Abstract

First a transmission-type optical element, which modulates the phase of the incident wave field in a 1-D manner, is proposed, then several such elements are combined in a cascade to construct an adaptive transmission-type 2-D phase modulator. The element consists of a deformable glass plate made by bonding an appropriately shaped PVDF piezoelectric film with transparent electrodes to a laminar glass plate, a plane glass plate, and a medium immersed between the two plates. The construction of the modulator, complementary methods to improve the characteristics, several basic experimental results, and its use to compensate for aberrations in an imaging system are given. The modulator constructed can generate wave fronts whose phase profile is expressed by the functional form ax2 + by2 + cxy.

© 1981 Optical Society of America

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References

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  1. D. C. Smith, Proc. IEEE 65, 1679 (1977).
    [CrossRef]
  2. Special issue on adaptive optics, J. Opt. Soc. Am. 67, 269–390 (1977).
  3. J. W. Hardy, Proc. IEEE 66, 651 (1978).
    [CrossRef]
  4. S. A. Kokorowski, J. Opt. Soc. Am. 69, 181 (1979).
    [CrossRef]
  5. T. Sato, H. Ishida, O. Ikeda, Appl. Opt. 19, 1430 (1980).
    [CrossRef] [PubMed]
  6. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 9.
  7. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]

1980 (1)

1979 (1)

1978 (1)

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

1977 (2)

1976 (1)

Hardy, J. W.

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

Ikeda, O.

Ishida, H.

Kokorowski, S. A.

Noll, R. J.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 9.

Sato, T.

Smith, D. C.

D. C. Smith, Proc. IEEE 65, 1679 (1977).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Proc. IEEE (2)

D. C. Smith, Proc. IEEE 65, 1679 (1977).
[CrossRef]

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

Other (1)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 9.

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Figures (14)

Fig. 1
Fig. 1

Structure of the phase modulating element.

Fig. 2
Fig. 2

Details of the deformable plate made by bonding an appropriately shaped PVDF piezoelectric film to a laminar glass plate.

Fig. 3
Fig. 3

Arrangement of three modulating elements to construct a transmission-type 2-D phase modulator.

Fig. 4
Fig. 4

Examples of the Zernike polynomials and construction of the phase modulators.

Fig. 5
Fig. 5

Schematic of the transmission-type 2-D phase modulating system.

Fig. 6
Fig. 6

Signal processing block diagram to generate the wave fronts expressed by the Zernike polynomials.

Fig. 7
Fig. 7

Improvement of the transmission characteristics by the refractive-index matching method. The results are intensity distributions of the diffraction fields in the Fourier transform plane.

Fig. 8
Fig. 8

Characteristics of the phase modulating element (I).

Fig. 9
Fig. 9

Characteristics of the phase modulating element (II).

Fig. 10
Fig. 10

Generation of 2-D wave fronts of circles.

Fig. 11
Fig. 11

Generation of 2-D wave fronts of ellipses.

Fig. 12
Fig. 12

Generation of 2-D wave fronts of hyperbolas.

Fig. 13
Fig. 13

Summary of generated wave fronts.

Fig. 14
Fig. 14

Use of the modulator to compensate for image distortions.

Equations (17)

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2 f ( x ; V ) / x 2 = K [ W ( x ) / W g ] V ,
f ( a ; V ) = f ( a ; V ) = 0 ,
g out ( x ; V ) = g in ( x ) h ( x ; V ) .
h ( x ; V ) = t 0 exp [ - j k ( n - 1 ) f ( x ; V ) ] ,
W ( x ) = W g ,
f ( x ; V ) = K ( x - a ) 2 V / 2.
h ( x ; V ) = t 0 exp [ - j k ( n - 1 ) K ( x - a ) 2 V / 2 ] .
f l ( V ) = [ ( n - 1 ) K V ] - 1             ( = r e R ) .
h ( x , y ; V 1 , V 2 , V 3 ) = t 0 3 i = 1 3 exp [ - j k ( n - 1 ) f ( ξ i ; V i ) ] ,
ξ i = x cos θ i + y sin θ i .
W ( ξ i ) = W g ,
f ( ξ i ; V i ) = K ξ i 2 V i / 2 ,
F ( x , y ) = a 1 x 2 + a 2 y 2 + a 3 x y ,
a 1 = - [ k ( n - 1 ) K / 2 ] i = 1 3 cos 2 θ i · V i , a 2 = - [ k ( n - 1 ) K / 2 ] i = 1 3 sin 2 θ i · V i , a 3 = - [ k ( n - 1 ) K / 2 ] i = 1 3 sin 2 θ i · V i ,
( V 1 V 2 V 3 ) = - 2 k ( n - 1 ) K × ( cos 2 θ 1 cos 2 θ 2 cos 2 θ 3 sin 2 θ 1 sin 2 θ 2 sin 2 θ 3 sin 2 θ 1 sin 2 θ 2 sin 2 θ 3 ) - 1 ( a 1 a 2 a 3 ) .
θ i [ det ( cos 2 θ 1 cos 2 θ 2 cos 2 θ 3 sin 2 θ 1 sin 2 θ 2 sin 2 θ 3 sin 2 θ 1 sin 2 θ 2 sin 2 θ 3 ) ] = 0 ,             i = 1 , 2 , 3 ,
θ i - θ j = 2 π / 3 ( or π / 3 )             i , j = 1 , 2 , 3.

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