ieee 1984 1065 stimulated raman forward scattering with a ... · pulse compressor with nonplanar...

IEEE JOURNAL O F QUANTUM ELECTRONICS, VOL. QE-20, NO. 9, SEPTEMBER 1984 1065

Stimulated Raman Forward Scattering with a Divergent or Convergent Pump Beam

HARTMUT GRUHL AND RICHARD SIGEL

Abstract-Stihulated Raman forward scattering is investigated for a divergent or convergent pump beam. The rate equations for this case are derived and analytically solved, including pump depletion. The solutions are compared to an experiment where the generation of a Stokes pulse from noise is investigated with a pump beam of variable divergence. Theoretical and experimental results are in good agreement. Stimulated Raman forward scattering thus may be controlled by the beam geometry, e.g., in Raman pulse compressors. An example is discussed in the Appendix.

I. INTRODUCTION

D URING recent years the interest in stimulated Raman scattering (SRS) [ 11 has broadened in the sense that

besides the phenomenon itself, also its application in science and technology as a means of manipulating the frequency and pulse shape of intense laser pulses is being investigated. In this context pulse propagation, in geometries different from the planar one (parallel beam) usually considered, becomes of interest. Here we investigate SRS for a divergent or convergent pump beam, both theoretically and experimentally.

More specifically, we consider in this paper the transformation of a pump (laser) beam into first Stokes radiation by stimulated Raman forward scattering. The basic situation is sketched in Fig. 1. The leading edge of a pump beam (either divergent, parallel, or convergent) enters a Raman cell. It generates by spontaneous Raman scattering photons at the first Stokes frequency, some of which travel along with the pump pulse and are exponentially amplified. After a distance where the amplification at the leading edge has reached a value of 8 , with A 35, the Stokes pulse will deplete the pump pulse ,i.e., the energy of the pump pulse is transferred to the Stokes pulse. Since in forward scattering each subsequent pump photon packet spontaneously generates its own initial Stokes photons, it suffers exactly the same fate as the leading edge of the pulse. This leads to the pulse transformation scheme shown in Fig. 1. An application of SRS which recently found considerable

interest in the laser fusion field is laser pulse compression by stimulated Raman backward scattering [2] -[4] . In this case Raman forward scattering is of interest as a competing process which may lead to a premature depletion of the pump pulse. Also, even if an intense backward pulse has been generated, undesirable energy losses could result by second Stokes generation, which again is a forward scattering process of the type

Manuscript received March 12, 1983; revised January 24,1984. This work was supported by Bundesministerium fur Forschung und Tech- nologic, and Euratom.

H. Gruhl was with the Max-Planck-Institut fur Quantenoptik, D-8046 Garching, West Germany. He is now with the Physikalisches Institut, University of Gottingen, D-3400 Gottingen, West Germany.

R. Sigel is with the Max-Planck-Institut fur Quantenoptik, D-8046 Garching, West Germany.

/

/ /

1 Fig. 1. Conversion of a pump pulse into a first Stokes pulse by stimulated

Raman forward scattering in a Raman cell.

discussed here. Since in such Raman compressors one does not necessarily have to propagate the pulses as parallel beams, this application is an example where the results of the present work might become useful. Stimulated Raman scattering in a nonplanar geometry is also of interest in the context of laser pulse conversion for isotope separation [SI.

We discuss here stimulated Raman forward scattering on the basis of the rate equations, i.e., neglecting coherent effects. In Section I1 we derive the rate equations for slightly divergent or convergent beams. These equations are integrated analytically and the corresponding solutions are given including pump depletion. It is shown that the solutions agree with numerical computations for a special case. We then give a de- tailed discussion of how Raman forward scattering can be controlled through the proper choice of the pump beam divergence or convergence. In Section I11 we describe experiments where we investigated the predicted threshold behavior of first Stokes generation as a function of the beam geometry. The experiments were carried out using 15 ns pump pulses from a frequency-doubled (X = 0.532 pm) Nd-YAG laser and Hz gas at 30 atm as the Raman medium. The divergence or convergence of the beam was controlled by lenses of various focal lengths (negative or positive) in front of the cell. Experimental and theoretical results are compared in Section IV and good agreement is found. In the Appendix we discuss a backward Raman pulse compressor with nonplanar beam geometry and the associated parasitic forward Stokes problem.

11. ANALYTICAL SOLUTION OF THE RATE EQUATIONS FOR A DIVERGENT O R CONVERGENT PUMP BEAM

A . Derivation of the Rate Equations for a Divergent or Convergent Pump Beam

In this section we derive the rate equations for a divergent or convergent pump beam. The rate equations have the general form of continuity equations for the different kinds of

0018-9197/84/0900-lO65$01.00 0 1984 IEEE

1066 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-20, NO. 9, SEPTEMBER 1984

photons with source terms due to stimulated scattering

Here Zi denotes the intensity of a Stokes beam of order i; the vector field zi determines the direction and spe.ed of propagation of the different kinds of photons at each point in space. The source term is denoted by Si.

The geometry under consideration is shown in Fig. 2. Diver- gence or convergence of an originally parallel pump beam is achieved by a lens with focal length f which is placed in front of the Raman cell. We consider only the case c'i = c', i.e., the pump and Stokes photons are assumed to follow the same optical paths. This assumption is further discussed below. In addition we assume = co (co =vacuum velocity of light), Le., we neglect dispersion in the medium. This is a good approximation for our experiments with H, gas as the Raman medium.

