Abstract

A confocal feedback system (CFS) has been modified in three ways to include space variance, time sampling, and a second feedback loop. The space-variant system performs analog solution of partial differential equations (PDE’s) with variable coefficients. The time-sampling system solves PDE’s in three dimensions. The CFS with a second feedback loop has a more flexible feedback transfer function and can solve an extended range of PDE’s. Finally, a combination of time sampling and a second feedback loop can solve four-dimensional problems. Experimental results verifying the abilities of each of these new confocal feedback systems have been obtained.

© 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. F. Walkup, “Space-variant coherent optical processing,” Opt. Eng. 19, 339–346 (1980).
    [Crossref]
  3. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 681–682.
  4. Ref. 3, pp. 688–689.
  5. H. Soodak, ed., Reactor Handbook (Interscience, New York, 1962), Vol. III, Part A, p. 138.
  6. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1966), p. 607.
  7. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), p. 222.
  8. J. Götz and et al., “Solving differential equations with TV-optical feedback,” in Proceedings of the International Optical Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1978), pp. 179–180.
  9. Ref. 3, pp. 842–846, 893.
  10. R. J. Marks, “Coherent optical extrapolation of 2-D bandlimited signals: processor theory,” Appl. Opt. 19, 1670–1672 (1980).
    [Crossref]
  11. Ref. 6, p. 382.

1980 (3)

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1966), p. 607.

Cederquist, J.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 681–682.

Götz, J.

J. Götz and et al., “Solving differential equations with TV-optical feedback,” in Proceedings of the International Optical Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1978), pp. 179–180.

Lee, S. H.

Marks, R. J.

Merzbacher, E.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), p. 222.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 681–682.

Walkup, J. F.

J. F. Walkup, “Space-variant coherent optical processing,” Opt. Eng. 19, 339–346 (1980).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

J. F. Walkup, “Space-variant coherent optical processing,” Opt. Eng. 19, 339–346 (1980).
[Crossref]

Other (8)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 681–682.

Ref. 3, pp. 688–689.

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

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1966), p. 607.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), p. 222.

J. Götz and et al., “Solving differential equations with TV-optical feedback,” in Proceedings of the International Optical Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1978), pp. 179–180.

Ref. 3, pp. 842–846, 893.

Ref. 6, p. 382.

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Figures (15)

Fig. 1
Fig. 1

Confocal feedback system showing interferometer mirrors M1 and M2 and imaging lenses L1, L2, and L3.

Fig. 2
Fig. 2

Confocal Fabry–Perot interferometer showing space-invariant frequency filters f ˜ and g ˜ and space-variant image filters ac and bc.

Fig. 3
Fig. 3

Optical solution of hyperbolic PDE. (a) Image filter ac. Output solution magnitude for (b) β = 0 and (c) β = π showing effect of feedback phase on location of solution maxima.

Fig. 4
Fig. 4

Optical solution of elliptic PDE. Cross sections of (a) image filter ac and (b) magnitude of output solution showing differentiating effect of filter f ˜ = (u2 + v2)/tc.

Fig. 5
Fig. 5

Optical solution of modified Helmholtz equation. (a) Image filter bc. (b) Pseudo-3-D plot of output solution ϕ versus x and y for input ϕo = ex. Collision density ϕ increases near scattering regions defined by bc. Coordinate axes are shown for reference.

Fig. 6
Fig. 6

Time-sampling feedback system. Thin wedge performs time sampling by spatial shifting of images. Fourier and image portions of midplane are reversed with respect to Fig. 2.

Fig. 7
Fig. 7

Optical solution of ∂ϕ/∂t + ϕ = ∂2ϕ/∂xy. (a) Input boundary condition ϕ(x,y,0) = sinc(x)sinc(y). (b)–(f) Magnitudes of time-sampled output solutions ϕ(x,y,nΔt) for n = 1 to n = 5. Each image is the previous one differentiated in x and y.

Fig. 8
Fig. 8

Optical solution of ∇2 ϕ − (∂2ϕ/∂t2) = 0. (a) Cross section of input boundary condition ϕ(x,y,0). (b)–(d) Outputs ao(x,y,n) for n = 1, 2, 3. Wave propagates from origin outward leaving a wake behind.

Fig. 9
Fig. 9

Multiple-feedback system for more flexible coherent transfer function. (a) System diagram showing inner and outer feedback loops and frequency filters f ˜ 1 , f ˜ 2 , g ˜ 1 and g ˜ 2. (b) Photograph of inner confocal Fabry–Perot showing mirrors, Invar rods, and liquid gate holding filters.

Fig. 10
Fig. 10

Optical solution of wave equation. (a) Driving force input s(x,t). (b) Output solution ϕ(x,t) showing wave propagation with damping. Coordinate axes are shown for reference. (c) Outer-loop spatial filter g ˜ 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.

Fig. 11
Fig. 11

Combination of multiple feedback with time sampling. Addition of second feedback loop increases the flexibility of the time-sampling system. Addition of a second wedge within the inner loop permits four-dimensional processing.

Fig. 12
Fig. 12

Time-sampled multiple feedback. (a) Input ai. (b)–(d) Magnitudes of outputs ao(x,y,n) for n = 1, 2, 3. Basic operation is ∂/∂y.

Fig. 13
Fig. 13

Four-dimensional processing. Input was a point, and no filters were used. Outputs ao(x,y,m,n) for m = 0 to m = 4 and n = 0 to n = 4 appear in array form.

Fig. 14
Fig. 14

Multiple-feedback interference fringes: (a) due to outer loop, (b) due to inner loop, and (c) combined effect. Peaks in (c) occur at points where both interferometers have positive feedback.

