Abstract

Holograms are made of a wavefront emerging from a lens having spherical aberration. The hologram, when used in combination with the lens, serves as a corrector plate for the lens. Experimental results are given, followed by a third-order analysis.

© 1966 Optical Society of America

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References

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  1. A. Marèchal, P. Croce, C. R. Acad. Sci., Paris 237, 607 (1953).
  2. E. L. O’Neill, Inst. Radio Engrs. Trans. 2, 56 (1956).
  3. J. Tsujiuchi, Progr. Opt. 2, 133 (1963). This paper contains a bibliography.
  4. Such phasing methods for this kind of application have been described in the literature, e.g., G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 193 (1952); Y. N. Denisyuk, Opt. Spectry., 15, 279 (1963); W. T. Cathey, J. Opt. Soc. Am. 55, 457 (1965).
    [CrossRef]
  5. P. J. Regus, J. Opt. Soc. Am. 54, 1407 (A) (1964); G. V. Conigleo, ibid. 54, 1407 ((A).
  6. R. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]

1965 (1)

1964 (1)

P. J. Regus, J. Opt. Soc. Am. 54, 1407 (A) (1964); G. V. Conigleo, ibid. 54, 1407 ((A).

1963 (1)

J. Tsujiuchi, Progr. Opt. 2, 133 (1963). This paper contains a bibliography.

1956 (1)

E. L. O’Neill, Inst. Radio Engrs. Trans. 2, 56 (1956).

1953 (1)

A. Marèchal, P. Croce, C. R. Acad. Sci., Paris 237, 607 (1953).

1952 (1)

Such phasing methods for this kind of application have been described in the literature, e.g., G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 193 (1952); Y. N. Denisyuk, Opt. Spectry., 15, 279 (1963); W. T. Cathey, J. Opt. Soc. Am. 55, 457 (1965).
[CrossRef]

Croce, P.

A. Marèchal, P. Croce, C. R. Acad. Sci., Paris 237, 607 (1953).

Marèchal, A.

A. Marèchal, P. Croce, C. R. Acad. Sci., Paris 237, 607 (1953).

Meier, R.

O’Neill, E. L.

E. L. O’Neill, Inst. Radio Engrs. Trans. 2, 56 (1956).

Regus, P. J.

P. J. Regus, J. Opt. Soc. Am. 54, 1407 (A) (1964); G. V. Conigleo, ibid. 54, 1407 ((A).

Rogers, G. L.

Such phasing methods for this kind of application have been described in the literature, e.g., G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 193 (1952); Y. N. Denisyuk, Opt. Spectry., 15, 279 (1963); W. T. Cathey, J. Opt. Soc. Am. 55, 457 (1965).
[CrossRef]

Tsujiuchi, J.

J. Tsujiuchi, Progr. Opt. 2, 133 (1963). This paper contains a bibliography.

C. R. Acad. Sci., Paris (1)

A. Marèchal, P. Croce, C. R. Acad. Sci., Paris 237, 607 (1953).

Inst. Radio Engrs. Trans. (1)

E. L. O’Neill, Inst. Radio Engrs. Trans. 2, 56 (1956).

J. Opt. Soc. Am. (2)

P. J. Regus, J. Opt. Soc. Am. 54, 1407 (A) (1964); G. V. Conigleo, ibid. 54, 1407 ((A).

R. Meier, J. Opt. Soc. Am. 55, 987 (1965).
[CrossRef]

Proc. Roy. Soc. Edinburgh (1)

Such phasing methods for this kind of application have been described in the literature, e.g., G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 193 (1952); Y. N. Denisyuk, Opt. Spectry., 15, 279 (1963); W. T. Cathey, J. Opt. Soc. Am. 55, 457 (1965).
[CrossRef]

Progr. Opt. (1)

J. Tsujiuchi, Progr. Opt. 2, 133 (1963). This paper contains a bibliography.

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Figures (7)

Fig. 1
Fig. 1

The basic system for making a corrector-plate hologram.

Fig. 2
Fig. 2

Use of the corrector plate in combination with the lens.

