Degenerate four-wave mixing for arbitrary pump and probe intensities

Transcription

1 95 J. Opt. Soc. Am. B/Vol. 16, No. 6/June 1999 Bratfalean et al. Degenerate four-wave mixing for arbitrary pump and probe intensities R. T. Bratfalean, G. M. Lloyd, and P. Ewart Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK Received August 5, 1998; revised manuscript received February, 1999 An analytical expression for the degenerate four-wave mixing signal produced in a medium composed of twolevel atoms by pump and probe lasers of arbitrary intensities and parallel linear polarization, suffering negligible absorption and depletion, has been derived. The Abrams Lind approach [Opt. Lett., 94and3, 05 (1978)] for deriving the phase-matched polarization has been extended to find all the other higher-order polarization terms contributing to the signal. The signal field amplitude is expressed as an infinite series whose convergence rate depends on the intensity of the input fields and their detuning from resonance. The solution is successfully verified by comparison with a full numerical nonperturbative calculation and is found to be equivalent to the Abrams Lind result in the limit of weak probe intensity. The result is used to calculate the saturation effects on spectral line shape that are relevant to simulation of molecular spectra obtained by degenerate four-wave mixing with saturating pump and probe fields. The result is extended to treat atomic motion in the case of forward geometry of the input beams Optical Society of America [S (99)00805-X] OCIS codes: , , , , INTRODUCTION Degenerate four-wave mixing (DFWM) is a nonlinear optical process that is now widely used for spectroscopic studies of solid, liquid, and gas phase media. 1 The process involves the interaction of two pump beams with a probe beam to generate a fourth, or signal, beam. Physically, the signal may be considered to result from the creation of interference patterns formed by the probe and each of the pump beams. The spatial modulation in the resulting light field leads to a grating-like modulation of the refractive index. This modulation arises from the nonlinear response of the medium, which may be due to a variety of physical mechanisms. The signal is generated by the Bragg scattering of each pump beam off the grating formed by the other pump and the probe. A large amount of theoretical effort has been expended during the past 0 years to understand the physics of DFWM. Of particular interest for spectroscopic applications of DFWM has been the dependence of the signal intensity and line shape on various parameters, such as saturation, 5 absorption, 6 9 laser bandwidth, 10,11 polarization effects, 1 14 and the effects of collisions 5,15,16 and atomic motion. 3 5,1,13,17 Many of these problems have been solved only by numerical methods, although in some circumstances analytical solutions have been possible. The standard theory of DFWM applicable to signal generation by saturable absorption has been presented by Abrams and Lind. This model treats the case of monochromatic pump and probe beams interacting with a twolevel, stationary, atomic medium with the additional assumption that absorption and depletion of the pumps can be ignored. The analysis uses a perturbation theory approach and so is restricted to the case of a weak, nonsaturating, probe beam but does provide an analytical expression for the signal intensity. A more sophisticated model has been developed by Attal-Tretout et al., who treat a three-level atomic model and use a radiative renormalization technique to solve the density-matrix equations. 18 They have found an analytical solution for the polarization, which allows the signal to be calculated for different polarization configurations and permits the effects of collisions to be calculated. The Doppler effect was included by numerical integration over the distribution of atomic velocity. This approach considered strong saturation by the pump fields but restricted the probe to nonsaturating levels and treated the forward phasematching geometry. The predicted line shapes showed a dip at line center for strongly saturating pumps, in contrast to the predictions of earlier perturbative models. For the case in which both pump and probe beams are saturating, a nonperturbative method was adopted by Lucht et al., 5 in which the relevant density-matrix equations determining the medium polarization are solved by direct numerical integration. This approach allowed saturation and collisional effects to be modeled for comparison with experimental observations. 