Abstract

We investigate the spontaneous emergence of transverse patterns in lasers by using both the standard two-level model and the so-called cubic approximation, which is generally valid in the threshold regions. The stationary intensity configurations fall into two distinct classes. The first includes solutions of the single-mode type with the frequency and spatial structure of one of the transverse resonances. The solutions of the second group involve the simultaneous oscillation of several cavity modes, operating in such a way as to produce a stationary intensity profile. The stationary character of these multimode configurations emerges from the fact that the transverse modes of the resonator lock onto a common frequency during the nonlinear transient. We call this phenomenon cooperative frequency locking.

© 1988 Optical Society of America

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References

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  1. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibri-um Systems (Wiley, New York, 1977).
  2. H. Haken, Synergetics—An Introduction (Springer-Verlag, Berlin, 1977).
  3. J. D. Murray, J. Theor. Bio. 88, 161 (1981).
    [CrossRef]
  4. A. M. Turing, Phil. Trans. R. Soc. London Ser. B 237, 37 (1952).
    [CrossRef]
  5. N. B. Abraham, L. A. Lugiato, and L. M. Narducci, eds., feature issue on instabilities in active optical media, J. Opt. Soc. Am. B 2 (1) (1985).
  6. F. T. Arecchi and R. Harrison, eds., Instabilities and Chaos in Quantum Optics (Springer-Verlag, Berlin, 1987).
    [CrossRef]
  7. L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
    [CrossRef] [PubMed]
  8. L. A. Lugiato, L. M. Narducci, and R. Lefever, in Lasers and Synergetics—A Volume in Honor of the 60th Birthday of Hermann Haken (Springer-Verlag, Berlin, 1987).
  9. L. A. Lugiato and R. Lefever, in Interaction of Radiation and Matter, a Volume in Honor of Adriano Gozzini (Quaderni della Scuola Normale Superiore, Pisa, 1987).
  10. L. A. Lugiato and C. Oldano, Phys. Rev. A (to be published).
  11. H. Haken, Light 2—Laser Light Dynamics (North-Holland, Amsterdam, 1985).
  12. H. Zeghlache and P. Mandel, J. Opt. Soc. Am. B 2, 18 (1985).
    [CrossRef]
  13. L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
    [CrossRef] [PubMed]
  14. Here we do not consider the case in which the coefficients fn have a random relative phase, which produces a stationary output intensity.
  15. L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, Opt. Commun. 64, 167 (1987).
    [CrossRef]
  16. D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
    [CrossRef] [PubMed]

1987 (2)

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, Opt. Commun. 64, 167 (1987).
[CrossRef]

1986 (1)

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[CrossRef] [PubMed]

1985 (3)

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, eds., feature issue on instabilities in active optical media, J. Opt. Soc. Am. B 2 (1) (1985).

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

H. Zeghlache and P. Mandel, J. Opt. Soc. Am. B 2, 18 (1985).
[CrossRef]

1981 (1)

J. D. Murray, J. Theor. Bio. 88, 161 (1981).
[CrossRef]

1952 (1)

A. M. Turing, Phil. Trans. R. Soc. London Ser. B 237, 37 (1952).
[CrossRef]

Abraham, N. B.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[CrossRef] [PubMed]

Bandy, D. K.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, Opt. Commun. 64, 167 (1987).
[CrossRef]

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[CrossRef] [PubMed]

Haken, H.

H. Haken, Light 2—Laser Light Dynamics (North-Holland, Amsterdam, 1985).

H. Haken, Synergetics—An Introduction (Springer-Verlag, Berlin, 1977).

Lefever, R.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

L. A. Lugiato, L. M. Narducci, and R. Lefever, in Lasers and Synergetics—A Volume in Honor of the 60th Birthday of Hermann Haken (Springer-Verlag, Berlin, 1987).

L. A. Lugiato and R. Lefever, in Interaction of Radiation and Matter, a Volume in Honor of Adriano Gozzini (Quaderni della Scuola Normale Superiore, Pisa, 1987).

Lugiato, L. A.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, Opt. Commun. 64, 167 (1987).
[CrossRef]

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[CrossRef] [PubMed]

L. A. Lugiato and C. Oldano, Phys. Rev. A (to be published).

L. A. Lugiato and R. Lefever, in Interaction of Radiation and Matter, a Volume in Honor of Adriano Gozzini (Quaderni della Scuola Normale Superiore, Pisa, 1987).

L. A. Lugiato, L. M. Narducci, and R. Lefever, in Lasers and Synergetics—A Volume in Honor of the 60th Birthday of Hermann Haken (Springer-Verlag, Berlin, 1987).

Mandel, P.

