Abstract

We develop a theory for understanding and modeling the effects of mid-spatial frequency (MSF) structures on the modulation transfer function (MTF) of an imaging system. We show that the MTF is related through Fourier transformation to what we call the pupil-difference probability density (PDPD) function. This relation is illustrated for several one-dimensional periodic groove patterns. We also consider two-dimensional pupils, particularly those presenting straight and circular concentric periodic surface errors, similar to those resulting from diamond turning and milling processes of freeform optical manufacturing, and develop simple approximations for the MTF based on the PDPD. Our approach provides an analytic way to understand the effects of tool profiles, as well as other freeform manufacturing parameters.

© 2017 Optical Society of America

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References

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  1. R. J. Noll, “Effect of Mid- and High-Spatial Frequencies on Optical Performance,” Opt. Eng. 18(2), 182137 (1979).
    [Crossref]
  2. D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and Control of Mid-Spatial Frequency Wavefront Errors in Optical Systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
    [Crossref]
  3. J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49, 4895 (2010).
  4. G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Optical Measurement Systems for Industrial Inspection IX, Proc. of SPIE9525, 95251B (2015).
  5. M. A. Alonso and G. W. Forbes, “The Strehl ratio as the Fourier transform of a probability density,” Opt. Lett. 41, 3735–3738 (2016).
    [Crossref] [PubMed]
  6. R. N. Youngsworth 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]
  7. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005), Chap. 6.
  8. G. D. Boremann, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE, 2001), pp. 85–88.
    [Crossref]
  9. L. He, C. J. Evans, and A. Davies, “Two-quadrant area structure function analysis for optical surface characterization,” Opt. Express 20, 23275–23280 (2012).
    [Crossref] [PubMed]
  10. L. He, C. J. Evans, and A. Davies, “Optical surface characterization with the area structure function,” CIRP Annals -Manufacturing Technology,  62(1), 539–542 (2013).
    [Crossref]
  11. R. E. Parks and M. T. Tuell, “The structure function as a metric for roughness and figure,” Proc. of SPIE 995199510 (2016).
  12. J. W. Goodman, Statistical Optics (John Wiley & Sons, 2015), Chap. 8.
  13. R. E. Hufnagel and N. R. Stanley, “Modulation Transfer Function associated with Image Transmission through Turbulent Media,” J. Opt. Soc. Am. 54, 52–61 (1964).
    [Crossref]
  14. A. Maréchal, “Etude des effets combinés de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. Opt. 2, 257–277 (1947).
  15. V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73, 860–861 (1983).
    [Crossref]
  16. J. V. Baliga and B. D. Cohn, “Simplified method for calculating encircled energy,” Proc. SPIE 892, 152 (1988).
    [Crossref]
  17. I. S. Gradshteyn and I. M. Ryxhik, Table of Integrals, Series, and Products (Academic, 1980), p. 739, ET II 360(14).

2016 (2)

M. A. Alonso and G. W. Forbes, “The Strehl ratio as the Fourier transform of a probability density,” Opt. Lett. 41, 3735–3738 (2016).
[Crossref] [PubMed]

R. E. Parks and M. T. Tuell, “The structure function as a metric for roughness and figure,” Proc. of SPIE 995199510 (2016).

2013 (1)

L. He, C. J. Evans, and A. Davies, “Optical surface characterization with the area structure function,” CIRP Annals -Manufacturing Technology,  62(1), 539–542 (2013).
[Crossref]

2012 (1)

2010 (1)

J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49, 4895 (2010).

2000 (1)

1988 (1)

J. V. Baliga and B. D. Cohn, “Simplified method for calculating encircled energy,” Proc. SPIE 892, 152 (1988).
[Crossref]

1983 (1)

1979 (1)

R. J. Noll, “Effect of Mid- and High-Spatial Frequencies on Optical Performance,” Opt. Eng. 18(2), 182137 (1979).
[Crossref]

1964 (1)

1947 (1)

A. Maréchal, “Etude des effets combinés de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. Opt. 2, 257–277 (1947).

Aikens, D.