16 Brown 7 numerically analyzed the case of orthogonally polarized pump beams interacting with a strong probe in the presence of absorption. The Abrams Lind model has been extended by Knize and by Ai and Knize to treat arbitrary intensities for all four beams. 19,0 The detailed derivation of this model divides the interaction region into a series of small, well-defined, identical zones determined by the fringes of the two gratings, which scatter the pump beams to produce the signal beam. An analytical solution was found for the average value of the induced polarization in each such zone. A numerical calculation was then carried out to solve the coupled-wave equations yielding the signal intensity. The aim of the present study is to derive an analytical /99/ $ Optical Society of America

2 Bratfalean et al. Vol. 16, No. 6/June 1999/J. Opt. Soc. Am. B 953 expression for the DFWM signal generated with pump and probe beams of arbitrary intensity and, in particular, with a saturating probe beam. This is important because, in many experimental situations, it is necessary to use intensities of both pump and probe beams that equal or exceed the saturation level to obtain a satisfactory signal-to-noise ratio, especially when trace species are being probed. Using the analogy of hologram generation we can see that the grating structure induced by interference between pump and probe beams will have maximum contrast, and that the scattered signal will thus be optimized, when both beams have the same intensity. In this paper a model of the DFWM interaction is presented that is less general than that of Ai and Knize 0 in the sense that the input beams are considered to be undepleted, and the resulting signal is weak in comparison. However, the advantage of the present model is that both the DFWM polarization and the DFWM signal can be expressed in analytical forms. In addition, the polarization responsible for the DFWM signal is a local value and is not an averaged value over a zone defined by the fringes of the grating structure of the induced polarization. 0 The present model is very convenient for the case of gas phase DFWM in which absorption and depletion of the pumps is negligible a situation commonly encountered in gas phase spectroscopy and applications in combustion or plasma diagnostics.. THEORY We consider the interaction of three plane-wave optical fields: E j rexpit A j expit k j r for j 1,, 3. (1) The pump fields are considered to have equal amplitudes, A 1 A A p, and A 3 is the probe amplitude. The signal wave vector, determined by the phase-matching condition, is given by k 4 k 1 k k 3. () Figure 1 shows the geometry of the four vectors that lie along the diagonals of a rectangular box, all pointing toward the same face, with k 1 and k, the pump wave vectors, pointing toward opposite corners. All the input fields are considered to be polarized in the same direction. This polarization condition can be rigorously satisfied when the pump beams are counterpropagating, i.e., in the limit of a zero c side of the rectangular box shown in Fig. 1, an arrangement usually known as the phaseconjugation geometry. It can also be approximately satisfied for other phase-matched arrangements if the probe wave vector, k 3, crosses the plane defined by k 1 and k at a small angle. The input fields are considered to suffer negligible absorption and depletion in the interaction region. The combined electric field of the input beams induces in the medium a polarization term oscillating at frequency, denoted by P(r)exp(i t). The total electric field, E(r)exp(i t), is coupled to P(r)exp(i t) through the general wave equation 1 Fig. 1. Spatial arrangement of the four-wave vectors in the phase-matched geometry for DFWM and the location of the site of signal detection relative to the interaction region. Erexpit 1 c t Erexpit 0 Prexpit, (3) t where c is the speed of light in vacuum and 0 is the vacuum magnetic permeability. Even though the source polarization arises in an interaction region of finite extent and will have a spatial modulation as a result of the