Abstract

The physical theory, of testing optics by the Ronchi sine grating interferometer is developed. In the experiment a sine grating splits the third-order wavefront of the zone plate under test into three displaced wavefronts, which then interfere with one another to form interference fringes. The intensity distribution of the interference pattern can be recorded for various positions of the sine grating. By subtracting the contribution of the inherent diffractive aberration of the sine grating, the value of the spherical aberration associated with the third-order wavefront of the zone plate under test can be determined. Theory and experiment are compared.

© 1990 Optical Society of America

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References

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  1. A. V. Baez, “Fresnel Zone Plate for Optical Image Formation Using Extreme Ultraviolet and Soft x Radiation,” J. Opt. Soc. Am. 51, 405–412 (1961).
    [CrossRef]
  2. H. H. Barrett, F. A. Horrigan, “Fresnel Zone Plate Image of Gamma Rays; Theory,” Appl. Opt. 12, 2686–2702 (1973).
    [CrossRef] [PubMed]
  3. M. J. Simpson, A. G. Michette, “Considerations of Zone Plate Optics for Soft X-Ray Microscopy,” Opt. Acta 31, 1417–1426 (1984).
    [CrossRef]
  4. T. Fujita, H. Nishihara, J. Koyama, “Blazed Gratings and Fresnel Lenses Fabricated by Electron-Beam Lithography,” Opt. Lett. 7, 578–580 (1982).
    [CrossRef] [PubMed]
  5. K. Kodate, T. Kamiya, Y. Okada, H. Takenaka, “Focusing Characteristics of High-Efficiency Fresnel Zone Plate Fabricated by Deep Ultraviolet Lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
    [CrossRef]
  6. V. Ronchi, “Forty Years of History of a Grating Interferometer,” Appl. Opt. 3, 437–451 (1964).
    [CrossRef]
  7. V. A. Komissaruk, “Investigation of Wave Front Aberration of Optical Systems Using Three-Beam Interference,” Opt. Spec-trosc. U.S.S.R. 16, 571–574 (1964).
  8. J. C. Wyant, “Double Frequency Grating Lateral Shear Interferometer,” Appl. Opt. 12, 2057–2060 (1973).
    [CrossRef] [PubMed]
  9. K. Patorski, “Talbot Interferometry with Increased Shear,” Appl. Opt. 24, 4448–4453 (1985).
    [CrossRef] [PubMed]
  10. K. Omura, T. Yatagai, “Phase Measuring Ronchi Test,” Appl. Opt. 27, 523–528 (1988).
    [CrossRef] [PubMed]
  11. J. A. Lin, J. M. Cowley, “Calibration of Operating Parameters for HB5 STEM Instrument,” Ultramicroscopy 19, 31–42 (1986).
    [CrossRef]
  12. J. A. Lin, J. Hsu, S. G. Shiue, “Quantitative Three-Beam Ronchi Test,” Appl. Opt. 29, 1912–1918 (1990).
    [CrossRef] [PubMed]
  13. R. N. Smartt, “Zone Plate Interferometer,” Appl. Opt. 13, 1093–1099 (1974).
    [CrossRef] [PubMed]
  14. C. Joenathan, V. Parthiban, R. S. Sirohi, “Shear Interferometry with Holographic Lenses,” Opt. Eng. 26, 359–364 (1987).
  15. A. W. Lohmann, “An Interferometer with a Zone Plate as Beam-Splitter,” Opt. Acta 32, 1465–1469 (1985).
    [CrossRef]
  16. N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A Dynamic Zone-Plate Interferometer for Measuring Aspherical Surfaces,” Opt. Commun. 54, 257–261 (1985).
    [CrossRef]
  17. R. W. Meier, “Magnification and Third-Order Aberrations in Holography,” J. Opt. Soc. Am. 55, 987–992 (1965).
  18. M. Young, “Zone Plates and Their Aberrations,” J. Opt. Soc. Am. 62, 972–976 (1972).
    [CrossRef]
  19. A. Engel, G. Herziger, “Computer Drawn Modulated Zone Plates,” Appl. Opt. 12, 471–479 (1973).
    [CrossRef] [PubMed]
  20. See R. W. Wood, Physical Optics (Macmillan, New York, 1934).
  21. K. Iizuka, Engineering Optics (Springer-Verlag, Berlin, 1985), pp. 99–102.
  22. J. E. Harvey, R. V. Shack, “Aberrations of Diffracted Wave Fields,” Appl. Opt. 17, 3003–3009 (1978).
    [CrossRef] [PubMed]
  23. Y. Cohen-Sabban, D. Joyeux, “Aberration-Free Nonparaxial Self-Imaging,” J. Opt. Soc. Am. 73, 707–719 (1983).
    [CrossRef]
  24. C. Mory, C. Colliex, J. M. Cowley, “Optimum Defocus For STEM Imaging and Microanalysis,” Ultramicroscopy 21, 171–178 (1987).
    [CrossRef]

