Abstract

A simple closed-loop system for estimating optical phase, called an interference phase loop, is presented. In this system the output intensity from an elementary interferometric phase sensor (e.g., Zernike phase contrast, homodyne, heterodyne, polarization, or shearing interferometer) is detected and used to drive a phase modulator in the path of the wavefront being measured. It is shown theoretically and experimentally that with large gain this configuration ignores amplitude fluctuations and unambiguously estimates phase at high speed over a dynamic range of multiple-π radians. When self-interference (e.g., Zernike phase contrast) is employed, monochromatic light is not required.

© 1979 Optical Society of America

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References

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  1. J. W. Hardy, Proc. IEEE 66, 651 (1978).
    [CrossRef]
  2. M. J. Lavan, G. E. Van Damme, W. K. Cadwallende, Opt. Eng. 15, 464 (1976).
  3. R. A. Sprague, B. J. Thompson. Appl. Opt. 11, 1469 (1972).
    [CrossRef] [PubMed]
  4. Y. Ichioka, M. Inuiya, Appl. Opt. 11, 1507 (1972).
    [CrossRef] [PubMed]
  5. G. W. Johnson, D. T. Moore, Proc. Soc. Photo-Opt. Instrum. Eng. 103, 76 (1977).
  6. W. T. Cathey, C. L. Hayes, W. C. Davis, V. F. Pizzurro, Appl. Opt. 9, 701 (1970).
    [CrossRef] [PubMed]
  7. G. Q. McDowell, D.Sc. thesis (MIT, Cambridge, Mass., 1971).
  8. P. W. Smith, E. H. Turner, Appl. Phys. Lett. 30, 280 (1977).
    [CrossRef]
  9. A. J. Viterbi, Principles of Coherent Communications (McGraw-Hill, New York, 1966).
  10. C. Warde, A. D. Fisher, D. M. Cocco, M. Y. Burmawi, Opt. Lett. 3, 196 (1978).
    [CrossRef] [PubMed]

1978 (2)

1977 (2)

P. W. Smith, E. H. Turner, Appl. Phys. Lett. 30, 280 (1977).
[CrossRef]

G. W. Johnson, D. T. Moore, Proc. Soc. Photo-Opt. Instrum. Eng. 103, 76 (1977).

1976 (1)

M. J. Lavan, G. E. Van Damme, W. K. Cadwallende, Opt. Eng. 15, 464 (1976).

1972 (2)

1970 (1)

Burmawi, M. Y.

Cadwallende, W. K.

M. J. Lavan, G. E. Van Damme, W. K. Cadwallende, Opt. Eng. 15, 464 (1976).

Cathey, W. T.

Cocco, D. M.

Davis, W. C.

Fisher, A. D.

Hardy, J. W.

J. W. Hardy, Proc. IEEE 66, 651 (1978).
[CrossRef]

Hayes, C. L.

Ichioka, Y.

Inuiya, M.

Johnson, G. W.

G. W. Johnson, D. T. Moore, Proc. Soc. Photo-Opt. Instrum. Eng. 103, 76 (1977).

Lavan, M. J.

M. J. Lavan, G. E. Van Damme, W. K. Cadwallende, Opt. Eng. 15, 464 (1976).

McDowell, G. Q.

G. Q. McDowell, D.Sc. thesis (MIT, Cambridge, Mass., 1971).

Moore, D. T.

G. W. Johnson, D. T. Moore, Proc. Soc. Photo-Opt. Instrum. Eng. 103, 76 (1977).

Pizzurro, V. F.

Smith, P. W.

P. W. Smith, E. H. Turner, Appl. Phys. Lett. 30, 280 (1977).
[CrossRef]

Sprague, R. A.

Thompson, B. J.

Turner, E. H.

P. W. Smith, E. H. Turner, Appl. Phys. Lett. 30, 280 (1977).
[CrossRef]

Van Damme, G. E.

M. J. Lavan, G. E. Van Damme, W. K. Cadwallende, Opt. Eng. 15, 464 (1976).

Viterbi, A. J.

A. J. Viterbi, Principles of Coherent Communications (McGraw-Hill, New York, 1966).

Warde, C.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

P. W. Smith, E. H. Turner, Appl. Phys. Lett. 30, 280 (1977).
[CrossRef]

Opt. Eng. (1)

M. J. Lavan, G. E. Van Damme, W. K. Cadwallende, Opt. Eng. 15, 464 (1976).

Opt. Lett. (1)

Proc. IEEE (1)

J. W. Hardy, Proc. IEEE 66, 651 (1978).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

G. W. Johnson, D. T. Moore, Proc. Soc. Photo-Opt. Instrum. Eng. 103, 76 (1977).

Other (2)

A. J. Viterbi, Principles of Coherent Communications (McGraw-Hill, New York, 1966).

G. Q. McDowell, D.Sc. thesis (MIT, Cambridge, Mass., 1971).

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Figures (4)

Fig. 1
Fig. 1

Homodyne–heterodyne implementation of the IPL. MOD is a spatial phase modulator. DET is a detector array. AMP is an amplifier array. Em and El are defined at the detector face.

Fig. 2
Fig. 2

(a) Modulator state–space trajectory ( ϕ ˙ m versus ϕm) from ϕ ˙ m = λ [ - ϕ m + G 0 - G 1 sin ( ϕ m - ϕ i ) ] with ϕ i = 0= 0, G0 = 1.8π, G1 = 7.3π, and ϕs = 3.8π. Stable equilibria are circled. (b) Phase estimation performance (ϕm versus ϕ i ).

Fig. 3
Fig. 3

Single-spatial-element homodyne JPL implemented with a Mach–Zehnder interferometer and with direct electrical readout of intensity |Ei(t)|2 and phase ϕi(t).

Fig. 4
Fig. 4

Experimental results: I is interferometer intensity. ϕe is derived from Eq. (2b). ϕm is the modulator phase, i.e., the phase estimate of ϕi. (a) Phase tracking (G1 ≈ 10π). (b) Closed-loop and open-loop response to wideband (0–150-Hz) phase fluctuations with G1 ≈ 20π and two different loop-filter bandwidths (λ1 ≈ 104 sec−1, λ2 ≈ 50 sec−1). (c) Immunity to amplitude |Ei| fluctuations (G1 ≈ 20π).

Equations (12)

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I = I 0 + I 1 cos [ ϕ i - ( ϕ m + ϕ 0 ) - ( ϕ l + π / 2 ) - ϕ 1 - ( ω i - ω l ) t ] ,
I 0 = α 2 E i 2 + E l 2
I 1 = 2 α E i E l .
I = I 0 + I 1 sin ϕ e .
ϕ e = ϕ i - ϕ m - ϕ b
i 0 = { 0 I I t g 0 ( I - I t ) I t I I s g 0 ( I s - I t ) I I s .
ϕ m = { 0 i 1 0 g 2 i 1 0 i 1 ϕ s / g 2 ϕ s i 1 ϕ s / g 2 .
I 1 ( s ) = g 1 H ( s ) I 0 ( s ) .
Φ m ( s ) = g 0 g 1 g 2 H ( s ) ( I 0 - I t + I 1 sin Φ e ) .
Φ m = H ( s ) [ G 0 + G 1 sin ( Φ i - Φ m - Φ b ) ] .
ϕ ˙ m = λ [ - ϕ m + G 0 - G 1 sin ( ϕ m - ϕ i + ϕ b ) ] ,
ϕ i = ϕ m + 2 n π + sin - 1 [ ( G 0 - ϕ m ) / G 1 ] .

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