field interference patterns, solutions to Eq. (3) usually involve approximating the signal field as a plane wave. An alternative method, used here, is to evaluate the field amplitude at a long distance from the interaction region and to consider the total polarization to be made up of a sum of contributions having different spatial amplitude and phase variations. The interaction region may be of arbitrary shape, centered on the point O, shown in Fig. 1, and the signal propagation axis is defined by the direction of k 4. We then determine the signal field at a point A on this axis at a large distance from the origin defined by the vector s OA. The magnitude, s s OA, is much larger than both the interaction region and the light wavelength, k 1, where k k 1 k k 3 k 4. The total polarization in the interaction region is given by Pr P k rexpik r, (4) k where the various vectors k are determined by combinations of the vectors k 1, k, and k 3. The strength of the signal field at point A is determined by the phasematched component, P k4 (r)exp(ik 4 r), of the total polarization, P(r). The radiated field amplitude at a large distance s is given in the dipole approximation by A 4 s k 4 0 s P k4 rdr 3, (5) where 0 is the vacuum electric permittivity and the triple integral is extended over the whole spatial domain of the interaction region. The phase-matched component of the polarization has a spatial phase dependence of

3 954 J. Opt. Soc. Am. B/Vol. 16, No. 6/June 1999 Bratfalean et al. the form exp(ik 4 r), which allows us to identify the relevant components of the total polarization, P(r). For a medium of two-level atoms, the total polarization, P(r), can be expressed as Pr E 1r E r E 3 re SAT N it / 1 E 1 r E r E 3 r, (6) where E j r E j r/e SAT for j 1,, 3, (7) and the total electric field is considered to be due only to the input fields, since the signal field is taken to be much weaker in comparison. The other quantities in Eq. (6) are as follows: E SAT is the amplitude of the saturation field, defined by E SAT /4 T T 1 ; (8) T and T 1 are the transverse and the longitudinal relaxation times, respectively; is the electric dipole moment; N is the equilibrium population difference between the upper and the lower levels; and 0 T (9) is the dimensionless detuning from the atomic resonance. Since the approach assumes a medium composed of twolevel atoms, the effects of level degeneracy, relative orientation of the input field polarization, and collision effects involving molecular angular-momentum sublevels are ignored. 18 Orthogonally polarized pump and probe beams are sometimes used in practical situations to enhance signal-to-noise ratios. However, in a molecular medium in which degeneracy of angular-momentum sublevels complicates the interaction, extension of the model to treat this geometrical arrangement would not be trivial. The denominator of P(r) can be rearranged to express P(r) as a convergent series (details are given in Appendix A): Pr P 0 n0 E 1 r E r E 3 r Br E 1r E re 3 r* c.c. Br B(r) is de- where c.c. denotes the complex conjugate. fined as n, (10) Br 1 E 3 4E p cos k r/, (11) where k k 1 k, (1) E 3 A 3 /E SAT, (13) E p A p /E SAT, (14) P 0 E SAT N it /. (15) Each term in the series of Eq. (10) contributes one phasematched component. After multiplication of E 1 (r) E (r) E 3 (r)/b(r) by the binomial expansion (E 1 (r) E (r)e 3 (r)* c.c. /B(r)) n, the phasematched terms can be identified as those having a spatial phase dependence of the form exp(ik 4 r). The spatial phase variation is determined by the product of the probe and the total pump field, E 1 (r) E (r) E 3 *(r), which leads to terms of the form expi(k 1 k k 3 ) r(1 cos k r). The cosine term describes the grating structure of the induced polarization. For the nth term in the summation series of Eq. (10), where n is odd (i.e., n 1 and m is a positive integer), the corresponding phase-matched term has the following form: P 0 1! m!m 1! cosk r/ Br E 1 3 E p expik 4 r. (16) For an even nth term (i.e., n, where m is a positive integer), the phase-matched term is expressed as! P 0 m 1!m 1! cosk r/ Br 1 E 1 3 E p expik 4 r. (17) So only terms like expressions (16) and (17) contribute to the phase-matched polarization. Note that these terms, for m 1,, 3..., represent the higher odd-order field contributions of the