1990 (1)

1988 (1)

1987 (2)

C. Joenathan, V. Parthiban, R. S. Sirohi, “Shear Interferometry with Holographic Lenses,” Opt. Eng. 26, 359–364 (1987).

C. Mory, C. Colliex, J. M. Cowley, “Optimum Defocus For STEM Imaging and Microanalysis,” Ultramicroscopy 21, 171–178 (1987).
[CrossRef]

1986 (2)

J. A. Lin, J. M. Cowley, “Calibration of Operating Parameters for HB5 STEM Instrument,” Ultramicroscopy 19, 31–42 (1986).
[CrossRef]

K. Kodate, T. Kamiya, Y. Okada, H. Takenaka, “Focusing Characteristics of High-Efficiency Fresnel Zone Plate Fabricated by Deep Ultraviolet Lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

1985 (3)

K. Patorski, “Talbot Interferometry with Increased Shear,” Appl. Opt. 24, 4448–4453 (1985).
[CrossRef] [PubMed]

A. W. Lohmann, “An Interferometer with a Zone Plate as Beam-Splitter,” Opt. Acta 32, 1465–1469 (1985).
[CrossRef]

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A Dynamic Zone-Plate Interferometer for Measuring Aspherical Surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

1984 (1)

M. J. Simpson, A. G. Michette, “Considerations of Zone Plate Optics for Soft X-Ray Microscopy,” Opt. Acta 31, 1417–1426 (1984).
[CrossRef]

1983 (1)

1982 (1)

1978 (1)

1974 (1)

1973 (3)

1972 (1)

1965 (1)

1964 (2)

V. Ronchi, “Forty Years of History of a Grating Interferometer,” Appl. Opt. 3, 437–451 (1964).
[CrossRef]

V. A. Komissaruk, “Investigation of Wave Front Aberration of Optical Systems Using Three-Beam Interference,” Opt. Spec-trosc. U.S.S.R. 16, 571–574 (1964).

1961 (1)

Baez, A. V.

Barrett, H. H.

Cohen-Sabban, Y.

Colliex, C.

C. Mory, C. Colliex, J. M. Cowley, “Optimum Defocus For STEM Imaging and Microanalysis,” Ultramicroscopy 21, 171–178 (1987).
[CrossRef]

Cowley, J. M.

C. Mory, C. Colliex, J. M. Cowley, “Optimum Defocus For STEM Imaging and Microanalysis,” Ultramicroscopy 21, 171–178 (1987).
[CrossRef]

J. A. Lin, J. M. Cowley, “Calibration of Operating Parameters for HB5 STEM Instrument,” Ultramicroscopy 19, 31–42 (1986).
[CrossRef]

Engel, A.

Fujita, T.

Harvey, J. E.

Herziger, G.

Honda, T.

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A Dynamic Zone-Plate Interferometer for Measuring Aspherical Surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Horrigan, F. A.

Hsu, J.

Ichimura, I.

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A Dynamic Zone-Plate Interferometer for Measuring Aspherical Surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Iizuka, K.

K. Iizuka, Engineering Optics (Springer-Verlag, Berlin, 1985), pp. 99–102.

Joenathan, C.