McLaughlin, D. W.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

Moloney, J. V.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

Murray, J. D.

J. D. Murray, J. Theor. Bio. 88, 161 (1981).
[CrossRef]

Narducci, L. M.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, Opt. Commun. 64, 167 (1987).
[CrossRef]

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[CrossRef] [PubMed]

L. A. Lugiato, L. M. Narducci, and R. Lefever, in Lasers and Synergetics—A Volume in Honor of the 60th Birthday of Hermann Haken (Springer-Verlag, Berlin, 1987).

Newell, A. C.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

Nicolis, G.

G. Nicolis and I. Prigogine, Self-Organization in Nonequilibri-um Systems (Wiley, New York, 1977).

Oldano, C.

L. A. Lugiato and C. Oldano, Phys. Rev. A (to be published).

Prati, F.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, Opt. Commun. 64, 167 (1987).
[CrossRef]

Prigogine, I.

G. Nicolis and I. Prigogine, Self-Organization in Nonequilibri-um Systems (Wiley, New York, 1977).

Ru, P.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, Opt. Commun. 64, 167 (1987).
[CrossRef]

Tredicce, J. R.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, Opt. Commun. 64, 167 (1987).
[CrossRef]

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[CrossRef] [PubMed]

Turing, A. M.

A. M. Turing, Phil. Trans. R. Soc. London Ser. B 237, 37 (1952).
[CrossRef]

Zeghlache, H.

J. Opt. Soc. Am. B (2)

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, eds., feature issue on instabilities in active optical media, J. Opt. Soc. Am. B 2 (1) (1985).

H. Zeghlache and P. Mandel, J. Opt. Soc. Am. B 2, 18 (1985).
[CrossRef]

J. Theor. Bio. (1)

J. D. Murray, J. Theor. Bio. 88, 161 (1981).
[CrossRef]

Opt. Commun. (1)

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, Opt. Commun. 64, 167 (1987).
[CrossRef]

Phil. Trans. R. Soc. London Ser. B (1)

A. M. Turing, Phil. Trans. R. Soc. London Ser. B 237, 37 (1952).
[CrossRef]

Phys. Rev. A (1)

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

Other (8)

L. A. Lugiato, L. M. Narducci, and R. Lefever, in Lasers and Synergetics—A Volume in Honor of the 60th Birthday of Hermann Haken (Springer-Verlag, Berlin, 1987).

L. A. Lugiato and R. Lefever, in Interaction of Radiation and Matter, a Volume in Honor of Adriano Gozzini (Quaderni della Scuola Normale Superiore, Pisa, 1987).

L. A. Lugiato and C. Oldano, Phys. Rev. A (to be published).

H. Haken, Light 2—Laser Light Dynamics (North-Holland, Amsterdam, 1985).

G. Nicolis and I. Prigogine, Self-Organization in Nonequilibri-um Systems (Wiley, New York, 1977).

H. Haken, Synergetics—An Introduction (Springer-Verlag, Berlin, 1977).

Here we do not consider the case in which the coefficients fn have a random relative phase, which produces a stationary output intensity.

F. T. Arecchi and R. Harrison, eds., Instabilities and Chaos in Quantum Optics (Springer-Verlag, Berlin, 1987).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of the Fabry–Perot cavity configuration discussed in this paper. The four mirrors have a transmittance coefficient T ≪ 1. The cavity is open in the y direction, which is also the direction of the magnetic field H.

Fig. 2
Fig. 2

Ring-cavity configuration. Domain of instability in the plane of the variables a(n) = 2n2 and C for θ = −1.

Fig. 3
Fig. 3

Instability domain in the plane of the variables a(1) = 2 and f02 for κ′= 0.1, γ/γ = 1, and (a), (b) θ = −0.6, (c) θ = −0.65, (d) θ = −0.67. (b) Is an expanded version of 3(a) for small values of a(1).

Fig. 4
Fig. 4

This figure illustrates qualitatively the regions A, B, and C discussed in the text.

Fig. 5
Fig. 5

Time evolution of the transverse intensity profile toward steady state for κ′= 0.1, γ/γ 1, θ = −0.6, f02 = 2, and a(1) = 0.65. (a) and (b) Show the evolution looking backward and forward in time, respectively.

Fig. 6
Fig. 6

Transverse profile of the stationary output intensity for κ′ = 0.1, γ/γ = 1, θ = −0.6, f02 = 2, and (a) a(1) = 0.65, (b) a(1) = 0.55, (c) a(1) = 0.12, (d) a(1) = 0.05.