We consider in this paper explicitly only the propagation of photons along straight light rays in the approximation of geo- metrical optics though our analysis is valid for more general cases (see discussion below). In cylindrical coordinates the vector 2 then has the components c' = {c, ~ c,-} with

c: + c: = c; ;

where a is the angle at which a light ray is inclined to the optical axis ( a < 0 for a negative lens, a > 0 for a positive lens). The trajectory of a photon in the cell as a function of time is given by

X(t) = c,t; ( 3 4

r(t) = y o + c,t = ro - c,t tan (Y (3b)

RAMAN CELL

Fig. 2 . Geometry of pump pulse propagation.

ing to note that the pump beams admitted within our approxi- mations have the property Z,(t) Y ' ( t ) = constant. This is readily seen by integrating (4a) with the scattering term neglected.

We now write down the rate equations for slightly divergent or convergent beams for the special case of the transformation of a pump pulse ( i = 0) into a first Stokes pulse (i = 1). The source terms are given by So = -goIoZl and S 1 = g l Z o Z l , respectively. These equations are

@(t) = - - = - 2; 2c0 tan Q 2co Y Y o - cot t a n a f - cot '

=-

Here we have introduced the total derivative d /d t = a jar t coa/dx . The coefficients go and g , are related by g , /go = w1 /ao where w o is the angular frequency of the pump radiation, and w 1 is the angular frequecy of the Stokes radiation. w o and w , are connected with the frequency oR of the Raman transition by the resonance condition WR = wo - wl. gl is related to the usual gain coefficient y for stimulated Raman scattering by y = g , / co .

These equations differ from the well-known planar equations [6] by an additional term which is due to the convergence or divergence of the beam.

where Y o is the radius at which the photon enters the cell. y o , B. Analytical Solution of the Rate Equations a, and f are connected by tan a = y o if.

Evaluating the divergence term in (1) one obtains We now derive an anlytical solution of the system of rate equations for a general function @(t) . Multiplication of the

= co - t - zi. az, 2; ax r

first equation by g l , the second equation by g o , and addition of the results gives

As we consider only slightly divergent or convergent beams with a << 1 we have neglected a term c,dIi/ar and replaced c, by co . For the dependence of c, onr andx we have assumed &(t) = @(t ) $(t).

With this definition we get

c, = - 4 ( x ) r / 2 , i.e., ac,/au = C,/Y = i / u . Here @(x) is an arbitrary function of x. The property c,- - Y for constant x ensures that

Integration of this equation gives

the tangents to the light rays in a plane normal to the axis intersect in a point on axis [(2b) with c, = co is of this form] . $ ( t ) = $(O) exp [ Ir @(t ' ) d t 'l . In general, however, this point will not be a single focal point for all planes in the cell as it is for the case shown in Fig. 2 and me first equation of the system (4) can be written described by (3a) and (3b). The derivation allows for the . more general case where beam diameter and divergence vary z o ( t ) - -goI l ( t ) + @(t) = @(tl +glzo( t ) - Q(t). along the celI in a rather arbitrary manner. It may be interest- Zo( t )

GRUHL AND SIGEL: STIMULATED RAMAN FORWARD SCATTERING 1067

or

This equation together with the equation defining $(t) forms a system of two equations for Io( t ) and I , ( t ) which can be resolved to give the analytical solution of the rate equation system (4):

=I&)

Here J, ( t ) is given by

and $ (0) by

J, (0) = $0 = g1Ioo + goI10 .

This analytical solution [7] describes small- and large-signal amplification for a first Stokes pulse in the forward direction. It is valid for an arbitrary function @(t) which is related to r( t ) by @(t) = -2i(t)/r(t). As we mentioned already the function ~ ( t ) may be specified arbitrarily and is by no means re- stricted to the special case described by (3a) and (3b). Thus, the present solutions describe pump and Stokes pulse evolution including pump depletion for rather general types of photon trajectories in the cell.

With respect to previous work on Raman forward scattering including pump depletion, we note the well known solutions of

the rate equations for the planar case [6] . They are obtained as a special case from our solutions with @(t) = 0.

The planar case with linear absorption can also be treated by our formalism setting #(t) = -Deo. Thiscase has been calculated numerically by D. von der Linde et al. [8] for CS2 as the Raman medium. We have compared our analytical solutions to his numerical results and found very good agreement (see Fig. 3). The following values for the coefficients were used: po = 3 X IO-’ me’, go/co = 1O-l’ m . W-’ , gl/co = wlgo/ ooco = 0.95 X IO-’’ m . W-’, wo = 2.7 X 10’’ s-’(ruby laser), oR = 1.24 X 1014 s-l , l o o = 3 X 10l2 W 1 m-2, I10 =

Stimulated Raman forward scattering of focused beams has been considered extensively in the literature [9] -[14]. Most of the studies are, however, limited to the small-signal regime whereas pump depletion is only taken into account numerically in [ 121 and [ 141 , In this last paper [ 141 , the amplification of a Gaussian Stokes beam by a Gaussian pump beam is considered. To compare to our results we define photon trajectories for a Gaussian beam by

r 2 = wi (1 t 1 2 )

I,.

where the dimensionless coordinate 6 , the radius at the beam waist wo, the confocal parameter b, and the wave vector k are related to our notation by = 2 (f- cot)/b and wi = b/k = b2r;/4f2. With this expression for Y it can be readily shown that (4) becomes identical to the equations used in [ 141 in the limit bo = bl = b and ko = k l = k. Thus, Gaussian beams can be treated within the frame used here only to the extent that the additional diffraction introduced by the wavelength change in the scattering process may be ignored; this is of course a consequece of our assumption that the photon trajectories of Stokes and pump photons be equal. The method of integration in [ 141 is basically similar to the one applied here although the present formulation is more readily applied to cases where linear absorption is present. We have therefore retained our method of integration as originally given in [7] .

