Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

Подождите немного. Документ загружается.

618 Part C Molecules

windows at each end. These long absorption cells lend

themselves naturally to electrical discharges for the

study of molecular ions. A measurement of the HCO

line at 1 THz exhibited a signal-to-noise ratio of 100 : 1

using a 1 s time constant; this is the same signal-to-

noise ratio that was obtained using the laser sideband

technique.

The instrumental resolution of the spectrometer is

limited by the combined frequency ﬂuctuation from each

laser (about 15 kHz). This is less than any Doppler-

limited line width and, therefore, does not limit the res-

olution. The entire data system is computerized to facil-

itate the data recording and optimize the data handling.

Improvements in this TuFIR technique may come

from either better diodes or detection schemes. The

nonlinearity of the MIM diode is extremely small, and

conversion efﬁciencies could be much larger if a more ef-

ﬁcient diode is discovered. Differential detection (which

requires more sensitive detectors) could also signiﬁ-

cantly improve the sensitivity and permit the detection

of weaker lines.

41.1.2 Laser Magnetic Resonance

Laser magnetic resonance (LMR) is performed by

magnetically tuning the transition frequencies of para-

94.0 cm

Coupler

Micrometer

drive

Quartz

spacer

Pump

beam

ZnSe

window

Quartz

spacer

Beam splitter

(Brewster angle)

Flow

Sample

volume

Quartz

tube

Magnet

Modulation

coil

Fixed mirror

Moveable

mirror

Far IR

output

Adjustable

iris

Gold-lined

Pyrex tube

(5 cm LD.)

Cylindrical

mirror

(10 cm LD.)

Quartz

spacer

Quartz

spacer

0 5 10 15 20 25 Centimeters

Fig. 41.2 40 µm to 1000 µm Far-infrared laser magnetic resonance (LMR) spectrometer using an optically pumped FIR

laser

magnetic species into coincidence with the ﬁxed

frequency radiation from a laser. LMR is a type of

Zeeman spectroscopy [41.27], and its chief forte is

its extreme sensitivity. Approximately one hundred

species have been observed in the SM-FIR region. These

include atoms, diatomics (especially hydrides), poly-

atomics, ions, metastables, metastable ions [41.28], and

many “ﬁrst observations” of free radicals. These ob-

servations are summarized in several review articles on

LMR [41.29–31].

The FIR LMR spectrometer at NIST, Boulder is

shown in Fig. 41.2. This spectrometer has an intracav-

ity paramagnetic sample in a variable magnetic ﬁeld

which is labeled “sample volume” in the ﬁgure. A reg-

ular 38 cm EPR magnet with a 7.5cm air gap is used.

The laser cavity is divided into two parts by a Brew-

ster angle polypropylene beam splitter about 12 µm

thick. A CO

laser optically pumps the FIR laser

gas. This pump beam makes many nearly perpendic-

ular passes reﬂected by the walls of the gold-lined

Pyrex tube. A 45

◦

coupler serves to couple out the

FIR radiation from the cavity to the helium-cooled

detector. Mirrors each with an 89 cm radius of cur-

vature are used in the nearly confocal geometry of

the 94 cm cavity. One of the gold-coated end mir-

rors is moved with a micrometer to tune the cavity

Part C 41.1

Laser Spectroscopy in the Submillimeter and Far-Infrared Regions 41.1 Experimental Techniques using Coherent SM-FIR Radiation 619

to resonance and also to determine the oscillating

wavelength by moving the end mirror several half-

wavelengths. The beam splitter is rotatable so that the

polarization of the laser can be varied with respect to

the magnetic ﬁeld. Quartz spacers are used to mini-

mize the thermal expansion of the cavity. This LMR

spectrometer oscillates at wavelengths between 40 and

1000 µm. The technology of FIR lasers has been re-

viewed by Douglas [41.32].

The LMR technique requires a close coincidence

(typically within 20 GHz) between the absorption line

and the frequency of the laser. Zero ﬁeld frequen-

cies which are 100 times more accurate than those

obtained from optical spectra are obtained from the

analysis of the Zeeman spectra observed (i. e. using

the laser frequency and the magnetic ﬁeld values). The

accuracy is within 1 or 2 MHz and has permitted the

far infrared astronomical observation of many of these

species.

Atomic FIR LMR spectra are due to ﬁne struc-

ture transitions and are magnetic dipole transitions;

hence, they are several orders of magnitude weaker

than electric dipole rotational spectra of molecules.