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and Control of Mid-Spatial Frequency Wavefront Errors in Optical Systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
[Crossref]

Alonso, M. A.

Baliga, J. V.

J. V. Baliga and B. D. Cohn, “Simplified method for calculating encircled energy,” Proc. SPIE 892, 152 (1988).
[Crossref]

Boremann, G. D.

G. D. Boremann, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE, 2001), pp. 85–88.
[Crossref]

Cohn, B. D.

J. V. Baliga and B. D. Cohn, “Simplified method for calculating encircled energy,” Proc. SPIE 892, 152 (1988).
[Crossref]

Dallas, W.

J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49, 4895 (2010).

Davies, A.

L. He, C. J. Evans, and A. Davies, “Optical surface characterization with the area structure function,” CIRP Annals -Manufacturing Technology,  62(1), 539–542 (2013).
[Crossref]

L. He, C. J. Evans, and A. Davies, “Two-quadrant area structure function analysis for optical surface characterization,” Opt. Express 20, 23275–23280 (2012).
[Crossref] [PubMed]

DeGroote, J. E.

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and Control of Mid-Spatial Frequency Wavefront Errors in Optical Systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
[Crossref]

Evans, C. J.

L. He, C. J. Evans, and A. Davies, “Optical surface characterization with the area structure function,” CIRP Annals -Manufacturing Technology,  62(1), 539–542 (2013).
[Crossref]

L. He, C. J. Evans, and A. Davies, “Two-quadrant area structure function analysis for optical surface characterization,” Opt. Express 20, 23275–23280 (2012).
[Crossref] [PubMed]

Forbes, G. W.

M. A. Alonso and G. W. Forbes, “The Strehl ratio as the Fourier transform of a probability density,” Opt. Lett. 41, 3735–3738 (2016).
[Crossref] [PubMed]

G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Optical Measurement Systems for Industrial Inspection IX, Proc. of SPIE9525, 95251B (2015).

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley & Sons, 2015), Chap. 8.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005), Chap. 6.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryxhik, Table of Integrals, Series, and Products (Academic, 1980), p. 739, ET II 360(14).

He, L.

L. He, C. J. Evans, and A. Davies, “Optical surface characterization with the area structure function,” CIRP Annals -Manufacturing Technology,  62(1), 539–542 (2013).
[Crossref]

L. He, C. J. Evans, and A. Davies, “Two-quadrant area structure function analysis for optical surface characterization,” Opt. Express 20, 23275–23280 (2012).
[Crossref] [PubMed]

Hufnagel, R. E.

Mahajan, V. N.

Maréchal, A.

A. Maréchal, “Etude des effets combinés de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. Opt. 2, 257–277 (1947).

Milster, T. D.

J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49, 4895 (2010).

Noll, R. J.

R. J. Noll, “Effect of Mid- and High-Spatial Frequencies on Optical Performance,” Opt. Eng. 18(2), 182137 (1979).
[Crossref]

Parks, R. E.

R. E. Parks and M. T. Tuell, “The structure function as a metric for roughness and figure,” Proc. of SPIE 995199510 (2016).

Ryxhik, I. M.

I. S. Gradshteyn and I. M. Ryxhik, Table of Integrals, Series, and Products (Academic, 1980), p. 739, ET II 360(14).

Stanley, N. R.

Stone, B. D.

Tamkin, J. M.

J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49, 4895 (2010).

Tuell, M. T.

R. E. Parks and M. T. Tuell, “The structure function as a metric for roughness and figure,” Proc. of SPIE 995199510 (2016).

Youngsworth, R. N.

Youngworth, R. N.

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and Control of Mid-Spatial Frequency Wavefront Errors in Optical Systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
[Crossref]

Appl. Opt. (2)

J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49, 4895 (2010).

