Abstract

The fields diffracted by planar one- or two-dimensional periodic objects, and in particular their Fourier and Fresnel self-images, can be computed with the aid of a ray-tracing technique based on the Fermat principle. This method (geometrical self-imaging) yields accurate results for any numerical aperture and image field. An analytical study of the image formation, carried out in the fourth-order approximation for the phase, leads to the definition of self-imaging aberrations. These aberrations are strongly dependent on spatial frequency and render the well-known relationships derived by Rayleigh for the location and magnification of self-images approximate at best. The aberrations can be described graphically by a phase diagram and a magnification diagram, which permit interpretation of the properties of high-aperture, large-field self-images and the prediction of optimal imaging conditions. In the case of large magnifications (100× and larger), we present a simple method to eliminate all fourth-order aberrations completely and even sixth-order ones partially. This method consists of introducing a compensating spherical aberration to the incident wave, e.g., by the insertion of a glass plate of appropriate index and thickness just before the object. Thus object spatial frequencies up to about 800 mm−1 can be imaged almost without aberration for several image periods.

© 1983 Optical Society of America

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References

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  1. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 403–405 (1836), Sec. 2.
  2. J. M. Cowley and A. F. Moodie, “Fourier images,” Parts I–III, Proc. Phys. Soc. B 70, 486–513 (1956).
    [CrossRef]
  3. M. Fujiwara, “Effects of spatial coherence on Fourier imaging of a periodic object,” Opt. Acta 21, 861–869 (1974).
    [CrossRef]
  4. S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta,  26, 439–446 (1979).
    [CrossRef]
  5. K. Patorski and G. Parfjanowicz, “Self-imaging phenomenon of a sinusoidal complex object,” Opt. Acta 28, 357–367 (1981).
    [CrossRef]
  6. J. J. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am 55, 373–381 (1965).
    [CrossRef]
  7. E. A. Hiedemann and M. A. Breazeale, “Secondary interference in the Fresnel zone of gratings,” J. Opt. Soc. Am. 49, 372–375 (1959).
    [CrossRef]
  8. A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971); “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
    [CrossRef]
  9. H. Damman, G. Groh, and M. Kock, “Restoration of faulty images of periodic objects by means of self imaging,” Appl. Opt. 10, 1454–1455 (1971).
    [CrossRef]
  10. A. Kalestynski and B. Smolinska, “Self restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
    [CrossRef]
  11. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416–419 (1973).
    [CrossRef]
  12. J. K. T. Eu and A. W. Lohmann, “Spatial filtering effects by means of hologram copying,” Opt. Commun. 8, 176–182 (1973).
    [CrossRef]
  13. J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
    [CrossRef]
  14. K. Patorski, S. Yokozeki, and T. Suzuki, “Image subtraction using Fourier imaging phenomenon,” Nouv. Rev. Opt. 6, 25–31 (1975).
    [CrossRef]
  15. D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1980–1982 (1971).
    [CrossRef]
  16. R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
    [CrossRef]
  17. W. D. Montgomery, “Self imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967); “Algebraic formulation of diffraction applied to self-imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968).
    [CrossRef]
  18. D. Joyeux and Y. Cohen-Sabban, “High magnification self-imaging,” Appl. Opt. 21, 625–627 (1982).
    [CrossRef] [PubMed]
  19. A. Rubinowicz, “The Miyamoto–Wolf diffraction wave,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1965), Vol. IV, pp. 199–240.
    [CrossRef]
  20. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  21. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 401–407.

1982 (1)

1981 (1)

K. Patorski and G. Parfjanowicz, “Self-imaging phenomenon of a sinusoidal complex object,” Opt. Acta 28, 357–367 (1981).
[CrossRef]

1979 (1)

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta,  26, 439–446 (1979).
[CrossRef]

1978 (1)

A. Kalestynski and B. Smolinska, “Self restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

1975 (1)

K. Patorski, S. Yokozeki, and T. Suzuki, “Image subtraction using Fourier imaging phenomenon,” Nouv. Rev. Opt. 6, 25–31 (1975).
[CrossRef]

1974 (1)

M. Fujiwara, “Effects of spatial coherence on Fourier imaging of a periodic object,” Opt. Acta 21, 861–869 (1974).
[CrossRef]

1973 (3)

O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416–419 (1973).
[CrossRef]

J. K. T. Eu and A. W. Lohmann, “Spatial filtering effects by means of hologram copying,” Opt. Commun. 8, 176–182 (1973).
[CrossRef]

