CARS

feature article A Review of the Theory and Application of Coherent Anti-Stokes Raman Spectroscopy (CARS) W. M. TOLLES,* J. W. N I B L E R , t J. R. McDONALD, and A. B. HARVEY Physical Chemistry Branch, Chemistry Division, Naval Research Laboratory, Washington, D.C. 20375

Coherent anti-Stokes Raman spectroscopy (CARS) is a relatively new kind of R a m a n spectroscopy which is based on a nonlinear conversion of two laser beams into a coherent, laser-like R a m a n beam of high intensity in the anti-Stokes region. The emission is often many orders of magnitude greater than normal Raman scattering and, because of the coherent and anti-Stokes character of radiation, the method is very useful for obtaining Raman spectra of fluorescing samples, gases in discharges, plasmas, combustion, atmospheric chemistry. In this paper we outline the basic theory behind CARS and describe its unusual effects and drawbacks. We review the research to date on various materials, and indicate the possible future direction, utility and applications of CARS such as surface studies, fluctuation phenomena, reaction dynamics, photochemistry, kinetics, relaxation, and energy transfer. Index: Headings: CARS; R a m a n spectroscopy; Nonlinear optics.

INTRODUCTION Coherent anti-Stokes Raman spectroscopy (CARS)$ is one of a number of different Raman processes that have developed with the availability of lasers in recent years. Examples include: Conventional, spontaneous Raman scattering' Resonance Raman effect2 Inverse Raman effect:~ Hyper-Raman effect4 Raman-induced Kerr effect'~ Coherent anti-Stokes Raman spectroscopy (CARS) 6 It is beyond the scope of this paper to discuss all of these Raman effects, but they are compared briefly in a recent review 7 and discussed in more detail in the references above. However, it is to be emphasized that each method is quite different from the others, and equipment required to carry out CARS can be readily adapted to conduct experiments based on the other techniques. It also may be stated that beyond normal Raman spectroscopy, CARS probably has the most general utility. CARS is potentially a powerful technique because of its promise in obtaining analytical and spectroscopic information pertaining to Raman active resonances in gases, liquids, and solids. With this process, spectral Received 15 April 1976. * Present address: Department of Physics and Chemistry, Naval Postgraduate School, Monterey, CA 93950. t On sabbatical leave from: Department of Chemistry, Oregon State University, Corvallis, OR 97330. $ Also referred to as coherent anti-Stokes R a m a n scattering.

Volume 31, Number 4, 1977

resonances corresponding to vibrational transitions may be observed by mixing together two visible laser beams. Unlike second-order processes, CARS, a thirdorder effect, is generally applicable to isotropic, as well as anisotropic media, and the conversion efficiency to (coherently) generated photons is considerably higher then with conventional, spontaneous Raman scattering. As such, it appears to be a superior tool for obtaining spectra of luminescent samples (fluorescent materials, impurities, combustion systems, and electric discharges) and of certain solutes in solution. By the CARS technique two relatively high-powered (typically pulsed) laser beams at angular frequencies oot and toe are focused together in a sample. As a result of mixing the two lasers, a coherent beam resembling a low intensity laser beam at frequency, ~o,,.~= 2 tot - oo~,is generated in the medium. The efficiency of the conversion to frequency ~o,.~depends critically upon the presence of molecular resonances at a frequency ~ot - oo.9,the laser intensities, the resonance line width, and the number density. Typically (os is varied to obtain a spectrum. As ~o~ - tos is swept over the molecular resonance, the intensity of the beam at oJ,,~changes. Recording this intensity as a function of tot - w~ constitutes a CARS spectrum. The first observation of this particular nonlinear mixing process was reported by Terhune s as a by-product of stimulated Raman emission. Later Maker and Terhune": used a 0.1 J pulse from a ruby laser and a second beam produced by stimulated Raman emission in benzene to observe the effect in several samples. Since these initial experiments, a number of investigations demonstrating the application of CARS in solids, '°-'" liquids, '7-~ gases, 34-4:* and for kinetics '~'~-5" have been reported. The high-powered, tunable, pulsed dye laser has made it possible to scan conveniently over a significant spectral region in order to observe the resonances. As we shall see, one of the most impressive properties of CARS is that very high conversion efficiencies are possible. Also, the process actually improves with higher resolution. Since lasers can be spectrally narrowed to rather small widths (< 10-:~ cm-', see Ref. 5) with only small sacrifices in laser output, high resolution spectra are readily attainable. Thus, it is safe to say that in the future CARS will be the preferred method of doing high resolution Raman spectroscopy. We shall also see that the coherent and anti-Stokes character of the emission