The production of a sufﬁcient atomic number den-

sity can be difﬁcult; however, “atomic ﬂames” have

been very effective sources, and the high sensitivity of

LMR has been the most successful spectroscopic tech-

nique for measuring these transitions. The atoms O, C,

metastable Mg, S, Si, Fe, Al, N

,Fe

,and

have been measured by FIR LMR. Atomic Zee-

man spectra are relatively simple to analyze, and ﬁne

structure frequencies accurate to within 1 MHz can be

determined.

In a FIR LMR spectrometer the sample is inside

the laser cavity; hence, sub-Doppler line widths can be

observed by operating the sample at low pressure and

observing saturation dips in the signals. This has per-

mitted the resolution of proton hyperﬁne structure in

a number of hydrides. The observation of the resolved

hyperﬁne structure is useful in the identiﬁcation of the

species involved and also yields accurate values of the

hyperﬁne splittings.

Rotational spectra of most hydrides lie in this spec-

tral region and many of the hydrides, such as OH, CH,

SH, and NH have been observed. Others, such as CaH,

MnH, TiH, and ZnH are excellent candidates for LMR

studies. Ions are much more difﬁcult because of their

low concentrations; however, the use of a special mi-

crowave discharge operating in the magnetic ﬁeld has

proved to be an extremely productive source of ions for

LMR studies [41.28].

The spectra of a number of polyatomic species

has been observed by LMR: for example, NO

,HCO,PH

,CH

,NH

,AsH

,HO

,HS

O, and CCH. The spectra of these polyatomic

species are more difﬁcult to analyze, resulting in

a somewhat less accurate prediction of the zero-ﬁeld

frequencies. Spectra of the extremely elusive CH

and CD

radicals have ﬁnally yielded to analysis

with the simultaneous observation of FIR LMR spec-

tra [41.33]andIR LMR spectra [41.34]. The data

yield rotational constants which predict the ground

state rotational transition frequencies and permit the

determination of the singlet–triplet splitting in that mol-

ecule [41.35].

In the last ten years, there has been an experimental

push to higher frequencies in FIR spectroscopy. In the

LMR spectrometer (shown in Fig. 41.2), this has been

achieved principally by reducing the internal diameter

of the gold-lined tube in the pumping region to 19 mm

so as to increase the overlap between the laser gain

medium and the FIR radiation ﬁeld in the laser cavity.

As a result, many new laser lines have been discov-

ered; the system operates down to below 40 µm. This

has enabled molecules with larger ﬁne-structure inter-

vals to be studied (e.g., FO and HF

[41.36, 37]) and

has also led to the ﬁrst detection of vibration-rotation

transitions in molecules with low-frequency bending

vibrations (CCN, HCCN, FeH

) [41.38–40].

The sensitivity of TuFIR is only about 1% of that of

laser magnetic resonance, but it is difﬁcult to compare

the two because the sample can be very large in the

TuFIR spectrometer and is limited to about 2 cm

in the

LMR spectrometer.

41.1.3 TuFIR and LMR Detectors

Four different detectors have been used in the TuFIR

and LMR spectrometers: (1) an InSb 4 K, liquid

He-

cooled, hot-electron bolometer, operating from 0.3

to 0.6 THz with a noise equivalent power (NEP)of

about 10

−13

√

Hz; (2) a gallium doped germanium

bolometer, cooled to the lambda point of liquid-helium,

operating from 0.6to8.5 THz with an NEP of about

−13

√

Hz; (3) a similar, but liquid

He-cooled

bolometer, with a NEP two orders of magnitude smaller;

and (4) an unstressed Ge:Ga photoconductor, cooled

to 4 K, with an NEP of 10

−14

√

Hz, operating

from 2.5to8.5 THz. The optimization of detector

systems for this new technique has been responsible

for a signiﬁcant improvement in sensitivity of these

spectrometers.

Part C 41.1

620 Part C Molecules

41.2 Submillimeter and FIR Astronomy

Submillimeter astronomy is reviewed in [41.6]. Many

atomic species (both neutrals and ions) and more than

100 molecules have been detected in interstellar space;

many were ﬁrst observed in the submillimeter and

FIR region. In the microwave and submillimeter re-

gion, radio (heterodyne) techniques are employed in the

receivers. At higher frequencies, interferometric tech-

niques are used [41.14]; however, heterodyne techniques

recently have been employed at these higher frequen-

cies and have resulted in the detection of CO [41.41]

and OH [41.42]at2.5 THz and a search for the IR

band of methylene [41.43] at 30 THz. Submillime-

ter radio astronomy observatories (at high altitudes

above much of the water vapor absorptions in our at-

mosphere) use laboratory determined frequencies. For

example, the Kuiper Airborne Observatory and the sub-

millimeter and FIR telescopes on Mauna Kea, Hawaii

require frequencies accurate to about 1 MHz in their

heterodyne receivers. For heterodyne detection, ﬁxed

frequency FIR gas lasers generally serve as local os-

cillators, and the frequencies of these lasers must be

known.

