Abstract

A procedure is developed to determine the transverse-mode structure of a cavity consisting of a dense, evanescently coupled, waveguide laser array, which, in addition, is externally coupled by feedback from an external cavity. The formalism is used to determine the loss and phasing properties of a multicore fiber array coupled to an external self-Fourier cavity. Best performance is predicted for linear arrays of up to five cores, or two-dimensional arrays of up to 25 cores. A low-loss, in-phase, fundamental array mode is predicted, which achieves better than 30  dB discrimination against higher-order modes at periodically spaced values of the array length. However, we show that a shift in operating wavelength of typically a few nanometers can bring about near-perfect phasing and loss operation over a continuum of fiber lengths. With increased fill factor, significantly more of the output power can be concentrated in the central lobe of the far field but at the penalty of increased loss in the fundamental eigenmode.

© 2007 Optical Society of America

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  1. E. M. Phillip-Rutz, "Spatially coherent radiation from an array of GaAs lasers," Appl. Phys. Lett. 26, 475-477 (1975).
    [CrossRef]
  2. A. A. Golubentsev, V. V. Likhanskii, and A. P. Napartovich, "Theory of phase locking of an array of lasers," Sov. Phys. JETP 66, 676-682 (1987).
  3. V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, V. V. Kuzminov, D. A. Mashkovsky, and A. M. Prokhorov, "Phase-locking of 2D structures," Opt. Express 4, 19-26 (1999).
    [CrossRef] [PubMed]
  4. D. Mehuys, W. Streifer, R. Waarts, and D. F. Welch, "Modal analysis of linear Talbot-cavity semiconductor lasers," Opt. Lett. 16, 823-825 (1991).
    [CrossRef] [PubMed]
  5. C. J. Corcoran and K. A. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A, Pure Appl. Opt. 7, L1-L7 (2005).
    [CrossRef]
  6. C. J. Corcoran and F. Durville, "Experimental demonstration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
    [CrossRef]
  7. C. J. Corcoran, F. Durville, K. A. Pasch, and E. J. Bochove, "Spatial filtering of large-mode area fibre lasers using a self-Fourier cavity for high power applications," J. Opt. A , Pure Appl. Opt. 9, 128-133 (2007).
    [CrossRef]
  8. P. K. Cheo, A. Liu, and G. G. King, "A high-brightness laser beam from a phase-locked multicore Yb-doped fiber laser array," IEEE Photon. Technol. Lett. 13, 439-441 (2001).
    [CrossRef]
  9. S. Riyopolous, "Randomly phase-locked microlaser arrays and fuzzy eigenmodes with stochastic phasing," Opt. Express 14, 10508-10521 (2006).
    [CrossRef]
  10. A. Hardy and W. Streifer, "Coupled modes of multiwaveguide systems and phased arrays," J. Lightwave Technol. 4, 90-99 (1986).
    [CrossRef]
  11. A. Hardy and W. Streifer, "Coupled mode theory of parallel waveguides," J. Lightwave Technol. LT-3, 1135-1146 (1985).
    [CrossRef]
  12. 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]
  13. J. K. Butler, D. E. Ackley, and D. Botez, "Coupled-mode analysis of phase-locked injection laser arrays," Appl. Phys. Lett. 44, 293-295 (1984).
    [CrossRef]
  14. M. J. Caola, "Self-Fourier functions," J. Phys. A 24L1143-LI144 (1991).
    [CrossRef]
  15. L. Liu, "Periodic self-Fourier-Fresnel functions," J. Phys. A 27, L285-L289 (1994).
    [CrossRef]
  16. L. Michaille, C. R. Bennett, D. M. Taylor, T. J. Shepherd, J. Broeng, H. R. Simonsen, and A. Petersson, "Phase locking and supermode selection in multicore photonic crystal fiber lasers with a large doped area," Opt. Lett. 30, 1668-1670 (2005).
    [CrossRef] [PubMed]
  17. L. Li, A. Schülzgen, S. Chen, V. Temyanko, J. V. Moloney, and N. Peyghambarian, "Phase locking and in-phase supermode selection in monolithic multicore fiber lasers," Opt. Lett. 31, 2577-2579 (2006).
    [CrossRef] [PubMed]

