Abstract

We demonstrate double phase-conjugation in pure and Tedoped Sn2P2S6, a semiconducting ferroelectric material, at the wavelength of 685 nm. We observe a phase conjugate reflectivity of more than 800% at an intensity ratio of the pump beams of 44 for Te-doped Sn 2P2S6. Using a laser diode bar emitting at 685 nm, we demonstrate double phase conjugation of three independent emitters of the laser diode bar with a single mode master laser. By adjusting the center wavelength of the master laser to the center wavelength of an emitter with an accuracy of less than 0.1 nm, locking of any emitter of the laser diode bar is demonstrated. We improve the spectral width of the emitter from 0.5nm to below 2.5·10-4 nm.

© 2008 Optical Society of America

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  1. S. Weiss, S. Sternklar, and B. Fischer, "Double phase-conjugate mirror - Analysis, demonstration, and applications," Opt. Lett. 12, 114-116 (1987).
    [CrossRef] [PubMed]
  2. D. Wang, Z. Zhang, Y. Zhu, S. Zhang, and P. Ye, "Observations on the coupling channel of two mutually incoherent beams without internal reflections in BaTiO3," Opt. Commun. 73, 495-500 (1989).
    [CrossRef]
  3. M. D. Ewbank, "Mechanism for photorefractive phase conjugation using incoherent beams," Opt. Lett. 13, 47-49 (1988).
    [CrossRef] [PubMed]
  4. R. W. Eason and A. M. C. Smout, "Bistability and noncommutative behavior of multiple-beam self-pulsing and self-pumping in BaTiO3," Opt. Lett. 12, 51-53 (1987).
    [CrossRef] [PubMed]
  5. M. D. Ewbank, R. A. Vazquez, R. R. Neurgaonkar, and J. Feinberg, "Mutually pumped phase conjugation in photorefractive strontium barium niobate: theory and experiment," J. Opt. Soc. Am. B 7, 2306-2316 (1990).
  6. C. Medrano, M. Zgonik, S. Berents, P. Bernasconi, and P. Günter, "Self-pumped and incoherent phase conjugation in Fe-doped KNbO3," J. Opt. Soc. Am. B 11, 1718-1726 (1994).
  7. M. P. Petrov, S. L. Sochava, and S. I. Stepanov, "Double phase conjugate mirror using a photorefractive Bi12TiO20 crystal," Opt. Lett. 14, 284-286 (1989).
    [CrossRef] [PubMed]
  8. K. Shcherbin, "Recent Progress in Semiconductor Photorefractive Crystals," in Photorefractive Materials and Their Applications II, P. Gunter and J.-P. Hiugnard, eds. (Springer-Verlag, New York, 2007), pp. 391-418.
  9. G. W. Ross and R. W. Eason, "Double phase-conjugate mirror with sixfold gain in photorefractive BaTiO3 at near-infrared wavelengths," Opt. Lett. 18, 571-573 (1993).
    [CrossRef] [PubMed]
  10. R. S. Cudney and M. Kaczmarek, "Optical poling in Rh:BaTiO3," in Trends in Optics and Photonics, Vol. 62, pp. 485-489 (2001).
  11. M. B. Klein, "Photorefractive Properties of BaTiO3," in Photorefractive materials and their applications I, P. G¨unter and J.-P. Hiugnard, eds. (Springer Verlag, Berlin, 1988), pp. 195-236.
  12. P. Yeh, Introduction to Photorefractive Nonlinear Optics (John Wiley and Sons, Inc., 1993).
  13. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, "Theory and applications of four-wave mixing in photorefractive media," IEEE J. Quantum Electron. QE-20, 12-30 (1984).
    [CrossRef]