It should be kept in mind that the formalism used above describes the Raman interaction in terms of local intensities of the interacting waves in the approximation of geometric optics and is thus limited in scope. Imbedded in a comprehensive study of Raman scattering of focused beams (including experiments [ 131 ) these limits have been discussed in [ 101 , [ 121 . Following [ 121 , “this approach is possible only for quasi-plane pump and Stokes waves which are specified at the entry into a medium.” Furthermore, it cannot afford an appropriate description if “diffraction effects are important and give rise to discrimination between the gains of the modes of an active waveguide channel formed by the pump wave.” These state- ments are supported by a recent investigation which shows that mode discrimination is strong for focused beams, but comparatively weak for collimated beams [ 151 . Also the problem of beam quality of pump and Stokes beams needs a treatment including diffraction effects [ 161 , [ 171 . Neverthe- less it is felt that the present solutions, being analytical in na- ture, may be useful in cases of practical interest. Potential applications may be found in Raman forward converters for isotope separation and communication purposes.

In Raman backward pulse compressors (the present study

1068 IEEE JOURNAL O F QUANTUM ELECTRONICS, VOL. QE-20, NO. 9, SEPTEMBER 1984

I , ( x 1 NUMERICAL SOLUTION !VON OER LINDEJ

I

!THIS WORK1 - ANALYTICAL SOLUTION

0 15 30 L5 6 0 x [chi

Fig. 3 . Comparison of a numerical solution (von der Linde et al., [ 8 ] ) of the rate equations with the analytical solution of this work for stimulated Raman forward scattering in CSz, including linear absorption in the medium (planar geometry).

was actually motivated by this application) convergent or divergent pump beams are also of potential interest. The parasitic forward Stokes generation then also occurs in this geometry making, at least in principle, the present solutions of interest for the design of much devices. In order to illustrate the potential benefits of a nonplanar beam geometry in Raman pulse compressors an example is discussed in the Appendix together with the corresponding parasitic Stokes problem. As a test of the typical situation in this application the threshold for forward Stokes generation is investigated for convergent and divergent pump beams in the experimental part of this paper. Nevertheless, because Stokes generation occurs from noise in this case and also because suppression rather than saturation of the Stokes pulse is of interest, the solutions given above apply here only in a more qualitative sense and the criteria given below could have been derived in a simpler manner, neglecting pump depletion. Clearly the potential of these solutions is not really exploited in the present paper.

C. Discussion of the Solutions We want to use the solutions derived above to analyze our ex-

periments with divergent and convergent pump beams where the threshold for first Stokes generation was measured as a function of beam geometry. In these experiments, the cell was always kept several Rayleigh ranges away from the focus or, in the experiments with a parallel beam, the cell was con- siderably shorter than the Rayleigh range. Under these conditions it is sufficient to describe the photon trajectories in the approximation of geometric optics by (3a) and (3b). The restriction to cases with the focal plane far from the cell is simply due to the fact that the high power pump pulse otherwise would have caused breakdown and possibly destruction of the cell window.

With r ( t ) given by (3b) one has $ = $O(r(0) / r ( t ) )2 . Calculat- ing Jii,(t)dt and substituting the results into the solution (5) one gets

In the experiment, the Raman medium is Hz (Raman shift 4155 cm-') at a pressure of 3 X lo6 N . mu'; the pump wavelength is 532 nm. In this case we have y = g , /co = 2.23 X IO-" m . W-' (from [ 181 after correction to our pump wavelength and molecule density), g, = 6.68 X m2 . J-' ,go = wogl / o1 = 8.58 X m2 . J- ' . For loo = 10" W . mu' andIlo =

W . m-' (i.e., A = ln(Ioo/Zlo) 2 35), Fig. 4 shows Io([) and I I ( t ) including pump depletion for different values off.

In Fig. 4(a) the time dependence ofZo(t) and ZI (t) is plotted for a convergent pump beam ( f > 0). As can be seen, the pump intensity grows after the pump pulse enters the Raman cell (t = 0). This is due to the convergence of the beam. In this first time interval the Stokes pulse has small-signal amplification. After about 3 ns the Stokes pulse reaches large-signal amplification and depletes the pump. We call the time when I l ( t ) becomes equal to [,(I) the generation time tg l of the Stokes pulse. Under the conditions of Fig. 4(a) the pump radiation converts into first Stokes radiation after traveling about 0.9 m in the Raman cell. As one can calculate with our solution including pump depletion, the time interval during which the Stokes pulse has large-signal amplification is only about 10 percent of the time interval during whch it has small-signal amplification. After t > 1 .OS tgl the pump is practically depleted and there is only the Stokes pulse which grows in intensity because of the convergence of the beams.

In Fig. 4(b) the case of a parallel pump beam (f = m) is shown. One can see that tgl is now greater than in the case of a parallel beam. In this particular case one obtains

1 1 ( t>t 'a = - = - E 0 1 g, - o.78 100 wo go

as expected from the Manley-Rowe relations. If one decreases f towards negative values (divergent beam)

tgl increases even more [see Fig. 4(c)J. The reason is simply that the area

JO

has to satisfy the condition A 35. Of special interest is the fact that a limiting (negative) focal

length f l , exists beyond whch no efficient Stokes generation can occur: I f f is in the interval j" < f < 0 (i.e., if the beam divergence exceeds a certain value), then for all times (i.e., even for an infinitely long Raman cell) one has I l ( t ) <Z,( t ) [see Fig. 4(d)] . One can show that for 0.77fL < f < 0 more than 99.9 percent of the pump energy is transmitted through the Raman cell (independent from the length of the cell). This means that pump depletion due to first Stokes generation in the forward direction does not occur.

