Abstract

The space–time dynamical behavior of multistripe index-guided semiconductor laser arrays is studied by using an extension of the usual phenomenological laser model to include transverse diffraction of the counterpropagating optical fields and transverse diffusion of the excited carriers. Our results confirm that evanescently coupled multistripe lasers are a fascinating manifestation of spatiotemporal complexity in spatially extended nonlinear systems. Stabilization of the laser output can be achieved by injection locking the array with a weak external injected signal, and we show that the stability of the externally driven array depends on the transverse profile of the injected signal. A numerical algorithm is presented that takes advantage of high-performance parallel computing architectures to solve the coupled partial differential equations describing the light–matter interaction in the laser structure. Both the model and the numerical algorithm are sufficiently flexible and modular to support arbitrary laser geometries and to allow for inclusion of important many-body semiconductor effects in future studies.

© 1993 Optical Society of America

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References

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  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 (1983).
    [CrossRef]
  2. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125–127 (1984).
    [CrossRef] [PubMed]
  3. S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774–1776 (1988).
    [CrossRef]
  4. H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semi-conductor laser arrays,” Appl. Phys. Lett. 52, 1894–1896 (1988).
    [CrossRef]
  5. G. P. Agrawal, “Fast-Fourier-transform based beam-propagation model for stripe-geometry semiconductor lasers: inclusion of axial effects,” J. Appl. Phys. 56, 3100–3109 (1984).
    [CrossRef]
  6. G. P. Agrawal, “Lateral-mode analysis of gain-guided and index-guided semiconductor-laser arrays,” J. Appl. Phys. 58, 2922–2931 (1985).
    [CrossRef]
  7. G. R. Hadley, J. P. Hohimer, and A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
    [CrossRef]
  8. Y. Twu, S. Wang, J. R. Whinnery, and A. Dienes, “Mode characteristics of phase-locked semiconductor laser arrays at and above threshold,” IEEE J. Quantum Electron. QE-23, 788–795 (1987).
  9. R. Binder, D. Scott, A. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
    [CrossRef]
  10. P. Jakobsen, J. V. Moloney, A. C. Newell, and R. A. Indik, “Space-time dynamics of wide gain section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
    [CrossRef] [PubMed]
  11. P. Ru, P. K. Jakobsen, J. V. Moloney, and R. A. Indik, “Generalized coupled-mode model for the multistripe index-guided laser arrays,” J. Opt. Soc. Am. B 10, 507–515 (1993).
    [CrossRef]
  12. R. A. Elliot, R. K. De Freez, T. L. Paoli, R. D. Burnham, and W. Streifer, “Dynamic characteristics of phase-locked multiple quantum well injection lasers,” IEEE J. Quantum Electron. QE-21, 598–602 (1985).
    [CrossRef]
  13. M. L. Tilton, G. C. Dente, A. H. Paxton, J. Cser, R. K. De Freez, C. E. Mueller, and D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
    [CrossRef]
  14. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).

1993 (1)

1992 (2)

R. Binder, D. Scott, A. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

P. Jakobsen, J. V. Moloney, A. C. Newell, and R. A. Indik, “Space-time dynamics of wide gain section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

1991 (1)

M. L. Tilton, G. C. Dente, A. H. Paxton, J. Cser, R. K. De Freez, C. E. Mueller, and D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

1988 (2)

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774–1776 (1988).
[CrossRef]

H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semi-conductor laser arrays,” Appl. Phys. Lett. 52, 1894–1896 (1988).
[CrossRef]

1987 (2)

G. R. Hadley, J. P. Hohimer, and A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
[CrossRef]

Y. Twu, S. Wang, J. R. Whinnery, and A. Dienes, “Mode characteristics of phase-locked semiconductor laser arrays at and above threshold,” IEEE J. Quantum Electron. QE-23, 788–795 (1987).

1985 (2)

G. P. Agrawal, “Lateral-mode analysis of gain-guided and index-guided semiconductor-laser arrays,” J. Appl. Phys. 58, 2922–2931 (1985).
[CrossRef]

R. A. Elliot, R. K. De Freez, T. L. Paoli, R. D. Burnham, and W. Streifer, “Dynamic characteristics of phase-locked multiple quantum well injection lasers,” IEEE J. Quantum Electron. QE-21, 598–602 (1985).
[CrossRef]

1984 (2)

G. P. Agrawal, “Fast-Fourier-transform based beam-propagation model for stripe-geometry semiconductor lasers: inclusion of axial effects,” J. Appl. Phys. 56, 3100–3109 (1984).
[CrossRef]

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125–127 (1984).
[CrossRef] [PubMed]

1983 (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 (1983).
[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 (1983).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, “Lateral-mode analysis of gain-guided and index-guided semiconductor-laser arrays,” J. Appl. Phys. 58, 2922–2931 (1985).
[CrossRef]

G. P. Agrawal, “Fast-Fourier-transform based beam-propagation model for stripe-geometry semiconductor lasers: inclusion of axial effects,” J. Appl. Phys. 56, 3100–3109 (1984).
[CrossRef]

Binder, R.

