1Laser Light

References

1. A. Siegman, Lasers (University Science Boks, Mill Valley, 1986), Chapter 1.

2. P. W. Milonni and J. H. Eberly (John Wiley and Sons, Inc., Hoboken, NJ,2010), Chapter 13.

Laser is an acronym for Light Amplification by Stimulated Emission of Radiation.The term light is used in a broad sense to include radiation at frequencies in theinfra-red, visible or ultraviolet regions of the electromagnetic wave spectrum. Incommon parlance the term laser refers more to a device based on this principlethan to the principle itself. The term laser action is often used when referring tothe process. Lasers are devices that generate coherent light.The physical principle (stimulated emission) responsible for laser action was intro-

duced by Albert Einstein in 1916. A device called MASER (microwave amplificationby stimulated emission of radiation) based on this principle was first operated inthe microwave regime. The laser is an extension of this principle to the visible partof the electromagnetic spectrum.A summary of principal developments in the field of laser follows.

1916: A. Einstein introduces stimulated emission as a fundamental processof light-matter interaction in addition to the already known processes of ab-sorption and spontaneous emission of light.

1924: Richard Tolman discusses negative absorption i.e. amplification,and explains that the emitted radiation would be coherent with the inputradiation.

928: Rudolph W. Landenburg confirms existence of stimulated emission.

1940: V. A. Fabrikant suggests method for producing population inversionin his PhD thesis. Population inversion is required for maser/laser operation.

1950: Alfred Kastler suggests a method of optical pumping for orientationof paramagnetic atoms or nuclei in the ground state. This was an importantstep on the way to the development of lasers for which Kastler received the1966 Nobel Prize in Physics.

1951: Edward Purcell and Robert Pound observe inverted populations ofstates in a nuclear magnetic resonance experiment. Population inversions is anecessary condition for maser and laser action.

1

2 Laser Physics

1952: Nikolay Basov and Alexander Prokhorov describe the principle of themaser (Microwave Amplification by Stimulated Emission of Radiation).

1954: C. H. Townes, J. P. Gordon, and H. J. Zeiger realize the first maserutilizing a beam of excited ammonia molecules to produce amplification ofmicrowaves by stimulated emission at a frequency of 24 gigahertz (GHz).

1958: Charles H. Townes and Arthur L. Schawlow introduce concept of thelaser.

1959: Gordon Gould introduces the term laser in a paper, The LASER:Light Amplification by Stimulated Emission of Radiation

1960: Laser action observed by T. H. Maiman in Ruby [Nature 187, 493(1960)]. It is now known to be one of the most dicult laser systems tooperate. Sorokin and Stevenson develop first four-level solid-state laser atIBM. Ali Javan, William Bennett, and Donald Herriott at Bell Labs developfirst helium neon (He:Ne) gas laser.

1961: Elias Snitzer reports the operation of a neodymium glass laser, cur-rently the prime candidate as a laser source for fusion. In the first medical useof the laser, Charles Campbell and Charles Koester destroy a retinal tumorwith the ruby laser. In the first example of ecient nonlinear optics, P. A.Franken, A. E. Hill, C. W. Peters and G. Weinreich demonstrate generationof second harmonic light by passing the pulses from a ruby laser through aquartz crystal, transforming red light into green.

1962: Scientists at Bell Labs report the first yttrium aluminum garnet (YAG)laser, which continues to dominate material processing applications. Scien-tists at General Electric, IBM, and MIT Lincoln Laboratory develop a galliumarsenide laser that converts electrical energy directly into infrared light. F.J. McClung and R. W. Hellwarth develop laser Q-switching technique to pro-duces laser pulses of short duration and high peak powers. Four groups inthe US (M. I. Nathan et al., R. N. Hall et al, T. M. Quist et al, N. Holonyakand S. F. Bevacqua) nearly simultaneously make first semiconductor diodelasers, which operate pulsed at liquid-nitrogen temperature. Semiconductordiode lasers are the first important step in the development of optical com-munication, optical storage, optical pumping of solid-state lasers and manyother applications.

1963: L. E. Hargrove, R. L. Fork, and M. A. Pollack report the first mode-locked operation of a laser in a helium-neon laser with an acousto-optic mod-ulator. Mode locking is the basis for the femtosecond pulsed laser. Her-bert Kroemer and the team of Rudolf Kazarinov and Zhores Alferov inde-pendently propose ideas to build semiconductor lasers from heterostructuredevices, which lead to their receiving the 2000 Nobel Prize in Physics. C. K.N. Patel develops first carbon dioxide laser at Bell Labs.

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1964: C. H. Townes, N. G. Basov and A. M. Prokhorov awarded the Nobelprize for their fundamental work in Quantum Electronics; Townes for demon-strating the ammonia (NH3) maser and subsequent work in masers and lasersand Basov and Prokhorov for contributing to the development masers andlasers. William B. Bridges develops first noble gas ion laser. J. E. Geusic,and H. M. Marcos, and L. G. Van Uitert develop neodymium-doped yttriumaluminum garnet (Nd: YAG) laser. This is the most widely used solid statelaser; from cutting and welding to medical applications and nonlinear optics.C. J. Koester and E. Snitzer develop neodymium-doped fiber amplification.Fiber amplifiers are used in communication and for high power lasers. ArnoPenzias and Robert Wilson use maser amplifier to observe 3K cosmic back-ground radiation proving the existence of the Big Bang. They are awardedthe Nobel Prize in Physics in 1978.

1965: George C. Pimentel and Jerome V. V. Kasper demonstrate the firstchemical laser. With output currently reaching megawatt levels, chemicallasers get their energy from chemical reactions and are some of the mostpowerful lasers in the world. James Russell invents the laser compact disk(CD player). Anthony J.DeMaria, D. A. Stetser, and H. A. Heynau reportthe first generation of picosecond laser pulses using a neodymium glass laserand a saturable absorber.

1966: Peter Sorokin and John R. Lankard built the first widely tunable or-ganic dye laser, now used in ultrafast science and spectroscopy. Charles K.Kao and George Hockham of Standard Telecommunications Laboratories inEngland publish landmark paper demonstrating that optical fiber can trans-mit laser signals and reduce loss if the glass strands are pure enough. AlfredKastler is awarded the Nobel Prize in Physics for the discovery and develop-ment of optical methods for studying Hertzian resonances in atoms.

1968: NASA launches the first satellite equipped with a laser.

1969: Led on Earth by American physicist Carroll Alley and using retrore-flectors placed on the moon by Neil Armstrong and Buzz Aldrin, NASAsLunar Laser Ranging experiments begin. Using these mirrors, scientists onEarth bounce lasers o the moon, measuring its orbital motions, and in theprocess determining fundamental gravitational and relativistic constants withextraordinary precision. D. J. Spencer, T. A. Jacobs, H. Mirels, and R. W. F.Gross develop the first continuous-wave chemical laser. High power chemicallasers generate megawatts of power, leading to proposals for laser weapons.The pulsed dye laser is invented.

1970: Nikolai Basov, V. A. Danilychev, and Yu. M. Popov of Lebedev Phys-ical Institute in Moscow develop the excimer lasers, which are important inphotolithography and laser eye surgery. Zhores Alferovs group at the IoePhysical Institute and Mort Panish and Izuo Hayashi at Bell Labs producethe first continuous-wave room-temperature semiconductor lasers, paving the

4 Laser Physics

way toward commercialization of fiber optics communications. The worldsfirst laser-driven lighthouse opens in Australia (Point Danger). Robert Mau-rer, Peter Schultz and Donald Keck at Corning Glass Work prepare the firstbatch of optical fiber hundreds of yards long capable of carrying optical signalover it. J. Beaulieu invents transversely excited atmospheric (TEA) pressureCO2 laser useful for the machining industry. O. G. Peterson, S. A. Tuccio, andB. B. Snavely develop CW dye laser leading to a revolution in spectroscopyand ultrafast science. Arthur Ashkin demonstrates the use of laser beams tomanipulate microparticles pioneering the field of optical tweezing and trap-ping, leading to important advances in physics and biology. Robert Maurerand his colleagues Donald Keck and Peter C. Schultz at Corning Glass Worksdesigned and produced the first fiber with optical losses low enough for use intelecommunications.

1971: Dennis Gabor was awarded the Nobel prize in physics for his inventionand development of the holographic method.

1974:The first product logged in a grocery store by a barcode scanner. E.P. Ippen and C. V. Shank develop the sub-picosecond mode-locked CW dyelaser, establishing ultrafast optical science.

1975: Laser Diode Labs develops first commercial continuous-wave semicon-ductor laser operating at room temperature. Continuous-wave operation al-lows the transmission of telephone conversations.

1976: John Madey and group at Stanford University demonstrate the first freeelectron laser (FEL). Instead of a gain medium, FELs use a beam of electronsaccelerated to near light speed, then passed through a series of alternatingmagnetic fields. The forced undulating motion results in the release of acoherent photon beams with widest tunable frequency range of any laser typedue to the tunable magnetic field.

1977: General Telephone and Electronics send first live telephone tracthrough fiber optics, 6 Mbit/s in Long Beach CA.

1981: N. Bloembergen and Arthur Schawlow were awarded the Nobel Prizein physics for their contributions to masers, nonlinear optics and spectroscopy.

1982: Kanti Jain publishes the first paper on excimer laser lithography usedextensively today to make microchips for the computer and electronics indus-try. P. F. Moulton develops titanium-sapphire laser, which has nearly replacedthe dye laser for tunable and ultrafast laser applications.

1985: Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips developmethods to cool and trap atoms with laser light. Their research helps to studyfundamental phenomena and measure important physical quantities with un-precedented precision. They are awarded the Nobel Prize in Physics in 1997.Grard Mourou and Donna Strickland demonstrate chirped pulse amplification

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or CPA. They used gratings to lengthen laser pulses before amplification, andthen again after the amplification to shorten them to their original length.This permits much higher powers without damaging the amplifying materialitself. CPA was later used to create ultrashort, very high-intensity (petawatt)laser pulses.