To determine f L we first calculate the dependence of the generation time tgl (Zoo, f ) on f and the pump intensity I ,


A l i I t ) 11012 Wlrn21 WEAKLY DIVERGENT

f = - 3 9 5 m c f L

O5I

cb) ( 4 Fig. 4. Graphic illustration of the analytical solution of the rate equa-

tions for convergent (a), parallel (b), weakly divergent (c), and divergent (d) beamgeometry. InitialconditionsareZoo = 1 X 10l2 W . m-2 andZlo = 1 X W .m-2 .

from the condition I o ( t g l ) = I , (tgl). From (7) one obtains

Noting that goIlo E exp (-35)w0g1Z,/wl <<glZoo and in- troducing the gain length Lg = co/glZoo (8) can be written in the form

This relation shows that the influence of the focussing of the beam on tgl becomes important when f is of the order of

Fig. 5 shows tgl with Zoo as a parameter. As can be seen, there exists for every curve with constant 1, a limiting value f L for which tgl becomes infinite. f L is obtained from the condition that the denominator in (8) becomes zero. One finds

35 Lg.

withgoI10 <<g,Ioo

f L ( l o o ) - co In ~ ~ 0 0 / ~ 1 0 ~ / ~ 1 ~ 0 0 *

For example, with Zoo = 10l2 W . m-2, g , /co = 2.23 X m . W-’ , and ln(Z,/llo) = 35, one finds f L = - 1.5 m. For a lens with - 1.5 m < f < 0 no strong Stokes signal would be generated, even in an infinitely long cell.

It may be of interest to note that derivation of this result does not require the full solution (7) but can be obtained essentially neglecting pump depletion. With this approximation we have for the Stokes gain

For a divergent pump beam (f< 0), A remains finite in the

I

I I I I I

i I I I

- 0.75 ti

- 1 -1.5 -3 00 3 f -DIVERGENT +CONVERGENT -

Fig. 5. Dependence of the generation time tgl of the first Stokes pulse on the focal length of the lens. The parameter is the pump pulse intensity Zoo. Zlo is kept constant at 1 X W . mW2. The circle marks the case shown in Fig. 4(c).

limit of an infinitely long Raman cell

lim A = - - Zoo f. g,

L’m CO

If one chooses as a criterion for the generation of a first Stokes pulse A ln(Zoo/Zlo) [this criterion is only slightly different from the above condition Il(tgl) = I o ( t g l ) ] one obtains the same result as above.

Finally, it may be noted that-as long as we are interested in the threshold behavior only-the assumption of equal pump and Stokes trajectories is not crucial for the above analysis. Whereas at threshold the generation of a “mode-matched’’ Stokes beam is expected on the basis of gain calculations for competing modes [ 151 and also observed experimentally (e.g., [ 131 ), we would like to illustrate in addition by an example that the result (8) depends only weakly on the structure of the Stokes beam. In this example we consider the amplification of a parallel Stokes beam by a convergent pump beam of constant power Po =Z,,(r) r 2 n (i.e., neglecting pump depletion). For a parallel Stokes beam we have @(t) = 0 and (4b) becomes

- =g,I,Po/m(t)2. d l , d t

With r = y o . (1 - c o t / f ) this equation is readily integrated

We now calculate a generation time tgl as before by setting I , (tgl) = Zo(tgl). With the notation r ( t g l ) = r1 , this relation becomes in the case under consideration Zl(tgl) = Zo(rl) = Po/nr: = I , ( ~ ~ / r , ) ~ . Combining these equations and re- solving for tgl now yields

Since 21111 (ro/rl)I 5 2 In 3.5 << In (Zoo/Zlo) 1 35 always, the result is essentially unchanged. The insensitivity of the result for tgl on the structure of the Stokes beam is obviously due to the fact that the intensity gain of the Stokes beam due to amplification is much more important than the gain due to its convergence (or divergence).


111. EXPERIMENTS A . bkxpevimental Setup

The criteria derived in the last section allow us to calculate whether for given conditions, i.e., pump intensity, Raman cell length, and pump divergence, Raman forward scattering can deplete the pump pulse or not. We checked the predicted behavior experimentally.

The experimental setup is shown in Fig. 6. It consists basically of a laser (not shown) for pump pulse generation, an in- terchangeable lens which determines the desired divergence or convergence of the pump beam, a Raman cell, and a fast photodiode with optical delay line and color filters for time- resolved measurement of the incident pump pulse, forward Stokes pulse, and transmitted pump pulse.

The pump pulse was generated by a commercial Quantel Nd-YAG pulse laser. The oscillator operated in a single trans- verse and longitudinal mode and delivered 20 ns pulses. For further amplification the pulse was passed through a YAG amplifier, a spatial filter, and three glass amplifiers. The last amplifier had a diameter of 45 mm. After frequency doubling in a KD*P crystal, pulses with energies of up to 2 J at h = 0.532 pm and a duration of 71/2 (FWHM) = 15 ns were available for the experiments. Before entering the KD*P crystal the beam diameter was reduced by an inverse telescope.

