Abstract

Aspheric and free-form surfaces are powerful surface forms that allow designers to achieve better performance with fewer lenses and smaller packages. Unlike spheres, these surfaces are fabricated with processes that leave a signature, or “structure,” that is primarily in the mid-spatial-frequency region. These structured surface errors create ripples in the modulation transfer function (MTF) profile. Using Fourier techniques with generalized functions, the drop in MTF is derived and shown to exhibit a nonlinear relationship with the peak-to-valley height of the structured surface error.

© 2010 Optical Society of America

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References

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  1. J. Bareau and P. P. Clark, “The optics of miniature digital camera modules,” in International Optical Design, OSA Technical Digest (CD) (Optical Society of America, 2006), paper WB3.
  2. G. W. Forbes and C. P. Brophy, “Asphere, O asphere, how shall we describe thee?,” Proc. SPIE 7100, 710002 (2008).
    [CrossRef]
  3. O. Cakmakci, G. E. Fasshauer, H. Foroosh, K. P. Thompson, and J. P. Rolland, “Meshfree approximation methods for free-form surface representation in optical design with applications to head-worn displays,” Proc. SPIE 7061, 70610D2008).
    [CrossRef]
  4. M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
    [CrossRef]
  5. C. R. Dunn and D. D. Walker, “Pseudo-random tool paths for CNC sub-aperture polishing and other applications,” Opt. Express 16, 18942-18949 (2008).
    [CrossRef]
  6. S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
    [CrossRef]
  7. A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835, 2008.
    [CrossRef]
  8. C. Heinzel, D. Grimme, and A. Moisan, “Modeling of surface generation in contour grinding of optical molds,” CIRP Ann. 55, 581-584 (2006).
    [CrossRef]
  9. R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18, 137-142 (1979).
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    [CrossRef]
  11. R. N. Youngworth and B. D. Stone, “Simple Estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. 39, 2198-2209 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. J. R. Rodgers, “Slope error tolerances for optical surfaces,” presented at the OptiFab Conference (SPIE, 2008), paper TD04-4, pp. 15-17.
  19. M. P. Rimmer, T. J. Bruegge, and T. G. Kuper, “MTF optimization in lens design,” Proc. SPIE 1354, 83-91 (1991).
    [CrossRef]
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  21. C. F. Cheung, L. B. Kong, W. B. Lee, and S. To, “Modelling and simulation of freeform surface generation in ultra-precision raster milling,” Proc. Inst. Mech. Eng. B 220, 1787-1801(2006).
    [CrossRef]
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  23. J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23, 858-865 (2006).
    [CrossRef]
  24. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

2010 (1)

2008 (7)

C. R. Dunn and D. D. Walker, “Pseudo-random tool paths for CNC sub-aperture polishing and other applications,” Opt. Express 16, 18942-18949 (2008).
[CrossRef]

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835, 2008.
[CrossRef]

G. W. Forbes and C. P. Brophy, “Asphere, O asphere, how shall we describe thee?,” Proc. SPIE 7100, 710002 (2008).
[CrossRef]

O. Cakmakci, G. E. Fasshauer, H. Foroosh, K. P. Thompson, and J. P. Rolland, “Meshfree approximation methods for free-form surface representation in optical design with applications to head-worn displays,” Proc. SPIE 7061, 70610D2008).
[CrossRef]

J. R. Rodgers, “Slope error tolerances for optical surfaces,” presented at the OptiFab Conference (SPIE, 2008), paper TD04-4, pp. 15-17.

P. Murphy, “Methods and challenges in quantifying mid-spatial frequencies,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA3.

2007 (1)

M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
[CrossRef]

2006 (4)

C. Heinzel, D. Grimme, and A. Moisan, “Modeling of surface generation in contour grinding of optical molds,” CIRP Ann. 55, 581-584 (2006).
[CrossRef]

J. Bareau and P. P. Clark, “The optics of miniature digital camera modules,” in International Optical Design, OSA Technical Digest (CD) (Optical Society of America, 2006), paper WB3.

C. F. Cheung, L. B. Kong, W. B. Lee, and S. To, “Modelling and simulation of freeform surface generation in ultra-precision raster milling,” Proc. Inst. Mech. Eng. B 220, 1787-1801(2006).
[CrossRef]

J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23, 858-865 (2006).
[CrossRef]

2005 (1)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

2000 (3)

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 2000).

