Abstract

A confocal Fabry-Perot interferometer is used to produce feedback within an optical processor. The resulting confocal feedback system (CFS) has a complex-valued feedback transfer function and performs analog solution of partial differential equations (PDE’s). The CFS is interfaced to a microcomputer via a laser scanner and video electronics to create a hybrid processor, thus taking a major step toward the practical use of the CFS for the high-speed solution of PDE’s. Solutions to the three types of second-order linear PDE’s in two dimensions—elliptic, hyperbolic, and parabolic—were obtained experimentally and analyzed.

© 1980 Optical Society of America

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References

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  1. J. Cederquist and S. H. Lee, “The Use of Feedback in Optical Information Processing,” Appl. Phys. 18, 311–319 (1979).
    [CrossRef]
  2. R. Akins, R. Athale, and S. H. Lee, “Feedback in Analog and Digital Optical Image Processing,” Opt. Eng. (to be published).
  3. M. Hercher, “The Spherical Mirror Fabry-Perot Interferometer,” Appl. Opt. 7, 951–966 (1968);Appl. Opt. 7, 1336 (E) (1968).
    [CrossRef] [PubMed]
  4. H. K. V. Lotsch, “The Confocal Resonator System,” Optik 30, 1–14, 181–201, 217–233, and 563–576 (1969).
  5. J. A. Clark, “Imaging Through a Confocal Optical Interference Filter,” Opt. Commun. 5, 163–167 (1972).
    [CrossRef]
  6. A. Riazi, O. P. Gandhi, and D. A. Christensen, “Imaging Characteristics of a Confocal Cavity,” Opt. Commun. 28, 163–165 (1979).
    [CrossRef]
  7. J. R. Leger and S. H. Lee, “Coherent Optical Implementation of Generalized Two-Dimensional Transforms,” Opt. Eng. 18, 518–523 (1979).
    [CrossRef]
  8. G. A. Bekey and W. J. Karplus, Hybrid Computation (Wiley, New York, 1968).
  9. S. P. McGrew, in New Concepts and Technologies in Parallel Information Processing, edited by E. R. Caianiello (Noordhoff, Leyden, 1975), pp. 55–73.
    [CrossRef]
  10. N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, “Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System,” Opt. Spectrosc. 40, 420–422 (1976).
  11. J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, “Solving Differential Equations with TV-Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.
  12. D. Jablonowski and S. H. Lee, “A Coherent Optical Feedback System for Optical Information Processing,” Appl. Phys. 8, 51–58 (1975).
    [CrossRef]
  13. J. Cederquist and S. H. Lee, “The Solution of Partial Differential Equations with Time-Varying Inhomogeneous Terms Using Coherent Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 181–184, IEEE Order No.CH1305-2/79/0000-0181.
  14. G. Dahlquist and Å Björck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 389–392.
  15. R. L. Burden, J. D. Faires, and A. C. Reynolds, Numerical Analysis (Prindle, Weber and Schmidt, Boston, 1978), p. 525.
  16. R. Akins and S. H. Lee, “Coherent Optical Image Amplification by an Injection-Locked Dye Amplifier at 632.8 nm,” Appl. Phys. Lett. 35, 660–663 (1979).
    [CrossRef]
  17. Reference 15, pp. 508–509.
  18. R. W. Hockney, Technical Report 53, Institute for Plasma Research, Stanford University, 1966 (unpublished).

1979 (4)

A. Riazi, O. P. Gandhi, and D. A. Christensen, “Imaging Characteristics of a Confocal Cavity,” Opt. Commun. 28, 163–165 (1979).
[CrossRef]

J. R. Leger and S. H. Lee, “Coherent Optical Implementation of Generalized Two-Dimensional Transforms,” Opt. Eng. 18, 518–523 (1979).
[CrossRef]

R. Akins and S. H. Lee, “Coherent Optical Image Amplification by an Injection-Locked Dye Amplifier at 632.8 nm,” Appl. Phys. Lett. 35, 660–663 (1979).
[CrossRef]

J. Cederquist and S. H. Lee, “The Use of Feedback in Optical Information Processing,” Appl. Phys. 18, 311–319 (1979).
[CrossRef]

1978 (2)

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, “Solving Differential Equations with TV-Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

J. Cederquist and S. H. Lee, “The Solution of Partial Differential Equations with Time-Varying Inhomogeneous Terms Using Coherent Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 181–184, IEEE Order No.CH1305-2/79/0000-0181.

1976 (1)

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, “Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System,” Opt. Spectrosc. 40, 420–422 (1976).

