Abstract

Matrix techniques are applied to the mode analysis of laser resonators having spherical mirrors and Gaussian profiles of the mirror reflectivity. These same analytical methods provide a useful approximation to the mode and loss characteristics of conventional resonators having an abrupt discontinuity of the reflectivity. Mode selection in apertured waveguides and resonators is also discussed.

© 1975 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, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  3. G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974) and references.
    [CrossRef]
  4. N. G. Vakhimov, Radio Eng. Electron. Phys. 10, 1439 (1965).
  5. H. Zuker, Bell Syst. Tech. J., 49, 2349 (1970).
  6. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  7. Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, Electron. Commun. Japan 51-B, 67 (1968).
  8. L. W. Casperson, Appl. Opt. 12, 2434 (1973).
    [CrossRef] [PubMed]
  9. L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
    [CrossRef]
  10. T. Li, Bell Syst. Tech. J. 44, 917 (1965).
  11. L. A. Vainshtein, Sov. Phys. JETP 17, 709 (1963).
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 67.
  13. O. R. Wood, IEEE Proc. 62, 355 (1974).
    [CrossRef]
  14. O. M. Stafsudd, UCLA; private communication.

1974 (3)

G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974) and references.
[CrossRef]

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

O. R. Wood, IEEE Proc. 62, 355 (1974).
[CrossRef]

1973 (1)

1970 (2)

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

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

1968 (1)

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, Electron. Commun. Japan 51-B, 67 (1968).

1966 (1)

1965 (2)

T. Li, Bell Syst. Tech. J. 44, 917 (1965).

N. G. Vakhimov, Radio Eng. Electron. Phys. 10, 1439 (1965).

1963 (1)

L. A. Vainshtein, Sov. Phys. JETP 17, 709 (1963).

1961 (1)

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 67.

Casperson, L. W.

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

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

Fox, A. G.

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

Iga, K.

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, Electron. Commun. Japan 51-B, 67 (1968).

Kitano, T.

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, Electron. Commun. Japan 51-B, 67 (1968).

Kogelnik, H.

Lacina, W. B.

G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974) and references.
[CrossRef]

Li, T.

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

T. Li, Bell Syst. Tech. J. 44, 917 (1965).

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

McAllister, G. L.

G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974) and references.
[CrossRef]

Miller, H. Y.

Nagashima, H.

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, Electron. Commun. Japan 51-B, 67 (1968).

Siegman, A. E.

Stafsudd, O. M.

O. M. Stafsudd, UCLA; private communication.

Steier, W. H.

G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974) and references.
[CrossRef]

Suematsu, Y.

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, Electron. Commun. Japan 51-B, 67 (1968).

Vainshtein, L. A.

L. A. Vainshtein, Sov. Phys. JETP 17, 709 (1963).

Vakhimov, N. G.

N. G. Vakhimov, Radio Eng. Electron. Phys. 10, 1439 (1965).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 67.

Wood, O. R.

O. R. Wood, IEEE Proc. 62, 355 (1974).
[CrossRef]

Zuker, H.

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

Appl. Opt. (3)

Bell Syst. Tech. J. (3)

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

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

T. Li, Bell Syst. Tech. J. 44, 917 (1965).

Electron. Commun. Japan (1)

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, Electron. Commun. Japan 51-B, 67 (1968).

IEEE J. Quantum Electron. (2)

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

G. L. McAllister, W. H. Steier, W. B. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974) and references.
[CrossRef]

IEEE Proc. (1)

O. R. Wood, IEEE Proc. 62, 355 (1974).
[CrossRef]

Radio Eng. Electron. Phys. (1)

N. G. Vakhimov, Radio Eng. Electron. Phys. 10, 1439 (1965).

Sov. Phys. JETP (1)

L. A. Vainshtein, Sov. Phys. JETP 17, 709 (1963).

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 67.

O. M. Stafsudd, UCLA; private communication.

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Figures (8)

Fig. 1
Fig. 1

Matrices for a Gaussian beam incident from the left.

Fig. 2
Fig. 2

Normalized spot size πω2d at the mirror as a function of the Fresnel number N = wm2d for various values of ρ = d/Rm.

Fig. 3
Fig. 3

Normalized phase front curvature R/d at the mirror as a function of the Fresnel number for various values of ρ = d/Rm.

Fig. 4
Fig. 4

Power transmission loss at the mirror as a function of the Fresnel number for various values of ρ = d/Rm. The solid lines indicate loss L00 for the fundamental mode, and the dashed lines show the loss L10 of the TEM10 mode.

Fig. 5
Fig. 5

Stability factor F = |A + B/q| as a function of the Fresnel number for various values of ρ = d/Rm. Stability is assured by F ≥ 1.

Fig. 6
Fig. 6

Power transmission L00 of the fundamental TEM00 mode as a function of the Fresnel number for various values of ρ = d/Rm. The exact numerical solutions are shown as solid lines, the analytic Gaussian mode approximations are shown as dashed lines, and a previous approximation is shown as a dotted line.

Fig. 7
Fig. 7

Power transmission L10 of the TEM10 mode as a function of the Fresnel number for various values of ρ = d/Rm. The exact numerical solutions are shown as solid lines, the analytic Gaussian mode approximations are shown as dashed lines, and a previous approximation is shown as a dotted line.

Fig. 8
Fig. 8

Mode discrimination parameter D = T10/T00 for a waveguide with Gaussian profiled transmission apertures (solid line). The corresponding results for abrupt apertures are also shown (dashed lines) for n = 10 and n = 1000.