R. N. Youngsworth 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]

CIRP Annals -Manufacturing Technology (1)

L. He, C. J. Evans, and A. Davies, “Optical surface characterization with the area structure function,” CIRP Annals -Manufacturing Technology,  62(1), 539–542 (2013).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

R. J. Noll, “Effect of Mid- and High-Spatial Frequencies on Optical Performance,” Opt. Eng. 18(2), 182137 (1979).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Proc. of SPIE (1)

R. E. Parks and M. T. Tuell, “The structure function as a metric for roughness and figure,” Proc. of SPIE 995199510 (2016).

Proc. SPIE (1)

J. V. Baliga and B. D. Cohn, “Simplified method for calculating encircled energy,” Proc. SPIE 892, 152 (1988).
[Crossref]

Rev. Opt. (1)

A. Maréchal, “Etude des effets combinés de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. Opt. 2, 257–277 (1947).

Other (6)

J. W. Goodman, Statistical Optics (John Wiley & Sons, 2015), Chap. 8.

I. S. Gradshteyn and I. M. Ryxhik, Table of Integrals, Series, and Products (Academic, 1980), p. 739, ET II 360(14).

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005), Chap. 6.

G. D. Boremann, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE, 2001), pp. 85–88.
[Crossref]

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and Control of Mid-Spatial Frequency Wavefront Errors in Optical Systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
[Crossref]

G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Optical Measurement Systems for Industrial Inspection IX, Proc. of SPIE9525, 95251B (2015).

Supplementary Material (3)

NameDescription
» Visualization 1       This shows the autocorrelating groove structures (and their difference), the PDPD, and the magnitude of the Fourier transform of the PDPD in the right, center, and left columns, respectively.
» Visualization 2       This shows the three different regions in the overlap region of two autocorrelating pupils, and the corresponding areas in the space with the change of coordinates.
» Visualization 3       This shows the comparison between the MTFs generated analytically and numerically. Also shown are the components to the analytic MTF, which relate to the three regions on the overlap region.

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Figures (5)

Fig. 1
Fig. 1 The PDPD functions for one-dimensional (a) rectangular, (b) triangular, (c) sinusoidal, and (d) piecewise parabolic periodic groove patterns. The left column shows two copies of the groove pattern (blue and orange) separated by ρ, and their difference (purple). The center and right columns are the corresponding PDPD and the modulus its Fourier transform (solid green). The dashed and dotted green lines show the approximations 1 − k2D(ρ)2/2 and exp[−k2D(ρ)2/2], described in Section 6. Here, we used ρ = T/6. Visualization 1 shows the same information for ρ varing over a complete period.
Fig. 2
Fig. 2 (a) Circular pupil whose error presents five parabolic grooves of height h across its diameter, aligned in the vertical direction. (b-d) MTF as a function of both kh and spatial frequency for different radial slices, corresponding to the directions of the orange, green, and blue arrows shown in (a). The axis variable ρ/R equals the normalized spatial frequency f/fc, where f is the spatial frequency in the direction of the radial slice and fc is the coherent cutoff frequency for the circular pupil.
Fig. 3
Fig. 3 Diagrams representing the various coordinates, for ρ < R (a,b) and ρR (c,d). In (a) and (c), two copies of the pupil with a rotationally symmetric periodic groove pattern are shown, mutually displaced by ρ. The distances to a given point q from the centers of the pupils are r1 and r2. The overlap region is mapped onto the ( ρ ¯ , r ) plane in (b) and (d), taking the shape of a polygon whose boundaries are given in Eqs. (10) and (11). The elliptic and hyperbolic (red and blue) regions in the (qx, qy) space over which both sets of grooves are nearly parallel map onto straight bands (red and blue) in the ( ρ ¯ , r ) space. Visualization 2 shows the same information for varying ρ.
Fig. 4
Fig. 4 MTF for four concentric circular groove shapes with R = 5T: (a) rectangular with a fill factor of 1/2, (b) triangular, (c) sinusoidal, and (d) piecewise parabolic. For each, the upper plot was generated from Eq. (17), and the lower plot resulted from a numerical calculation.
Fig. 5
Fig. 5 MTF for the same circular groove shapes used in Fig. 4, for kh = 1.6. In each figure, the orange dots correspond to a numerical computation, of the MTF, the black curve corresponds to the estimate in Eq. (17), and the blue, red and green curves correspond to the contributions from each of the terms in Eq. (17), in that order. The gray curves show the baseline MTF perf ( ρ ) P ˜ 0 ( k ). Visualization 3 shows the same information for varying kh.