1987: Ophthalmologist Steven Trokel performs the first laser eye surgeryusing an excimer laser. Emmanuel Desurvire, David Payne, and P.J. Mearsdemonstrate optical amplifiers that are built into the fiber-optic cable itself.

1988: Samuel Blum, Rangaswamy Srinivasan, and James Wynne observedthe eect of the ultraviolet excimer laser on biological materials. Further in-vestigations revealed that the laser made clean, precise cuts ideal for delicatesurgeries. First transatlantic fiber cable is laid with glass so transparent thatamplifiers are only needed about every 40 miles. Double clad fiber laser devel-oped by E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum.These high power solid-state lasers are used for machining.

1989: Norman F. Ramsey was awarded the Nobel Prize for the inventionof the separated oscillatory fields method and its use in the hydrogen maserand other atomic clocks and Hans G. Dehmelt and Wolfgang Paul for thedevelopment of the ion trap technique.

1992: Eric Betzig, Ray Wolfe, Mike Gyorgy, Jay Trautman, and Pat Flynndevelop a magneto-optic data storage technique that can squeeze 45 billionbits of data into a square-inch of disk space.

1994: First proposed in 1971 by Rudy Kazarinov and Robert Suris, the firstquantum cascade laser was demonstrated by Jerome Faist, Federico Capasso,Deborah Sivco, Carlo Sirtori, Albert Hutchinson, and Alfred Cho of Bell Labs.

1996: S. Nakamura and coworkers develop GaN (Gallium nitride) and InGaN(Indium gallium nitride) semiconductor lasers.

1997: Steven Chu, Claude Cohen-Tannoudji and William D. Phillips awardedthe NObel Prize for the development of methods to cool and trap atoms withlaser light. Researchers at MIT create the first atom laser.

2000: Zhores I. Alferov and Herbert Kroemer are awarded the Nobel Prizein Physics for basic work on information and communication technologyand for developing semiconductor heterostructures used in high-speed- andopto-electronics. John Hall and Theodor Hansch develop optical frequencycomb technique used in research as well as in precision metrology and timemeasurement. This work leads to their receiving the 2005 Nobel Prize inPhysics.

2001: Eric A. Cornell, Wolfgang Ketterle and Carl E. Wieman were awardedthe Noble Prize for the achievement of Bose-Einstein condensation in dilute

6 Laser Physics

gases of alkali atoms, and for early fundamental studies of the properties ofthe condensates.

2005: J. L. Hall and T. W. Hansch were awarded the Nobel prize for theircontributions to the development of laser-based precision spectroscopy, in-cluding the optical frequency comb technique and to Roy Glauber for hiscontribution to the quantum theory of optical coherence. INTEL creates achip containing eight continuous Raman lasers by using fairly standard sil-icon processes rather than the somewhat expensive materials and processesrequired for making lasers today.

2009: Charles Kao was awarded the Nobel Prize for physics for his work infiber optics along with Willard S. Boyle and George E. Smith of Bell Labs whodeveloped the CCD (charge coupled device) which made digital photographypossible. The SLAC Linac Coherent Light Source produces the first evercoherent, hard X-ray beam. The ultrafast laser pulses are powerful enough tomake images of single molecules or atoms in motion.

There are three essential elements of a laser:

1. A gain or amplifying medium consisting of atoms, molecules, ions, or chargedcarriers along with a pumping mechanism to excite these species to their higherquantum mechanical energy states. The energy stored in the excitation canbe emitted as light spontaneously or stimulated by pre-existing light leadingto an amplification of light energy.

2. A suitable arrangement of optical elements (lenses, mirrors, prisms, etc. )or some other mechanism to allow multiple passage of light through the gainmedium (feedback).

3. A loss mechanism to extract light energy from the device. In addition tothis desirable or essential loss, nonessential but unavoidable losses of lightenergy due to absorption, diraction, scattering, and transmission throughmirrors and other optical elements are also present.

These three elements come in a great variety of forms and shapes and provide thebasis for classifying lasers.

1.1 Survey of Laser Elements

A wide variety of laser gain media and pumps are used to generate radiation rangingfrom far infra-red (far-IR) to soft X-rays.

Laser Light 7

PumpMirror

(feedback)

Gain medium

Loss

Mirror

(feedback)

FIGURE 1.1Basic elements of a laser.

1.1.1 Gain Media

Important laser gain media and wavelengths include,

HCN far-IR laser (311, 337, 545, 676, 744 m)

H2O far-IR laser (28,48, 120 m)

CO2 laser (9.6-10.6 m)

CO laser (5.1-6.5 m)

HF chemical laser (2.7-3.0 m)

Nd:YAG laser (1.06 m)

He:Ne laser (1.15 m, 633 nm)

Ga-As semiconductor laser (870 nm)

Ruby laser (694 nm)

Rhodamine 6G dye laser (560-640 nm)

Argon-ion laser (488-514 nm)

Pulsed N2 discharge laser (337 nm)

Pulsed H2 discharge laser (160 nm)

1.1.1.1 Laser Pump

Depending on the gain medium, many dierent types of pump mechanisms are usedto supply energy to the gain media. These include

Gas discharge including dc, radio-frequency, and pulsed electrical dischargesinvolving both direct electron excitation and two-stage collision pumping.

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Optical pumping using flash lamps, arc lamps, semiconductor LEDs (lightemitting diodes), other lasers and even direct sunlight.

Chemical reactions including chemical mixing, photolysis, and combustion.

Direct electrical pumping includes high-voltage electron beams directed intohigh-pressure gas cells and direct current injection into semiconductor injec-tion lasers.

Nuclear pumping of gases by nuclear fission fragments when a gas laser tubeis placed in close proximity of a nuclear reactor.

Supersonic expansion of gases, usually pre-heated by chemical reaction orelectrical discharge, through supersonic expansion nozzles, to create the so-called gas-dynamic lasers.

Plasma pumping in hot dense plasmas, created by plasma pinches, focusedhigh-power laser pulses, or electrical pulses. There are reports of X-ray laseraction in some laser materials pumped by the explosion of a nuclear bomb.

1.1.2 Optical feedback

The principle mechanism for providing feedback in lasers involves optical cavitiesor resonators. Resonators store electromagnetic energy. At microwave frequenciesthese resonators are closed metallic boxes but at optical frequencies they can beopen. The need for open resonators arises because we want only a few modesinteracting with the atoms to grow1. The presence of boundaries in resonators(boundary conditions) allows only certain field configurations (frequency and spatialvariation) to exist inside the resonators. These allowed field configurations are calledmodes of the resonator. For open resonators modes corresponding to propagationaway from the resonator axis will have very large losses because any light emittedinto such modes will be quickly lost. Only a group of paraxial modes where energyis localized near the axis will experience build up. Even these modes will experiencelosses due to diraction, absorption, and transmission by the end mirrors. As aresult mode definition in the sense of stationary field configurations can not beused. However, modes with quasi-stationary (long lived) patterns do exist. Thesemodes experience very little loss of light energy in one cavity round trip so that anyenergy emitted into these modes will remain in the cavity for a long time.

1.1.3 Losses

Mechanisms that lead to a loss of light energy stored in the cavity includeDiraction by optical elements inside the resonator,

1At optical frequencies the density (# per unit frequency band per unit volume of the resonator)of modes near a frequency is very large 82/c3. If the atoms interacts with modes in a band (typically 109 Hz or more), their number 82/c3 is enormous. Such a situation is not conduciveto the growth of any mode amplitude to a significant level.

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Absorption inside the gain medium and mirrors, coating,Scattering, andTransmission through the mirrors.Although the loss is detrimental to achieving laser action, not all loss is undesir-

able. In fact the loss of light energy stored in the cavity via mirror transmission iswhat emerges as the beam of light that has made the laser into such a useful toolthat hardly any aspect of our life has remain untouched by it.A coupling of gain, loss, and feedback mechanisms via the electromagnetic field

makes laser action possible and imparts to the light generated by lasers certainextreme characteristics. The long lived modes interact with the gain medium andextract energy from it. The gain medium can either emit light spontaneously intoany of the modes - long or short lived - or it can be stimulated to emit into aparticular mode by the light energy stored in that mode. Stimulated emission oflight into a particular mode increases as the light energy of the mode increases.Clearly the light energy in long lived modes is more likely to be amplified providedthey can overcome their energy losses. Thus it is the competition between gain dueto stimulated emission and loss due to diraction, transmission, absorption, etc.that determines whether light amplification by stimulated emission of radiation cantake place or not. If gain exceeds loss, laser action can occur.An understanding of lasers will require us to study an interacting matter and

light system inside a resonator. It is an open system as energy can be added to orextracted from the system. This may seem like a formidable problem. We will see,however, that this seemingly complex problem can be dealt with quite eectively.We will first study atoms and light separately by ignoring their interaction and

then couple the two. The atoms are described by Schrodinger equation. Light, beingan electromagnetic wave phenomenon, is described by Maxwells equations. Itspropagation as rays or waves, diraction, and interference, all follow from Maxwellsequations. These equations admit wave-like solutions. Plane waves are the bestknown of these solutions. For lasers we require beam like wave solutions which,like plane waves, have a pre-dominant direction of propagation but they have finiteextent in directions perpendicular to the direction of propagation. In view of theremarks in the preceding paragraph one might suspect that these solutions are theright type to fit the boundary conditions imposed by open optical resonators. Wewill see that this is indeed the case.

1.2 Laser Light Characteristics

The light output from a laser is electromagnetic radiation and is not fundamentallydierent from the light emitted by other sources of electromagnetic radiation. Thereare, however, several important dierences in detail between laser light and the lightemitted by thermal sources. The output beam produced by lasers have much morein common with the output of conventional low-frequency electronic oscillators than

10 Laser Physics

they do with any kind of thermal light sources. We will briefly review laser beamcharacteristics that distinguish them from other light sources. The numbers statedbelow for lasers should not be taken as the final word. They change as progress ismade in improving the performance of lasers.

