1. Introduction

Two-beam coupling (TBC) has a rich history in nonlinear optics. Early studies established two conditions for one-way energy transfer between two electromagnetic waves in a Kerr medium: 1) the frequencies must be nondegenerate, and 2) the nonlinearity of the medium must have a finite response time [1]. The nonlinear refractive index is usually assumed to obey a simple Debye relaxation model. Energy transfer under these conditions is always from the high frequency beam to the low frequency beam for a positive Kerr nonlinearity. Silberberg and Joseph showed that counter-propagating beams in such a Kerr medium can exhibit optical instabilities and self-oscillation [1,2], while Yeh gave an exact solution for energy transfer between two co-propagating beams including the effect of linear absorption [3]. In each of these studies, both beams had non-zero input values.

TBC also describes several stimulated scattering phenomena, such as stimulated Raman (SRS), Brillouin (SBS), and Rayleigh-wing scattering (SRWS). The Stokes (down-shifted frequency) wave in these phenomena arises internally from scattered light and experiences gain at the expense of the incident laser beam. The Stokes frequency shift is a normal mode of the medium (e.g., vibrational) for SRS, an acoustic frequency for SBS, and the inverse reorientation time of an anisotropic molecule for SRWS. SRS, SBS, and SRWS are resonant phenomena, indicating that the nonlinear medium has a finite response time. In each of these stimulated scattering effects the small-signal gain of the Stokes wave is proportional to the laser intensity [4].

Recently, He et al. reported the observation of stimulated backscattering in a two-photon absorption (TPA) medium, where the frequency of the stimulated wave was identical to the incident laser frequency (within the resolution of their interferometer), and the small-signal gain was quadratic in the incident laser intensity [5,6], in contrast to the stimulated light scattering phenomena described above. They considered a stimulated thermal Rayleigh scattering model based on TPA-enhanced temperature and density fluctuations, but ruled this out due to (a) the broad linewidth of their pump laser, which would severely reduce the gain of the stimulated wave, and (b) the fact that the peak gain in this theory is for an anti-Stokes-shifted wave, contrary to their experimental results. They concluded that the stimulated wave was due to a Bragg grating formed by the superposition of the incident laser beam and an elastically (Rayleigh) backscattered wave, creating an index grating via a TPA-resonance enhanced nonlinear index coefficient n ₂, i.e., a Kerr effect. The stimulated backscattered wave is then due to reflection of the laser from this grating.

The model proposed by He et al. would appear to violate the two conditions previously established for one-way energy transfer by TBC in a Kerr medium. The frequencies are degenerate, and the third order susceptibility related to TPA is due to electronic polarization, which has a virtually instantaneous response for nanosecond laser pulses. Although, under the condition of a TPA resonance, this mechanism may indeed have a finite response time, the intensity dependence of the Kerr effect in TBC is inconsistent with the experimental results reported by He et al., as I discuss below.

Here I present a simple model that is in agreement with the main results of the observed stimulated backscattering in TPA media: the frequency of the stimulated wave is unshifted with respect to that of the incident laser, and the small-signal gain is quadratic in the laser intensity. Two assumptions are made: 1) the incident laser has a nonlinear time dependent phase, consistent with a relatively broad linewidth, and 2) TPA produces a non-negligible population of an excited state of the medium [7,8].

2. Model

Consider a field with a time dependent amplitude and phase, E(t) ~ A(t)exp[-i φ(t)]. A Taylor series expansion of the phase yields (ignoring a constant term) φ(t) =ωt + bt ² +…, where ω is the central frequency of the wave, and b is a linear chirp coefficient. For simplicity, I will ignore higher order terms. Co-propagating TBC with chirped pulses has been examined in Kerr media with a finite response time [9–11]. However, none of these previous studies have considered the interaction of counter-propagating waves through TPA-populated excited states.

