Abstract

Carrier and gain depletion due to cross-cavity interactions in actively coupled vertical-cavity surface-emitting laser arrays cause nonlinear-cavity frequency pulling. As a result the collective mode structure departs significantly from earlier studied linearly coupled eigenmodes. Theory and simulations find that array eigenmodes have uniform amplitude and density distribution over the interior array sites, accompanied with boundary-layer formation at the edges. Exact zero or π phasing among adjacent sites is achieved, depending on the coupling-strength sign. In contrast, in linearly coupled array modes neglecting cross-cavity hole burning, the phase difference cannot be exactly zero or π, yielding a sinusoidal amplitude distribution over the array sites. The exact zero phasing and the uniform intensity distribution in nonlinearly coupled arrays offer advantages for practical applications.

© 2006 Optical Society of America

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  1. J. K. Butler, D. E. Ackley, and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
    [CrossRef]
  2. H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y.-S. Kwon, "Array mode analysis of two-dimensional phased arrays of VCSELs," IEEE J. Quantum Electron. 26, 1039-1051 (1990).
    [CrossRef]
  3. S. Riyopoulos, "Coherent phase locking, collective oscillations and stability in coupled VCSEL arrays," Phys. Rev. A 66, 53820 (2002).
    [CrossRef]
  4. S. Riyopoulos, "Simulations of boundary layers and point defects in coupled VCSEL arrays," IEEE J. Sel. Top. Quantum Electron. (to be published).
  5. D. Zhou, L. J. Mawst, and Z. Dai, "Modal properties of two-dimensional antiguided VCSEL arrays," IEEE J. Quantum Electron. 38, 652-663 (2002).
    [CrossRef]
  6. D. Botez and A. P. Napartovich, "Phase-locked arrays of anti-guides: analytical theory," IEEE J. Quantum Electron. 30, 975-979 (1994).
    [CrossRef]
  7. D. Botez and A. P. Napartovich, "Corrections to 'Phase-locked arrays of anti-guides: analytical theory'," IEEE J. Quantum Electron. 32, 2175 (1996)
    [CrossRef]
  8. G. Hergenhan, B. Lucke, and U. Brauch, "Coherent coupling of VCSELs arrays and efficient beam combining by diffractive optical elements: concept and experimental verification," Appl. Opt. 42, 1667-1680 (2003).
    [CrossRef] [PubMed]
  9. F. Prati, D. Vecchione, and G. Vendramin, "Frequency locking of supermodes and stability of the out of phase locked state in 1-D and 2-D arrays of VCSELs," Opt. Lett. 22, 1633-1635 (1997).
    [CrossRef]
  10. P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, and B. E. Hammons, "Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays," Appl. Phys. Lett. 58, 890-892 (1991).
    [CrossRef]
  11. M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, "Large two-dimensional arrays of phase-locked VCSELs," Appl. Phys. Lett. 60, 1535-1537 (1992).
    [CrossRef]
  12. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, "High-power coherently coupled 8×8 VCSEL array," Appl. Phys. Lett. 61, 1160-1162 (1992).
    [CrossRef]
  13. P. Debernardi, G. P. Bava, F. Monti diSopra, and B. M. Willemsen, "Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory," IEEE J. Quantum Electron. 39, 109-119 (2003).
    [CrossRef]

2003 (2)

P. Debernardi, G. P. Bava, F. Monti diSopra, and B. M. Willemsen, "Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory," IEEE J. Quantum Electron. 39, 109-119 (2003).
[CrossRef]

G. Hergenhan, B. Lucke, and U. Brauch, "Coherent coupling of VCSELs arrays and efficient beam combining by diffractive optical elements: concept and experimental verification," Appl. Opt. 42, 1667-1680 (2003).
[CrossRef] [PubMed]

2002 (2)

S. Riyopoulos, "Coherent phase locking, collective oscillations and stability in coupled VCSEL arrays," Phys. Rev. A 66, 53820 (2002).
[CrossRef]