Tables (2)

Tables Icon

Table 1 Four groove shapes with their PDPD (second column) and the PDPD Fourier transform (third column).

Tables Icon

Table 2 Fourier expansion coefficients (second and third columns) for the functions P ˜ ( k , ρ ) in Table 1, corresponding to the groove structures shown in the first column. The fourth column shows an estimate of how many Fourier terms are needed (m ≤ Max(0, m10%)) for the truncated Fourier series to achieve an error below 10%.

Equations (57)

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OTF ( k , ρ ) = A ( q ρ / 2 ) e i k W ( q ρ / 2 ) A ( q + ρ / 2 ) e i k W ( q + ρ / 2 ) d 2 q A 2 ( q ) d 2 q ,
OTF ( k , ρ ) = 1 a O ( ρ ) e i k Δ W ( q , ρ ) d 2 q ,
P ( η , ρ ) = O ( ρ ) δ [ η Δ W ( q , ρ ) ] d 2 q a OTF perf ( ρ ) ,
OTF ( k , ρ ) = 1 a O ( ρ ) δ [ η Δ W ( q , ρ ) ] d 2 q e i k η d η = OTF perf ( ρ ) P ˜ ( k , ρ ) ,
P ˜ ( k , ρ ) = P ( η , ρ ) e i k η d η .
MTF ( k , ρ ) = MTF perf ( ρ ) | P ˜ ( k , ρ ) | , MTF perf ( ρ ) = | OTF perf ( ρ ) | .
MTF ( k , ρ ) MTF perf ( ρ ) | P ˜ 1 ( k , ρ x ) | .
r = r 1 + r 2 2 , ρ ¯ = r 2 r 1 .
J ( r , ρ ¯ ) = r 2 ( ρ / 2 ) 2 ρ 2 ρ ¯ 2 + 1 4 ρ 2 ρ ¯ 2 r 2 ( ρ / 2 ) 2 .
ρ 2 r R | ρ ¯ | 2 and 0 ρ ¯ R | R ρ | ,
0 ρ ¯ Min [ ρ , 2 ( R r ) ] and ρ 2 r R ,
P ( η , ρ ) = 1 F ( ρ ) O ( ρ ) J ( r , ρ ¯ ) δ ¯ ( η , r , ρ ¯ ) d r d ρ ¯ ,
P 1 ( η , ρ ¯ ) = 1 T 0 T δ ¯ ( η , r , ρ ¯ ) d r .
P ¯ 1 ( η , r ) = 1 2 T 0 2 T δ ¯ ( η , r , ρ ¯ ) d ρ ¯ .
P 0 ( η ) = 1 T 0 T P 1 ( η , ρ ¯ ) d ρ ¯ = 2 T 0 T / 2 P ¯ ( η , r ) d r ,
P ( η , ρ ) 1 F ( ρ ) [ A ( ρ ) 2 Q ^ t P 1 ( η , ρ ) + B ( ρ ) 2 Q ^ τ P ¯ 1 ( η , ρ / 2 ) + E ( ρ ) P 0 ( η ) ] ,
OTF ( k , ρ ) 4 [ A ( ρ ) 2 Q ^ t P ˜ 1 ( k , ρ ) + B ( ρ ) 2 Q ^ τ P ¯ ˜ 1 ( k , ρ / 2 ) + E ( ρ ) P ˜ 0 ( k ) ] .
Q ^ x f ( ρ ) = 0 1 f ( ρ x v ) v d v .
f ( ρ ) = 1 2 a 0 + m = 1 a m cos ( 2 π m ρ T ) ,