J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
[CrossRef]

1971 (3)

D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1980–1982 (1971).
[CrossRef]

A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971); “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

H. Damman, G. Groh, and M. Kock, “Restoration of faulty images of periodic objects by means of self imaging,” Appl. Opt. 10, 1454–1455 (1971).
[CrossRef]

1969 (1)

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
[CrossRef]

1967 (1)

1965 (1)

J. J. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am 55, 373–381 (1965).
[CrossRef]

1962 (1)

1959 (1)

1956 (1)

J. M. Cowley and A. F. Moodie, “Fourier images,” Parts I–III, Proc. Phys. Soc. B 70, 486–513 (1956).
[CrossRef]

1836 (1)

F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 403–405 (1836), Sec. 2.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 401–407.

Breazeale, M. A.

Bryngdahl, O.

Cohen-Sabban, Y.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier images,” Parts I–III, Proc. Phys. Soc. B 70, 486–513 (1956).
[CrossRef]

Damman, H.

Edgar, R. F.

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
[CrossRef]

Eu, J. K. T.

J. K. T. Eu and A. W. Lohmann, “Spatial filtering effects by means of hologram copying,” Opt. Commun. 8, 176–182 (1973).
[CrossRef]

Eu, J. T. K.

J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
[CrossRef]

Fujiwara, M.

M. Fujiwara, “Effects of spatial coherence on Fourier imaging of a periodic object,” Opt. Acta 21, 861–869 (1974).
[CrossRef]

Groh, G.

Hiedemann, E. A.

Joyeux, D.

Kalestynski, A.

A. Kalestynski and B. Smolinska, “Self restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

Keller, J. B.

Kock, M.

Liu, C. Y. C.

J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
[CrossRef]

Lohmann, A. W.

J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
[CrossRef]

J. K. T. Eu and A. W. Lohmann, “Spatial filtering effects by means of hologram copying,” Opt. Commun. 8, 176–182 (1973).
[CrossRef]

A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971); “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

Montgomery, W. D.

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “Fourier images,” Parts I–III, Proc. Phys. Soc. B 70, 486–513 (1956).
[CrossRef]

Parfjanowicz, G.

K. Patorski and G. Parfjanowicz, “Self-imaging phenomenon of a sinusoidal complex object,” Opt. Acta 28, 357–367 (1981).
[CrossRef]

Patorski, K.

K. Patorski and G. Parfjanowicz, “Self-imaging phenomenon of a sinusoidal complex object,” Opt. Acta 28, 357–367 (1981).
[CrossRef]

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta,  26, 439–446 (1979).
[CrossRef]

K. Patorski, S. Yokozeki, and T. Suzuki, “Image subtraction using Fourier imaging phenomenon,” Nouv. Rev. Opt. 6, 25–31 (1975).
[CrossRef]

Rubinowicz, A.

A. Rubinowicz, “The Miyamoto–Wolf diffraction wave,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1965), Vol. IV, pp. 199–240.
[CrossRef]

Silva, D. E.

A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971); “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1980–1982 (1971).
[CrossRef]

Smolinska, B.

A. Kalestynski and B. Smolinska, “Self restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

Suzuki, T.

K. Patorski, S. Yokozeki, and T. Suzuki, “Image subtraction using Fourier imaging phenomenon,” Nouv. Rev. Opt. 6, 25–31 (1975).
[CrossRef]

Szapiel, S.

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta,  26, 439–446 (1979).
[CrossRef]

Talbot, F.

F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 403–405 (1836), Sec. 2.

Winthrop, J. J.

J. J. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am 55, 373–381 (1965).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 401–407.

Worthington, C. R.

J. J. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am 55, 373–381 (1965).
[CrossRef]

Yokozeki, S.