1975 (1)

D. Jablonowski and S. H. Lee, “A Coherent Optical Feedback System for Optical Information Processing,” Appl. Phys. 8, 51–58 (1975).
[CrossRef]

1972 (1)

J. A. Clark, “Imaging Through a Confocal Optical Interference Filter,” Opt. Commun. 5, 163–167 (1972).
[CrossRef]

1969 (1)

H. K. V. Lotsch, “The Confocal Resonator System,” Optik 30, 1–14, 181–201, 217–233, and 563–576 (1969).

1968 (1)

Akins, R.

R. Akins and S. H. Lee, “Coherent Optical Image Amplification by an Injection-Locked Dye Amplifier at 632.8 nm,” Appl. Phys. Lett. 35, 660–663 (1979).
[CrossRef]

R. Akins, R. Athale, and S. H. Lee, “Feedback in Analog and Digital Optical Image Processing,” Opt. Eng. (to be published).

Athale, R.

R. Akins, R. Athale, and S. H. Lee, “Feedback in Analog and Digital Optical Image Processing,” Opt. Eng. (to be published).

Bekey, G. A.

G. A. Bekey and W. J. Karplus, Hybrid Computation (Wiley, New York, 1968).

Björck, Å

G. Dahlquist and Å Björck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 389–392.

Burden, R. L.

R. L. Burden, J. D. Faires, and A. C. Reynolds, Numerical Analysis (Prindle, Weber and Schmidt, Boston, 1978), p. 525.

Cederquist, J.

J. Cederquist and S. H. Lee, “The Use of Feedback in Optical Information Processing,” Appl. Phys. 18, 311–319 (1979).
[CrossRef]

J. Cederquist and S. H. Lee, “The Solution of Partial Differential Equations with Time-Varying Inhomogeneous Terms Using Coherent Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 181–184, IEEE Order No.CH1305-2/79/0000-0181.

Christensen, D. A.

A. Riazi, O. P. Gandhi, and D. A. Christensen, “Imaging Characteristics of a Confocal Cavity,” Opt. Commun. 28, 163–165 (1979).
[CrossRef]

Clark, J. A.

J. A. Clark, “Imaging Through a Confocal Optical Interference Filter,” Opt. Commun. 5, 163–167 (1972).
[CrossRef]

Dahlquist, G.

G. Dahlquist and Å Björck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 389–392.

Faires, J. D.

R. L. Burden, J. D. Faires, and A. C. Reynolds, Numerical Analysis (Prindle, Weber and Schmidt, Boston, 1978), p. 525.

Gandhi, O. P.

A. Riazi, O. P. Gandhi, and D. A. Christensen, “Imaging Characteristics of a Confocal Cavity,” Opt. Commun. 28, 163–165 (1979).
[CrossRef]

Götz, J.

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, “Solving Differential Equations with TV-Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

Häusler, G.

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, “Solving Differential Equations with TV-Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

Hercher, M.

Hockney, R. W.

R. W. Hockney, Technical Report 53, Institute for Plasma Research, Stanford University, 1966 (unpublished).

Jablonowski, D.

D. Jablonowski and S. H. Lee, “A Coherent Optical Feedback System for Optical Information Processing,” Appl. Phys. 8, 51–58 (1975).
[CrossRef]

Karplus, W. J.

G. A. Bekey and W. J. Karplus, Hybrid Computation (Wiley, New York, 1968).

Kurashov, V. N.

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, “Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System,” Opt. Spectrosc. 40, 420–422 (1976).

Kuzmenko, A. V.

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, “Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System,” Opt. Spectrosc. 40, 420–422 (1976).

Lee, S. H.

J. R. Leger and S. H. Lee, “Coherent Optical Implementation of Generalized Two-Dimensional Transforms,” Opt. Eng. 18, 518–523 (1979).
[CrossRef]

J. Cederquist and S. H. Lee, “The Use of Feedback in Optical Information Processing,” Appl. Phys. 18, 311–319 (1979).
[CrossRef]

R. Akins and S. H. Lee, “Coherent Optical Image Amplification by an Injection-Locked Dye Amplifier at 632.8 nm,” Appl. Phys. Lett. 35, 660–663 (1979).
[CrossRef]

J. Cederquist and S. H. Lee, “The Solution of Partial Differential Equations with Time-Varying Inhomogeneous Terms Using Coherent Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 181–184, IEEE Order No.CH1305-2/79/0000-0181.