C. J. Progler and A. K. K. Wong, “Zernike coefficients: are they really enough?,” Proc. SPIE 4000, 40-52 (2000).
[CrossRef]

R. N. Youngworth and B. D. Stone, “Simple Estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. 39, 2198-2209 (2000).
[CrossRef]

1991 (1)

M. P. Rimmer, T. J. Bruegge, and T. G. Kuper, “MTF optimization in lens design,” Proc. SPIE 1354, 83-91 (1991).
[CrossRef]

1983 (1)

J. P. Marioge and S. Slansky, “Effect of figure and waviness on image quality,” J. Opt. 14, 189-198 (1983).
[CrossRef]

1979 (1)

R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18, 137-142 (1979).

1978 (1)

M. Rimmer, “A tolerancing procedure based on modulation transfer function (MTF),” Proc. SPIE 147, 66-70 (1978).

1975 (1)

1974 (1)

Bareau, J.

J. Bareau and P. P. Clark, “The optics of miniature digital camera modules,” in International Optical Design, OSA Technical Digest (CD) (Optical Society of America, 2006), paper WB3.

Beaucamp, A.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835, 2008.
[CrossRef]

Bogunovic, D.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 2000).

Brophy, C. P.

G. W. Forbes and C. P. Brophy, “Asphere, O asphere, how shall we describe thee?,” Proc. SPIE 7100, 710002 (2008).
[CrossRef]

Bruegge, T. J.

M. P. Rimmer, T. J. Bruegge, and T. G. Kuper, “MTF optimization in lens design,” Proc. SPIE 1354, 83-91 (1991).
[CrossRef]

Cakmakci, O.

O. Cakmakci, G. E. Fasshauer, H. Foroosh, K. P. Thompson, and J. P. Rolland, “Meshfree approximation methods for free-form surface representation in optical design with applications to head-worn displays,” Proc. SPIE 7061, 70610D2008).
[CrossRef]

Cheng, M. N.

M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
[CrossRef]

Cheung, C. F.

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
[CrossRef]

C. F. Cheung, L. B. Kong, W. B. Lee, and S. To, “Modelling and simulation of freeform surface generation in ultra-precision raster milling,” Proc. Inst. Mech. Eng. B 220, 1787-1801(2006).
[CrossRef]

Church, E. L.

Clark, P. P.

J. Bareau and P. P. Clark, “The optics of miniature digital camera modules,” in International Optical Design, OSA Technical Digest (CD) (Optical Society of America, 2006), paper WB3.

Dallas, W. J.

Dunn, C. R.

Fasshauer, G. E.

O. Cakmakci, G. E. Fasshauer, H. Foroosh, K. P. Thompson, and J. P. Rolland, “Meshfree approximation methods for free-form surface representation in optical design with applications to head-worn displays,” Proc. SPIE 7061, 70610D2008).
[CrossRef]

Forbes, G. W.

G. W. Forbes and C. P. Brophy, “Asphere, O asphere, how shall we describe thee?,” Proc. SPIE 7100, 710002 (2008).
[CrossRef]

Foroosh, H.

O. Cakmakci, G. E. Fasshauer, H. Foroosh, K. P. Thompson, and J. P. Rolland, “Meshfree approximation methods for free-form surface representation in optical design with applications to head-worn displays,” Proc. SPIE 7061, 70610D2008).
[CrossRef]

Freeman, R.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835, 2008.
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Grimme, D.

C. Heinzel, D. Grimme, and A. Moisan, “Modeling of surface generation in contour grinding of optical molds,” CIRP Ann. 55, 581-584 (2006).
[CrossRef]

Harvey, J. E.

Heinzel, C.

C. Heinzel, D. Grimme, and A. Moisan, “Modeling of surface generation in contour grinding of optical molds,” CIRP Ann. 55, 581-584 (2006).
[CrossRef]

Kong, L. B.

C. F. Cheung, L. B. Kong, W. B. Lee, and S. To, “Modelling and simulation of freeform surface generation in ultra-precision raster milling,” Proc. Inst. Mech. Eng. B 220, 1787-1801(2006).
[CrossRef]

Krywonos, A.