APPLIED SPECTROSCOPY

253

m a k e s the technique v e r y useful for m a n y applications where fluorescence is a problem or where only small viewing apertures are available. In this paper we shall a t t e m p t to outline the basic t h e o r y behind CARS in order to explain its properties, to review some of the recent progress, and to project s o m e w h a t into the f u t u r e in order to suggest problems t h a t need solving and to point out applications. I. O U T L I N E OF T H E D E V E L O P M E N T THEORY

O F CARS

A brief discussion of the basic t h e o r y behind CARS is offered here in order to explain the process and describe its properties. A more complete account of the t h e o r y m a y be found elsewhere, 9, ~3, 5s-~4 and a review on nonlinear effects including CARS m a y be found in a recent account by S h e n 2 5 Conversion of two laser beams into the anti-Stokes component at (Oas = 2oil - cog is a direct consequence of the n o n l i n e a r dielectric properties of materials. The efficiency of this process is considered for both plane wave and focused beam cases and an expression is developed which relates this efficiency to the n o r m a l R a m a n cross section. A. N o n l i n e a r Mixing. The polarization of a m e d i u m in an electric field m a y generally be expressed as a power series: P (~o)

(1)

= X(~) (o~) E (o~) + X(~) (o~) E ~ (~o) + X(~) (o~) E ~ where P is the macroscopic polarization vector, X") is the dielectric susceptibility tensor of r a n k i + 1 associated with the ith order of the electric field, and E is the applied electric field. The subscripts denoting the tensor properties of the susceptibility have been eliminated for simplicity. U n d e r low i n t e n s i t y fields, only the firstorder t e r m is i m p o r t a n t and is the basic for classical, linear optics including n o r m a l R a m a n scattering. The h i g h e r order t e r m s become i m p o r t a n t as the field s t r e n g t h s approach v e r y high levels, a fact which explains why the whole field of n o n l i n e a r optics achieved exponential growth i m m e d i a t e l y following the a d v e n t of the laser. The first n o n l i n e a r t e r m depends on the square of the field s t r e n g t h and is responsible for second harmonic generation (doubling of laser frequencies), sum and difference frequency generation, h y p e r - R a m a n effect and p a r a m e t r i c oscillation. The third-order t e r m is responsible for third harmonic g e n e r a t i o n (3oo~ ~ ~o~) and other processes, e.g., 2o~1 + oo~--~ oo~, 2o~ - o~ ~ ~o~, etc. It is the last m e n t i o n e d process, namely, 2o~ - ~o~ ~o~, which is designated as CARS. An isotropic m e d i u m such as a liquid or gas exhibits inversion s y m m e t r y in its macroscopic dielectric properties. E x a m i n a t i o n of Eq. (1) u n d e r the inversion operation reveals t h a t X(~ m u s t be identically equal to zero for such a medium. Hence, the lowest order n o n l i n e a r i t y which m a y be present in a liquid or gas is due to the third-order susceptibility, X(~. The electric field is, of course, a vector. In w h a t follows, we shall assume t h a t the direction of all electric fields involved are along the same axis and t r e a t each field as a scalar quantity. Using complex notation, we 254

V o l u m e 31, N u m b e r 4, 1 9 7 7

m a y express the m a g n i t u d e of the electric field at angular frequency oJi (= 2 zrc/h~) as: E~ (oJi) = ~1 [$iei(k,z_~,t) + c.c.]

(2)

where $i is the amplitude, k~ is the m o m e n t u m vector equal to o~n~/c, t is time, z is distance along the direction of propagation, n~ is the index of refraction at toi and c.c. is the complex conjugate. In general, E m a y consist of a sum of a n u m b e r of different frequencies. For the process designated as CARS, only two frequencies at col and 002 are introduced, and the consequent polarization at frequency 2to~ - oJ2 will be examined. Conventions differ in the l i t e r a t u r e r e g a r d i n g the introduction of a factor of 3 at this point. I f E = E~ + E2, the third-order t e r m in Eq. (1) has a component of m a g n i t u d e 3E12E2. The r e a d e r is cautioned t h a t this factor of 3 is the source of some confusion w h e n r e l a t i n g results from various articles. For clarity, the relationships derived here explicitly show this factor of 3 as a multiplicative coefficient associated with the value for susceptibility. Thus, r e t a i n i n g only those t e r m s in which the polarization varies by 003 = 2ool - 002, 1 p(a) = ~ [3 X(3) (-co3, oJ,, oo,, - o~2)]

{$12 $2*

(3)

ei[2kl-k2]z-(2°Jl-(°2)t + C.C.} ,

where the usual notation of Bloembergen 6° is used such t h a t X(3/ (-~oa, coo, ooc, ~o~1)r e p r e s e n t s the susceptibility for the process in which oo, = (oh + ooe + co~. Substituting Eq. (3) into Maxwell's equations gives the gain equation for plane waves; 45"46 d~as -