D. Zhou, L. J. Mawst, and Z. Dai, "Modal properties of two-dimensional antiguided VCSEL arrays," IEEE J. Quantum Electron. 38, 652-663 (2002).
[CrossRef]

1997 (1)

1996 (1)

D. Botez and A. P. Napartovich, "Corrections to 'Phase-locked arrays of anti-guides: analytical theory'," IEEE J. Quantum Electron. 32, 2175 (1996)
[CrossRef]

1994 (1)

D. Botez and A. P. Napartovich, "Phase-locked arrays of anti-guides: analytical theory," IEEE J. Quantum Electron. 30, 975-979 (1994).
[CrossRef]

1992 (2)

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, "Large two-dimensional arrays of phase-locked VCSELs," Appl. Phys. Lett. 60, 1535-1537 (1992).
[CrossRef]

A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, "High-power coherently coupled 8×8 VCSEL array," Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

1991 (1)

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, and B. E. Hammons, "Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays," Appl. Phys. Lett. 58, 890-892 (1991).
[CrossRef]

1990 (2)

J. K. Butler, D. E. Ackley, and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
[CrossRef]

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y.-S. Kwon, "Array mode analysis of two-dimensional phased arrays of VCSELs," IEEE J. Quantum Electron. 26, 1039-1051 (1990).
[CrossRef]

Ackley, D. E.

J. K. Butler, D. E. Ackley, and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
[CrossRef]

Asom, M.

A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, "High-power coherently coupled 8×8 VCSEL array," Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Bava, G. P.

P. Debernardi, G. P. Bava, F. Monti diSopra, and B. M. Willemsen, "Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory," IEEE J. Quantum Electron. 39, 109-119 (2003).
[CrossRef]

Botez, D.

D. Botez and A. P. Napartovich, "Corrections to 'Phase-locked arrays of anti-guides: analytical theory'," IEEE J. Quantum Electron. 32, 2175 (1996)
[CrossRef]

D. Botez and A. P. Napartovich, "Phase-locked arrays of anti-guides: analytical theory," IEEE J. Quantum Electron. 30, 975-979 (1994).
[CrossRef]

Brauch, U.

Brennan, T. M.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, and B. E. Hammons, "Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays," Appl. Phys. Lett. 58, 890-892 (1991).
[CrossRef]

Butler, J. K.

J. K. Butler, D. E. Ackley, and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
[CrossRef]

Dai, Z.

D. Zhou, L. J. Mawst, and Z. Dai, "Modal properties of two-dimensional antiguided VCSEL arrays," IEEE J. Quantum Electron. 38, 652-663 (2002).
[CrossRef]

Debernardi, P.

P. Debernardi, G. P. Bava, F. Monti diSopra, and B. M. Willemsen, "Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory," IEEE J. Quantum Electron. 39, 109-119 (2003).
[CrossRef]

diSopra, F. Monti

P. Debernardi, G. P. Bava, F. Monti diSopra, and B. M. Willemsen, "Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory," IEEE J. Quantum Electron. 39, 109-119 (2003).
[CrossRef]

Ettenberg, M.

J. K. Butler, D. E. Ackley, and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
[CrossRef]

Florez, L. T.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, "Large two-dimensional arrays of phase-locked VCSELs," Appl. Phys. Lett. 60, 1535-1537 (1992).
[CrossRef]

Focht, M. W.

A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, "High-power coherently coupled 8×8 VCSEL array," Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Gourley, P. L.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, and B. E. Hammons, "Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays," Appl. Phys. Lett. 58, 890-892 (1991).
[CrossRef]

Guth, G. D.