1.2.1 Monochromaticity

The light emitted by a laser has high degree of spectral purity. This means it has rel-atively well-defined frequency or wavelength so that the output signal from an ideallaser is very nearly a constant amplitude sinusoidal wave. Two factors contribute tothe spectral purity of laser light. First, light emission from atoms occurs in a narrowrange of frequencies around atomic transition frequency o = (E2 E1)/h. Thisrange of frequencies defines the atomic linewidth. Consequently only EM waveswith frequencies close to the atomic transition frequency can interact strongly withthe atoms and be amplified. Second, the laser cavity forms a resonant structure.

!c!

Laser emission

Atomic emission lineCavityresonances

!o

"!L

Laser Light 11

moderately stabilized lasers to L = 10 Hz or less in well stabilized lasers. Sincevisible light has frequencies of order = 5 1014 Hz, the spectral purity of laserlight is L/L = 2 1013. The ratio Q = LL is a measure of the qualityfactor (Q-factor) of the laser oscillator. Such large values of Q are dicult toachieve in mechanical or electronic oscillators. Thermal sources such as the sunand incandescent solids generally emit a broadband spectrum of light. There are,however, some thermal sources such as discharge lamps, that emit only a few spectrallines or narrow bands of wavelengths, but the spectral widths of the light emitted byeven the best such sources are still limited by the linewidths of the atomic transitionsin the discharge which range from 108 1011 Hz. Table (1.1) shows a comparisonof the spectral purity of light emitted by dierent sources of light [FWHM=FullWidth at Half Maximum].

TABLE 1.1A comparison of spectral purity of dierent light sources

Light Source Peak Wavelength FWHM () FWHM ()He:Ne Laser 633 nm 108 nm 7.5 103 HzCadmium low pressure lamp 644 nm 103 nm 9.4 108 HzSodium discharge lamp 590 nm 0.1 nm 9 1010 HzBlackbody radiator at 5800 K 500 nm 600 nm 1014 Hz

The ultimate limit on laser spectral purity is set by quantum noise fluctuations dueto spontaneous emission from the atoms in the gain medium. This limit, however,can be reached with great diculty only on the very best and highly stabilizedlasers.

1.2.2 Coherence

If we picture the light wave from the laser as a sine wave, its amplitude and phasewill in fact change with position and time. Coherence refers to how well the phaseand amplitude of light wave at one space-time point stay correlated to the phaseand amplitude at some other space-time point. There are two types of coherences.

Temporal coherence refers to strong correlations between the amplitude and/orphase of light wave at dierent times. A measure of temporal coherence is thelength of time c over which the phase and amplitude of the sine wave are correlated.We may picture that the phase of the sine wave is interrupted at random, at anaverage rate 1/c. Laser light has high degree of temporal coherence. This meansthe amplitude and phase of the sine wave representing the light from the laser ispredictable over time intervals of order c, which can span millions of optical cycles.For stationary beams (whose statistical properties do not change wit time), the time

12 Laser Physics

interval c, called the coherence time, is related to the spectral width

c 12 (1.1)The distance traversed by light in one coherence time c is called the (longitudinal)coherence length

c = cc . (1.2)

Coherence time c may be thought of as the average duration for which the laserlight can be thought of as a pure sine wave and c may then be thought of as theaverage length of perfect sine wave emitted by the laser. For lasers c can rangefrom 300 m to 104 m or even larger.

Source c (s) c (m)He:Ne laser 104 102 3 104 3 1010Sodium lamp 1010 3 102

Spatial coherence refers to correlation between laser field at dierent points in aplane transverse to the direction of wave propagation. For this reason, times alsoreferred to as transverse coherence, We can think of transverse coherence in terms ofa two slit interference experiment. Its the largest separation between two pinholes,which, when illuminated by the same light wave will produce interference fringes.Another way to look at transverse spatial coherence is to think of it is the transversespatial extent over which the field can be considered to be a part of the same phasefront. At the output of a laser oscillator, laser beam has almost perfect transversespatial coherence. For thermal sources transverse spatial coherence does not extendover distances much larger than a few wavelengths. Transverse spatial coherence oflaser light is also a consequence of the presence of a laser cavity.

1.2.3 Directionality or Collimation

Thermal sources emit light in random directions over a broad wavelength range. Wecan capture some fraction of this radiation and collimate it with a lens or mirroras in a searchlight or flashlight. The resulting degree of collimation (amount ofradiation emitted per unit solid angle) is still much smaller than that for a laser.Consequently, thermal beams spread very rapidly with propagation.A single-transverse mode laser oscillator, on the other hand, can produce a beam

that can propagate for sizable distances with very little diraction spread. Theangular divergence (half angle) in radians of a laser beam in the far zone (far frombeam waist) is given by

wo

(1.3)

where wo is the radius of laser beam spot at its waist (location where the beam hasnarrowest transverse size). The distance over which this beam stays approximatelycollimated before diraction spreading significantly increases is given by

b 2w2o

. (1.4)

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This is a direct consequence of the fact that laser beam comes from a resonant

z0

w(z)

Beam waist

!="/#wowo

FIGURE 1.3Divergence of a laser beam from its waist.

cavity where only the rays propagating close to the cavity axis can pass throughthe gain medium multiple times and thus grow in energy.For laser light of wavelength = 1.06 103 mm, w0 = 3 mm,

=

wo=

1.06 1063 103 = 1.1 10

4 rad = 0.006o .

For a small Helium Neon (He:Ne) laser emitting at = 633 nm, the beam waistmight be w0 0.5 mm. This corresponds to an angular divergence of

0.633 106

0.5 103 = 4 104 radians = 0.02o

and a collimation distance of b = 2.3 m. For an Argon-ion (Ar-ion) laser at 514 nm,the waist might be 5 mm corresponding to angular divergence of = 3 105 anda collimation distance of b = 310 m.Comparing these values to a normal flashlight for which the divergence is about

25o or a searchlight that has a typical divergence angle of 10o, the high directionalityof laser light is obvious.

1.2.4 Laser Beam Focusing

A laser beam can also be focused by a lens to a small spot only a few laser wave-lengths in diameter. The diameter of the focused spot is given by the formula

w wo

f (1.5)

14 Laser Physics

where f is the lens focal length. If the laser beam fills the lens aperture, the ratiof/w0 is simply the f -number of the lens. For best lenses this number is of orderone. It follows that a laser beam can be focused to spots which are only a fewwavelengths in diameter.The directionality of laser beam is also a consequence of the presence of a res-

onator.

1.2.5 Brightness

Brightness B of a source is defined as the power eux (power emitted per unit areaof the emitting surface per unit solid angle). Its units are W/m2sr. Spectral Bbrightness is power emitted per unit area of the source per unit solid angle per unitbandwidth. Its units are W/m2srHz. The idea of brightness can be understoodby considering a source that emits through a surface area S. Each area elementcan emit light into a solid angle 2 steradian. Then, if a surface element S emitspower P into a solid angle , the brightness of the source is given by

B =2P

S , [B] = W/m2sr .

The power emitted by a black body at temperature T is given by Stefan-Boltzmann

!S="wo2

!S

r2!#!#

Emission from athermal source

(a)

(b)

Emission from alaser cavity

FIGURE 1.4(a) Emission from the surface of a thermal source. (b) Emission from a laser cavity.

law to beI = T 4 , [I] = W/m2 ,

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where = 5.6705 108 W/m2K4 is the Stefan-Boltzman constant. For the sunwith T=6000 K this gives an eux of

Isun = 5.6705 108(6 103)4 = 7 107W/m2

A small He:Ne laser of modest power P = 1 mW and beam waist of 0.5 mm willproduce an eux

Ilaser =P

w2o=

103

(0.5 103)2 = 1.3 103W/m2

This looks even more impressive when we take into account the directionality. Theeux from the sun is emitted isotropically into a solid angle of 2 steradian (sr)giving a brightness of Bsun = Isun/2 107W/m2sr, because of the directionalityof the laser beam, the laser emits its power into a solid angle 2 = (/wo)2 =2/w2o . This leads to a brightness for the laser

BL =P

w202=

P

2=

103

(0.6328 106)2 = 2.5 108W/m2sr .

This exceeds the brightness of the sun. This comparison looks even more impressiveif we examine the spectral brightness. The sun radiates like a blackbody over widespectral bandwidth. Let us compare the spectral brightness of the sun near the peak(yellow) of visible spectrum. The spectral energy density [energy per unit volumeper unit bandwidth (J/m3Hz)] of a black-body radiator is given by

() =82

c3h

eh 1 , =1

kBT. (1.6)

The spectral intensity then is 12c() (W/m2Hz), where the factor of half accounts

for the fact that only half of the radiation is propagating outward toward the radiat-ing surface of the the black body. Since each surface element radiates isotropically(into and out of the source) only the radiation into the external solid angle = 2escapes the black body. Hence the spectral brightness of a black body radiator isgiven by

B =12()c

=12

82

c3h

eh 1c

2=

22

c2

h

eh 1.

For the sun, which radiates as a black body at a temperature of T 6000 K,spectral brightness at a wavelength in the yellow region [h = 2.5 eV] we have

kBT = kB 300T

300=

140 20 1

2eV

h

kBT 5 , eh = e5 150

B =2

(633 109)22.5 1.6 1019

150 1 2 108W/m2-sr-Hz

= 2 1012W/cm2srHz .

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For a l mW He:Ne laser ( = 633 nm) of spectral width = 104 Hz and spot size0.5 mm, the spectral brightness will be

B =P

2=

1 103(0.633 106)2 104 2.5 10

5W/m2srHz= 25W/cm2srHz

For a Neodymium glass (Nd-glass) laser with a power of P = 104 MW, = 1.06,and bandwidth limited by pulse duration of 30 ps, we have

=1

2p=

12 3 1011 5 10

9Hz

B =1 104 106

(106)2 5 109 2 1012W/m2.sr.Hz

= 2 108W/cm2.sr.Hz

1.2.6 Laser Performance Records

1. Wide power range: Continuous wave (CW) powers of up to hundreds of kilo-watts are available from certain IR chemical lasers. In pulsed mode peakpowers in excess of 1013 watts, which exceeds the total electrical power gen-erated in the U.S. for very short times (pico-seconds) are available.