K. Patorski, S. Yokozeki, and T. Suzuki, “Image subtraction using Fourier imaging phenomenon,” Nouv. Rev. Opt. 6, 25–31 (1975).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am (1)

J. J. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am 55, 373–381 (1965).
[CrossRef]

J. Opt. Soc. Am. (4)

Nouv. Rev. Opt. (1)

K. Patorski, S. Yokozeki, and T. Suzuki, “Image subtraction using Fourier imaging phenomenon,” Nouv. Rev. Opt. 6, 25–31 (1975).
[CrossRef]

Opt. Acta (5)

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
[CrossRef]

A. Kalestynski and B. Smolinska, “Self restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

M. Fujiwara, “Effects of spatial coherence on Fourier imaging of a periodic object,” Opt. Acta 21, 861–869 (1974).
[CrossRef]

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta,  26, 439–446 (1979).
[CrossRef]

K. Patorski and G. Parfjanowicz, “Self-imaging phenomenon of a sinusoidal complex object,” Opt. Acta 28, 357–367 (1981).
[CrossRef]

Opt. Commun. (3)

A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971); “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

J. K. T. Eu and A. W. Lohmann, “Spatial filtering effects by means of hologram copying,” Opt. Commun. 8, 176–182 (1973).
[CrossRef]

J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
[CrossRef]

Philos. Mag. (1)

F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 403–405 (1836), Sec. 2.

Proc. Phys. Soc. B (1)

J. M. Cowley and A. F. Moodie, “Fourier images,” Parts I–III, Proc. Phys. Soc. B 70, 486–513 (1956).
[CrossRef]

Other (2)

A. Rubinowicz, “The Miyamoto–Wolf diffraction wave,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1965), Vol. IV, pp. 199–240.
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 401–407.

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Figures (13)

Fig. 1
Fig. 1

Optical setup for observing magnified self-images of a periodic thin object.

Fig. 2
Fig. 2

The self-imaging diagrams in the paraxial approximation: (a) magnification and (b) phase diagram. Each line in (b) shows the variation of the propagation phase at a given observation point versus the source distance d for a given object harmonic. The origin of the abscissa is d = d0, the in-focus Rayleigh position. The general aspect of these paraxial diagrams is completely independent of the actual parameter values.

Fig. 3
Fig. 3

Ray-tracing geometry. (a) The field at an observation point R is the interference sum of contributions carried by a finite number of rays (one for each nonevanescent order). (b) Notation: rp and lp are optical paths to and from the object, respectively, along a ray contained in the xz plane. θ0 and θ are the ray-to-normal angles before and after diffraction.

Fig. 4
Fig. 4

The geometrical approximation: the diffracted beams propagate undisturbed within the corresponding geometrical projection of the pupil and do not contribute to points outside it.

Fig. 5
Fig. 5

Nonparaxial self-imaging diagrams computed by GSI: (a) magnification and (b) phase. The diagrams were computed using the following parameter values: object period a = 10 μm, observation distance z = 100 mm, image order N = 4, λ = 0.6328 μm. Figure 5 is to be compared with Fig. 2.

Fig. 6
Fig. 6

Same as Fig. 5, except for the use of a glass correction plate. Its optical thickness is computed from Eq. (24) (fourth-order correction). Thickness e = 1.71, index n = 1.511. Note the change in d scale between Figs. 5(b) and 6(b). Comparison of Figs. 5 and 6 shows a slight overcorrection of the initial phase aberrations.

Fig. 7
Fig. 7

Computed photometric section of self-images of a Dirac comb (central and first off-axis image periods). Object period a = 10 μm, pupil size 2w = 56 periods, λ = 0.6328 μm, observation distance z = 100 mm, source-to-object distance d0 is determined by paraxial conjugation formulas [Eq. (4) with N = 4]. (a) Exact computation (by Kirchhoff integral), (b) computation by GSI, (c) truncated Fourier development of the Dirac comb (orders −6 to +6, corresponding to the pupil size).

Fig. 8
Fig. 8

Computed photometric sections of self-images of the same object as in Fig. 7 but with an optimal source-to-object distance, dopt = d0 − 6 μm determined by the phase diagram in Fig. 5(b). (a) Exact, (b) GSI, (c) truncated Fourier development.

Fig. 9
Fig. 9

Self-images of a binary grating with white/period ratio 4/10. Parameter values are the same as in Fig. 7 unless specified. (a) Truncated Fourier development (orders −6 to +6). (b) Self-images of (a) computed by GSI at optimal source distance (dopt = d0 −6 μm). (c) Self-images of same object, produced with a glass correcting plate (thickness e = 1.71 mm, index n = 1.511) at optimal source distance found from Fig. 6(b) (dopt = d0 + 1 μm).

Fig. 10
Fig. 10

(a) Magnification and (b) phase diagrams computed with optimal glass-plate thickness (e = 1.59 mm, n = 1.511). Other parameters are the same as in Fig. 6. To be compared with Fig. 6.

