Abstract

In this paper, mode patterns and losses are determined for unstable laser resonators with finite, rectangular reflectors of spherical curvature. An analysis of the uniform intensity mode, based upon the Cornu spiral is given. For more general calculations, gaussian quadrature integration is used to convert the integral equation for the modes into a matrix equation. The latter is solved using ALLMAT on a digital computer, thereby simultaneously determining many modes. The mode competition effect reported by Siegman and Arrathoon is shown to occur because the unstable modes do not generally retain their ordering, according to relative loss, as the reflector size changes. We also discuss a perturbation calculation in which the infinite mirror solutions of Bergstein are used as expansion functions.

© 1969 Optical Society of America

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References

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  1. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  2. A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
    [Crossref]
  3. A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
    [Crossref]
  4. A. E. Siegman, Stanford University, has also noted this behavior; private communication.
  5. R. L. Sanderson, W. Streifer, Appl. Opt. 8, 131 (1969);it has been called to our attention that H. K. V. Lotsch has also employed gaussian quadrature integration in resonator problems in Z. Naturforsch. 20a, 38 (1965).
    [Crossref] [PubMed]
  6. L. Bergstein, Appl. Opt. 7, 495 (1968).
    [Crossref] [PubMed]
  7. W. Streifer, IEEE J. Quantum Electron. QE-4, 229 (1968).
    [Crossref]
  8. A. E. Siegman, Proc. IEEE 53, 277 (1965).
    [Crossref]
  9. R. S. Longhurst, Geometrical and Physical Optics (Longmans, Green and Co. Ltd., London, 1964), Chap. 13.
  10. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., Oxford, 1965), p. 432.
  11. See M. Abramowitz, I. A. Stegun, Handbook of Math. Functions, Natl. Bur. Std., Appl. Math. Ser. No. 55, see Eqs. 7.3.9, 10, 27, 28, pp. 301, 302.
  12. L. Bergstein, H. Schachter, J. Opt. Soc. Amer. 55, 1226 (1965).
    [Crossref]

1969 (1)

1968 (2)

L. Bergstein, Appl. Opt. 7, 495 (1968).
[Crossref] [PubMed]

W. Streifer, IEEE J. Quantum Electron. QE-4, 229 (1968).
[Crossref]

1967 (1)

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[Crossref]

1965 (2)

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[Crossref]

L. Bergstein, H. Schachter, J. Opt. Soc. Amer. 55, 1226 (1965).
[Crossref]

1963 (1)

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

1961 (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Abramowitz, M.

See M. Abramowitz, I. A. Stegun, Handbook of Math. Functions, Natl. Bur. Std., Appl. Math. Ser. No. 55, see Eqs. 7.3.9, 10, 27, 28, pp. 301, 302.

Arrathoon, R.

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[Crossref]

Bergstein, L.

L. Bergstein, Appl. Opt. 7, 495 (1968).
[Crossref] [PubMed]

L. Bergstein, H. Schachter, J. Opt. Soc. Amer. 55, 1226 (1965).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., Oxford, 1965), p. 432.

Fox, A. G.

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Li, T.

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics (Longmans, Green and Co. Ltd., London, 1964), Chap. 13.

Sanderson, R. L.

Schachter, H.

L. Bergstein, H. Schachter, J. Opt. Soc. Amer. 55, 1226 (1965).
[Crossref]

Siegman, A. E.

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[Crossref]

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[Crossref]

A. E. Siegman, Stanford University, has also noted this behavior; private communication.

Stegun, I. A.

See M. Abramowitz, I. A. Stegun, Handbook of Math. Functions, Natl. Bur. Std., Appl. Math. Ser. No. 55, see Eqs. 7.3.9, 10, 27, 28, pp. 301, 302.

Streifer, W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., Oxford, 1965), p. 432.

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

IEEE J. Quantum Electron. (2)

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[Crossref]

W. Streifer, IEEE J. Quantum Electron. QE-4, 229 (1968).
[Crossref]

J. Opt. Soc. Amer. (1)

L. Bergstein, H. Schachter, J. Opt. Soc. Amer. 55, 1226 (1965).
[Crossref]

Proc. IEEE (2)

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[Crossref]

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

Other (4)

A. E. Siegman, Stanford University, has also noted this behavior; private communication.

R. S. Longhurst, Geometrical and Physical Optics (Longmans, Green and Co. Ltd., London, 1964), Chap. 13.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., Oxford, 1965), p. 432.

See M. Abramowitz, I. A. Stegun, Handbook of Math. Functions, Natl. Bur. Std., Appl. Math. Ser. No. 55, see Eqs. 7.3.9, 10, 27, 28, pp. 301, 302.

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Figures (21)

Fig. 1
Fig. 1

Resonator geometry.

Fig. 2
Fig. 2

Geometry used in spherical wave approximation of uniform intensity mode.

Fig. 3
Fig. 3

Cornu spiral.

Fig. 4
Fig. 4

Uniform intensity mode loss as determined from Cornu spiral analysis with ξ = 6.0.

Fig. 5
Fig. 5

Loss for even symmetric α, β, γ modes with ξ = 1.2.

Fig. 6
Fig. 6

Complex eigenvalue loci for even symmetric α, β, γ modes with ξ = 1.2 and 1 ≤ c ≤ 32. Arrows indicate the direction of increasing c.

Fig. 7
Fig. 7

Loss for odd symmetric α′, β′, γ′ modes with ξ = 1.2.

