effective method of anti-stokes generation by quasi-phase-matched stimulated raman scattering

N. S. Makarov and V. G. Bespalov Vol. 22, No. 4 /April 2005 /J. Opt. Soc. Am. B 835

Effective method of anti-Stokes generation byquasi-phase-matched stimulated Raman scattering

Nikolai S. Makarov

Saint-Petersburg State University of Informational Technologies, Mechanics and Optics, Sablinskaya str., 14,Saint-Petersburg 197101, Russia

Victor G. Bespalov

Russian Research Center S.I. Vavilov State Optical Institute, Birzhevaya 12, Saint-Petersburg 199034, Russia

Received March 24, 2004; revised manuscript received November 6, 2004; accepted November 9, 2004

We have proposed a new method of enhancing the efficiency of pulsed anti-Stokes stimulated Raman genera-tion by using quasi-phase matching in a stack of alternating layers with different Raman properties. We havederived a system of coupled differential equations describing a multifrequency stimulated Raman process withinteracting waves propagating in both the forward and the backward direction. We show that the efficiency ofanti-Stokes generation can reach 20–30%. We also show that an optimized layered structure can perform fora broad range of pump parameters, such as pulse duration, intensity, and Fresnel number. © 2005 OpticalSociety of America

OCIS codes: 190.2620, 190.4360, 290.5910.

1. INTRODUCTIONStimulated Raman scattering (SRS) is widely used fordiscrete frequency conversion of pulsed and continuous-wave lasers.1–5 As a rule, SRS gives rise to a series oflines shifted both to the Stokes and to the anti-Stokesspectral regions, with respect to the frequency of thepump radiation. It has been shown that the highest con-version efficiency of the pump into the first-order Stokescomponent for focused Gaussian beams in a homogeneousRaman medium can reach the limit set by the Manley–Rowe relation.4 On the other hand, the conversion effi-ciency into the first-order anti-Stokes component atsimple focusing into the medium does not exceed 3–6%.6,7

To increase this efficiency, the authors of Ref. 8 used com-plex noncollinear four-wave phase-matching systems.However, in these systems experimental anti-Stokes con-version efficiency does not exceed 15%.

Quasi-phase matching (QPM) was first proposed in1962 by Armstrong et al. for efficient second-harmonicgeneration9,10 in a medium with a periodic variation ofthe second-order nonlinear susceptibility @x (2)# along thepropagation coordinate. Today QPM is widely used forx (2)-based frequency conversion,11 including harmonicgeneration, sum-frequency generation, and parametricgeneration in x (2).

Recently Gibson and co-authors12 used a hollow wave-guide with a periodically modulated diameter for en-hanced generation of coherent x rays, which was a QPM-like technique.

So far, only a few studies of the possibility of QPM in amedium with alternation of the third-order susceptibility@x (3)# along a longitudinal coordinate have been per-formed. The study of anti-Stokes SRS QPM generationis topical from the point of view of both the developmentof the conception of physics of the SRS processes and the

0740-3224/2005/040835-09$15.00 ©

creation of frequency-discrete, tuned, high-coherencepulse sources.

In a previous paper13 we proposed to use the QPM-liketechnique to increase the efficiency of anti-Stokes SRSgeneration and showed some peculiarities of this process.We showed that, by optimizing the lengths of Raman-active and Raman-passive layers, it is possible to createconditions for efficient anti-Stokes generation in eachRaman-active layer that allows for an increase of anti-Stokes generation efficiency up to 30%. In this paper wediscuss a novel method of more accurate modeling of theanti-Stokes SRS generation in the conditions of QPM andpay more attention to its adequacy.

In Section 2 we present a theory of backward and for-ward multifrequency SRS and its simulations. In Sec-tion 3 we present the results of simulations of differentregimes of QPM SRS.

2. THEORYA. Derivation of Master Equations for SimultaneousBackward and Forward Multifrequency StimulatedRaman ScatteringLet us consider a scalar wave equation:

S ]2

]z21

]2

]x21

]2

]y22

1

c2

]2

]t2D «~z, x, y, t ! 2 m0

]2PLN

]t2

5 m0

]2PNL

]t2, (1)

where the paraxial solution is sought in the form of thesum of interacting linearly polarized waves:

2005 Optical Society of America

836 J. Opt. Soc. Am. B/Vol. 22, No. 4 /April 2005 N. S. Makarov and V. G. Bespalov

«~z, x, y, t ! 51

2 H(j

Ej1 exp@i~v jt 2 kj

1z !#

1 (j

Ej2 exp@i~v jt 2 kj

2z !#J 1 c.c.