Fig. 11
Fig. 11

Experimental and computed photometric sections of self-images. The object is a binary grating with period a = 10 μm and white/period ratio 0.308, numerical aperture 0.55 (orders −8 to +8), z = 200 mm, N = 4, d0 = 634 μm, λ = 0.6328 μm. Experiment: (a) no correction, d = d0 − 1 μm; (c) with a glass correcting plate (e = 1.61 mm, n = 1.511), d = d0 − 2 μm. Theory: (b) same as (a), (d) same as (c).

Fig. 12
Fig. 12

Diffraction by a 2-D periodic object. The ray travels in a straight line from P1(x1, y1, z1) to the object and therefrom to P2(x2, y2, z2). The relationship between the directions of ê1 and ê2 can be found from the principle of Fermat. These relationships are reduced to the simple grating equation under certain restrictive conditions.

Fig. 13
Fig. 13

Self-imaging with a glass correcting plate (thickness e, index n). The parameters η and x can be determined numerically as functions of x1 by using the stationarity of the optical path function OP(η, x, x1).

Equations (61)

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t ( x ) = p = - C p t p ( x ) , t p ( x ) = exp ( 2 π i p x a ) .
u p ( R ) = B exp [ - i π λ p 2 ( 1 / z + 1 / d ) a 2 ] exp ( 2 π i x 1 p G 0 a ) ,
G 0 = 1 + z / d .
π λ p 2 ( 1 / z + 1 / d ) a 2 = π m p ,             m p = any integer .
1 d + 1 z = λ N a 2 ,             N = m 1 = any integer .
sin θ 0 + p λ / a = sin θ
ϕ ( x , 0 + ) = ϕ ( Q ) = ϕ ( x , 0 - ) + 2 π p x a = 2 π λ ( r p + λ x p a ) .
x 1 = x + z tan θ ,
ϕ ( R ) = ϕ ( Q ) + 2 π λ l p = 2 π λ { ( x 2 + d 2 ) 1 / 2 + λ p x a + [ ( x 1 - x ) 2 + z 2 ) 1 / 2 } ,
u p ( R ) = u 0 j p ( z , d , θ 0 , θ ) ,
j p ( z , d , θ 0 , θ p ) = [ ( d cos 2 θ 0 + cos θ 0 cos 3 θ ) × ( d cos θ 0 + z cos θ ) ] - 1 / 2 1 d + z .
- 1 sin θ p 1.
u ( R ) = p = p 1 p 2 u p ( R ) = u 0 p = p 1 p 2 c p j p × exp [ 2 π i λ ( l p + r p + p λ x a ) ] .
ϕ p ( R ) = 2 π λ { ( d 2 + x 2 ) 1 / 2 + p λ a x + [ z 2 + ( x 1 - x ) 2 ] 1 / 2 } .
ϕ p ( R ) = 2 π λ { ( z + d ) + x 1 2 2 ( z + d ) [ 1 - 1 4 ( x 1 z ) 2 ] - 1 2 ( 1 z + 1 d ) - 1 ( p λ a ) 2 [ 1 + 1 4 ( p λ a ) 2 + 3 4 ( x 1 z ) 2 ] + ( 1 z + 1 d ) - 1 p λ a x 1 z [ 1 + 1 4 ( p λ a ) ] } + δ ϕ .
u ( R ) = p = p 1 p 2 C p u p ( R ) = u 0 exp [ i ϕ 0 ( R ) ] p = p 1 p 2 C p × exp ( 2 π i x 1 p G a ) exp [ i ϕ p propag ( R ) ] ,