- i zr oJ~, ~t 2 ~ , (3 X(s)) e "~kt-k'-ko`)z d Z 2 cnas

(4)

where the previous subscripts 1, 2, and 3 are replaced by l, s and as to r e p r e s e n t the laser frequency, Stokes frequency, and the anti-Stokes frequency, respectively. Introducing the usual relationship between electric field and intensity [I~ = (c/8 ~-) 15~]2], integration of Eq. (4) gives: I~

P°~

I~

P, = (4 ~.2 o ~ 2

(5) r sin (hkL/2)~ 2

where e is the efficiency of the process, P is the power of the respective beams, L is the length over which the beams are mixed t h r o u g h the sample and hk = 2kt - k~ - k,~. The m o m e n t u m m i s m a t c h h k is a direct result of the fact t h a t the propagating waves move in and out of phase because of dispersion in the medium. As can be seen from Fig. 1, conversion efficiency for a length of path L is m u c h h i g h e r for h k = 0, the phase-matched condition. Since dispersion in gases is usually quite small, particularly for low-pressure conditions, phase m a t c h i n g over moderate path lengths (many centimeters) is readily achieved. For condensed media, Ak ¢ 0 even over small paths. In this case, phase m a t c h i n g is achieved, however, if the beams are crossed at angle 0,

[

I.O

(4xz] ]

d(z) ~ = d o 2 1 + \~-~/o~] j

CARS CONVERSION EFFICIENCY VS PHASE MATCHING, Aft (L constonl )

(7)

where z is the longitudinal distance from the focal point and do is the minimum diameter of the beam waist. For a lens of focal length f and an unfocused beam diameter d and wavelength X, it can be shown ss that:

0.8

do = 4kf/~'d

(8)

The beam diameter described here represents the 1/e value of the electric field or 1/e~ value of the intensity. If the intensity is considered to be a constant in a beam profile of diameter d(z), an approximate solution to Eq. (4) may be obtained for the conversion efficiency:

"~ 0.6

0.4

,/2 L

k\

Pa8 [ 2 t a n - ' ~ d J ) ] 2 (4~c°~"~13X'a~12Pl~ ~= P-~-~ kna~ j ',, c ]

0.2

(9)

where X is some average wavelength taken to be k~. This relation indicates that 71% of the maximum conversion takes place over length L equal to 2 ~rdo2/k. Implicitly -0

ko

2.0

s.o

4.0

5.0

6.0

7.0

Ak L / 2 Fie. 1. C A R S conversion efficiency vs p h a s e m a t c h i n g (Ak) for constant interaction length.

the phase matching angle. This angle is determined from the geometry illustrated in Fig. 2. Note that when this is done, the CARS beam emerges at a second angle 0', which aids in spatial filtering. Of course, if the beams are crossed, the interaction length is now limited by the beam walk-off. The phase matching angle, 0, is usually quite small (-1 ° for benzene at 992 cm-') but increases rapidly with Raman shift or for conditions near electronic absorption as encountered in resonance CARS enhancement. If the beams are not crossed and Ak ¢ 0, then according to Eq. (5), the CARS signal will vary sinusoidally with path length. The path length to reach maximum conversion efficiency we define as the coherence length, L Lc which occurs when hkLc/2 = zr/2 or Ak = ~'/Lc. In Fig. 3, we show how conversion efficiency varies with path length for collinear beams and various coherence lengths. It should be mentioned that some authors 68"~7 define coherence length as the path length between successive maxima (Lc = 2 ~-/hk). We should now consider the question of the effects of focusing on conversion efficiency. In one treatment 48the focal region is assumed to be a cylinder of plane waves and because of fortuitous cancellation of terms, the resulting expression (Ak = 0) for the efficiency is independent of focal length:

k2 2~

I -

wz

= w3

Fro. 2. W a v e vector d i a g r a m for p h a s e m a t c h i n g (Ak = 0). 5.0

CARS CONVERSION EFFICIENCY VS INTERACTION LENGTH FOR VARIOUS COHERENCE LENGTHS 4.0

=

=

(2)= ~ \(4 n . , c ~ ] 13x,3,1=Pl2

5.0

5 L c = (Jo-~

55 v

2.0

I.O

= i00

(6) , =60

In another approach to the problem, one assumes a functional form for the beam area but maintains the plane wave approximation for the field. The beam diameter d(z) can then be expressed in usual gaussian form6s as:

0~ 0

20

40 60 80 I00 120 140 INTERACTION LENGTH, L (arbitrary units}

t60

180

FIo. 3. C A R S conversion efficiency as a f u n c t i o n of i n t e r a c t i o n l e n g t h for v a r i o u s coherence l e n g t h s .