2007 (1)

C. J. Corcoran, F. Durville, K. A. Pasch, and E. J. Bochove, "Spatial filtering of large-mode area fibre lasers using a self-Fourier cavity for high power applications," J. Opt. A , Pure Appl. Opt. 9, 128-133 (2007).
[CrossRef]

2006 (2)

2005 (3)

L. Michaille, C. R. Bennett, D. M. Taylor, T. J. Shepherd, J. Broeng, H. R. Simonsen, and A. Petersson, "Phase locking and supermode selection in multicore photonic crystal fiber lasers with a large doped area," Opt. Lett. 30, 1668-1670 (2005).
[CrossRef] [PubMed]

C. J. Corcoran and K. A. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A, Pure Appl. Opt. 7, L1-L7 (2005).
[CrossRef]

C. J. Corcoran and F. Durville, "Experimental demonstration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

2001 (1)

P. K. Cheo, A. Liu, and G. G. King, "A high-brightness laser beam from a phase-locked multicore Yb-doped fiber laser array," IEEE Photon. Technol. Lett. 13, 439-441 (2001).
[CrossRef]

1999 (1)

1994 (1)

L. Liu, "Periodic self-Fourier-Fresnel functions," J. Phys. A 27, L285-L289 (1994).
[CrossRef]

1991 (2)

1988 (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]

1987 (1)

A. A. Golubentsev, V. V. Likhanskii, and A. P. Napartovich, "Theory of phase locking of an array of lasers," Sov. Phys. JETP 66, 676-682 (1987).

1986 (1)

A. Hardy and W. Streifer, "Coupled modes of multiwaveguide systems and phased arrays," J. Lightwave Technol. 4, 90-99 (1986).
[CrossRef]

1985 (1)

A. Hardy and W. Streifer, "Coupled mode theory of parallel waveguides," J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

1984 (1)

J. K. Butler, D. E. Ackley, and D. Botez, "Coupled-mode analysis of phase-locked injection laser arrays," Appl. Phys. Lett. 44, 293-295 (1984).
[CrossRef]

1975 (1)

E. M. Phillip-Rutz, "Spatially coherent radiation from an array of GaAs lasers," Appl. Phys. Lett. 26, 475-477 (1975).
[CrossRef]

Ackley, D. E.

J. K. Butler, D. E. Ackley, and D. Botez, "Coupled-mode analysis of phase-locked injection laser arrays," Appl. Phys. Lett. 44, 293-295 (1984).
[CrossRef]

Apollonov, V. V.

Bennett, C. R.

Bochove, E. J.

C. J. Corcoran, F. Durville, K. A. Pasch, and E. J. Bochove, "Spatial filtering of large-mode area fibre lasers using a self-Fourier cavity for high power applications," J. Opt. A , Pure Appl. Opt. 9, 128-133 (2007).
[CrossRef]

Botez, D.

J. K. Butler, D. E. Ackley, and D. Botez, "Coupled-mode analysis of phase-locked injection laser arrays," Appl. Phys. Lett. 44, 293-295 (1984).
[CrossRef]

Broeng, J.

Butler, J. K.

J. K. Butler, D. E. Ackley, and D. Botez, "Coupled-mode analysis of phase-locked injection laser arrays," Appl. Phys. Lett. 44, 293-295 (1984).
[CrossRef]

Caola, M. J.

M. J. Caola, "Self-Fourier functions," J. Phys. A 24L1143-LI144 (1991).
[CrossRef]

Chen, S.

Cheo, P. K.

P. K. Cheo, A. Liu, and G. G. King, "A high-brightness laser beam from a phase-locked multicore Yb-doped fiber laser array," IEEE Photon. Technol. Lett. 13, 439-441 (2001).
[CrossRef]

Corcoran, C. J.