Fig. 7
Fig. 7

(a) Plot of the stationary modal intensities for n = 0 and n = 1 as functions of a(1) over the domain labeled B in Fig. 4. The remaining parameters are the same as in Fig. 6. (b) Is the same as Fig. 7(a) but for modes n = 2 and n = 3.

Fig. 8
Fig. 8

Behavior of the oscillation period T = 2π/δ over the same domain of a(1) considered in Fig. 7.

Equations (93)

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L b
F = b 2 λ L ,
a = 1 2 π F T
E x cos ( k x x ) sin ( k z z ) ,
E z sin ( k x x ) cos ( k z z ) ,
H y cos ( k x x ) cos ( k z z ) ,
k x = π n b , k z = π n z L ,
k x k z .
E z / E x K x / k z ,
cos ( π b n x ) sin ( π L n z z ) .
cos ( π b n x ) exp ( i 2 π L n z z ) ,
a = 1 4 π F T .
α L 1 , T 1 ,
C α L σ 2 T
γ L 2 π c 1 ,
E x = E ( x , t ) exp ( i 2 π L m z z ) exp ( i ω R t ) + c . c . ,
E ( x , t ) = n = 0 f n ( t ) cos ( π n x ) ,
x = x / b .
E t = κ [ ( 1 + i θ ) E 2 C P i a 2 E x 2 ] ,
P t = γ [ ( 1 + i Δ ) P E D ] ,
D t = γ [ D 1 + ½ ( E * P + E P * ) ] ,
κ = c T L .
θ = ω C ω R κ , Δ = ω A ω R γ ,
ω R = κ ω A + γ ω C κ + γ
Δ = θ ,
| E | 2 = 2 C 1 θ 2 , P = 1 + i θ 2 C E , D = 1 + θ 2 2 C ,
κ γ , γ
κ 1 E t = E ( 1 + i θ ) ( 1 2 C 1 + θ 2 + | E | 2 ) + i a 2 E x 2 .
1 1 + θ 2 + | E | 2 1 1 + θ 2 | E | 2 ( 1 + θ 2 ) 2 ,
E = ( 2 C ) 1 / 2 1 + θ 2 E , r = 2 C 1 + θ 2 1
κ 1 E t = E ( 1 + i θ ) ( | E | 2 r ) + i a 2 E x 2 .
| E | 2 = r .
τ = κ r t , E = E / r 1 / 2 , a = a / r ,
E τ = E ( 1 + i θ ) ( | E | 2 1 ) + i a 2 E x 2 .
E x = E ( x , t ) cos ( π L m z z ) exp ( i ω 0 t ) + c . c .
E t = κ [ ( 1 + i θ ) E 2 C × π + π d η cos η P ( x , η , t ) i a 2 E x 2 ] ,
P t = γ [ ( 1 + i Δ ) P 2 E D cos η ] ,
D t = γ [ D 1 + ( E * P + E P * ) cos η ] ,
κ = c T 2 L , C = α L σ T
P ( x , t ) = n = 0 p n ( t ) cos ( π n x ) ,
D ( x , t ) = n = 0 d n ( t ) cos ( π n x ) .
0 1 d x cos ( π n x ) cos ( π m x ) = ½ ( δ n , 0 δ m , 0 + δ n , m ) .
d f n d t = κ [ ( 1 + i θ ) f n 2 C p n + i a ( n ) f n ] ,
d p n d t = γ [ ( 1 i θ ) p n 1 2 ( 1 + δ n , 0 ) m 1 0 f * m d 1 ] ,
d d n d t = γ [ d n δ n , 0 + 1 4 ( 1 + δ n , 0 ) × m 1 0 ( f m p 1 * + f m * p 1 ) * ] ,
a ( n ) = a π 2 n 2
| f n | 2 = ( 2 C 1 θ 2 ) δ n , 0 , p n = 1 + i θ 2 C f n , d n = 1 + θ 2 2 C δ n , 0 ,
E ( x , t ) = n = 0 f n ( t ) cos ( π n x )
d f n d τ = ( 1 + i θ ) [ f n 1 4 ( 1 + δ n , 0 ) m 1 s 0 f * m f 1 f s * ] i a ( n ) f n ,
a ( n ) = a π 2 n 2
| f n | 2 = δ n , 0 ,
ω n = c ( k x 2 + k z 2 ) 1 / 2 ω 0 + c k x 2 2 k z ,
