Abstract

A coherent processing system for obtaining complex addition of two or more patterns in real time is described. In this system, a sinusoidal diffraction grating is used in the spatial-frequency plane. The input patterns to be added are displayed symmetrically about the optical axis in the object plane. The complex amplitude addition appears in th zero-order location in the output plane. Complex amplitude subtraction may be obtained by adding a 180° phase plate behind one of the input patterns. Displacing the sinusoidal diffraction grating by half a line pitch is also shown to change the performance of the system from addition to subtraction for two input functions, without the need for a phase plate. Experimental verification of the performance is presented. Analysis of the system for two input patterns is first given, then extended to the complex addition of multiple inputs.

© 1970 Optical Society of America

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References

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  1. D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, Phys. Letters 18, 116 (1965).
    [Crossref]
  2. K. Bromley, M. A. Monahan, J. F. Bryant, and B. J. Thompson, Appl. Phys. Letters 14, 67 (1969).
    [Crossref]
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), Sec. 2.1.
  4. R. Kraushaar, J. Opt. Soc. Am. 40, 480 (1950).
    [Crossref]
  5. A. R. Maddox and R. C. Binder, Appl. Opt. 8, 2191 (1969).
    [Crossref] [PubMed]

1969 (2)

K. Bromley, M. A. Monahan, J. F. Bryant, and B. J. Thompson, Appl. Phys. Letters 14, 67 (1969).
[Crossref]

A. R. Maddox and R. C. Binder, Appl. Opt. 8, 2191 (1969).
[Crossref] [PubMed]

1965 (1)

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, Phys. Letters 18, 116 (1965).
[Crossref]

1950 (1)

Binder, R. C.

Bromley, K.

K. Bromley, M. A. Monahan, J. F. Bryant, and B. J. Thompson, Appl. Phys. Letters 14, 67 (1969).
[Crossref]

Brumm, D.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, Phys. Letters 18, 116 (1965).
[Crossref]

Bryant, J. F.

K. Bromley, M. A. Monahan, J. F. Bryant, and B. J. Thompson, Appl. Phys. Letters 14, 67 (1969).
[Crossref]

Funkhouser, A.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, Phys. Letters 18, 116 (1965).
[Crossref]

Gabor, D.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, Phys. Letters 18, 116 (1965).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), Sec. 2.1.

Kraushaar, R.

Maddox, A. R.

Monahan, M. A.

K. Bromley, M. A. Monahan, J. F. Bryant, and B. J. Thompson, Appl. Phys. Letters 14, 67 (1969).
[Crossref]

Restrick, R.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, Phys. Letters 18, 116 (1965).
[Crossref]

Stroke, G. W.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, Phys. Letters 18, 116 (1965).
[Crossref]

Thompson, B. J.

K. Bromley, M. A. Monahan, J. F. Bryant, and B. J. Thompson, Appl. Phys. Letters 14, 67 (1969).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Letters (1)

K. Bromley, M. A. Monahan, J. F. Bryant, and B. J. Thompson, Appl. Phys. Letters 14, 67 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Letters (1)

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, Phys. Letters 18, 116 (1965).
[Crossref]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), Sec. 2.1.

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Figures (9)

Fig. 1
Fig. 1

System for complex addition or subtraction of functions f1 and f2 in real time by a sinusoidal grating in the spatial-filtering plane, P2

Fig. 2
Fig. 2

Schematic of the output plane (P3) showing complex addition of two complex pattern functions f1 and f2, at position 3. f2 is at position 1; 2f2 is at 2; 2f1 appears at 4, and f1 appears at 5.

Fig. 3
Fig. 3

Complex amplitude subtraction. Functions f1 = CIT and f2 = IT with the diffraction patterns successfully subtracted at the center of the output plane.

Fig. 4
Fig. 4

Complex amplitude addition. Functions f1 = CIT and f2 = IT with the diffraction patterns successfully added as shown by the radiance of IT in the composite output.

Fig. 5
Fig. 5

Schematic of the output plane (P3) showing complex addition of three complex pattern functions given by Eq. (10). f−1 is at position 1; 2 f - 1 + 1 2 f 0 is at 2; f−1 + f0 + f1 is at 3, 1 2 f 0 + 2 f 1 is at 4; and f1 is at 5.

Fig. 6
Fig. 6

Complex amplitude addition of three nonoverlapping patterns. Functions f1 = CIT, f0 =MU, and f−1 = IT. Behind the function f−1, a 180° phase plate has been inserted.