The discovery and direct frequency measurement of

FIR laser lines has continued apace since the last ma-

jor review of this topic was published in 1986 [41.44].

Almost 1000 lines were listed in that publication; the

present number is at least twice this. Two main fac-

tors have fuelled this progress. First, the design of

lasers has continued to develop so that now

a single laser can produce 275 individual transitions

from the regular, hot [41.45], and sequence [41.46]

bands of CO

with power levels from 2 to 30 W. Sec-

ondly, many new, short-wavelength laser lines have

been discovered in a cavity specially designed to pro-

mote them. As a result, many high frequency lines

have now been characterized, several in the range

of 26 to 45 µm. The main lasing molecules used

in this work are CH

OH [41.47–50] and hydrazine,

[41.51].

Recent observations of FIR transitions in molecules

in the interstellar medium have been made from satellite-

based platforms such as the Infrared Space Observatory

(ISO) [41.52] using Fourier Transform methods. Atten-

tion has been paid to the measurement of less abundant

isotopic forms, such as

OH and

OH, because of

the information that they provide about star formation

processes.

There are four main goals of laboratory SM-FIR

spectroscopy which serve the needs of the SM-FIR radio

astronomy ﬁeld:

1. to provide accurate frequencies of SM-FIR species

for their detection,

2. to ﬁnd new far infrared active species,

3. to measure the frequencies of FIR species which can

be used to calibrate Fourier transform spectrometers,

and

4. to measure frequencies of far-infrared lasers for use

as local oscillators in radio astronomy receivers (and

to be used in the analysis of laboratory LMR data).

41.3 Upper Atmospheric Studies

A very impressive set of SM-FIR spectra of our

upper atmosphere has been observed using balloon-

borne FT spectrometers ﬂown at altitudes where

the lines are narrow and the spectrometer is above

the “black”, heavily absorbing water vapor transi-

tions [41.7]. A number of very important species with

strong SM-FIR spectra have been observed. They in-

clude: O

O, NO, ClO, OH, HO

,andO.

Numerous lines have not yet been identiﬁed. It is

difﬁcult to calibrate these instruments absolutely be-

cause the spectra come from the species emitting at

temperatures of about 200 K, and from an indetermi-

nate path length. The high sensitivity of LMR might

provide an alternate way of measuring the paramag-

netic species in our upper atmosphere by ﬂying an

LMR spectrometer to high altitudes. A light-weight

solenoid magnet could be used to increase the path

length. Absolute concentrations of the paramagnetic

species such as OH, HO

, NO, and O could be

obtained.

Part C 41.3

Laser Spectroscopy in the Submillimeter and Far-Infrared Regions References 621

References

41.1 A.Crocker, H. A. Gebbie, M. F. Kimmitt, L. E. S. Math-

ias: Nature 201,250(1964)

41.2 E. E. Haller: Infrared Phys. Technol. 35,127–146

(1994)

41.3 J.Strong,G.Vanasse:J.Opt.Soc.Amer.49, 884

(1959)

41.4 K. M. Evenson, H. P. Broida, J. S. Wells, R. J. Mahler:

Phys. Rev. Lett. 21, 1038 (1968)

41.5 B. F. J. Zuidberg, A. Dymanus: Appl. Phys. Lett. 32,

367 (1978)

41.6 T. G. Phillips: Techniques of submillimeter Astron-

omy. In: Millimetre and Submillimetre Astronomy,

ed. by W. B. Burton (Kluwer, Doordrecht 1988) p. 1

41.7 W. A. Traub, K. V. Chance, D. G. Johnson, K. W. Jucks:

Proc. Soc. Photo-Opt. Instrum. Engrs. 1491,298

(1991)

41.8 D. A. Jennings, K. M. Evenson, D. J. E. Knight: Proc.

IEEE 74, 168 (1986)

41.9 T. D. Varberg, K. M. Evenson: IEEE Trans. Instrum.

Meas. 42, 412 (1993)

41.10 I. G. Nolt, J. V. Radostitz, G. DiLonardo, K. M. Even-

son, D. A. Jennings, K. R. Leopold, M. D. Vanek,

L. R. Zink, A. Hinz, K. V. Chance: J. Mol. Spectrosc.

125, 274–287 (1987)

41.11 D. A. Jennings, K. M. Evenson, L. R. Zink, C. De-

muynck, J. L. Destombes, B. Lemoine: J. Mol.

