Power-Dependent Dispersion in Far-Off-Resonant Raman Scattering

We investigate the saturation effects of power broadening, Stark shifting, and population transfer on Stokes conversion in stimulated Raman scattering. We do not make the usual rotating wave approximation because the detuning from the next electronic state is assumed to be in the optical regime. Retaining the counter-rotating terms allows an exact determination of the pump and Stokes indexes of refraction. Steady-state solutions for the Stokes intensity and phase are obtained and the effects of making the rotating wave approximation (RWA) are discussed. Finally, we examine the behavior of these solutions for Stokes conversion in hydrogen gas when geometric propagation is appropriate.


I INTRODUCTION
T IS generally hoped that relatively unaberrated Stokes beams can be produced by pumping a Raman active medium appropriately [1]- [5].However, a variety of phasedegrading mechanisms exist.If the beams have more than one longitudinal mode, the pump mode phases can alter the Stokes gain and the Stokes phase [1]-[ 111.Saturated dispersion causes Stokes phase degradation when the Raman level is populated significantly by a strong pump/ Stokes combination [5].The two populated vibrational levels define two different indexes of refraction and result in power-dependent dispersion terms in the field differential equations [ 121, [ 131.If the pump/Stokes frequencies are not tuned exactly to the two-photon Raman resonance, the field equations will contain normal offresonant dispersion terms as well.Even if the pump and Stokes fields are tuned correctly, the dynamic Stark effect will shift the molecular levels out of resonance [14], resulting in power-dependent off-resonant dispersion terms [5].We will investigate all of these dispersion effects in the regime where the rotating wave approximation (RWA) [ 151 cannot be made.
In the following section, we derive the time dependent density matrix and electric field equations for first Stokes conversion using only the adiabatic and slowly-varying amplitude approximations.Following this development we obtain analytic expressions for the steady-state intensities and phases as functions of z.In the final section these equations are applied to geometric propagation of input beams with flat phases and Gaussian intensity pro-Manuscript received May 2 , 1986; revised October 10, 1986.The authors are with the Air Force Weapons Laboratory, Quantum Op-IEEE Log Number 8613303.tics Branch, Kirtland AFB, NM 87117.files in hydrogen.This then allows the prediction of intensities and phases as functions of the beam radius.
This approach exhibits several important features manifested in the steady-state Stokes intensity and phase solutions.We show that the terms which drive the powerdependent intensity, and hence reduce the gain, are due to the coupling between Stark broadening and the population decay time.We also derive a simple formula allowing calculation of the power-dependent extraction from the low-power equations.Furthermore, we show that the Stokes phase is also driven by the same decay rate coupled through its index of refraction.This gives a formula for the intensity dependent longtudinal and transverse asymptotic Stokes phase.
EQUATIONS OF MOTION We consider the three-level atomic or molecular system depicted in Fig. 1, with pump and Stokes beams detuned from resonance as shown.The detuning from the Raman level I 3 ) will be considered very small.Since we will eventually consider the hydrogen molecule, with Ap = w21up -3.8wp, the adiabatic approximation is justified [ 141.Specifically, the upper level 1 2 ) time integration is performed to first order in f/ Ap (where f is any nonoptical frequency) to obtain the equations of motion for the effective two-level system between state I 1 ) and 13 ) P I , 1141.