  14. D. Engin, M. Segev, S. Orlov, and A. Yariv, "Double phase conjugation," J. Opt. Soc. Am. B 11, 1708-1717 (1994).
  15. N. Wolffer, P. Gravey, J. Y. Moisan, C. Laulan, and J. C. Launay, "Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu," Opt. Commun. 73, 351-356 (1989).
    [CrossRef]
  16. S. MacCormack, J. Feinberg, and M. H. Garret, "Injection locking a laser-diode array with a phase conjugate beam," Opt. Lett. 19, 120-122 (1994).
    [CrossRef] [PubMed]
  17. K. Iida, H. Horiuchi, O. Matoba, T. Omatsu, T. Shimura, and K. Kuroda, "Injection locking of a broad-area diode lasers through a double phase conjugate mirror," Opt. Commun. 146, 6-10 (1998).
    [CrossRef]
  18. F. Wang, A. Hermerschmidt, and H. J. Eichler, "High-power narrowed-beandwidth output of a broad-area multiple-stripe diode laser with photorefractive phase-conjugated injection," Opt. Commun. 209, 391-395 (2002).
    [CrossRef]
  19. A. A. Grabar, M. Jazbinsek, A. N. Shumelyuk, Y. M. Vysochanskii, G. Montemezzani, and P. Günter, "Photorefractive effects in Sn2P2S6," in Photorefractive Materials and Their Applications II, P. Günter and J.-P. Huignard, eds. (Springer-Verlag, New York, 2007), pp. 327-362.
  20. S. G. Odoulov, A. N. Shumelyuk, U. Hellwig, R. A. Rupp, and A. A. Grabar, "Photorefractive beam coupling in tin hypothiodiphosphate in the near infrared," Opt. Lett. 21, 752-754 (1996).
    [CrossRef] [PubMed]
  21. M. Jazbinsek, G Montemezzani, P. G¨unter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Fast nearinfrared self-pumped phase conjugation with photorefractive Sn2P2S6," J. Opt. Soc. Am. B 20, 1241-1246 (2003).
  22. M. Jazbinšek, D. Haertle, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Wavelength dependence of visible and near infrared photorefraction and phase conjugation in Sn2P2S6," J. Opt. Soc. Am. B 22, 2459-2467 (2005).
    [CrossRef]
  23. R. Mosimann, P. Marty, T. Bach, F. Juvalta, M. Jazbinsek, P. Günter, and A. A. Grabar, "High-speed photorefraction at telecommunication wavelength 1.55 μm in Sn2 P2S6:Te," Opt. Lett. 32, 3230-3232 (2007).
    [CrossRef] [PubMed]
  24. T. Bach, M. Jazbinsek, P. Gunter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Self pumped optical phase conjugation at 1.06 μm in Te-doped Sn2P2S6," Opt. Express 13, 9890-9896 (2005).
    [CrossRef] [PubMed]
  25. T. Bach, M. Jazbinsek, G. Montemezzani, P. Günter, A. A. Grabar, and Y. M. Vysochanskii, "Tailoring of infrared photorefractive properties of Sn2P2S6 crystals by Te and Sb doping," J. Opt. Soc. Am. B 24, 1535-1541 (2007).
  26. A. A. Grabar, I. V. Kedyk, M. I. Gurzan, I. M. Stoika, A. A. Molnar, and Y. M. Vysochanskii, "Enhanced photorefractive properties of modified Sn2P2S6," Opt. Commun. 188, 187-194 (2001).
    [CrossRef]

2007 (2)

2005 (2)

2003 (1)

2002 (1)

F. Wang, A. Hermerschmidt, and H. J. Eichler, "High-power narrowed-beandwidth output of a broad-area multiple-stripe diode laser with photorefractive phase-conjugated injection," Opt. Commun. 209, 391-395 (2002).
[CrossRef]

2001 (1)

A. A. Grabar, I. V. Kedyk, M. I. Gurzan, I. M. Stoika, A. A. Molnar, and Y. M. Vysochanskii, "Enhanced photorefractive properties of modified Sn2P2S6," Opt. Commun. 188, 187-194 (2001).
[CrossRef]

1998 (1)