nonlinear polarization, 3, 5, 7..., etc. Taking this into account, we can express P k4 (r) as 4 P k4 rp 0 m 1!E 1 3 E p m1 m!m1! E 3 m 1Br 1 cosk r/. Br (18) Substituting Eq. (18) into Eq. (5), which gives the integrated polarization over the interaction region, we obtain E 4 s A 4 s/e SAT 0k i s m1 4 m 1!E 3 1 E p m!m 1! m 1Br 1 E 3 cosk r/ Br d 3 r, (19) where 0 is the line-center small-signal-field absorption coefficient given by 0 NT k. (0) 0 From Eq. (19) we can see that finding the signal amplitude requires an integral over the interaction volume. It is convenient to define the x axis of our coordinate system as lying along the k vector so that the integrand is constant along any direction perpendicular to and periodic along this axis. We further assume that the interaction length will cover many periods of the pump-induced grat-

4 Bratfalean et al. Vol. 16, No. 6/June 1999/J. Opt. Soc. Am. B 955 ing structure determined by 1/k. These symmetry and periodicity properties allow us to reexpress Eq. (19) as E 4 s 0k ivk s 0 m1 /k E 3 coskx/ Bx 4 m 1!E 3 1 E p m!m 1! m 1Bx 1 dx, (1) where k k, B(x) 1 E 3 4E p cos (kx/), and V is the volume of the interaction region. Equation (1) can be rearranged and written in terms of spatial integrals 1, to yield an analytical expression for the signal field: where E 4 s 0kV i s m1 m 1!E 3 1 E p m!m 1!B 1f, m E 3 m 1B f, m, () B 1 E p E 3, (3) and the spatial integrals are f E p /B (4) 1 f, m 0 f, m 0 1 cos x m dx, (5) 1 f cos x 1 cos x m dx. (6) 1 1 f cos x The analytical forms of 1 (f, m) and (f, m) are derived and given in Appendix B. The signal intensity can be simply related to the field amplitude by I 4 s 0c n E 4s E SAT, (7) where n is the refractive index of the medium at the observation point A. Hence from Eq. () we find that the signal intensity, I 4 (s), may be expressed as I 4 s 0 0 ck V 1 I SAT ns 4 m1 m 1!I 3 m1/ I p m m!m 1!B 1f, m I 3 m m 1B f,, (8) where I SAT E SAT and the normalized probe and pump intensities are given by I 3 A 3 /I SAT and I p A p /I SAT, respectively. Equations () and (8) may be used to calculate the signal amplitude and intensity, respectively, as a function of detuning of a monochromatic laser from a single transition. DFWM signals involving more than one transition may be readily simulated by the addition of signal contributions from different transitions with the correct relative phase, as given in Eq. (). The integrals 1 (f, m) and (f, m) can be calculated either numerically or with the analytical forms given in Appendix B. In practice, since f can take only values between 0 and 1 and, as will be shown below, only a relatively small number of orders, m, are needed for accurate calculation, these functions can be calculated for a range of values of f and stored in a look-up table. Thus the DFWM signal strengths and simulated spectra can be rapidly calculated for arbitrary pump and probe intensity. 3. DISCUSSION We have derived an analytical expression for the DFWM signal generated by pump and probe fields of arbitrary intensity. Note that the result applies to any phasematched geometry; i.e., it describes both the phase conjugate and the forward geometry, including folded boxcar arrangements involving small crossing angles. The result for the DFWM signal amplitude is expressed as a sum to infinity of a convergent series. In practice, the accuracy of the result will depend on the number of terms included and on the convergence rate of the series, which is determined by the magnitude of the input fields and by the dimensionless detuning, ( 0 )T. The dependence of the convergence rate can be investigated more conveniently by consideration of a thin-layer interaction region. For convenience, we consider a layer that is much thinner than the width of a fringe of the grating formed by the pumps and which is situated in the very center of such a fringe, where the pump field intensity is maximum. The layer is disposed parallel to the fringes of the pump grating such that the effect of the spatial integrals is removed but the dependence on input field intensity and detuning is retained. Then the signal field, given by Eq. (), can be expressed in a simpler form, as E 4 s 0k iv s m1 m 1! E 1 3 E p m!m 1! B E 3 m 1B 1, (9)