Q ^ x f ( ρ ) = a 0 + T | x | m = 1 a m m [ C ( 2 m | x | T ) cos ( 2 π m ρ T ) + sgn ( x ) S ( 2 m | x | T ) sin ( 2 π m ρ T ) ] ,
P ( η , ρ ) = ( P 1 P 2 P n ) ( η , ρ )
MTF ( k , ρ ) = MTF perf ( ρ ) i = 1 n | P ˜ i ( k , ρ ) | .
D ( ρ ) = O ( ρ ) Δ W 2 ( q , ρ ) d 2 q = η 2 P ( η , ρ ) d η .
MTF ( k , ρ ) MTF perf ( ρ ) | P ( η , ρ ) ( 1 + i k η k 2 η 2 2 ) d η | = MTF perf ( ρ ) [ 1 k 2 D ( ρ ) 2 ] ,
MTF ( k , ρ ) MTF perf ( ρ ) exp [ k 2 D ( ρ ) 2 ] .
EE ( k , ρ ) = 1 OTF ( k , 0 ) S O ( 0 ) OTF ( k , ρ ) S O ( ρ ) S D ( ρ ) d 2 ρ ,
I 1 = 0 R | R ρ | 1 ρ 2 ρ ¯ 2 ρ / 2 R ρ ¯ / 2 δ ¯ ( η , r , ρ ¯ ) r 2 ( ρ / 2 ) 2 d r d ρ ¯ ,
I 2 = 1 4 ρ / 2 R 1 r 2 ( ρ / 2 ) 2 0 Min [ ρ , 2 ( R r ) ] δ ¯ ( η , r , ρ ¯ ) ρ 2 ρ ¯ 2 d ρ ¯ d r .
ρ / 2 R ρ ¯ / 2 δ ¯ ( η , r , ρ ¯ ) r 2 ( ρ / 2 ) 2 d r 0 T δ ¯ ( η , ρ / 2 + v , ρ ¯ ) n = 0 N 1 r n 2 ( ρ / 2 ) 2 d v ,
n = 0 N 1 r n 2 ( ρ / 2 ) 2 1 T ρ / 2 R ρ ¯ / 2 r 2 ( ρ / 2 ) 2 d r .
I 1 0 R | R ρ | P 1 ( η , ρ ¯ ) α ( ρ , ρ ¯ ) ρ ρ ¯ d ρ ¯ , α ( ρ , ρ ¯ ) = ρ / 2 R ρ ¯ / 2 r 2 ( ρ / 2 ) 2 ρ + ρ ¯ d r .
I 2 ρ / 2 R P ¯ 1 ( η , r ) β ( ρ , r ) r ρ / 2 d r , β ( ρ , r ) = 1 4 0 Min [ ρ , 2 ( R r ) ] ρ 2 ρ ¯ 2 r + ρ / 2 d ρ ¯ .
I 2 0 T / 2 P ¯ 1 ( η , ρ / 2 + u ) m = 0 M 1 β ( ρ , r m ) r m ρ / 2 d u .
I 2 0 T / 2 P ¯ 1 ( η , ρ / 2 + u ) u β ( ρ , ρ / 2 + u ) d u + 0 T / 2 P ¯ 1 ( η , ρ / 2 + u ) m = 1 M 1 β ( ρ , r m ) r m ρ / 2 d u β ( ρ , ρ / 2 ) 0 T / 2 P ¯ 1 ( η , ρ / 2 + u ) u d u + 0 T / 2 P ¯ 1 ( η , ρ / 2 + u ) 2 T ρ / 2 + T / 2 R β ( ρ , r ) r ρ / 2 d r d u = β ( ρ , ρ / 2 ) 0 T / 2 P ¯ 1 ( η , ρ / 2 + u ) u d u + P 0 ( η ) ρ / 2 + T / 2 R β ( ρ , r ) r ρ / 2 d r ,
I 1 { α ( ρ , ρ ) 0 T P 1 ( η , ρ u ) u d u + P 0 ( η ) 0 ρ T α ( ρ , ρ ¯ ) ρ , ρ ¯ d ρ ¯ , ρ R , P 0 ( η ) 0 2 R ρ α ( ρ , ρ ¯ ) ρ , ρ ¯ d ρ ¯ , ρ > R ,