K. Iida, H. Horiuchi, O. Matoba, T. Omatsu, T. Shimura, and K. Kuroda, "Injection locking of a broad-area diode lasers through a double phase conjugate mirror," Opt. Commun. 146, 6-10 (1998).
[CrossRef]

1996 (1)

1994 (3)

1993 (1)

1990 (1)

1989 (3)

N. Wolffer, P. Gravey, J. Y. Moisan, C. Laulan, and J. C. Launay, "Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu," Opt. Commun. 73, 351-356 (1989).
[CrossRef]

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, and P. Ye, "Observations on the coupling channel of two mutually incoherent beams without internal reflections in BaTiO3," Opt. Commun. 73, 495-500 (1989).
[CrossRef]

M. P. Petrov, S. L. Sochava, and S. I. Stepanov, "Double phase conjugate mirror using a photorefractive Bi12TiO20 crystal," Opt. Lett. 14, 284-286 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (2)

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, "Theory and applications of four-wave mixing in photorefractive media," IEEE J. Quantum Electron. QE-20, 12-30 (1984).
[CrossRef]

Bach, T.

Berents, S.

Bernasconi, P.

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, "Theory and applications of four-wave mixing in photorefractive media," IEEE J. Quantum Electron. QE-20, 12-30 (1984).
[CrossRef]

Eason, R. W.

Eichler, H. J.

F. Wang, A. Hermerschmidt, and H. J. Eichler, "High-power narrowed-beandwidth output of a broad-area multiple-stripe diode laser with photorefractive phase-conjugated injection," Opt. Commun. 209, 391-395 (2002).
[CrossRef]

Engin, D.

Ewbank, M. D.

Feinberg, J.

Fischer, B.

S. Weiss, S. Sternklar, and B. Fischer, "Double phase-conjugate mirror - Analysis, demonstration, and applications," Opt. Lett. 12, 114-116 (1987).
[CrossRef] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, "Theory and applications of four-wave mixing in photorefractive media," IEEE J. Quantum Electron. QE-20, 12-30 (1984).
[CrossRef]

G¨unter, P.

Garret, M. H.

Grabar, A. A.

T. Bach, M. Jazbinsek, G. Montemezzani, P. Günter, A. A. Grabar, and Y. M. Vysochanskii, "Tailoring of infrared photorefractive properties of Sn2P2S6 crystals by Te and Sb doping," J. Opt. Soc. Am. B 24, 1535-1541 (2007).

R. Mosimann, P. Marty, T. Bach, F. Juvalta, M. Jazbinsek, P. Günter, and A. A. Grabar, "High-speed photorefraction at telecommunication wavelength 1.55 μm in Sn2 P2S6:Te," Opt. Lett. 32, 3230-3232 (2007).
[CrossRef] [PubMed]

M. Jazbinšek, D. Haertle, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Wavelength dependence of visible and near infrared photorefraction and phase conjugation in Sn2P2S6," J. Opt. Soc. Am. B 22, 2459-2467 (2005).
[CrossRef]

T. Bach, M. Jazbinsek, P. Gunter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Self pumped optical phase conjugation at 1.06 μm in Te-doped Sn2P2S6," Opt. Express 13, 9890-9896 (2005).
[CrossRef] [PubMed]

M. Jazbinsek, G Montemezzani, P. G¨unter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Fast nearinfrared self-pumped phase conjugation with photorefractive Sn2P2S6," J. Opt. Soc. Am. B 20, 1241-1246 (2003).

A. A. Grabar, I. V. Kedyk, M. I. Gurzan, I. M. Stoika, A. A. Molnar, and Y. M. Vysochanskii, "Enhanced photorefractive properties of modified Sn2P2S6," Opt. Commun. 188, 187-194 (2001).
[CrossRef]

S. G. Odoulov, A. N. Shumelyuk, U. Hellwig, R. A. Rupp, and A. A. Grabar, "Photorefractive beam coupling in tin hypothiodiphosphate in the near infrared," Opt. Lett. 21, 752-754 (1996).
[CrossRef] [PubMed]

Gravey, P.

