Abstract

We study the near- and far-fields of a linear array of fiber lasers in an external Talbot cavity. Each emitter has a random optical path difference (OPD)phase due to length and dispersion differences. The individual emitter fields are described by forward and reverse differential equations in the Rigrod approximation with the Talbot cavity coupling all emitters through boundary conditions. We analytically determine the effect of the rms phase on the increase in the threshold, the decrease in the emitter amplitude, and the decrease in the far-field intensity. These results are confirmed numerically by using a Monte Carlo technique for the phase. This leads to a locking probability, a coherence function, and the on-axis intensity as functions of the rms phase. Another issue which we investigate is the cavity performance for inter-cavity and external cavity phasing and find the latter preferable. We also determine the strong coupling limit for the fill factor.

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References

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  1. D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, "Supermode control in diffraction-coupled semiconductor laser arrays," Appl. Phys. Lett. 53, 1165-1167 (1988).
    [CrossRef]
  2. David Mehuys, William Streifer, Robert G. Waarts. and Davie F. Welsh, "Modal analysis of linear Talbot-cavity semiconductor lasers," Opt. Lett. 16, 823-825 (1991).
    [CrossRef] [PubMed]
  3. Robert Waarts, David Mehuys, Derek Nam, David Welch, William Streifer, "High-power, cw, diffraction-limited GaAlAs laser diode array in an external Talbot cavity," Appl. Phys. Lett. 58, 2586-2588 (1991).
    [CrossRef]
  4. James R. Leger, "Lateral mode control of an AlGaAs laser array in a Talbot cavity," Appl. Phys. Lett. 55, 334-336 (1989).
    [CrossRef]
  5. James R. Leger, Miles L. Scott, and Wilfrid B. Veldkamp, "Coherent operation of AlGaAS lasers using microlenses and diffractive coupling," Appl. Phys. Lett. 52, 1771-1773 (1988).
    [CrossRef]
  6. R. G. Waarts, D. W. Nam, D. F. Welch, D. Mehuys, W. Cassarly, J. C. Ehlert, J. M. Finlan, K. M. Flood, "Semiconductor laser array in an external Talbot cavity," Laser Doiode Technology and Applications, SPIE 1634, 288-298 (1992).
  7. William J. Cassarly, John C. Ehlert, J. Michael Finlan, Kevin M. Flood, Robert Waarts, Davie Mehuys, Derek Nam, and Davie Welch, "Intercavity phase correction of an external Talbot cavity laser with the use of liquid crystals," Opt. Lett. 17, 607-609 (1992).
    [CrossRef] [PubMed]
  8. W. J. Cassarly, J. C. Ehlert, S. H. Chakmakjian, D. Harnesberger, J. M. Finlan, K. M. Flood, R. Waarts, D. Nam, D. Welch "Automatec two-dimensional phase sensing and control using phase contrast imaging," Laser Diode Technology and Applications, SPIE 1634, 299-3091992).
  9. V. P. Kandidov, A. V. Kondrat'ev,"Collective modes of laser arrays in Talbot cavities of various geometries," Quantum Electronics 27, 234-238 (1997).
    [CrossRef]
  10. V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Koval. V. V. Kuz'minov, D. A. Mashkovskii, A. M. Prokhorov, "Investigations of linear and two-dimensional arrays of semiconductor laser diodes in an external cavity," Quantum electronics 27, 850-854 (1997).
    [CrossRef]
  11. V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Koval. V. V. Kuz'minov, D. A. Mashkovskii, A. M. Prokhorov, "Spatial phase locking of linear arrays of 4 and 12 wide-aperture semiconductor laser diodes in an external cavity," Quantum electronics 28, 257-263 (1998).
    [CrossRef]
  12. V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Kocal. V. V. Kuz'minov, D. A. Mashkovskii, A. M. Prokhorov, "Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity," Quantum electronics 28, 344-346 (1998).
    [CrossRef]
  13. V. P. Kandidov, A. V. Kondrat'ev,"Influence of the Talbot cavity selectivity on the evolution of collective operation of diffraction-coupled laser arrays," Quantum Electronics 28, 972-976 (1998).
    [CrossRef]
  14. V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, V. V. Kuzminov, D. A. Mashkovsky, A,. M. Prokhorov,"Phase-locking of the 2D structures," Opt. Express 4, 19-26 (1999), http://www.opticsexpress.org/oearchive/source/8312.htm
    [CrossRef]
  15. M. Wrange, P. Glas, D. Fischer, M. Leitner, D. V. Vysotsky, A. P. Napartovich, "Phase locking in a nulticore fiber laser by means of a Talbot resonator," Opt. Lett. 25, 1436-1438 (2000).
    [CrossRef]
  16. B. R. Frieden, Probability, Statistical Optics, and Testing, (Springer-Verlag, New York, 1983), pp.186.
  17. Murray Sargent III, Marlan O. Scully, Willis E. Lamb, Laser Physics, (Addisonn-Wesley Reading Mass., 1977), pp. 86.