C. Joenathan, V. Parthiban, R. S. Sirohi, “Shear Interferometry with Holographic Lenses,” Opt. Eng. 26, 359–364 (1987).

Joyeux, D.

Kamiya, T.

K. Kodate, T. Kamiya, Y. Okada, H. Takenaka, “Focusing Characteristics of High-Efficiency Fresnel Zone Plate Fabricated by Deep Ultraviolet Lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

Kodate, K.

K. Kodate, T. Kamiya, Y. Okada, H. Takenaka, “Focusing Characteristics of High-Efficiency Fresnel Zone Plate Fabricated by Deep Ultraviolet Lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

Komissaruk, V. A.

V. A. Komissaruk, “Investigation of Wave Front Aberration of Optical Systems Using Three-Beam Interference,” Opt. Spec-trosc. U.S.S.R. 16, 571–574 (1964).

Koyama, J.

Lin, J. A.

J. A. Lin, J. Hsu, S. G. Shiue, “Quantitative Three-Beam Ronchi Test,” Appl. Opt. 29, 1912–1918 (1990).
[CrossRef] [PubMed]

J. A. Lin, J. M. Cowley, “Calibration of Operating Parameters for HB5 STEM Instrument,” Ultramicroscopy 19, 31–42 (1986).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “An Interferometer with a Zone Plate as Beam-Splitter,” Opt. Acta 32, 1465–1469 (1985).
[CrossRef]

Meier, R. W.

Michette, A. G.

M. J. Simpson, A. G. Michette, “Considerations of Zone Plate Optics for Soft X-Ray Microscopy,” Opt. Acta 31, 1417–1426 (1984).
[CrossRef]

Mory, C.

C. Mory, C. Colliex, J. M. Cowley, “Optimum Defocus For STEM Imaging and Microanalysis,” Ultramicroscopy 21, 171–178 (1987).
[CrossRef]

Nishihara, H.

Ohyama, N.

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A Dynamic Zone-Plate Interferometer for Measuring Aspherical Surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Okada, Y.

K. Kodate, T. Kamiya, Y. Okada, H. Takenaka, “Focusing Characteristics of High-Efficiency Fresnel Zone Plate Fabricated by Deep Ultraviolet Lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

Omura, K.

Parthiban, V.

C. Joenathan, V. Parthiban, R. S. Sirohi, “Shear Interferometry with Holographic Lenses,” Opt. Eng. 26, 359–364 (1987).

Patorski, K.

Ronchi, V.

Shack, R. V.

Shiue, S. G.

Simpson, M. J.

M. J. Simpson, A. G. Michette, “Considerations of Zone Plate Optics for Soft X-Ray Microscopy,” Opt. Acta 31, 1417–1426 (1984).
[CrossRef]

Sirohi, R. S.

C. Joenathan, V. Parthiban, R. S. Sirohi, “Shear Interferometry with Holographic Lenses,” Opt. Eng. 26, 359–364 (1987).

Smartt, R. N.

Takenaka, H.

K. Kodate, T. Kamiya, Y. Okada, H. Takenaka, “Focusing Characteristics of High-Efficiency Fresnel Zone Plate Fabricated by Deep Ultraviolet Lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

Tsujiuchi, J.

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A Dynamic Zone-Plate Interferometer for Measuring Aspherical Surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Wood, R. W.

See R. W. Wood, Physical Optics (Macmillan, New York, 1934).

Wyant, J. C.

Yamaguchi, I.

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A Dynamic Zone-Plate Interferometer for Measuring Aspherical Surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Yatagai, T.

Young, M.