N. Wolffer, P. Gravey, J. Y. Moisan, C. Laulan, and J. C. Launay, "Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu," Opt. Commun. 73, 351-356 (1989).
[CrossRef]

Gunter, P.

Günter, P.

Gurzan, M. I.

A. A. Grabar, I. V. Kedyk, M. I. Gurzan, I. M. Stoika, A. A. Molnar, and Y. M. Vysochanskii, "Enhanced photorefractive properties of modified Sn2P2S6," Opt. Commun. 188, 187-194 (2001).
[CrossRef]

Haertle, D.

Hellwig, U.

Hermerschmidt, A.

F. Wang, A. Hermerschmidt, and H. J. Eichler, "High-power narrowed-beandwidth output of a broad-area multiple-stripe diode laser with photorefractive phase-conjugated injection," Opt. Commun. 209, 391-395 (2002).
[CrossRef]

Horiuchi, H.

K. Iida, H. Horiuchi, O. Matoba, T. Omatsu, T. Shimura, and K. Kuroda, "Injection locking of a broad-area diode lasers through a double phase conjugate mirror," Opt. Commun. 146, 6-10 (1998).
[CrossRef]

Iida, K.

K. Iida, H. Horiuchi, O. Matoba, T. Omatsu, T. Shimura, and K. Kuroda, "Injection locking of a broad-area diode lasers through a double phase conjugate mirror," Opt. Commun. 146, 6-10 (1998).
[CrossRef]

Jazbinsek, M.

Jazbinšek, M.

Juvalta, F.

Kedyk, I. V.

A. A. Grabar, I. V. Kedyk, M. I. Gurzan, I. M. Stoika, A. A. Molnar, and Y. M. Vysochanskii, "Enhanced photorefractive properties of modified Sn2P2S6," Opt. Commun. 188, 187-194 (2001).
[CrossRef]

Kuroda, K.

K. Iida, H. Horiuchi, O. Matoba, T. Omatsu, T. Shimura, and K. Kuroda, "Injection locking of a broad-area diode lasers through a double phase conjugate mirror," Opt. Commun. 146, 6-10 (1998).
[CrossRef]

Laulan, C.

N. Wolffer, P. Gravey, J. Y. Moisan, C. Laulan, and J. C. Launay, "Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu," Opt. Commun. 73, 351-356 (1989).
[CrossRef]

Launay, J. C.

N. Wolffer, P. Gravey, J. Y. Moisan, C. Laulan, and J. C. Launay, "Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu," Opt. Commun. 73, 351-356 (1989).
[CrossRef]

MacCormack, S.

Marty, P.

Matoba, O.

K. Iida, H. Horiuchi, O. Matoba, T. Omatsu, T. Shimura, and K. Kuroda, "Injection locking of a broad-area diode lasers through a double phase conjugate mirror," Opt. Commun. 146, 6-10 (1998).
[CrossRef]

Medrano, C.

Moisan, J. Y.

N. Wolffer, P. Gravey, J. Y. Moisan, C. Laulan, and J. C. Launay, "Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu," Opt. Commun. 73, 351-356 (1989).
[CrossRef]

Molnar, A. A.

A. A. Grabar, I. V. Kedyk, M. I. Gurzan, I. M. Stoika, A. A. Molnar, and Y. M. Vysochanskii, "Enhanced photorefractive properties of modified Sn2P2S6," Opt. Commun. 188, 187-194 (2001).
[CrossRef]

Montemezzani, G

Montemezzani, G.

Mosimann, R.

Neurgaonkar, R. R.

Odoulov, S. G.

Omatsu, T.

K. Iida, H. Horiuchi, O. Matoba, T. Omatsu, T. Shimura, and K. Kuroda, "Injection locking of a broad-area diode lasers through a double phase conjugate mirror," Opt. Commun. 146, 6-10 (1998).
[CrossRef]

Orlov, S.