D. Jablonowski and S. H. Lee, “A Coherent Optical Feedback System for Optical Information Processing,” Appl. Phys. 8, 51–58 (1975).
[CrossRef]

R. Akins, R. Athale, and S. H. Lee, “Feedback in Analog and Digital Optical Image Processing,” Opt. Eng. (to be published).

Leger, J. R.

J. R. Leger and S. H. Lee, “Coherent Optical Implementation of Generalized Two-Dimensional Transforms,” Opt. Eng. 18, 518–523 (1979).
[CrossRef]

Lohmann, A. W.

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, “Solving Differential Equations with TV-Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

Lotsch, H. K. V.

H. K. V. Lotsch, “The Confocal Resonator System,” Optik 30, 1–14, 181–201, 217–233, and 563–576 (1969).

McGrew, S. P.

S. P. McGrew, in New Concepts and Technologies in Parallel Information Processing, edited by E. R. Caianiello (Noordhoff, Leyden, 1975), pp. 55–73.
[CrossRef]

Nakhodkin, N. G.

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, “Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System,” Opt. Spectrosc. 40, 420–422 (1976).

Pochernyaev, I. M.

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, “Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System,” Opt. Spectrosc. 40, 420–422 (1976).

Reynolds, A. C.

R. L. Burden, J. D. Faires, and A. C. Reynolds, Numerical Analysis (Prindle, Weber and Schmidt, Boston, 1978), p. 525.

Riazi, A.

A. Riazi, O. P. Gandhi, and D. A. Christensen, “Imaging Characteristics of a Confocal Cavity,” Opt. Commun. 28, 163–165 (1979).
[CrossRef]

Simon, M.

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, “Solving Differential Equations with TV-Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

Appl. Opt. (1)

Appl. Phys. (2)

J. Cederquist and S. H. Lee, “The Use of Feedback in Optical Information Processing,” Appl. Phys. 18, 311–319 (1979).
[CrossRef]

D. Jablonowski and S. H. Lee, “A Coherent Optical Feedback System for Optical Information Processing,” Appl. Phys. 8, 51–58 (1975).
[CrossRef]

Appl. Phys. Lett. (1)

R. Akins and S. H. Lee, “Coherent Optical Image Amplification by an Injection-Locked Dye Amplifier at 632.8 nm,” Appl. Phys. Lett. 35, 660–663 (1979).
[CrossRef]

Opt. Commun. (2)

J. A. Clark, “Imaging Through a Confocal Optical Interference Filter,” Opt. Commun. 5, 163–167 (1972).
[CrossRef]

A. Riazi, O. P. Gandhi, and D. A. Christensen, “Imaging Characteristics of a Confocal Cavity,” Opt. Commun. 28, 163–165 (1979).
[CrossRef]

Opt. Eng. (1)

J. R. Leger and S. H. Lee, “Coherent Optical Implementation of Generalized Two-Dimensional Transforms,” Opt. Eng. 18, 518–523 (1979).
[CrossRef]

Opt. Spectrosc. (1)

N. G. Nakhodkin, A. V. Kuzmenko, V. N. Kurashov, and I. M. Pochernyaev, “Operator Method for Solving Differential Equations in a Digital Computer-Coherent Optical Device Hybrid System,” Opt. Spectrosc. 40, 420–422 (1976).

Optik (1)

H. K. V. Lotsch, “The Confocal Resonator System,” Optik 30, 1–14, 181–201, 217–233, and 563–576 (1969).

Proc. Int. Opt. Comput. Conf., London (2)

J. Götz, G. Häusler, A. W. Lohmann, and M. Simon, “Solving Differential Equations with TV-Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 179–180, IEEE Order No. CH1305-2/79/0000-0179.

J. Cederquist and S. H. Lee, “The Solution of Partial Differential Equations with Time-Varying Inhomogeneous Terms Using Coherent Optical Feedback,” Proc. Int. Opt. Comput. Conf., London, 1978, pp. 181–184, IEEE Order No.CH1305-2/79/0000-0181.

Other (7)

G. Dahlquist and Å Björck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 389–392.

R. L. Burden, J. D. Faires, and A. C. Reynolds, Numerical Analysis (Prindle, Weber and Schmidt, Boston, 1978), p. 525.

Reference 15, pp. 508–509.

R. W. Hockney, Technical Report 53, Institute for Plasma Research, Stanford University, 1966 (unpublished).

R. Akins, R. Athale, and S. H. Lee, “Feedback in Analog and Digital Optical Image Processing,” Opt. Eng. (to be published).

