Abstract

The passive Fabry-Perot cavity is shown to be a good practical approach to the match-filter optimization for the sensitive detection of mode-locked laser signals. Doppler measurements of relative motion over a wide range of velocities are possible simply by measuring the cavity length for a peak output.

© 1973 Optical Society of America

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References

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  1. L. E. Hargrove, R. L. Fork, M. A. Pollack, Appl. Phys. Lett. 5, 4 (1964).
    [CrossRef]
  2. W. H. Steier, Proc. IEEE 54, 1604 (1966).
    [CrossRef]
  3. R. L. Fork, D. R. Herriott, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
    [CrossRef]
  4. H. Kogelnik, A. Yariv, Proc. IEEE 52, 165 (1964).
    [CrossRef]
  5. R. W. Uhlhorn, D. F. Holshouser, IEEE J. Quantum Electron. QE-6, 775 (1970).
    [CrossRef]
  6. C. M. Sonnenschien, F. A. Horrigan, Appl. Opt. 10, 1600 (1971).
    [CrossRef]
  7. M. J. Rudd, J. Phys. E 2, 55 (1969).
    [CrossRef]
  8. Equation (1) is a generalization, to include frequency variations of t1, t2, r1, and r2 of the well known expression for cavity transmissivity in a Fabry-Perot, e.g., M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970), pp. 62, 327.
  9. W. M. Brown, Analysis of Linear Time-Invariant Systems (McGraw-Hill, New York, 1963), p. 245.
  10. A. Blaquiere, Ann. Radioelectr. 8, 36 (1953).
  11. J. P. Gordon, H. J. Ziegler, C. H. Townes, Phys. Rev. 99, 1264 (1955).
    [CrossRef]
  12. C. Freed, H. A. Haus, Appl. Phys. Lett. 6, 85, (1965).
    [CrossRef]
  13. P. Grivet, A. Blaquiere, in Symposium on Optical Masers, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1963), p. 69.
  14. A. Yariv, Quantum Electronics (John Wiley, New York, 1967), p. 409.
  15. C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965). p. 225.

1971

1970

R. W. Uhlhorn, D. F. Holshouser, IEEE J. Quantum Electron. QE-6, 775 (1970).
[CrossRef]

1969

M. J. Rudd, J. Phys. E 2, 55 (1969).
[CrossRef]

1966

W. H. Steier, Proc. IEEE 54, 1604 (1966).
[CrossRef]

1965

C. Freed, H. A. Haus, Appl. Phys. Lett. 6, 85, (1965).
[CrossRef]

1964

R. L. Fork, D. R. Herriott, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
[CrossRef]

L. E. Hargrove, R. L. Fork, M. A. Pollack, Appl. Phys. Lett. 5, 4 (1964).
[CrossRef]

H. Kogelnik, A. Yariv, Proc. IEEE 52, 165 (1964).
[CrossRef]

1955

J. P. Gordon, H. J. Ziegler, C. H. Townes, Phys. Rev. 99, 1264 (1955).
[CrossRef]

1953

A. Blaquiere, Ann. Radioelectr. 8, 36 (1953).

Blaquiere, A.

A. Blaquiere, Ann. Radioelectr. 8, 36 (1953).

P. Grivet, A. Blaquiere, in Symposium on Optical Masers, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1963), p. 69.

Born, M.

Equation (1) is a generalization, to include frequency variations of t1, t2, r1, and r2 of the well known expression for cavity transmissivity in a Fabry-Perot, e.g., M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970), pp. 62, 327.

Brown, W. M.

W. M. Brown, Analysis of Linear Time-Invariant Systems (McGraw-Hill, New York, 1963), p. 245.

Fork, R. L.

L. E. Hargrove, R. L. Fork, M. A. Pollack, Appl. Phys. Lett. 5, 4 (1964).
[CrossRef]

R. L. Fork, D. R. Herriott, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
[CrossRef]

Freed, C.

C. Freed, H. A. Haus, Appl. Phys. Lett. 6, 85, (1965).
[CrossRef]

Gordon, J. P.

J. P. Gordon, H. J. Ziegler, C. H. Townes, Phys. Rev. 99, 1264 (1955).
[CrossRef]

Grivet, P.

P. Grivet, A. Blaquiere, in Symposium on Optical Masers, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1963), p. 69.

Hargrove, L. E.

L. E. Hargrove, R. L. Fork, M. A. Pollack, Appl. Phys. Lett. 5, 4 (1964).
[CrossRef]

Haus, H. A.

C. Freed, H. A. Haus, Appl. Phys. Lett. 6, 85, (1965).
[CrossRef]

Herriott, D. R.

Holshouser, D. F.

R. W. Uhlhorn, D. F. Holshouser, IEEE J. Quantum Electron. QE-6, 775 (1970).
[CrossRef]

Horrigan, F. A.

Kogelnik, H.

Papas, C. H.

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965). p. 225.

Pollack, M. A.

L. E. Hargrove, R. L. Fork, M. A. Pollack, Appl. Phys. Lett. 5, 4 (1964).
[CrossRef]

Rudd, M. J.