Spectrosc. 122, 477 (1987)

41.12 F. Matsushima, K. M. Evenson, L. R. Zink: J. Mol.

Spectrosc. 164, 517 (1994)

41.13 L.-H. Xu, R. M. Lees, K. M. Evenson, C.-C. Chou,

J.-T. Shy, E. C. C. Vasconcellos: Can. J. Phys. 72, 1155

(1994)

41.14 J. W. V. Storey, D. M. Watson, C. H. Townes: Int. J.

Infrared Millimeter Waves 1, 15 (1980)

41.15 H.-P. Röser: Infrared Phys. 32, 385 (1991)

41.16 P. Helminger, J. K. Messer, F. C. DeLucia: Appl. Phys.

Lett. 42, 309 (1983)

41.17 S. P. Belov, L. I. Gershstein, A. F. Krupnov,

A. V. Maslovsky, V. Spirko, D. Papousek: J. Mol.

Spectrosc. 84, 288 (1980)

41.18 G. Winnewisser: Vib. Spectrosc. 8, 241 (1995)

41.19 B. F. J. Zuidberg, A. Dymanus: Appl. Phys. Lett. 32,

367 (1978)

41.20 K. M. Evenson, D. A. Jennings, F. R. Petersen: Appl.

Phys. Lett. 44,576(1984)

41.21 P. Verhoeve, E. Zwart, M. Versluis, M. Drabbels,

J. J. ter Meulen, W. Leo Meerts, A. Dymanus: Rev.

Sci. Instrum. 61, 1612 (1990)

41.22 R. C. Cohen, K. L. Busarow, K. B. Laughlin,

G. A. Blake, M. Havenith, Y. T. Lee, R. J. Saykally:

J. Chem. Phys. 89, 4494 (1988)

41.23 F. Lewen, E. Michael, R. Gendriesch, J. Stutzki,

G. Winnewisser: J. Molec. Spectrosc. 183, 207 (1997)

41.24 G. A. Blake, K. B. Laughlin, R. C. Cohen, K. L. Busa-

row, D.-H. Gwo, C. A. Schmuttenmaer, D. W. Stey-

ert, R. J. Saykally: Rev. Sci. Instrum. 62, 1693, 1701

(1991)

41.25 E. N. Grossman: Infrared Phys. 29, 875 (1989)

41.26 K. V. Chance, D. A. Jennings, K. M. Evenson,

M. D. Vanek, I. G. Nolt, J. V. Radostitz, K. Park:

J. Mol. Spectrosc. 146, 375 (1991)

41.27 J. S. Wells, K. M. Evenson: Rev. Sci. Instrum. 41, 226

(1970)

41.28 T. D. Varberg, K. M. Evenson, J. M. Brown: J. Chem.

Phys. 100, 2487 (1994)

41.29 K. M. Evenson: Faraday Discussions of The Royal

Society of Chemistry 71,7(1981)

41.30 D. K. Russell: Laser magnetic resonance spec-

troscopy. In: Electron Spin Resonance, Specialist

Periodical Reports, Vol. 16, ed. by M. Symons (The

Royal Sociey of Chemistry, London 1990) p. 64

41.31 A. I. Chichinin: Magnetic Resonance: Laser Mag-

netic Resonance. In: Encyclopedia of Spectroscopy

and Spectrometry, ed. by J. Linton, G. Tranter,

J. Holmes (Academic, London 2000) pp. 1133–1140

41.32 N. G. Douglas: Millimetre and Submillimetre

Wavelength Lasers (Springer, Berlin, Heidelberg

1990)

41.33 T. J. Sears, P. R. Bunker, A. R. W. McKellar,

K. M. Evenson, D. A. Jennings, J. M. Brown: J. Chem.

Phys. 77, 5348 (1982)

41.34 T. J. Sears, P. R. Bunker, A. R. W. McKellar: J. Chem.

Phys. 75, 4731 (1981)

41.35 A. R. W. McKellar, P. R. Bunker, T. J. Sears,

K. M. Evenson, R. J. Saykally, S. R. Langhoff:

J. Chem. Phys. 79, 5251 (1983)

41.36 F. Tamassia, J. M. Brown, K. M. Evenson: J. Chem.

Phys. 110, 7273 (1999)

41.37 M. D. Allen, K. M. Evenson, J. M. Brown: J. Mol.

Spectrosc. 227, 13 (2004)

41.38 M. D. Allen, K. M. Evenson, D. A. Gillett, J. M. Brown:

J. Mol. Spectrosc. 201, 18 (2000)

41.39 M. D. Allen, K. M. Evenson, J. M. Brown: J. Mol.

Spectrosc. 209, 143 (2001)

41.40 H. Körsgen, K. M. Evenson, J. M. Brown: J. Chem.

Phys. 107, 1025 (1997)