2. Extreme frequency stability: The short term frequency stability of a highlystabilized laser can be as good as one part in1013. A He:Ne laser operatingat 3.39 m stabilized against a methane absorption line has absolute repro-ducibility of 1 part in 1010. Frequency stabilities of = 102 Hz have beenachieved.

3. Wide tunability: Most common lasers are limited to sharply defined discretefrequencies. However, widely tunable sources of coherent radiation includingdye lasers, titanium-doped sapphire (Ti-sapphire) lasers and optical paramet-ric oscillators (OPOs) can provide tunable radiation up to bandwidths of order 300 nm. This corresponds to a frequency bandwidth of 10131014Hz.

4. Ultra-short pulses : Mode-locked laser pulses shorter than 1 ps (picosecond)are routine. Mode-locked compressed dye laser pulses of only a few femtosec-ond long (FWHM) (few optical cycles) have been produced.

5. Very ecient: Power conversion eciency is defined to be the ratio of opticalpower radiated by the laser to the power supplied to operate the laser. Powerconversion eciencies of lasers range from 0.001 to 0.1% for gas lasers, 1 to 2 %for solid state lasers and 50-70% for carbon dioxide (CO2) and semiconductorlasers.

Laser Light 17

1.3 Electromagnetic Waves in Homogeneous Media

We will use the complex analytic representation of the fields so that the real physicalfields are given by

F (r, t) = ReF(r, t) , (1.7)

where F (r, t) represents any of the components of the fields. In a homogeneoustransparent isotropic medium characterized by dielectric permittivity and mag-netic permeability the constitutive relations are simple proportionalitiesD(r, t) =E(r, t) and B(r, t) = H(r, t). These relations hold for arbitrarily rapid variationof the fields as long as the most significant part of the field spectrum lies in thetransparency range of the medium (, real).In the presence of atoms or molecules that interact strongly with the field the

constitutive relations are modified to read

D(r, t) = E(r, t) +Pat(r, t) , (1.8a)B(r, t) = H(r, t) , (1.8b)J (r, t) = E(r, t) , (1.8c)

where J represents the conduction current density and is the conductivity of themedium representing dissipation of electromagnetic energy and Pat is the atomicpolarization induced by the interaction of light with the atom.In writing the constitutive relation between D and E, we assumed that the

strongly interacting atoms (active atoms) are embedded in a host medium. Letus denote the response of the host medium to the field by the polarization Pm andthat of the active atoms by Pat. Now the interaction of the field and the hostmedium is weak and we can take medium response to be a linear function of thefield: P = oeE, where e is the linear dielectric susceptibility of the host medium.In contrast, the response Pat of the strongly interacting active atoms cannot be ex-pected to be a linear function of the field. Indeed we shall see that the nonlinearresponse of the active atoms is essential for the working of a laser. Thus we havewritten the overall polarization as the sum of two parts P = Pm +Pat so that theconstitutive relation for the electric fields becomes

D(r, t) = oE(r, t) +P = o(1 + e)E(r, t) +Pat(r, t) . (1.9)

This relation then defines the dielectric constant = o(1 + e) in Eq. (1.8a). Westill need to compute Pat using quantum mechanics before proceeding further.In writing the preceding relations we have assumed that the magnetic response of

the atoms is negligible compared to their electrical response (magnetizationMat =0), which holds for most atomic media. We have also assumed that the fields are

18 Laser Physics

quasi-monochromatic2. Maxwells equations then read

D(r, t) = 0 , (1.10a) B(r, t) = 0 , (1.10b) E(r, t) = B(r, t)

t, (1.10c)

B(r, t) = J (r, t) + D(r, t)t

. (1.10d)

These are coupled first-order partial dierential equations. By eliminating the mag-netic (electric) field we can obtain a closed equation for the electric (magnetic) field.For example, on taking the curl of Eq. (1.10c) we obtain

( E) = tB . (1.11a)

Using the identity ( E) = ( E) 2E together with E = 0 andeliminating B with the help of Eq. (1.10d) and the constitutive relations, we findthat the electric field satisfies the equation

2E Et

2Et2

= 2Patt2

. (1.11)

This is driven damped vector wave equation. The second term represents the damp-ing of the wave amplitude due to loss of electromagnetic energy into heat. Theterm on the right hand side represents the source term involving atomic polariza-tion which is the source of the electromagnetic wave. To make further progress withthis equation, we need to know Pat. To make progress we deal with this problem inseveral stages. The idea is to start with a simpler system without loss and sourceterms and solve for the fields. The eect of loss and source terms is then included bymultiplying the solution in the absence of loss and source terms by an appropriatespace and/or time dependent factors.So we first consider lossless ( = 0 ) and source free region (Pat = 0), where the

equation satisfied by the field becomes2

2

t2

E(r, t) = 0. (1.12)

It is easily checked that in this case, the magnetic field B also satisfies the sameequation. The quantity 1/ has the dimensions of square of a speed v given by

v =

1

=

100

00

=c

n, (1.13)

2For quasi-monochromatic fields we can write F (r, t) = F o(r, t)eit, where the envelope F o(r, t)changes negligibly in times of order 2/. In terms of Fourier decomposition of the fields, we cansay that the field contains frequencies in a narrow band centered around the carrier frequency ( ). Parameters and can then be taken to represent the values of dielectric permittivityand magnetic permeability at the carrier frequency .

Laser Light 19

where c is the speed of light in free space and n is the refractive index of the medium,

c = 2.99792458 108 m/s 3.00 108 m/s. (1.14)n =

00. (1.15)

Equation (1.12) is then readily identified as homogeneous (source free) wave equa-tion with v as the wave speed.

1.4 Solutions of the Wave Equation

To gain some insight into these wave-like solutions of Maxwells equations, we con-sider a few special cases of the scalar wave equation for fields that depend only ona single spatial variable

2

z2 1v2

2

t2

F(z, t) = 0 , (1.16)

where F stands for any of the three Cartesian components of the field. By meansof the change of variables = t z/v and = t+ z/v, the derivatives in the waveequation can be transformed to

2

z2=1v

+

1v

1v

+

1v

, (1.17)

1v2

2

t2=

1v2

+

+

. (1.18)

In terms of and the wave equation becomes

1v2

2

G(, ) = 0 , (1.19)

where G(, ) is the function F(z, t) expressed in terms of and . The solution tothis equation are of the form

G(, ) = K1f() +K2g() , (1.20)where K1 and K2 are some constants and f() and g() are the two independentsolutions of Eq. (1.20). Transforming back to z and t we find the solutions to thewave equation are of the form

F(z, t) = C1f(t z/v) + C2g(t+ z/v) , (1.21)where C1 and C2 are constants. We see that F(z, t) is not an arbitrary function ofz and t but in it the space and time variables occur in the combination t z/v or

20 Laser Physics

F(z,t)

zz1z2

P1P1

P2 P2

Fo

F(z,t+dt)

!

!

FIGURE 1.5Pulse profile at times t and t+ dt for F (z, t) = ReF(z, t) Ref(t z/v).

t+z/v. Examples of acceptable waveforms are periodic functions such as ei(tz/v)or functions of finite duration such as ea(tz/v)2 , where and a are constants havingdimensions of frequency and frequency squared, respectively.The function f(t z/v) represents a disturbance propagating in the +z directionwith speed v and g(t+z/v) represents a disturbance propagating in the z directionwith speed v. To see this consider a real valued disturbance F (z, t) = f(t z/v) inthe form of a pulse consisting only of the first term. A plot of F (z, t) as functionof z when t is held fixed represents pulse shape or profile. Figure (1.5) showsa profile of this pulse at times t. To see what happens to this pulse at a latertime, consider a point, such as P1, where the pulse amplitude has a value F0. Ata later time t + dt, the pulse will have this same value F0 at point z2 such thatF (z2, t + dt) = F0 = f(z1, t). For this to happen the argument of f at time t andt + dt must be the same. This means t z1/v = t + dt z2/v or z2 = z1 + vdt.Hence at time t, the disturbance has value F0 at position z1 and at time t + dt ithas the same value at point z2 = z1 + vdt. If dt is an infinitesimal interval, z2 andz1 will dier by an infinitesimal amount dz so that dz = vdt. Similar considerationsfor other points on the pulse profile at time t show that at time t+ dt the pulse hasmoved to the right, without change of shape, by an amount dz = vdt. It is clearthat the disturbance propagates with speed v = dz/dt in the positive z-direction.Similar considerations show that F (z, t) = g(t+z/v) represents a wave propagatingin the z direction with constant speed v. We have considered a simple case wherewave propagates without change of shape. Propagation with change of pulse shapeis possible when the speed of the wave depends on frequency.In general, a wave propagating in the direction specified by the unit vector =

k/|k| is given byF(r, t) = C1f(t r/v) + C2g(t+ r/v) . (1.22)

The term plane wave is used for such solutions because the surfaces of constantwave amplitude, called wavefronts, are planes. For example, the wavefronts forF(r, t) = Cf(t r/v) are given by t r/v = const, which for dierent valuesof the constant defines a family of planes perpendicular to . These wave frontsmove with constant velocity given by dr/dt = v. If an energy density is associated

Laser Light 21

Spherical phasefronts r = const

rays

(c)

k

Plane phase frontsz = const

(a)

rays

k

Plane phase fronts!" r = const

(b)

FIGURE 1.6Wavefronts and rays for (a) plane and (b) spherical waves.

with the modulus squared of F(r, t) = Cf(t r/v), transport of energy occursalong trajectories called rays, which for the waves considered here are straight linesparallel to and perpendicular to the wavefronts. For example,

F(z, t) = Aei[(tz/v)o] , (1.23)and F(r, t) = Aei[(tr/v)o] , (1.24)

where A is a constant amplitude and o is a constant phase angle, represent singlefrequency plane waves propagating, respectively, in +z and directions. Wave(1.23) has an amplitude A and phase (z, t) = (t z/v) o. The surfaceson which wave has a definite amplitude and phase, at a fixed time t, are planes,z = vt+const, perpendicular to the direction of propagation [Fig. 1.6(a)]. Similarly,for the wave in Eq. (1.24), the wavefronts3 are planes perpendicular to the directionof wave propagation [Fig. 1.6(b)].It is possible to have other types of scalar solutions. For example solutions of the

form F(r, t) = F(r, t) exist which satisfy2 1

v22

t2

F(r, t) = 0. (1.25)

Since F is assumed to depend only on r and t, we can use the identity 2F(r) =(1/r)(2/r2)rF(r) in source free regions. Then the wave equation reduces to

2

r2 1v2

2

t2

rF(r, t) = 0 . (1.26)

3For a plane wave, the surfaces of constant phase are also the surfaces of constant wave amplitude.This is not the case in general. Strictly speaking, the term wavefront should be used only in theformer case. Nevertheless, it is commonly used in situations where phase front (surfaces of constantphase) is actually implied.

