## Two-wave mixing in a broad-area semiconductor amplifier

Optics Express, Vol. 14, Issue 25, pp. 12373-12379 (2006)

http://dx.doi.org/10.1364/OE.14.012373

Acrobat PDF (177 KB)

### Abstract

The two-wave mixing in the broad-area semiconductor amplifier was investigated, both theoretically and experimentally. In detail we investigated how the optical gain is affected by the presence of the two-wave mixing interference grating. In the experimental setup we are able to turn on and off the interference pattern in the semiconductor amplifier. This arrangement allows us to determine the two-wave mixing gain. The coupled-wave equations of two-wave mixing were derived based on the Maxwell’s wave equation and rate equation of the carrier density. The analytical solutions of the coupled-wave equations were obtained in the condition of small signal and the total intensity is far below the saturation intensity of the amplifier. The results show that when the amplifier is operated below transparency we obtain an increase in the optical gain, and when the amplifier is operated above transparency we obtain a decrease in the optical gain. The experimental results obtained in an 810 nm, 200 µm wide GaAlAs amplifier show good agreement with the theory. A diffusion length of 2.0 µm is determined from the experiment.

© 2006 Optical Society of America

## 1. Introduction

^{11. H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. 22, 1349–1354 (1986). [CrossRef] ,22. 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] }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.

^{3–53. 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] }Recently, we suggested that the gain and index grating created in broad-area semiconductor amplifier by four-wave mixing may be used to produce novel diode laser systems with better beam quality.

^{66. 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] }Although two-wave mixing (TWM) has been intensively investigated in photorefractive materials,

^{77. P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications I and II, (Springer-Verlag, Berlin, 1988, 1989). [CrossRef] }only few works was done in gain media.

^{88. A. Brignon and J.-P. Huignard, “Two-wave mixing in Nd:YAG by gain saturation,” Opt. Lett. 18, 1639–1641 (1993). [CrossRef] [PubMed] }To our knowledge, this is the first time that the influence of the TWM grating on the optical gain is investigated in broad-area semiconductor amplifier.

## 2. Theory of two-wave mixing in broad-area semiconductor amplifier

*A*

_{1}and signal beam of the amplitude

*A*

_{2}are coupled into the broad-area amplifier. Both beams are at the same frequency

*ω*and linearly polarized. The two beams interfere in the medium and a modulation of the carrier density in the active medium is caused, thus both a gain and a phase gratings are created. The nonlinear interaction in the gain media is governed by the wave equation:

*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:

^{99. 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:

^{99. G. P. Agrawal, “Four-wave mixing and phase conjugation in semiconductor laser media,” Opt. Lett. 12, 260–262 (1987). [CrossRef] [PubMed] }

^{99. 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 material 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 equation

^{66. 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, and |

*E*|

^{2}=|

*A*

_{1}|

^{2}+|

*A*

_{2}|

^{2}the total intensity. 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 gratings may be written as:

*N*

_{B}is the average carrier density,

*ΔN*is the induced carrier modulation.

*K*=

*K*

_{2}-

*K*

_{1}=4

*π*sin(

*θ*/2)/

*λ*is the grating vector, where

*θ*is the angle between the two beams and

*λ*is the wavelength. In the following perturbation analysis it is assumed that

*ΔN*≪

*N*

_{B}. Inserting Eqs. (2) and (6) into Eq. (5), we find after some simple calculations that the average carrier density

*N*

_{B}and the carrier modulation

*ΔN*are given by:

*P*=(

_{s}*ħω*)/(Γ

*aτ*) is the saturation intensity.

*ΔN*, after some calculations, the coupled-wave equations for two-wave mixing are obtained:

*α*=

*Γa*(

*Iτ*/

*qV*-

*N*

_{0})/2 is the small-signal gain coefficient of the amplifier. 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.

*A*

_{2}|

^{2}≪|

*A*

_{1}|

^{2}≪

*P*, the terms accounting for saturation in the denominators of the coupled-wave equations and the term accounting for the coupling in Eq. (9) are 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*is a parameter defined as:

*g*

_{TWM}as:

*z*

_{0}is the length of the semiconductor amplifier. In the experiment, the coherent pump and the non-coherent pump is achieved by changing the polarization of the pump beam. Eq. (13) shows that the

*g*

_{TWM}is negative when the amplifier is operated above the transparency, is positive when it is operated below the transparency, and is zero when it is operated at transparency. It agrees with the analyse above. Eq. (13) also shows that the

*g*

_{TWM}decreases 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 grating as the angle between the two beams increases. These analyses will be verified by experiments of TWM in a semiconductor amplifier below.

