## Nonlinear gain amplification due to two-wave mixing in a broad-area semiconductor amplifier with moving gratings

Optics Express, Vol. 16, Issue 8, pp. 5565-5571 (2008)

http://dx.doi.org/10.1364/OE.16.005565

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### Abstract

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

## 1. Introduction

1. H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. **22**, 1349–1354 (1986). [CrossRef]

2. P. Kürz, R. Nagar, and T. Mukai, “Highly efficient phase conjugation using spatially nondegenerate fourwave mixing in a broad-area laser diode,” Appl. Phys. Lett. **68**, 1180–1182 (1996). [CrossRef]

3. M. Lucente, G. M. Carter, and J. G. Fujimoto, “Nonlinear mixing and phase conjugation in broad-area diode lasers,” Appl. Phys. Lett. **53**, 467–469 (1988). [CrossRef]

6. P. M. Petersen, E. Samsøe, S. B. Jensen, and P. E. Andersen, “Guiding of laser modes based on self-pumped four-wave mixing in a semiconductor amplifier,” Opt. Express **13**, 3340–3347 (2005). [CrossRef] [PubMed]

7. P. Günter and J.-P. Huignard, eds., *Photorefractive Materials and Their Applications I and II*, (Springer-Verlag, Berlin, 1988, 1989). [CrossRef]

8. A. Brignon and J.-P. Huignard, “Two-wave mixing in Nd:YAG by gain saturation,” Opt. Lett. **18**, 1639–1641 (1993). [CrossRef] [PubMed]

9. M. Chi, S. B. Jensen, J.-P. Huignard, and P. M. Petersen, “Two-wave mixing in a broad-area semiconductor amplifier,” Opt. Express **14**, 12373–12379 (2006). [CrossRef] [PubMed]

9. M. Chi, S. B. Jensen, J.-P. Huignard, and P. M. Petersen, “Two-wave mixing in a broad-area semiconductor amplifier,” Opt. Express **14**, 12373–12379 (2006). [CrossRef] [PubMed]

*P*, the diffusion constant

_{s}*D*, [9

9. M. Chi, S. B. Jensen, J.-P. Huignard, and P. M. Petersen, “Two-wave mixing in a broad-area semiconductor amplifier,” Opt. Express **14**, 12373–12379 (2006). [CrossRef] [PubMed]

*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

*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.:

*n*is the refractive index of the semiconductor material, and

*c*is the velocity of light in vacuum, and the

*ε*

_{0}is the vacuum permittivity. The total electric field is given by: [9

**14**, 12373–12379 (2006). [CrossRef] [PubMed]

10. G. P. Agrawal, “Four-wave mixing and phase conjugation in semiconductor laser media,” Opt. Lett. **12**, 260–262 (1987). [CrossRef] [PubMed]

*K*

_{1}and

*K*

_{2}are the wave vectors of the pump and signal beam in the amplifier.

*P*is the induced polarization in the semiconductor amplifier. It is given by: [9

**14**, 12373–12379 (2006). [CrossRef] [PubMed]

10. G. P. Agrawal, “Four-wave mixing and phase conjugation in semiconductor laser media,” Opt. Lett. **12**, 260–262 (1987). [CrossRef] [PubMed]

**14**, 12373–12379 (2006). [CrossRef] [PubMed]

10. G. P. Agrawal, “Four-wave mixing and phase conjugation in semiconductor laser media,” Opt. Lett. **12**, 260–262 (1987). [CrossRef] [PubMed]

*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.

*N*is governed by the following rate Eq. [6

6. P. M. Petersen, E. Samsøe, S. B. Jensen, and P. E. Andersen, “Guiding of laser modes based on self-pumped four-wave mixing in a semiconductor amplifier,” Opt. Express **13**, 3340–3347 (2005). [CrossRef] [PubMed]

*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:

*N*is the average carrier density,

_{B}*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*≪

*N*. Inserting Eqs. (2) and (6) into Eq. (5), we find after some simple calculations that the average carrier density

_{B}*N*and the carrier modulation

_{B}*ΔN*are given by:

*E*

_{0}|

^{2}=|

*A*

_{1}|

^{2}+|

*A*

_{2}|

^{2}is the average intensity, and

*P*=(

_{s}*ħω*)/(Γ

*aτ*) is the saturation intensity.

*N*and carrier modulation

_{B}*ΔN*, after some calculations, the coupled-wave equations for two-wave mixing with moving gratings are obtained:

*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;[9

**14**, 12373–12379 (2006). [CrossRef] [PubMed]

**14**, 12373–12379 (2006). [CrossRef] [PubMed]

*A*

_{2}|

^{2}≪|

*A*

_{1}|

^{2}≪

*P*, 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:

_{s}*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:

*g*as the natural logarithm of the ratio of the output intensity of signal with the coherent pump to that with the non-coherent pump:

_{TWM}*z*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. [9

_{0}**14**, 12373–12379 (2006). [CrossRef] [PubMed]

*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. [9

**14**, 12373–12379 (2006). [CrossRef] [PubMed]

*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. [9

**14**, 12373–12379 (2006). [CrossRef] [PubMed]

## 3. Calculation and discussion

*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; [9

**14**, 12373–12379 (2006). [CrossRef] [PubMed]

*A*

_{1}(

*z*

_{0})|

^{2}=48.8 mW, |

*A*

_{10}|

^{2}=9.1 mW,

*P*=220 mW,

_{s}*Dτ*=4.1 µm

^{2},

*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.