22 Laser Physics

Noting the similarity of this equation with the one-dimensional wave equation (1.16),its solution can be written down at once as rF(r, t) = C1f(t r/v) +C2g(t+ r/v).This can be rewritten in the form

F(r, t) = C1 f(t r/v)r

+ C2g(t+ r/v)

r. (1.27)

Here the first term represents a spherically diverging wave and the second termrepresents a spherically converging wave. Note that for F(r, t) = C1f(t r/v)/rthe surfaces of constant wave amplitude (wavefronts) are spheres centered at theorigin and energy transport occurs along radial lines diverging from the origin. Thewave amplitude falls o as 1/r so that the energy of the wave as it propagatesremains constant. More complex scalar spherical wave solutions which behave likeoutgoing or incoming waves far from the origin are

F(r, t) =const h(1)

(kr)Y m (,) e

it , outgoing waveconst h(2)

(kr)Y m (,) e

it , incoming wave(1.28)

where h(1)(kr) and h(2)

(kr) are spherical Hankel functions of the first and second

kind.In cylindrical coordinates we have two-dimensional cylindrical waves

F(, t) = C1 f(t /v)

+ C2g(t+ /v)

, (1.29)

where the wavefronts are (surfaces of constant wave amplitude) are cylinders coaxialwith the z-axis and rays are radial lines diverging from the axis. Other solutions thatinvolve Hankel functions and behave like outgoing or incoming cylindrical waves farfrom the z-axis also exist. We have mentioned only some of the simplest travelingwave solutions of the scalar wave equation. Many other solutions with more complexwavefronts representing standing or traveling waves are possible.Vector waves predicted by Maxwells equations can be constructed from the solu-

tions of the scalar wave equation. If the vector wave field has a fixed direction e inspace, then plane wave solutions of the form F(r, t) = ef(t r/v) exist. Thusa monochromatic vector plane wave propagating in the direction of wave vector kconstructed from the solutions of the scalar wave function has the form

F(r, t) = Foei(krt+o) , (1.30a)where k =

v , k k =

2

v2. (1.30b)

Plane wave like electric and magnetic fields satisfying Maxwells equations will alsobe of this form. Maxwells equations place further restrictions on the amplitudesand relative orientations of wave vector k and the electric and magnetic field vectors.Thus a plane wave solution of Maxwells equations is given by

E(r, t) = Eoei(krt+o) , k E = 0 , (1.31a)B(r, t) = k E

=k Eo

ei(krt+o) , k B = 0 . (1.31b)

Laser Light 23

Note that the electric vector can be expressed in terms of magnetic vector (usingAmpere-Maxwell equation and = 1/v2) as E = k Bv2/ = Bv. Theseequations imply that in a transparent medium, electric, magnetic and propagationvectors form a right handed triad of vectors. Furthermore, the time-averaged electricand magnetic energy densities ue and um are equal and contribute equally to theoverall energy density uem associated with the wave:

ue =12Re

12E E

=

14 |Eo|2 , (1.32a)

um =12Re

12B B

=

14

k2

2|Eo|2 = 14 |Eo|

2 = ue , (1.32b)

uem ue + um = 12 |Eo|2 . (1.32c)

Time-averaged Poynting vector (energy flux density vector) describing power flowin the wave is given by

S =12Re

E B

=

12Re

E (k E)

,

=12|Eo|2v = uemv I . (1.32d)

Poynting vector thus points in the direction of wave propagation, i.e., power flowoccurs in the direction of wave propagation and the direction of Poynting vectoris the ray direction in the wave. The magnitude of Poynting vector, I |S| =12|Eo|2v, referred to as the wave intensity in physics (irradiance in radiometry),has units of J/s/m2 (W/m2). For a nonmagnetic medium ( = o) the expressionfor wave intensity reduces to I = 12on|Eo|2c with the refractive index given byn =

/o.

Vector spherical or cylindrical wave solutions are more complex even in the sim-plest case. For example, the simplest vector spherical wave allowed by Maxwellsequations has the form

F(r, t) = constei(tr/v)

kr ie

i(tr/v)

(kr)2

sin e , k = /v , (1.33)

where is a frequency. We encounter this and other types of vector spherical wavesin the context of scattering and radiation problems in electrodynamics.Thus Maxwells equations admit solutions that also satisfy the wave equation.

However, not all solutions of the wave equation are admissible as solutions ofMaxwells equations; only those that also satisfy the constraints imposed by theMaxwells equations describe electromagnetic waves. This must be kept in mindeven though there are situations where vector character of the field can be ignored.

24 Laser Physics

1.5 Solutions in a Cavity: Mode Density

Wave like solutions of Maxwells equations also exist in the presence of boundaries.In addition to satisfying the wave equation and Maxwells equations, such solutionsmust also satisfy certain boundary conditions. This situation arises naturally in thecontext of lasers.Let us recall that the amplification of a propagating light signal requires its re-

peated passage (feedback) through a collection of excited atoms (gain medium) withwhich it interacts strongly. We also know that the atoms will interact strongly withsignal frequencies in a small range centered at one of their transition frequencies.We may take the FWHM (full width at half maximum) of atomic line to be ameasure of this range of frequencies. An excited population of atoms is necessarybut not sucient to produce amplification and build up of optical signals.Opticalresonators that provide feedback (multiple passage through the gain medium) arean essential element of lasers and are needed to

(i) Store and build up light energy at the frequency of interest since the rateof stimulated emission is proportional to light energy density at a particularfrequency.

(ii) Act as filters (spatial and frequency) responding selectively to field with pre-scribed spatial variation and frequency. Spatial filtering is responsible forthe collimation properties (directionality) of amplified optical signals and fre-quency filtering is responsible for their narrow bandwidth.

The ability of a resonator to perform these two tasks is measured by a figure ofmerit, the quality factor Q. Let us examine the field configurations and frequenciesthat an optical resonator will support. These field configurations are referred to asmodes of the resonator.For a rectangular cavity with perfectly conducting walls and each side of length

L, the electric field satisfies the wave equation and the boundary condition that itstangential component (component parallel to the wall) vanishes at the walls x = 0,x = L, y = 0 , y = L, z = 0 and z = L. Only the normal component of the electricfield can be nonzero at the surface of a perfect conductor. This electric field for amonochromatic wave has the form E = u(r)eit, where the vector function u(r)satisfying the wave equation and the boundary conditions is given by4

ux = A01 cosm1x

L

sin

m2yL

sin

m3zL

, (1.34)

uy = A02 sinm1x

L

cos

m2yL

sin

m3zL

, (1.35)

uz = A03 sinm1x

L

sin

m2yL

cos

m3zL

, (1.36)

4For simplicity of writing the labeling of mode functions ux, uy, uz, mode amplitudes Aoi, frequen-cies, etc. by the integer indices m1, m2, m3 will be suppressed.

Laser Light 25

(0,0,0)

x

y

x=L

y=L

z=L

FIGURE 1.7A rectangular cavity with conducting walls will support a discrete set of modes toallow the electric and magnetic field vectors to satisfy certain boundary conditionsat the walls.

where A01, A02, A03 are some constants and m1, m2, m3 are a set of nonnegativeintegers. In terms of these integers, the mode frequency is given by

2

v2=

2

L2m21 +m

22 +m

23

. (1.37)

Note that all three functions ux and uy and uz must be labeled by the same threeinteger indices (m1, m2, m3) for the field to satisfy Maxwells equation and no morethan one of the mode indices may be zero to allow for nonzero field solutions. Inaddition, since the divergence of the electric field must vanish ( E = 0) in thecharge-free interior of the box, we must have

m1A01 +m2A02 +m3A03 = 0 . (1.38)

The last equation implies that for given set of integers (m1, m2, m3) only two ofthe constants A0i can be chosen independently. The set of integers (m1, m2, m3)and the corresponding electric field define a mode of the cavity. The magnetic fieldassociated with this electric field can be calculated using Maxwells equations.If we introduce the wave vector k and the electric field amplitude Eo of a mode

characterized by integers (m1, m2, m3), respectively, as

k k1ex + k2ey + k3ez = L[m1ex +m2ey +m3ez] , (1.39)

Eo Eoxex + Eoyey + Eozez = Ao1ex +Ao2ey +Ao3ez , (1.40)

26 Laser Physics

we can re-write Eqs. (1.37) and (1.38) as

2

v2= k k = k2 , (1.41)

2

=kv

2=

v

2L

m21 +m22 +m23 , (1.42)

k E0 = 0 . (1.43)Let us recall that in writing these equations we have suppressed mode indices onwave vector, field amplitude and frequency for simplicity of writing. These relationsare analogous to Eqs. (1.30b) and (1.31a) for the solutions in unbounded space.From these equations we see that monochromatic solutions exist only for fre-

quencies given by Eq. (1.42). These frequencies can be represented by a point inthree-dimensional space as shown in Fig. (1.8). In addition, the vanishing diver-gence of the electric field in charge free region [Eq. (1.43)] requires that for a givenmode (m1, m2, m3) only two of the three constants A01i can be chosen indepen-dently. Thus for each mode there (only) two independent solutions (polarizations).The mode functions satisfy the usual orthogonality relation