ω 0 = ω C = c k z = 2 π c m z / L .
ω n ω 0 κ = a π 2 n 2 = a ( n ) .
d δ f n d t = κ [ ( 1 + i θ ) δ f n 2 C δ p n + i a ( n ) δ f n ] ,
d δ p n d t = γ [ ( 1 i θ ) δ p n f 0 δ d n 1 + θ 2 2 C δ f n ] ,
d δ d n d t = γ ( δ d n + ½ f 0 { δ p n + δ p n * + 1 2 C [ ( 1 + i θ ) δ f n * + ( 1 i θ ) δ f n ] } ) ,
f 0 = ( 2 C 1 θ 2 ) 1 / 2 ,
[ δ f n ( t ) δ f n * ( t ) δ p n ( t ) δ p n * ( t ) δ d n ( t ) ] = e λ t [ b n c n u n υ n w n ]
λ 5 + a 4 ( n ) λ 4 + a 3 ( n ) λ 3 + a 2 ( n ) λ 2 + a 1 ( n ) λ + a 0 ( n ) = 0 .
a ( n ) 2 θ f 0 2 1 + θ 2 + f 0 2 = θ C ( 2 C 1 θ 2 ) .
ω n ω R < ω R ω 0 .
λ 2 + 2 λ + a ( n ) [ a ( n ) + 2 θ ] = 0 ,
a ( n ) 2 θ or a ( n ) 2 θ r .
E ( x , t ) = exp ( i δ κ t ) E s ( x ) ,
P ( x , t ) = exp ( i δ κ t ) P s ( x ) ,
D ( x , t ) = D s ( x ) .
i δ E s = ( 1 + i θ ) E s 2 C P s i a 2 E s x 2 ,
P s = E s 1 + i ( θ + δ κ ) 1 + ( θ + δ κ ) 2 + | E s | 2 ,
D s = 1 + i ( θ + δ κ ) 2 1 + i ( θ + δ κ ) 2 + | E s | 2 ,
κ = κ / γ .
i δ | E s | 2 = ( 1 + θ ) | E s | 2 2 C | E s | 2 × 1 + i ( θ + δ κ ) 1 + ( θ + δ κ ) 2 + | E s | 2 i a 2 E s x 2 E s * .
0 = | E s | 2 [ 1 2 C 1 + ( θ + δ κ ) 2 + | E s | 2 ] + i a 2 x × ( E s E s * x E s * E s x ) ,
0 = | E s | 2 [ θ δ 2 C ( θ + δ κ ) 1 + ( θ δ κ ) 2 + | E s | 2 ] i a 2 x × ( E s E s * x + E s * E s x ) + a | E s x | 2 .
| E s | 2 = 2 C | E s | 2 1 + ( θ + δ κ ) 2 + | E s | 2 ,
| E s | 2 ( θ δ ) = 2 C ( θ + δ κ ) | E s | 2 1 + ( θ + δ κ ) 2 + | E s | 2 a | E s x | 2 .
δ = a 1 + κ | E s x | 2 | E s | 2 .
E s ( x ) = n = 0 f n cos ( π n x ) ,
δ = 1 1 + κ n = 1 | f n | 2 a ( n ) 2 | f 0 | 2 + n = 1 | f n | 2 ,
| E s | 2 = | E s | 4 .
P s ( x ) = n = 0 p n cos ( π n x ) ,
D s ( x ) = n = 0 d n cos ( π n x ) ,
| f k | 2 = 4 3 { 2 C [ 1 + ( θ + δ κ ) 2 ] } ,
δ = a ( k ) 1 + κ .
ω k = ω R + δ κ = κ ω A + γ ω k κ + γ ,
| f k | 2 = 4 3 and δ = a ( k )
n f n g n ( x ) exp ( i ω n t ) + c . c . ,
E x = n f n cos ( π n x ) exp ( i 2 π L m z z ) × exp [ i ( ω R + δ κ ) t ] + c . c .
a 4 ( n ) = 2 γ + γ + 2 κ ,
a 3 ( n ) = 2 γ γ + γ γ f 0 2 + 2 κ ( 2 γ + γ ) + ( κ γ ) ( 1 + θ 2 ) + κ 2 a ( n ) [ a ( n ) + 2 θ ] ,
a 2 ( n ) = γ ( 1 + θ 2 ) ( κ + γ ) 2 + γ γ ( κ + γ ) f 0 2 + 2 κ γ γ ( 2 + f 0 2 ) + κ 2 a ( n ) ( 2 + γ + γ ) [ a ( n ) + 2 θ ] ,
a 1 ( n ) = 2 κ γ γ ( κ + γ ) f 0 2 + κ 2 γ a ( n ) [ a ( n ) × ( 2 γ + γ + θ 2 + γ f 0 2 ) + θ γ ( 4 + f 0 2 ) ] ,
a 0 2 = κ 2 γ 2 γ a ( n ) [ a ( n ) ( 1 + θ 2 + f 0 2 ) + 2 θ f 0 2 ] ,

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