.19 with interfaces of similar, but uncorrelated, roughness. The scattered light is s-polarized and the incident light is incident normally and polarized normal to the scattering plane.

960 980 1000 1020 1040

Wavelength (nm)

0

10

20

30

40

50

60

70

80

Inci

dent

ang

le (

°)

Log transmittance (dB)

–10 .. 0

–20 .. –10

–30 .. –20

–40 .. –30

–50 .. –40

–60 .. –50

–70 .. –60

–80 .. –70

–90 .. –80

–100 .. –90

Figure 9.23 Contour plot of the s-polarized transmittance in dB of the narrowband filter of Figure 9.22. The incident angle is measured in glass corresponding to the transmittance out of the coating into the glass substrate. The easy directions are echoed in Figure 9.22.


the region of the plot next to the wavelength axis, where it corresponds to zero inci-dence. The ease of emergence is essentially given by the whole plot. If each row of the plot is modulated by the varying transmittance along the wavelength axis, then a variation akin to the plot of Figure 9.22 is obtained.

There is nothing mysterious about the behavior of coatings in systems. Good models are available for all aspects of performance, but it is certainly necessary to know of the effects and how to calculate them.

9.12 Future Possibilities

An area that is still very much under development is that of composite materi-als [20,21]. Composite materials have been studied and employed since the time of the early Egyptians, who used them to produce durable pigments based on ground colored glass. Anyone who has visited the ancient Egyptian tombs can testify to their durability. Venetian glass has long used metal dispersions to produce deep, vivid colors. Metal and/or semiconductor dispersions in dielectric materials yield proper-ties that are outside the range possible with the individual constituents. Blacks, i.e., wideband absorbing materials, color filters, and enhanced nonlinearities, are all pos-sible with composite materials. Theory is well developed, but there are many theo-ries. The correct one depends on the details of the eventual microstructure, and so we have no completely reliable way of determining the appropriate model until we have prepared and investigated the material to determine its microstructure.

Deliberately regular structures also have interesting properties. Such struc-tures date back several centuries with ruled gratings used for decorative purposes. Recently, advances in lithography have revolutionized manufacturing techniques particularly in the manufacture of metal grid polarizers that now work into the vis-ible and near ultraviolet [22–24]. These can be combined with conventional thin films to improve still further their properties. The 1960s and early 1970s saw the development of various grid structures for use as infrared filters of different kinds [25,26]. We can expect these also to penetrate the visible and ultraviolet, with thin-film enhanced properties. Structures that scatter into and out of surface plasmons can yield, at normal incidence, properties that are available to conventional thin films only at very oblique incidence [27]. An example of this is a corrugated thin metal structure that presents properties similar to metal-dielectric induced transmission fil-ter [28]. There are many more possibilities. Indeed, the use of surface plasmons is expanding in many different directions, especially in their use in sensitive detectors of specific materials such as complex proteins [29,30].

A related area is that of photonic crystals [31,32]. Here, the distribution of the inclusions is also completely regular so that the material should appear to light photons as a crystalline material appears to an electron. The hope is that materials might present an optical band structure similar to an electronic one. The technique has had great success in two-dimensional form in optical fibers. Unfortunately, one-dimensional structures of this kind are indistinguishable from normal optical


coatings, a fact that appeared to escape many early workers in this field and caused great confusion in the early days of the subject.

Less regular structures, known often as moth eye because they recall similar structures that reduce reflection from the eyes of moths, presenting properties simi-lar to those of an inhomogeneous thin antireflecting film, can be created on the sur-faces of components by techniques of etching [33,34]. The microscopic appearance is of a forest of narrow spikes that taper from substrate to outer surface. The basic idea is not new, but the technology is constantly advancing. Such treated surfaces are mechanically rather weak and recently ways of strengthening them by the addition of thin conventional coatings have been devised [33]. Also there are many more sealed applications nowadays, where environmental resistance is of less importance than for an externally exposed coating.