41.41 R. T. Boreiko, A. L. Betz: Astrophys. J. 346,L97

(1989)

41.42 A. L. Betz, R. T. Boreiko: Astrophys. J. 346, L101

(1989)

41.43 D. M. Goldhaber, A. L. Betz, J. J. Ottusch: Astrophys.

J. 314,356(1987)

41.44 M. Inguscio, G. Moruzzi, K. M. Evenson, D. A. Jen-

nings: J. Appl. Phys. 60, R161 (1986), Table 5

41.45 A. G. Maki, C.-C. Chou, K. M. Evenson, L. R. Zink,

J.-T. Shy: J. Mol. Spectrosc. 167, 211 (1994)

Part C 41

622 Part C Molecules

41.46 C.-C. Chou, A. G. Maki, S. J. Tochitsky, J.-T. Shy,

K. M. Evenson, L. R. Zink: J. Mol. Spectrosc. 172,233

(1995)

41.47 S. C. Zerbetto, E. C. C. Vasconcellos: Int. J. IR &

mmwaves 15, 889 (1994)

41.48 D. Pereira, E. M. Telles, F. Strumia: Int. J. IR &

mmwaves 15, 1 (1994)

41.49 S. C. Zerbetto, L. R. Zink, K. M. Evenson, E. C. C. Vas-

concellos: Int. J. IR & mmwaves 17, 1049 (1996)

41.50 E. M. Telles, H. Odashima, L. R. Zink, K. M. Evenson:

J. Mol. Spectrosc. 195, 360 (1999)

41.51 E. C. C. Vascocellos, S. C. Zerbetto, L. R. Zink,

K. M. Evenson: J. Opt. Soc. Am. B 15,1839

(1998)

41.52 C. Ceccarelli, J.-P. Baluteau, M. Walmsley,

B. M. Swinyard, E. Caux, S. D. Sidher, P. Cox, C. Gry,

M. Kessler, T. Prusti: Astron. Astrophys. 383,603

(2002)

Part C 41

623

Spectroscopic

42. Spectroscopic Techniques: Lasers

As a primary research tool, the laser plays a fun-

damental role in the spectroscopic study of atomic

and molecular systems. This Chapter describes

the basic operating principles, conﬁgurations,

and characteristic parameters of lasers. Laser de-

signs are discussed and then the details of the

interaction of the laser light with matter delin-

eated. The reader is also referred to Chapts. 70

(Laser Priciples) and 71 (Types of Lasers) for further

information.

42.1 Laser Basics......................................... 623

42.1.1 Stimulated Emission .................. 623

42.1.2 Laser Conﬁgurations .................. 623

42.1.3 Gain ........................................ 623

42.1.4 Laser Light................................ 624

42.2 Laser Designs ...................................... 625

42.2.1 Cavities .................................... 625

42.2.2 Pumping .................................. 626

42.3 Interaction of Laser Light with Matter ... 628

42.3.1 Linear Absorption...................... 628

42.3.2 Multiphoton Absorption ............. 628

42.3.3 Level Shifts............................... 629

42.3.4 Hole Burning ............................ 629

42.3.5 Nonlinear Optics ....................... 629

42.3.6 Raman Scattering...................... 630

42.4 Recent Developments........................... 630

References .................................................. 631

42.1 Laser Basics

42.1.1 Stimulated Emission

A cross section σ

for absorption of radiation by

a lower state 1 engenders a balancing cross section σ

for emission stimulated by radiation interacting with an

upper state 2. Detailed balance relates these two cross

sections according to

= g

, (42.1)

where g

and g

are the statistical degeneracies of their

respective states [42.1].

For a collection of emitting and absorbing states

with densities n

and n

, ampliﬁcation may occur when

> n

, which leads to a requirement for an

inversion of the state populations:

> g

. (42.2)

The rate of spontaneous emission at frequency ν can be

modeled itself by stimulated emission induced by a noise

source of the magnitude of the density of states ρ(ν)

(ν) = σ

(ν)ρ(ν)/c = σ

(ν)8πλ

−2

. (42.3)

42.1.2 Laser Conﬁgurations

A practical laser combines a population inversion with

a means for controlling the radiation.

The basic laser source is the laser oscillator, an am-

pliﬁer possessing positive feedback. The usual form is

simply a piece of active gain medium placed inside a res-

onant optical cavity (Fig. 42.1). Tunability is produced if

the resonant cavity is frequency selective and adjustable

(Fig. 42.2). Many laser sources use an ampliﬁer after the

oscillator.