ϕ p ( R ) = ϕ 0 ( R ) + ϕ p image ( R ) + ϕ p propag ( R ) .
ϕ p propag = π m p             for all p .
ϕ p ( R ) image = ( 1 z + 1 d ) - 1 p λ a x 1 z [ 1 + 1 4 ( p λ a ) 2 ]
ϕ p propag ( R ) = π λ p 2 ( 1 z + 1 d ) a 2 [ 1 + 1 4 ( p λ a ) 2 + 3 2 ( x 1 z ) 2 ] .
G = G 0 1 + 1 4 ( p λ a ) 2 + .
1 z + 1 d = λ N a 2 [ 1 + 1 4 ( p λ a ) 2 + 3 2 ( x 1 z ) 2 ] ,
ϕ ( 4 ) = - 2 π λ [ d 8 tan 4 θ 0 + z 8 tan 4 θ ] ,
tan θ 0 = x d x 1 z + d - z z + d p λ a , tan θ = x 1 - x z x 1 z + d + d z + d p λ a .
x 1 = 0 { ( d tan 4 θ 0 / 8 ) p = d ( z z + d ) 4 ( λ a ) 4 = z 4 d 3 1 G 0 4 ( λ a ) 4 ( z tan 4 θ / 8 ) p = z ( d z + d ) 4 ( λ a ) 4 = z ( 1 G 0 ) 4 ( λ a ) 4 .
δ ϕ plate = + 2 π λ e 8 n 2 - 1 n 3 tan 4 θ 0 .
Δ d - e n 2 - 1 n 3 = 0.
ϕ p propag ( R ) = - 1 2 2 π λ ( 1 z + 1 d ) - 1 ( p λ a ) 2 × { 1 + ( z z + d ) 3 [ 1 4 ( Δ d + ( d z ) 3 ) ( p λ a ) 2 + 3 2 ( Δ d + d z ) ( x 1 z ) 2 ] } ,
ϕ p image ( R ) = 2 π λ ( 1 z + 1 d ) - 1 p λ a x 1 z × { 1 + ( z z + d ) 3 [ 1 2 ( Δ d - ( d z ) 3 ) ( p λ a ) 2 + 1 2 ( Δ d - 1 ) ( x 1 z ) 2 ] } .
1 2 ( 1 z + 1 d ) - 1 p λ a ( x 1 z ) 3 λ 10
- d 8 ( 1 z + 1 d ) - 1 ( p λ a ) 4 1.7 λ
t ( x ) = p 2 N C p exp ( 2 π i x p a ) ,             N = set of all integer numbers .
d / z - 1
u ( x , y , 0 - ) = A ( x , y ) exp [ i ϕ ( x , y , 0 - ) ] ,
t p q ( x , y ) = exp [ 2 π i ( p x a + q y b ) ] .
u ( x , y , 0 + ) = u ( x , y , 0 - ) t p q ( x , y ) = A ( x , y ) exp [ i ϕ p q ( x , y , 0 + ) ,
ϕ p q ( x , y , 0 + ) = ϕ ( x , y , 0 - ) + 2 π ( p x a + q y b ) .
ϕ ( x , y , 0 - ) = ϕ ( x 1 , y 1 , z 1 ) + k r = ϕ ( x 1 , y 1 , z 1 ) + 2 π λ [ ( x 1 - x ) 2 + ( y 1 - y ) 2 + z 1 2 ] 1 / 2 , ϕ p q ( x 2 , y 2 , z 2 ) = ϕ p q ( x , y , 0 + ) + k l = ϕ p q ( x , y , 0 + ) + 2 π λ [ ( x 2 - x ) 2 + ( y 2 - y ) 2 + z 2 2 ] 1 / 2 .
0 = ϕ p q x = 2 π λ { x - x 1 [ ( x - x 1 ) 2 + ( y - y 1 ) 2 + z 1 2 ] 1 / 2 + p λ a - x 2 - x [ ( x 2 - x ) 2 + ( y 2 - y ) 2 + z 2 2 ] 1 / 2 } .
0 = l 1 x + p λ a - l 2 x = sin α 1 cos β 1 + p λ a - sin α 2 cos β 2 ,
0 = l 1 y - q λ b - l 2 y = sin α 1 sin β 1 + q λ b - sin α 2 sin β 2 .
sin 2 α 2 = sin 2 α 1 + ( p λ a ) 2 + ( q λ b ) 2 + 2 sin α 1 ( cos β 1 p λ a + sin β 1 q λ b ) , tan β 2 = sin α 1 sin β 1 + q λ / b sin α 1 cos β 1 + p λ / a .
