Abstract

The transfer of a delta-function pulse by parallel-sided optical cavities is considered. It is shown that the spectral characteristics of the cavities can be easily derived by calculating the Fourier transform of this impulse response. This method of calculation appears to be simpler and more intuitive than the usual standing-wave approach. The method is applied to a number of examples including cavities whose dimensions change with time.

© 1973 Optical Society of America

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References

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  1. O. Kafri, S. Kimel, J. Shamir, IEEE J. Quantum Electron. QE-8, 295 (1972).
    [CrossRef]
  2. L. W. Davis, Phys. Rev. A5, 2594 (1972).
  3. E. Delano, R. J. Pegis, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1969), Vol. 7, Ch. 2.
    [CrossRef]
  4. M. Bertolotti, D. Sette, F. Wanderlingh, Nuovo Cimento 48, 301 (1967).
    [CrossRef]
  5. D. Roess, Opto-Electronics 2, 183 (1970).
    [CrossRef]
  6. S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., London, 1969), p. 230.
  7. H. D. Polster, J. Opt. Soc. Am. 42, 21 (1952).
    [CrossRef]
  8. O. Kafri, S. Speiser, S. Kimel, IEEE J. Quantum Electron. QE-7, 122 (1971).
    [CrossRef]

1972

O. Kafri, S. Kimel, J. Shamir, IEEE J. Quantum Electron. QE-8, 295 (1972).
[CrossRef]

L. W. Davis, Phys. Rev. A5, 2594 (1972).

1971

O. Kafri, S. Speiser, S. Kimel, IEEE J. Quantum Electron. QE-7, 122 (1971).
[CrossRef]

1970

D. Roess, Opto-Electronics 2, 183 (1970).
[CrossRef]

1967

M. Bertolotti, D. Sette, F. Wanderlingh, Nuovo Cimento 48, 301 (1967).
[CrossRef]

1952

Bertolotti, M.

M. Bertolotti, D. Sette, F. Wanderlingh, Nuovo Cimento 48, 301 (1967).
[CrossRef]

Davis, L. W.

L. W. Davis, Phys. Rev. A5, 2594 (1972).

Delano, E.

E. Delano, R. J. Pegis, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1969), Vol. 7, Ch. 2.
[CrossRef]

Kafri, O.

O. Kafri, S. Kimel, J. Shamir, IEEE J. Quantum Electron. QE-8, 295 (1972).
[CrossRef]

O. Kafri, S. Speiser, S. Kimel, IEEE J. Quantum Electron. QE-7, 122 (1971).
[CrossRef]

Kimel, S.

O. Kafri, S. Kimel, J. Shamir, IEEE J. Quantum Electron. QE-8, 295 (1972).
[CrossRef]

O. Kafri, S. Speiser, S. Kimel, IEEE J. Quantum Electron. QE-7, 122 (1971).
[CrossRef]

Lipson, H.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., London, 1969), p. 230.

Lipson, S. G.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., London, 1969), p. 230.

Pegis, R. J.

E. Delano, R. J. Pegis, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1969), Vol. 7, Ch. 2.
[CrossRef]

Polster, H. D.

Roess, D.

D. Roess, Opto-Electronics 2, 183 (1970).
[CrossRef]

Sette, D.

M. Bertolotti, D. Sette, F. Wanderlingh, Nuovo Cimento 48, 301 (1967).
[CrossRef]

Shamir, J.

O. Kafri, S. Kimel, J. Shamir, IEEE J. Quantum Electron. QE-8, 295 (1972).
[CrossRef]

Speiser, S.

O. Kafri, S. Speiser, S. Kimel, IEEE J. Quantum Electron. QE-7, 122 (1971).
[CrossRef]

Wanderlingh, F.

M. Bertolotti, D. Sette, F. Wanderlingh, Nuovo Cimento 48, 301 (1967).
[CrossRef]

IEEE J. Quantum Electron.

O. Kafri, S. Kimel, J. Shamir, IEEE J. Quantum Electron. QE-8, 295 (1972).
[CrossRef]

O. Kafri, S. Speiser, S. Kimel, IEEE J. Quantum Electron. QE-7, 122 (1971).
[CrossRef]

J. Opt. Soc. Am.

Nuovo Cimento

M. Bertolotti, D. Sette, F. Wanderlingh, Nuovo Cimento 48, 301 (1967).
[CrossRef]

Opto-Electronics

D. Roess, Opto-Electronics 2, 183 (1970).
[CrossRef]

Phys. Rev.

L. W. Davis, Phys. Rev. A5, 2594 (1972).

Other

E. Delano, R. J. Pegis, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1969), Vol. 7, Ch. 2.
[CrossRef]

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., London, 1969), p. 230.

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of a Fabry-Perot cavity in the time domain.