Other

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, "Supermode control in diffraction-coupled semiconductor laser arrays," Appl. Phys. Lett. 53, 1165-1167 (1988).
[CrossRef]

David Mehuys, William Streifer, Robert G. Waarts. and Davie F. Welsh, "Modal analysis of linear Talbot-cavity semiconductor lasers," Opt. Lett. 16, 823-825 (1991).
[CrossRef] [PubMed]

Robert Waarts, David Mehuys, Derek Nam, David Welch, William Streifer, "High-power, cw, diffraction-limited GaAlAs laser diode array in an external Talbot cavity," Appl. Phys. Lett. 58, 2586-2588 (1991).
[CrossRef]

James R. Leger, "Lateral mode control of an AlGaAs laser array in a Talbot cavity," Appl. Phys. Lett. 55, 334-336 (1989).
[CrossRef]

James R. Leger, Miles L. Scott, and Wilfrid B. Veldkamp, "Coherent operation of AlGaAS lasers using microlenses and diffractive coupling," Appl. Phys. Lett. 52, 1771-1773 (1988).
[CrossRef]

R. G. Waarts, D. W. Nam, D. F. Welch, D. Mehuys, W. Cassarly, J. C. Ehlert, J. M. Finlan, K. M. Flood, "Semiconductor laser array in an external Talbot cavity," Laser Doiode Technology and Applications, SPIE 1634, 288-298 (1992).

William J. Cassarly, John C. Ehlert, J. Michael Finlan, Kevin M. Flood, Robert Waarts, Davie Mehuys, Derek Nam, and Davie Welch, "Intercavity phase correction of an external Talbot cavity laser with the use of liquid crystals," Opt. Lett. 17, 607-609 (1992).
[CrossRef] [PubMed]

W. J. Cassarly, J. C. Ehlert, S. H. Chakmakjian, D. Harnesberger, J. M. Finlan, K. M. Flood, R. Waarts, D. Nam, D. Welch "Automatec two-dimensional phase sensing and control using phase contrast imaging," Laser Diode Technology and Applications, SPIE 1634, 299-3091992).

V. P. Kandidov, A. V. Kondrat'ev,"Collective modes of laser arrays in Talbot cavities of various geometries," Quantum Electronics 27, 234-238 (1997).
[CrossRef]

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Koval. V. V. Kuz'minov, D. A. Mashkovskii, A. M. Prokhorov, "Investigations of linear and two-dimensional arrays of semiconductor laser diodes in an external cavity," Quantum electronics 27, 850-854 (1997).
[CrossRef]

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Koval. V. V. Kuz'minov, D. A. Mashkovskii, A. M. Prokhorov, "Spatial phase locking of linear arrays of 4 and 12 wide-aperture semiconductor laser diodes in an external cavity," Quantum electronics 28, 257-263 (1998).
[CrossRef]

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Kocal. V. V. Kuz'minov, D. A. Mashkovskii, A. M. Prokhorov, "Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity," Quantum electronics 28, 344-346 (1998).
[CrossRef]

V. P. Kandidov, A. V. Kondrat'ev,"Influence of the Talbot cavity selectivity on the evolution of collective operation of diffraction-coupled laser arrays," Quantum Electronics 28, 972-976 (1998).
[CrossRef]

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, V. V. Kuzminov, D. A. Mashkovsky, A,. M. Prokhorov,"Phase-locking of the 2D structures," Opt. Express 4, 19-26 (1999), http://www.opticsexpress.org/oearchive/source/8312.htm
[CrossRef]

M. Wrange, P. Glas, D. Fischer, M. Leitner, D. V. Vysotsky, A. P. Napartovich, "Phase locking in a nulticore fiber laser by means of a Talbot resonator," Opt. Lett. 25, 1436-1438 (2000).
[CrossRef]

B. R. Frieden, Probability, Statistical Optics, and Testing, (Springer-Verlag, New York, 1983), pp.186.

Murray Sargent III, Marlan O. Scully, Willis E. Lamb, Laser Physics, (Addisonn-Wesley Reading Mass., 1977), pp. 86.