Petrov, M. P.

Ross, G. W.

Rupp, R. A.

Segev, M.

Shimura, T.

K. Iida, H. Horiuchi, O. Matoba, T. Omatsu, T. Shimura, and K. Kuroda, "Injection locking of a broad-area diode lasers through a double phase conjugate mirror," Opt. Commun. 146, 6-10 (1998).
[CrossRef]

Shumelyuk, A. N.

Smout, A. M. C.

Sochava, S. L.

Stepanov, S. I.

Sternklar, S.

Stoika, I. M.

Vazquez, R. A.

Vysochanskii, Y. M.

Wang, D.

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, and P. Ye, "Observations on the coupling channel of two mutually incoherent beams without internal reflections in BaTiO3," Opt. Commun. 73, 495-500 (1989).
[CrossRef]

Wang, F.

F. Wang, A. Hermerschmidt, and H. J. Eichler, "High-power narrowed-beandwidth output of a broad-area multiple-stripe diode laser with photorefractive phase-conjugated injection," Opt. Commun. 209, 391-395 (2002).
[CrossRef]

Weiss, S.

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, "Theory and applications of four-wave mixing in photorefractive media," IEEE J. Quantum Electron. QE-20, 12-30 (1984).
[CrossRef]

Wolffer, N.

N. Wolffer, P. Gravey, J. Y. Moisan, C. Laulan, and J. C. Launay, "Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu," Opt. Commun. 73, 351-356 (1989).
[CrossRef]

Yariv, A.

D. Engin, M. Segev, S. Orlov, and A. Yariv, "Double phase conjugation," J. Opt. Soc. Am. B 11, 1708-1717 (1994).

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, "Theory and applications of four-wave mixing in photorefractive media," IEEE J. Quantum Electron. QE-20, 12-30 (1984).
[CrossRef]

Ye, P.

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, and P. Ye, "Observations on the coupling channel of two mutually incoherent beams without internal reflections in BaTiO3," Opt. Commun. 73, 495-500 (1989).
[CrossRef]

Zgonik, M.

Zhang, S.

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, and P. Ye, "Observations on the coupling channel of two mutually incoherent beams without internal reflections in BaTiO3," Opt. Commun. 73, 495-500 (1989).
[CrossRef]

Zhang, Z.

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, and P. Ye, "Observations on the coupling channel of two mutually incoherent beams without internal reflections in BaTiO3," Opt. Commun. 73, 495-500 (1989).
[CrossRef]

Zhu, Y.

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, and P. Ye, "Observations on the coupling channel of two mutually incoherent beams without internal reflections in BaTiO3," Opt. Commun. 73, 495-500 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, "Theory and applications of four-wave mixing in photorefractive media," IEEE J. Quantum Electron. QE-20, 12-30 (1984).
[CrossRef]

J. Opt. Soc. Am. B (6)

Opt. Commun. (5)

N. Wolffer, P. Gravey, J. Y. Moisan, C. Laulan, and J. C. Launay, "Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu," Opt. Commun. 73, 351-356 (1989).
[CrossRef]

A. A. Grabar, I. V. Kedyk, M. I. Gurzan, I. M. Stoika, A. A. Molnar, and Y. M. Vysochanskii, "Enhanced photorefractive properties of modified Sn2P2S6," Opt. Commun. 188, 187-194 (2001).
[CrossRef]

K. Iida, H. Horiuchi, O. Matoba, T. Omatsu, T. Shimura, and K. Kuroda, "Injection locking of a broad-area diode lasers through a double phase conjugate mirror," Opt. Commun. 146, 6-10 (1998).
[CrossRef]

F. Wang, A. Hermerschmidt, and H. J. Eichler, "High-power narrowed-beandwidth output of a broad-area multiple-stripe diode laser with photorefractive phase-conjugated injection," Opt. Commun. 209, 391-395 (2002).
[CrossRef]

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, and P. Ye, "Observations on the coupling channel of two mutually incoherent beams without internal reflections in BaTiO3," Opt. Commun. 73, 495-500 (1989).
[CrossRef]

Opt. Express (1)

Opt. Lett. (8)

Other (5)

K. Shcherbin, "Recent Progress in Semiconductor Photorefractive Crystals," in Photorefractive Materials and Their Applications II, P. Gunter and J.-P. Hiugnard, eds. (Springer-Verlag, New York, 2007), pp. 391-418.