42.1.3 Gain

The fundamental gain per pass is given by

G = J/J

= exp[(κ −µ)L] , (42.4)

Gain medium

100%

mirror

Multipass cavity

Partial

mirror

Output

Fig. 42.1 Simple laser oscillator and beam parameters. Dis-

tances z are generally measured from the minimum beam

waist w

. The beam appears with a far-ﬁeld divergence

angle θ

Part C 42

624 Part C Molecules

a) b)

Grating

Beam expander

Grating

Mirror

Fig. 42.2a–d Tunable laser oscillator geometries. (a) Fabry–

Perot: tuning is usually done by changing the cavity length,

although changing the index of refraction by changing the

temperature or current is common with solid state laser

diodes.

(b) Littrow prism line selector: typical of atomic

ion lasers capable of multiple line output.

ing tuning: common in pulsed dye lasers with high gain

(> 10) per pass. Telescope increases resolution by ﬁlling,

and reducing angular divergence at the grating.

(d) Grazing

incidence, mirror tuned, grating mount

where J/J

is the ratio of light output to input, κ is the

gain coefﬁcient, µ is the nonradiative loss rate, and L is

the path length. The gain coefﬁcient

κ = n

∗

(42.5)

depends upon the net inversion n

∗

= n

−(g

. (42.6)

If λ is the wavelength of radiation, F

is the emission

line shape function normalized over frequency ν, τ

the lifetime of the transition, and f

is the branching

ratio for the upper state to undergo this transition, then

the stimulated emission cross section is

(ν)

8πτ

. (42.7)

For a Lorentzian lifetime-broadened line, the cross sec-

tion for stimulated emission at the line center becomes

4π

, (42.8)

where Γ

is the full width at half maximum of the line.

42.1.4 Laser Light

Lasers are inherently bright sources of radiation: the ra-

diation ﬁeld within a practical laser must be high enough

for stimulated emission to compete with spontaneous

emission. The effective source of spontaneous

ﬂuctuations approximates that of the density of states.

In terms of the beam parameters photon ﬂux J per solid

angle Ω, and frequency ν,thisis

dΩ dν

2ν



. (42.9)

Beam quality is given by the product of the angular

divergence times the beam width. Highest beam quality

is associated with diffraction limited light emitted from

a Gaussian spot. For circular laser beams traveling in

the z-direction, this corresponds to a solution of the

electromagnetic wave equation

u(r,φ,z) =ψ(r, z) exp(−ikz),

(42.10)

where u is a polarization component of the ﬁeld. For high

values of k = 2π/λ, corresponding to short wavelength,

the adiabatic radial solution is also Gaussian:

ψ(r, z) = exp



−i



P +kr

/(2q)



, (42.11)

where the complex phase shift P, beam parameter q,

and beam radius w are functions of z:

P(z) =−iln



1 −iλz/



πw



=−iln



1 +



λz/



πw



−tan

−1



λz/



πw



(42.12)

q(z) =iπw

/λ +z , (42.13)

(z) =w





1 +



λz

πw







(42.14)

Here, w

is the beam waist parameter, the minimum

width of the Gaussian beam at a focused spot. For Gaus-

sian beams, the product of the minimum beam waist and

beam divergence angle θ

is given by

= λ/π . (42.15)

The beam waist and divergence follow optical imaging

according to paraxial ray theory.

Higher order circular modes with p radial nodes

and l angular node planes are speciﬁed by multiply-

ing (42.11)and(42.12) by angular and radial factors to

obtain

(r,φ,z) =



√

2r/w







ilφ

ψ(r, z),

(42.16)

(z) =(2p+l+1) P(z). (42.17)

Here, the functions L

(x) are the Laguerre polynomials

as deﬁned in Sect. 9.4.2. The radial phase shifts produce

Part C 42.1

Spectroscopic Techniques: Lasers 42.2 Laser Designs 625

a wave front curvature of effective radius

R = z+



(2p+l +1)/2 +w

π/λ



π/z .

(42.18)

Modes with the same values of 2 p+l have identi-

cal axial and radial phase shifts. The two polarization

components of the electromagnetic ﬁeld double the de-

generacies of all modes considered here. Often these

degeneracies are split in practice by optical inhomo-

geneities of the medium through which they pass. More

details can be found in the summary of Kogelnick

and Li [42.2], or in the texts by Verdeyen [42.3]or

Svelto [42.4].

Some applications require knowledge of the elec-

tric ﬁeld in additionto the ﬂux density J.For

purely sinusoidal single mode beams, the rms ﬁeld

E=



hν J

c



1/2

. (42.19)

Nonlinear effects are often expressed in terms of powers

of the ﬁeld by

E

=2

(n−1)/2

E

(42.20)

for single mode and multimode radiation of random fre-

quency spacings. For m equally spaced modes, this is

increased by m!/(m−n)!.