R. Binder, D. Scott, A. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[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 (1983).
[CrossRef]

Burnham, R. D.

R. A. Elliot, R. K. De Freez, T. L. Paoli, R. D. Burnham, and W. Streifer, “Dynamic characteristics of phase-locked multiple quantum well injection lasers,” IEEE J. Quantum Electron. QE-21, 598–602 (1985).
[CrossRef]

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 (1983).
[CrossRef]

Cser, J.

M. L. Tilton, G. C. Dente, A. H. Paxton, J. Cser, R. K. De Freez, C. E. Mueller, and D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

De Freez, R. K.

M. L. Tilton, G. C. Dente, A. H. Paxton, J. Cser, R. K. De Freez, C. E. Mueller, and D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

R. A. Elliot, R. K. De Freez, T. L. Paoli, R. D. Burnham, and W. Streifer, “Dynamic characteristics of phase-locked multiple quantum well injection lasers,” IEEE J. Quantum Electron. QE-21, 598–602 (1985).
[CrossRef]

Dente, G. C.

M. L. Tilton, G. C. Dente, A. H. Paxton, J. Cser, R. K. De Freez, C. E. Mueller, and D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Depatie, D.

M. L. Tilton, G. C. Dente, A. H. Paxton, J. Cser, R. K. De Freez, C. E. Mueller, and D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Dienes, A.

Y. Twu, S. Wang, J. R. Whinnery, and A. Dienes, “Mode characteristics of phase-locked semiconductor laser arrays at and above threshold,” IEEE J. Quantum Electron. QE-23, 788–795 (1987).

Elliot, R. A.

R. A. Elliot, R. K. De Freez, T. L. Paoli, R. D. Burnham, and W. Streifer, “Dynamic characteristics of phase-locked multiple quantum well injection lasers,” IEEE J. Quantum Electron. QE-21, 598–602 (1985).
[CrossRef]

Hadley, G. R.

G. R. Hadley, J. P. Hohimer, and A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
[CrossRef]

Haug, H.

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).

Henneberger, K.

R. Binder, D. Scott, A. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

Hohimer, J. P.

G. R. Hadley, J. P. Hohimer, and A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
[CrossRef]

Indik, R. A.

P. Ru, P. K. Jakobsen, J. V. Moloney, and R. A. Indik, “Generalized coupled-mode model for the multistripe index-guided laser arrays,” J. Opt. Soc. Am. B 10, 507–515 (1993).
[CrossRef]

P. Jakobsen, J. V. Moloney, A. C. Newell, and R. A. Indik, “Space-time dynamics of wide gain section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Jakobsen, P.

P. Jakobsen, J. V. Moloney, A. C. Newell, and R. A. Indik, “Space-time dynamics of wide gain section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Jakobsen, P. K.

Kapon, E.

Katz, J.

Koch, S. W.

R. Binder, D. Scott, A. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).

Lindberg, M.

R. Binder, D. Scott, A. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

Moloney, J. V.

P. Ru, P. K. Jakobsen, J. V. Moloney, and R. A. Indik, “Generalized coupled-mode model for the multistripe index-guided laser arrays,” J. Opt. Soc. Am. B 10, 507–515 (1993).
[CrossRef]

P. Jakobsen, J. V. Moloney, A. C. Newell, and R. A. Indik, “Space-time dynamics of wide gain section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Mueller, C. E.

M. L. Tilton, G. C. Dente, A. H. Paxton, J. Cser, R. K. De Freez, C. E. Mueller, and D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Newell, A. C.

P. Jakobsen, J. V. Moloney, A. C. Newell, and R. A. Indik, “Space-time dynamics of wide gain section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Owyoung, A.

G. R. Hadley, J. P. Hohimer, and A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
[CrossRef]

Paoli, T. L.

R. A. Elliot, R. K. De Freez, T. L. Paoli, R. D. Burnham, and W. Streifer, “Dynamic characteristics of phase-locked multiple quantum well injection lasers,” IEEE J. Quantum Electron. QE-21, 598–602 (1985).
[CrossRef]

Paul, A.

R. Binder, D. Scott, A. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

Paxton, A. H.

M. L. Tilton, G. C. Dente, A. H. Paxton, J. Cser, R. K. De Freez, C. E. Mueller, and D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Ru, P.