(2)

Here Ej1 and Ej

2 are the slowly varying amplitudes offorward and backward SRS components, relative to thepump that is propagating in the positive direction of the zaxis; kj

6 are the forward and backward propagation con-stants; x, y are the coordinates transverse to propagationdirection z; PLN and PNL are linear and nonlinear on-fieldcomponents of media polarization; « is the electric field ofthe light wave; c is the speed of light; and m0 is the per-meability of vacuum:

kj2 5 2kj

1 5 2v jn~v j!

c, (3)

where the negative values of j correspond to the StokesSRS components, the zero value corresponds to the pumpwave, the positive values correspond to the anti-StokesSRS components (v j 5 v0 1 jvn), n(v j) is the refractiveindex, and vn is the Raman transition frequency.

By assumption, with the linear polarization as

PLN 51

2 (j

PLN1j exp@i~v jt 2 kj

1z !#

11

2 (j

PLN2j exp@i~v jt 2 kj

2z !# 1 c.c., (4)

using the expression

1

c2Ej

6 1 m0PLN6j 5

n2~v j!

c2Ej

6, (5)

and substituting the derivatives of the electric field intoEq. (1), we obtain

S ]2

]z21

]2

]x21

]2

]y22

1

c2

]2

]t2D «~z, x, y, t ! 2 m0

]2PLN

]t2

' c.c. 1 (j

exp@i~v jt 2 kj1z !#

3 H 2ikj1F ]

]z1

n~v j!

c

]

]tG1

1

2 S ]2

]x21

]2

]y2D J Ej1

1 (j

exp@i~v jt 2 kj2z !#H 2ikj

2F ]

]z2

n~v j!

c

]

]tG1

1

2 S ]2

]x21

]2

]y2D J Ej2. (6)

The nonlinear polarization response of dielectric mediawith the assumption of zero-level quantum noise can bedescribed by a system of material equations14:

PNL 5 «0x~3 !Q«;]2Q

]t21

2

T2

]Q

]t1 vn

2Q 5 gn«2,

(7)

where Q is the phonon wave describing molecular oscilla-tions, T2 is the dephasing time, x (3) is the third-ordernonlinear susceptibility, and gn characterizes the disper-sion of the nonlinear polarization response.

By neglecting all nonlinear effects, except the resonantinteraction of third-order nonlinearity x (3), we obtain

PNLRES 5

1

2 (j

PjNR1 1

1

2 (j

PjNR2 1 c.c., (8)

where PjNR6 are the forward and backward components

of resonant nonlinear polarization.For a phonon wave, using the expression

Q 51

2q exp@i~vnt 2 knz !# 1 c.c., (9)

where q is the slowly varying amplitude of the phononwave and kn is the phonon wave number,

kn 5 k01 2 k21

1, (10)

we can recast the expressions for nonlinear resonant po-larization in the form

PjNR1 5

«0x~3 !

2@Ej11

1q* exp~izD j111!

1 Ej211q exp~2izD j

1!#exp@i~v jt 2 kj1z !#

1«0x~3 !

2@Ej11

2q* exp~izD j112!

1 Ej212q exp~2izD j

3!#exp@i~v jt 2 kj1z !#,

PjNR2 5

«0x~3 !

2@Ej11

1q* exp~izD j113!

1 Ej211q exp~2izD j

2!#exp@i~v jt 2 kj2z !#

1«0x~3 !

2@Ej11

2q* exp~izD j114!