The interaction of the two chirped waves is illustrated schematically in Fig. 1. Let the total field in a medium of length d be described by

where k = nω/c, n is the linear refractive index, and 2τ= 2n(d - z)/c is the relative time delay at position z and time t between the forward propagating laser wave (L) and the backward propagating scattered wave (S). I assume that the scattered wave originates at z = d by some elastic scattering process, so A_S (d,t) = √A_L (d,t) where η is a constant << 1. Consequently, the scattered wave has the same spectral composition as the incident laser wave. However, for z ≠ d a lower frequency part of the scattered wave is always interacting with a higher frequency part of the incident wave. The total polarization of the medium is given by

where χ ⁽ⁿ⁾ is the n-th order susceptibility, and the angular brackets indicate an average over a time longer than an optical period but shorter than (2bτ)^-1. I have assumed that TPA produces a single excited state of number density N_e << N, the total number density of the nonlinear chromophore. I have also assumed an isotropic medium with linearly polarized light for simplicity, and both Δχ⁽¹⁾ =${\chi }_e^{\left(1\right)}$ - ${\chi }_g^{\left(1\right)}$ and χ ⁽³⁾ are complex quantities (g signifies the ground state of the medium). The excited state decays back to the ground state with a time constant T_e , so N_e obeys the following kinetic equation:

where σ ₂ = β/N is the TPA cross section (β is the TPA coefficient), and I(z,t) = 2ε ₀ nc〈E ²(z,t〉 is the total intensity. Let the amplitudes be slowly varying in time compared to T_e . Equation (3) can then be integrated to yield

 

Fig. 1. Schematic diagram of the chirped pulse interaction in a nonlinear medium. The backscattered pulse is assumed to be derived from the incident laser pulse at the back of the medium where it has the same frequency as the incident pulse. The color schematically indicates the frequency chirp, with blue representing higher frequencies and red representing lower frequencies.

Download Full Size | PDF

where ε ₀ is the free-space permittivity. I will make the assumption that (4bτT_e )² << 1.

Following a procedure directly analogous to that of Yeh [3] and Boyd [4] for non-degenerate TBC, Eqs. (4a)–(4d) are substituted into Eq. (2), and then both Eqs. (1) and (2) are substituted into Maxwell’s wave equation in the slowly varying amplitude approximation. Matching up synchronous terms, the following coupled-wave equations for the laser and backscattered intensities are derived:

where g and γ_eff are the backward wave gain and effective three-photon absorption (3PA) coefficients, respectively, with

where Δ${\chi }_R^{\left(1\right)}$ = Re(Δχ ⁽¹⁾), Δσ = σ_e -σ_g is the difference between the linear absorption cross sections of the excited and ground states, and I_sat = (2ħω/σ ₂ T_e )^1/2 is the two-photon saturation intensity. Note that g = 0 when b = 0 (no chirp). Hence, without chirp there can be no growth of the backward scattered wave. The gain g also carries the sign of b (positive or negative chirp). For a negatively chirped pulse, the scattered wave will be attenuated. For the rest of the paper, I will assume that b > 0.

3. Results and discussion

Let us examine Eqs. (5a) and (5b) in the case when I_S << I_L ≈ constant. It can be seen that the backward stimulated wave will experience exponential growth when $I_L>\frac{2\beta }{\left(\frac{1}{2}g-3{\gamma }_{\mathit{eff}}\right)}$. This defines the threshold condition for stimulated backscattering when there is no linear absorption. From Eq. (5b), the small signal gain G, given in terms of ΔI_S = I_S (0)-I_S (d), is G = ΔI_S /I_S (d) ∝ $I_L^2$ for input intensities where the gain dominates TPA. Also, the reflectance of the scattered wave is defined by R = I_S (0)/I_L (0), and the change ΔR=R-η = ΔI_S /I_L ∝I_S (d)I_L ∝GI_S (d)/I_L . These results are in agreement with the data presented by He et al. for the measured small-signal gain and reflectance in a 0.01-M solution of PRL 802 in THF [6].

Under conditions where the nonlinear absorption terms (β and γ_eff ) can be ignored, Eqs. (5a) and (5b) can be solved analytically. The result can be expressed as

where η ₀ = I_S (d)/I_L (0), and

Note that in general η ₀ ≠ η, although when the gain is sufficiently small η ₀ ≈ η. Also, when the gain is not too large so that (1 - R) >> η ₀, Eq. (8) can be simplified to the following approximation:

It is interesting to compare this result to the case of SBS, for which the denominator of Eq. (10) becomes exp(Γ)-,R with Γ→(1-R)g_BI_L (0)d , where g_B is the Brillouin gain coefficient [4]. Thus, in the case of negligible loss by absorption, the backscattered reflectance as a function of incident laser intensity will look similar to the Brillouin reflectance. A plot of the reflectance given by Eq. (10) for a constant input I_S (d)/I_L (0), over a range of Γ where the approximation is valid, is shown in Fig. 2.

