Two-wave mixing (TWM) is an interesting area in nonlinear optics and has been intensively investigated in the past three decades. TWM can take place in many different nonlinear media, such as second-order nonlinear media like photorefractive materials (Marrakchi et al., 1981; Huignard & Marrakchi, 1981 ; Yeh, 1983; 1989; Garrett et al., 1992), third-order nonlinear materials like Kerr media (Silberberg &Bar-Joseph, 1982; 1984; Yeh, 1986; 1989; Grandclément et al., 1987; McGraw, 1992), and in gain media like YAG (Brignon & Huignard, 1993). The microcosmic physical process for TWM in different nonlinear media is different. But in general the TWM process can be explained as: two coherent beams are incident on a nonlinear medium and a interference pattern is formed in the medium, such a pattern is characterized by a periodic spatial variation of the intensity; thus a refractive index and/or a gain (absorption) periodic variations will be induced because of the nonlinear response of the medium, and these refractive index and gain variations are usually called volume refractive index (or phase) grating and gain (or absorption) grating; the two beams propagate through the volume gratings formed by them and they undergo Bragg scattering (the Bragg condition is satisfied automatically); one beam scatters into the other and vice versa, so the energy and phase exchanges may occur between these two beams, i.e., the TWM takes place.
Nonlinear four-wave mixing in narrow-stripe and broad-area semiconductor lasers and amplifiers is of interest as a method to obtain high phase conjugate reflectivity (Nakajima & Frey, 1985; 1986; Frey, 1986; Agrawal, 1987; Kürz et al., 1996). The nonlinear four-wave mixing can also be used to measure carrier dynamics and gain behaviour directly in the devices, as well as for understanding device physics and application (Lucente et al., 1988a; 1988b; Zhu, 1997a; 1997b; 1997c). The gain and refractive index gratings created in broad-area semiconductor lasers by coherent four-wave mixing are very interesting nonlinear interactions which may be applied to realize high brightness semiconductor lasers as well as to study the carrier dynamics and the physics of the devices (Petersen et al., 2005). But no work on TWM was done in broad-area semiconductor amplifiers previously.
In this chapter, we present both the theoretical and experimental results of TWM in broad-area semiconductor amplifier. For the generality, we assume that the frequencies of the pump beam and the signal beam are different, i.e., a moving gain grating and a moving refractive index grating are induced in the broad-area semiconductor amplifier. The coupled-wave equations of TWM are derived based on 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 light 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. The different contributions from both the refractive index grating and the gain grating to the TWM gain are analyzed. Depending on the moving direction of the gratings and the anti-guiding parameter, the optical gain of the amplifier may increase or decrease due to the TWM.
As a special case, the degenerate TWM (the frequencies of the pump beam and the signal beam are the same, i.e., a static gain grating and a static refractive index grating are induced in the semiconductor amplifier) in an 810 nm, 2 mm long and 200 m wide GaAlAs broad-area amplifier is investigated experimentally. In this case, the theoretical results show that when the amplifier is operated below transparency the optical gain of both beams is increased due to the induced gain grating, and when the amplifier is operated above transparency the optical gain of both beams is decreased due to the gain grating. The refractive index grating does not affect the optical gain of both beams; and there is no energy exchange between the pump and the signal beams. The dependence of the TWM gain on the output power of the pump and angle between the two beams is measured. The experimental results show good agreement with the theory. A diffusion length of 2.0 m for the carrier is determined from the experiment.
2. Theory of TWM in broad-area semiconductor amplifier
The TWM geometry is shown in Figure 1, the pump beam of amplitude
Inserting Eqs. (2) and (3) into Eq. (1), and using the obtained results of the average carrier density
In the small signal approximation, and if we assume that the total intensity of the two beams is much less than the saturation intensity, i.e.,
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
3. Experiment of the degenerate TWM in a broad-area amplifier
In order to verify the theory described in Section 2, a special case of TWM, i.e., degenerate TWM, in a broad-area semiconductor amplifier is investigated experimentally. For this case, the frequencies of the pump and the signal are the same, i.e.,
The equations show that the coupling term between the two beams decreases the optical gain (above transparency) or absorption (below transparency) for both beams simultaneously. This is different to the situation in photorefractive materials, where one beam is amplified and the other is decreased at the same time (Marrakchi et al., 1981; Huignard & Marrakchi, 1981; Yeh, 1983; 1989). This is also different to the situation in Kerr media, where the intensity of one beam is not affected by the other beam in the degenerate TWM case (Yeh, 1986; 1989; Chi et al., 2009).