1 Ej212q exp~2izD j

4!#exp@i~v jt 2 kj2z !#,

(11)

where D ji are wave mismatchings:

H D j111 5 kj11

1 2 kj1 2 kn D j11

2 5 kj112 2 kj

1 2 kn

D j113 5 kj11

1 2 kj2 2 kn D j11

4 5 kj112 2 kj

2 2 kn

D11 5 D. (12)

Recasting the Eqs. (7) for a phonon wave, we obtain

]q

]t S 1 11

ivnT2D exp@i~vnt 2 knz !# 1

q

T2

3 exp@i~vnt 2 knz !# 5gn«2

ivn

. (13)

Finally by, substituting Eqs. (13), (11), and (6) into Eq.(1) and neglecting nonresonant interaction, we arrive at a


system of coupled differential equations that describe theevolution of complex amplitudes of interacting waves:

F ]

]z1

n~v j!

c

]

]t1

i

2kj1 S ]2

]x21

]2

]y2D GEj1

5kj

1x~3 !

4n~v j!2i

Ej111 q* exp~izD j11

1!

1kj

1x~3 !

4n~v j!2i

@Ej112q* exp~izD j11

2 !

1 Ej212 q exp~2izD j

3! 1 Ej211 q exp~2izD j

1!]

F2]

]z1

n~v j!

c

]

]t1

i

2kj1 S ]2

]x21

]2

]y2D GEj2

5kj

1x~3 !

4n~v j!2i

Ej112 q* exp~izD j11

4 !

1kj

1x~3 !

4n~v j!2j

@Ej111 q* exp~izD j11

3 !


2! 1 Ej212 q exp~2izD j

4!],

]q

]t S 1 11

ivnT2D 5 2

q

T21

gn

4ivn(

jEj

1Ej211* exp~izD j

1!

1gn

4ivnF(

jEj

2Ej211* exp~izD j

2!

1 (j

Ej1Ej21

2* exp~izD j3!

1 (j

Ej2Ej21

2* exp~izD j4!G . (14)

By normalizing the amplitudes of interacting wavesand the phonon wave and using the expression for thesteady-state Raman gain coefficient g 5 g0

1,

g01 5

v21x~3 !T2gn

4n2«0c2vn

; E ~new! 5 S «0cn

2 D 1/2

E,

q ~new! 5 q ~old!2vn«0cn

T2gn

, (15)

we can rewrite the system of Eqs. (14) in the form

F ]

]z1

n~v j!

c

]

]t1

i

2kj1 S ]2

]x21

]2

]y2D GEj1

5gj

1v j

2v21i@Ej11

1 q* exp~izD j111 ! 1 Ej21

1 q exp~2izD j1!#

1gj

1v j

2v21i@Ej11

2 q* exp~izD j112!


3!#

F2]

]z1

n~v j!

c

]

]t1

i

2kj1 S ]2

]x21

]2

]y2D GEj2

5gj

2v j

2v21i@Ej11

2 q* exp~izD j114 ! 1 Ej21

2 q exp~2izD j4!#

1gj

2v j

2v21i@Ej11

1 q* exp~izD j113 !

1 Ej211 q exp~izD j

2!#,

]q

]t5

1

iT2F2iq 1 (

jEj

1Ej211* exp~izD j

1!

1 (j

Ej2Ej21

2* exp~izD j4!

1 (j

Ej2Ej21

1* exp~izD j2!

1 (j

Ej1Ej21

2* exp~izD j3!G . (16)

This system describes the generation of an arbitrarynumber of forward and backward SRS components. Toensure that this system adequately models the SRS pro-cesses, we compared the results of simulations with ex-perimental data15,16 and received quantity and qualityagreement between them. However, our simulationsshowed that high-intensity pulses require simulations ofthe full system, whereas for our purposes of low intenseinteraction we can use the plane-wave model.

The generalized system of Eqs. (16) can be reduced towell-known SRS models.8,17,18 To reduce Eqs. (16), forexample, to the simplest model, describing backward andforward SRS,18 let us assume that Ej [ 0 ( j . 0, j, 21), gj

6 5 g, q 5 q1 1 q2 exp(2iz2k211) and that

items with fast oscillating exponents are small enough toneglect them:

F ]

]z1

n~v0!

c

]

]tGE0 5ig

2

v0

v21~q1E21

1 1 q2E212!,

F ]

]z1

n~v21!

c

]

]tGE211 5

ig

2q1* E0 ,

F2]

]z1

n~v21!

c

]

]tGE212 5

ig

2q2* E0 ,

]q6

]t5

1

T2~2q6 1 iE21

6* E0!.