42.2 Laser Designs

42.2.1 Cavities

The simple Fabry–Perot cavity consists of two spherical

mirrors facing one another. The surfaces are chosen to be

constant phase surfaces for the desired modes (42.18).

Stability criteria are shown in Fig. 42.3. A cavity is stable

when initial angles θ and displacements r of parax-

ial rays transform during a round trip into θ



and r



satisfying

−2 <

∂θ



∂θ

∂r



∂r

< 2 .

(42.21)

At frequencies for which the round-trip phase change

per passage

δ(ν) = 2π(z

−z

)ν/c+2



) − P

)



≈ 2π(z

−z

)/λ +π(2 p+l +1) (42.22)

is an integer multiple of 2π, the phases from differ-

ent passages interfere constructively, giving longitudinal

modes. (Here P

) gives the additional phase shift for

higher transverse modes at mirrors i = 1, 2.)

For a particular radial mode structure in an empty

Fabry–Perot cavity, the ratio of the maximum cavity

decay time for these standing waves to the minimum

cavity decay time for frequencies between longitudinal

modes is

(1 +r)

/(1 −r)

. (42.23)

Here r is the reﬂectivity of the end mirrors; for cavi-

ties with mirrors having different reﬂectivities, one may

use the square root of the product of their reﬂectivities.

A simple Fabry–Perot cavity may be tuned by changing

–2

–4

0.0 1.0 2.0 3.0 4.0

Unstable

Plano r

Unstable

Plano r

Unstable

Fig. 42.3 Stability parameters for simple two-mirror laser cavi-

ties of length L and mirror radii of curvature R

and R

. Here,

x =

(L/R

−L/R

) is the mean curvature difference of the two

mirrors; y =

(L/R

+L/R

) is the mean curvature of the two

mirrors. Cavities with parameters in the unshaded region are stable

Part C 42.2

626 Part C Molecules

the cavity length or by changing the index of refrac-

tion of the cavity material. Since both of these may be

properties of temperature, temperature tuning may be

possible. The index of refraction of a material may also

be sensitive to the intensity of excitation. Diode lasers

consisting of a semiconductor die with polished, reﬂect-

ing faces are often tuned by changing temperature and

pumping current.

For lasers with a dispersive optical element within

the cavity, highest selectivity is obtained when the

light has low angular divergence and impinges upon

the dispersive element as nearly plane waves. A beam

expander may reduce the angular spread while simulta-

neously increasing the beam width. For a grating used

as a mirror in a Littrow mount, the dispersion equation

∆λ = (d/n) cos φ ∆φ,

(42.24)

where n is the diffraction order, d is the distance between

lines, and φ is the angle of incidence off normal.

Cavities with prisms or gratings can be conveniently

tuned by rotating the angle of the dispersive element.

For a grating used as a mirror in a Littrow mount, the

grating equation is

λ = (2d/n) sin φ.

(42.25)

Often, more than one longitudinal cavity mode op-

erates within the selected frequency band, and tuning

consists of “hopping” from mode to mode, rather than

smoothly sweeping a single line across a band of fre-

quencies. Smooth frequency tuning can be achieved, for

example, in a design by Wallenstein and Hänsch [42.5],

in which the grating and Fabry–Perot cavity are placed

together inside a chamber. The whole laser is then tuned

by changing the index of refraction of the gas inside by

varying its pressure.

The current trend with pulsed lasers is to use

a very lossy, short oscillator cavity in which the lon-

gitudinal modes are nearly absent, making up for the

cavity losses with a very high gain lasing medium. The

front, output mirror of such a cavity may actually be

antireﬂection coated, with a reﬂectivity of only a percent

or less.

Low gain, continuous wave (cw) lasers often use

a combination of cavity length tuning along with dis-

persive element rotation, such as a prism or Lyot ﬁlter.

Often, lasing on a traveling wave in a ring conﬁgura-

tion is used to avoid longitudinal modes, as illustrated

in Fig. 42.4. Some commonly used gain media are listed

in Tables 42.1 and 42.2.

Pump beam

Gain

Faraday

rotator

Etalon

Output

Lyot filter

λ /2

Fig. 42.4 Ring laser. The Faraday rotator and half-wave

plate permit circulation of the cavity ﬁelds in only one

direction

42.2.2 Pumping

Many methods, including electrical discharges and

ﬂashlamps, have been used to pump the gain media of

lasers. Generally, the best pump is another laser.