Fig. 3
Fig. 3

A practical system for recording the aberrated wavefront emerging from a lens. L2 is the lens to be corrected; L3 and L4 form a unity-magnification telescopic system.

Fig. 4
Fig. 4

Point spread function of (a) the lens by itself, (b) the lens and hologram corrector plate together at same magnification as (a), and (c) the lens and hologram corrector plate magnified 25 times relative to (a).

Fig. 5
Fig. 5

The image of a transparency (a) using the lens alone, and (b) using the lens and corrector plate together.

Fig. 6
Fig. 6

Diagram for analysis of hologram-making process. The hologram is made at Plane P2, and the test signal and the reference beam emanate from point sources in Plane P0.

Fig. 7
Fig. 7

Diagram for analysis of the hologram corrector plate.

Equations (14)

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I = Φ ( u , v ) 2 = a 0 e j α u + a 1 e j ϕ 2 = a 0 2 + a 1 2 + 2 a 0 a 1 cos ( α u - ϕ ) ,
Φ ( u , v ) = a 1 exp ( j k l 1 ) + a 2 exp { j k [ l 2 + ϕ ( u , v ) ] } ,
Φ ( u , v ) 2 = a 1 2 + a 2 2 + a 1 a 2 exp { j k [ l 1 - l 2 - ϕ ( u , v ) ] } + a 1 a 2 exp { - j k [ l 1 - l 2 - ϕ ( u , v ) ] } ,
l 1 2 = f 1 2 + ( u - b ) 2 + v 2
l 2 2 = f 1 2 + u 2 + v 2 .
o h ¯ 2 = f 2 2 + ( x - u ) 2 + ( y - v ) 2 .
ϕ 1 ( u , v ) = o h ¯ + l 1 - l 2 .
h i ¯ 2 = ( f 1 - r tan γ ) 2 + [ u - ( r - c ) ] 2 + ( v - s ) 2 .
ϕ ( u , v ) = o h ¯ - h i ¯ + l 1 - l 2 = x 2 + y 2 + u 2 + v 2 2 f 2 - ( u x + v y ) f 2 + ( u x + v y ) ( x 2 + y 2 + v 2 ) 2 f 2 3 - ( u x + v y ) 2 2 f 2 3 - ( u 2 + v 2 + x 2 + y ) 2 2 f 2 3 - ( f 2 - r tan γ ) { 1 + [ r 2 + s 2 + ( u + c ) 2 + v 2 ] 2 f 2 2 - [ ( u + c ) r + v s ] f 2 2 + [ ( u + c ) r + s ] [ r 2 + s 2 + ( u + c ) 2 + v 2 ] 2 f 2 4 - [ ( u + c ) r + r s ] 2 2 f 2 4 - [ ( u + c ) 2 + v 2 + r 2 + s 2 ] 2 8 f 2 4 } + ( u + b ) 2 + v 2 2 f 1 - [ ( u + b ) 2 + v 2 ] 2 8 f 2 4 - u 2 + v 2 2 f 1 + ( u 2 + v 2 ) 2 8 f 1 3 .
C = ( - b ω 3 cos ϕ / 2 f 1 ) [ ( 1 / f 1 2 ) - ( 1 / f 2 2 ) ] ;
A = - b ω 2 cos ϕ ( x cos ϕ + y sin ϕ ) f 2 2 f 1 - b 2 ω 2 cos 2 ϕ 2 f 1 2 ( 1 f 1 - 1 f 2 ) ;
C F = x ω 2 2 f 2 ( tan γ - b f 1 ) + b 2 ω 2 4 f 1 2 ( 1 f 2 - 1 f 1 ) ;
D = - ω x ( x cos ϕ + y sin ϕ ) f 2 2 ( tan γ - b f 1 ) + b x ω cos ϕ f 1 f 2 ( tan γ - b f 1 ) + b ( x 2 + y 2 ) ω cos ϕ 2 f 2 2 f 1 - b 2 ω ( x cos ϕ + y sin ϕ ) 2 f 2 f 1 2 .
tan γ = b / f 1

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