Fig. 15
Fig. 15

Resolution test of multiple-feedback system. Space–bandwidth product of image is greater than 128 × 128.

Equations (47)

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a o 1 ( x , y ) = a c ( x , y ) a i ( x , y ) + t c e i β a c ( x , y ) × ( g ( x , y ) { b c ( x , y ) [ f ( x , y ) a o 1 ( x , y ) ] } )
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 ) = 1 t c 2 [ a c ( x , y ) a i ( x , y ) ] x y + e i β 2 [ a c ( x , y ) a o 2 ( x , y ) ] x y .
e i β 2 [ a c ( x , y ) ϕ ( x , y ) ] x y = ϕ ( x , y ) + s ( x , y ) .
a o 2 ( x , y ) = - 1 t c 2 [ a c ( x , y ) a i ( x , y ) ] - 2 [ a c ( x , y ) a o 2 ( x , y ) ] .
2 [ a c ( x , y ) ϕ ( x , y ) ] + ϕ ( x , y ) = s ( x , y ) .
2 ϕ ( x , y ) - ϕ ( x , y ) = - v ( x , y ) ϕ ( x , y ) .
ϕ ( x , y ) = ϕ 0 ( x , y ) + K 0 [ ( x 2 + y 2 ) 1 / 2 ] [ v ( x , y ) ϕ ( x , y ) ] ,
2 ϕ o - ϕ o = 0
a o 1 ( x , y ) = a i ( x , y ) + K 0 [ ( x 2 + y 2 ) 1 / 2 ] [ t c b c ( x , y ) a o 1 ( x , y ) ] .
a o ( x , y , n + 1 ) = t c g ( x , y ) a o ( x , y , n ) ,
a o ( x , y , 0 ) = a i ( x , y ) .
ψ t = { ψ } ,
ψ ( x , y , 0 ) = s ( x , y ) ,
ψ = ϕ e t / Δ t .
ϕ t + ϕ Δ t = { ϕ } .
ϕ ( x , y , 0 ) = s ( x , y ) .
ϕ [ x , y , ( n + 1 ) Δ t ] - ϕ ( x , y , n Δ t ) Δ t + ϕ ( x , y , n Δ t ) Δ t = ϕ [ x , y , ( n + 1 ) Δ t ] Δ t = a o ( x , y , n + 1 ) Δ t .
a o ( x , y , n + 1 ) = Δ t { a o ( x , y , n ) } ,
a o ( x , y , 0 ) = s ( x , y ) .
ψ t = 2 ψ x y ,
ψ ( x , y , 0 ) = sinc ( x ) sinc ( y ) .
2 ϕ - 2 ϕ t 2 = 0 ,
ϕ ( x , y , 0 ) = s ( x , y ) ,
ϕ ( x , y , t ) t | t = 0 = 0.
ϕ [ x , y , ( n + 1 ) Δ t ] = p ( x , y , Δ t ) ϕ ( x , y , t ) t | t = n Δ t - p ( x , y , t ) t | t = Δ t ϕ ( x , y , n Δ t ) ,
p ( x , y , t ) = step [ t - ( x 2 + y 2 ) 1 / 2 ] [ t 2 - ( x 2 + y 2 ) ] 1 / 2 .
a o ( x , y , n + 1 ) = - p t | t = Δ t a o ( x , y , n ) ,
h ˜ 2 = 1 1 - t c 2 e i β 2 f ˜ 2 g ˜ 2 ,
h ˜ = h ˜ 2 f ˜ 1 1 - t c 1 e i β 1 h ˜ 2 f ˜ 1 g ˜ 1
= f ˜ 1 1 - t c 1 e i β 1 f ˜ 1 g ˜ 1 - t c 2 e i β 2 f ˜ 2 g ˜ 2 ,
2 ϕ t 2 + ϕ t + ϕ - 2 ϕ x 2 = s ( x , t ) ,
ϕ ( x , 0 ) = 0 ,
ϕ ( x , t ) t | t = 0 = 0.
ϕ ˜ ( u , v ) = s ˜ ( u , v ) 1 - v 2 + u 2 - i v .
h ˜ = 1 1 - v 2 + u 2 - i v .
f ˜ 1 = 1 ,             g ˜ 1 = v / t c 1 ,             β 1 = π / 2
f ˜ 2 = 1 ,             g ˜ 2 = ( v 2 - u 2 ) / t c 2 ,             β 2 = 0.
g ˜ 1 = f ˜ 2 1 - t c 2 e i β 2 f ˜ 2 g ˜ 2 ,
g ˜ 1 = v / t c 2 1 - i v .
0 K 0 ( r ) J 0 ( ρ r ) r d r ,
( 2 - 1 ) K 0 [ ( x 2 + y 2 ) 1 / 2 ] = - δ ( x , y ) .
( - u 2 - v 2 - 1 ) K ˜ 0 [ ( u 2 + v 2 ) 1 / 2 ] = - 1 ,
K ˜ 0 [ ( u 2 + v 2 ) 1 / 2 ] = 1 1 + u 2 + v 2 .
I 0 step ( t - r ) ( t 2 - r 2 ) 1 / 2 J 0 ( ρ r ) r d r .
I = 0 t J 0 ( ρ r ) r d r ( t 2 - r 2 ) 1 / 2 .
I = t 0 π / 2 J 0 ( t ρ cos θ ) cos θ d θ = sin ( t ρ ) ρ = sin [ t ( u 2 + v 2 ) 1 / 2 ] ( u 2 + v 2 ) 1 / 2 .