Equations (37)

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1 q ( 2 ) = C + D / q ( 1 ) A + B / q ( 1 ) .
1 q = 1 R i λ π w 2 ,
1 q s = C + D / q s A + B / q s .
1 q s = D A 2 B ± 1 B [ ( A + D 2 ) 2 1 ] 1 / 2 ,
t ( r ) = t 0 exp ( r 2 / w m 2 ) ,
E ( r ) = E 0 exp ( r 2 / w i 2 ) ,
1 w t 2 = 1 w i 2 + 1 w m 2 .
( 1 0 i λ π w m 2 1 ) .
G ( x , y , z ) = exp i [ Q x ( z ) x 2 2 + Q y ( z ) y 2 2 + S x ( z ) x + S y ( z ) y + P ( z ) ] ,
S x ( 2 ) = S x ( 1 ) A + B / q x ( 1 ) .
d xa ( 2 ) = d xa ( 1 ) { 1 + [ w x ( 1 ) / w m ] 2 } 1 .
( A B C D ) = ( 1 d 0 1 ) × ( 1 0 2 R m i λ π w m 2 1 ) = ( 1 2 d R m i λ d π w m 2 d 2 R m i λ π w m 2 1 ) .
d q = d R m + i λ d 2 π w m 2 ± [ ( 1 d R m i λ d 2 π w m 2 ) 2 1 ] 1 / 2 .
d R i λ d π w 2 = ρ + i 2 π N ± [ ( 1 ρ i 2 π N ) 2 1 ] 1 / 2 ,
( a + ib ) 1 / 2 = ± 2 1 / 2 { [ ( a 2 + b 2 ) 1 / 2 + a ] 1 / 2 + i [ ( a 2 + b 2 ) 1 / 2 a ] 1 / 2 } .
L 00 = P t P i = I 0 0 2 π 0 exp ( 2 r 2 / w 2 ) [ 1 exp ( 2 r 2 / w m 2 ) ] rdrd θ I 0 0 2 π 0 exp ( 2 r 2 / w 2 ) rdrd θ ,
L 00 = ( 1 + w m 2 / w 2 ) 1 = [ 1 + π N ( λ d / π w 2 ) ] 1 .
L 10 = I 0 0 2 π 0 cos 2 θ exp ( 2 r 2 / w 2 ) [ 1 exp ( 2 r 2 / w m 2 ) ] r 3 drd θ I 0 0 2 π 0 cos 2 θ exp ( 2 r 2 / w 2 ) r 3 drd θ
= 1 ( 1 + w 2 / w m 2 ) 2 = 1 [ 1 + ( π N ) 1 ( π w 2 / λ d ) ] 2 .
F = | A + B / q | .
L 00 = P t P i = I 0 0 2 π w m exp ( 2 r 2 / w 2 ) rdrd θ I 0 0 2 π w m exp ( 2 r 2 / w 2 ) rdrd θ .
L 00 = exp ( 2 w m 2 / w 2 ) = exp [ 2 π N ( λ d / π w 2 ) ] .
L 10 = I 0 0 2 π w m cos 2 θ exp ( 2 r 2 / w 2 ) r 3 drd θ I 0 0 2 π 0 cos 2 θ exp ( 2 r 2 / w 2 ) r 3 drd θ
= ( 1 + 2 w m 2 w 2 ) exp ( 2 w m 2 w 2 ) = [ 1 + 2 π N ( λ d π w 2 ) ] exp [ 2 π N ( λ d π w 2 ) ] .
L nm = 8 ν n m 2 β ( M + β ) [ ( M + β ) 2 + β 2 ] 2 ,
( A B C D ) = ( 1 d nf i λ d n π w m 2 d n 1 f i λ π w m 2 1 ) .
( A B C D ) n = 1 sin θ { A sin ( n θ ) sin [ ( n 1 ) θ ] B sin ( n θ ) C sin ( n θ ) D sin ( n θ ) sin [ ( n 1 ) θ ] } ,
θ = ( d nf + i λ d n π w m 2 ) 1 / 2 .
( A B C D ) n 1 sin θ { 2 sin ( θ 2 ) cos [ ( n 1 2 ) θ ] B sin ( n θ ) C sin ( n θ ) 2 sin ( θ 2 ) cos [ ( n 1 2 ) θ ] } .
( A B C D ) n [ cos ( n θ ) B θ sin ( n θ ) C θ sin ( n θ ) cos ( n θ ) ] = { cos [ ( n fd + in λ π w m 2 d ) 1 / 2 d ] ( n fd + in λ π w m 2 d ) 1 / 2 sin [ ( n fd + in λ π w m 2 d ) 1 / 2 d ] ( n fd + in λ π w m 2 d ) 1 / 2 sin [ ( n fd + in λ π w m 2 d ) 1 / 2 d ] cos [ ( n fd + in λ π w m 2 d ) 1 / 2 d ] } .
β 2 = 2 π n / λ df ,
α 2 = 2 n / w m 2 d ,
D = T 10 T 00 = ( 1 L 10 ) n ( 1 L 00 ) n
D = T 00 = ( 1 + w 2 / w m 2 ) n = [ 1 + ( π N ) 1 ( π n w 2 / λ d ) ] n .
D [ 1 + ( 2 / π N ) 1 / 2 ] n exp [ n ( 2 / π N ) 1 / 2 ] .
D = [ 1 ( ν 10 / ν 00 ) 2 L 00 1 L 00 ] n .
D = T 00 1 [ 1 ( ν 10 / ν 00 ) 2 ( 1 T 00 1 / n ) ] n .

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