Abstract

A system has been developed to accurately detect phase signals produced in optical interferometric sensors. The system employs optical heterodyning and synchronously detects optical phase by feeding back an error signal to a phase modulator in the reference leg of the interferometer. This system is seen to have properties similar to a phase-locked loop. The system is mathematically analyzed and a simple second-order model developed which accurately predicts the system response.

© 1983 Optical Society of America

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References

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  1. J. A. Bucaro, H. D. Dardy, E. F. Carome, Appl. Opt. 16, 1761 (1977).
    [CrossRef] [PubMed]
  2. F. R. Trowbridge, R. L. Phillips, Opt. Lett. 6, 636 (1981).
    [CrossRef] [PubMed]
  3. G. B. Hocker, Appl. Opt. 18, 1445 (1979).
    [CrossRef] [PubMed]
  4. J. M. Martin, J. T. Winkler, Proc. Soc. Photo-Opt. Instrum. Eng. 139, 98 (1978).
  5. A. Dandridge, A. B. Tveten, Appl. Opt. 20, 2337 (1981).
    [CrossRef] [PubMed]
  6. K. Bohm, P. Russer, E. Weidel, R. Ulrick, Opt. Lett. 6, 64 (1981).
    [CrossRef] [PubMed]
  7. R. Ulrick, Opt. Lett. 5, 173 (1980).
    [CrossRef]
  8. R. F. Cahill, E. Udd, Opt. Lett. 4, 93 (1979).
    [CrossRef] [PubMed]
  9. N. W. McLachlin, Theory and Application of Mathieu Functions (Dover, New York, 1964).
  10. S. G. Gopani, R. L. Phillips, L. C. Andrews, IEEE Trans. Acoust. Speech Signal Process. ASSP-31,202 (1983).
    [CrossRef]

1983

S. G. Gopani, R. L. Phillips, L. C. Andrews, IEEE Trans. Acoust. Speech Signal Process. ASSP-31,202 (1983).
[CrossRef]

1981

1980

1979

1978

J. M. Martin, J. T. Winkler, Proc. Soc. Photo-Opt. Instrum. Eng. 139, 98 (1978).

1977

Andrews, L. C.

S. G. Gopani, R. L. Phillips, L. C. Andrews, IEEE Trans. Acoust. Speech Signal Process. ASSP-31,202 (1983).
[CrossRef]

Bohm, K.

Bucaro, J. A.

Cahill, R. F.

Carome, E. F.

Dandridge, A.

Dardy, H. D.

Gopani, S. G.

S. G. Gopani, R. L. Phillips, L. C. Andrews, IEEE Trans. Acoust. Speech Signal Process. ASSP-31,202 (1983).
[CrossRef]

Hocker, G. B.

Martin, J. M.

J. M. Martin, J. T. Winkler, Proc. Soc. Photo-Opt. Instrum. Eng. 139, 98 (1978).

McLachlin, N. W.

N. W. McLachlin, Theory and Application of Mathieu Functions (Dover, New York, 1964).

Phillips, R. L.

S. G. Gopani, R. L. Phillips, L. C. Andrews, IEEE Trans. Acoust. Speech Signal Process. ASSP-31,202 (1983).
[CrossRef]

F. R. Trowbridge, R. L. Phillips, Opt. Lett. 6, 636 (1981).
[CrossRef] [PubMed]

Russer, P.

Trowbridge, F. R.

Tveten, A. B.

Udd, E.

Ulrick, R.

Weidel, E.

Winkler, J. T.

J. M. Martin, J. T. Winkler, Proc. Soc. Photo-Opt. Instrum. Eng. 139, 98 (1978).

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

S. G. Gopani, R. L. Phillips, L. C. Andrews, IEEE Trans. Acoust. Speech Signal Process. ASSP-31,202 (1983).
[CrossRef]

Opt. Lett.

Proc. Soc. Photo-Opt. Instrum. Eng.

J. M. Martin, J. T. Winkler, Proc. Soc. Photo-Opt. Instrum. Eng. 139, 98 (1978).

Other

N. W. McLachlin, Theory and Application of Mathieu Functions (Dover, New York, 1964).

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Figures (11)

Fig. 1
Fig. 1

HPF-OFI configured for acoustic sensing. The direct couplers are evanescent wave fiber to fiber couplers.

Fig. 2
Fig. 2

HPF-OFI configured for a Sagnac interferometer rate sensor.

Fig. 3
Fig. 3

Graph of the parameter values showing regions of stability for Mathieu's equation, Eq. (13). For small values of amplitude β of carrier signal the operating point lies on operating line in lower-left stable region.

Fig. 4
Fig. 4

Amplitude and phase response for various values of system gain G0 with λ = 2100.