APPLIED

SPECTROSCOPY

255

a s s u m e d in this d e r i v a t i o n is the p l a n e w a v e approxim a t i o n a n d a b e a m at was which h a s the s a m e spatial distribution as t h a t of the b e a m a t w, a n d w,. A rigorous d e r i v a t i o n of conversion efficiency for focused b e a m s h a s b e e n carried out by Regnier, 4~ Bjorklund, "°-7' and S h a u b F 2 In Fig. 4 the functional f o r m of Eq. (9) is c o m p a r e d with a n u m e r i c a l solution o b t a i n e d by Regnier. Because Regnier's results include a n undefined scale factor, a n absolute c o m p a r i s o n of his calculation w i t h Eq. (9) is not possible. Nonetheless, Fig. 4 shows t h a t both h a v e the s a m e g e n e r a l form and t h a t the s a m e m a x i m u m is r e a c h e d r e g a r d l e s s of t h e focal l e n g t h (for s h o r t e r focal length lens, this m a x i m u m is r e a c h e d in a s h o r t e r m i x i n g path, L). T h e m o r e rigorous calculations ,9-~2 show significant increase in conversion efficiency for t i g h t e r focusing. B. M a g n i t u d e o f X(3). T h e effect of a n intense field on m a t t e r is to polarize it in a n o n l i n e a r fashion. The m a g n i t u d e of the t h i r d - o r d e r n o n l i n e a r coefficient or susceptibility shown in Eq. (1) is a m e a s u r e of the conversion efficiency of t h e process 2wz - ws = w,~. This conversion of l a s e r power into a n t i - S t o k e s r a d i a t i o n t a k e s place in a n y m e d i u m , including noble gases. However, w h e n t h e r e is a v i b r a t i o n a l r e s o n a n c e present, the conversion efficiency increases rapidly w h e n w~ - ws = w~. The total susceptibility is t h e n a s u m of a frequencyd e p e n d e n t r e s o n a n t p a r t a n d a n e a r l y frequency-indep e n d e n t n o n r e s o n a n t part: X (3) ~

X :

X I'(~S -~- X NR

(10)

where for simplicity the s u p e r s c r i p t "3" has been removed, and t h r o u g h o u t this text it is a s s u m e d t h a t we are concerned with the t h i r d - o r d e r susceptibility associated with the process 2wt - ~o.~ = Was. It is to be e m p h a sized t h a t w h e n Xr~s is r e l a t i v e l y small because t h e r e are no n e a r b y r e s o n a n c e s or if the n u m b e r density of reson a n t m a t e r i a l s is v e r y low, t h e n the CARS emission is governed by the n o n r e s o n a n t susceptibility, which usually arises f r o m the solvent or diluent gas. It is the presence of this n o n r e s o n a n t t e r m XNR, which limits the sensitivity of the technique, because one can record R a m a n spectra only to an e x t e n t t h a t Xr~" exceeds XNR. We shall r e t u r n to this point l a t e r w h e n we discuss the limits of sensitivity for the technique. For now, how-

I°0

--

Pas Pas(max)0.6-

ever, we shall show how X res is r e l a t e d to spectral properties of n o r m a l R a m a n transitions. Expressions for the r e s o n a n t p a r t of X m a y be derived from classical or q u a n t u m m e c h a n i c a l developments. The results allow X to be e x p r e s s e d in t e r m s of the n o r m a l R a m a n cross section. In addition, the various resonances p r e s e n t in X are explicitly d e m o n s t r a t e d in the q u a n t u m m e c h a n i c a l results. The h a r m o n i c oscillator serves as a model for the classical approach. Placzek's d e v e l o p m e n t 73 r e l a t e s the polarizability, a, of a molecule to a bond s t r e t c h i n g coordinate, q: a = o~ +

q + ...

The force produced on the oscillator by a field because of this polarizability is:

F=~

o

The e q u a t i o n of motion for the simple d a m p e d harmonic oscillator is thus: 6:~

1 (Oa) E 2 0+F0+w'2q+~m

3 X(3) (was) =

(14)

m\0q/0 w,2 - ( w t ws)2 - i F ( w l w~) w h e r e Nhj, the difference in population in the lower and u p p e r states for a p a r t i c u l a r t r a n s i t i o n j , h a s been introduced to account for statistical distribution (see Refs. 40, 42, and 74), t h a t is, hj = I for low t e m p e r a t u r e , h~ = 0 for infinite t e m p e r a t u r e . T h e factor, As, is an i n t e g r a l p a r t of the q u a n t u m m e c h a n i c a l a p p r o a c h by density m a t r i x methods. .~o E l e c t r o m a g n e t i c s c a t t e r i n g t h e o r y gives t h e differential s c a t t e r i n g cross section, do-~dR, of a molecule exposed to e l e c t r o m a g n e t i c radiation: 74"75 - (a')2

(15)

where, for the R a m a n Stokes frequency:

0.2

[ Oa~

(a') ~ = ~ q /

L (cm) Fro. 4. Effects of focusing laser beams using plane wave approximation. The ratio of power in the CARS beam to maximum conversion is plotted as a function of cell length, L. The upper curve, A, represents computed data of Regnier25 The lower curve, B, is calculated from Eq. (9). Volume

(13)

~q o

where F is a d a m p i n g c o n s t a n t a n d wv t h e r e s o n a n t frequency. With the application of t h e f i e l d E -- E 1 + E~, w h e r e Ei h a s the form given in Eq. (2), it is seen t h a t q is driven not only at 6ol a n d ~os b u t also at cot - oo~. The solution of Eq. (13) for q(wl - w~) gives rise to a polarization at the a n t i - S t o k e s frequency due to the relation: P = xE = N(Oo~/Oq)o q(wl - w~) Et, w h e r e N is the m o l e c u l a r n u m b e r density. A direct c o m p a r i s o n of these relations t h e n yields the following expression for the susceptibility: ~3

da

2.56

(11)

0

31, N u m b e r 4, 1 9 7 7



(16)

This r e l a t i o n ignores contributions to do-~dR f r o m anisotropic scattering, for which a 5 to 10% correction can be included w h e n desired. 75 The e l e m e n t a r y q u a n t u m m e c h a n i c a l expression of < q2 > for a h a r m o n i c oscillator is:

< q2 > _

(17)

2moJv

By combining the t h r e e previous expressions, we get: do4/ d~-2mcov

( 0 a ~ oj__~ ~q/ c 4

(18)

The polarizability derivative in this expression m a y be substituted into Eq. (14) to obtain X(3) in t e r m s of the n o r m a l R a m a n differential cross section: 3 X(3) =

2 N c4 ( do ") ~4

~

oJv Aj ~

(19)

- (o2t - ~o~)2 - i F (cot - toe)

Note t h a t the r e s o n a n t third-order susceptibility is a complex q u a n t i t y which can be separated into a real and an i m a g i n a r y part. These characteristics, the added n o n r e s o n a n t contribution, and the squared dependence of susceptibility on CARS signals produce u n u s u a l spectral properties compared with normal, spontaneous Ram a n effect. These features are diseussed f u r t h e r in Section IIB. If we define dioo -= (or - (coz - co,) and a s s u m e t h a t AoJ/ oo~ ~ 1, which is usually a v e r y valid a p p r o x i m a t i o n in the vicinity of a resonance, t h e n Eq. (19) simplifies to:

[ 2 A o o -i

3~a)=A

iF

1

where -

go~. 4

~

Aj

(20)

According to Eqs. (5), (6), and (9), the CARS signal or efficiency is proportional to the square of the modulus of X"}) or: A~ 13 x(~) l~ -

4hco2 + F,2

(21)

For the integrated area over the CARS band, Eq. (21) becomes f [ 3 X(3) ix d ( ± ~ ) - 7r 2A F2

(22)

While at the peak of the emission (A o~ = 0), the relation is: A2

i3 X(3) l~=o - F2

ever, it is also clear t h a t resonance e n h a n c e m e n t is very complex and m a y not be a benefit in e v e r y case. In Section IIIC, we refer to a recent e x p e r i m e n t where such e n h a n c e m e n t has been observed and was of definite advantage. C. S a t u r a t i o n Effects. The basic relation for conversion efficiency Eqs. (6) and (9) indicates h i g h e r conversion into the anti-Stokes component with increasing power. This is t r u e until various s a t u r a t i o n effects become important. The derivation of Eq. (9) was based on an assumption t h a t the power at oJ(,.~was sufficiently small t h a t it had negligible effect on the beams at oat and oJs and t h a t the power of the b e a m s at oJi and o~s was not a t t e n u a t e d in the m e d i u m or in the conversion process. This approximation is valid if the power at ~o,s is less t h a n about 1% of the power of e i t h e r of the other two beams. In addition to this limitation, at high-power levels the population of the e n e r g y states involved in the CARS process m a y be altered significantly from equilibrium values. This s a t u r a t i o n process has been discussed by Maier et al. TM F r o m the harmonic oscillator model used in obtaining differential Eq. (13), it follows t h a t the presence of two electric fields at frequencies oo~ and o~, will drive the oscillator at a frequency equal to o~ - (os. This effect may, in fact, be used to populate upper vibrational levels of h o m o n u c l e a r diatomic molecules and has been used in studies of relaxation times between excited species and admixed gases. If oot - oo~ = ~o,,, the equation describing the time dependence of the population difference, h, between the two lowest vibration levels is:

(23)