We begin with the system wave function where propagation is along the z-axis.Here Awj is the energy of the eigenstate I j ) having amplitude uj ( t ) , and i?J, &, wJ are the polarization unit vector, complex amplitude, and circular frequency of the J = Stokes, pymp field.T_he spatial plane wave phase in (2) is exp where 'I: is the internal atomic coordinate, and R is the location of the atom.First Stokes conversion is indepentent of the macroscopic position of the atom, allowing R to be neglected from the outset.Also, in the dipole approximation Y is set to zero.The quantities u l , phases of the fields are incorporated in the complex field amplitudes.The initial state coefficients are ul( 0 ) = 1, a 2 ( 0 ) = a 3 ( 0 ) = 0. Inserting the above in Schrodinger's equation leads to the coupled equations with the Rabi frequencies defined by QqJ = < i I ji * CJ 1 j ) E J / h = iiij -P J ( J / h , and ji13 = 0.Here ji is the dipole operator and wij = wi -wj.In the sums, i, j = 1, 2, 3 and J = p , s.We will choose the phases of the wave functions such that ( iiij 2JEJ )* = j i i j * CJgf.We will also assume the pump and the Stokes fields are polarized identically so that 2 , = zs and we may write Gij sim- The adiabatic approximation entails integrating the equation for u 2 to first order in Q v J / A , and assuming ul( t ) , a3( t ) , and the Rabi frequencies are slowly varying so that they can be removed from the integral [14].Integrating a 2 in this way then gives ply as piJ. f The equations for u 1 and a3 are obtained by substituting the above equation into the right-hand side of ( 3 ) with i = 1 , 3 and retaining only the terms which have exponential oscillations of exp ( *iA, t ) (where A, = ~3 1 -up + w,) or slower.This approximation is equivalent to ignoring terms of order Q2i,J/APw31 in the a l ( t ) and u3( t ) solutions.Since the adiabatic approximation is also good only to first order in OqJ/Ap, the dropping of rapidly oscillating terms at this point does not impose any further restrictions on the validity of the solutions.With the change of variables, a3( t ) = b3( t ) exp (iA,t), ( 3 ) becomes where 6Ji = aii ( o J ) / 4 h 7 6 ; , = ai3( w p ) / 4 A , 6 ; , = ay3( 0,) /4h, and Since the overall detuning A$ is small, we will let ai3 = ar3 = a13 and define a ,, = a 1 3 ( up) /4A.The aq's are the isotropic components of the three-level atom polarizability.
Finally, the density matrix equations are obtained by defining the complex transition amplitude Q = 2 a l b$ and the population difference W = I b3 1 ' -I al 1 ' .Combining the above equations yields where Ae = A s + S,lEJ + 6,IE,I2 6, = 6,1 -8,3 (6c) and the decay rates have been added phenomenologically.F is the dephasing rate (HWHM in rad/ s), y in the population decay rate (FWHM in rad / s), and A is the steady-state population decay rate (FWHM in rad/ s) [5].In passing we mention that the RWA form of these equations is recovered by dropping all the counterrotating terms.That is, if only the first terms in a I 1 ( w p ) , a I l ( w s ) , and aI3(wp) are retained, Valley's expressions for the Stark shift A, and the power broadening !le are obtained [5].
The final developmental step is to couple the above equations to the field equations.In the slowly-varying envelope approximation (SVEA), the field equations are where the pJ ( J = p , s ) are the slowly-varying parts of the polarization defined by The polarization is evaluated using , where N is the atomic number density.A straightforward calculation yields where K = NnoSal3( up) / e .In these expressions the indexes of refraction associated with the i + 2 ( i = 1, 3 ) transitions have been defined as [16] n&) -1 = 2nNCY,(o).