A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, "High-power coherently coupled 8×8 VCSEL array," Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Hadley, G. R.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, and B. E. Hammons, "Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays," Appl. Phys. Lett. 58, 890-892 (1991).
[CrossRef]

Hammons, B. E.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, and B. E. Hammons, "Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays," Appl. Phys. Lett. 58, 890-892 (1991).
[CrossRef]

Harbison, J. P.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, "Large two-dimensional arrays of phase-locked VCSELs," Appl. Phys. Lett. 60, 1535-1537 (1992).
[CrossRef]

Hayes, J. R.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y.-S. Kwon, "Array mode analysis of two-dimensional phased arrays of VCSELs," IEEE J. Quantum Electron. 26, 1039-1051 (1990).
[CrossRef]

Hergenhan, G.

Kapon, E.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, "Large two-dimensional arrays of phase-locked VCSELs," Appl. Phys. Lett. 60, 1535-1537 (1992).
[CrossRef]

Kojima, K.

A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, "High-power coherently coupled 8×8 VCSEL array," Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Kwon, Y.-S.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y.-S. Kwon, "Array mode analysis of two-dimensional phased arrays of VCSELs," IEEE J. Quantum Electron. 26, 1039-1051 (1990).
[CrossRef]

Leibenguth, R. E.

A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, "High-power coherently coupled 8×8 VCSEL array," Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Lucke, B.

Mawst, L. J.

D. Zhou, L. J. Mawst, and Z. Dai, "Modal properties of two-dimensional antiguided VCSEL arrays," IEEE J. Quantum Electron. 38, 652-663 (2002).
[CrossRef]

Morgan, A.

A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, "High-power coherently coupled 8×8 VCSEL array," Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Mullally, T.

A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, "High-power coherently coupled 8×8 VCSEL array," Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Napartovich, A. P.

D. Botez and A. P. Napartovich, "Corrections to 'Phase-locked arrays of anti-guides: analytical theory'," IEEE J. Quantum Electron. 32, 2175 (1996)
[CrossRef]

D. Botez and A. P. Napartovich, "Phase-locked arrays of anti-guides: analytical theory," IEEE J. Quantum Electron. 30, 975-979 (1994).
[CrossRef]

Orenstein, M.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, "Large two-dimensional arrays of phase-locked VCSELs," Appl. Phys. Lett. 60, 1535-1537 (1992).
[CrossRef]

Paek, E. G.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y.-S. Kwon, "Array mode analysis of two-dimensional phased arrays of VCSELs," IEEE J. Quantum Electron. 26, 1039-1051 (1990).
[CrossRef]

Prati, F.

Riyopoulos, S.

S. Riyopoulos, "Coherent phase locking, collective oscillations and stability in coupled VCSEL arrays," Phys. Rev. A 66, 53820 (2002).
[CrossRef]

S. Riyopoulos, "Simulations of boundary layers and point defects in coupled VCSEL arrays," IEEE J. Sel. Top. Quantum Electron. (to be published).

Scherer, A.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y.-S. Kwon, "Array mode analysis of two-dimensional phased arrays of VCSELs," IEEE J. Quantum Electron. 26, 1039-1051 (1990).
[CrossRef]

Stoffel, N. G.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, "Large two-dimensional arrays of phase-locked VCSELs," Appl. Phys. Lett. 60, 1535-1537 (1992).
[CrossRef]

Vawter, G. A.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, and B. E. Hammons, "Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays," Appl. Phys. Lett. 58, 890-892 (1991).
[CrossRef]

Vecchione, D.

Vendramin, G.

Warren, M. E.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, and B. E. Hammons, "Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays," Appl. Phys. Lett. 58, 890-892 (1991).
[CrossRef]

Willemsen, B. M.

P. Debernardi, G. P. Bava, F. Monti diSopra, and B. M. Willemsen, "Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory," IEEE J. Quantum Electron. 39, 109-119 (2003).
[CrossRef]

Yoo, H.-J.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y.-S. Kwon, "Array mode analysis of two-dimensional phased arrays of VCSELs," IEEE J. Quantum Electron. 26, 1039-1051 (1990).
[CrossRef]

Zhou, D.