C. J. Corcoran, F. Durville, K. A. Pasch, and E. J. Bochove, "Spatial filtering of large-mode area fibre lasers using a self-Fourier cavity for high power applications," J. Opt. A , Pure Appl. Opt. 9, 128-133 (2007).
[CrossRef]

C. J. Corcoran and F. Durville, "Experimental demonstration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

C. J. Corcoran and K. A. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A, Pure Appl. Opt. 7, L1-L7 (2005).
[CrossRef]

Derzhavin, S. I.

Durville, F.

C. J. Corcoran, F. Durville, K. A. Pasch, and E. J. Bochove, "Spatial filtering of large-mode area fibre lasers using a self-Fourier cavity for high power applications," J. Opt. A , Pure Appl. Opt. 9, 128-133 (2007).
[CrossRef]

C. J. Corcoran and F. Durville, "Experimental demonstration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

Eng, L.

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]

Golubentsev, A. A.

A. A. Golubentsev, V. V. Likhanskii, and A. P. Napartovich, "Theory of phase locking of an array of lasers," Sov. Phys. JETP 66, 676-682 (1987).

Hardy, A.

A. Hardy and W. Streifer, "Coupled modes of multiwaveguide systems and phased arrays," J. Lightwave Technol. 4, 90-99 (1986).
[CrossRef]

A. Hardy and W. Streifer, "Coupled mode theory of parallel waveguides," J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

King, G. G.

P. K. Cheo, A. Liu, and G. G. King, "A high-brightness laser beam from a phase-locked multicore Yb-doped fiber laser array," IEEE Photon. Technol. Lett. 13, 439-441 (2001).
[CrossRef]

Kislov, V. I.

Kuzminov, V. V.

Li, L.

Likhanskii, V. V.

A. A. Golubentsev, V. V. Likhanskii, and A. P. Napartovich, "Theory of phase locking of an array of lasers," Sov. Phys. JETP 66, 676-682 (1987).

Liu, A.

P. K. Cheo, A. Liu, and G. G. King, "A high-brightness laser beam from a phase-locked multicore Yb-doped fiber laser array," IEEE Photon. Technol. Lett. 13, 439-441 (2001).
[CrossRef]

Liu, L.

L. Liu, "Periodic self-Fourier-Fresnel functions," J. Phys. A 27, L285-L289 (1994).
[CrossRef]

Marshall, W. K.

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]

Mashkovsky, D. A.

Mehuys, D.

D. Mehuys, W. Streifer, R. Waarts, and D. F. Welch, "Modal analysis of linear Talbot-cavity semiconductor lasers," Opt. Lett. 16, 823-825 (1991).
[CrossRef] [PubMed]

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]

Michaille, L.

Mitsunaga, K.

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]

Moloney, J. V.

Napartovich, A. P.

A. A. Golubentsev, V. V. Likhanskii, and A. P. Napartovich, "Theory of phase locking of an array of lasers," Sov. Phys. JETP 66, 676-682 (1987).

Pasch, K. A.

C. J. Corcoran, F. Durville, K. A. Pasch, and E. J. Bochove, "Spatial filtering of large-mode area fibre lasers using a self-Fourier cavity for high power applications," J. Opt. A , Pure Appl. Opt. 9, 128-133 (2007).
[CrossRef]

C. J. Corcoran and K. A. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A, Pure Appl. Opt. 7, L1-L7 (2005).
[CrossRef]

Petersson, A.

Peyghambarian, N.

Phillip-Rutz, E. M.

E. M. Phillip-Rutz, "Spatially coherent radiation from an array of GaAs lasers," Appl. Phys. Lett. 26, 475-477 (1975).
[CrossRef]

Prokhorov, A. M.

Riyopolous, S.

Schülzgen, A.

Shepherd, T. J.

Simonsen, H. R.

Streifer, W.