(10)
The contributions of the indexes of refraction depend on the population of the lower state through N ( 1 k W ) .The ( nl -1 ) terms in (9a) and the ( n3 -1 ) term in (9b) are due to counterrotating terms in a2( t ) , as is evidenced by the nonresonant denominators in nl( os) and n3( u p ) .
Also, in deriving ( 9) and (10) the field polarization has not been averaged and the normalization condition I al 1 ' + I b3 = 1 has been used, since 1 a2 l2 is of order ( s2 /A,)' I ai 1 1" .

STEADY-STATE SOLUTIONS
The steady-state equations have been solved previously for the Stokes phase in the RWA [5].However, we wish to evaluate the phases and intensities when the only assumptions are the adiabatic approximation and the SVEA.
In this section we present these solutions following well established techniques.In the steady state the density matrix equations ( 5 ) have the solutions where we have let A = y; the subscript ss denotes steady state, and These solutions require the fields to be constant over a time T > 1 /y, l/r.In hydrogen, 1 /I? -1 ns while 1 /y -6 ps.To relax the T > 1 /y restriction, (5b) must be solved more generally.In the Appendix, we find W as a function of time and show that for intensities less than 30 MW /cm2, W ( t ) = Wss.Thus, the solutions given below hold even if l/r c T e l / ~.
When these steady-state solutions are substituted into (9), the complex field equations become Again, the ( n -1 ) terms describe the power-dependent propagation of each field through a medium having two possible indexes of refraction, nl and n3.The last two terms in (13) corresponds to the solution obtained in a two-level atom calculation.The A , / D terms represent normal dispersion for the two-photon transition modified by the power-dependent Stark shift.The I ' /D term is the power-dependent gain for the Raman transition and reduces to the usual small signal gain in the limit of small pump intensities.
Equation (13) is easily solved by introducing the sum and difference variables [5] where tJ ( 2 , t ) = eJ ( z , t ) exp [i+., ( z , t ) ] , J = s , p .Here 7 is proportional to the total photon density and S is proportional to the difference in photon densities between the pump and Stokes fields.Casting (13) in terms of these variables gives ar where the denominator is where we used (15b) to equate [ E ; ( Z ) -E ; ( O ) ] / W , to   -[ E ; ( Z ) -E : ( O ) ] / W , .Here the S,, 6, terms arise from the dynamic Stark shift, the S , , term is due to power broadening, and the A, terms contribute if the pump/ Stokes frequencies do not match the two-photon transition between the two ground states.
Returning to (16), we note that if all of the 760 terms on the right-hand side are zero (no Stark shift or power broadening), ( 16) can be solved for I, ( z ) = C E ; ( z )   (17) were the two logarithmic functions in (16).The effect of the last term in ( 16) is to hold off extraction.This term arises from the I Qe l2 contribution in the denominator D of (12c).From (13) the power-dependent effective gain geR is related to the lowpower gain g by 8,ff = g / ( l + &/r2 + 1 QeI2/yF) (18) and for Zp = Z, = 50 MW /cm2, geff = g / 1.29.The reduction in geff is due to 1 Q, l2 / yr largely because y >> Following this theme we can obtain an approximate formula for the increase in z for a specific Z, ( z ) / I, (z) ratio due to the lower extraction.When 6, wp = 6, w, and Z,, /Zpo is very small, the first terms in ( 16) combine and we are left with r.The first term on the right hand side is just the powerindependent z value where I,( z ) = I,( z ) , which would be obtained from (17) under the above approximations.Hence, the separation between this value and that obtained by keeping all power contributions is In other words, when the Stark shifting and power broadening effects are included, the z value at which Z, ( z ) = Zp(z) is increased by A z beyond the low-power value.
For Zpo = 50 MW/cm', A z = 2.3 cm.Since the leading term in (21) depends only on known field and atomic frequencies ( a ,, / K is independent of dipole moments), measuring A z may afford a technique to determine y .
Finally, we close this section with a discussion of the pump and Stokes phases.With tJ = E J exp ( i @ J ) , the complex field (13), and the sum and difference variables Equations (23a) and (23b) are nearly identical.The appearance of [ nl( wJ ) -1 ] kJz in GJ and the lack of a similar [n3( w J ) -1 ) kJz term is due to the initial condition a3( t = 0 ) = 0.These dispersion terms linear in z represent propagation in the absence of induced power effects.Since population is transferred to state 13 ) only through the two-photon Rabi frequency, the n3 index will manifest itself only in the power-dependent terms.The origin of the ( nl -1 ) term in (23a) may be explained along with the ( n3 -nl ) contribution by first recalling the relationship between classical field energy density and field photon number a: a / V = E' / 8aAw, where V is the quantization volume.The Stokes photon density created in a distanceLis then o , / V = [ E : ( L ) -E;(O)]/STAO,, and equals the density of excited molecules.Consequently, a, / N V is the fraction of molecules in state I 3 ) and 1os / N V is the fraction of molecules in state 1 1 ) .Thus, the Stokes phase accumulated in the length L = c / y in which the molecules remain excited is ( n3 -1 )( a, / N V ) k,L + ( nl -1 )( 1 -a, / N V ) k,L which can be rewritten as ( n3 As mentioned earlier, the A, term is just the normal dispersion present in a detuned two-level atom and leads to part of the logarithmic dependence on E : .The Stark shift dispersion separates into both a logarithmic dependence as well as a linear dependence on E : .An identical discussion may be given for the analogous terms in the pump phase aP (z).