(17)

B. Methods of SimulationsTo simulate the simple models,8,17 we used the well-known finite-difference methods; however, they cannot beused to simulate the model of Eqs. (16) and the plane-wave case because half of the initial conditions is given atthe medium input whereas the other half is given at themedium output. To simulate forward and backward in-teraction, partially implicit methods were used,19 but it


became too intricate for simulations of interaction of anarbitrary number of waves. So we had to construct thisexplicit scheme for our simulations:

q~t 1 Dt, z ! 5Dt

iT2F2iq~t, z !

1 (j

Ej1~t, z !Ej21

1* ~t, z !exp~izD j1!

1 (j

Ej2~t, z !Ej21

2* ~t, z !exp~izD j4!G

1Dt

iT2F(

jEj

2~t, z !Ej211* ~t, z !

3 exp~izD j2! 1 (

jEj

1~t, z !

3 Ej212* ~t, z !exp~izD j

3!G 1 q~t, z !,

Ej6~t 1 Dt, z ! 5 C2

jEj6~t, z ! 1 C3

jgj

6v j

2v21iq* ~t, z !

3 @Ej112 ~t, z !exp~izD j11

2,4 !

1 Ej111 ~t, z !exp~izD j11

1,3 !#

1 C3jgj

6v j

2v21iq~t, z !

3 @Ej211 ~t, z !exp~2izD j

1,2!

1 Ej212 ~t, z !exp~2izD j

3,4!#

1 C1jEj

6~t 1 Dt, z 7 Dz !, (18)

where

C1j 5

cDt

cDt 1 n~v j!Dz, C2

j 5n~v j!Dz

cDt 1 n~v j!Dz,

C3j 5

cDtDz

cDt 1 n~v j!Dz. (19)

For calculating the steady-state Raman gain coefficientat different wavelengths we used

gj6 5

A6

l jS B6 21

l j112 D 2 , j , 0;

g06 5 g1

6 5 g216; gj

6 5A6

l jS B6 21

l j212 D 2 ,

j . 1; g01 5 g, (20)

where, for hydrogen, A1 5 A2 5 1.41 3 1016 GW21/cm22 and B1 5 B2 5 7.19 3 109 l/cm2 (Ref. 20). Forbarium nitrate, A1 5 1.04 3 1015 GW21/cm22, A2

5 0 GW21/cm22, and B1 5 B2 5 9.82 3 108 l/cm2.21

One possibility to create Raman and non-Raman alter-nation of layers in gaseous media is to use the opticalStark effect in a direct electric field. The layers with anelectric field sufficient for frequency shift from Ramanresonance will be the passive layers with no amplificationof Stokes radiation.22

To realize an alternation of active and passive layers ina crystalline Raman-active medium, one has to use morecomplex methods, such as electron-beam lithography, dryetching techniques,23 or other methods to createquantum-size structures.

C. Optimization of Layer LengthsBecause we cannot use periodically changing birefrin-gence in x (3) media to alter the sign of nonlinear polariza-tion x (3), we propose to use a layered media with alter-nating Raman-active @x (3) Þ 0# and Raman-passive @x (3)

5 0# layers along the propagation coordinate z. Let usconsider the propagation of light through the firstRaman-active layer. Along a small length, although thephase mismatching is small, we have some generation ofanti-Stokes radiation (see zoom area Fig. 1); and the con-dition of maximum anti-Stokes efficiency at the output ofthe first layer allows us to calculate its length. Then weuse the Raman-passive layer to compensate phase mis-matching by the dispersion of group velocities of interact-ing waves and allow again an efficient anti-Stokes gen-eration in the next Raman-active layer. And so we havetwo layers (Raman active and passive) in which the effi-ciency of anti-Stokes generation does not decrease (anti-Stokes component growth only), and we can now considera new pair of layers as the first one. This procedure canbe repeated until the energy of the pump wave is depleted(our calculations show that 40–60 pairs of layers are usu-ally enough). Using this procedure we can achieve anti-Stokes generation efficiency of '1–2% in each layer, so forthe whole media the efficiency will be '30%. Figure 1shows an example of the choice of lengths of the fifthRaman-active and Raman-passive layers. The graycurve demonstrates the efficiency of anti-Stokes genera-tion in the Raman-active medium when if we do not usepassive layers to compensate for the wave mismatching(e.g., if we consider a long Raman-active layer without apassive one). It can be seen that there is a maximum ofanti-Stokes efficiency in this Raman-active layer [Fig.1(b)]. And if we choose the length of our fifth Raman-active layer corresponding to this maximum @La(opt)],then we can start with the fifth Raman-passive layer tocompensate wave mismatching that arises in the fifthRaman-active layer. The length of the fifth Raman-passive layer is selected to receive an increase of anti-Stokes efficiency in the beginning of next Raman-activelayer [Fig. 1(a)].