Numerical solutions of Eqs. (5a) and (5b) are also given in Fig. 2. These illustrate the agreement of the approximation given by Eq. (10) with the exact result, and the effect of including the nonlinear absorption terms. Figure 3 gives I_S (0) as a function of I_L (0). Parameters have been adjusted to yield results close to the experimental data of He et al [5,6]. The coefficients used for the solid curve in Fig. 3 are β = 9.46 cm/GW (the value quoted in Ref 6), η = 0.04, g = 1800 cm³/GW², and γ_eff = 120 cm³/GW², with d = 1 cm. In Figure 4 the corresponding plot for the laser transmittance is given. Here the transmittance is the ratio of output power to input power. To compute this, I assumed a Gaussian dependence for the input intensity and then integrated the output intensity over the area of the cylindrically symmetric beam. The result is compared with what would be expected for pure TPA (no TBC or ESA). The departure from pure TPA in this case is due primarily to energy transfer from the laser to the backscattered beam. Figure 3 also shows a somewhat different result (dashed curve) yielding comparable backscattered intensity. Often the effective TPA cross section measured in the nanosecond regime is a factor ~ 100 larger than the intrinsic cross section (usually measured in the femtosecond regime) [7]. Thus, for these calculations I chose β = 0.09 cm/GW. The other parameters are η = 0.02, g = 2860 cm³/GW², and γ_eff = 300 cm³/GW². TPA is low in this case, but the loss due to ESA significantly slows down the growth of the backscattered wave at higher intensities.

To get an order of magnitude of the numbers involved, consider the experiment of He et al. [5,6] and the results shown in Figs. 3 and 4 for β= 9.46 cm/GW. For a laser wavelength of 532 nm and T_e ~ 1 ns [5,8], I_sat ~ 700 MW/cm². For a severely chirped pulse the laser linewidth is Δω _L ~ 2bτ_L where τ_L is the laser pulse width. For Δω _L/2π~ 24 GHz (0.8 cm^-1), τ_L ~ 10 ns, and N = 6 × 10¹⁸ cm^-3 (0.01-M concentration), Eqs. (6) and (7) yield Δ${\chi }_R^{\left(1\right)}$ ~ 4 × 10^-3 and Δσ ~ 1 × 10^-17 cm². I note that this numerical example contravenes the approximation made earlier for which (4bτT_e )² is small compared to 1. To account for deviations due to this, the gain and ESA terms in Eqs. (5a) and (5b) would need to be modified by the inclusion of the factor [1 + (4b τT_e )²]-1. For example, in the negligible nonlinear absorption case, Eq. (9) would be multiplied by a factor ln(1 + α)/α, where α= (4bT_end/c)², which is ~ 1 for α<< 1. In the present numerical example, this factor is ~ 0.5. Consequently, the estimates for Δ${\chi }_R^{\left(1\right)}$ and Δσ should be increased by a factor ~ 2.

 

Fig. 2. Backscattered reflectance as a function of exponential gain factor for various values of nonlinear absorption coefficients. I_S (d)/I_L (0) = 10^-2 in all cases. Circles give values for R calculated by the approximation in Eq. (10).

Download Full Size | PDF

 

Fig. 3. Backscattered intensity as a function of incident laser intensity.

Download Full Size | PDF

 

Fig. 4. Nonlinear transmittance of the incident laser as a function of incident intensity.

Download Full Size | PDF

Notice that Re(χ ⁽³⁾), or n ₂, does not appear in the gain coefficient of Eq. (6). The reason for this is that χ ⁽³⁾ was assumed to have an instantaneous response [Im(χ ⁽³⁾) ∝ β]. There is no two-beam coupling in a Kerr medium with an instantaneous response [3,4]. It is possible however, under TPA resonant conditions, that n ₂ could have a finite response time (i.e., χ ⁽³⁾ is complex with a finite damping coefficient [4]). However, if this is included in the development of the coupled-wave intensity equations, it would lead to a term in the gain G that is proportional to I_L , not $I_L^2$, which would be inconsistent with the experimental results of He et al. [6]. It is also quite likely that the Kerr refractive term would be much smaller than the term proportional to Δ${\chi }_R^{\left(1\right)}$.