## 3. Experiment

*g*

_{TWM}decreases linearly with the output pump power. Fitting the experimental data with Eq. (13), the two parameters: the input power of the pump |

*A*

_{10}|

^{2}and 1/(1+

*DτK*

^{2})

*P*are obtained. The |

_{s}*A*

_{10}|

^{2}is round 9.1 mW, corresponding to a coupling efficiency of 43%; and using the results of

*Dτ*obtained later, the saturation power

*P*is found to be around 220 mW, which is much larger than the output pump power in this experiment. Using the value of |

_{s}*A*

_{10}|

^{2}and the Eq. (11), the optical gain of 1.7 is obtained for highest output power of the pump. The

*g*

_{TWM}is about 5% of the optical gain.

*g*

_{TWM}on the grating vector is also measured by changing the angle between the two beams. The direction of the pump beam is fixed during the experiment; the angle is changed by changing the direction of the signal beam. The injected powers of the pump and the signal measured before the aspherical lens are 21.0 and 4.1 mW; the output power of the pump is around 35 mW. The experimental results are shown in Fig. 4. Fitting the experimental data with Eq. (13),

*Dτ*is obtained to be 4.1 µm

^{2}, leading to a diffusion length

^{1212. J. R. Marciante and G. P. Agrawal, “Nonlinear mechanisms of filamentation in broad-area semiconductor lasers,” J. Quantum Electron. 32, 590–596 (1996). [CrossRef] }D is calculated to be 8.2 cm

^{2}/s. This is in good agreement with the direct measured value of 9.5 cm

^{2}/s.

^{2}We should mention that the output power of pump beam is decreased a little when the polarization direction of it is changed from perpendicular to the chip to parallel to the ship. We do not know the reason of this decrease but the effect of this decrease on the measured

*g*

_{TWM}is small.

^{1212. J. R. Marciante and G. P. Agrawal, “Nonlinear mechanisms of filamentation in broad-area semiconductor lasers,” J. Quantum Electron. 32, 590–596 (1996). [CrossRef] }The second is the linear variation of material gain

*g*(

*N*) on the carrier density. The transparent current of the amplifier used here is around 1.1 A, and the highest current used in our experiment is 1.8 A, according to Eq. (7), the carrier density

*N*

_{B}is calculated to be around 1.5

*N*

_{0}, not much higher than the transparent carrier density. The third assumption is the small population modulation in Eq. (6). With the injected current of 1.8 A, according to Eq. (8), |Δ

*N*| is calculated to be around 2% of

*N*

_{0}(≈1.3% of

*N*

_{B}), it is much less than the carrier density

*N*

_{B}. Therefore, we believe the assumptions made in the theory are valid in our experiment.

## 4. Conclusion

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

10. | M. Chi, O. B. Jensen, J. Holm, C. Pedersen, P. E. Andersen, G. Erbert, B. Bumpf, and P. M. Petersen, “Tunable high-power narrow-linewidth semiconductor laser based on an external-cavity tapered amplifier,” Opt. Express |

11. | L. Goldberg, D. Mehuys, M. R. Surette, and D. C. Hall, “High-power, near-diffraction-limited large-area traveling-wave semiconductor amplifiers,” J. Quantum Electron. |

12. | 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:**

Nonlinear Optics

**History**

Original Manuscript: October 6, 2006

Revised Manuscript: November 20, 2006

Manuscript Accepted: November 20, 2006

Published: December 11, 2006

**Citation**

Mingjun Chi, Søren B. Jensen, Jean-Pierre Huignard, and Paul Michael Petersen, "Two-wave mixing in a broad-area semiconductor amplifier," Opt. Express **14**, 12373-12379 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12373

Sort: Year | Journal | Reset

### 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]
- G. P. Agrawal, "Four-wave mixing and phase conjugation in semiconductor laser media," Opt. Lett. 12, 260-262 (1987). [CrossRef] [PubMed]
- M. Chi, O. B. Jensen, J. Holm, C. Pedersen, P. E. Andersen, G. Erbert, B. Bumpf, and P. M. Petersen, "Tunable high-power narrow-linewidth semiconductor laser based on an external-cavity tapered amplifier," Opt. Express 13, 10589-10596 (2005). [CrossRef] [PubMed]
- L. Goldberg, D. Mehuys, M. R. Surette, and D. C. Hall, "High-power, near-diffraction-limited large-area traveling-wave semiconductor amplifiers," J. Quantum Electron. 29, 2028-2042 (1993). [CrossRef]
- J. R. Marciante and G. P. Agrawal, "Nonlinear mechanisms of filamentation in broad-area semiconductor lasers," J. Quantum Electron. 32, 590-596 (1996). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.