In a n alternative approach, one derives an expression for X(~) from q u a n t u m mechanical considerations. Armstrong et al. '~s and M a k e r and T e r h u n e 9 obtain expressions using a third-order p e r t u r b a t i o n development. DeWitt et al. ~9 h a v e derived expressions for the thirdorder susceptibility by the density m a t r i x method. F r o m all of these developments, it is a p p a r e n t t h a t there are r e s o n a n t denominators which show t h a t X(3) is enhanced as the anti-Stokes signal or the laser frequency spectrally approaches a n electronic resonance. Such a n effect is analogous to resonance R a m a n effect in conventional R a m a n spectroscopy. 2' 76 F r o m the theory, 59 how-

Ot

h 2c ~F

Ill, A + . . .

z-'A

(24)

where additional t e r m s including relaxation effects have been excluded in the first approximation. The intensity, I, or power density at the focus of a diffraction-limited gaussian b e a m can be obtained from Eq. (8): I -

zrP

4 k ~ (f/d) ~

(25)

where f/d is the ratio of focal length to b e a m d i a m e t e r and P is the power of the laser. The power density at the focus will be of the order of 10 '° W/cm 2 for modest powers (100 kW) and a typical value of 30 for f/d. The value of z for H2 gas is t h e n of the order of 10 -9 s, indicating rapid equilibration of the v = 0 and v = 1 levels (> 99% after t = 6 z). Since the CARS e x p e r i m e n t is typically 10 to 15 nsec, r must be kept as large as possible to avoid saturation effects. To an extent, this can be done by increasing the f/d ratio, although the sample length must be conc u r r e n t l y increased to m a i n t a i n the same CARS signal (Eq. (9)). At relatively high laser powers (P~ > 1.0 MW, Ps = 0.1 MW) for H2 and for spectral laser widths comparable to the H2 line widths, s a t u r a t i o n becomes certain over much of the focal volume. F r o m Eq. (24), one sees t h a t the v = 0 n u m b e r density is halved (in the absence of f u r t h e r r e s o n a n t p u m p i n g from v = 1 or rotational relaxation into v = 0 or out of v = 1). In addition, the appearance of hj in Eqs. (14 and 19) m e a n s APPLIED

SPECTROSCOPY

2,57

that there is a limit to the amount of CARS signal which can be generated in a region of saturation. The surrounding nonsaturated regions of the focal zone thus contribute a greater proportionate share of the total CARS signal than is the case for a low-power, nonsaturated experiment. This means that a more complicated analysis will be required for number density measurements at high laser powers. In particular, care must be taken in interpreting relative intensities in terms of vibrational or rotational temperatures since according to Eq. (24) the saturation level will vary for lines of different X values, which depend on Raman cross section and line width. It is important to note t h a t the total magnitude of CARS signal will continue to increase with laser power even under saturation conditions due to the effective increase in sampling volume. Thus, for frequency measurements at low pressures or concentrations, the use of high laser powers is desirable. Some broadening of CARS lines would be expected under saturation conditions. Thus, plots of line widths (as well as CARS power) vs Pt 2P~ might serve to identify power regions in which saturation effects are occurring. II. COMPARISON OF CARS WITH NORMAL RAMAN SCATTERING As was shown in Section IB, CARS is related to the normal Raman cross section and, therefore, all molecular vibrations which are active in normal Raman spectroscopy are CARS active. However, there are several differences between CARS and normal Raman spectroscopy which need to be discussed. A. Efficiencies. First, it is useful to compare the efficiency of normal Raman scattering with CARS. In principle, this should be a straightforward exercise, but in practice the comparison is difficult since normal Raman scattering is usually performed with cw sources, whereas CARS is better carried out with intense, pulsed sources. However, for simplicity, let us assume that both Raman and CARS experiments use the same laser, i.e., a 5 MW, pulsed (20 ns), Nd:YAG laser operating on the second harmonic (532 nm and 0.03 cm -1 line width) at 10 pps. This laser yields an average power of about 1 W or about the same cw laser power used in many normal Raman experiments. If a portion of this laser is used to pump a dye laser and dye laser amplifier, then typical laser powers for CARS experiments might be: 2 MW at 532 nm and ~0.2 MW of dye laser power. A CARS arrangement similar to this is used for studying gases at the Naval Research Laboratory (NRL) (see Section IIIB and Fig. 18). These power levels are probably high enough to saturate the CARS signal in certain gases (see previous section), but for the purposes of this comparison, we shall neglect this effect. Now, combining Eq. (23) for the peak CARS conversion with Eq. (9) for CARS efficiency (with long paths and focused beams), we get for anti-Stokes power: = i(167rXs4~ \hch~] N [FA-

( ~]d°- '-'

where centimeter-gram-second units are used and F is full width at half-maximum in circular measure. For