Kuper, T. G.

M. P. Rimmer, T. J. Bruegge, and T. G. Kuper, “MTF optimization in lens design,” Proc. SPIE 1354, 83-91 (1991).
[CrossRef]

Lee, W. B.

M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
[CrossRef]

C. F. Cheung, L. B. Kong, W. B. Lee, and S. To, “Modelling and simulation of freeform surface generation in ultra-precision raster milling,” Proc. Inst. Mech. Eng. B 220, 1787-1801(2006).
[CrossRef]

Li, B.

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

Marioge, J. P.

J. P. Marioge and S. Slansky, “Effect of figure and waviness on image quality,” J. Opt. 14, 189-198 (1983).
[CrossRef]

Milster, T. D.

Moisan, A.

C. Heinzel, D. Grimme, and A. Moisan, “Modeling of surface generation in contour grinding of optical molds,” CIRP Ann. 55, 581-584 (2006).
[CrossRef]

Morton, R.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835, 2008.
[CrossRef]

Murphy, P.

P. Murphy, “Methods and challenges in quantifying mid-spatial frequencies,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA3.

Noll, R. J.

R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18, 137-142 (1979).

Ponudurai, K.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835, 2008.
[CrossRef]

Progler, C. J.

C. J. Progler and A. K. K. Wong, “Zernike coefficients: are they really enough?,” Proc. SPIE 4000, 40-52 (2000).
[CrossRef]

Rimmer, M.

M. Rimmer, “A tolerancing procedure based on modulation transfer function (MTF),” Proc. SPIE 147, 66-70 (1978).

Rimmer, M. P.

M. P. Rimmer, T. J. Bruegge, and T. G. Kuper, “MTF optimization in lens design,” Proc. SPIE 1354, 83-91 (1991).
[CrossRef]

Rodgers, J. R.

J. R. Rodgers, “Slope error tolerances for optical surfaces,” presented at the OptiFab Conference (SPIE, 2008), paper TD04-4, pp. 15-17.

Rolland, J. P.

O. Cakmakci, G. E. Fasshauer, H. Foroosh, K. P. Thompson, and J. P. Rolland, “Meshfree approximation methods for free-form surface representation in optical design with applications to head-worn displays,” Proc. SPIE 7061, 70610D2008).
[CrossRef]

Saito, T. T.

Simmons, L. B.

Slansky, S.

J. P. Marioge and S. Slansky, “Effect of figure and waviness on image quality,” J. Opt. 14, 189-198 (1983).
[CrossRef]

Stone, B. D.

Tamkin, J. M.

Thompson, K. P.

O. Cakmakci, G. E. Fasshauer, H. Foroosh, K. P. Thompson, and J. P. Rolland, “Meshfree approximation methods for free-form surface representation in optical design with applications to head-worn displays,” Proc. SPIE 7061, 70610D2008).
[CrossRef]

To, S.

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

C. F. Cheung, L. B. Kong, W. B. Lee, and S. To, “Modelling and simulation of freeform surface generation in ultra-precision raster milling,” Proc. Inst. Mech. Eng. B 220, 1787-1801(2006).
[CrossRef]

Walker, D. D.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835, 2008.
[CrossRef]

C. R. Dunn and D. D. Walker, “Pseudo-random tool paths for CNC sub-aperture polishing and other applications,” Opt. Express 16, 18942-18949 (2008).
[CrossRef]

Wang, H.

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

Wong, A. K. K.

C. J. Progler and A. K. K. Wong, “Zernike coefficients: are they really enough?,” Proc. SPIE 4000, 40-52 (2000).
[CrossRef]

Youngworth, R. N.

Zavada, J. M.

Appl. Opt. (4)

CIRP Ann. (1)

C. Heinzel, D. Grimme, and A. Moisan, “Modeling of surface generation in contour grinding of optical molds,” CIRP Ann. 55, 581-584 (2006).
[CrossRef]

J. Opt. (1)

J. P. Marioge and S. Slansky, “Effect of figure and waviness on image quality,” J. Opt. 14, 189-198 (1983).
[CrossRef]

J. Opt. Soc. Am. A (1)

Key Eng. Mater. (2)

M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
[CrossRef]

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

Opt. Eng. (1)

R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18, 137-142 (1979).