D. Mehuys, W. Streifer, R. Waarts, and D. F. Welch, "Modal analysis of linear Talbot-cavity semiconductor lasers," Opt. Lett. 16, 823-825 (1991).
[CrossRef] [PubMed]

A. Hardy and W. Streifer, "Coupled modes of multiwaveguide systems and phased arrays," J. Lightwave Technol. 4, 90-99 (1986).
[CrossRef]

A. Hardy and W. Streifer, "Coupled mode theory of parallel waveguides," J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

Taylor, D. M.

Temyanko, V.

Waarts, R.

Welch, D. F.

Yariv, A.

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]

Appl. Phys. Lett. (4)

E. M. Phillip-Rutz, "Spatially coherent radiation from an array of GaAs lasers," Appl. Phys. Lett. 26, 475-477 (1975).
[CrossRef]

C. J. Corcoran and F. Durville, "Experimental demonstration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

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]

J. K. Butler, D. E. Ackley, and D. Botez, "Coupled-mode analysis of phase-locked injection laser arrays," Appl. Phys. Lett. 44, 293-295 (1984).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

P. K. Cheo, A. Liu, and G. G. King, "A high-brightness laser beam from a phase-locked multicore Yb-doped fiber laser array," IEEE Photon. Technol. Lett. 13, 439-441 (2001).
[CrossRef]

J. Lightwave Technol. (2)

A. Hardy and W. Streifer, "Coupled modes of multiwaveguide systems and phased arrays," J. Lightwave Technol. 4, 90-99 (1986).
[CrossRef]

A. Hardy and W. Streifer, "Coupled mode theory of parallel waveguides," J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

J. Opt. A (1)

C. J. Corcoran, F. Durville, K. A. Pasch, and E. J. Bochove, "Spatial filtering of large-mode area fibre lasers using a self-Fourier cavity for high power applications," J. Opt. A , Pure Appl. Opt. 9, 128-133 (2007).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

C. J. Corcoran and K. A. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A, Pure Appl. Opt. 7, L1-L7 (2005).
[CrossRef]

J. Phys. A (2)

M. J. Caola, "Self-Fourier functions," J. Phys. A 24L1143-LI144 (1991).
[CrossRef]

L. Liu, "Periodic self-Fourier-Fresnel functions," J. Phys. A 27, L285-L289 (1994).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Sov. Phys. JETP (1)

A. A. Golubentsev, V. V. Likhanskii, and A. P. Napartovich, "Theory of phase locking of an array of lasers," Sov. Phys. JETP 66, 676-682 (1987).

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Figures (8)

Fig. 1
Fig. 1

(Color online) Diagram of the resonator consisting of a laser array coupled to an external SF cavity. AR, array subcavity; EC, external subcavity; M, 100% feedback mirror; P, input plane to the EC; O, outcoupling mirror; f, effective round-trip focal length, which approximately equals the length of the EC. The z axis is defined to follow a round trip in the AR subcavity, unfolded at the feedback mirror M. The plots ψ 1 , … ,  ψ N illustrate the individual fundamental mode of each waveguide channel.

Fig. 2
Fig. 2

(a) Eigenvalue associated with the fundamental SF-cavity supermode plotted for several array sizes N as function of the normalized separation d / w . The lattice separation constant d is given by Eq. (25).

Fig. 3
Fig. 3

Eigenvalue of the fundamental SF-cavity supermode of a seven-element array of Gaussian emitters, as calculated using (i) the “correct” matrix C 1 κ , and (ii) the incorrect matrix κ, which neglects individual emitter mode overlap. We assumed 2 π z R / λ = 6 , and d / z R = 0.5 , where z R is the Rayleigh distance of the fundamental channel modes.

Fig. 4
Fig. 4

Round-trip loss and phasing of five-element slab waveguide array as function of AR round-trip distance, L, where 2 a = waveguide width. Plots (i) (dashed) and (ii) (solid) give the AR-EC coupling loss of the fundamental mode, calculated rigorously and by coupled-mode analysis, respectively. (iii) Loss of next higher-order mode as predicted by exact and (iv) coupled-mode analysis. (v) Phasing function 1 σ of the fundamental mode; where σ = 1 corresponds to perfect phasing. The assumed effective focal length is f = 400 a .