To clarify this phenomenon, the relative position of the intensity pattern, the carrier density grating, the refractive index grating and the gain grating is shown in Figure 2 when the amplifier is operated above the transparency. Because of the spatial hole-burning effect, the carrier density grating is
Eq. (17) shows that the
The experimental set-up is shown in Figure 3. The set-up is arranged like a Mach-Zehnder interferometer. The pump beam
The broad-area amplifier is an 810 nm, 2 mm long and 200 m wide GaAlAs amplifier. It was grown by the Metallorganic Chemical Vapor Phase Deposition (MOCVD) technique on a GaAs substrate by Alcatel Thales III-V Lab. The structure contains a Large Optical Cavity (LOC), which has a thickness of approximately 1 µm, and which consists of a tensile-strained GaInP quantum well, two GaInP barriers and two AlGaInP claddings. Both facets of the amplifier are antireflection coated; the reflectivity is less than 0.1%.
First, the dependence of the
The dependence of the
power of the pump is around 35 mW. The experimental results are shown in Figure 5. Fitting the experimental data with Eq. (17),
To obtain the coupled-wave equations of TWM, three assumptions are made. Here we should discuss the validity of these assumptions in our experiment. The first is the plane-wave assumption. In the experiment, since the two beams are coupled into the amplifier from an external laser, the mode of the two beams in the slow axis is determined by the external laser and the focusing optics. The two beams are nearly Gaussian beams in the slow axis, they are collimated by the aspherical lens and the width of the beams is around 140 μm. We believe the plane-wave assumption is a good approximation for these two beams in this direction. The wave guiding mode of the field distribution in the fast axis does not affect the derivation of the equations (Marciante & Agrawal, 1996). The second is the linear variation of the material gain
In conclusion, the degenerate TWM in broad-area semiconductor amplifier is investigated experimentally. The experimental results show good agreement with the theory. The validity of the theory is discussed.
4. Calculations and discussion
Unlike the condition of degenerate TWM, where only static gratings are generated; the coupling term between the two beams has different contribution to the optical gain of these two beams for the nondegenerate TWM (Chi et al., 2008). The nondegenerate TWM may increase the power of one beam and decrease the power of another beam in this case, i.e., energy exchange occurs.
According to Eq. (14), the dependence of
The modulation part
Eq. (19) shows, because of the hole-burning effect and the finite response time of the broad-area amplifier, there is a phase difference
The refractive index grating is
The relative position of the interference pattern, the carrier density grating, the refractive index grating and the gain grating formed in the broad-area amplifier is shown in Figure 7.
The TWM gain caused by the gain grating
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 the signal beams simultaneously, thus it will not cause the energy exchange between the two beams. The TWM gain caused by the phase grating
In conclusion, the TWM in broad-area semiconductor amplifier in nondegenerate condition 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 signal, but much less than the saturation intensity of the amplifier. A special case of TWM, degenerate TWM, is investigated experimentally in a GaAlAs broad-area semiconductor amplifier. The experimental results show good agreement with the theory, and the validity of the theory for this experiment is discussed. A diffusion length of 2.0 μm is determined from the experiment. The TWM gain in broad-area semiconductor amplifier is calculated as a function of the frequency difference between the pump and the signal based on the data obtained from the degenerate TWM experiment; and the calculated results are discussed based on the different contributions from the refractive index grating and the gain grating to the TWM gain. Depending on δ and β, the TWM gain in semiconductor broad-area amplifier can be positive or negative. The energy exchange between the pump and signal beams occurs when δ ≠ 0.Acknowledgements
The authors acknowledge the financial support of the European community through the project WWW.BRIGHTER.EU (Grant No. FP6-IST-035266). Mingjun Chi wishes to acknowledge the Danish Research Agency under grant no. 26-04-0229.
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