Appl. Opt. (9)

J. Opt. Soc. Am. (4)

Jpn. J. Appl. Phys. (1)

K. Kodate, T. Kamiya, Y. Okada, H. Takenaka, “Focusing Characteristics of High-Efficiency Fresnel Zone Plate Fabricated by Deep Ultraviolet Lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

Opt. Acta (2)

M. J. Simpson, A. G. Michette, “Considerations of Zone Plate Optics for Soft X-Ray Microscopy,” Opt. Acta 31, 1417–1426 (1984).
[CrossRef]

A. W. Lohmann, “An Interferometer with a Zone Plate as Beam-Splitter,” Opt. Acta 32, 1465–1469 (1985).
[CrossRef]

Opt. Commun. (1)

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A Dynamic Zone-Plate Interferometer for Measuring Aspherical Surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Opt. Eng. (1)

C. Joenathan, V. Parthiban, R. S. Sirohi, “Shear Interferometry with Holographic Lenses,” Opt. Eng. 26, 359–364 (1987).

Opt. Lett. (1)

Opt. Spec-trosc. U.S.S.R. (1)

V. A. Komissaruk, “Investigation of Wave Front Aberration of Optical Systems Using Three-Beam Interference,” Opt. Spec-trosc. U.S.S.R. 16, 571–574 (1964).

Ultramicroscopy (2)

J. A. Lin, J. M. Cowley, “Calibration of Operating Parameters for HB5 STEM Instrument,” Ultramicroscopy 19, 31–42 (1986).
[CrossRef]

C. Mory, C. Colliex, J. M. Cowley, “Optimum Defocus For STEM Imaging and Microanalysis,” Ultramicroscopy 21, 171–178 (1987).
[CrossRef]

Other (2)

See R. W. Wood, Physical Optics (Macmillan, New York, 1934).

K. Iizuka, Engineering Optics (Springer-Verlag, Berlin, 1985), pp. 99–102.

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Figures (7)

Fig. 1
Fig. 1

Ray diagram illustrating that the zone plate performs as both positive and negative lenses: SF, p, q1, and q3 denote the spatial filter, point source distance, first-order focusing distance, and third-order focusing distance, respectively.

Fig. 2
Fig. 2

Ray diagram illustrating the experimental setup of the Ronchi test of zone plates. P, ϕ, θ, Δ, G, and z represent pinhole filter, semiangle of illumination on the sine grating, diffraction angle of the first order, distance of the sine grating from the focus, sine grating, and distance between the grating and the film plane, respectively.

Fig. 3
Fig. 3

Ray diagram used to explain the inherent diffractive aberrations of (a) divergent beam illumination and (b) convergent beam illumination.

Fig. 4
Fig. 4

Photographs of the edge of the zone plates showing the 3.5-μm outermost zone: (a) amplitude zone plate and (b) phase zone plate.

Fig. 5
Fig. 5

Interferogram of the amplitude zone plate under test: Δ = 0.87 mm, p = 77 cm, f = 1.33 cm, q3 = 1.35 cm, and z = 3.5 cm.

Fig. 6
Fig. 6

Interferogram of the phase zone plate under test: p = 57 cm, q3 = 1.36 cm, and z = 3.5 cm; (a) Δ = 0.05-mm inside focus, (b) Δ = 0.67-mm outside focus, (c) Δ = 1.07-mm outside focus, (d) Δ = 1.47-mm outside focus, (e) Δ = 1.87-mm outside focus.

Fig. 7
Fig. 7

Overexposed spots of Fig. 6, which are centers of three interfering wavefronts diffracted by the sine grating, can be used together with the semiaperture of any interferogram in Fig. 6 to calculate the normalized shear for Figs. 6(a)–(e): (a) Δ = 0.05-mm inside focus, (b) Δ = 0.67-mm outside focus, (c) Δ = 1.07-mm outside focus, (d) Δ = 1.47-mm outside focus, (e) Δ = 1.87-mm outside focus.

Tables (1)

Tables Icon

Table I Inherent Aberrations for the Sine Grating at Four Outside Focus Positions

Equations (48)