Two notable pump lasers have dominated the ﬁeld

of tunable, visible lasers: pulsed neodymium YAG ,

frequency doubled to ≈ 503 nm, and cw Ar II ion at

514.5 nm. Both are extremely effective at exciting the

highly efﬁcient rhodamine class dyes in the red–yellow

portion of the visible spectrum. The typical pump beam

of a Nd/YAG laser enters the amplifying dye cell trans-

versely along one on the faces of the cell. The typical

cw pump beam enters the dye almost collinearly with

the laser axis.

By temporarily “spoiling” the Q by making the

laser cavity lossy, lasing can be held off until the gain

medium stores a greater energy density than the min-

imum required for lasing. Rapidly switching off the

loss mechanism releases this energy in one giant pulse.

An electronic optical shutter, such as a Pockels cell, or

a saturable dye inside the laser cavity, designed to pho-

tobleach from the spontaneous emission just before the

laser reaches threshold, are commonly used.

Periodically spoiling the laser gain or the cavity Q

at the period of a round trip produces intense, short

pulses. Viewed from the frequency domain, the phases of

individual longitudinal modes are “locked” together to

produce a light packet circulating at the frequency of the

reciprocal of the mode spacing. Extremely short pulses

(< 1 ps) can be produced by mode locking. Practically,

mode locking can be achieved by using a thin, saturable

absorber near one of the cavity mirrors [42.6], or by

acoustically modulating an optical element in the cavity

– even an end mirror itself. One of the best ways of mode

locking is to pump a short lifetime gain medium, such as

a dye, with mode locked laser light, such as from a mode

locked argon ion laser [42.7].

Part C 42.2

Spectroscopic Techniques: Lasers 42.2 Laser Designs 627

Table 42.1 Fixed frequency lasers

Laser Wa velength (nm) Excitation method

ArCl ≈170 (band) Pulsed, gas discharge

ArF ≈193 Pulsed, gas discharge

KrCl ≈222 Pulsed, gas discharge

KrF ≈248 Pulsed, gas discharge

XeBr ≈282 Pulsed, gas discharge

XeCl ≈308 Pulsed, gas discharge

XeF ≈351, 353 Pulsed, gas discharge

≈337.1 (other bands) Pulsed (1-10 ns), gasdischarge

488.0, 514.5 cw, gas discharge

(454.4, 457.9, 465.8, 472.7,

476.5, 501.7, 528.7)

351.1, 363.8 cw, gas discharge, high magnetic ﬁeld

568.2, 647.1, (476.2, 520.8, 530.9) cw, gas discharge

350.7, 356.4, 406.7 cw, gas discharge, high magnetic ﬁeld

Ne/He 632.8, 1152.3, 3390 (others) cw, gas discharge

-Ruby 694.3 Pulsed, ﬂashlamp

-YAG ≈1065, ≈1300 Pulsed: ﬂashlamp; cw: lamp, LED or laser diode

-glass(various) ≈920, ≈1060, ≈1370 Pulsed, ﬂashlamp

-glass ≈1060 Pulsed, ﬂashlamp

-glass(various) ≈1540, ≈1536, ≈1543, ≈1550 Pulsed: ﬂashlamp; cw: lamp, LED or laser diode

Xe 3507 cw gas discharge

≈10 600 (lines) cw: gas discharge or gas dynamic

Molecular vibration and rotation 1×10

–3 × 10

(numerous lines) IR laser (CO

,CO,···)

Table 42.2 Approximate tuning ranges for tunable lasers

Laser Wavelengths (nm) Notes

Dye solution <330 – >1200 [≈10 each dye] Pulsed: laser, ﬂashlamp; cw: laser

Alexandrite 700– 820 Pulsed: lamp, laser; cw: laser

/Sapphire 680– 1100 Pulsed: laser, lamp; cw: laser

GaAs 840–900 pn junction

InGaAlP/GaAs 630– 700 [1–10] pn junction

GaAsP 550– 880 pn junction

AlGaAs, AlGaAs/GaAs ≈820 [1–20], 720– 880 pn junction, temp. & current tuning

InP ≈900 pn junction, temp. & current tuning

GaInAs 906–3100 pn junction, temp. & current tuning

InGaAlAs/GaAs 800– 1100 pn junction, temp. & current tuning

InPAs 900– 4000 [1– 50] pn junction, temp. & current tuning

InGaAsP/InP 1200– 1650 pn junction, temp. & current tuning

InAs ≈3100 pn junction, temp. & current tuning

InSb ≈5200 pn junction, temp. & current tuning

PbS ≈4300 pn junction, temp. & current tuning

PbTe ≈6500 pn junction, temp. & current tuning

PbSe ≈8500 pn junction, temp. & current tuning

PbSnSeTe “Lead Salt Diode” 4500– 15 000 [1– 50] pn junction, temp. & current tuning

Part C 42.2