G. A. Bekey and W. J. Karplus, Hybrid Computation (Wiley, New York, 1968).

S. P. McGrew, in New Concepts and Technologies in Parallel Information Processing, edited by E. R. Caianiello (Noordhoff, Leyden, 1975), pp. 55–73.
[CrossRef]

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Figures (16)

FIG. 1
FIG. 1

(a) Confocal feedback system showing mirrors M1 and M2 and imaging lenses L1, L2, and L3. (b) A detailed drawing of the confocal Fabry–Perot interferometer showing the location of the spatial frequency filters f and g .

FIG. 2
FIG. 2

Photograph of the confocal Fabry–Perot showing the invar mirror mounting system. The spatial filters are held in the liquid gate in the center. The mirror on the right is mounted on a piezoelectric translator.

FIG. 3
FIG. 3

Block diagram of the image analysis system.

FIG. 4
FIG. 4

Example of a spline-fitted calibration curve for converting eight-bit digitized video values into relative optical amplitudes.

FIG. 5
FIG. 5

Amplitude transmittance of a laser-scanner-generated test pattern (as measured by the image analysis system) versus the desired transmittance. The rms deviation of the data points from the straight line is 2% of the maximum value.

FIG. 6
FIG. 6

Spatial filters for the solution of Poisson’s equation. On the left is f , the forward path filter, on the right, g , the feedback path filter. Coordinate axes are provided for reference.

FIG. 7
FIG. 7

Experimentally measured ratio of the transfer function h with positive feedback to that obtained without feedback versus spatial frequency (solid line). The dynamic range of h is increased at low frequencies by a factor of 2.86. Also shown is the curve that would have been found if feedback had had no effect (dotted line).

FIG. 8
FIG. 8

Optical solutions of Poisson’s equation, (a)–(c) Charge distributions ρ(x,y) used as optical inputs. (d)–(f) Corresponding optical outputs representing potentials ϕ(x,y). (g)–(i) Corresponding pseudo–three-dimensional plots of outputs. Coordinate axes (middle) are shown for reference.

FIG. 9
FIG. 9

Negative of the finite difference Laplacian of an optically generated potential ϕ(x,y) (upper plot) compared with the optically input charge distribution ρ(x,y) (lower plot). The rms difference is 8% of the maximum value of ρ.

FIG. 10
FIG. 10

Spatial filters for the solution of the wave equation. On the left is f , the forward path filter, on the right g , the feedpack path filter. See Fig. 6 for coordinate axes.

FIG. 11
FIG. 11

Optical solutions of the wave equation. (a) Solution for the boundary condition ∂ϕ/∂tt=0 = tri(x). (b) Solution with the boundary condition on ϕ(x,0) given along the x axis. In each case, the upper plot is the optical solution, the lower plot a numerical solution for comparison. See Fig. 8 for coordinate axes. The rms differences are 8% and 15%, respectively.

FIG. 12
FIG. 12

Spatial filters for the solution of a parabolic equation. On the left is f , the forward path filter; on the right, g , the feedback path filter. See Fig. 6 for coordinate axes.

FIG. 13
FIG. 13

Absolute value of the ratio of the transfer function h ( 0 , v ) to h ( 0 , 0 ) versus spatial frequency v for positive, imaginary, and negative feedback (β = 0, π/2, and π). β was varied using the piezoelectric translator. For the solution of the parabolic equation, β = π/2.

FIG. 14
FIG. 14

Optical solution of a parabolic equation. The boundary condition on ϕ(x,0) is given along the x axis. The upper plot is the optical solution, the lower plot a numerical solution for comparison. See Fig. 8 for coordinate axes. The rms difference is 8%.

FIG. 15
FIG. 15

Path of a single ray traveling at angle θ through a confocal Fabry–Perot interferometer.

FIG. 16
FIG. 16

Error Δβ in the phase of the feedback β versus the dimensionless distance x of the ray from the optical axis for exactly confocal spacing of the mirrors ( = 0) and for the experimentally used spacing ( = −117 μm). This choice for reduces the error in β to 5% of 2π.