EXAMPLES
This section is devoted to evaluating the above equations for Z, ( z ) and +$ ( z ) with various input intensities Zpo and ZSo.We will also find the transverse phase and intensity by assuming geometric propagation.To do this we give both input beams a Gaussian transverse intensity structuregivenby I, = ZJoexp ( --r 2 / 2 o 2 ) , J = s , p , with flat phase.Zpo is chosen to be 75 MW /cm2 so that Stark effects are prominent.Additionally, we assume amplification in H2 with Zs0 = 1 .O x lop5 MW / cm2.16).The dashed curve shows the low-power limit, (17).Curves drawn from the RWA equations would lie exactly on top of these curves due to the use of the same numbers for the terms identified as isotropic polarizabilities.Comparing Fig. 2(a), 2(b), and 2(c) shows that for Zpo less than about 25 MW /cm2 the power-dependent formula approaches the low-power expression.However, for larger input intensities, such as that shown in Fig. 2 Additionally, the separation A z between the solid and dashed curves is again due to the last term in (16) and is approximately A z = (Zs(z) -Z,o)6p,8.?r1013/~~-y where Z, ( z ) is calculated from (17).Thus, one can obtain an approximation to the power-dependent extraction equation by simply translating the low-power solution, (17), a distance Az.
The Stokes phase as a function of the propagation length is presented in Fig. 3, again for the same input intensities Z , , as for Fig. 2. The solid line shows the nonlinear phase = +,(z) -( n l -1 ) k,z with the counter-rotating terms included while the chained curve is with the RWA.The corresponding phase curves for the low-power limit are not shown because they are zero.This can be seen, for example, by setting all the 6's equal to zero in (23a).We mentioned at the end of the previous section that the phase is dominated by the last term in (24) and, indeed, the curves in Fig. 3 have the functional form of Z, ( 2 ) -Zs0.Furthermore, the asymptotic behavior for the solid curve is +sNL = -0.77ZP0/2awaves and the asymptotic RWA phase is 12.2 times larger.The great difference in these asymptotic values is due to the fact that the dominant term is proportional to n3( 0,) -nl ( w , ~) = 1.28 X lop4 in our case and to n3( as) -1 = 1.56 x lop3 in the RWA case.
The transverse characteristics are illustrated in Figs. 4  and 5. Fig. 4 shows two graphs of the radial Stokes intensity as a function of r / a at four propagation distances z = 25, 50, 75, and 100 cm.These are generated with (16), (17) when the initial conditions are given by ZJo exp ( -( r/o)' /2).The labeling of the two curves in each graph is consistent with our previous notation.The most outstanding feature as a function of increasing z is that the near-axis intensity flattens due to saturation of the smaller intensities thereby causing the flat portion to move to larger radii.In Fig. 4(a), for z = 25 cm, the Stokes intensity is a little narrower than the inputs with a 1 / e point at r / B = 1.08; at this distance there is as yet no saturation.At z = 50 cm, Z, has a 1 / e point occurring at r / a = 1.6.By z = 100 cm, the 1 / e point has moved to r / a = 2.After z = 100 cm the intensity continues to broaden until the intensity is so low that there is no difference between ( 16) and (17).
Fig. ( 5 ) shows the transverse Stokes phase difference A%, = [ @$( r , z ) -as( r = 0, z ) ] in four different z planes.These graphs dramatically show the effect of the RWA on the phase.Specifically, after the near-axis intensity saturates, the peak-to-valley phase variation is about 9.2 waves with the counter-rotating terms included, and 112 waves when the RWA is invoked.This and other features can be explained through the last and dominant term in (24).In the region of saturation this term yields a transverse Stokes phase difference of A @$ = -0.77 ) / 2 7 r waves (without the RWA).This explains the inverted Gaussian shape of the curves.For 99 percent saturation, (17) gives 2gZp0 exp [ -( r / /2] z = 20.3.Thus, for z = 100 cm, A+, 1s an inverted Gaussian for r / a < 1.5, which is consistent with Fig. 5(d).
Furthermore, in the near-axial region A%,y(r) = -0.77ZPo( r / 0 ) ~/ 4 n and represents a focusing effect.The steep slope between this value and larger values of r / a is due to the exponential growth region shown in Fig. 4(d).For r / B >> 1, the asymptotic phase difference A+s = -0.77ZPo/27r and equals 9.2 waves for Zp0 = 75 MW/cm2, as noted before.Notice that the shapes of the two curves with and without the RWA are nearly identical (deviations from an inverted Gaussian occurring at the same place).This is due to the fact that the gain used in each case was the same.