Fig. 5
Fig. 5

Low-frequency region of amplitude and phase response for increasing G0. The error is reduced with increasing G0.

Fig. 6
Fig. 6

Two-pole system filter with shunt resistance R3 for compensation.

Fig. 7
Fig. 7

Amplitude response numerically calculated for the compensated system with a gain of G0 = 1500 and λ = λ1 = λ2 = 200.

Fig. 8
Fig. 8

Comparison of the step response of the linear and the numerical nonlinear models of the compensated HPF-OFI for a step input with G0 = 2500, λ = 1000, and various δ values.

Fig. 9
Fig. 9

Ramp response of compensated HPF-OFI for various damping ratios and G0 = 2000, λ = 400.

Fig. 10
Fig. 10

System response for various compensations: λ1 = λ2 = 500, G0 = 1200; (a) K = 0, δ = 0.029; (b) K = 0.0031, δ = 0.082; (c) K = 0.0077, δ = 0.162; (d) K = 0.015, δ = 0.295; vertical axis—10 dB/div, horizontal axis—500 Hz/div.

Fig. 11
Fig. 11

Maximum amplitude of sinusoidal signal for which HPF-OFI can maintain phase lock with G0 = 435 and λ = 1800.

Equations (47)

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E s = E 0 exp [ i ( ω t + ϕ s ) ] ,
E r = E 0 exp [ i ( ω t + ϕ f + ϕ if ) ] .
I = I 0 + I 0 cos ( ϕ s ϕ f ϕ if ) ,
ϕ if = β sin ( ω m t ) ,
I = I 0 + I 0 cos [ β sin ( ω m t ) ϕ e ] .
I = I 0 + I 0 { [ J 0 ( β ) + 2 k = 1 J 2 k ( β ) cos ( 2 k ω m t ) ] cos ϕ e + [ 2 k = 0 J 2 k + 1 ( β ) sin [ ( 2 k + 1 ) ω m t ] ] sin ϕ e } .
I = 2 I 0 sin ( ω m t ) { cos ϕ e k = 1 J 2 k ( β ) cos ( 2 k ω m t ) + sin ϕ e k = 0 J 2 k + 1 ( β ) sin [ ( 2 k + 1 ) ω m t ] } .
H ( s ) = λ 2 / ( s + λ ) 2 ,
d 2 ϕ f dt 2 + 2 λ d ϕ dt + λ 2 ϕ f = G λ 2 I 0 2 k = 1 J 2 k ( β ) sin ( ω m t ) × cos ( 2 k ω m t ) cos ϕ e + G λ 2 I 0 2 k = 0 × J 2 k + 1 ( β ) sin ( ω m t ) × sin [ ( 2 k + 1 ) ω m t ] sin ϕ e ,
d 2 ϕ f dt 2 + 2 λ d ϕ f dt + λ 2 { 1 + 2 G I 0 k = 0 J 2 k + 1 ( β ) × sin ( ω m t ) sin [ ( 2 k + 1 ) ω m t ] } ϕ f = ϕ s 2 G λ 2 I 0 k = 0 J 2 k + 1 ( β ) sin ( ω m t ) sin [ ( 2 k + 1 ) ω m t ] ϕ s + 2 G λ 2 I 0 k = 1 J 2 k ( β ) sin ( ω m t ) cos ( 2 k ω m t ) .
ϕ f = U exp ( λ t ) ,
d 2 U dt 2 + G I 0 λ 2 { J 1 ( β ) [ J 1 ( β ) J 3 ( β ) cos ( 2 ω m t ) ] } U = exp ( λ t ) G I 0 λ 2 ( { J 1 ( β ) [ J 1 ( β ) J 3 ( β ) ] cos ( ω m t ) } ϕ s J 2 ( β ) sin ( ω m t ) + J 2 ( β ) sin ( 3 ω m t ) ) .
d 2 U d τ 2 + ( a + 2 q cos 2 τ ) U = f ( τ ) ,
τ = ω m t , a = GI 0 λ 2 J 1 ( β ) / ω m 2 , q = ½ G I 0 λ 2 [ J 1 ( β ) J 3 ( β ) ] / ω m 2 ,