5 956 J. Opt. Soc. Am. B/Vol. 16, No. 6/June 1999 Bratfalean et al. where B 1 4E p E 3. (30) The convergence rate of the series can be studied more easily in the form of Eq. (9) rather than Eq. (), since the former does not involve the spatial integrals. The convergence rate is characterized by the percentage change in the result of summing the series to order m relative to that for order (m 1). A limit set by a typical PC allows calculation to the first 38 terms of the series, and this is used in all the results presented here. We find that the convergence rate decreases with the magnitude of the input fields and increases with the dimensionless detuning, ( 0 )T. This result is illustrated by consideration of the following situations, in which the input pump and probe fields are of equal amplitude characterized by the value relative to E SAT : (1) E p E 3 10, 0, () E p E 3 1, 0, (3) E p E 3 0.1, 0, (4) E p E 3 0.1, 1. With inclusion of the first 38 terms the relative error is found to be less than 10 5 % and % for the first and the second cases, respectively. Thus, even for strong saturation (case 1), the series converges more than adequately. For case 3, which corresponds to nonsaturating fields, the convergence rate is even faster and reaches a relative error of % after only the 7th order (m 7). When the detuning is increased from zero to 1 the convergence rate again increases such that, for field strengths equal to that in case 3, the series converges to a relative error of % for terms higher than m 6. The analytical expression given in Eq. () may be compared with the result obtained by Abrams and Lind. As shown in Appendix C, our result is equivalent to that obtained by Abrams and Lind in the limit of weak, nonsaturating, probe fields. It is also worth verifying our analytical solution against a nonperturbative numerical calculation of the signal field. Good agreement between the analytical result and a full numerical calculation would confirm not only the accuracy of the present method but also that the phasematched terms have been identified correctly in P(r). One can also perform this comparison more conveniently for a thin-layer interaction region, using the analytical result of Eq. (9), where the pump beams are considered to be counterpropagating along the x axis. The interaction region is again considered to be of small extent relative to the distance to the observation point. So within the dipole approximation, the signal field amplitude may be expressed as an integral over the total polarization, E 4 ss 0 Vk i 1 L y0 L y E p E 3 expik 3y yexpik 3y y 1 E p E 3 expik 3y y dy, (31) where k 3y is the y component of the k 3 vector and L y is the transverse width of the interaction region along the y Fig.. Relative deviation of the analytical result from that of the nonperturbative numerical solution of the DFWM signal intensity versus L y. Fig. 3. Normalized signal spectrum for unsaturating fields, E 3 E p 0.1 (dashed curve) and saturating fields, E 3 E p 1 (solid curve) in units of the saturation field strength E SAT. axis. The deviation of the numerical calculation of E 4 (s)s/ 0 Vk( i), as given by Eq. (31), relative to that of the analytic result of Eq. (9) is calculated as a function of L y, for a range covering the first 500. The result is shown in Fig. for the case in which A p A 3 E SAT, 0, and the integration step for the numerical value of E 4 (s)s/ 0 Vk( i) is /100. In this comparison the pump probe crossing angle has been arbitrarily chosen to be 6 deg. The oscillations of the numerical value around the analytical result arise from the non-phase-matched terms, and these oscillations will be washed out as L y increases. These oscillations consist of approximately % of the mean signal field, at L y 500, and less than 0.% at L y Such oscillations, which are due to the non-phase-matched terms, have been previously reported by Lucht et al. 5 Our analytical result, expressed by Eq. (), may now be used to investigate the characteristics of DFWM by means of arbitrary pump and probe intensity. Power broadening of the DFWM spectrum generated by a frequency-scanned monochromatic laser is illustrated in Fig. 3, which shows the line shape calculated for nonsaturating and saturating input fields of amplitude 0.1E SAT and E SAT, respectively. For strongly saturating fields, 10E SAT, the spectrum develops the characteristic dip at line center, as shown in Fig. 4.