Opt. Express (1)

Proc. Inst. Mech. Eng. B (1)

C. F. Cheung, L. B. Kong, W. B. Lee, and S. To, “Modelling and simulation of freeform surface generation in ultra-precision raster milling,” Proc. Inst. Mech. Eng. B 220, 1787-1801(2006).
[CrossRef]

Proc. SPIE (6)

M. P. Rimmer, T. J. Bruegge, and T. G. Kuper, “MTF optimization in lens design,” Proc. SPIE 1354, 83-91 (1991).
[CrossRef]

M. Rimmer, “A tolerancing procedure based on modulation transfer function (MTF),” Proc. SPIE 147, 66-70 (1978).

C. J. Progler and A. K. K. Wong, “Zernike coefficients: are they really enough?,” Proc. SPIE 4000, 40-52 (2000).
[CrossRef]

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835, 2008.
[CrossRef]

G. W. Forbes and C. P. Brophy, “Asphere, O asphere, how shall we describe thee?,” Proc. SPIE 7100, 710002 (2008).
[CrossRef]

O. Cakmakci, G. E. Fasshauer, H. Foroosh, K. P. Thompson, and J. P. Rolland, “Meshfree approximation methods for free-form surface representation in optical design with applications to head-worn displays,” Proc. SPIE 7061, 70610D2008).
[CrossRef]

Other (6)

P. Murphy, “Methods and challenges in quantifying mid-spatial frequencies,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA3.

CODE V 10.2 Reference Manual, Optical Research Associates.

J. R. Rodgers, “Slope error tolerances for optical surfaces,” presented at the OptiFab Conference (SPIE, 2008), paper TD04-4, pp. 15-17.

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 2000).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

J. Bareau and P. P. Clark, “The optics of miniature digital camera modules,” in International Optical Design, OSA Technical Digest (CD) (Optical Society of America, 2006), paper WB3.

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Figures (11)

Fig. 1
Fig. 1

Example of an optical system with a thin phase plate at the stop, including beam footprints for surface S4 and stop surface S8 across four field positions. Note that the size of the footprint for the edge of the field is smaller than the stop, indicating pupil distortion.

Fig. 2
Fig. 2

MTF of the system in Fig. 1 with 0.103 waves PV sinusoidal phase error on surface S8. The optical system has 0.005 PV wave error with no MSF component.

Fig. 3
Fig. 3

Peak MTF contrast loss versus sinusoidal PV surface height phase difference. Computational results are calculated from CODE V simulation of thin phase skin at stop S4 in the Fig. 1 lens. Analytical results are from Eq. (19).

Fig. 4
Fig. 4

Pupil map of rotational cosine MSF error on surface S4; see Fig. 1.

Fig. 5
Fig. 5

PSF of rotational cosine across field: a, 0 ° ; b, 2 ° ; c, 5 ° ; d, 10 ° . Surface PV phase = 0.2 waves. Scales are the same for each image.

Fig. 6
Fig. 6

MTF across field for radial sinusoidal MSF error: 0.5 mm period, 0.2 wave PV surface height.

Fig. 7
Fig. 7

PSF of rotational cosine across field: a, 0 ° ; b, 2 ° ; c, 5 ° ; d, 10 ° . Surface PV height = 0.76 waves. Scales are the same for each image.

Fig. 8
Fig. 8

Comparison of PSF [Eq. (11)] with PSF approximation [Eq. (13)]. φ β = 1.18   rad (PV height = 0.36 waves). a, one cycle of sinusoidal ripple across pupil; b, two cycles across pupil.

Fig. 9
Fig. 9

Plot of Eq. (21) for M = 1 , 2, 3. MTF contrast drop for a single MSF sinusoid. The Bessel function series can be truncated at m = 1 to 1 for good approximation when the peak phase error ϕ is less than 1.1 rad , which corresponds to a surface height of 0.35 waves PV.

Fig. 10
Fig. 10

MTF response for two surfaces with MSF errors. a, MSF error of 0.134 waves PV on one surface gives a 15% drop in MTF; b, MSF error of 0.134 waves PV on two surfaces with the same spatial period and phase cause a 55% drop in MTF; c, MSF error of 0.134 waves PV on two surfaces with different frequencies (5 and 10 cycles/aperture) add linearly (to first order).