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r n 2 = n f 1 λ + ( n λ ) 2 / 4 ,
t ( r ) = { 1 if sin ( π r 2 / λ f 1 ) < 0 , 0 if sin ( π r 2 / λ f 1 ) 0 ,
t ( r ) = j = - ( odd or zero ) T j exp [ π i ( x 2 + y 2 ) / f j λ ] ,
t ( r ) = j = - ( odd or zero ) T j exp [ π i ( x 2 + y 2 ) / f j λ ] exp [ 2 π i W j ( x , y ) / λ ] ,
T ( x , y ) = n = - 1 1 F n exp [ 2 π i n x / d ] ,
I ( x 3 , y 3 ) = C ( { δ ( x 0 , y 0 ) * exp [ - π i ( x 0 2 + y 0 2 ) / p λ ] × exp [ π i ( x 1 2 + y 1 2 ) / f j λ ] Φ ( x 1 , y 1 ) } * exp [ - π i ( x 1 2 + y 1 2 ) / q j λ ] * exp [ - π i ( x f 2 + y f 2 ) / Δ λ ] × T ( x 2 , y 2 ) ) * exp [ - π i ( x 2 2 + y 2 2 ) / z λ ] 2 ,
I ( x 3 , y 3 ) = C ϕ ( x f / q j λ , y f / q j λ ) × T ( x 2 , y 2 ) exp [ - π i ( x 2 2 + y 2 2 ) ( 1 / z + 1 / Δ ) / λ ] × exp { 2 π i [ x 2 ( x 3 / z + x f / Δ ) / λ + y 2 ( y 3 / z + y f / Δ ) / λ ] } × exp [ - π i ( x f 2 + y f 2 ) ( 1 / q j + 1 / Δ ) / λ ] d x f d y f d x 2 d y 2 2 ,
ϕ ( x f / q j λ , y f / q j λ ) = Φ ( x 1 , y 1 ) exp [ 2 π i ( x f x 1 + y f y 1 ) / q j λ ] d x 1 d y 1 .
I ( x 3 , y 3 ) = C | n F n exp [ π i z Δ λ ( n / d + x 3 / z λ ) 2 / ( z + Δ ) ] × ϕ ( x f / q j λ , y f / q j λ ) exp { - π i ( x f 2 + y f 2 ) [ 1 / q j + 1 / ( z + Δ ) ] / λ } × exp { 2 π i z [ x f ( x 3 / z λ + n / d ) + y f ( y 3 / z λ ) ] / ( z + Δ ) } | 2 .
I ( x 3 , y 3 ) = C | n F n exp [ π i z Δ λ ( n / d + x 3 / z λ ) 2 / ( z + Δ ) ] × Φ [ - ( x 3 + n λ z / d ) / M , - y 3 / M ] | 2 ,
I ( x 3 , y 3 ) = C | n F n exp { 2 π i ( 1 + Δ / z ) ( Δ R 2 / 2 q j 2 λ ) [ - ( x 3 + n λ z / d ) / M R ] 2 } × A ( - x 3 / M - n λ z / d M , - y 3 / M ) × exp [ 2 π i W j ( - x 3 / M - n λ z / d M , - y 3 / M ) / λ ] | 2 .
I ( x , y ) = C | n F n A ( - x - n s , - y ) × exp { 2 π i D [ ( - x - n s ) 2 + ( - y ) 2 ] / λ } × exp [ 2 π i W j ( - x - n s , - y ) / λ ] | 2 ,
I ( x , y ) = C | n F n A ( - x - n s , - y ) exp [ 2 π i W j ( - x - n s , - y ) / λ ] | 2 = C A ( - x , - y ) exp [ 2 π i W j ( - x , - y ) / λ ] + σ exp ( i ) A ( - x - s , - y ) exp [ 2 π i W j ( - x - s , - y ) / λ ] + σ exp ( - i ) A ( - x + s , - y ) exp [ 2 π i W j ( - x + s , - y ) / λ ] 2 ,