a = γ q ,
γ = 2 J 1 ( β ) / [ J 1 ( β ) J 3 ( β ) ] .
ce ν ( τ , q ) = cos ν τ q 4 [ cos ( ν + 2 ) τ ( ν + 2 ) cos ( ν 2 ) τ ν 1 ] + 1 32 q 2 [ cos ( ν + 4 ) τ ( ν + 1 ) ( ν + 2 ) + cos ( ν 4 ) τ ( ν 1 ) ( ν 2 ) ] + 0 ( q 3 ) ,
s e ν ( τ , q ) = sin ν τ q 4 [ sin ( ν + 2 ) τ ( ν + 1 ) sin ( ν 2 ) τ ( ν 1 ) ] + 1 32 q 2 [ sin ( ν + 4 ) τ ( ν + 1 ) ( ν + 2 ) + sin ( ν 4 ) τ ( ν 1 ) ( ν 2 ) ] + 0 ( q 3 ) ,
a = ν 2 + 1 2 ( ν 2 1 ) q 2 + 0 ( q 4 ) .
ϕ f = exp ( λ t ) { Ace ν ( τ q ) + Bse ν ( τ , q ) 1 C 2 [ ce ν ( τ , q ) se ν ( z , q ) g ( z ) dz se ν ( τ , q ) ce ν ( z , q ) g ( z ) dz ] } ,
d 2 U dt 2 + a U = f ( t ) ,
d 2 ϕ f dt 2 + 2 λ d ϕ f dt + λ 2 ϕ n f = G λ 2 sin ( ϕ s ϕ f ) ,
ϕ f = G 0 sin ( ϕ s ϕ f ) ,
ϕ f ( max ) = 2 G I 0 J 1 ( β ) .
ϕ f / G 0 = sin ( ϕ s ϕ f ) ,
ω 2 A sin ω t + 2 λ A sin ω t + λ 2 A sin ω t = G 0 λ 2 sin ( ϕ s A sin ω t ) ,
sin ( ϕ s A sin ω ) = A ω 2 G 0 λ 2 ( λ 2 ω 2 1 ) sin ω t + 2 A G 0 λ ω cos ω t .
d ϕ f dt + 2 λ d ϕ f dt + λ 2 ( 1 + G 0 ) ϕ f = G 0 λ 2 ϕ s .
[ S 2 + 2 S λ + λ 2 ( 1 + G 0 ) ] ϕ f ( S ) = G 0 λ 2 ϕ s ( S ) ,
H ( S ) = ϕ f ( S ) ϕ s ( S ) = G 0 λ 2 S 2 + 2 S λ + λ 2 ( 1 + G 0 ) .
| H ( j ω ) | = G 0 λ 2 [ 4 ω 2 λ 2 + [ λ 2 ( 1 + G 0 ) ω 2 ] 2 ] 1 / 2 ,
H ( j ω ) = tan 1 [ 2 ω λ λ 2 ( 1 + G 0 ) ω 2 ] .
ω n = λ ( G 0 1 ) 1 / 2 or f n = λ ( G 0 1 ) 1 / 2 2 .
f c = λ / 2 π [ G 0 1 + ( 2 G 0 2 4 G 0 ) 1 / 2 ] 1 / 2 .
f c = 1.55 f n .
δ = 1 / ( 1 + G 0 ) 1 / 2 .
H ( f n ) = ( G 0 / 2 ) 1 / 2 .
L c ( S ) = K λ 1 ( S + λ 3 ) ( S + λ 1 ) ( S + λ 2 ) ,
H c ( S ) = G 0 λ 1 K ( S + λ 3 ) S 2 + S ( λ 1 + λ 2 + λ 1 G 0 K ) + λ 1 ( λ 2 + G K λ 3 ) .
| H c ( S = j ω ) | = G 0 λ 1 K ( l 2 + m 2 ) 1 / 2 c 2 + d 2 ,
c = 2 ω 2 λ 1 ( 2 + G 0 K ) , d = 2 ω ( ω 2 λ 1 2 G 0 K λ 3 ) , l = 2 c λ 3 2 d ω , m = 2 d λ 3 2 c ω , λ 3 = λ 2
H c ( j ω ) = tan 1 ( l / m ) .
δ c = λ 1 ( 1 + G 0 K ) + λ 2 2 ( λ 1 ( λ 2 + G 0 K λ 3 ) 1 / 2 .
d 2 ϕ f dt 2 + 2 d ϕ f dt + λ 2 ϕ f = G 0 λ 2 sin ( ϕ s ϕ f ) + G 0 λ K d dt sin ( ϕ s ϕ f ) ,
ϕ f = G 0 sin ( ϕ s ϕ f ) and ϕ f ( max ) = G 0 .
r c ( t ) = K G 0 λ 1 λ 3 B + exp At [ z g cos ( m t ) f g sin ( m t ) ] ,
A = λ 1 ( 2 + G 0 K ) / 2 , B = λ 1 2 ( 1 + G 0 ) , m = ( B A 2 ) 1 / 2 a = K G 0 λ 1 ( λ 3 A ) , b = K G 0 λ 1 m , c = m 2 , d = Am z = ac + bd , e = ac bd , f = bd ad , g = c 2 + d 2 .

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