D. Zhou, L. J. Mawst, and Z. Dai, "Modal properties of two-dimensional antiguided VCSEL arrays," IEEE J. Quantum Electron. 38, 652-663 (2002).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, and B. E. Hammons, "Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays," Appl. Phys. Lett. 58, 890-892 (1991).
[CrossRef]

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, "Large two-dimensional arrays of phase-locked VCSELs," Appl. Phys. Lett. 60, 1535-1537 (1992).
[CrossRef]

A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, "High-power coherently coupled 8×8 VCSEL array," Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

IEEE J. Quantum Electron. (6)

P. Debernardi, G. P. Bava, F. Monti diSopra, and B. M. Willemsen, "Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory," IEEE J. Quantum Electron. 39, 109-119 (2003).
[CrossRef]

J. K. Butler, D. E. Ackley, and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
[CrossRef]

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y.-S. Kwon, "Array mode analysis of two-dimensional phased arrays of VCSELs," IEEE J. Quantum Electron. 26, 1039-1051 (1990).
[CrossRef]

D. Zhou, L. J. Mawst, and Z. Dai, "Modal properties of two-dimensional antiguided VCSEL arrays," IEEE J. Quantum Electron. 38, 652-663 (2002).
[CrossRef]

D. Botez and A. P. Napartovich, "Phase-locked arrays of anti-guides: analytical theory," IEEE J. Quantum Electron. 30, 975-979 (1994).
[CrossRef]

D. Botez and A. P. Napartovich, "Corrections to 'Phase-locked arrays of anti-guides: analytical theory'," IEEE J. Quantum Electron. 32, 2175 (1996)
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

S. Riyopoulos, "Simulations of boundary layers and point defects in coupled VCSEL arrays," IEEE J. Sel. Top. Quantum Electron. (to be published).

Opt. Lett. (1)

Phys. Rev. A (1)

S. Riyopoulos, "Coherent phase locking, collective oscillations and stability in coupled VCSEL arrays," Phys. Rev. A 66, 53820 (2002).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(Color online) Bargraph showing the peak amplitude distribution E n = E o sin n Φ in a 1 × 11 linearly coupled array for (a) the lowest “in-phase-like” (b) the highest “antiphase-like” eigenmodes.

Fig. 2
Fig. 2

(Color online) Bargraph showing the peak amplitude distribution E n = E ( n ) cos n Φ from the simulation of a 1 × 11 nonlinearly (actively) coupled array for (a) in-phase (b) antiphase modes. The corresponding phase deviations from exact 0 or π phasing at a given instant t, shown in (c) and (d) respectively, are smaller than 0.002 and are caused by spontaneous noise fluctuations.

Fig. 3
Fig. 3

Time evolution of the phase differences Φ n φ n φ n 1 among neighbor cavities, for all 11 sites, during the simulations that produced the steady-state array patterns shown in Fig. 2a, 2b, respectively. The dashed lines indicate the locked-phase values for the lowest and highest modes in linearly coupled arrays.

Fig. 4
Fig. 4

(Color online) Nonlinear array eigenmode structure derived from simulations of an 8 × 8 actively coupled array. The left and right column results correspond, respectively, to in- and antiphase locking, Φ = 0 , Φ = π for coupling strengths Υ = 0.006 and Υ = 0.006 . (a) and (b) show bargraph distribution of normalized carrier density N m n over array sites, (c) and (d) show bargraph distribution of peak (center) cavity amplitudes E m n = E m cos ( m Φ x + n Φ y ) , and (e) and (f) show corresponding distribution of phase difference among x-direction neighbors Φ m n x φ m n φ m 1 , n .