exp [ i Φ 1 ( x 2 , y 2 ) ] = exp [ - i k ( x 2 2 + y 2 2 + Δ 2 - Δ ) ] exp [ 2 π i g s x 2 ] ,
exp [ i Φ 1 ( x 2 , y 2 ) ] exp ( - i k { [ ( x 2 - Δ λ g s ) 2 + y 2 2 ] / 2 Δ - ( x 2 2 + y 2 2 ) / 8 Δ 3 } ) × exp [ i k Δ λ 2 g s 2 / 2 ] .
exp [ i Φ 1 ( x 2 , y 2 ) ] = exp [ - i k ( N Q ¯ - N O ¯ ) ] exp [ - i k W N ( x 2 , y 2 ) ] = exp [ - i k ( ( x 2 - Δ λ g s ) 2 + y 2 2 + Δ 2 - Δ 2 + Δ 2 λ 2 g s 2 ) ] × exp [ - i k W N ( x 2 , y 2 ) ] .
exp [ i Φ 1 ( x 2 , y 2 ) ] exp [ - i k ( x 2 2 + y 2 2 + Δ 2 λ 2 g s 2 - 2 Δ λ g s x 2 ) / 2 Δ ] × exp [ i k ( x 2 2 + y 2 2 + Δ 2 λ 2 g s 2 - 2 Δ λ g s x 2 ) 2 / 8 Δ 3 ] × exp [ i k ( Δ λ 2 g s 2 / 2 - Δ λ 4 g s 4 / 8 ) ] exp [ - i k W N ( x 2 , y 2 ) ] .
W N ( x 2 , y 2 ) = λ 2 g s 2 ( x 2 2 + y 2 2 ) / 4 Δ + λ 2 g s 2 x 2 2 / 2 Δ - λ g s x 2 ( x 2 2 + y 2 2 ) / 2 Δ 2 - λ 3 g s 3 x 2 / 2.
W N ( x , y ) = - W 31 x ( x 2 + y 2 ) + W 20 ( x 2 + y 2 ) + W 22 x 2 - W 11 x ,
W 20 = Δ tan 2 ϕ ( λ g s ) 2 / 4 , W 22 = Δ tan 2 ϕ ( λ g s ) 2 / 2 W 31 = Δ tan 3 ϕ ( λ g s ) / 2 , W 11 = Δ tan ϕ ( λ g s ) 3 / 2.
W P ( x , y ) = W 31 x ( x 2 + y 2 ) + W 20 ( x 2 + y 2 ) + W 22 x 2 + W 11 x .
I ( x , y ) = C exp [ 2 π i W ( - x , - y ) / λ ] + σ exp ( i ) × exp [ 2 π i W ( - x - s , - y ) / λ - 2 π i W N ( x + s , y ) / λ ] + σ exp ( - i ) exp [ 2 π i W ( - x + s , - y ) / λ - 2 π i W P ( x - s , y ) / λ ] 2 .
I ( x , y ) 1 + 4 σ cos [ 2 π E ( x , y ; s ) / λ ] cos [ 2 π O ( x , y ; s ) / λ + ] ,
E ( x , y ; s ) = ( 6 A s 2 - W 20 - W 22 + 3 W 31 s ) x 2 + ( 2 A s 2 - W 20 + W 31 s ) y 2 + W 31 s 3 + W 11 s + A s 4 + ( D - W 22 - W 20 ) s 2 ;
O ( x , y ; s ) = ( 4 A s + W 31 ) x ( x 2 + y 2 ) + ( 4 A s 3 + 2 D s - 2 W 20 s - 2 W 22 s + W 11 + 3 W 31 s 2 ) x .
α 1 x 2 + α 2 y 2 + β = ( 2 n + 1 ) λ / 4 ,
x 2 a n 2 + y 2 b n 2 = 1.
a n 2 = [ ( 2 n + 1 ) λ - 4 β ] / 4 α 1 ,             b n 2 = [ ( 2 n + 1 ) λ - 4 β ] / 4 α 2 .
b n + 1 2 - b n 2 = λ / ( 4 A s 2 - 2 W 20 + 2 W 31 s ) ,
A = [ λ + 2 ( W 20 - W 31 s ) ( b n + 1 2 - b n 2 ) ] / [ 4 s 2 ( b n + 1 2 - b n 2 ) ] .
A = [ λ - 2 W 20 ( b n + 1 2 - b n 2 ) ] / [ 4 s 2 ( b n + 1 2 - b n 2 ) ] .