3. SIMULATION RESULTSIn our simulations we optimize the lengths of the alter-nating Raman-active and Raman-passive layers to maxi-mize the efficiency of anti-Stokes generation in eachRaman-active layer (Fig. 1). The accuracy of our simula-tions was '0.3%.


The pump wavelength was 532 nm (the second-harmonic Nd:YAG laser), and the pump intensity changedin the range of 10 MW/cm2 up to 10 GW/cm2. The mosteffective Raman-active media of compressed hydrogen20

and a crystal of barium nitrate @Ba(NO3)2] (Ref. 21) wereused as a model for the calculations. To realize an effec-tive anti-Stokes QPM SRS generation, a relatively strong

Fig. 1. Illustration of optimal layer lengths for maximization ofanti-Stokes generation efficiency. The example here is for thefifth active and passive layers in hydrogen. The black curve cor-responds to anti-Stokes efficiency at the beginning of the sixthactive layer, and the gray curve corresponds to anti-Stokes effi-ciency at the end of the fifth active layer (without the fifth pas-sive layer). (a) Zoomed changes of efficiency after the passivelayer and (b) efficiency in the active and passive layers.

Fig. 2. Comparison of anti-Stokes intensity growth for a QPMcondition, phase matching, and simple focusing in compressedhydrogen (white layers, active; gray layers, passive) 1, QPM (D5 3.84 rad/cm); 2, simple focusing in compressed hydrogen (D5 3.84 rad/cm); 3, phase matching (D 5 0.025 rad/cm).

Stokes seed pulse is required, and that is why we can ne-glect the quantum noise in our SRS system.

Our comparison of anti-Stokes intensity growth for con-ditions of QPM, phase matching (D 5 0.025 rad/cm), andsimple focusing in compressed hydrogen (Fig. 2) showedthat for phase matching the resulting intensity is higherthan for QPM. However, the full phase matching in hy-drogen at a pressure of 30 atm is impossible as the Stokesand pump beams must cross at an angle ;1° with an ac-curacy of 8 mrad (Ref. 8) and must interleave through thewhole Raman medium, which cannot be technicallyachieved. These simulations of the simplest anti-StokesSRS model showed that the realization of quasi-phase-matched conditions is possible both in compressed hydro-gen and in barium nitrate.

For compressed hydrogen ( g 5 4.42 cm/GW, D5 3.84 rad/cm, p 5 30 atm) and barium nitrate ( g5 47.42 cm/GW, D 5 77.84 rad/cm), we simulated themultifrequency SRS taking and not taking into accountthe dispersion of the steady-state Raman gain coefficient(Fig. 3). From our simulations and the curves in Fig. 3, itcan be seen that the difference between second Stokesgeneration efficiencies with and without the dispersion ofthe steady-state Raman gain coefficient is 30% for hydro-gen and 15% for barium nitrate. And since in realRaman-active media this dispersion is present, we took itinto account in all our following simulations to increasetheir accuracy. Furthermore, our simulations showedthat, to increase its accuracy, it is necessary to take intoaccount the generation of at least four to five Stokes andanti-Stokes waves.

