Abstract

A new method for recording Fresnel transformations of two- and three-dimensional scenes illuminated by spatially incoherent light is described. The technique is based on the properties of the triangular interferometer and the afocal optical system. Experimental results with one- and two-point objects have verified the basic principles of the method.

© 1966 Optical Society of America

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References

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  1. D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).
    [Crossref]
  2. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962); J. Opt. Soc. Am. 53, 1377 (1963); J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  3. L. Mertz and N. O. Young, Proc. ICO Conf. Opt. Instr., London, 1961, p. 305; N. O. Young, Sky and Telescope, Jan.1963.
  4. L. Mertz, J. Opt. Soc. Am. 54, No. 10, Advertisement iv (1964).
  5. A. Lohmann, J. Opt. Soc. Am. 55, 1555 (1965).
    [Crossref]
  6. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 383.
  7. P. Hariharan and D. Sen, J. Sci. Instr. 38, 428 (1961).
    [Crossref]
  8. M. V. R. K. Murty, Appl. Opt. 3, 853 (1964).
    [Crossref]
  9. Since the writing of this paper, further experiments by P. Peters at Conductron have produced a successful hologram of the word Conductron and the reconstruction is excellent.

1965 (1)

1964 (2)

L. Mertz, J. Opt. Soc. Am. 54, No. 10, Advertisement iv (1964).

M. V. R. K. Murty, Appl. Opt. 3, 853 (1964).
[Crossref]

1962 (1)

1961 (1)

P. Hariharan and D. Sen, J. Sci. Instr. 38, 428 (1961).
[Crossref]

1948 (1)

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 383.

Gabor, D.

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).
[Crossref]

Hariharan, P.

P. Hariharan and D. Sen, J. Sci. Instr. 38, 428 (1961).
[Crossref]

Leith, E. N.

Lohmann, A.

Mertz, L.

L. Mertz, J. Opt. Soc. Am. 54, No. 10, Advertisement iv (1964).

L. Mertz and N. O. Young, Proc. ICO Conf. Opt. Instr., London, 1961, p. 305; N. O. Young, Sky and Telescope, Jan.1963.

Murty, M. V. R. K.

Sen, D.

P. Hariharan and D. Sen, J. Sci. Instr. 38, 428 (1961).
[Crossref]

Upatnieks, J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 383.

Young, N. O.

L. Mertz and N. O. Young, Proc. ICO Conf. Opt. Instr., London, 1961, p. 305; N. O. Young, Sky and Telescope, Jan.1963.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

J. Sci. Instr. (1)

P. Hariharan and D. Sen, J. Sci. Instr. 38, 428 (1961).
[Crossref]

Nature (1)

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).
[Crossref]

Other (3)

Since the writing of this paper, further experiments by P. Peters at Conductron have produced a successful hologram of the word Conductron and the reconstruction is excellent.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 383.

L. Mertz and N. O. Young, Proc. ICO Conf. Opt. Instr., London, 1961, p. 305; N. O. Young, Sky and Telescope, Jan.1963.

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Figures (6)

Fig. 1
Fig. 1

Amplitude distribution from point source.

Fig. 2
Fig. 2

Afocal optical system.

Fig. 3
Fig. 3

Double afocal system.

Fig. 4
Fig. 4

Interferometric arrangement of double afocal system.

Fig. 5
Fig. 5

Incoherent hologram generated by one point target.

Fig. 6
Fig. 6

Incoherent hologram generated by two point targets.

Equations (27)

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u = a exp [ i ( 2 π / λ ) r ] ,
r 2 = ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 .
r r 0 ( x 0 x + y 0 y ) / r 0 + ( x 2 + y 2 ) / 2 r 0 ( x 0 x + y 0 y ) 2 / 2 r 0 3 ,
r 0 2 = x 0 2 + y 0 2 + z 0 2 .
f = E u ( α x , α y ) + G u ( β x , β y ) ,
| f | 2 = a 2 { E 2 + G 2 + 2 E G cos ( 2 π / λ ) [ r ( α x , α y ) r ( β x , β y ) ] } .
r ( α x , α y ) r ( β x , β y ) ( α β ) ( x 0 x + y 0 y ) / r 0 + ( α 2 β 2 ) ( x 2 + y 2 ) / 2 r 0 ( α 2 β 2 ) ( x 0 x + y 0 y ) 2 / 2 r 0 3 .
x 1 = x 0 / ( α + β ) y 1 = y 0 / ( α + β ) r 10 = r 0 / ( α + β ) z 1 = z 0 / ( α + β )
α β = 1.
r ( α x , α y ) r ( β x , β y ) ( x 1 x + y 1 y ) / r 10 + ( x 2 + y 2 ) / 2 r 10 ( x 1 x + y 1 y ) 2 / 2 r 10 3 r 1 ( x , y ) r 10 ,
r 10 2 = x 1 2 + y 1 2 + z 1 2 .
υ ( x , y ) = u ( x , y ) T ( x , y ; x , y ) d x d y ,
υ ( x , y ) = u ( x , y ) δ [ x + ( f 1 / f 2 ) x , y + ( f 1 / f 2 ) y ] d x d y = u [ ( f 1 / f 2 ) x , ( f 1 / f 2 ) y ] .
υ ( x , y ) = u ( x / m , y / m ) .
α 1 / m = f 1 / f 2 and β m = f 2 / f 1 .
R 2 = m 2 ( m + 1 / m ) 2 T 2 = ( 1 / m 2 ) ( m + 1 / m ) 2 .
E 2 = R 2 / m 2 G 2 = T 2 m 2 .
| f | = 2 a 2 ( m + 1 / m ) 2 [ 1 + cos ( 2 π / λ ) r 1 ] .
I = 2 ( m + 1 / m ) 2 i = 0 N a i 2 [ 1 + cos ( 2 π / λ ) r i ] .
I c = | i = 0 N u i + A | 2 = i , j u i u j * + | A | 2 + 2 A i = 0 N a i cos ( 2 π / λ ) r i ,
I i = B [ i = 0 N a i 2 + i = 0 N a i 2 cos ( 2 π / λ ) r i ] .
A 2 i = 0 N a i 2 .
I i C [ 1 + i = 0 N ( a i / A ) 2 cos ( 2 π / λ ) r i ] , I c D [ 1 + i = 0 N ( a i / A ) cos ( 2 π / λ ) r i ] .
[ i = 0 N ( a i / A ) cos ( 2 π / λ ) r i ] 2 Av 1 2 = i = 0 N j = 0 N ( a i a j / A 2 ) cos ( 2 π / λ ) r i cos ( 2 π / λ ) r j Av 1 2 = i = 0 N ( a i 2 / A 2 ) cos 2 ( 2 π / λ ) r i + i = 0 i j N j = 0 N ( a i a j / A 2 ) cos ( 2 π / λ ) r i cos ( 2 π / λ ) r j Av 1 2 .
i = 0 N ( a i 2 / A 2 ) cos 2 ( 2 π / λ ) r i Av = ( 1 2 ) i = 0 N ( a 0 2 / A 2 ) = ( 1 2 ) N · ( a 0 2 / A 2 ) = 1 2 ,
A 2 i = 0 N a 0 2 = N a 0 2
i = 0 N ( a i 4 / A 4 ) cos 2 ( 2 π / λ r i ) Av = 1 2 N ( a 0 4 / A 4 ) = 1 / ( 2 N ) .