CONCLUSION
Using only the adiabatic approximation we have derived the density matrix equations and coupled them with the time dependent SVEA field equations.In this procedure the RWA is not used and consequently the powerdependent indexes of refraction take the correct form when the counter-rotating terms are retained.The theoretical analysis deals with obtaining the analytic solution for the intensities and phases.
The remainder of the paper is concerned with calculating the intensity and phase with and without the RWA, as well as illustrating the contribution of the power-dependent Stark effect.Specifically, we consider Raman scattering of w,, = 5.34 X 1015 into w, = 4.56 x lOI5/s in Hz with an upper level detuning of A,, = 3 . 8~~ and a ratio of depopulating rate to dephasing rate of yI2r = 4.  y = 26.8X lo3 X 27rls.We show that the Stark terms couple through y and create a considerable departure from the typical low power solution for input intensities greater than about 50 MW /cm2.In fact, we derive a simple equation expressing the hold-off in extraction due to the power dependent effects.
For 50 MW /cm2 the power-dependent effects increase the length required to reach 50 percent extraction by about 11 percent [see Fig.

2(b)]
. Measuring this A z may also provide an experimental proceedure for determining the population decay rate Y.
We have calculated the transverse Stokes intensity and phase for Gaussian inputs with flat phase when geometric propagation is appropriate.As one would expect, I, ( r ) is flat in the region of saturation with a rapid decrease to the low-power solution.The radius of the flat region can be estimated from gIp ( r ) z = 20.Finally, we show that the transverse phase difference between the axial r = 0 value and arbitrary r is very sensitive to the RWA.In the RWA, nl( w,) = 1 results in a factor of 12 difference in the dominant term.Near the axial region the Stokes beam is focused and the RWA focusing is about 12 times as strong as that with the counter-rotating terms.We further note that the focusing aberration coefficient is proportional to the input pump intensity.

