Abstract

The analysis of the losses in a Fabry-Perot resonator, worked out by Bergstein and Schachter, is considered and commented upon. It is shown that, at large values of the Fresnel number, it yields results in agreement with those of other authors, if the mathematics involved is correctly developed.

© 1970 Optical Society of America

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References

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  1. L. A. Wainstein, Sov. Phys.—JETP 17, 709 (1963).
  2. A. G. Fox, T. Li, Bell System Tech. J. 40, 453 (1961).
  3. S. R. Barone, J. Appl. Phys. 34, 831 (1963).
    [CrossRef]
  4. H. Risken, Z. Physik 180, 150 (1964).
    [CrossRef]
  5. L. Bergstein, H. Schachter, J. Opt. Soc. Amer. 54, 887 (1964).
    [CrossRef]
  6. L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 16 (1966).
    [CrossRef]
  7. H. Ogura, Y. Yoshida, Jap. J. Appl. Phys. 3, 546 (1964).
    [CrossRef]
  8. H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Japan 22, 1434 (1967).
    [CrossRef]
  9. L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 1139 (1966).
    [CrossRef]
  10. S. P. Morgan, T. Li, J. Opt. Soc. Amer. 55, 1128 (1965).
    [CrossRef]
  11. S. P. Morgan, T. Li, J. Opt. Soc. Amer. 56, 1138 (1966).
    [CrossRef]

1967

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Japan 22, 1434 (1967).
[CrossRef]

1966

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 1139 (1966).
[CrossRef]

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 16 (1966).
[CrossRef]

S. P. Morgan, T. Li, J. Opt. Soc. Amer. 56, 1138 (1966).
[CrossRef]

1965

S. P. Morgan, T. Li, J. Opt. Soc. Amer. 55, 1128 (1965).
[CrossRef]

1964

H. Ogura, Y. Yoshida, Jap. J. Appl. Phys. 3, 546 (1964).
[CrossRef]

H. Risken, Z. Physik 180, 150 (1964).
[CrossRef]

L. Bergstein, H. Schachter, J. Opt. Soc. Amer. 54, 887 (1964).
[CrossRef]

1963

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

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[CrossRef]

1961

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

Barone, S. R.

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[CrossRef]

Bergstein, L.

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 16 (1966).
[CrossRef]

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 1139 (1966).
[CrossRef]

L. Bergstein, H. Schachter, J. Opt. Soc. Amer. 54, 887 (1964).
[CrossRef]

Fox, A. G.

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

Ikenoue, J.-I.

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Japan 22, 1434 (1967).
[CrossRef]

Li, T.

S. P. Morgan, T. Li, J. Opt. Soc. Amer. 56, 1138 (1966).
[CrossRef]

S. P. Morgan, T. Li, J. Opt. Soc. Amer. 55, 1128 (1965).
[CrossRef]

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

Marom, E.

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 16 (1966).
[CrossRef]

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 1139 (1966).
[CrossRef]

Morgan, S. P.

S. P. Morgan, T. Li, J. Opt. Soc. Amer. 56, 1138 (1966).
[CrossRef]

S. P. Morgan, T. Li, J. Opt. Soc. Amer. 55, 1128 (1965).
[CrossRef]

Ogura, H.

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Japan 22, 1434 (1967).
[CrossRef]

H. Ogura, Y. Yoshida, Jap. J. Appl. Phys. 3, 546 (1964).
[CrossRef]

Risken, H.

H. Risken, Z. Physik 180, 150 (1964).
[CrossRef]

Schachter, H.

L. Bergstein, H. Schachter, J. Opt. Soc. Amer. 54, 887 (1964).
[CrossRef]

Wainstein, L. A.

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

Yoshida, Y.

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Japan 22, 1434 (1967).
[CrossRef]

H. Ogura, Y. Yoshida, Jap. J. Appl. Phys. 3, 546 (1964).
[CrossRef]

Bell System Tech. J.

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

J. Appl. Phys.

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[CrossRef]

J. Opt. Soc. Amer.

L. Bergstein, H. Schachter, J. Opt. Soc. Amer. 54, 887 (1964).
[CrossRef]

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 16 (1966).
[CrossRef]

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 1139 (1966).
[CrossRef]

S. P. Morgan, T. Li, J. Opt. Soc. Amer. 55, 1128 (1965).
[CrossRef]

S. P. Morgan, T. Li, J. Opt. Soc. Amer. 56, 1138 (1966).
[CrossRef]

J. Phys. Soc. Japan

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Japan 22, 1434 (1967).
[CrossRef]

Jap. J. Appl. Phys.