β ( ρ , ρ / 2 ) = 1 4 0 Min [ ρ , 2 ( R r ) ] ρ 2 ρ ¯ 2 ρ d ρ ¯ 1 2 T ρ / 2 ρ / 2 + T / 2 0 Min [ ρ , 2 ( R r ) ] J ( r , ρ ¯ ) d ρ ¯ d r = B ( ρ ) 2 T ,
α ( ρ , ρ ) A ( ρ ) 2 T ,
α ( ρ , ρ ) A ( ρ ) 2 ρ .
β ( ρ , ρ / 2 ) B ( ρ ) 2 ( R ρ / 2 ) .
A ( ρ ) = 1 2 [ V U X X 2 U 2 U V arccosh ( X U ) + R 2 arccos ( X R ) X R 2 X 2 ] ,
B ( ρ ) = π 4 U + V + ,
A ( ρ ) = 0 ,
B ( ρ ) = 1 2 [ V + U + X + U + 2 X + 2 + U + V + arcsin ( X + U + ) + R 2 arccos ( X + R ) X R 2 X + 2 ] ,
U ± = ρ ± T 2 , V ± = T ( 2 ρ ± T ) 2 , X ± = [ 2 R ( ρ ± T ) ] ( ρ ± T ) 2 ρ , X ± = X ± + ρ 2 .
F ( ρ ) = 1 2 [ R 2 arccos ( ρ 2 R ) ρ 2 R 2 ( ρ 2 ) 2 ] ,
E ( ρ ) = F ( ρ ) A ( ρ ) B ( ρ ) .
a m = 2 0 1 P ˜ 1 ( k , u T ) cos ( 2 π m u ) d u ,
a m = 0 1 sin [ 4 k h u ( 1 u ) ] 2 k h u ( 1 u ) cos ( 2 π m u ) d u ,
1 u ( 1 u ) = 1 u + 1 1 u .
a m = 0 1 sin [ 4 k h u ( 1 u ) ] 2 k h u cos ( 2 π m u ) d u + 0 1 sin [ 4 k h u ( 1 u ) ] 2 k h ( 1 u ) cos ( 2 π m u ) d u .
a m = 0 1 sin [ 4 k h u ( 1 u ) ] 2 k h u cos ( 2 π m u ) d u + 0 1 sin [ 4 k h u ( 1 u ) ] 2 k h u cos [ 2 π m ( 1 u ) d u .
a m = 0 1 sin [ 4 k h u ( 1 u ) ] k h u cos ( 2 π m u ) d u = 1 1 sin [ 4 k h u ( 1 | u | ) ] 2 k h u cos ( 2 π m u ) d u .
a m = 1 1 1 + | u | 1 | u | exp ( i 4 k h u u ) d u cos ( 2 π m u ) d u .
a m = 1 2 1 1 1 1 exp [ i k h ( u 1 2 u 2 2 ) ] cos [ π m ( u 1 + u 2 ) ] d u 1 d u 2 .
a m = 1 2 { 1 1 exp [ i k h ( u 1 2 u 2 2 ) ] cos ( π m u 1 ) cos ( π m u 2 ) d u 1 d u 2 1 1 exp [ i k h ( u 1 2 u 2 2 ) ] sin ( π m u 1 ) sin ( π m u 2 ) d u 1 d u 2 } = 1 2 [ | 1 1 exp ( i k h u 2 ) cos ( π m u ) d u | 2 | 1 1 exp ( i k h u 2 ) sin ( π m u ) d u | 2 ] = 1 2 | 1 1 exp ( i k h u 2 ) cos ( π m u ) d u | 2 ,
a m = π 8 | k h | | Erfi [ ( 1 + i ) ( 2 k h m π ) 4 k h / 2 ] + Erfi [ ( 1 + i ) ( 2 k h + m π ) 4 k h / 2 ] | 2 ,
a 0 = π 2 k h [ C 2 ( 2 k h π ) + S 2 ( 2 k h π ) ] .

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