Fig. 5
Fig. 5

(Color online) (a)–(d) Loss plots (i) and (ii) of the first two eigenmodes, and phasing plot (iii) of the fundamental mode for a three-element slab-geometry array as function of wavelength. The solid arrows indicate the fixed wavelength λ 0 at which the separation d is determined. In (a), (b) L = 10.18   m , as determined by requiring coincidence of λ 0 = 1080   nm with a point of minimum loss. In (c), (d) the value L = 10.40   m has been chosen arbitrarily, as are the values of λ 0 in both figures. The dashed arrows indicate possible operating wavelengths.

Fig. 6
Fig. 6

(Color online) Phasing and loss plots of the array of Fig. 4, with effective focal length f = 220 a . The plotted range is increased in order to exhibit the performance decline as L increases. The labels (i)–(v) have the same meaning as in Fig. 4.

Fig. 7
Fig. 7

(a), (b) Near-field and far-field plots of the five-element array of Fig. 6 for L / a = 281, 000 , calculated using the RM. The fundamental mode coupling loss is 70.4%, and that of the next higher-order mode is 99.97%.

Fig. 8
Fig. 8

(Color online) Phasing and loss plots of 25-element ( 5 × 5 ) 2D array of circular cores. The structural parameters are the same as those of the slab array of Fig. 4. (i) Plot of phasing function 1 σ , (ii) fundamental mode loss, (iii) next higher-order mode loss.

Equations (28)

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d = f λ ,
W w = f λ / π ,
N ( d w ) D .
L c ( π λ ) 1 A N 2 / D 1 .
E ( x , y , 0 ) Λ { E ( x , y , L ) } ,
E ( x , y , L ) = n A n ψ n ( x , y ) + H ( x , y ) ,
( ψ n , H ) = 0 ,
E ( x , y , 0 ) = n A n ψ n ( x , y ) + H ( x , y ) ,
E ( x , y , 0 ) = n A n ψ n ( x , y ) + F ( x , y ) ,
( ψ n , F ) = 0.
C A = κ A + H ,
A = R e c A + Δ , where   R e c = C 1 κ ,
l e c = 1 ( E ˜ , E ˜ ) / ( E , E ) ,
l e c = 1 C 1 R e c + C R e c ,
l j e c = 1 | ξ j e c | 2 .
d U ( z ) d z = i K ( z ) U ( z ) .
K = C 1 ( B C + κ ˜ ) ,
κ ˜ l m = π λ n ( n 2 ( x , y ) n l 2 ( x , y ) ) ψ l ( x , y ) ψ m ( x , y ) d x d y ,
R = R a r R e c ,
l = 1 C 1 R + C R ,
h ( u ) = 2 1 / 4 π 1 / 4 w 1 / 2 e u 2 / w 2
Λ { ψ n ( x , y ) } = 1 f λ e i θ h ( x x n ) e i k x x / f d x h ( y ) ,
κ n m = ( f 2 z R + z R 2 f ) ( 1 / 2 ) exp { i ( θ π 4 ) k [ 4 i f x n x m + 2 z R ( x n 2 + x m 2 ) ] 4 ( f 2 + z R 2 ) } ,
C n m = exp [ k ( x n x m ) 2 4 z R ] .
d 0 ( f λ + π 2 w 4 f λ ) 1 / 2 ,
κ n m = ( f 2 z R + z R 2 f ) ( 1 / 2 ) exp [ i ( θ π 4 ) k z R d 2 ( n 2 + m 2 ) 2 ( f 2 + z R 2 ) ] d = d 0 ( f 2 z R + z R 2 f ) ( 1 / 2 ) × exp [ i ( θ π 4 ) π z R ( n 2 + m 2 ) f ] .
σ | i A 1 i | / i | A 1 i | .
L ( N. A. ) 2 O L f 1 w 2

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