M. J. Rudd, J. Phys. E 2, 55 (1969).
[CrossRef]

Sonnenschien, C. M.

Steier, W. H.

W. H. Steier, Proc. IEEE 54, 1604 (1966).
[CrossRef]

Townes, C. H.

J. P. Gordon, H. J. Ziegler, C. H. Townes, Phys. Rev. 99, 1264 (1955).
[CrossRef]

Uhlhorn, R. W.

R. W. Uhlhorn, D. F. Holshouser, IEEE J. Quantum Electron. QE-6, 775 (1970).
[CrossRef]

Wolf, E.

Equation (1) is a generalization, to include frequency variations of t1, t2, r1, and r2 of the well known expression for cavity transmissivity in a Fabry-Perot, e.g., M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970), pp. 62, 327.

Yariv, A.

H. Kogelnik, A. Yariv, Proc. IEEE 52, 165 (1964).
[CrossRef]

A. Yariv, Quantum Electronics (John Wiley, New York, 1967), p. 409.

Ziegler, H. J.

J. P. Gordon, H. J. Ziegler, C. H. Townes, Phys. Rev. 99, 1264 (1955).
[CrossRef]

Ann. Radioelectr.

A. Blaquiere, Ann. Radioelectr. 8, 36 (1953).

Appl. Opt.

Appl. Phys. Lett.

L. E. Hargrove, R. L. Fork, M. A. Pollack, Appl. Phys. Lett. 5, 4 (1964).
[CrossRef]

C. Freed, H. A. Haus, Appl. Phys. Lett. 6, 85, (1965).
[CrossRef]

IEEE J. Quantum Electron.

R. W. Uhlhorn, D. F. Holshouser, IEEE J. Quantum Electron. QE-6, 775 (1970).
[CrossRef]

J. Phys. E

M. J. Rudd, J. Phys. E 2, 55 (1969).
[CrossRef]

Phys. Rev.

J. P. Gordon, H. J. Ziegler, C. H. Townes, Phys. Rev. 99, 1264 (1955).
[CrossRef]

Proc. IEEE

W. H. Steier, Proc. IEEE 54, 1604 (1966).
[CrossRef]

H. Kogelnik, A. Yariv, Proc. IEEE 52, 165 (1964).
[CrossRef]

Other

Equation (1) is a generalization, to include frequency variations of t1, t2, r1, and r2 of the well known expression for cavity transmissivity in a Fabry-Perot, e.g., M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970), pp. 62, 327.

W. M. Brown, Analysis of Linear Time-Invariant Systems (McGraw-Hill, New York, 1963), p. 245.

P. Grivet, A. Blaquiere, in Symposium on Optical Masers, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1963), p. 69.

A. Yariv, Quantum Electronics (John Wiley, New York, 1967), p. 409.

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965). p. 225.

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Figures (5)

Fig. 1
Fig. 1

Passive cavity receiver: T laser transmitter; M3,M4 laser mirrors; h laser cavity length; E signal; V velocity of laser relative to receiver; No noise; F coarse bandpass filter; M1,M2 passive cavity mirrors; ho passive cavity length; D detector; A detector electronics; C mirror control.

Fig. 2
Fig. 2

Departure from matched-filter vs relative line widths.

Fig. 3
Fig. 3

Signal-to-noise as a function of time and relative cavity length.

Fig. 4
Fig. 4

Fraction of the maximum output vs time in nanoseconds and vs the number of complete bounces M, parameterized for various Rα = r1r2 exp(−2αh).

Fig. 5
Fig. 5

Signal-to-noise improvement with passive cavity.

Equations (37)

