Abstract

Confocal feedback systems can perform analog optical solutions of two-dimensional integral equations. Methods for solution of Fredholm equations by space-variant feedback and of Volterra equations using multiple feedback have been found and experimental results obtained. Two-dimensional problems of this type are difficult and time consuming to solve digitally, but the optical solution is obtained at high speed.

© 1981 Optical Society of America

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References

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  1. J. Cederquist and S. H. Lee, “Coherent optical feedback for the analog solution of partial differential equations,” J. Opt. Soc. Am. 70, 944–953 (1980).
    [Crossref]
  2. J. Cederquist and S. H. Lee, “Confocal feedback systems with space variance, time sampling, and secondary-feedback loops,” J. Opt. Soc. Am. 71, 643–650 (1981).
    [Crossref]
  3. H. Soodak, ed., Reactor Handbook (Interscience, New York, 1962), Vol. III, Part A, p. 138.
  4. Ref. 3, pp. 112–113.
  5. Ref. 3, pp. 138–139.
  6. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 202.

1981 (1)

1980 (1)

J. Opt. Soc. Am. (2)

Other (4)

H. Soodak, ed., Reactor Handbook (Interscience, New York, 1962), Vol. III, Part A, p. 138.

Ref. 3, pp. 112–113.

Ref. 3, pp. 138–139.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 202.

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Figures (5)

Fig. 1
Fig. 1

Space-variant confocal feedback system for solution of Fredholm integral equations. M1 and M2 are interferometer mirrors, f ˜ and g ˜ are spatial-frequency filters, and ac and bc are image filters.

Fig. 2
Fig. 2

Optical solution of neutron-diffusion integral equation. (a) Space-variant image filter ac(x, y) with moderator (white disk), radioactive fuel (black ring), and reflector (white ring). (b) Output solution for thermal neutron flux.

Fig. 3
Fig. 3

Optical solution of neutron-transport integral equation. (a) Image filter ac(x,y) with radioactive fuel rods (black disks) in moderator (white background). (b) Output solution for neutron-collision density. The image has been shifted relative to (a) to show a cross section.

Fig. 4
Fig. 4

Space-variant feedback system with a second feedback loop for solution of Volterra integral equations. M1, M2, M3, and M4 are interferometer mirrors, f ˜ 1 and g ˜ 2 are spatial-frequency filters, and ac is an image filter.

Fig. 5
Fig. 5

Optical solution of time-dependent neutron-diffusion integral equation. (a) Image filter ac(x,y) with alternating regions of high and low absorption. (b) Optical solution for neutron flux. Coordinate axes are shown for reference. (c) Outer-loop spatial filter f ˜ 1 (left) and inner-loop filter g ˜ 2 (right). The filters are scaled according to the focal lengths of the outer and inner interferometer mirrors.

Equations (19)

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a o 2 ( x , y ) = b c ( x , y ) { f ( x , y ) [ a c ( x , y ) a i ( x , y ) ] } + t c e i β b c ( x , y ) ( f ( x , y ) { a c ( x , y ) [ g ( x , y ) a o 2 ( x , y ) ] } ) ,
a o 2 ( x , y ) = - - f ( x - x , y - y ) × a c ( x , y ) a i ( x , y ) d x d y + t c e i β × - - f ( x - x , y - y ) a c ( x , y ) × a o 2 ( x , y ) d x d y ,
ϕ ( x , y ) = - - K 0 { [ ( x - x ) 2 + ( y - y ) 2 ] 1 / 2 } × [ s ( x , y ) + μ ( x , y ) ϕ ( x , y ) ] d x d y ,
ϕ ( x , y ) = - - ( 1 K 0 { [ ( x - x ) 2 + ( y - y ) 2 ] 1 / 2 w } d w ) × [ s ( x , y ) + μ ( x , y ) ϕ ( x , y ) ] d x d y ,
f ˜ ( u , v ) = 1 ( u 2 + v 2 ) 1 / 2 { π 2 - tan - 1 [ 1 ( u 2 + v 2 ) 1 / 2 ] } ,
h ˜ 2 ( u , v ) = 1 1 - t c 2 e i β 2 g ˜ 2 ( u , v ) ,
a o ( x , y ) = - - h 1 ( x - x , y - y ) [ a c ( x , y ) a i ( x , y ) + t c 1 e i β 1 a c ( x , y ) a o ( x , y ) ] d x d y ,
a o ( x , y ) = - y - h 1 ( x - y , y - y ) [ a c ( x , y ) a i ( x , y ) + t c 1 e i β 1 a c ( x , y ) a o ( x , y ) ] d x d y ,
ϕ ( x , t ) = - t - exp [ - ( t - t ) ] [ 4 π ( t - t ) ] 1 / 2 exp / [ - ( x - x ) 2 4 ( t - t ) ] × [ s ( x , t ) + μ ( x , t ) ϕ ( x , t ) ] d x d t ,
f 1 ( x , y ) h 2 ( x , y ) = e - y ( 4 π y ) 1 / 2 exp [ - ( x 2 / 4 y ) ] step ( y ) .
f ˜ 1 h ˜ 2 = 1 1 + u 2 - i v .
f ( x , y ) = 1 K 0 [ ( x 2 + y 2 ) 1 / 2 w ] d w
f ˜ ( ρ ) = 0 [ 1 K 0 ( r w ) d w ] J 0 ( ρ r ) r d r = 1 [ 0 K 0 ( w r ) J 0 ( ρ r ) r d r ] d w ,
1 d w w 2 + ρ 2 = 1 ρ tan - 1 ( w ρ ) | 1 ,
f ˜ ( ρ ) = 1 ρ [ π 2 - tan - 1 1 ρ ] .
I - - e - y ( 4 π y ) 1 / 2 exp [ - ( x 2 / 4 y ) ] step ( y ) × exp [ i ( x u + y v ) ] d x d y .
I = - e - y ( 4 π y ) 1 / 2 step ( y ) e i y v × { - exp [ - ( x 2 / 4 y ) ] e i x u d x } d y = - e - y e - u 2 y step ( y ) e i y v d y .
I = - exp [ - ( 1 + u 2 ) y ] step [ ( 1 + u 2 ) y ] e i y v d y .
I = 1 1 + u 2 × 1 1 - i v 1 + u 2 = 1 1 + u 2 - i v .