6 Bratfalean et al. Vol. 16, No. 6/June 1999/J. Opt. Soc. Am. B 957 The saturation behavior of the signal as a function of both pump and probe amplitudes may also be calculated from Eq. (). This is illustrated in Fig. 5, which shows a Fig. 7. Spectral surface for a saturating probe, E 3 1. Fig. 4. Normalized signal spectrum for strongly saturating fields, E 3 E p 10 in units of the saturation field strength E SAT. Fig. 8. Spectral surface for a strongly saturating probe, E 3 6. Fig. 5. Saturation surface representing the dependence of the signal intensity on the magnitude of both pump and probe fields. The maximum signal is obtained when pump and probe amplitudes are equal and have a value of 1.04 E SAT. saturation surface for 0. The peak signal is found in this case to occur for E p E , in units of E SAT. The pump and probe fields are found to be equal to better than one part in 10 9 at the maximum of the signal amplitude. The effect on the signal spectrum for increasing pump field strength, for fixed probe strength, is illustrated in Figs The spectral surface, indicating the line shape as a function of pump field amplitude, is shown for a weak probe, E 3 0.1, in Fig. 6. This case corresponds to the situation treated in the perturbation analysis performed by Abrams and Lind and again shows power broadening, but no dip ever appears at line center even for strongly saturating pumps. For a probe amplitude equal to the saturation value, E 3 1.0, a dip appears at line center after the saturation peak is exceeded, as shown in Fig. 7. In the case in which the probe is strongly saturating, the signal spectrum displays a dip even for nonsaturating pump fields, and this is shown in Fig. 8 for a probe field E Fig. 6. Spectral surface representing the line shape (signal amplitude versus normalized detuning ) as a function of pump field strength in the case of a weak, unsaturating probe, E ATOMIC MOTION IN THE FORWARD GEOMETRY The analytical expression for the DFWM signal is easily extended to include the effect of atomic motion in the forward-geometry case, i.e., k 4 k 1 k k 3. One

7 958 J. Opt. Soc. Am. B/Vol. 16, No. 6/June 1999 Bratfalean et al. achieves this by forming a convolution of the expression for the signal field, given in Eq. (), with a Gaussian Doppler profile, which yields the following expression: E 4 s 0kV s m1 m 1!E 3 1 E p m!m 1! kb T/M kv iexp T m Mv k E 3 1f v, m f v, B m 1B v dv, (3) B v where T is the temperature of the medium; M and v are the individual atomic or molecular mass and velocity, respectively; and k B is Boltzmann s constant. Also, B v 1 kv E p E 3, (33) f v E p /B v. (34) 5. CONCLUSIONS In this paper we have derived an analytical expression for the DFWM signal produced by input fields of arbitrary intensity appropriate for stationary media and for situations in which the input beams do not suffer significant absorption or depletion. In diagnostic applications of DFWM spectroscopy of minor species in hostile environments it is often desirable to use equal pump and probe beams that equal or exceed the saturation intensity. By retaining all higher-order phase-matched terms in the nonlinear polarization, we found an expression for the signal amplitude as the sum of an infinite series that converged at a rate determined by the input field strengths and the dimensionless detuning ( 0 )T. We validated the result by finding that it is exactly equivalent to the standard perturbation theory result obtained by Abrams and Lind in the limit of weak probe fields. Validation of the result for saturating probe fields was obtained by comparison of the predictions of the analytical expression with those of a nonperturbative numerical calculation. Furthermore, agreement between the numerical and the analytical predictions confirmed that the correct higher-order phase-matched terms had been identified in the total nonlinear polarization. Using the analytical result, we calculated the saturation and spectral characteristics of the signal. The maximum signal strength was found to be obtained for zero detuning and equal pump and probe field amplitudes of approximately 1.04 E SAT. The analytical expression is a general result applying to all phase-matching arrangements. Extension of the result to account for atomic motion was readily achieved only for the case of the forward geometry. In the case of monochromatic lasers and counterpropagating pump beams the spectral response is modified by the Doppler effect, since nondegenerate