b = { sin 2 α 2 = sin 2 α 1 + ( p λ a ) 2 + 2 sin α 1 cos β 1 p λ a tan β 2 = sin α 1 sin β 1 sin α 1 cos β 1 + p λ / a .
b = β 1 = 0 } { sin α 2 = sin α 1 + p λ a , β 2 = 0.
0 < a , b < β 1 = 0 } { sin 2 α 2 = ( sin α 1 + p λ a ) 2 + ( q λ b ) 2 , tan β 2 = q λ / b sin α 1 + p λ / a .
O P ( x 0 + Δ x , x 1 ) - O P ( x 0 , x 1 ) = 1 2 2 ( O P ) x 2 ( Δ x ) 2 + O [ ( Δ x ) 3 ] .
O P ( x , x 1 ) = ( d 2 + x 2 ) 1 / 2 + p λ a x + [ z 2 + ( x 1 - x ) 2 ] 1 / 2 .
( O P ) x = x ( x 2 + d 2 ) 1 / 2 + p λ a - x 1 - x [ ( x 1 - x ) 2 + z 2 ] 1 / 2 = sin θ 0 + p λ a - sin θ ,
2 ( O P ) x 2 = d 2 ( x 2 + d 2 ) 3 / 2 + z 2 [ ( x 1 - x ) 2 + z 2 ] 3 / 2 = cos 3 θ 0 d + cos 3 θ z 1 d + 1 z .
x ˜ 0 = x 1 G 0 - z G 0 p λ a = x 0 + Δ x , Δ x d 2 ( x ˜ 0 d ) 3 = d tan 3 θ 0 2 .
ϕ p ( R ) = 2 π λ [ O P ( x ˜ 0 , x 1 ) - d 8 ( x ˜ 0 d ) 6 d + z z ] + O [ ( x ˜ 0 d ) 6 ] .
O P ( x 0 , x 1 ) = d [ 1 + 1 2 ( x ˜ 0 d ) 2 - 1 8 ( x ˜ 0 d ) 4 + 1 16 ( x ˜ 0 d ) 6 ] + p λ a x ˜ 0 + z [ 1 + 1 2 ( x 1 - x ˜ 0 d ) 2 - 1 8 ( x 1 - x ˜ 0 d ) 4 + 1 16 ( x 1 - x ˜ 0 d ) 6 ] ,
ϕ p ( R ) = ϕ ( 0 ) + ϕ ( 2 ) + ϕ ( 4 ) + ϕ ( 6 ) .
ϕ ( 2 ) = 2 π λ ( d 2 tan 2 θ 0 + z 2 tan 2 θ + p λ a x ˜ 0 ) ,
ϕ ( 4 ) = 2 π λ ( - d 8 tan 4 θ 0 - z 8 tan 2 θ ) .
ϕ ( 6 ) = 2 π λ ( d 16 tan 6 θ 0 + z 16 tan 6 θ - d + z z tan 6 θ 0 ) 2 π 10 .
ϕ p ( R ) = 2 π λ { ( z + d ) + x 1 2 2 ( z + d ) - x 1 4 8 ( z + d ) 3 - 1 2 ( 1 z + 1 d ) - 1 ( p λ a ) 2 [ 1 + 1 4 d 3 + z 3 ( z + d ) 3 ( p λ a ) 2 + 3 2 z 2 ( z + d ) 2 ( x 1 z ) 2 ] + ( 1 z + 1 d ) - 1 × p λ a x 1 z [ 1 + 1 2 ( z - d ) z ( z + d ) 2 ( p λ a ) 2 ] + δ ϕ .
ϕ p ( R ) = 2 π λ { ( z + d ) + x 1 2 2 ( z + d ) [ 1 - 1 4 ( x 1 z ) 2 ] - 1 2 ( 1 z + 1 d ) - 1 ( p λ a ) 2 [ 1 + 1 4 ( p λ a ) 2 + 3 2 ( x 1 z ) 2 ] + ( 1 z + 1 d ) - 1 p λ a x 1 z [ 1 + 1 2 ( p λ a ) 2 ] } + δ ϕ .
O P ( η , x , x 1 ) = [ ( d - e / n ) 2 + η 2 ] 1 / 2 + n [ e 2 + ( x - η ) 2 ] 1 / 2 + p λ a x + [ z 2 + ( x 1 - x ) 2 ] 1 / 2 .
ϕ p ( R ) = 2 π λ { ( z + d ) + x 1 2 2 ( z + d ) [ 1 - 1 4 ( x 1 z ) 2 ] - 1 2 ( 1 z + 1 d ) - 1 ( p λ a ) 2 [ 1 + 1 4 ( Δ d + ( d z ) 3 ) × ( p λ a ) 2 + 3 4 ( Δ d + d z ) ( x 1 z ) 2 ] + ( 1 z + 1 d ) - 1 p λ a x 1 z [ 1 + 1 2 ( Δ d - ( d z ) 2 ) × ( p λ a ) 2 + 3 2 ( Δ - d ) ( x 1 z ) 2 ] } + δ ϕ .
Δ = d - e n 2 - 1 n 3 .