APPENDIX
Solving the Bloch equations in the steady state requires that the field pulse length 7 be greater than the maximum of l / y or l/r.In hydrogen, l / r = 1 ns and 1 / y = 6 ,us.For r >> y, Q reaches the steady state long before W. The restriction on 7 may be relaxed to T > 1 /I' if we solve d Q / d t = 0 while retaining the time-dependence in W ( t ) .Once again holding the fields constant, ( 5 ) gives W ( t > = Wss -(1 + %s> exp ( ~t / K , > ( A l l where 01 = D /( Af + r 2 ) and W,, is the steady-state population difference given by (12b).Using the values given in the paper for hydrogen, we find A: < 1.2 X 108Z2 / s2 and I Q, l2 < 3.4 X 10'oZ2 /s2, where Z is the number of MW /cm2 in the initial pump intensity.For Z = 30, W,, = -0.9 and W ( t ) never gets very far from its initial value of -1.So in hydrogen, for intensities less than 30 MW / cm', W,, = -1 and W ( t ) = W,, indicating that our solutions are good even for 1 /I' < 7 < 1 / y.
During his last two years of graduate work he was an adjunct faculty member of the Math Department of the Rochester Institute of Technology, where he taught several courses in undergraduate algebrahigonometry and calculus.In February 1984 he joined the Air Force Weapons Laboratory where he continued his research in theoretical quantum optics and nonlinear optics.
A. Gavrielides was born in Salonika, Greece, on March 25, 1943.He received the B.S. degree in electrical engineering from University of Illi-nois, Urbana, in 1967.In 1974 he received the Ph.D. degree in physics from the University of Minnesota, Minneapolis.His dissertation was on particle field theory on anomalous decays of strange particles.
Following his work at the University of Minnesota he held a postdoctorate position at Purdue University, Lafayette, Indiana, for two years.In 1979 he joined the Air Force Weapons Laboratory and spent the first year at the Optical Sciences Center at the University of Arizona, Tucson.Thereafter he returned to the Air Force Weapons Laboratory and worked on C 0 2 laser kinetics, resonators, and time-dependent thermal blooming.He is currently conducting research on nonlinear optics.

r and 7 ,
the differential equations for the Stokes and pump phases in the steady state are I kPThese equations are trivially integrated to give + 6,ws -6,w, Ej(Z) -Ej(0, -6,wp E j ( Z ) -Ej(0)

Fig. 2
Fig.2shows the Stokes intensity as a function of z for input pump intensities ZPo = 75 MW/cm2, 0.736ZpO, 0.368Zp0, and input Stokes intensities ZSo = 1.0 X lop5 MW /cm2, 0.736ZsO, 0.386Z,To corresponding to sampling the Gaussian at r/cr = 0.0, 0.782, and 1.41.Each graph shows two curves.The solid curve arises from the powerdependent extraction formula,(16).The dashed curve shows the low-power limit,(17).Curves drawn from the RWA equations would lie exactly on top of these curves due to the use of the same numbers for the terms identified as isotropic polarizabilities.Comparing Fig.2(a), 2(b), and 2(c) shows that for Zpo less than about 25 MW /cm2 the power-dependent formula approaches the low-power expression.However, for larger input intensities, such as that shown in Fig.2(a), inclusion of the Stark terms holds off extraction.In fact, the input intensity Zpo at which Stark effects start to dominate can be estimated from (20).This (a), inclusion of the Stark terms holds off extraction.In fact, the input intensity Zpo at which Stark effects start to dominate can be estimated from (20).This occurs when the last term in (20) is about & of the leading term, which is 14.Solving for Zpo gives Zpo = [ 1 4 ( O .l ) ~r / 2 ] ' / ~ c x 10p'3/(87r6p,) = 61 MW/cm2.

Fig. 3 .
Fig. 2. Stokes intensity in MW /cm2 as a function of z, normalized to the total input intensity I = I,( 0 ) + wsIp ( 0 ) / u p .The solid curve is the power-dependent solution and the dashed curve is the low-power limit solution.

Fig. 4 .Fig. 5 .
Fig. 4. Stokes intensity in MW/cm2 as a function of the radius r , normalized to the total input intensity I = I,( 0 ) + usIp ( 0 ) /up., The solid curve is the power-dependent solution and the dashed curve IS the lowpower limit solution.