exp [ i Φ ( x 2 , y 2 ) ] = exp [ i k ( x 2 2 + y 2 2 + Δ 2 - Δ ) ] exp [ 2 π i g s x 2 ] exp ( i k { [ ( x 2 + Δ λ g s ) 2 + y 2 2 ] / 2 Δ - ( x 2 2 + y 2 2 ) 2 / 8 Δ 3 } ) × exp [ - i k Δ λ 2 g s 2 / 2 ] .
exp [ i Φ ( x 2 , y 2 ) ] = exp [ i k ( N Q ¯ - N O ¯ ) ] exp [ i k W N ( x 2 , y 2 ) ] = exp [ i k ( ( x 2 + Δ λ g s ) 2 + y 2 2 + Δ 2 - Δ 2 + Δ 2 λ 2 g s 2 ) ] × exp [ i k W N ( x 2 , y 2 ) ] .
exp [ i Φ ( x 2 , y 2 ) ] = exp [ i k ( x 2 2 + y 2 2 + λ 2 Δ 2 g s 2 + 2 Δ λ g s x 2 ) / 2 Δ ] × exp [ - i k ( x 2 2 + y 2 2 + λ 2 Δ 2 g s 2 + 2 Δ λ g s x 2 ) 2 / 8 Δ 3 ] × exp [ - i k ( Δ λ 2 g s 2 / 2 + Δ λ 2 g s 4 / 8 ) ] exp [ i k W N ( x 2 , y 2 ) ] .
W N ( x 2 , y 2 ) = λ 2 g s 2 ( x 2 2 + y 2 2 ) / 4 Δ + λ 2 g s 2 x 2 2 / 2 Δ + λ g s x 2 ( x 2 2 + y 2 2 ) / 2 Δ 2 + λ 3 g s 3 x 2 / 2.
W N ( x , y ) = W 31 x ( x 2 + y 2 ) + W 20 ( x 2 + y 2 ) + W 22 x 2 + W 11 x .
W P ( x , y ) = - W 31 x ( x 2 + y 2 ) + W 20 ( x 2 + y 2 ) + W 22 x 2 - W 11 x .
I ( x , y ) = C exp [ 2 π i W ( - x , - y ) / λ ] + σ exp ( i ) × exp [ 2 π i W ( - x - s , - y ) / λ + 2 π i W N ( - x - s , - y ) / λ ] + σ exp ( - i ) exp [ 2 π i W ( - x + s , - y ) / λ + 2 π i W P ( - x + s , - y ) / λ ] 2 .
x 2 a n 2 + y 2 b n 2 = 1 ,
a n 2 = [ ( 2 n + 1 ) λ - 4 β ] / 4 α 1 ,             b n 2 = [ ( 2 n + 1 ) λ - 4 β ] / 4 α 2 , α 1 = 6 A s 2 + W 20 + W 22 - 3 W 31 s ,             α 2 = 2 A s 2 + W 20 - W 31 s , β = - W 31 s 3 - W 11 s + A s 4 + ( D + W 22 + W 20 ) s 2 .
A = [ λ - 2 ( W 20 - W 31 s ) ( b n + 1 2 - b n 2 ) ] / [ 4 s 2 ( b n + 1 2 - b n 2 ) ] .
A = [ λ + 2 W 20 ( b n + 1 2 - b n 2 ) ] / [ 4 s 2 ( b n + 1 2 - b n 2 ) ] .
A = λ 4 s 2 ( b n + 1 2 - b n 2 ) .
1 p + 1 q j = 1 f j .
tan ϕ = ( λ g s ) / s = 0.271 ,             g 0 = sin ϕ / λ = 413 lines / mm .
R 2 = N f 1 λ + N 2 λ 2 / 4.
OPD = p + q 3 - ( p 2 + R 2 ) - ( q 3 2 + R 2 ) - R 2 ( 1 / p + 1 / q 3 ) / 2 + R 4 ( 1 / p 3 + 1 / q 3 3 ) / 8 = - R 2 / ( 2 f 3 ) + R 4 ( 1 / p 3 + 1 / q 3 3 ) / 8 = - 3 N λ / 2 - N 2 λ 2 / 8 f 3 + R 4 ( 1 / p 3 + 1 / q 3 3 ) / 8.
A = R 4 ( 1 / p 3 + 1 / q 3 3 ) / 8 - N 2 λ 2 / ( 8 f 3 ) .

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