Abstract

An analytical solution to the problem of the misaligned optical resonator with a Gaussian reflectivity mirror is presented. It is shown that the fundamental mode of such a resonator remains a Gaussian beam which propagates along a tilted axis in the misalignment plane. The exact analytical solution yields simple expressions that characterize the misalignment sensitivity of the resonator by giving the beam-steering angle and the loss variations.

© 1983 Optical Society of America

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References

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  1. H. Zucker, Bell Syst. Tech. J. 49, 2349 (1970).
  2. A. G. Fox, T. Li, Bell Syst.Tech. J. 40, 453 (1961).
  3. A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  4. Y.A. Anan'ev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
    [CrossRef]
  5. L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
    [CrossRef]
  6. G. Giuliani, Y. K. Park, R. L. Byer, Opt. Lett. 5, 491 (1980).
    [CrossRef] [PubMed]
  7. A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
    [CrossRef]
  8. L. W. Casperson, S. D. Lunnam, Appl. Opt. 14, 1193 (1975).
    [CrossRef] [PubMed]
  9. J. A. Arnaud, Appl. Opt. 8, 1909 (1969).
    [CrossRef] [PubMed]
  10. A. Hardy, Appl. Phys. 18, 223 (1979).
    [CrossRef]
  11. L. W. Casperson, Appl. Opt. 20, 223 (1979).
  12. A. Chester, IEEE J. Quantum Electron. QE-9, 209 (1973).
    [CrossRef]
  13. N. Kumagai, H. Mori, T. Shiozawa, Electron. Commun. Jpn. 49, 1 (1966).
  14. L. W. Casperson, Appl. Opt. 12, 2434 (1973).
    [CrossRef] [PubMed]
  15. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  16. W. F. Krupke, W. R. Sooy, IEEE J. Quantum Electron. QE-5, 575 (1969).
    [CrossRef]
  17. R. Hauck, H. P. Kortz, H. Weber, Appl. Opt. 19, 598 (1980).
    [CrossRef] [PubMed]

1980 (2)

1979 (2)

A. Hardy, Appl. Phys. 18, 223 (1979).
[CrossRef]

L. W. Casperson, Appl. Opt. 20, 223 (1979).

1975 (2)

1974 (1)

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

1973 (2)

A. Chester, IEEE J. Quantum Electron. QE-9, 209 (1973).
[CrossRef]

L. W. Casperson, Appl. Opt. 12, 2434 (1973).
[CrossRef] [PubMed]

1971 (1)

Y.A. Anan'ev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
[CrossRef]

1970 (2)

H. Zucker, Bell Syst. Tech. J. 49, 2349 (1970).

A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

1969 (2)

J. A. Arnaud, Appl. Opt. 8, 1909 (1969).
[CrossRef] [PubMed]

W. F. Krupke, W. R. Sooy, IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

1966 (2)

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

N. Kumagai, H. Mori, T. Shiozawa, Electron. Commun. Jpn. 49, 1 (1966).

1961 (1)

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

Anan'ev, Y.A.

Y.A. Anan'ev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
[CrossRef]

Arnaud, J. A.

Byer, R. L.

Casperson, L. W.

L. W. Casperson, Appl. Opt. 20, 223 (1979).

L. W. Casperson, S. D. Lunnam, Appl. Opt. 14, 1193 (1975).
[CrossRef] [PubMed]

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

L. W. Casperson, Appl. Opt. 12, 2434 (1973).
[CrossRef] [PubMed]

Chester, A.

A. Chester, IEEE J. Quantum Electron. QE-9, 209 (1973).
[CrossRef]

Fox, A. G.

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

Giuliani, G.

Hardy, A.

A. Hardy, Appl. Phys. 18, 223 (1979).
[CrossRef]

Hauck, R.

Kogelnik, H.

Kortz, H. P.

Krupke, W. F.

W. F. Krupke, W. R. Sooy, IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

Kumagai, N.

N. Kumagai, H. Mori, T. Shiozawa, Electron. Commun. Jpn. 49, 1 (1966).

Li, T.

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

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

Lunnam, S. D.

Miller, H. Y.

Mori, H.

N. Kumagai, H. Mori, T. Shiozawa, Electron. Commun. Jpn. 49, 1 (1966).

Park, Y. K.

Sherstobitov, V. E.

Y.A. Anan'ev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
[CrossRef]

Shiozawa, T.

N. Kumagai, H. Mori, T. Shiozawa, Electron. Commun. Jpn. 49, 1 (1966).

Siegman, A. E.

Sooy, W. R.

W. F. Krupke, W. R. Sooy, IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

Weber, H.

Yariv, A.

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

Yeh, P.

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

Zucker, H.

H. Zucker, Bell Syst. Tech. J. 49, 2349 (1970).

Appl. Opt. (7)

Appl. Phys. (1)

A. Hardy, Appl. Phys. 18, 223 (1979).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Zucker, Bell Syst. Tech. J. 49, 2349 (1970).

Bell Syst.Tech. J. (1)

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

Electron. Commun. Jpn. (1)

N. Kumagai, H. Mori, T. Shiozawa, Electron. Commun. Jpn. 49, 1 (1966).

IEEE J. Quantum Electron. (3)

A. Chester, IEEE J. Quantum Electron. QE-9, 209 (1973).
[CrossRef]

W. F. Krupke, W. R. Sooy, IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

Opt. Commun. (1)

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

Opt. Lett. (1)

Sov. J. Quantum Electron. (1)

Y.A. Anan'ev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Misaligned resonator with a Gaussian reflectivity mirror: (A) resonator geometry, (B) equivalent periodic lens waveguide.

