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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.
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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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Laser Light 3
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
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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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Laser Light 5
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
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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.
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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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8 Laser Physics
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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Laser Light 9
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
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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
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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
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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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Laser Light 13
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)
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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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Laser Light 15
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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16 Laser Physics
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.
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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
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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 .
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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
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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
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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.
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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)
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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.
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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.
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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)
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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)
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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
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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).
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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
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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
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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
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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.
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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.
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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/.
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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 = ,
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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.
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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)
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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)
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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,
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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
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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
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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)
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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
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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
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