terms contribute to the signal. This more complex situation has not been treated here. The use of a two-level atom model and restriction to plane-parallel input beams indicate that the model may not apply to arbitrary polarization arrangements, which are sometimes used to enhance signal-to-noise ratio in luminous or scattering environments. The extension of the model to treat degenerate molecular states and collisional relaxation would not be trivial. However, in the appropriate circumstances, the analytical expression derived in this study is suitable for rapid simulation of complex molecular spectra recorded by means of DFWM with saturating pump and probe fields. DFWM spectra of C in an oxy-acetylene flame have been successfully simulated with this model, and results will be presented elsewhere. APPENDIX A The denominator of P(r) can be rearranged to show that P(r) is proportional to P 0 E 1 r E r E 3 r 1 E 1 r E r E 3 r E 1 r E re 3 r* 1 c.c. 1 1 E 1 r E r. E 3 r (A1) After denoting B(r) (1 ) E 1 (r) E (r) E 3 (r), one can see that P(r) can be expanded into a convergent series, given by Eq. (10), if E 1 (r) E (r) E 3 (r)* c.c. /B(r) 1. To verify this inequality we start with the identity E 1 (r) E (r) E 3 (r)] c.c. 0, which leads to E 1 r E r E 3 r E 1 r E re 3 r* c.c.. (A) This in turn implies that E 1 (r) E (r)e 3 (r)* c.c. /B(r) 1, so the inequality has been demonstrated. APPENDIX B We consider the binomial expansion m m! 1 cos x m t0 m t!t! cos xt. (B1) As f E p /(1 ) E p E 3 1, we may write the following convergent Taylor series: 1 1 f cos x a f n0 n a n 1! cos x n. a 1!n! (B)

8 Bratfalean et al. Vol. 16, No. 6/June 1999/J. Opt. Soc. Am. B 959 The product of (B1) and (B) gives 1 cos x m 1 f cos x a f n0 n a n 1! cos x n a 1!n! m t0 m! m t!t! cos xt. (B3) APPENDIX C The perturbation treatment given by Abrams and Lind considers only a weak probe intensity. In this case only 1 (f, 1) must be retained in Eq. (). Then E 4 s 0kV ie 3 E p sb 0 1 cos x 1 f cos x dx. (C1) We define u n t. Then Eq. (B3) can be rewritten as Using the change of variable, w x/, we can write the integral in Eq. (C1), as 1 cos x m 1 f cos x a u0 cos x u m! a 1! minu, m f ut a u t 1! t0, m t!t!u t! (B4) where min(u, m) stands for the minimum of u and m. Using the standard form integral, 3 we have 0 u 1!! cos x u, (B5) u!! where n!! n(n )(n 4)... and so on, up to the lowest positive integer. Taking into account Eq. (B5) when integrating Eq. (B4) from 0 to, we obtain 0 1 cos x m 1 f cos x a dx u0 u 1!! u!! m! a 1! / cos w 40 sin w cos dw w 3, (C) where 1 f and 1 f and where the result of the integral in Eq. (C) may be found from a table of standard integrals. 3 Substituting Eq. (C) in Eq. (C1) and considering again that E 3 is small, we finally obtain E 4 L 0kV i E 3 E p (1 4E p s 1 4E p 1 1/ ) 1, (C3) which is equivalent to the result obtained by Abrams and Lind. ACKNOWLEDGMENTS R. T. Bratfalean and G. M. Lloyd are grateful to the University of Oxford and to the Engineering and Physical Sciences Research Council (UK), respectively, for personal financial support. minu, m f ut a u t 1! t0. m t!t!u t! (B6) By replacing the parameter a in Eq. (B6) with either or ( 1) we obtain analytical expressions for 1 ( f, m) and ( f, m), as follows: m! 1 f, m u 1!! 1! u0 u!! f, m minu, m t0 m!! u0 minu, m t0 f ut u t 1 u t!m t!t!, u 1!! u!! f ut u t u t!m t!t!. (B7) (B8) REFERENCES 1. R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).. R. L. Abrams and R. C. Lind, Opt. Lett., and 3, 05 (1978). 3. D. Bloch and M. Ducloy, J. Opt. Soc. Am. 73, (1983) and 73, (1983). 4. G. Grynberg, M. Pinard, and P. Verkerk, Opt. Commun. 50, (1984). 5. R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, J. Opt. Soc. Am. B 10, (1993). 6. R. G. Caro and M. C. Gower, IEEE J. Quantum Electron. QE-18, (198). 7. W. P. Brown, J. Opt. Soc. Am. 73, (1983). 8. P. Ewart and S. V. O Leary, J. Phys. B 17, (1984). 9. M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, J. Opt. Soc. Am., (1985). 10. J. Cooper, A. Charlton, D. Meacher, P. Ewart, and G. Alber, Phys. Rev. A 40, (1989). 11. D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, Phys. Rev. A 46, (199). 1. M. Ducloy, F. A. M. de Oliveira, and D. Bloch, Phys. Rev. A 3, (1985). 13. P. Verkerk, P. Pinard, and G. Grynberg, Phys. Rev. A 35, (1987). 14. S. Williams, R. N. Zare, and L. A. Rahn, J. Chem. Phys. 101, (1994). 15. D. G. Steel, R. C. Lind, and J. F. Lam, Phys. Rev. A 3, (1981).

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