Scott, D.

R. Binder, D. Scott, A. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

Streifer, W.

R. A. Elliot, R. K. De Freez, T. L. Paoli, R. D. Burnham, and W. Streifer, “Dynamic characteristics of phase-locked multiple quantum well injection lasers,” IEEE J. Quantum Electron. QE-21, 598–602 (1985).
[CrossRef]

Tilton, M. L.

M. L. Tilton, G. C. Dente, A. H. Paxton, J. Cser, R. K. De Freez, C. E. Mueller, and D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Twu, Y.

Y. Twu, S. Wang, J. R. Whinnery, and A. Dienes, “Mode characteristics of phase-locked semiconductor laser arrays at and above threshold,” IEEE J. Quantum Electron. QE-23, 788–795 (1987).

Wang, S.

Y. Twu, S. Wang, J. R. Whinnery, and A. Dienes, “Mode characteristics of phase-locked semiconductor laser arrays at and above threshold,” IEEE J. Quantum Electron. QE-23, 788–795 (1987).

Wang, S. S.

H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semi-conductor laser arrays,” Appl. Phys. Lett. 52, 1894–1896 (1988).
[CrossRef]

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774–1776 (1988).
[CrossRef]

Whinnery, J. R.

Y. Twu, S. Wang, J. R. Whinnery, and A. Dienes, “Mode characteristics of phase-locked semiconductor laser arrays at and above threshold,” IEEE J. Quantum Electron. QE-23, 788–795 (1987).

Winful, H. G.

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774–1776 (1988).
[CrossRef]

H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semi-conductor laser arrays,” Appl. Phys. Lett. 52, 1894–1896 (1988).
[CrossRef]

Yariv, A.

Appl. Phys. Lett. (3)

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774–1776 (1988).
[CrossRef]

H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semi-conductor laser arrays,” Appl. Phys. Lett. 52, 1894–1896 (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 (1983).
[CrossRef]

IEEE J. Quantum Electron. (4)

G. R. Hadley, J. P. Hohimer, and A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
[CrossRef]

Y. Twu, S. Wang, J. R. Whinnery, and A. Dienes, “Mode characteristics of phase-locked semiconductor laser arrays at and above threshold,” IEEE J. Quantum Electron. QE-23, 788–795 (1987).

R. A. Elliot, R. K. De Freez, T. L. Paoli, R. D. Burnham, and W. Streifer, “Dynamic characteristics of phase-locked multiple quantum well injection lasers,” IEEE J. Quantum Electron. QE-21, 598–602 (1985).
[CrossRef]

M. L. Tilton, G. C. Dente, A. H. Paxton, J. Cser, R. K. De Freez, C. E. Mueller, and D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

J. Appl. Phys. (2)

G. P. Agrawal, “Fast-Fourier-transform based beam-propagation model for stripe-geometry semiconductor lasers: inclusion of axial effects,” J. Appl. Phys. 56, 3100–3109 (1984).
[CrossRef]

G. P. Agrawal, “Lateral-mode analysis of gain-guided and index-guided semiconductor-laser arrays,” J. Appl. Phys. 58, 2922–2931 (1985).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Rev. A (1)

P. Jakobsen, J. V. Moloney, A. C. Newell, and R. A. Indik, “Space-time dynamics of wide gain section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (1)

R. Binder, D. Scott, A. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

Other (1)

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).

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Figures (8)

Fig. 1
Fig. 1

Diagram of a multistripe index-guided laser array. Current stripes are laid above each rib, and carriers generated under each rib within the active layer can diffuse transversely (in x) and recombine with holes to generate the light field. Laser output is through the low reflectivity facet; csw, current stripe width.

Fig. 2
Fig. 2

(a) Graph of the total power output and total carrier density for a 50-mA drive curent in a 5-μm stripe as a function of time, showing damped relaxation oscillations. (b) Contour plot of the near-field emission from the 5-μm-wide single emitter over a 12-ns time interval. The light regions depict maximum intensity excursions, showing a sequence of decaying relaxation oscillations to a steady output emission. The small bar at the bottom and top of each picture indicates the emitter transverse dimension (5 μm) on the scale of the picture.

Fig. 3
Fig. 3

(a) Contour plot of the time evolution of the near-field emission for a twin-stripe laser array shown over a total time interval of 18 ns. Each stripe is 5 μm wide and is separated from the next stripe by a 6-μm gap. After the transient relaxation oscillations have decayed, the output never settles down but pulsates randomly in a chaotic fashion. The drive current per emitter is 30 mA. (b) Contour plot of the time evolution of the far field, corresponding to the near-field emission shown in (a) over the same time interval. Dynamic beam steering evident with a mean period in the tens of picoseconds.