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T ( ω ) = L exp ( i ψ ( ω ) ) / [ 1 + P 2 sin 2 ( ω h 0 c ) ] 1 / 2
P 2 = [ 4 r 1 r 2 exp ( - 2 α h 0 ) ] / [ 1 - r 1 r 2 exp ( - 2 α h 0 ) ] 2
ψ ( ω ) = - ( ω / c ) h 0 - tan - 1 { r 1 r 2 sin [ 2 ω ( h 0 / c ) ] 1 - r 1 r 2 cos [ 2 ω ( h 0 / c ) ] }
L = [ t 1 t 2 exp ( - α h 0 ) ] / [ 1 - r 1 r 2 exp ( - 2 α h 0 ) ] .
T ( ω ) A = L / [ 1 + ( 2 Δ ω Δ ω p ) 2 ] 1 / 2
F ( ω ) = [ S n ( ω ) L ] / A * p = N N 1 / [ 1 + ( 2 Δ ω Δ ω p ) 2 ] 1 / 2 .
E 1 ( t ) = p = - N N exp [ i ( ω 0 + p ω c ) t ] = exp ( i ω 0 t ) { sin [ ( 2 N + 1 ) ω c t 2 ] / sin ( ω c t 2 ) }
E ( ω ) = 2 π p = - N N δ [ ω - ( ω 0 + p ω c ) ]
P 1 = [ 4 exp ( - 2 h 0 ) ] / { 1 - [ exp ( - 2 h 0 ) ] 2 } .
E ( ω ) = t 1 t 2 [ 2 ( r 1 r 2 ) 1 / 2 ] P 1 { 1 / [ 1 + P 1 sin 2 ω ( h 0 / c ) ] 1 / 2 } .
E ( ω ) = ( t 1 t 2 ) / [ 2 ( r 1 r 2 ) 1 / 2 ] P 1 p = - N N 1 / [ 1 + ( 2 Δ ω Δ ω 1 ) 2 ] 1 / 2
S N R ( t ) = T 1 E 1 2 / R h n ( 0 )
R h n ( 0 ) = 1 2 π - S n ( ω ) T ( ω ) 2 d ω .
| T 1 E 1 | 2 = | 1 2 π - T ( ω ) E ( ω ) exp ( i ω t ) d ω | 2
| T 1 E 1 | 2 = | L t 1 t 2 2 ( r 1 r 2 ) 1 / 2 P 1 - × exp ( i ω t ) d ω [ 1 + P 1 sin 2 ω ( h 0 / c ) ] 1 / 2 [ 1 + P 2 sin 2 ω ( h 0 / c ) ] 1 / 2 | .
| T 1 E 1 | 2 = ( q / 2 π ) 2 L 2 [ t 1 t 2 2 ( r 1 r 2 ) 1 / 2 ] 2 ( 2 c / h 0 ) 2 K 2 ( m )
K ( m ) = 0 π / 2 [ ( d θ ) / ( 1 - m 2 sin 2 θ ) 1 / 2 ]
m = [ ( 1 / P 2 ) - ( 1 / P 1 ) ] 1 / 2 / ( 1 / P 2 ) 1 / 2 .
R h n ( 0 ) = L 2 2 π q [ ( π c ) / h 0 ] S n ( ω ) d ω 1 + P 2 sin 2 [ ω ( h 0 / c ) ] = L 2 N 0 2 π q π c h 0 1 ( 1 + P 2 ) 1 / 2
SNR p = 4 ( q / 2 π ) [ t 1 t 2 / 2 ( r 1 r 2 ) 1 / 2 ] 2 ( c / h 0 ) ( 1 + P 2 ) 1 / 2 K 2 ( m ) / ( π N 0 ) .
SNR 1 = 4 ( q / 2 π ) [ ( t 1 t 2 ) / [ 2 ( r 1 r 2 ) 1 / 2 ] ] 2 ( c / h 0 ) [ tan - 1 ( P 1 ) ] 2 × ( 1 + P 1 ) 1 / 2 / ( π N 0 ) .
SNR p / SNR 1 = ( 1 + P 2 ) 1 / 2 / ( 1 + P 1 ) 1 / 2 [ K 2 ( m ) / [ tan - 1 ( P 1 ) ] 2 .
SNR p / SNR 1 = ( 2 / π ) 2 ( Δ ω 1 / Δ ω p ) K 2 ( m ) .
SNR with / SNR without = ( 1 + P 2 ) 1 / 2 .
G 1 ( t ) = p = - N N [ cos ( a - b ) ] / ( 1 + P 2 sin 2 b ) 1 / 2
T ( s ) = exp ( - a s ) T ( s ) = [ T α exp ( - a s ) ] / [ 1 - R α exp ( - 2 a s ) ]
T ( s ) = T α a m = 0 ( s - σ ) / [ ( s - σ ) 2 + ω m 2 ]
E ( s ) = L [ Real E 1 ( t ) ] = p = - N N [ s / ( s 2 + ω p 2 ) ] .
G ( s ) = exp ( - a s ) G ( s ) = exp ( - a s ) E ( s ) T ( s )
L - 1 [ s s 2 + ω p 2 · ( s - σ ) ( s - σ ) 2 + ω m 2 ] = [ 1 + σ 2 ( σ 2 + ω m 2 - ω p 2 ) 2 + 4 σ 2 ω p 2 ] 1 / 2 × [ ω p sin ( ω p t + ψ 1 ) + ω m sin ( ω m t + ψ 2 ) ] .
f = 1 - exp [ σ ( τ - a ) ] u ( τ - a ) = ( 1 - R α ( τ - a ) / 2 a ) u ( τ - a )
u ( x ) = { 1 x > 0 0 d < 0 .
τ = - 1 / σ = - 2 a [ 1 / ( ln R α ) ] 2 ( h 0 / c ) R α / ( 1 - R α )
τ = ( h 0 / c ) P R a = ( 2 R α ) / ( Δ ω p ) = R α / ( π Δ ν p )
τ Δ ν p = R α / π .
E 1 ( t ) = ( { sin [ ( 2 N + 1 ) ω c γ ( 1 + v / c ) t / 2 ] } / { sin [ ω c γ × ( 1 + v / c ) t ] / 2 } ) · exp [ i ω 0 γ ( 1 + v / c ) t ] .
h 0 = π c / { ω c γ [ 1 + ( v / c ) ] } = h / { γ [ 1 + ( v / c ) ] } .

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