Fig. 8
Fig. 8

Loss for even symmetric α and β modes with ξ = 6.0.

Fig. 9
Fig. 9

Complex eigenvalue loci for even symmetric α and β modes with ξ = 6.0 and 0.17 ≤ c ≤ 3.2. Arrows indicate the direction of increasing c.

Fig. 10
Fig. 10

Loss for even symmetric α, β, γ, and δ modes with ξ = 1.8.

Fig. 11
Fig. 11

Complex eigenvalue loci for even symmetric α, β, γ, and δ modes with ξ = 1.8 and 10 ≤ c ≤ 29. Arrows indicate the direction of increasing c and the point ceq = 37 is indicated on the curves.

Fig. 12
Fig. 12

Expanded complex eigenvalue loci for even symmetric modes α and β in the vicinity of the eigenvalue degeneracy at c ≃ 5.45 and ξ ≃ 3.47. Indicated points correspond to c = 5.4, 5.42, 5.45. 5.47, 5.5, and the arrows indicate the direction of increasing c.

Fig. 13
Fig. 13

Intensity of the E0, E2, E4, E6 modes and the infinite mirror Bergstein mode with ξ = 1.8 and ceq = 34.

Fig. 14
Fig. 14

Residual phase shift of the E0, E2, E4, E6 modes and the infinite mirror Bergstein mode with ξ = 1.8 and ceq = 34.

Fig. 15
Fig. 15

Intensity of the E0 and E2 modes with ξ = 1.8 and ceq = 36.

Fig. 16
Fig. 16

Residual phase shift of the E0 and E2 modes with ξ = 1.8 and ceq = 36.

Fig. 17
Fig. 17

Intensity of the E0 and E2 modes with ξ = 1.8 and ceq = 37.

Fig. 18
Fig. 18

Residual phase shift of the E0 and E2 modes with ξ = 1.8 and ceq = 37.

Fig. 19
Fig. 19

Intensity of the E0 and E2 modes with ξ = 1.8 and Ceq = 40.

Fig. 20
Fig. 20

Residual phase shift of the E0 and E2 modes with ξ = 1.8 and ceq = 40.

Fig. 21
Fig. 21

Intensity of the E0 and E2 modes with ξ = 1.8 and ceq = 18.

Tables (1)

Tables Icon

Table I Matrix Diagonalization Times for ξ = 1.8 on IBM 360 Model 65 Under the HASP Operating System

Equations (24)

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ξ = 1 d / b ,
γ E ( s ) = ( i e ikd 2 π ) 1 2 ( c ) 1 2 ( c ) 1 2 exp { i [ ( ξ / 2 ) ( s 2 + t 2 ) s t ] } × E ( t ) d t .
PL = 1 | γ | 2 .
| γ 0 | | γ 2 | | γ 4 | . . . ,
| γ 1 | | γ 3 | | γ 5 | . . . ,
E 2 ( x R ) { i exp [ ikd ( r 1 + 1 ) ] 2 M } exp [ ( i k / 2 d ) ] ρ x R 2 s = s s = s + exp [ i ( π / 2 ) S 2 ] d S ,
s = ( c M / π ) 1 2 x A / a ,
s + = ( c M / π ) 1 2 [ 1 ( 1 / M ) ( x R / a ) ] ,
s = ( c M / π ) 1 2 [ 1 + ( 1 / M ) ( x R / a ) ] ,
r 1 = ρ ξ + 1 2 ( ξ 1 ) ,
M = ( ξ + 1 ) 1 2 + ( ξ 1 ) 1 2 ( ξ + 1 ) 1 2 ( ξ 1 ) 1 2 = ρ + ξ .
Y = s = s s = s + exp [ ( i π / 2 ) S 2 ] d S = C ( S + ) C ( S ) i [ S ( S + ) S ( S ) ] ,
C ( S ) = 0 s cos [ ( π / 2 ) t 2 ] d t ,
S ( S ) = 0 s sin [ ( π / 2 ) t 2 ] d t .
E 2 ( x R ) = { i exp [ ikd ( r 1 + 1 ) ] 2 M } 1 2 exp [ ( i k / 2 d ) ρ x R 2 ] + exp [ i ( π / 2 ) S 2 ] d S = { i exp [ ikd ( r 1 + 1 ) ] M } 1 2 exp [ ( i k / 2 d ) ρ x R 2 ] ,
E 2 ( x R ) { i exp [ ikd ( r 1 + 1 ) ] 2 M } 1 2 exp [ ( i k / 2 d ) ρ x R 2 ] s 1 + s 1 exp [ i ( π / 2 ) S 2 ] d S ,
| γ | 1 ( 2 M ) 1 2 | s 1 + s 1 exp [ i ( π / 2 ) S 2 ] d S | ,
sin [ ( π / 2 ) S 1 2 ] = cos [ ( π / 2 ) S 1 2 ] ,
c e q = 3 π / 2 + n π , n = 0 , 1 , 2 , . . . .
| γ α | > | γ β | > | γ γ | > . . . .
K ( s , t ) = n K n n ϕ n ( s ) ϕ n ( t ) ,
E ( t ) = n n ϕ n ( t ) ,
ϕ n ( t ) = H n [ ( i ρ ) 1 2 t ] exp ( i ρ t 2 / 2 ) ,
D n m = ( i c ρ ) 1 2 ( i c ρ ) 1 2 e u 2 H n ( u ) H m ( u ) d u .

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