H. Ogura, Y. Yoshida, Jap. J. Appl. Phys. 3, 546 (1964).
[CrossRef]

Sov. Phys.—JETP

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

Z. Physik

H. Risken, Z. Physik 180, 150 (1964).
[CrossRef]

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Equations (35)

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P m = 0.128 N 1.5 m 2 .
P m = 0.059 N 1.5 m 2 ,
n ( A m n σ δ m n ) c n = 0.
A m n = ( H 2 i ) 1 2 1 1 1 1 exp [ i π 2 H ( ξ ξ ) 2 ] × sin [ n π 2 ( ξ + 1 ) ] sin [ m π 2 ( ξ + 1 ) ] d ξ d ξ ,
det | A m n σ δ m n | = 0.
N ( x ) = 0 x exp ( i π 2 x 2 ) d x .
A j k = A k j = ( 2 i ) 1 2 π j k k 2 j 2 [ f ( j ) j f ( k ) k ] ,
f ( k ) = exp ( i π k 2 / 8 H ) { N [ 2 H 1 2 k 2 ( H ) 1 2 ] N [ 2 ( H ) 1 2 + k 2 ( H ) 1 2 ] + 2 N [ k 2 ( H ) 1 2 ] } ,
A k k = 1 ( 2 i ) 1 2 { g ( k ) + ( i π k k 4 H ) f ( k ) i π ( H ) 1 2 [ 1 exp ( i π k + 2 i π H ) ] } ,
g ( k ) = exp ( i π k 2 / 8 H ) { N [ 2 ( H ) 1 2 k 2 ( H ) 1 2 ] + N [ 2 ( H ) 1 2 + k 2 ( H ) 1 2 ] } .
σ m = A m m .
A m m exp ( i π m 2 / 8 H ) { 1 ( 1 i ) m 2 12 H ( H ) 1 2 + 0 [ m 4 H 2 ( H ) 1 2 ] } ,
( A 11 σ 1 ) c 1 = n A 1 n c n ,
n = 3 A j n c n + ( A j j σ 1 ) c j = A j 1 c 1 ( j 1 ) ,
A j k = ( 2 i ) 1 2 k j 12 H ( H ) 1 2 + ,
A j j σ 1 A j j A 11 ( i π / 8 H ) ( j 2 1 )
( 2 i ) 1 2 j 12 H ( H ) 1 2 3 n ¯ n c n i π 8 H ( j 2 1 ) c j = ( 2 i ) 1 2 j 12 H ( H ) 1 2 c 1 ,
c n = n n 2 1 K ( H ) 1 2 c 1 for n odd
K ( H ) 1 2 [ n n n 2 1 A j n + j j 2 1 ( A j j A 11 ) ] = A j 1 ( j 1 ) ,
K ( H ) 1 2 { ( 2 i ) 1 2 π n n 2 n 2 1 1 n 2 j 2 [ f ( j ) j f ( n ) n ] + A j j A 11 j 2 1 } = ( 2 i ) 1 2 π 1 j 2 1 [ f ( j ) j f ( 1 ) ] .
n n 2 n 2 1 1 n 2 j 2 = j 2 + 1 ( j 2 1 ) 2 ,
n n n 2 1 1 n 2 j 2 f ( n ) = 1 2 ( j 2 1 ) { i π 2 ( 2 i ) 1 2 ( A j j A 11 ) 1 2 [ f ( j ) j f ( 1 ) ] i π 8 H [ j f ( j ) f ( 1 ) ] + ( 1 2 + 2 j 2 1 ) f ( 1 ) ( 1 2 2 j 2 j 2 1 ) f ( j ) j } .
n 2 n 2 1 1 n 2 j 2 = 1 2 ( j 2 1 ) ( 1 n 1 1 n + 1 + j n + j + j j n ) .
n n 2 1 1 n 2 j 2 = 1 2 ( j 2 1 ) ( 1 n 1 + 1 n + 1 1 j + n + 1 j n ) . f ( n ) = 4 i ( H ) 1 2 0 1 e 2 i π H x 2 sin π n x d x ,
m = 1 1 m sin 2 π m x = π 2 π x .
n n 2 n 2 1 1 n 2 j 2 [ f ( j ) j f ( n ) n ] = 1 2 ( j 2 1 ) × { i 2 π ( 2 i ) 1 2 ( A j j A 11 ) + j 2 + 1 j 2 1 [ f ( j ) j f ( 1 ) ] + i π 8 H [ j f ( j ) f ( 1 ) ] } .
K 2 ( H ) 1 2 { ( A j j A 11 ) + ( 2 i ) 1 2 π j 2 + 1 j 2 1 [ f ( j ) j f ( 1 ) ] + i ( 2 i ) 1 2 8 H [ j f ( j ) f ( 1 ) ] } = ( 2 i ) 1 2 π [ f ( j ) j f ( 1 ) ] .
K = 4 3 ( 2 i ) 1 2 / π ,
u ( x ) = n c n sin { n ( π / 2 ) [ ( x / a ) + 1 ] } ,
3 A 1 n c n = K ( 2 i ) 1 2 π ( H ) 1 2 3 n 2 ( n 2 1 ) 2 [ f ( 1 ) f ( n ) n ] c 1 = K ( 2 i ) 1 2 ( H ) 1 2 π 32 H e i π / 8 H ( 1 + i ) c 1 .
1 1 m 2 cos 2 π m x = 1 1 m 2 2 π 0 x ( 1 1 m sin 2 π m x ) d x .
A 11 σ 1 = 4 π ( 2 i ) 1 2 3 ( 2 i ) 1 2 H ( H ) 1 2 32 × e i π / 8 H ( 1 + i ) = e i π / 8 H 12 ( H ) H 1 2 ( 1 i )
σ 1 = e i π / 8 H [ 1 1 i 12 H ( H ) 1 2 1 i 12 H ( H ) 1 2 ] .
P 1 = 1 | σ 1 2 | = 1 3 H ( H ) 1 2 = 1 6 ( 2 ) 1 2 N 1.5 = 0.118 N 1.5 .
P m = 0.118 m 2 N 1.5 .

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