The central feature of this theory is contained in the term 4bτT_e = ΔωT_e , where Δω = 4bτ represents the instantaneous difference in the frequency between the forward and backward propagating waves at position z in the medium. Δω varies continuously from 0 at z = d to a maximum of 4bnd/c at z = 0. The interference of the forward and backward propagating waves sets up an interference pattern that is traveling to the left if b > 0. This intensity pattern forms a population grating via TPA, which lags behind the interference pattern due to the finite response time T_e . By examining Eqs. (4c), (4a), and (2), we see that ΔωT_e ≠ 0 results in a contribution to the imaginary part of the total refractive index of the medium due to the presence of the two waves. As Boyd has observed in the context of nondegenerate TBC in a Kerr medium, the only way to obtain a complex refractive index (apart from pure absorption), and thus to achieve energy coupling between the two waves, is that the product ΔωT_e not vanish [4]. In the present theory, this requires a non-zero chirp (b ≠0). This can also be explained by considering the phase difference Δφ = 4bτT_e between the polarization and the field. The rate of energy transfer per unit volume to or from an electric field E by a polarization P is 2Re[iω∣E∣∣P∣exp(iΔφ)], which is 0 if Δφ = 0 [12]. Thus, although backscattering by an unchirped wave could also lead to a population grating, the grating would not transfer energy to the scattered wave because the field and polarization would be in phase. There would thus be no increase in the index modulation as the incident intensity increases and no exponential growth of the scattered wave, i.e., no stimulated scattering. This has an analogy in SBS and SRS. In both cases, the gain is related to the imaginary part of the nonlinear susceptibility, which vanishes when the difference between the laser and scattered wave frequencies shrinks to zero. In addition, the scattered wave in the present theory is attenuated for a negative chirp, analogous to the attenuation of the anti-Stokes wave in SBS and SRS [4].

It should be noted at this stage that linear chirp is not a unique requirement for the energy transfer between incident laser and elastically backscattered waves. Higher order chirp terms in the nonlinear time dependent phase have been ignored for simplicity but will make additional contributions to the beam coupling. Another potential mechanism involves multimode beams. In this case, though, the spectral nature of the backscattered beam will not match that of the incident beam. Take the simple case where the incident laser wave consists of two modes: ω ₀ and ω ₁ = ω ₀ + δω > ω ₀ (δω << ω ₀). Employing the mechanism involving a TPA-populated excited state described above, there will be energy transfer from the ω ₁ mode of the incident laser beam to the ω ₀ mode of the backscattered beam [4]. Likewise, the ω ₁ mode of the backscattered wave will yield its energy to the ω ₀ mode of the laser wave. The backscattered wave will thus be single-mode and its spectrum conspicuously different from the incident laser. However, if the modes in the incident laser beam are equally chirped, both modes of the backscattered wave will be amplified, but not equally. When the laser linewidth is determined primarily by the linear chirp (2bτ_L >> δω), the spectrum of the backscattered wave will superficially resemble that of the incident laser, but the energy distribution amongst the modes will differ. This will generally be the case also when the number of modes is greater than two.

4. Conclusions

In summary, I have presented a model of two-beam coupling in a nonlinear absorption medium whereby energy is transferred from an incident laser beam to an elastically backscattered beam. In the mechanism proposed, the incident laser has a nonlinear time dependent phase and populates an excited state of the medium by two-photon absorption. The incident and backscattered waves superpose to form an interference pattern, leading to the formation of a Bragg grating. The Bragg grating consists of an index modulation resulting from a modulation of the excited state population. This grating is out of phase with the interference pattern which forms it due to the finite lifetime of the two-photon-populated excited state and the frequency chirp of the two waves. Energy flows one-way from the higher frequency part to the lower frequency part of the coupled waves. Complete energy conversion to the backscattered wave is prohibited, however, in part because of loss due to two-photon and excited state absorption. Nevertheless, the power spectrum of the backscattered wave can be nearly identical to that of the incident wave.