Fig. 7
Fig. 7

Schematic of the output plane showing complex addition of five pattern functions given by Eq. (12). f−2 is at position 1; f−1 + f−2 is at 2; 1 4 f 0 + f - 1 + 4 f - 2 is at 3; 1 4 f 0 + f 1 + 4 f - 1 + f - 2 is at 4; f0 + f1 + f2 + f−1 + f−2 is at 5; 1 4 f 0 + 4 f 1 + f - 1 + f 2 is at 6; 1 4 f 0 + f 1 + 4 f 2 is at 7; f1 + f2 is at 8; and f2 is at 9.

Fig. 8
Fig. 8

Schematic of the output plane showing complex addition of five pattern functions given by Eq. (13b). f3 is at position 1; f2 + f3 is at 2; 1 4 f 0 + 4 f 3 is at 3; f1 + f3 is at 4; f2 is at 5; 1 4 f 0 + 4 f 2 is at 6; f0 + f1 + f2 + f3 + f4 is at 7; 1 4 f 0 + 4 f 1 is at 8; f1 is at 9; f2 + f4 is at 10; 1 4 f 0 + 4 f 4 is at 11; f1 + f4 is at 12; and f4 is at 13.

Fig. 9
Fig. 9

Transverse displacement of the input plane.

Equations (21)

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t 1 ( x 1 ) = f 1 ( x 1 - b ) + f 2 ( x 1 + b ) .
A P 2 - ( v ) = F 1 ( v ) e - i 2 π v · b + F 2 ( v ) e i 2 π v · b ,
t ( v ) = t b + β ( a 2 + a e i 2 π v · b + a e - i 2 π v · b ) ,
t ( v ) = 2 + e i 2 π v · b + e - i 2 π v · b .
A P 2 + ( v ) = F 1 ( v ) e - i 4 π v · b + 2 F 1 ( v ) e - i 2 π v · b + F 1 ( v ) + F 2 ( v ) + 2 F 2 ( v ) e i 2 π v · b + F 2 ( v ) e i 4 π v · b .
u 3 ( x 3 ) = f 1 ( x 3 - 2 b ) + 2 f 1 ( x 3 - b ) + f 1 ( x 3 ) + f 2 ( x 3 ) + 2 f 2 ( x 3 + b ) + f 2 ( x 3 + 2 b ) .
f 1 ( x ) = f 1 ( x ) e i ϕ 1 ( x ) f 2 ( x ) = f 2 ( x ) e i ϕ 2 ( x ) .
f 1 ( x ) + f 2 ( x ) = [ f 1 ( x ) + f 2 ( x ) e i π ] e i ϕ 1 ( x ) f 1 ( x ) - f 2 ( x ) .
f 1 ( x ) + f 2 ( x ) = [ f 1 ( x ) + f 2 ( x ) ] e i ϕ 1 ( x ) f 1 ( x ) + f 2 ( x ) .
f 1 ( x - b ) = e i ( ϕ 0 + π ) f 2 ( x + b ) = e i [ ϕ 0 + ϕ 2 ( x ) ] ,
f 1 ( x - b ) + f 2 ( x + b ) = e i ϕ 0 ( - 1 + e i ϕ 2 ( x ) ) ϕ 2 ( x )
t 1 ( x ) = f 1 ( x 1 - b ) + 1 2 f 0 ( x 1 ) + f - 1 ( x 1 + b ) .
t ( v ) = 1 + e i 2 π v · b 1 2 + 1 + e i 2 π v · b 2 2 = 4 + e i 2 π v · b 1 + e - i 2 π v · b 1 + e i 2 π v · b 2 + e - i 2 π v · b 2 .
b 2 = 2 b 1 = 2 b 1 a y
t 1 ( x 1 ) = 1 4 f 0 ( x 1 ) + f 1 ( x 1 - b 1 ) + f - 1 ( x 1 + b 1 ) + f 2 ( x 1 - 2 b 1 ) + f - 2 ( x 1 + 2 b 1 ) .
b 1 = b 1 a y ,             b 2 = b 1 a x
t 1 ( x 1 ) = 1 4 f 0 ( x 1 ) + f 1 ( x 1 - b 1 ) + f 2 ( x 1 + b 1 ) + f 3 ( x 1 - b 2 ) + f 4 ( x 4 + b 2 ) .
t 1 ( x 1 ) = f 1 ( x 1 - b - Δ ) + f 2 ( x 1 + b - Δ ) .
A p 2 - ( v ) A p 2 - ( v ) e - i 2 π v · Δ A p 2 + ( v ) A p 2 + ( v ) e - i 2 π v · Δ .
t ( v ) = 2 + ( e i ( 2 π v · b + ϕ d ) - e - i ( 2 π v · b + ϕ d ) ) .
u 3 ( x 3 ) = f 1 ( x 3 ) e i ϕ d + f 2 ( x 3 ) e - i ϕ d .