Fig. 2
Fig. 2

The gain as a function of time (a) for cw lasers, (b) for pulsed lasers.

Fig. 3
Fig. 3

Laser output as a function of time (a) for cw lasers, (b) for pulsed lasers.

Fig. 4
Fig. 4

Amplitude transmission coefficients θi and reflection coefficients ri and r ¯ i by layers with optical thickness li.

Equations (29)

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δ ( t , r ) = ( 1 / 2 π ) 4 - exp ( i ω t ) d ω - exp ( i k · r ) d 3 k .
δ ( t , z ) = ( 1 / 2 π ) 2 - exp ( i ω t ) d ω - exp ( - i k z z ) d k z .
ψ ( t , z 0 ) = m = 0 A m δ ( t - t m - z 0 / c ) ,
Ψ ( ω ) = F [ ψ ( t , z 0 ) ] = exp ( i ω z 0 / c ) - exp ( i ω t ) m A m δ ( t - t m ) d t .
A m = θ 1 θ 2 ( r 1 r 2 ) m = Θ R m             ( R 1 ) .
ψ ( t ) = m = 0 Θ R m δ ( t - m τ - z 0 / c ) ,
ψ ( t ) = Θ exp [ ( t / τ ) ln R ] m = 0 δ ( t - m τ - z 0 / c ) .
Ψ ( ω ) = Θ [ 0 exp ( α t ) exp ( - i ω t ) d t ] * [ - m = - δ ( t - m τ ) exp ( - i ω t ) d t ] = [ 4 π 2 Θ / τ ( α - i ω ) ] * p = - δ ( ω - 2 π p / τ ) .
α = ( 1 / τ ) ln R ( α 0 if R 1 ) .
I ( ω ) = Ψ ( ω ) * Ψ ( ω ) = ( 4 π 2 ) 2 Θ 2 τ 2 ( α 2 + ω 2 ) * p = - δ ( ω - 2 π p / τ ) .
A m + 1 / A m = R G .
R G = K exp ( - β t ) ,
exp [ ( t / τ ) ln R G ] = exp [ ln 2 K / 4 β τ ] exp [ - β ( t - t 0 ) 2 / τ ] ,
ψ n ( t ) = δ ( t ) r n + θ n 2 { ψ n - 1 ( t ) * δ ( t - τ n ) + r ¯ n [ ψ n - 1 ( t ) * ] 2 δ ( t - 2 τ n ) + + r ¯ n j - 1 [ ψ n - 1 ( t ) * ] j δ ( t - j τ n ) + }
Ψ n ( ω ) = r n + θ n 2 [ Ψ n - 1 ( ω ) exp ( - i ω τ n ) + + r ¯ n j - 1 Ψ n - 1 j ( ω ) exp ( - j i ω τ n ) + ] = r n + { θ n 2 Ψ n - 1 ( ω ) exp ( - i ω τ n ) / [ 1 - r ¯ n Ψ n - 1 ( ω ) exp ( - i ω τ n ) ] } .
Ψ 2 ( ω ) = r 2 + { θ 2 2 r 1 exp ( - i ω τ ) / [ 1 - r ¯ 2 r 1 exp ( - i ω τ ) ] } .
I ( ω ) = 1 - Ψ 2 ( ω ) 2 = θ 1 2 θ 2 2 / ( 1 + 2 r 1 r 2 cos ω τ + r 1 2 r 2 2 ) ,
L ( t ) = L 0 + v t .
Δ t n = 2 ( L 0 + v t n ) / c .
t n = 2 L 0 n / c + L 0 n ( n - 1 ) v / c 2 = n t 0 + n ( n - 1 ) δ t / 2 ,
δ t = t n + 1 - t n = 2 v L 0 / c 2 .
ψ ( t ) = n = 0 N δ ( t - t n ) .
Ψ ( ω ) = n = 0 N exp ( i ω τ n ) = n = 0 N exp [ i ω ( n t 0 + 1 2 n ( n - 1 ) δ t ) ] = n = 0 N exp ( i ω n 2 δ t / 2 ) exp [ i ω n ( t 0 - δ t / 2 ) ] .
Ψ ( ω ) = { 1 - exp [ i ω N ( t 0 - δ t / 2 ) ] } / { 1 - exp [ i ω ( t 0 - δ t / 2 ) ] } ( when ω N 2 δ t / 2 1 )
n m 2 ω δ t / 2 = π / 2 ,
Ψ ( ω ) = { 1 - exp [ i ω n m ( t 0 - δ t / 2 ) ] } / { 1 - exp [ i ω ( t 0 - δ t / 2 ) ] } ( when ω N 2 δ t / 2 / ) .
δ ω = Δ ω / N or Δ ω / n m
Δ ω = 2 π ( t 0 - δ t / 2 ) .
ω t 0 ( 1 - v / 2 c ) = 2 r π             ( r - integer ) .

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