Fig. 5
Fig. 5

(Color online) Bargraph showing the peak amplitude distribution E m n = E o sin ( m Φ x + n Φ y ) in a 8 × 8 linearly coupled array for (a) lowest “in-phase-like” Φ x = Φ y = π 9 (b) highest “antiphase-like” eigenmodes Φ x = Φ y = 8 π 9 .

Fig. 6
Fig. 6

Contour plot amplitude distributions corresponding to (a) the nonlinear in-phase locked array mode of Fig. 5c; (b) the linear in-phase locked mode of Fig. 6a.

Fig. 7
Fig. 7

(Color online) Schematic illustration of the analogies between coupled cavity arrays with linearly coupled masses and coupled nonlinear oscillators. The electric field E n = E n exp ( i φ n ) corresponds to the “displacement” and the coupling strength Υ to the external spring constant. The nonlinear oscillators posses an “internal spring” with adjustable spring constant K Δ ω n E n .

Fig. 8
Fig. 8

(Color online) Bargraphs showing the mode structure in a in-phase 1 × 11 nonlinearly (actively) coupled array, including mirror interference coupling among adjacent cavities (a) the peak amplitude distribution E m (b) neighbor phase difference Φ m , for Υ = V = 0.0060 . (c) and (d) are as (a) and (b) but for Υ = 0 , V = 0.012 .

Fig. 9
Fig. 9

(Color online) Numerically obtained steady state for an in-phase, mirror-only coupled 8 × 8 array with V = 0.0061 . (a) Bargraph showing the peak amplitude distribution E m n ; (b) near-field contour plot obtained by a superposition of Gaussian eigenmodes of height E m n and phase φ m n from the simulation.

Equations (16)

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

t E m n = v g ζ g o E m n v g μ E m n + v g ζ g o Υ m = m ± 1 n = n ± 1 E m n .
Υ d r 2 U ( r ) χ ( r ) U ( r ± b ) ,
E ( r , t ) = exp ( i Δ ω t ) m , n = 1 N ( E m n + E m n ) U ( r R m n ) ,
Γ = v g ζ g r ln N ̂ v g μ + 2 v g ζ g r Υ cos Φ ,
Δ ω = v g ζ g i ln N ̂ + 2 v g ζ g i Υ sin Φ .
E ( r , t ) = exp ( i Δ ω t ) 1 2 i n = 1 N [ E o exp ( i φ n ) c c ] U ( r R n ) = exp ( i Δ ω t ) n = 1 N E o sin n Φ U ( r R n ) .
g o g m n = g ̂ o ln N ̂ m n ,
N m n t = J m n e d w γ N m n B N m n 2 ξ g ̂ r ln N ̂ m n f E m n E m n * ξ g ̂ r ln N ̂ m n 2 f Υ m , n E m n E m n * .
E = exp ( i Δ ω t ) n = 1 , m = 1 M , N E m n U ( r R m n ) = exp ( i Δ ω t ) n = 1 , m = 1 M , N E m n cos ( m Φ + n Φ ) U ( r R m n ) .
N ̂ m n N ̂ o ln N ̂ o S m n { Λ + 2 Υ cos Φ + Π + 2 V cos Φ } ,
F m n = F o + ( γ ̃ N o ξ g r ) S m n { Λ + 2 Υ cos Φ + Π + 2 V cos Φ } + S m n { Π + 2 V cos Φ } .
N ̂ o N o N tr = exp [ μ ζ g r ] ,
F o = ( J o e d w γ ̃ N o ) v g μ
Δ ω m n φ m n t = v g ζ g i 2 ln N ̂ m n + Υ m = m ± 1 n = n ± 1 v g ζ ( ln N ̂ m n + ln N ̂ m n ) E m n E m n [ g ̂ r 2 sin ( φ m n φ m n ) + g ̂ i 2 cos ( φ m n φ m n ) ] .
Φ m n ; m n t φ m n t φ m n t = Δ ω m n Δ ω m n = 0.
+ v g μ V m = m ± 1 n = n ± 1 E m n ,

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