The dependence of the intensity at the entrance of the cell (= plane of the lens) on the radius and time was approximately Gaussian, i .e ., of the form

with a = (2.9 ? 0.2) X m and T~ = ~ ~ / ~ / ( 2 m ) = 9 x s. roo can be calculated from a , T~ and the measured

pulse energy E, by f, = E L / ( 7 r 3 1 2 ~ 0 a 2 ) for this particular pulse shape.

As Raman medium in the cell we used Hz gas at room temperature and a molecule density of 8 X loz6 mT3 (corresponding to a pressure of 3 X lo6 N . m-2 or 30 atm). With the pump at 532 nm and a vibrational frequency of the Q,, (1) line of 4155 cm-' , first Stokes radiation at 683 nm was generated. Note that under our conditions the laser linewidth (2 X 1 0-4 cm-' for a transform limited pulse) is much smaller than the width of the Raman line of g 4 X lo-* cm-' [ 181. The Raman cell had a length of 2.5 m and an inner diameter of 4 X

After the cell the beam was recollimated by an auxiliary lens (not shown in Fig. 6), split by a beam splitter and color filters into Stokes and transmitted pump radiation, and, after passing through an optical delay line, finally recombined with the incident pump pulse on a single Valvo XA 1003 photodiode. In conjunction with a Tektronix 5 19 oscilloscope, a time resolu- tion of g0 .4 ns was achieved. The optical delay line consisted of a set of planar mirrors suitably distributed around the lab- oratory. In addition, the energy of all pulses was monitored in each shot.

Without going into all details of the experimental setup, we would like to mention one observation made during tests of the apparatus which is important for the present investigation. We observed that the divergence of the pump beam could

m.

RAMAN CELL LENS 7y/ , PUMP PULSE

RAMAN CELL

,<biP

PULSE .___. . -

COLOUR FILTER- -INCIDENT

PUMP PULSE STOKES PULSE

TRANSMITTED PUMP PULSE I

.. .. OPTICAL DELAY"

-INCIDENT y, PUMP PULSE

STOKES PULSE

Fig. 6 . Experimental scheme.

change between shots by an amount which was of the same order of magnitude as the divergence (or convergence) intention- ally introduced by the lens at the entrance of the cell. It was found that this effect was due to thermal effects in the laser rods and depended on the time interval (cooling time) between shots. When the initially cold laser chain was fired at equal intervals of 5 min (see Fig. 7), it was found that the divergence of the pump pulse strongly changed during the first few shots but stabilized after a warmup period of -5 shots. For this measurement the beam divergence was determined from the beam impact on burn paper several meters behind a Hartmann plate. Using the inverse telescope the output beam of the warm chain was then adjusted for zero divergence. All final measurements were made with the warm laser chain fired at 5 min intervals.

B. Measurements Fig. 8 shows the signals o f the fast photodiode for 3 shots.

Fig. 8(a) shows a calibration shot with empty cell and color filters removed. The pump pulse is recorded three times in succession; in this way the time relation between the three channels is established (see arrows). The modulation of the pump pulse due to longitudinal mode beating in the oscillator (mode distance 0.003 cm-') is useful in this shot because it gives sharp time marks.

Fig. 8(b) shows a shot under normal conditions. One sees the sharp onset of the Stokes pulse (generated at the spatial center of the beams) and depletion of the transmitted pump pulse, The Stokes photons at the leading edge of the Stokes pulse copropagate with pump photons whose intensity at the entrance of the cell is readily found from the time calibration established in Fig. 8(a) [the leading edge of the Stokes pulse and the copropagating packet of pump photons are connected by the arrow in Fig. 8(b); its length comes from 8(a)] . This intensity obviously corresponds to the threshold pump intensity Igin necessary for first Stokes generation in our Raman cell. It is readily determined from the oscillogram in terms of f o 0

and hence absolutely (see discussion on I , above). In the particular shot shown in Fig. 8(b) a f = - 2 m lens was used; the measured energies of incident pump, Stokes, and transmitted pump pulses were 0.98, 0.17, and 0.71 J, respectively.

Fig. 8(c) demonstrates again the sharp onset of Stokes generation. Only those spikes of this strongly modulated pulse whose intensity is about a well-defined value (dash-dotted line) generate first Stokes spikes.

IV. COMPARISON OF THE EXPERIMENTAL RESULTS TO THEORY

In Section I1 we defined a generation time tgl ( Ioo , f) for first Stokes generation. Obviously, first Stokes generation in a


0 *-c 0 10 20 30

t [ m ~ n ]

Fig. 7. Change of the angular divergence of the pump beam during the warmup period. The laser is fired .at intervals of 5 min. After -4 shots the beam has become parallel (a = 0).

INCI~ENT FIRST T R ~ N S M I T T E O PUMP STOKES PUMP

Fig. 8. Incident pump pulse, forward first Stokes pulse, and transmitted pump pulse as recorded by the fast photodiode. (a) Calibration shot (empty Raman cell, color filters removed) to establish the temporal relation between pulses (note horizontal arrows). (b) Typical shot (pump pulse 0.98 J, Stokes pulse 0.17 J, transmitted pump pulse 0.71 J) showing the sudden onset of the Stokes pulse and partial depletion of the transmitted pump pulse. (c) With a strongly modulated pump pulse only spikes exceeding the threshold (dashed line) appear in the Stokes pulse.

cell of length L can only occur if tgl 5 L/co. We call I$" the pump intensity where Stokes generation just sets in at the end of the cell. From (8) we then obtain a relation between f, I$h, and L

(9)

This relation can be compared directly to the experiments where I,$" was measured as a function o f f for constant L . For all experiments we had (CY./ < 5 X 1 0-3 to satisfy the condition /a[ << 1 used in the theory.