*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:

*N*

_{m}of the carrier density for the generating of gain and phase gratings is:

*π*-

*θ*between the interference pattern and the carrier density grating. Since the gain varies linearly with carrier density, the gain grating Δ

*g*is also

*π*-

*θ*out of phase with the intensity pattern, i.e.,

*n*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.

*g*

_{gain}is:

*g*

_{phase}is: [7

7. P. Günter and J.-P. Huignard, eds., *Photorefractive Materials and Their Applications I and II*, (Springer-Verlag, Berlin, 1988, 1989). [CrossRef]

*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}.

*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.

*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

## 4. Conclusion

*δ*and

*β,*the TWM gain can be positive or negative. The energy exchange between the pump and signal beams occurs when

*δ*≠0.

## Acknowledgment

## References and Links

1. | H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. |

2. | P. Kürz, R. Nagar, and T. Mukai, “Highly efficient phase conjugation using spatially nondegenerate fourwave mixing in a broad-area laser diode,” Appl. Phys. Lett. |

3. | M. Lucente, G. M. Carter, and J. G. Fujimoto, “Nonlinear mixing and phase conjugation in broad-area diode lasers,” Appl. Phys. Lett. |

4. | M. Lucente, J. G. Fujimoto, and G. M. Carter, “Spatial and frequency dependence of four-wave mixing in broad-area diode lasers,” Appl. Phys. Lett. |

5. | D. X. Zhu, S. Dubovitsky, W. H. Steier, K. Uppal, D. Tishinin, J. Burger, and P. D. Dapkus, “Noncollinear four-wave mixing in a broad area semiconductor optical amplifier,” Appl. Phys. Lett. |

6. | P. M. Petersen, E. Samsøe, S. B. Jensen, and P. E. Andersen, “Guiding of laser modes based on self-pumped four-wave mixing in a semiconductor amplifier,” Opt. Express |

7. | P. Günter and J.-P. Huignard, eds., |

8. | A. Brignon and J.-P. Huignard, “Two-wave mixing in Nd:YAG by gain saturation,” Opt. Lett. |

9. | M. Chi, S. B. Jensen, J.-P. Huignard, and P. M. Petersen, “Two-wave mixing in a broad-area semiconductor amplifier,” Opt. Express |

10. | G. P. Agrawal, “Four-wave mixing and phase conjugation in semiconductor laser media,” Opt. Lett. |

11. | J. R. Marciante and G. P. Agrawal, “Nonlinear mechanisms of filamentation in broad-area semiconductor lasers,” J. Quantum Electron. |

**OCIS Codes**

(140.3280) Lasers and laser optics : Laser amplifiers

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.7070) Nonlinear optics : Two-wave mixing

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: February 15, 2008

Revised Manuscript: March 28, 2008

Manuscript Accepted: March 31, 2008

Published: April 4, 2008

**Citation**

Mingjun Chi, Jean-Pierre Huignard, and Paul Michael Petersen, "Nonlinear gain amplification due to two-wave mixing in a broad-area semiconductor amplifier with moving gratings," Opt. Express **16**, 5565-5571 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5565

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### References

- H. Nakajima and R. Frey, "Collinear nearly degenerate four-wave mixing in intracavity amplifying media," IEEE J. Quantum Electron. 22, 1349-1354 (1986). [CrossRef]
- P. Kürz, R. Nagar, and T. Mukai, "Highly efficient phase conjugation using spatially nondegenerate four-wave mixing in a broad-area laser diode," Appl. Phys. Lett. 68, 1180-1182 (1996). [CrossRef]
- M. Lucente, G. M. Carter, and J. G. Fujimoto, "Nonlinear mixing and phase conjugation in broad-area diode lasers," Appl. Phys. Lett. 53, 467-469 (1988). [CrossRef]
- M. Lucente, J. G. Fujimoto, and G. M. Carter, "Spatial and frequency dependence of four-wave mixing in broad-area diode lasers," Appl. Phys. Lett. 53, 1897-1899 (1988). [CrossRef]
- D. X. Zhu, S. Dubovitsky, W. H. Steier, K. Uppal, D. Tishinin, J. Burger, and P. D. Dapkus, "Noncollinear four-wave mixing in a broad area semiconductor optical amplifier," Appl. Phys. Lett. 70, 2082-2084 (1997). [CrossRef]
- P. M. Petersen, E. Samsøe, S. B. Jensen, and P. E. Andersen, "Guiding of laser modes based on self-pumped four-wave mixing in a semiconductor amplifier," Opt. Express 13, 3340-3347 (2005). [CrossRef] [PubMed]
- P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications I and II, (Springer-Verlag, Berlin, 1988, 1989). [CrossRef]
- A. Brignon and J.-P. Huignard, "Two-wave mixing in Nd:YAG by gain saturation," Opt. Lett. 18, 1639-1641 (1993). [CrossRef] [PubMed]
- M. Chi, S. B. Jensen, J.-P. Huignard, and P. M. Petersen, "Two-wave mixing in a broad-area semiconductor amplifier," Opt. Express 14, 12373-12379 (2006). [CrossRef] [PubMed]
- G. P. Agrawal, "Four-wave mixing and phase conjugation in semiconductor laser media," Opt. Lett. 12, 260-262 (1987). [CrossRef] [PubMed]
- Q1. J. R. Marciante and G. P. Agrawal, "Nonlinear mechanisms of filamentation in broad-area semiconductor lasers," J. Quantum Electron. 32, 590-596 (1996). [CrossRef]

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