Vum1m2m3 um1m2m3d3r = m1m1m2m2m3m3 . (1.44)

The modes of a rectangular can be represented by discrete points in a three-dimensional space spanned by (m1,m2,m3) axes as shown in Fig. 1.7. Each pointis associated a unique wave vector and represents two modes corresponding to twoindependent polarizations associated with a given wave vector.With this mode structure, we can think of the frequency [Eq. (1.42)] as rep-

resenting the distance of a point (m1,m2,m3) from the origin. Then the numberof modes in the frequency interval and + d (which is the same as the num-ber of modes whose k vector has a magnitude between k 2/v and k + dk 2( + d)/v) is the number of points inside a spherical shell of radius and thick-ness d

dN =18(volume of spherical shell)

volume associated with one mode 2 = 1

84k2dk(/L)3

2 = 2v

2 2dv

(/L)3

,

=8n32dV

c3, v = c/n (1.45)

where V = L3 is the volume of the cavity and the factor 18 in the first line of thisequation accounts for the fact that mode indices are positive integers. Hence onlythe points (m1,m2,m3) in the positive octant of the sphere in the m1,m2,m3 spacecount toward the number of modes. The factor of 2 at the end takes into accountthe two polarization degree of freedom for light for each wave number km1m2m3 .If such a resonator is used at an optical frequency with an inverted gain medium

inside the resonator, the number of resonator modes p falling under the laser tran-sition will be

p =82

c3V at = 8

V

3

ato

, (1.46)

Laser Light 27

m1!/L

m2!/L

m3!/L

(m1, m2, m3)!/L

!/L

!/L

!/L

FIGURE 1.8Each mode of a rectangular cavity, characterized by three positive integers m1,m2and m3, can be represented as a point in the positive octant of (m1,m2,m3) space.Note that the points on the axes do not represent allowed modes, since no morethan one of mode indices (m1,m2,m3) may be zero. Each point is counted twicecorresponding to two polarization states of the field associated with a given wavevector.

where we have put the refractive index n = 1. For a frequency = 5 1014 Hz( = 600 nm), at = 1.5 109 Hz and a cavity of volume V=1 cm3, we find thenumber of modes interacting with the atom is p = 3.5108. For a closed resonator,all of these modes will have access to atomic gain. They will have similar lossesand feedback so that oscillation would occur at a very large number of frequencies.Such a behavior would be highly undesirable because it would result in light fromthe laser being emitted in a wide spectral range and in all directions (and hence nocollimation).How do we reduce the number of modes? One possibility (suggested by p being

proportional to cavity volume V ) is to make a small cavity. Suppose we want p = 1at =600 nm. This will require a cavity of volume

V = p3

8oat

= 1 (0.6m)3

85 10141 5 109 = 14 (m)

3 ! (1.47)

Such a cavity, although not impossible nowadays, is not practical because we needsome room to place the amplifying medium. Moreover, even if we could accommo-date the medium, the gain itself will be very small.Problems discussed above can be overcome to a large extent by employing open

resonators. In open resonators only those modes that correspond to a superpositionof waves traveling very nearly parallel to the resonator axis will have low enoughlosses for fields to build up. Energy in all other modes will be lost in a few traver-sals. These modes will have a very low Q. Open resonators were first suggested by

28 Laser Physics

Schawlow and Townes and Prokhorov.5 In open resonators there are no conductingside walls. The active medium is placed between two mirrors carefully aligned. Insuch resonators photons traveling along the axis are trapped. Those photons thattravel at an angle eventually escape. One can improve things a bit by using curvedmirrors so that due to the focusing action of mirrors only photons making smallangles with the axis are trapped. For such photons we can write

k1k

=m1/L

(/L)m21 +m22 +m23

, (1.48a)

k2k

=(m2/L)

(/L)m21 +m22 +m23

, (1.48b)

k3k

=(m3/L)

(/L)m21 +m22 +m23

. (1.48c)

With k1, k2 k, k3 and using the relation k = 2 = Lm21 +m22 +m23, we can

write k1k =m12L , k2 =

m22L , k3 =

n32L , m1,m2 m3. Hence for each value of m3 we

have a small group of modes that have a frequency close to m3 = kv =kcn m3 c2nL

that will be amplified. We call m3 =m3c2nL a resonance frequency. These resonance

frequencies are separated from each other by

= m3+1 m3 =c

2nL.

These groups of frequencies are referred to as quasi-modes of the resonator becausefor open cavities true modes (stationary modes) are not defined. It is clear that thelight coming out from open resonators will have beam-like quality. This is becauseonly k1, k2 k3 modes will be populated. There is another change that occursdue to finite mirror apertures; even paraxial modes suer some energy loss due todiraction as every time a wave hits the end mirrors, energy will be lost due totheir finite size. Because of decay of field energy one cannot use the concept oftrue modes (stationary configurations) for open resonators. However quasi-modes(field configurations lasting millions of optical cycles) do exist, which have transverseextent that falls o rapidly with distance from the axis. These fields have space-timestructure

E(r, t) = EoU(r)eitet/2c , (1.49)where Eo is the field amplitude at time t = 0 and c is a characteristic time scaleon which the amplitude of the field decays. It is clear that in the absence of am-plification, quasi-mode field amplitude decays to zero in time. In the rest of thiscourse when we talk about modes of open resonators, we will be referring to thesequasi-modes.In order to see what kind of quasi-modes are possible in open resonators, we

use the complementary description of wave propagation in terms of rays, which as

5A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958); A. M. Prokhorov, Sov. Phys.JETP 1, 1140(1958).

Laser Light 29

we have seen in the preceding section, are geometrical curves, orthogonal to phasefronts, along which electromagnetic energy is transported. In essence, we will belooking for ray trajectories that stay confined in open resonator structures. Suchrays correspond to phase fronts that are stable spatial structures.

1.6 Ray Optics

In ray optics6, optical energy is regarded as being transported along certain curvescalled light rays, which as we have seen in Sec. 1.4, are geometrical curves orthogonalto phase fronts along which light energy is transported. Since Poynting vectordescribes the local energy flow in an electromagnetic wave, rays represent the localdirection of Poynting vector.We will show that for small wavelength the field locally has the same general

character as that of a plane wave and the laws of reflection and refraction establishedfor plane waves incident upon a plane boundary remain valid under more generalconditions. Hence if a light ray falls on a sharp boundary (for example the surfaceof a lens) it is split into a reflected ray and transmitted ray obeying the laws ofreflection and refraction. The preceding remarks imply that, when wavelength issmall enough, optical phenomena may be deduced from geometrical considerationsby determining the path of the light rays.In this limit, wavelike monochromatic solutions of Maxwells equations are of the

form

E(r, t) = e(r) ei(tko(r)) , B(r, t) = b(r) ei(tko(r)) . (1.50)Here the envelope functions e(r) and b(r) are some slowly varying7 functions ofposition and (r) is a real scalar function which remains to be determined. Sub-stituting these into Maxwells equations for transparent media ( and real),

E = 0 B = 0 E = iB B = iE

(1.51)

we find, with (r, t) = t k0(r), [e ln + e+ ik0e (r)] ei(r,t) = 0

[ b+ ik0b (r)] ei(r,t) = 0[ e+ ik0(r) e] ei(r,t) = ib ei(r,t)

1[ ln b+ b+ ik0(r) b] ei(r,t) = ie ei(r,t)

(1.52)

6The branch of optics characterized by the neglect of wavelength in comparison to the character-istic length of the problem - for example, the length scale on which the refractive index changessignificantly or the radius of curvature of interfaces between media - is known as geometrical optics.In this approximation the laws of optics may be formulated in the language of geometry.7Fractional change in their values is negligible over distances of the order of a few wavelengths

30 Laser Physics

Rearranging these we find

e (r) = 1ik0

[e ln + e] (1.53a)

b (r) = 1ik0

[ b] (1.53b)

(r) e cb = 1ik0

[ e] (1.53c)

(r) b+ ce = 1ik0

[( ln) b+ b] (1.53d)

If the changes in , and the envelopes e and b over distances of the order of afew wavelengths are small we can ignore the terms on the right hand side of eachof these equations,

e(r) (r) = 0 (1.54a)b(r) (r) = 0 (1.54b)

(r) e(r) cb(r) = 0 (1.54c)(r) b(r) + ce(r) = 0 (1.54d)

Note that the first two equations follow from the last two by taking scalar productwith (r). It follows from Eqs.(1.54) that e(r), b(r), and (r) form a righthanded triad of mutually orthogonal vectors at each point r. Furthermore, since thevector(r) is perpendicular to the surface (r)=const, vectors e(r) and b(r) aretangential to the surface (r)=const. This surface may be called the geometricaloptics wave surface or the geometrical wavefront.By eliminating e or b from Eqs. (1.37c) and (1.37d), we find that the condition

for nontrivial solutions of Eqs.(1.54) is

[(r)]2 = c2 = n2(r) , (1.55)

where n(r) is the refractive index of the medium

n(r) =

(r)(r)00

. (1.56)

In general, n(r) is a function of position because and are functions of position.Equation (1.55), known as the eikonal (from German Eikonal, which is from Greek, image) equation, is the basic equation of geometrical optics. The function is known as the eikonal. It follows from the eikonal equation that the vector

s =(r)n(r)

(1.57)

has unit magnitude and is perpendicular to the surface (r) =const. The surfaces(r) =const are called the geometrical wavefronts

Laser Light 31

s

Phase frontsurface!(r)=const

Wavefronts

(a) (b)

!(r)+d!(r)=const

!(r)=const

FIGURE 1.9(a) Geometrical wavefront and the direction of the unit vector s. (b) Rays aredirected (pointing in the direction of energy flow) trajectories perpendicular tophasefronts.