In Fig. 9 I$ is plotted (dash-dotted lines) as a function of f for L = 2'5 m. The curves are limiting curves between areas where first Stokes generation is either possible (dotted areas) or impossible (white areas). In evaluating (9) we took I lo = 10-3 W . m-2 and adjusted g , = g o w l ioo to the experimental results, as we discuss below. It should be mentioned that all results are very insensitive against variation in I l o . If I l o is changed by one order of magnitude, all results change only by a relative amount of 6 percent.

The experimental results for 50 shots are plotted in Fig. 9.

- DIVERGENT+CONVERGENT-

Fig. 9. ExpeFimental results: dependence of the threshold pump intensity I$" on the focal length of the lens. Each experimental point is an average over about 6 shots. The dashdotted lines represent the theoretictilly expected dependence according to relation (9). The length of the Raman cell is 2.5 m.

Seven different lenses between - 10 m < f < 10 m were used; in addition, shots with a parallel beam (no lens) were also made. Each experimental point is an average of about -6 shots and for all 50 shots the pump pulse had Gaussian shape in time and space.

It is evident in Fig. 9 that the experimental points are very much in keeping with theoretical expectations. We conclude therefore that first Stokes generation by forward Raman scattering as a function of the divergence or convergence of the pump beam can be accurately predicted on the basis of the rate equations.

Finally, we note that it is possible to obtain a correct value for the gain coefficient y from our measurements, showing the consistency of the evaluation. To this end we plotted (9) in Fig. 10 with the approximationgoIlo <<glI$" in the form

We can fit a straight line through the experimental points and obtain from its slope y L = (4.2 rf: 0.7) X 10-l' m2 . W-' . Again, we assumed I l o = W . m-'. With L = 2.5 m and systematic errors one obtains y = (1.7 k 0.6) X lo-" m W-' where the error margins are mainly determined by the systematic error in the determination of the pump beam intensity. Within the error bars this value of y agrees with [ 181 .

V. SUMMARY In this paper we have investigated the onset of stimulated

Raman forward scattering as a function of the divergence or convergence of the pump beam. We have derived analytical solutions of the rate equations which allow calculation of the distance (or generation time) which a packet of pump photons of given intensity and divergence will travel in a Raman cell before pump depletion occurs. The theoretical results were compared with experiments where the onset of forward scat-


f L i f RAMAN CELl

-2 c '\ \ \'\ \

Fig. 10. All -50 experimental points in a representation (sce text) which allows determination of the gain coefficient y from the slope of the measured curve.

tering in a H2 Raman cell was measured for a frequency-doubled pulsed Nd-YAG laser beam of variable divergence. The experiment fully confirmed the calculations. As a result of this investigation it seems possible to predict with confidence the influence of the divergence or convergence of the pump beam on the onset of forward Raman scattering and to suppress it by the proper choice of the beam geometry. A possible application of these results is in the design of Raman pulse compressors.

APPENDIX A RAMAN PULSE COMPRESSOR WITH NONI 'LANAK

BEAM GEOMETKY One of the limitations of a backward Raman pulse com-

pressor is due to parasitic forward Stokes generation [3] , [4] , Forward Stokes generation may lead to energy losses due to premature pump depletion; also, the backward propagating Stokes pulse may decay into higher order Stokes pulses by forward scattering. One is interested in minimizing the problem by a proper choice of the beam geometry in the cell (a conical amplifier has been mentioned in [4] ). As an example we discuss here a Raman compressor and the associated parasitic Stokes problem for a particular nonplanar beam geometry which a p pears advantageous in several respects. However, a general discussion of the optimization of Raman compressors with respect to beam geometry is not intended here.

The scheme is shown in Fig. 1 1. A pump pulse and a Stokes pulse are injected from opposite sides into a Raman cell where the photons counterpropagate along the same trajectories. The beam cross section in the cell is not constant but varies continuously in a prescribed manner; experimentally this variation could be approximated by using a series of lenses in the cell. The variaticn of the beam cross section along the cell is determined by the postulate that the energy density of the Stokes pulse at the entrance of the cell as well as throughout the cell be equal to the saturation energy density. Because a Stokes pulse with the saturation energy density will saturate the pump as it travels along the cell, the increase in pulse energy must be campensated by an increase in beam cross section in order to maintain the Stokes pulse at saturation energy density. Ob- viously, for a constant power fully saturated pump, the linear increase of the energy of the Stokes pulse with distance must be compensated by a linear increase of the beam area, i.e., an

Pig. 11. Scheme of a backward Raman pulse compressor with a convergent pump pulse.

increase of the beam radius with the square root of the distance. The advantages gained by the variation of the beam cross

section are a high conversion efficiency (due to the continuous saturation of the pump throughout the cell) with a low Stokes pulse energy (due to the small beam cross section at the entrance of the Stokes pulse into the cell). At the same time the gain for second Stokes generation is kept minimal (for a case with pump depletion throughout the cell) and, as we will show by an example, parasitic Stokes conversion can be avoided for cases of interest.

In the following we outline briefly the design of such a compressor. We introduce the saturation energy density es and saturation intensity 1s which are connected by

= 2p/go = 1; T2 (AI?

where p is the gain ratio between forward and backward scattering and T2 is the relaxation time for back-scattering.