R. S. Cudney and M. Kaczmarek, "Optical poling in Rh:BaTiO3," in Trends in Optics and Photonics, Vol. 62, pp. 485-489 (2001).

M. B. Klein, "Photorefractive Properties of BaTiO3," in Photorefractive materials and their applications I, P. G¨unter and J.-P. Hiugnard, eds. (Springer Verlag, Berlin, 1988), pp. 195-236.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (John Wiley and Sons, Inc., 1993).

A. A. Grabar, M. Jazbinsek, A. N. Shumelyuk, Y. M. Vysochanskii, G. Montemezzani, and P. Günter, "Photorefractive effects in Sn2P2S6," in Photorefractive Materials and Their Applications II, P. Günter and J.-P. Huignard, eds. (Springer-Verlag, New York, 2007), pp. 327-362.

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Figures (7)

Fig. 1.
Fig. 1.

Experimental set-up for double phase conjugation characterization with a Te-doped Sn2P2S6 crystal at 685 nm. The phase conjugated beam A 4 is partly reflected by a glass plate and measured by a photodiode. The intensity ratio can be changed by putting filters in the path of beam A 1 or A 3. The beams are linearly polarized in the plane of incidence.

Fig. 2.
Fig. 2.

Experimental set-up for phase conjugate injection locking with laser diodes at around 685 nm and Sn2P2S6 crystal. The laser beams are linearly polarized in the plane of incidence.

Fig. 3.
Fig. 3.

Long-term measurement of the phase-conjugate reflectivity of beam 4 for an intensity ratio q=I 10/I 3L =0.9 of the pump beams. The phase conjugate beam 4 is measured for 450 s to check the stability of the double phase conjugate mirror.

Fig. 4.
Fig. 4.

Saturated phase conjugated reflectivity R as a function of the intensity ratio q=I 10/I 3L . Measured data (points) is compared with calculations. Dotted/dashed curves are based on Eq. (1) for the gain coefficients Γ=10cm-1, Γ=10.5cm-1 and Γ=11.5cm-1. The solid line includes a correction term ξ for the intensity ratio q due the absorption loss in the crystal (see Eq. (6)).

Fig. 5.
Fig. 5.

Saturated phase conjugate reflectivity R as a function of the total intensity I 0. Dotted/ dashed curves represent calculations based on Eq. (1) considering the background illumination I β leading to a modified parameter ρ according to Eq. (7). The solid curve additionally considers the absorption loss in the crystal in the correction parameter ξ, as described by Eq. (6).

Fig. 6.
Fig. 6.

Comparison of the free running mode and the locked mode spectrum of the slave laser output measured with a spectrometer. One diode of the laser bar is coupled with the master laser (IM =0.20W/cm2) and the intensity of the slave laser beam before entering the crystal is IS =0.36W/cm2.

Fig. 7.
Fig. 7.

Double phase conjugated image observed on the screen of three independent emitters of a laser diode bar.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

s = I 0 tan h ( κ L ) ,
s = σ 2 + ρ 0 2 ( I 0 + σ ) 2 ,
σ = I 0 1 q 1 + q ,
κ = s Γ * 4 I 0 .
ρ 0 2 = A 4 ( 0 ) A 1 * ( 0 ) 2 .
σ = I 0 1 q ξ 1 + q ξ
κ = s Γ * 4 ( I 0 + I β ) .

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