With the help of Eqs. (1.37c) and (1.37d) we can express the electric and magneticfield vectors as

b(r) =s e(r)

v, (1.58a)

e(r) = vs b(r) , (1.58b)where v = c/n is the wave speed in the medium. The time-averaged electric andmagnetic energy densities are then

ue =14|e|2 , (1.59a)

um =14|b|2

=14|e|2v2

=14|e|2v2

=14|e|2 = ue . (1.59b)

Thus in the limit of geometrical optics, the time averaged electric and magneticenergy densities associated with a monochromatic wave in a transparent mediumare equal.The time averaged Poynting vector (energy flux density vector) is given by

S =12Re

e b

=

12Re

e (ns e)

c

=

12c

Re [ns(e e) e(e ns)]

= cn

12e e

s = vuems . (1.60)

The average Poynting vector thus is in the direction of normal to the geometricalwavefront (r) =const. Its magnitude is equal to the product of the average energy

32 Laser Physics

density and speed v = c/n of the wave in the medium. It follows from Eqs. (1.58)-(1.60) that the fields in the geometrical optics limit have the same local characteras a plane wave.

1.7 Ray Propagation

We can now define the geometrical light rays as the orthogonal trajectories to thegeometrical wavefronts (r) = const. We regard them as directed curves whosedirection coincides everywhere with the direction of the average Poynting vector.We may then say that in geometrical optics light energy is transported along thelight rays. The dierential equation obeyed by the ray is easily derived as follows.Let r(s) denote the position vector of a point P on a ray, considered as a function ofthe arc length s along the ray measured from some fixed point on it. Then the unitvector dr/ds = ds is tangential to the ray in the direction of energy flow. Using therelation of s to (r), the equation for the ray can be written as

dr

ds= s (r)

n= ndr

ds=(r) . (1.61)

This equation is purely formal as it specifies rays in terms of (r) which mustbe determined from eikonal equation. We can derive a dierential equation forthe rays directly in terms of the refractive index n(r) which is much more useful.Dierentiating the equation for the ray with respect to arc length we obtain

d

dsndr

ds=

d

ds . (1.62)

Now the right hand side of this equation can be expressed in terms of n(r) as

r(s)r(s+ds)

drds

O

P

s

FIGURE 1.10Rays are curves along which energy of a wave is transported.

Laser Light 33

d

ds = (s )

=

n

= 1

2n()2 = 1

2n(n2) =n . (1.63)

Using this result in Eq. (1.62), we find the equation of a ray in terms of the variationof the refractive index

d

dsndr

ds=n . (1.64)

We can gain some insight into what this equation says by writing this in yet anotherform. We first note that s = /n [Eq. (1.57)] is a unit vector so that s s = 1.By dierentiating this with respect to s we find s (ds/ds) = 0 which means thatds/ds is a vector perpendicular to s. In fact from dierential geometry

ds

ds=

R=d2r

ds2, (1.65)

where R is the radius of curvature of the trajectory and is a unit vector alongthe principal normal. Using these results we find the equation of a ray can also bewritten as

dn

dss+ n

R=n . (1.66)

Rewriting this equation as

R=

1n

n sdn

ds

, (1.67)

and taking the scalar product with we find

1R

=

nn = lnn (1.68)

This equation says that a ray is bent toward the region of higher refractive index.We are familiar with a special case of this in Snells law: when a light ray enters adenser medium from a rarer medium, it is bent toward the normal.

1.7.1 Homogeneous Medium

As an application of this equation we consider a homogeneous medium n = const.In this case the equation for the ray becomes

d2r

ds2= 0 . (1.69)

On integrating this equation we obtain

r (s) = r or(s) = r os+ ro .

(1.70)

where r 0 is the initial slope of the ray at point r0. This is the equation of a straightline.

34 Laser Physics

r=ro+ s

ro

ro!

ro!

s

O

FIGURE 1.11In a homogeneous medium rays are straight lines.

1.7.2 Rays in a Duct

As another example we consider an axially symmetric medium with quadratic indexvariation in directions perpendicular to the axis of symmetry (taken to be the z-axis) n = n0 12n2(x2 + y2) with n2 > 0. Then in the paraxial approximationds = dz, the equation for paraxial rays becomes

ro

ro!

n(r)

FIGURE 1.12Rays in a duct with radially decreasing refractive index.

d2r

dz2+n2n0r =

d2r

dz2+ 2r = 0 , =

n2/n0. (1.71)

with solution

r(z) = r0 cos(z) +r0

sin(z) (1.72)

where ro is the ray displacement from the z axis and ro is the initial slope of theray at z = 0. Thus the ray oscillates up and down about the z-axis. We note alsothat a family of parallel rays (rays with the same slope but dierent displacement)periodically converge in planes z = (4n+ 1)/2 at points at a height rf = (ro/)from the z-axis and emerge from these planes with their original slope. These planesare separated from each other by zf = 2/.

Laser Light 35

Exercise 1:

Show that a section of the quadratic medium of length surrounded by a mediumof refractive index nm acts as a lens. [Hint: Show that a family of parallel raysentering at z = 0 with dierent displacement converge after emerging at z = to acommon focus at a distance f = 1nm cot.]

Exercise 2:

Find the equation for paraxial rays when n = n0 + 12n2(x2 + y2).

The answer is

r(z) = r0 coshz +r0

sinhz , =

n2/n0 .

1.7.2.1 Laws of Reflection and Refraction

In a two dimensional region (Y Z plane) where the refractive index depends on yonly, the ray equation becomes (br = yey + zez)

d

dsndz

ds= 0 ndz

ds= const . (1.73)

Using dzds = cos = sin [Fig. 1.13], we can write this as n cos = constant. For

ds

O

dydz

!dzds =cos!= sin"dydz = tan!

Y

Z

ray

"

!1

"2

n1

n2

n3

"1

"3

!2"2

!3

(a) (b)

s1

s2

s3

FIGURE 1.13(a) A ray in a continuous inhomogeneous medium; (b) Ray path in a layered (piece-wise homogeneous) medium.

a layered medium this leads to n1 cos1 = n2 cos2 = n3 cos3 = . Expressingangle in terms of the angle that the ray makes with the normal to the interface,we have

n1 sin 1 = n2 sin 2 = n3 sin 3 = . (1.74)This is precisely the Snells law for a layered medium.The laws of refraction and reflection at a surface across which the refractive index

changes abruptly can be derived more rigorously using the fact that ns = ,

36 Laser Physics

!h

t1

t2

n12

a

tn12

at=a"n12= t2=#t1

P1

Q1

Q2P2

n1

n2

FIGURE 1.14Ray transformation across a surface of discontinuity of refractive index.

which implies ns = 0. To see this we replace the surface of discontinuity bya thin transition layer in which n changes rapidly but smoothly from its value n1on one side of the surface of discontinuity to its value n2 on the other side [Fig.1.14]. Then integrating the normal component of ns over the open surfaceof a small rectangle P1Q1Q2P2P1 straddling the interface between two media [Fig.1.14], using Stokes theorem to convert the surface integral into a line integral andletting the thickness of the rectangle h (transition layer thickness) shrink to zero,we obtain

[n1s1 t1 + n2s2 t2] = 0 , (1.75)where is the line element in which the rectangle intersects the surface. Expressingthe unit vectors t1 and t2 in terms the unit tangent vector t along the surface [Fig.1.14] we obtain a[n12(n2s2n1s1)] = 0. Since the orientation of the rectangle andtherefore the unit vector a is arbitrary, we conclude that the tangential componentof the ray vector is continuous across the surface of discontinuity.

n12 (n2s2 n1s1) = 0 = n12 n2s2 = n12 n1s1 (1.76)

where n12 is a unit normal to the interface directed from medium 1 to 2. Thisequation implies that the normal to the boundary n12 and the ray vectors s1 and s2are coplanar and the angles the ray vectors makes with the normal to the boundaryare related by [Fig. 1.15]

n1 sin 1 = n2 sin 2 . (1.77)

Here 1 and 2 are the angles that the rays in the two media make with n12. Asimilar procedure for the reflected ray leads to the laws of reflection [Fig. 1.15(b)].These laws are usually derived for plane waves incident on a refracting plane

surface. Here we find that they are valid for more general waves and refractingsurface provided that the wavelength is suciently small. This means the radius ofcurvature of the incident wave and of the interface must be large compared to thewavelength of the incident light.

Laser Light 37

!1

!2n12

n1

n2 s1

s2

!1

!2

n12

n1

n2 s1s2

FIGURE 1.15Laws of (a) refraction and (b) reflection. Note that for reflection n2 = n1 so thatEq. (1.77) gives sin 2 = sin 1, which implies 2 = 1, which is the law ofreflection.

Thus in the limit of short wavelength Maxwells equations lead to a descriptionof light propagation in terms of rays which are geometric curves along which lightenergy is transported. We also see that in this limit wave nature of light is maskedand phenomena such as diraction are neglected. It may seem a drastic simplifica-tion but it is very useful in instrument optics and we will see that laws of ray opticsare very useful even for understanding diraction eects. In what follows we willlimit our discussion to paraxial rays.

1.7.3 Paraxial Rays

In problems involving instrument optics or laser resonators we are interested in raysthat stay close to the optical axis, usually taken to be the z axis, of the system.Such rays are called paraxial rays. They make small angles with the optical axis suchthat the sine and tangent of the angle can be approximated by the angle (expressedin radians) itself,

tan sin . (1.78)This approximation is good to within 3% for angles less than about 18o (0.31 1/radian). For such rays the arc length ds dz and the equation for the ray becomes

d

dzndr

dz=n (1.79)

For media with cylindrical symmetry we can write r = er + ezz, where r is thelateral displacement of the ray from the z axis, we find the equation of a ray becomes

d

dzndr

dz= e n . (1.80)

38 Laser Physics

r(z)dz

ds!

r(z+dz)

dr

z z+dz

r"(z)

FIGURE 1.16A paraxial ray makes small angles with the optical axis (usually the zaxis) andstays close to it during propagation.