The simulations of the quasi-phase-matched anti-Stokes generation showed that, by an optimal choice ofthe lengths of the active and passive layers, it is possibleto create a layered structure, providing the efficiency ofanti-Stokes generation of more than 20%. The depen-dencies of layer lengths for this structure are shown inFig. 4. Our calculations show that, in comparison with aunidirectional assumption, taking into account backwardSRS generation results in a slight decrease of the activelayer lengths and a slight increase of the passive layerlengths at the beginning of the QPM structure and aslight increase of the active layer lengths and a slight de-crease of the passive layer lengths at its end. The effi-ciency of the anti-Stokes conversion lowers from 30% to25%, and the efficiency of backward SRS is a negligiblevalue. The study of the dependence of the anti-Stokesgeneration efficiency on the accuracy of the choice of thelayer lengths showed that the changes of layer lengths by5% result in the practically negligible efficiency changes.The maximum possible allowable random centered errorin the choice of layer lengths—that for practical purposesdoes not significantly influence conversion efficiency—was calculated to be 15%. It gives us a basis to considerthat the realization of the necessarily layered structure ispossible and some errors of layer lengths will not undulyprohibit the realization of QPM conditions.

For the determination of the optimal conditions of anti-Stokes SRS generation, we studied the dependence of theoptimal ratio between input Stokes and pump wave inten-sities on the wave mismatching and the steady-state Ra-man gain coefficient. For the fixed wave mismatching,


Fig. 3. Influence of the dispersion of the steady-state Raman gain coefficient on multifrequency SRS in hydrogen. (a) Results withoutthe dispersion and (b) results with the dispersion. 1, pump; 2, first Stokes; 3, first anti-Stokes; 4, second Stokes. The plots are analo-gous to barium nitrate.

we determined that the optimal ratio decreases with anincrease of the steady-state Raman gain coefficient; forthe fixed steady-state Raman gain coefficient it does notdepend on the wave mismatching (Fig. 5). The depen-dence of the ratio can be approximated by the expression(I21

1/I01)opt 5 0.1359g22.6146, where the steady-state Ra-

man gain coefficient is measured in centimeters per giga-watt.

For the model media, the dependence of peak conver-sion efficiency from input pump intensity was investi-gated. It was determined that with an increase of pumpintensity, the peak conversion efficiency increases, how-ever, for each medium there is a critical value of pump in-tensity at which the efficiency of anti-Stokes conversionessentially decreases. The presence of a critical value isconnected with an increase of the Stokes intensity in thefirst active layer and further generation of high-orderStokes waves that completely depletes the energy of thepump wave, and an increase of anti-Stokes intensity inthe following layer does not occur.

From Fig. 6 it can be seen that at a fixed wave mis-matching the critical intensity of the pump wave is in-versely proportional to the steady-state Raman gain coef-ficient, and at a fixed steady-state Raman gain coefficientit depends linearly on wave mismatching. The depen-dence of the critical pump intensity on the steady-stateRaman gain coefficient and wave mismatching can be ap-proximated by the formula I0,cr 5 0.4022D0.9977g21,where the intensity is measured in gigawatts per squarecentimeter, wave mismatching is measured in radians percentimeter, and the steady-state Raman gain coefficient ismeasured in centimeters per gigawatts.

The simulations of the diffractive multifrequency SRSin hydrogen and barium nitrate showed that for eachvalue of the beam diameter there is an optimal QPMstructure in which a high efficiency of anti-Stokes genera-tion is reached. For the determination of the optimumgeneration conditions, we studied the dependence of theanti-Stokes generation efficiency on the Fresnel number(F 5 ka2L21, where k is the wave number, a is the beamdiameter, and L is the length of the Raman-active me-dium; Fig. 7) and the position of the beam waist (Fig. 8)within a fixed layered structure. Our simulationsshowed that, with an increase of Fresnel number, the ef-

ficiency of anti-Stokes generation in a layered medium in-creases and reaches its maximum value at a Fresnelnumber of 3. With a further increase of Fresnel number,the efficiency of anti-Stokes generation in a layered me-dium does not practically change. The maximum anti-Stokes generation efficiency reaches 32% for hydrogenand 21% for barium nitrate. It was determined that themaximum efficiency of anti-Stokes SRS generation ofbeams with an input diameter of 1 mm reaches in thecase of when the beam waist is located within the Raman-active medium (at a distance of '80% of Lm for hydrogenand '100% of Lm for barium nitrate, where Lm is alength of the Raman-active medium). At the same time,with the changing of the beam-waist location within 630

Fig. 4. Dependencies of the lengths of (a) active and (b) passivelayers of an optimal layered structure, providing the effectivegeneration of quasi-phase-matched anti-Stokes SRS radiation inhydrogen (I0

1 5 0.2 GW/cm2, I211 5 0.01 GW/cm2) on the lay-

er’s ordinal number (the comparison of a unidirectional assump-tion with a bidirectional one). The gray curves correspond to aforward SRS approximation and the black curves correspond to asimultaneous forward and backward SRS.


cm (in front of the medium, within it, or behind it), theanti-Stokes generation efficiency does not practicallychange. Positive values of dL of the beam with r0 radiuscorrespond to the case of laser positioning in front of the

Fig. 5. Dependence of the optimum ratio of the input Stokes in-tensity to the input pump intensity on the steady-state Ramangain coefficient.