258

Volume 31, Number 4, 1977

normal Raman emission the power scattered in the laser beam is: PRaman =

NL~ ~

P~

(27)

where, as before, N is the number density, L is the length of the focal region, Vt is the solid angle collected, and (do./d~) is the differential Raman cross section. These equations indicate that CARS signals climb rapidly with laser power (cubic dependence on laser power), whereas normal Raman scattering increases linearly with laser power. These equations also indicate that CARS output drops rapidly with number density, an effect which can be counterbalanced with increased laser power to the point of saturation. We are now prepared to calculate the efficiency of a CARS signal in some typical laboratory eases to be compared with normal Raman scattering using the same laser source. Suppose the gas to be detected is H2 at 0.1 Torr. Let us also assume that the laser power is typical of the apparatus described earlier (2 MW at 532 nm, 0.2 MW of dye laser power). Normal Raman cross sections and, particularly, line widths are not precisely known; however, we shall assume that for H2, F = 0.03 cm-' and do./d~ ~ 2 x 10 -:~° em2/sr for Qo,. ' Using Eq. (26) we calculate a CARS, single-pulse output of - 3 kW. If it is assumed that the CARS pulse narrows to - 1 0 ns, this output corresponds to an average power of - 3 x 10 4 W for 10 pps repetition rate. This enormous signal compares with - 2 x 10-~ W single pulse, or - 4 x 10 15 W average power for normal Raman scattering calculated from Eq. (27), where we have assumed a laser power of 5 MW, scattering into 1 sr and a 1 cm long focal region. Thus, CARS produces signals in 0.1 Torr of H2 which are - 1 0 " times greater than normal Raman signals. Of course, at high laser powers, saturation effects will reduce the CARS efficiency as determined in Section IC. Inhomogeneities in the laser beams, unoptimized spatial and temporal overlap, etc. will also adversely affect the CARS output. Moreover, other gases with smaller scattering cross sections which are distributed among a complieated set of transitions or with broader line widths will result in lower CARS emission. Nonetheless, under favorable conditions CARS efficiencies close to theoretical have been observed. 'a' 34 For low pressure gases (

Xj

-I

Fro. 8. This is a vector display of susceptibility in the complex plane. A vector d r a w n from the origin to a point on the circle represents the modulus of the susceptibility. The q u a n t i t y A is defined in Eq. (20) and is proportional to the normal Raman cross section and n u m b e r density.

APPLIED SPECTROSCOPY

261

2.0

2.0I 1.0

i.o

.~.

o.o

0.0

-I .C 991 2.0

I I000

I010 r =4.0

cm -I

1.0 ~

.

o.o -I.(

a

990

-I .0 990 2"0II

t

iO00

I010

¢rn - I

I

I000

I010

2:1 INT RATIO

F = 4 cm - I 6 cm - I SEPARATION

1.0

i'i - .

990

I

I000

I010

cm-I

FIG. 9. These synthetically drawn curves illustrate the slightly better "resolving" power inherent in CARS. a. Two overlapping resonances of equal normal Raman cross section, number density, line width (4 cm-]), and separated by 4 cm -~ are drawn as they are expected in a normal Raman spectrometer, where the signal is proportional to X" compared with the upper curve which represents the expected CARS trace, b. The traces in Fig. 9a are redrawn for an unequal number density or normal Raman cross section (2:1). Note the slight improvement in discerning the two overlapped peaks by CARS brought about by the negative contribution of the real part of the susceptibility not present in normal Raman mechanisms. Note also the fact t h a t cross terms in the CARS process moderate the squared dependence on number density and cross section for strongly overlapped peaks.

mental arrangements have been used to generate coherent anti-Stokes signals. Early experiments 9"32, ~6-~9,44.-48 utilized stimulated Raman scattering to create a Stokes beam which could be mixed with the pump laser beam. With the development of the dye laser, the ability to scan continuously the wavelength of one of the beams enabled spectra to be measured. 6' 1~-~6,18-22,26-29,4o-4~,49 The pioneering work of Maker and Terhune 9' 33 made use of two fixed frequency beams, one from a ruby laser and the second generated by stimulated Raman scattering in benzene giving a Stokes shifted beam. These two beams, separated by 992 cm -1, were useful only for measuring resonant CARS signals from benzene or nonresonant cross sections of other materials. Other liquids were employed to generate stimulated Stokes beams at frequencies appropriate for specific studies. Obviously, however, this was not a very convenient means of recording spectra. The method utilizing stimulated Raman scattering has also been used many times in the 262

