Abstract

A system for extracting the phase components of an optical traveling wave field has been constructed. This system consists of the Mach-Zehnder interferometer to capture the wave field as the hologram, a photoelectronic scanning system, and an analog computing circuit. The detected phase signal is immediately displayed on a cathode ray tube as continuous-tone patterns. The method has been confirmed through the experimental results.

© 1972 Optical Society of America

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References

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  1. C. B. Burckhardt, L. H. Enloe, Bell Syst. Tech. J. 48, 1927 (1969).
  2. M. Bertolotti, Appl. Opt. 10, 42 (1971).
    [CrossRef] [PubMed]
  3. R. V. Shack, J. Opt. Soc. Am. 61, 655A (1971).
  4. L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
    [CrossRef]
  5. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [CrossRef]
  6. J. A. Jordan et al., Appl. Opt. 9, 1883 (1970).
    [PubMed]

1971 (2)

R. V. Shack, J. Opt. Soc. Am. 61, 655A (1971).

M. Bertolotti, Appl. Opt. 10, 42 (1971).
[CrossRef] [PubMed]

1970 (1)

1969 (2)

C. B. Burckhardt, L. H. Enloe, Bell Syst. Tech. J. 48, 1927 (1969).

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

1962 (1)

Bertolotti, M.

Burckhardt, C. B.

C. B. Burckhardt, L. H. Enloe, Bell Syst. Tech. J. 48, 1927 (1969).

Enloe, L. H.

C. B. Burckhardt, L. H. Enloe, Bell Syst. Tech. J. 48, 1927 (1969).

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Jordan, J. A.

J. A. Jordan et al., Appl. Opt. 9, 1883 (1970).
[PubMed]

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Leith, E. N.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Shack, R. V.

R. V. Shack, J. Opt. Soc. Am. 61, 655A (1971).

Upatnieks, J.

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

C. B. Burckhardt, L. H. Enloe, Bell Syst. Tech. J. 48, 1927 (1969).

IBM J. Res. Dev. (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

J. Opt. Soc. Am. (2)

R. V. Shack, J. Opt. Soc. Am. 61, 655A (1971).

E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

System composing a hologram.

Fig. 2
Fig. 2

Spectrum of a hologram.

Fig. 3
Fig. 3

Block diagram of a direct phase detecting system. This system uses the method that can yield the amplitude and phase by taking off the carrier frequency of the filtered signal 2A(t)B cos[ωtϕ(t)].

Fig. 4
Fig. 4

Spectrum of the signal C(t) or S(t).

Fig. 5
Fig. 5

Mach-Zehnder interferometer composing a hologram.

Fig. 6
Fig. 6

(a) Tchebycheff filter and (b) characteristic curves of the filter.

Fig. 7
Fig. 7

Schematic diagram of the circuit to extract the phase component.

Fig. 8
Fig. 8

Waveforms in individual steps of operations of the phase demodulation circuit.

Fig. 9
Fig. 9

Waveforms in individual steps of operations of the constructed phase demodulating circuit corresponding to the arrangement in Fig. 8.

Fig. 10
Fig. 10

Experimental results in the simulation of the phase demodulation: (a) ω′ ≃ 15 kHz; (b) ω′ ≃ 16 kHz; (c) ω′ ≃ 18 kHz; (d) ω′ ≃ 23 kHz; (e) ω′ ≃ 24 kHz; (b′) waveforms of the detected phase signal corresponding to (b), (d′) that corresponding to (d).

Fig. 11
Fig. 11

Test object (a), the display of the test object on TV monitor (b), and detected phase distribution of the test object (c)–(f). The inner carrier frequency ω/2π is changed, and it becomes gradually large through (c) to (f) but in (f) ω = ω′.

Fig. 12
Fig. 12

Continuous-tone display of the phase distribution of the tilted plane wave: (a) (c) interference patterns on the TV monitor; (b) (d) detected phase patterns corresponding to (a) (c).

Fig. 13
Fig. 13

Cross section of phase distribution of the converging spherical wave.

Fig. 14
Fig. 14

Hologram of the converging spherical wave and display of the detected phase distribution: (a) (c) holograms of the converging spherical wave; (b) (d) detected phase patterns corresponding to holograms shown in (a) (c).

Fig. 15
Fig. 15

Detection of the phase of a transparency: (a) object before bleaching; (b) hologram of the bleached object; (c) phase distribution detected by the constructed system.

Equations (16)

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I ( x , y ) = | A ( x , y ) exp [ i ϕ ( x , y ) + B exp ( i k x sin θ ) ] | 2 + | A ( x , y ) | 2 + B 2 + 2 A ( x , y ) B cos [ k x sin θ ϕ ( x , y ) ] .
I ( t ) = A 2 ( t ) + B 2 + 2 A ( t ) B cos [ ω t ϕ ( t ) ] ,
C ( t ) = I ( t ) cos ω t = [ A 2 ( t ) + B 2 ] cos ω t + A ( t ) B cos [ 2 ω t ϕ ( t ) ] + A ( t ) B cos ϕ ( t ) , S ( t ) = I ( t ) sin ω t = [ A 2 ( t ) + B 2 ] sin ω t + A ( t ) B sin [ 2 ω t ϕ ( t ) ] + A ( t ) B sin ϕ ( t ) .
C ( t ) ~ A ( t ) cos ϕ ( t ) , S ( t ) ~ A ( t ) sin ϕ ( t ) ,
A ( t ) 2 = C 2 ( t ) + S 2 ( t ) , ϕ ( t ) ~ tan 1 [ S ( t ) / C ( t ) ] .
I ( t , y ) = A ( t , y ) cos [ ω t ϕ ( t , y ) ] ,
I ( t , y ) sin ω t = A ( t , y ) sin [ 2 ω t ϕ ( t , y ) ] + A ( t , y ) sin ϕ ( t , y ) , I ( t , y ) cos ω t = A ( t , y ) cos [ 2 ω t ϕ ( t , y ) ] + A ( t , y ) cos ϕ ( t , y ) ,
A 2 = ( A cos ϕ ) 2 + ( A sin ϕ ) 2 , ϕ = tan 1 ( A sin ϕ / A cos ϕ ) .
f ( t ) = A sin ω t .
f ( t ) = A sin [ ω t ( ω ω ) t ] .
f ( x , y ) A ( 1 + sin ω x ) .
f ( x , y ) A { 1 + sin [ ω x ( ω ω ) x ] } .
f ( x , y ) = A exp ( i k x sin θ ) ,
ϕ ( x , y ) = k x sin θ ( mod 2 π ) .
f ( x , y ) = A exp [ i k x 2 + y 2 2 ( f z ) ] ,
ϕ ( x , y ) = [ k x 2 + y 2 2 ( f z ) ] ( mod 2 π ) .

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