This equation determines a ray given the lateral displacement ro and slope ro =[dr/dz]z=zo of a ray at some fixed point zo. In what follows we will use the abbre-viation r = dr/dz to denote the slope.Example 1. In a homogeneous medium n =const so that the equation of a raybecomes

d

dzndr

dz= 0 . (1.81)

Integrating this equation from zo to z, we find the ray is a straight line given by

r(z) = ro ,r(z) = ro + ro(z zo) ,

(1.82)

where ro and ro are, respectively, the ray displacement and slope at z = zo.If n changes, as, for example, happens when a ray is incident from one homoge-

neous medium (refractive index n1) with slope r1 and displacement r1 onto anotherhomogeneous medium (refractive index n2), an integration of the ray equation acrossthe interface (z = 0) into the second medium gives

r2 =n1n2

r1 ,

r2 = r1 +n1n2

r1z .(1.83)

The first of these equations is Snells law n11 = n22 in the paraxial approximation.The second equation gives the ray displacement in the second medium in terms ofthe ray displacement r1 and slope r2 = (n1/n2) r1 at the boundary just inside thesecond medium.The study of ray propagation is important in its own right in instrument optics.

We will see shortly, that although the treatment of propagation of light in terms ofray bundles ignores diraction, laws of paraxial ray propagation turn out to be veryuseful in understanding the full diractive propagation of light in optical resonatorsand laser beams.In paraxial optics a ray is specified by its displacement r from the optical axis

and its slope r = dr/dz. Both of these quantities vary with z as the ray propagates

Laser Light 39

through an optical system. If we introduce a column matrix with r and r itselements by

r =rr

, (1.84)

we can describe the eect of an optical element on ray parameters (r, r) by a 2 2matrix of the form

M =A BC D

. (1.85)

For most purposes we have to know the transformation properties of three basicelements:

(i) free propgation in a homogeneous medium of length L and refractive index n;

(ii) reflection from a curved surface of radius of curvature R;

(iii) refraction at a curved interface (radius of curvature R) between two mediawith refractive indices n1 and n2 when a ray is incident from medium n1.

Matrices for most others elements can be derived from these.

1.7.3.1 Propagation in a homogeneous medium

rin

r!"

d

rout

r!out

in

nz z+d

FIGURE 1.17Propagation in a homogeneous medium of length d.

Consider a ray propagating from plane z to plane z+d in a homogeneous mediumof refractive index n, the ray parameters at the input and output faces are relatedby [Eq. (1.82)]

rout = rin + d rin ,rout = r

in

A BC D

=1 d0 1

(1.86)

1.7.3.2 Reflection at a curved surface

From Fig. (1.18) it is clear that the ray displacement remains unchanged in reflec-tion.

rout = rin (1.87)

40 Laser Physics

!in

!out

routrin

C"

P#1#2

FIGURE 1.18According to the laws of reflection the angles of incidence and reflection are equal2 = 1. Note that rout is negative so that rout = out.

To find a relation between the input and output ray slopes we use the law ofreflection, which leads to

2 = 1 (law of reflection)or out = inor out = 2 in (1.88)To relate these angles to ray slopes [Fig. 1.18] we note that rin is positive whereasrout is negative,

rin = in ,rout = out , =

rinR

,

(1.89)

where we consider R to be positive for a concave mirror (reflecting surface). Usingthese, we find that the exit ray slope is given by

rout = 2rinR

+ rin (1.90)

With the help of Eqs. (1.87) and (1.90) we obtain the ABCD matrix for a reflectingsurface

A BC D

=1 0 2R 1

(1.91)

1.7.3.3 Refraction at a curved interface.

From Fig. (1.19) we see that the input an output displacements are the same

rout = rin . (1.92)

To relate input and output ray slopes we consider the relation between angles,

Laser Light 41

!in"

rin rout

C

Rn1 n2

""

#1#2

!in

!out

FIGURE 1.19Refraction at a curved interface between two media. Note that in the figure rout ispositive so that rout = out.

n11 = n22 Snells lawor n1(in + ) = n2(+ out)

or out = (n2 n1)n2

+n1n2

in (1.93)

Now noting rin = in and rout = out and = rin/R where we choose R positivefor a convex refracting surface. This leads us to

rout = (n2 n1)

n2

rinR

+n1n2rin (1.94)

From Eqs. (1.92) and (1.94) we find the ABCD matrix

A BC D

=

1 0n2 n1n2

1R

n1n2

(1.95)For a thin lens we can then find the ABCD matrix by multiplying the ABCD matrixfor each of its surfaces,

A BC D

=

1 0n1 n2n1

1R2

n2n1

1 0n2 n1n2

1R1

n1n2

=

1 0n2 n1n1

1R1 1R2

1

1 0 1f 1. (1.96)

We can, of course, derive the matrix of a thin lens of focal length f directly withthe help of Fig. 1.20 by recalling the thin lens formula . Let rin and rin denote the

42 Laser Physics

zo

zi

rin

r!in

rout

r!out

f

FIGURE 1.20Relation between input and output ray parameters for a thin lens.

displacement and slope of the incident ray just before the lens and rout and routtheir values just after the lens. Then it follows

rout = rin . (1.97)

To determine the slope after the lens we note that the incident ray may be thought ofas coming from the axial point zo and the emergent ray may thought of as proceedingtowards the point zi. These distances are related by the thin lens formula

1zo

+1zi

=1f. (1.98)

Multiplying both sides by rin and noting that rin/do = rin and rin/zi = rout (theemergent ray has negative slope) and rearranging the terms we find the slope of theemergent ray

rout = rin 1frin . (1.99)

From the preceding examples, it is clear that we can write the relation betweenthe input and output ray parameters in matrix form as

routrout

=A BC D

rinrin

(1.100)

This relation represents a transformation of input ray parameters into putput rayparameters. The matrix of transformation, also called the ABCD matrix, dependson the nature of the optical element inside the black box. For example the ABCDmatrix for propagation over a section of length L in a homogeneous medium is

A BC D

=1 L0 1

(1.101)

ABCD matrices for a number of common optical elements are given in Table 1.3.Once these basic matrices are known we can calculate the overall ABCD matrix

Laser Light 43

TABLE 1.2ABCD matrices for paraxial rays for three basic optical elements.

Straight section of length d in a homogeneous medium of refractive index n

n

d

1 d0 1

Dielectric interface, radius of curvature R (+ for convex and for concave refractingsurface), arbitrary angle of incidencenT =

n2cos 1

n1cos 2

,

nS = n2 cos 2 n1 cos 1 .

n1

n2

!1

Incident axis

Exit axis

!2

R

cos 2cos 1 0nTn2R

n1 cos 1n2 cos 2

plane of incidence(tangential plane) 1 0nS

n2R

n1n2

perpendicular to theplane of incidence(sagittal plane)

Spherical mirror of radius of curvature R (+ for concave mirror and for convexmirror), arbitrary angle of incidence

!

!

Incident axis

Exit axis

1 0

2R cos

1

plane of incidence(tangential plane)

1 0

2 cos R

1

perpendicular to theplane of incidence(sagittal plane)

44 Laser Physics

TABLE 1.3ABCD matrices of some common optical elements.

Plate of thickness d, refractive index n in a medium of refractive index nm, nearnormal incidence

nm n

d

nm

1d

n0 1

Thin lens of focal length f in a medium of refractive index nm, normal incidence

f

nm

R1

R2

nm

n

1 0 1f1

1f=n nmnm

1R1 1R2

Dielectric interface with radius of curvature R (+ for convex and for concaveinterface).

n1

n2

R

1 0n2 n1

n2

1R

n1n2

Dielectric interface at Brewsters angle.

n1

n2

!"Input

axis

Output

axis

n2n1

0

0n21n22

plane of incidence(tangential plane)1 00n1n2

perpendicular to theplane of incidence(sagittal plane)

Spherical mirror of radius of curvature R, normal incidence (R > 0 for concavemirror and R < 0 for convex mirror)

R

1 0

2R

1

Laser Light 45

1 2 3 4 5

Input

plane

6

Exit

plane

FIGURE 1.21The overall transformation matrix for ray propagation through an optical systemis obtained by multiplying the ABCD matrices for each optical element, includingfree space sections, in correct order [see the text].

for any system of optical elements. Consider, for example, the passage of a raythrough a sequence of optical elements shown in Fig. [ a thin lens, followed by freespace and a dilectric slab(refractive index n)]. By labeling various optical elelmentsas 1 , 2 , 3 in the order in which they are encountered, we can write the ABCDmatrix as

A BC D

=M6 M5 M4 M3 M2 M1 . (1.102)

Note that the matrices are written from right to left in the order in which they areencountered by the ray.

1.7.4 Periodic Focusing System

An interesting and important application of ray matrices comes in the analyses ofperiodic focusing systems in which the same sequence of elements is repeated manytimes down a cascaded chain. An optical resonator can be modeled by such aniterated periodic focusing system because propagation through repeated round tripsin the resonator is physically equivalent to propagation through repeated sectionsof a periodic lens guide.As an example consider an optical resonator shown in Fig. 1.22 formed by two

spherical mirrors of radii of curvature R1 and R2 placed a distance L apart. Imaginea ray propagating to the right starting at the left end of the resonator. After a roundtrip, this ray will have been transformed by a straight section of length L, a sphericalmirror of radius of curvature R1 another section of length L, and finally a sphericalmirror of radius of curvature R2. In each roundtrip, the ray encounters the sametransformation. This is equivalent to a lens waveguide where lenses of focal lengthf1 = R1/2 and f2 = R2/2 are placed alternately separated by L. The ABCD matrixM describing the ray transformation in a roundtrip through the resonator is givenby M = M1 ML M2 ML, where M1 and M2 are the ray matrices of mirrors R1and R2, respectively, and ML is the ray matrix of a section of length L. Note thatthe matrices are written from right to left in the order in which the optical element

46