Thin films deposited at oblique incidence retain their columnar structure but are usually still further enhanced [35–37]. The low packing density reduces the refractive index and with such films new types of more efficient antireflection coatings can be produced [38,39]. Also the enhanced columnar structure gives rise to enhanced bire-fringence, and this permits polarization-sensitive coatings to be constructed at normal incidence. Further, thin films with chiral properties can now be constructed [40]. The small area and environmental weakness of such coatings is not a barrier to their use in the small protected environments that are becoming more and more common.

Organic light-emitting diodes are thin-film structures that include light emis-sion. Although the thin-film theory is well developed, conventional theory does not include the generation of light. This is, therefore, a new area for thin-film theory and the field has responded with some useful models [41]. More will certainly be required in the future.

An opposite application is the deliberate absorption of light in a structure. Of course, light absorbers and especially selective light absorbers for efficient trap-ping and utilization of the thermal energy in sunlight have been studied for some time. A newer topic is the direct conversion of the light into electron-hole pairs in photocell structures. Such structures are not new, but usually their operation has been divorced from the properties of the optical coatings deposited on their entry surfaces to reduce reflection loss. Nowadays, we are including the entire cell in the optical thin-film model so that the field distributions can be designed with the object of assuring that carrier pairs should be most efficiently produced in the optimum regions of the cell. Such are the rewards for more efficient cells that much of this work is still proprietary, but we can expect the modeling of optoelectronic structures to expand still further.

These are just some of the exciting areas of current development. Some aspects of thin films may be considered mature, but there is great activity at the still advancing and expanding edge of the subject.

Acknowledgments

The figures in this article were all provided by courtesy of Thin Film Center Inc., and the author is particularly grateful to Chris Clark of that company for his valuable help.


Further Reading

There are a number of books that deal with the subject of optical coatings in more detail. A small selection follows.P.W. Baumeister, Optical Coating Technology. 2004, SPIE Press: Bellingham, WA.J.A. Dobrowolski, Optical properties of films and coatings, in: Handbook of Optics, M. Bass,

et al. (Eds.) 1995, McGraw-Hill: New York, NY. pp. 42.1-42.130.F.R. Flory, Thin films for optical systems. first ed. Optical Engineering, vol. 49 (Ed.) B.J.

Thomson. 1995, Marcel Dekker: New York, NY., 585 pp.I.J. Hodgkinson, Q.H. Wu, Birefringent Thin Films and Polarizing Elements. first ed. 1997,

World Scientific Publishing: Singapore. 379 pp.R.E. Hummel, K.H. Guenther, Handbook of Optical Properties: Thin Films for Optical

Coatings. vol. 1. 1995, CRC Press: Boca Raton, FL., 361 pp.N. Kaiser, H.K. Pulker (Eds.) Optical Interference Coatings. Optical sciences, (Ed.) W.T.

Rhodes. 2003, Springer-Verlag: Berlin/Heidelberg/New York., 500 pp.H.A. Macleod, Thin-Film Optical Filters. fourth ed. 2010, CRC Press: Boca Raton/London/

New York., 782 pp.H.K. Pulker, Coatings on Glass (Thin Films Science and Technology). second ed. 1999,

Elsevier: Amsterdam., 548 pp.J.D. Rancourt, Optical Thin Films: Users’ Handbook. Optical and electro-optical engineering,

(Eds.) W.J. Smith and R. Fischer 1987, Macmillan: New York, NY., 289 pp. (Now avail-able from SPIE Press: Bellingham, WA).

A. Thelen, Design of optical interference coatings. first ed. McGraw-Hill Optical and Electro-Optical Engineering Series, (Eds.) R.E. Fischer and W.J. Smith 1988, McGraw-Hill: New York, NY., 255 pp. (Out of print. Can be downloaded from: http://www.alfredthelen.com/dooic.pdf).

R.R. Willey, Practical design and production of optical thin films. second ed. Optical Engineering, (Ed.) B. Thompson 2002, Marcel Dekker: New York, Basel., 547 pp.

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Date post:	23-Dec-2016
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Handbook of Thin Film Deposition || Optical Thin Films

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