The two-wave mixing in a broad-area semiconductor amplifier with moving gratings is investigated theoretically, where a pump beam and a signal beam with different frequencies are considered, thus both a moving phase grating and a moving gain grating are induced in the amplifier. The coupled-wave equations of two-wave mixing are derived based on the Maxwell’s wave equation and rate equation of the carrier density. The analytical solutions of the coupled-wave equations are obtained in the condition of small signal when the total intensity is far below the saturation intensity of the amplifier. The results show that the optical gain of the amplifier is affected by both the moving phase grating and the moving gain grating, and there is energy exchange between the pump and signal beams. Depending on the moving direction of the gratings and the anti-guiding parameter, the optical gain may increase or decrease due to the two-wave mixing.
©2008 Optical Society of America
Nonlinear four-wave mixing in narrow-stripe and broad-area semiconductor lasers is of interest as a method to obtain high phase conjugate reflectivity. [1,2] The nonlinear wave mixing can also be used to measure carrier dynamics and gain behavior directly in the device, as well as for understanding device physics and application. [3–5] The gain and index gratings created in broad-area semiconductor amplifiers by coherent wave mixing are very interesting nonlinear interactions which may apply to realize high brightness semiconductor lasers as well as to study the carriers dynamics and the physics of the devices.  Although two-wave mixing (TWM) has been intensively investigated in photorefractive materials,  only few works was done in gain media.  Recently, we investigated the TWM in a broad-area semiconductor amplifier, both theoretically and experimentally, where the frequency of the pump beam and the signal beam are the same, thus a static phase grating and a static gain grating in the semiconductor amplifier are induced.  The results show that the optical gain of both beams is decreased due to the induced gain grating, the phase grating does not affect the optical gain; there is no energy exchange between the pump and signal beams. 
Here, we present the theoretical results of TWM in a broad-area semiconductor amplifier with moving gratings, i.e., the frequencies of the pump beam and the signal beam are different. Unlike the condition of static gratings, both a moving phase gating and a moving gain grating affect the optical gain of the amplifier. The coupled-wave equations of two-wave mixing are derived. The analytical solutions to the coupled-wave equations are obtained in the condition of small signal and the total intensity is far below the saturation intensity. We find the optical gain can be increased or decreased due to the TWM, and there is energy exchange between the pump and signal beams. The aim of the presented theory is to study the physics of the device, and to analyze the different contributions from phase grating and gain grating to the TWM gain. The parameters of the device, such as the saturation intensity of the device Ps, the diffusion constant D,  the anti-guiding parameter β and the spontaneous recombination lifetime τ, can be obtained by fitting the experimental result with the analytical results.
2. Theory of TWM in broad-area semiconductor amplifier with moving gratings
The TWM geometry is shown in Fig. 1, the pump beam of amplitude A 1 and signal beam of the amplitude A 2 are coupled into the broad-area amplifier. Both beams are linearly polarized, and the frequencies are ω 1 and ω 2 respectively. The two beams interfere in the medium to form a moving interference pattern, and a moving modulation of the carrier density in the active media is caused, thus both a moving gain and a moving phase gratings are created. The nonlinear interaction in the gain media is governed by the wave Eq.:
the quantity β is the anti-guiding parameter accounting for the carrier-induced index change in semiconductor amplifier, and g(N) is the gain that is assumed to vary linearly with carrier density N, i.e., g(N)=Γa(N-N 0) where a is the gain cross-section, Γ is the confinement factor, and N 0 is the carrier density at transparency.
The carrier density N is governed by the following rate Eq. :
where I is the injected current, q is the electron charge, V is the active volume, τ is the spontaneous recombination lifetime, D is the ambipolar diffusion constant. In the TWM configuration the origin of the gain and index gratings is the modulation of the carrier density due to the interference between A 1 and A 2. Thus the carrier density that leads to the formation of the moving gratings may be written as:
where NB is the average carrier density, ΔN is the induced carrier modulation. K=K 2-K 1=4πsin((θ 1-θ 2)/2) is the grating vector; θ 1 is the angle between the pump beam and Z axis, and θ 2 is the angle between the signal beam and Z axis; we assume θ 1=-θ 2, thus the direction of the grating vector is in the X direction. δ=ω 2-ω 1 is the frequency difference between the single and pump beams. In the following perturbation analysis it is assumed that ΔN≪NB. Inserting Eqs. (2) and (6) into Eq. (5), we find after some simple calculations that the average carrier density NB and the carrier modulation ΔN are given by:
where |E 0|2=|A 1|2+|A 2|2 is the average intensity, and Ps=(ħω)/(Γaτ) is the saturation intensity.
Inserting Eqs. (2) and (3) into Eq. (1), and using the obtained results of the average carrier density NB and carrier modulation ΔN, after some calculations, the coupled-wave equations for two-wave mixing with moving gratings are obtained:
where α=Γa(Iτ/qV-N 0)/2 is the small-signal gain coefficient of the amplifier. Since the refractive index of the semiconductor material is high, normally the angles θ 1(or θ 2) is less than 2° in experiment; so the cosine factor in Eqs. (9) and (10) is neglected below. Unlike the condition when the pump and signal beams have the same frequency, and only static gratings are generated; the coupling term between the two beams has different contribution to the optical gain of this two beams. 