Fig. 6. Dependence of the critical pump wave intensity on wavemismatching and the steady-state Raman gain coefficient.

Fig. 7. Diffraction influence on anti-Stokes SRS generation ef-ficiency: (a) hydrogen and (b) barium nitrate.

medium at a distance dL between them. Negative val-ues correspond to the case of laser positioning directly infront of the medium, and the beam with R radius is fo-cused within the medium by the lens with focal length F,where

dL 5Fp2R4

p2R4 1 F2l2; r0 5

FlR

~p2R4 1 F2l2!1/2.

Fig. 8. Influence of beam focusing on anti-Stokes SRS genera-tion efficiency: (a) hydrogen and (b) barium nitrate. Positivevalues of dL correspond to the case of the beam-waist position infront of the medium, and negative values of dL correspond to thecase of the beam-waist position within and behind the medium.

Fig. 9. Influence of the pump (a) intensity and (b) duration onanti-Stokes generation efficiency.


For the layered structure in hydrogen, consisting of 40layers, we studied the influence of the intensity and theduration of the pump (Fig. 9) and the Stokes seed pulse(Fig. 10) on the anti-Stokes SRS generation efficiency.Our simulations show that there is a wide enough set ofpump parameters (T0

1 > 1 ns and 0.15 GW/cm2 < I01

< 0.6 GW/cm2), at which the efficiency of anti-Stokesconversion is high enough, that allows realization of thetuned source of anti-Stokes SRS generation. Moreover,effective generation occurs within a wide enough set ofStokes seed pulse parameters (5 ns < T21

1 < 20 ns and0.001 GW/cm2 < I21

1 < 0.025 GW/cm2), at which the ef-ficiency exceeds 20%. This fact reduces the requirementson the system of preliminary Stokes seed generation,which is required for efficient anti-Stokes SRS genera-tion. These simulations also show that transitivity sig-nificantly lowers the efficiency of anti-Stokes QPM SRSgeneration.

We studied the dependence of anti-Stokes generationefficiency in a fixed layered structure on the pump wave-

Fig. 10. Influence of the (a) Stokes seed intensity and (b) dura-tion of the anti-Stokes generation efficiency.

Fig. 11. Influence of the pump wavelength on anti-Stokes gen-eration efficiency in a fixed QPM structure.

length (Fig. 11). The simulations showed that the QPMstructure has a high sensitivity to the wavelength varia-tions. This leads to a sufficient decrease of the anti-Stokes generation efficiency even at small wavelengthvariations of '0.75%.

4. CONCLUSIONSWe derived the generalized system of connected differen-tial equations, describing forward and backward multifre-quency stimulated Raman scattering (SRS) in quasi-phase-matched conditions. Simulations of the anti-Stokes SRS generation in different regimes showed thatfor each Raman-active medium there is an optimal valueof the input Stokes seed intensity at which the maximalefficiency of generation is reached. It is determined that,with an increase of the pump wave intensity, the effi-ciency of the anti-Stokes generation increases; however,for each medium there is a critical value of pump inten-sity at which the efficiency essentially decreases. Simu-lations showed that the maximum standard deviation inthe choice of layer lengths is 15%. We discovered that ata Fresnel number of more than 3 the diffraction influenceon a QPM realization can be neglected. The studyshowed that there is a wide enough set of input radiationparameters in which the anti-Stokes SRS generation effi-ciency does not practically change and exceeds 20%.

N. S. Makarov’s current address is Department ofPhysics, Montana State University, Bozeman, Montana59717. His e-mail address is [email protected].

V. G. Bespalov’s e-mail address [email protected].

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effective method of anti-stokes generation by quasi-phase-matched stimulated raman scattering

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