Fig. 11
Fig. 11

Rotational MSF errors on S4 and S8 (stop) in Fig. 1: a, on axis; b, off axis. PSFs are shown in a and c; pupil maps are in b and d. Phase error of 0.1 μm on each surface with an annular period of 0.5 mm .

Tables (1)

Tables Icon

Table 1 Lens with Radial Sinusoid on L2

Equations (22)

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U o ( x s ) = A o exp ( i ϕ ( x s ) ) ,
T ( x s ) = rect ( x s D ) ,
U s ( x s ) = T ( x s ) U o ( x s ) = A o rect ( x s D ) exp ( i ϕ ( x s ) ) ,
h ( x s ) = β 2 sin ( 2 π x s ξ o ) ,
ϕ ( x s ) = 2 π h ( x s ) ( n 1 ) λ ,
φ β = 2 π [ ( n 1 ) λ ] β 2 ,
U s ( x s ) = A o rect ( x s D ) exp [ i φ β sin ( 2 π x s ξ o ) ] .
U s ( x s ) = A o rect ( x s D ) m = J m ( φ β ) exp ( i m 2 π x s ξ o ) .
U i ( ξ ) = A o D m = J m ( φ β ) sinc [ ( ξ m ξ o ) D ] .
ξ i = x i λ z i ,
PSF ( ξ ) = | U i ( ξ ) | 2 = { A o D m 1 = J m 1 ( φ β ) sinc [ ( ξ m 1 ξ o ) D ] } × { A o D m 2 = J m 2 ( φ β ) sinc [ ( ξ m 2 ξ o ) D ] } = A o 2 D 2 { { m = J m 2 ( φ β ) sinc 2 [ ( ξ m ξ o ) D ] } + m 1 = m 1 m 2 m 2 = J m 1 ( φ β ) J m 2 ( φ β ) × sinc [ ( ξ m 1 ξ o ) D ] sinc [ ( ξ m 2 ξ o ) D ] } | ξ = x i λ z i .
S 0 , ± 1 ( ξ D ) = J 0 ( φ β ) J 1 ( φ β ) sinc ( ξ D ) sinc [ ( ξ ξ o ) D ] .
I ( φ β ) = J 0 ( φ β ) J 1 ( φ β ) .
PSF ( ξ ) A o 2 D 2 m = J m 2 ( φ β ) sinc 2 [ ( ξ m ξ o ) D ] | ξ = x i λ z i .
PSF ( x i ) = A o 2 D 2 sinc 2 ( x i λ z i D ) * m = J m 2 ( φ β ) δ ( x i m λ z i ξ o D ) ,
MTF ( ξ i ) = mod [ F { PSF ( x i ) } ] A o 2 D 2 = mod [ tri ( λ z i ξ i D ) × { m = J m 2 ( φ β ) exp ( i m 2 π λ z i ξ i ξ o ) } ] ,
MTF 1 ( ξ i ) tri ( λ z i ξ i D ) × | m = 1 1 J m ( φ β ) 2 exp ( i m 2 π λ z i ξ i ξ o ) | = | J 0 ( φ β ) 2 + J 1 ( φ β ) 2 [ exp ( i m 2 π λ z i ξ i ξ o ) + exp ( i m 2 π λ z i ξ i ξ o ) ] | = | J 0 ( φ β ) 2 + 2 J 1 ( φ β ) 2 cos ( 2 π λ z i m ξ i ξ o ) | .
MTF ( ξ i ) = | tri ( λ z i ξ i D ) × { J 0 2 ( φ β ) + 2 m = 1 J m 2 ( φ β ) cos ( 2 π λ z i m ξ i ξ o ) } | .
MTF ( ξ i ) = | tri ( λ z i ξ i D ) { m = J m 2 ( φ β ) cos ( 2 π λ z i m ξ i ξ o ) } | .
ξ i min = 1 2 λ z i m ξ o .
Δ MTF ( φ β ) = | m = N N J m 2 ( φ β ) cos ( π m ) | = | m = N N J m 2 ( φ β ) | ,
1.00 × 10 + 18

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