Equations (38)

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ã o = t 2 r 2 t p 3 f ã i + r 4 t p 4 f g e i β ã o ,
h = t 2 r 2 t p 3 f / ( 1 r 4 t p 4 f g e i β ) f / ( 1 t c f g e i β ) ,
h max = t 2 r 2 t p 3 / ( 1 t c ) .
h min = t 2 r 2 t p 3 / d ( 1 + t c / d ) .
d CFS = d ( 1 + t c / d ) / ( 1 t c ) .
N = ( ( 2 δ / λ R ) { [ δ 2 / 4 + ( 8 p λ R 3 ) 1 / 2 ] 1 / 2 δ } ) 2
L { ϕ ( x , y ) } = b ( x , y ) ,
ϕ ( u , υ ) = b ( u , υ ) / p ( u , υ ) ,
ã o ( u , υ ) = h ( u , υ ) ã i ( u , υ ) ,
h ( u , υ ) 1 / p ( u , υ )
f 1 t c f g e i β 1 p .
0 f x e i ( x u + y υ ) d x d y = [ f ( x , y ) e i x u | 0 i u 0 f ( x , y ) e i x u d x ] e i y υ d y = f ( 0 , y ) e i y υ d y i u f ( u , υ ) ,
2 ϕ ( x , y ) x 2 + 2 ϕ ( x , y ) y 2 = ρ ( x , y ) .
ϕ ( u , υ ) = p ( u , υ ) u 2 + υ 2 .
f = 1 / ( u 2 + υ 2 ) , g = 0 , β = 0 for u 2 + υ 2 1
f = 1 , g = 1 ( u 2 + υ 2 ) , β = 0 for u 2 + υ 2 < 1 .
2 ϕ ( x , t ) x 2 2 ϕ ( x , t ) t 2 = 0
ϕ ( x , o ) = b 1 ( x ) , ϕ t | t = 0 = b 2 ( x ) .
0 = u 2 ϕ ( ϕ t e i t υ | 0 i υ ϕ e i t υ | 0 υ 2 0 ϕ e i t υ d t ) e i x u d x = u 2 ϕ ( b 2 ( x ) + i υ b 1 ( x ) υ 2 0 ϕ e i t υ d t ) e i x u d x = u 2 ϕ ( u , υ ) [ b 2 ( u ) + i υ b 1 ( u ) υ 2 ϕ ( u , υ ) ] ,
ϕ = i υ b 1 ( u ) + b 2 ( u ) u 2 υ 2 .
f = 1 / ( u 2 υ 2 ) , g = 0 , β = 0 for | u 2 υ 2 | 1 , f = 1 , g = 1 ( u 2 υ 2 ) , β = 0 for 0 u 2 υ 2 < 1 ,
f = 1 , g = [ 1 ( υ 2 u 2 ) ] , β = 0 for 1 < u 2 υ 2 < 0 .
2 ϕ ( x , t ) x 2 ϕ ( x , t ) = ϕ ( x , t ) t
ϕ ( x , 0 ) = b ( x ) .
u 2 ϕ ϕ = ( ϕ e i t υ | 0 i υ 0 ϕ e i t υ d t ) e i x u d x = ( b ( x ) i υ 0 ϕ e i t υ d t ) e i x u d x = b ( u ) i υ ϕ ,
ϕ = b ( u ) 1 + u 2 i υ .
f = 1 / ( 1 + u 2 ) , g = υ / t c , β = π / 2 .
l = 4 ( R + ) ρ 1 2 ρ 2 2 R 3 2 ( ρ 1 2 + ρ 2 2 ) R 2 + higher-order terms ,
δ = ρ 2 ρ 1 , x = ρ 1 / ( ρ 2 ρ 1 ) ,
ρ 1 = x δ , ρ 2 = ( x + 1 ) δ .
l = 4 ( R + ) x 2 ( x + 1 ) 2 δ 4 R 3 2 δ 2 ( 2 x 2 + 2 x + 1 ) R 2 = 4 ( R + ) 2 δ 2 R 2 δ 4 R 3 [ x ( x + 1 ) ( x 2 + x + 4 R δ 2 ) ] .
Δ β = 2 π δ 4 λ R 3 [ x ( x + 1 ) ( x 2 + x + 4 R δ 2 ) ] .
x max = 1 2 + ( 1 4 4 R δ 2 ) 1 / 2 .
Δ β min = 2 π λ ( 4 2 R ) .
Δ β min / 2 = 2 π p
= ( p λ R / 2 ) 1 / 2 .
ρ max x max δ = δ / 2 + [ δ 2 / 4 + ( 8 p λ R 3 ) 1 / 2 ] 1 / 2
N = [ ( ρ max ρ min ) 2 δ / λ R ] 2 = ( { δ + [ δ 2 / 4 + ( 8 p R 3 ) 1 / 2 ] 1 / 2 } 2 δ / λ R ) 2