In our design we have fixed the backward Stokes intensity to the saturation intensity throughout the cell. In order to avoid decay of the backward first Stokes pulse into a second Stokes pulse, the condition

G2 =g,1;z,/c, 5 35 ( 4

must be fulfilled. This limits the maximum cell length to

where (All has been used. With the cell length also the maximum pump pulse length T~.+, = 2L,,,/c and the maximum compression ratio Q = 7,,,/T2 are fixed. For a given laser energy E,, the beam radius Y,,, in the plane where the com- pressed pulse exits the cell is also fixed because the energy density eo of the laser pulse with energy E , must match the energy density of the Stokes pulse:

From this relation we have with (AI) and g1 = wlgo/oo

All these quantities are determined by the Raman medium. It is a question of a suitable Raman medium, therefore, whether given conditions, Le., laser frequency, energy, pulse duration, and desired compression ratio can be met in a compact (small v,,,) compressor.

So far we have not exactly fixed the beam geometry in the cell. According to our assumptions, the energy of a fully sat-


urated Stokes pulse increases in the cell according to

where T~ is the pump pulse duration. Furthermore we have El ( t) = nr(t)2ef and El , = 7rr&&. With these last two relations we obtain from (A7)

7 and rmin are interconnected by

(A9)

We thus still have one open parameter, either 77 or r,h. If we fix one of these parameters, the required Stokes input energy is also fixed by E,, = nrkh&S.

En principle, one would like to make rmh and hence EIo very small. On the other hand, with decreasing rmh the gain for first Stokes amplification by the pump pulse in forward direction increases, leading eventually to premature pump depletion. The condition for the maximum permissible gain of this process fixes the remaining parameter. For a pump pulse of the form Z o ( t ) = Z00r&ax/r2(t) (i.e., neglecting pump depletion) this condition is

= p In (1 t qL/co ) 5 35 (A1 0)

where we have used (A9). This condition can also be written in the form

77 ,< 2 (exp ( 3 5 / p } - 1). (A1 1)

With the restriction for 7, rmh and E,, are also fixed. As an example we consider a pulse compressor for pump

radiation with a wavelength of 1.064 pm, operating at a H2 gas molecule number density of 8 X m-3 (corresponding to 30 atm at room temperature). After frequency correction following [l J we take for the gain factors go = 3.0 X m2 . J-’ , g , = 1.7 X low3 m2 . J-’ , g, = 3.5 X m2 . J-’ . AC- cording to [ 181 and [ 191 , the forward/backward gain ratio is p = 1.5 and the relaxation time T2 = 0.1 6 ns. The saturation energy density then becomes ef = 1000 J . m-’ and the saturation intensity If = 6.25 X 10l2 W . m-’. From (A3) we find for the maximum cell length LmaX = 4.8 m, for the maximum pulse length I-,,, = 32 ns, and for the maximum compression ratio Q = 200. For a laser pulse energy of 5 J we find rmax = 3.0 X IO-’ m. The condition for 77 becomes77 < 8.5 X s-’. The very small valdes allowed by this condition for rmin and E,, are hardly practical because, from (A8), the convergence angle of the beams tan CY = rmh77/2c0 at the entrance of the Stokes pulse would become exceedingly large. If we choose 77 = 5 X 10” s-’ we obtain tan 01 = 9 X rmh = 1.1 X

m, and E,, = 3.8 X J , values which are still quite practical.

ACKNOWLEDGMENT This paper is an extract from the Diploma Thesis of H. Gruhl,

submitted in January 1981 to Ludwig-Maximilians-Universitat, Munchen. He would like to thank Prof. Dr. H. Walther for his interest in and for his promotion of this work at the University, and also Dr. S . Witkowski for his interest and for affording the possibility to perform this work in his division at MPQ Garching.

Both authors would like to thank P. Sachsenmaier, E. Wanka, and C. Kraft for their helpful technical assistance during the experiments and Dr. K. Witte for a critical reading of the manuscript and discussions.

REFERENCES W. Kaiser and M. Maier, “Stimulated Rayleigh, Brillouin and Raman spectroscopy,” in Laser Handbook,vol. E-2, F. T. Arecchi, Ed. Amsterdam, The Netherlands: North-Holland, 1972, pp.

M. Maier, W. Kaiser, and J. A. Giordmaine, “Intense light bursts in the stimulated Raman effect.” Phys. Rev. Lett., vol. 17, pp.

1077-1150.

. . 1275-1277,1966. A. J. Glass, “Design considerations for Raman lasers.”IEEE J. Quantum Electron:, vol. QE-3, pp. 516-520, 1967. J . R. Murray, J. Goldhar, D. Eimcrl, and A. Szoke, “Raman pulse compression of excimer lasers for application to laser fusion,” IEEEJ. Quantum EkCtTOFl., vol. QE-15, pp. 342-368, 1979. W . R . Trutna and R. L. Byer, “Multiple-pass Raman gain cell,”