In the small signal approximation, and if we assume that the total intensity of the beams is much less than the saturation intensity, i.e., |A 2|2≪|A 1|2≪Ps, the terms accounting for saturation in the denominator and the term accounting for the coupling in Eq. (9) may be neglected. Thus the coupled-wave equations can be solved analytically. The solutions are:
where A 10 and A 20 are the amplitudes of pump and signal beam at the front facet of the amplifier. γ 1 is a parameter defined as:
The first term in Eq. (13) is for the saturation effect, the second term is for the beam coupling.
Define the TWM gain of the signal beam gTWM as the natural logarithm of the ratio of the output intensity of signal with the coherent pump to that with the non-coherent pump:
where z0 is the length of the semiconductor amplifier. The non-coherent pump means the pump beam is not coherent with the signal beam, but the intensity is the same as the coherent pump, thus the term accounting for saturation in Eq. (14) vanishes. In experiment, the coherent pump and the non-coherent pump can be achieved by changing the polarization of the pump beam.  Equation (14) shows that g TWM changes linearly with the output intensity (power) of the pump, and it decreases quickly when the angle between the two beams increases because the diffusion of carriers washes out the gratings as the angle between the two beams increases. This is the same as the situation when the pump and signal beams have the same frequency, and only static gratings are generated.  Equation (14) also shows that depending on the detuning frequency δ, the TWM gain can be positive or negative no matter the amplifier is operated above or below the transparency (i.e., |A 1(z 0)|2=|A 10|2). This is different from the situation of static gratings.  This new phenomenon will be discussed below.
3. Calculation and discussion
Figure 2 shows the calculated results of the dependence of g TWM on the frequency difference δ with different anti-guiding parameter β, here we assume that the amplifier is operated above the transparent current. In the calculation, we use the same parameters used in and obtained from the TWM experiment in a GaAlAs broad-area amplifier with static gratings;  i.e., |A 1(z 0)|2=48.8 mW, |A 10|2=9.1 mW, Ps=220 mW, Dτ=4.1 µm2, K=0.51 µm-1 (the K number corresponds to a 1.2° angle between the two beams). Assuming that τ is 5 ns.11 From Fig. 2 we can find that when δ=0, the g TWM is negative and independent of β; if β=0, the g TWM is always negative and the curve of TWM gain versus δ is symmetric around the axis of δ=0. If β≠0, however, the g TWM is negative when δ>0, and the g TWM can be negative or positive when δ<0. These properties can be explained by using the relative position of the interference pattern, the carrier density grating, the index grating and gain grating formed in the broad-area amplifier.
Since the frequencies of the pump and signal are different, a moving interference pattern is generated in the amplifier: |E|2=|E 0|2+(A 1 A*2 e i(-kx+δt)+c.c.). Inserting Eq. (8) into Eq. (6), the carrier density is obtained:
The modulation part N m of the carrier density for the generating of gain and phase gratings is:
Eq. (15) shows, because of the hole-burning effect and the finite response time of the material, there is a phase difference π-θ between the interference pattern and the carrier density grating. Since the gain varies linearly with carrier density, the gain grating Δg is also π-θ out of phase with the intensity pattern, i.e.,
The refractive index grating is π out of phase with the gain grating because of the anti-guiding effect, so the refractive index grating Δn is −θ out of phase with the intensity pattern and proportional to the anti-guiding parameter β, i.e.,
λ is wavelength of the incident beams. The relative position of the interference pattern, the carrier density grating, the index grating and gain grating formed in the broad-area amplifier is shown in Fig. 3.
The two-wave mixing gain caused by the gain grating g gain is:
Here we should notice that the effect of the gain grating is the same for both beams, i.e., to increase (below transparent current) or decrease (above transparent current) the intensity of the pump and signal beams simultaneously, thus it will not cause the energy exchange between two beams. The two-wave mixing gain caused by the phase grating g phase is: 
When δ≠0, the refractive index grating will cause energy exchange between two beams, since there is a phase difference -θ (θ≠0) between the intensity pattern and the refractive index grating.7 The two-wave mixing gain gTWM is the sum of g gain and g phase.
When δ=0, (i.e., static gratings are induced in the amplifier), θ is equal to zero; thus the gain grating is π out of phase with the interference pattern, and the phase grating is in phase with the interference pattern. According to Eqs (19) and (20), the gain of the phase grating g phase is zero; and the g TWM equal to g gain, is negative and independent of β.9 If β=0, only the gain grating is generated, according to Eqs. (16) and (19), the two wave mixing gain g TWM is always negative and is symmetric around the axis of δ=0. If β≠0, both a gain and a phase grating are generated; when δ>0 (θ>0), according to Eqs. (19) and (20), both g gain and g phase are negative, so the g TWM is negative; when δ<0 (θ<0), the g gain is negative and the g phase is positive, so g TWM can be positive or negative.
The parameters β and τ can be obtained by fitting the measuring results of g TWM versus δ. The optimal δ to achieve the maximum TWM gain depends on the device parameters τ, D, β and the grating vector K. From Eq. (14), the optimal δ is .
In conclusion, the two-wave mixing in broad-area semiconductor amplifier with moving gratings is investigated theoretically. The coupled-wave equations are derived and analytical solutions are obtained when the intensity of the pump is much larger than that of the single, but much less than the saturation intensity of the amplifier. The two-wave mixing gain is discussed based on the phase of the intensity pattern, carrier density grating, gain grating and refractive index grating. Depending on δ and β, the TWM gain can be positive or negative. The energy exchange between the pump and signal beams occurs when δ≠0.
The authors wish to acknowledge the financial support of the European Community through the FP-6 project WWW.BRIGHTER.EU. Mingjun Chi wishes to acknowledge the Danish Research Agency under grant no. 26-04-0229.
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