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Figures (4)

Fig. 1.
Fig. 1.

First (black), second (red), and third (green) nearest neighbor coupling amplitudes as a function of fill factor.

Fig. 2.
Fig. 2.

On-axis intensity as a function of cavity length normalized to the Talbot distance. The black curve is without correction, the red is with external phasing, the green is with internal phasing. (a) is for a gain of 8 and (b) is for a gain of 4.

Fig. 3.
Fig. 3.

(a) the amplitude of emitter number 3, (b) the locking probability, and (c) the coherence function all as function of the rms phase in the half-Talbot plane.

Fig. 4.
Fig. 4.

(a) the amplitude of emitter number 3, (b) the locking probability, and (c) the coherence function all as function of the rms phase in the quarter-Talbot plane.

Equations (21)

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d E j ± d z = ± g j 1 + E j ± 2 + E j 2 E j ± ± i k j E j ±
E j ± ( z ) = A j ± ( z ) exp ( ϕ j ± ( z ) ) .
ϕ j ± ( L j ) ϕ j ± ( 0 ) = ± k j L j
C j = A j + ( z ) A j ( z ) = A j + ( L j ) A j ( L j ) = A j + ( 0 ) A j ( 0 ) .
d A j ± d z = ± g j 1 + A j ± 2 + C j A j ± 2 A j ±
ln A j + ( L j ) A j + ( 0 ) + 1 2 [ A j + 2 ( L j ) A j + 2 ( 0 ) ] C j 2 [ 1 A j + 2 ( L j ) 1 A j + 2 ( 0 ) ] = g j L j .
1 r i A i + 2 ( 0 ) A i + ( L i ) exp ( i ϕ i ( L i ) ) = Σ R i , j A j + ( L j ) exp ( i ϕ j + ( L j ) ) exp ( i k j z + ϕ 0 ) .
E i + ( L ) = r exp ( 2 g L ) Σ R i , j E j + ( L ) exp [ i ( ϕ j ϕ i ) ]
< E i + ( L ) > = r exp ( 2 g L ) Σ R i , j < E j + ( L ) ) > < exp [ i ( ϕ j ϕ i ) ] > .
< E i + ( L ) > = r exp ( 2 g L ) [ R i , i < E i ( L ) > + exp ( σ 2 ) Σ j i N R i , j < E j + ( L ) > ] .
A j + 2 ( 0 ) A j + ( L ) A j + ( L ) 1 2 r g L + r A j + 2 ( L ) 1 + r A j + 2 ( L ) .
E i + ( L ) 1 + r E i + ( L ) 2 1 2 r g L + r E i + ( L ) 2 r Σ R i , j E j + ( L ) exp [ i ( ϕ j ϕ i ) ] ,
C = Σ E j ( 0 ) 2 Σ E j ( 0 ) 2
< C > = 1 N < i , j N exp [ i ( ϕ j ϕ i ) ] > = 1 N [ N + N ( N 1 ) exp ( σ 2 ) ] ,
E 0 ( x ) = m = 0 N E m exp [ ( x m d ) 2 ω 0 2 ] .
E P ( x , z ) = i exp ( i k z ) z E 0 ( x , y ) exp ( i k 2 z [ ( x x ) 2 + ( y y ) 2 ] ) d x d y
E P ( x , z ) = exp ( i k z ) 1 1 + i z z 0 m = 0 N E m exp [ i k 2 ( z i z 0 ) ( x m d ) 2 ]
Σ E m G m ( x , y , z )
R i , j = G i ( x , y , 0 ) * G j ( x , y , z ) d x d y G i ( x , y , 0 ) * G i ( x , y , 0 ) d x d y
R i , j = exp ( i k z ) 1 1 + i z 2 z 0 exp [ i k 2 ( z i 2 z 0 ) ( x i x j ) 2 ) ]
R i , j exp ( i k z ) 1 1 + i z / 2 z 0 exp [ i π 2 ( i j ) 2 z t z ] .

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