Volume 31, Number 4, 1977

study of hydrogen gas. .~"-39, 44-48 A diagram of such an experimental setup is shown in Fig. 12. The stimulated Raman generation cell, C, contains hydrogen at 10 to 20 atm. The filter, F, blocks the stimulated anti-Stokes line, leaving the frequencies oJ~ and oos. The combined beam is split, half being focused into the sample under investigation, the remaining half is focused into a reference called RC. With suitable filters and detectors the frequency oJa, due to the sample is detected. Because amplitude fluctuations and spatial inhomogeneities of the ruby laser introduce considerable fluctuation in the magnitude of the CARS signal, the reference cell is used to generate a reference signal at a nonresonant frequency of some reference gas. What is measured rather than the CARS signal, P(CARS), then is the ratio, R = P(CARS)/P (Ref) of the CARS and reference signals. This ratio is approximately independent of the Pt2Ps (pump and Stokes shifted) intensity fluctuations and reduces effects of filter factors, etc.

which may be translated into the beam, aids in the alignment of the coincident beams. The use of nitrogen laser-pumped dye lasers has both advantages and disadvantages. Among the advantages is the fact that all the components needed can be obtained commercially and will operate perhaps with less attention than current high-power solid state lasers. Also, nitrogen lasers can operate at higher repetition rates which can make experiments easier to perform and allow spectra to be recorded in short periods of time. On the other hand, because of the need for transverse pumping, the spatial quality of the nitrogen laser-pumped dye laser beams is invariably less than ideal. Such dye laser beams, tailored sufficiently to be acceptable for narrow band width

N =lXresl : IO.O

2.O F : 1.0 cm-' I.O

~'O,O

-I .O

-2,0

-4'0

-2'0

0:0

20, z~oJ(cm")

LASER

40

.

C

L I FI

Fro. 10. This curve is a log plot of the curve illustrated in Fig. 6a for N a IX('~)I = 10.0. Note that this type of plot accentuates the minimum which goes almost unnoticed in the linear plot except where this region is expanded as in Fig. 6b. The difference in frequency between the maximum and minimum is a measure of XN".

2 ~ ( ~ '/~

o(1)

6(3C-4OC

10C -

-

-

-

S

F2

PMz

Y Fxa. 12. Experimental arrangement; the ruby laser emits a diffraction-limited signal t-frequency pulse; strong Raman-shifted sidebands are generated in the high-pressure stimulated Raman cell C filled with H2. Filters F1 (4-mm-thick Schott OG 590 + 1-cm-thick water cell) cut off the anti-Stokes lines and the second Stokes line, in that order; filters F2 (interference filter + Schott VG 14 glass) admit the anti-Stokes into photomultipliers PM~ and PM2 with S-11 photocathodes. RC is the reference cell; S is the gas sample (here, a flame) (taken from Regnier and Taran 47 with permission of the publisher, American Institute of Physics).

100C. . . . .

20C -

L2

-

22-2 ¢ - - - -

1 c - 6-

NITROGEN LASER

i__

~1

1---

DYE LASERS

06 04

\

-

~

% M2

M1

02

4170

4160

4150

4140

4130

4120

4110

4100

%_% (cm-') Fro. 11. Spectrum of the susceptibility of pure H2 at STP; the quantity ( P , / P , ~ O "2 is plotted in arbitrary units; the solid line is theoretical, calculated after Eq. (2), assuming XNR= 4 x 10-~s esu/6 d o - / d l l = 9.7 x 10-3~ cm2/sr, T/27rc = 2 x 10-2 cm -~, a monochromatic ruby laser, and a gaussian dye laser line of width 0.3 cm-~; this curve has been normalized to give a good fit with the experimental data between the Q (1) and Q (2) lines (taken from Moya et al. 4o by permission of the publisher, North-Holland Publishing Company).

C SCAN CONTROL

KNIFE EOGETEST,EL._

I

) L2

(u1 > w 2 SAMPLE CELL

REFERENCE L mt DETECTOR r

/ ~ 2 ~I,¢:2~3~ L3

An experimental arrangement employing two dye lasers pumped by a single nitrogen laser generates two beams of individually tunable wavelengths at power levels of from 10 to 100 kW in each dye beam. This arrangement, as used by Wynne, t3'14 Levenson et a l . , 26-2s Itzkan and Leonard, 2~ Begley et a l . , 6 Chabay et a l . / ° and Klauminzer, 22 is illustrated in Fig. 13. The two dye laser beams can be split, forming a double, nonlinear mixing arrangement to generate a reference signal for the purpose of reducing amplitude fluctuations. The introduction of two orthogonal knife edges,

PULSE RATIO PHOTOMETER

[

I

CHARTRECORDER J

MONOCHR~,~

FiG. 13. CARS optical schematic, M~, M2, and M~ are mirrors; L~, L~, and La are lenses; BS is a beam splitter which reflects 33% of the light. The focal length of Li is 500 to 800 mm, that of L2 is 200 mm. The crossing angle is 2a (taken from Chabay et al. z° with permission from the publisher, American Institute of Physics).

APPLIED

SPECTROSCOPY

263

(0.03 cm-') and high spatial quality needed for gas phase applications, will have at most