Fig. 4
Fig. 4

(a) Contour plot of the near-field emission of a four-stripe laser array. Each stripe is 5 μm wide and is separated from the next stripe by a 6-μm gap. After transient relaxation oscillations have decayed, the system continues to pulsate chaotically in time. The drive current per emitter is 30 mA. (b) Contour plot of the far-field emission corresponding to Fig. 5, shown over the same time interval. Switching between phase-locked and phase-antilocked emission is evident from the picture.

Fig. 5
Fig. 5

(a) Near-field emission from a five-stripe index-guided array over an 18-ns time interval. The 5-μm stripes are separated by a 6-μm gap. (b) Near-field emission from a five-stripe index-guided array with each 5-μm stripe separated by a 10-μm gap. (c) Near-field emission from a five-stripe index-guided array with each 5-μm stripe separated by 3 μm. The drive current per emitter for each case is 30 mA.

Fig. 6
Fig. 6

Nonlinear dispersion curves [(k, w) plots] for free-running twin-stripe (a), four-stripe (b), and 10-stripe (c) arrays. These curves are used for the choice of the optimum external drive frequencies to injection lock the arrays to stable outputs with single-lobed far-field emissions. The spatiotemporal characteristics of the randomly pulsing laser outputs can be gauged from inspection of such curves.

Fig. 7
Fig. 7

(a) Near-field emission from a 10-stripe free-running index-guided array; each 5-μm stripe is separated from the next by a 6-μm gap over 18 ns. The drive current per emitter is 30 mA. (b) Far-field emission from a 10-stripe free-running index-guided array corresponding to the case in (a). The array outputs predominantly in an out-of-phase mode.

Fig. 8
Fig. 8

(a) Near-field emission from the 10-stripe index-guided array when a square signal with a total power of 2 mW is injected into the array whose free-running near-field emission is shown in Fig. 7(a). The square injected signal spans the entire array (width, 104 μm). The array locks to a stable emission if the injected power is increased to 3 mW (b) Far-field emission for the almost-locked 10-stripe index-guided array whose corresponding near-field emission is shown in (a). When the array is transiently locked, most of the power is confined to a single lobe.

Tables (2)

Tables Icon

Table 1 Physical Parameter Values Used in Computations

Tables Icon

Table 2 Steady-State Output Power of a Single-Stripe Laser

Equations (12)

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

F t + F z = i D p 2 F x 2 - i Δ ( x ) F + κ Γ ( x ) [ g ( N ) - i R N ] F ,
B t - B z = i D p 2 B x 2 - i Δ ( x ) F + κ Γ ( x ) [ g ( N ) - i R N ] B ,
N t = D f 2 N x 2 + J ( x ) - γ N - Γ ( x ) g ( N ) ( F 2 + B 2 ) ,
F ( t , z = 0 , x ) = - R 1 B ( t , z = 0 , x ) , B ( t , z = 1 , x ) = - R 2 F ( t , z = 1 , x ) .
F t + F z ~ F i n + 1 - F i n Δ t + F i + 1 n - F i - 1 n 2 Δ z - Δ t 2 F i + 1 n - 2 F i n + F i - 1 n Δ z 2 .
F t + F z - Δ t 2 2 F z 2 = right - hand side .
F z + F t = i D p 2 F x 2 - α ( x ) F + κ Γ ( x ) ( R ¯ N - 1 ) F ,
B z + B t = i D p 2 B x 2 - α ( x ) B + κ Γ ( x ) ( R ¯ N - 1 ) B ,
N t = D f 2 N x 2 + J ( x ) - γ N - Γ ( x ) ( N - 1 ) × ( F 2 + B 2 ) ,
F i , j n + 1 = F i , j n + ρ 2 ( ρ - 1 ) F i + 1 , j n + 1 + ρ 2 ( ρ + 1 ) F i - 1 , j n + 1 + D p σ ( F i , j + 1 n + 1 + F i , j - 1 n + 1 ) 1 + ρ 2 + 2 D p σ + Δ t α j - Δ t κ Γ j ( R ¯ N i , j n + 1 - 1 ) ,
N i , j n + 1 = N i , j n + D f σ ( N i , j + 1 n + 1 + N i , j - 1 n + 1 ) + Δ t J j + Δ t Γ j I i , j n + 1 1 + 2 D f σ + Δ t γ + Δ t Γ j I i , j n + 1 ,
F 1 , j n / n + 1 = R 1 B 1 , j n / n + 1 at the left mirror , B M , j n / n + 1 = - R 2 F M , j n / n + 1 at the right mirror .

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