Fig. 2
Fig. 2

Fundamental mode behavior in a misaligned GM resonator.

Fig. 3
Fig. 3

Geometry of a Gaussian beam which propagates along an axis that is tilted relative to the reference z axis in the y-z plane [Δz = (zz1)β2/2].

Fig. 4
Fig. 4

Ratio of the fundamental mode beam-steering angle β to the mirror misalignment angle α for different values of the GM Fresnel number N1: (A) in a Cassegrain resonator as a function of the magnification M: …, geometrical limits; (B) in a half-symmetric resonator as a function of the GM parameter g1.

Fig. 5
Fig. 5

Fundamental mode losses vs relative tilt angle α normalized to the half-angle αm (= wm/d) sustained by the Gaussian mirror for different values of N1: (A) plane–plane; (B) Cassegrain with M = 2; and (C) Cassegrain with M = 5 resonators. The reflectivity at the mirror center r 0 2 is assumed to be unity.

Fig. 6
Fig. 6

Misalignment sensitivity of (A) Cassegrain resonators as a function of magnification M, (B) half-symmetric resonators as a function of the Gaussian mirror parameter g1, for different values of N1.

Equations (29)

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r = r 0 exp ( x 2 + y 2 λ d N 1 ) ,
Δ phase ( y ) = exp ( i 2 k α y ) ,
γ u 1 ( x , y ) = P 1 plane K ( x , x 1 ; y , y 1 ) u 1 ( x 1 , y 1 ) d x 1 d y 1 ,
K ( x , x 1 ; y , y 1 ) = i 2 λ d g 2 exp { i π λ d [ ( x 2 + y 2 ) ( G 1 1 2 g 2 ) + 2 y α d g 2 2 α 2 d 2 g 2 ] } × exp { i π λ d × [ ( x 1 2 + y 1 2 ) ( G 1 1 2 g 2 ) x x 1 g 2 y 1 g 2 ( y 2 α d ) ] } ,
u 1 ( x 1 , y 1 ) = exp { i π λ d [ x 1 2 ( d R x i π A x ) + d R y ( y 1 ρ ) 2 i π A y ( y 1 η ) 2 ] }
( d R x i π A x ) 2 = ( d R y i π A y ) 2 = a + i b ,
η d = α D g 2 2 ( 1 + A 2 N 1 ) ,
ρ R = α D g 2 2 ( g 2 D + g 1 1 g 2 + d R ) ,
a = g 1 2 g 1 g 2 1 4 π 2 N 1 2 , b = 1 π N 1 ( g 1 1 2 g 2 ) , D = ( g 1 1 g 2 + d R ) 2 + 1 π 2 ( 1 A + 1 2 N 1 ) 2 .
d R = ± 1 2 ( a 2 + b 2 + a ) 1 / 2 ( + if b 0 )
1 π A = 1 2 ( a 2 + b 2 a ) 1 / 2 ,
u r ( x , y ) = A exp { i π λ [ x 2 + ( y ρ r ) 2 R r i λ π x 2 + ( y η r ) 2 w r 2 ] } ,
A = r 0 exp [ η 0 2 w 0 2 ( 1 w r 2 w 0 2 ) i π λ ρ 0 2 R 0 ( 1 R r R 0 ) ] ,
1 R r = 1 R 0 2 R m ,
1 w r 2 = 1 w 0 2 + 1 w m 2 ,
η r w r 2 = η 0 w 0 2 ,
ρ r R r = ρ 0 R 0 .
u ( x , y , z ) = exp [ i ( P ( z ) + π λ { x 2 q ( z ) + [ y ρ ( z ) ] 2 R ( z ) i λ π w ( z ) 2 [ y η ( z ) ] 2 } ) ] ,
2 u x 2 + 2 u y 2 2 i k u z = 0 .
1 q ( z ) = 1 R ( z ) i λ π w ( z ) 2 = 1 q ( z 1 ) + ( z z 1 ) ,
η ( z ) ρ ( z ) R ( z ) = η ( z 1 ) ρ ( z 1 ) R ( z 1 ) = β ,
η ( z ) = η ( z 1 ) + ( z z 1 ) β ,
exp [ i P ( z ) ] = q ( z 1 ) q ( z ) exp { i π λ β 2 [ z z 1 + R ( z 1 ) R ( z ) ] } .
u ( x , y , z ) = q 0 q ( z ) exp ( i π λ { x 2 + [ y η ( z ) ] 2 q ( z ) + 2 β [ y η ( z ) ] + z β 2 } ) ,
β α 2 M M 1 ,
L 00 = 1 [ | u r ( x , y ) | 2 dxdy | u 0 ( x , y ) | 2 dxdy ] 1 / 2 ,
L 00 = 1 ( r 0 2 1 + A 0 / N 1 ) 1 / 2 exp [ ( α α m ) 2 1 g 2 4 D 2 ( 1 + A 0 N 1 ) ] ,
L 00 1 r 0 M exp [ ( α α m ) 2 16 M 2 ( M 1 ) 4 ] .
S = 1 ( α / α m ) d L 00 d ( α / α m ) | ( α / α m = 0 ) = 2 r 0 D 2 g 2 4 1 + A 0 / N 1 .

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