W. H. Culver and E. J. Seppi, “Characteristics of an ideal Raman oscillator-amplifier,” J. Appl. Phys., vol. 35, pp. 3421-3422, 1964. H. Gruhl, “Stimulierte Ramanstreuung bei divergenter oder konvcrgentcr Pumpstrahlung,” Diploma thesis, Ludwig-Maximil- ians-Universitat, Munchen, West Germany, Jan. 1981. D. von der Linde, M. Maier, and W. Kaiser, “Quantitative investi- gations of the stimulated Raman effect using subnanosecond light pulses,”Phys. Rev., vol. 178, pp. 11-17, 1969. G. K. Boyd, W. D. Johnston, and I. P. Kaminow, “Optimization of the stimulated Raman scattering threshold,” IEEE J. Quantum Electron., vol. QE-5, pp. 203-206, 1969. A. A. Betin and G. A. Pasmanik, “Stimulated scattering of focussed light beams,” Sov. J . Quantum Electron., vol. 3, pp. 312-316, 1974. D. Cotter, D. C. Hanna,and R. Wyatt, “Infrared stimulated Raman generation: Effects of gain focussing on threshold and tuning behaviour,”Appl. Phys., vol. 8, pp. 333-340, 1975. A. A. Betin, G. A. Pasmanik, and L. V. Piskunova, “Stimulated Ralnan scattering of light beams under saturation conditions,” Sov. J. Quantum Electron.,vol. 5,pp. 1309-1313,1976. N. P. Andreev, V. I . Bespalov, A. M. Kiselev, and G. A. Pasmanik, “Experimental investigation of the spatial structure of the first Stokes component of stimulated Raman scattering,” Sov. J. Quantum Electron., vol. 9, pp. 585-589, 1979. R. T. V. Kunn, “Multiple pass stimulated Raman conversion with

Appl. Opt., V O ~ . 19 ,pP. 301-312, 1980.

pump depletibn,” IEEE J . Quantum Electron., vol. QE-17, pp. 509-513,1981. -, “Higher order mode Raman gain coefficients,” IEEE J. Quantum Electi‘on.,vol. QE-18,pp. 1323-1325, 1982. B. Y. Zel’dovich and V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Sov. J. Quantum Electron., vol. 7,pp. 610-615. 1977. R.-T. V. Kung’and J . H. Hammond, “Phase front reproduction in Raman conversion,” IEEE J. Quantum Electron., vol. QE-18,

N. Bloembergen, G . Bret, P. Lallemand, A. Pine, and P. Simova, “Controlled stimulated Raman amplification and oscillation in hydrogengas,” IEEE J. Quantum Electron., vol. QE-3, pp.

P. Lallemand, P. Simova, and G. Bret, “Pressure-induced line shift and collisional narrowing in hydrogen gas determined by stimulated Raman emission,”Phys. Rev. Lett., vol. 17, pp. 1239- 1241,1966.

pp. 1306-1310,1982.

197-201,1967.

1074 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-20, NO. 9, SEPTEMBER 1984

Continuously Tunable Sum-Frequency Generation Involving Sodium Rydberg States

ROBERT W. BOYD, DANIEL J. GAUTHIER, JERZY KRASINSKI, AND MICHELLE S. MALCUIT

Abstract-Broadly tunable sum-frequency generation has been observed in a vapor of atomic sodium in the presence of a dc electric field. This field induces a x(’) nonlinearity which is resonantly enhanced when the sum frequency corresponds to the energy separation between the ground state and an atomic Rydberg state. In a vapor of number density 4 X l O I 4 ~ m - ~ , we obtain an energy conversion efficiency as large as 3 X lo4 and a x ( 2 ) as large as 1.2 X ESU. We have also observed sum-frequency generation in the absence of an applied dc field, and we relate these observations to mechanisms that have been proposed to explain this effect.

T INTRODUCTION

HERE is currentIy great interest in sum-frequency generation as a means of producing tunable ultraviolet radiation

[l] -[ 101 . Most of the mixing processes that have been dem- onstrated thus far make use of a third-order or optical nonlinearity, in which case three input fields combine to pro- duce the output frequency. To avoid the inconvenience of using three separate laser sources, such mixing experiments usually derive at least two of the input fields from a single laser. In such cases, at least one of the intermediate steps of the mixing process is usually nonresonant, which limits the efficiency with which the ultraviolet frequency can be produced.

In this paper, we describe a nonlinear mixing process utiliz- ing two input lasers in which each intermediate step is resonantly enhanced [ 1 11 , [ 121 . As shown in Fig. 1, an incident laser field at frequency w connects the sodium 32S1/2 ground

Manuscript received January 16, 1984; revised April 24, 1984. This work was supported by the U.S. Army Research Office and the Air Force Office of Scientific Research.

R. W. Boyd, D. J. Gauthier, and M. S. Malcuit are with the Institute of Optics, University of Rochester, Rochester, NY 14627.

J. Krasinski is with Allied Corporation, Mt. Bethel, NJ 07060.

Fig. 1. Optical waves at frequencies w1, w2, and w g = w1 t- w2 interact by means of the nonlinear response of an atom with eaergy eigenstates li), l i ) , and Ik). In the experimental work, these states correspond to the 3 2S1p, 3 2 P 3 p , and a Rydberglevel, respectively.

state to the 3 2 P 3 / 2 excited state while a second laser field at frequency o2 completes a two-photon resonance with a sodium Rydberg level. In addition, a dc electric field is applied to the atomic system. This field breaks the inversion symme- try of the sodium atom, and thus allows the existence of an electric dipole moment oscdlating at the sum frequency w3 = o + w2 [ 131 . The resulting nonlinear polarization is related to the applied field strengths E(w,) and E ( w z ) through the relation

P(W3) = 2X(2’(W3 = 0 1 -I- W 2 ) E(U1) E(w2) (1)

where the nonlinear susceptibility describing the mixing process is given by [ 1 1 ]

x q w 3 = w1 + 0 2 )

0018-9197/84/0900-1074$01.00 0 1984 IEEE

ieee 1984 1065 stimulated raman forward scattering with a ... · pulse compressor with nonplanar...

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