The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems

Peter R.T. Munro; Konstantin Ignatyev; Robert D. Speller; Alessandro Olivo

doi:10.1364/OE.18.004103

The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems

Peter R.T. Munro, Konstantin Ignatyev, Robert D. Speller, and Alessandro Olivo

Optics Express
Vol. 18,
Issue 5,
pp. 4103-4117
(2010)
•https://doi.org/10.1364/OE.18.004103

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Peter R.T. Munro, Konstantin Ignatyev, Robert D. Speller, and Alessandro Olivo, "The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems," Opt. Express 18, 4103-4117 (2010)

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Abstract

X-ray phase contrast imaging is a very promising technique which may lead to significant advancements in medical imaging. One of the impediments to the clinical implementation of the technique is the general requirement to have an x-ray source of high coherence. The radiation physics group at UCL is currently developing an x-ray phase contrast imaging technique which works with laboratory x-ray sources. Validation of the system requires extensive modelling of relatively large samples of tissue. To aid this, we have undertaken a study of when geometrical optics may be employed to model the system in order to avoid the need to perform a computationally expensive wave optics calculation. In this paper, we derive the relationship between the geometrical and wave optics model for our system imaging an infinite cylinder. From this model we are able to draw conclusions regarding the general applicability of the geometrical optics approximation.

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References

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Publication

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004).
[Crossref] [PubMed]
E. Castelli, F. Arfelli, D. Dreossi, R. Longo, T. Rokvic, M. Cova, E. Quaia, M. Tonutti, F. Zanconati, A. Abrami, V. Chenda, R. Menk, E. Quai, G. Tromba, P. Bregant, and F. de Guarrini, “Clinical mammography at the SYRMEP beam line,” Nucl. Instrum. Meth. A 572(1), 237–240 (2007).
[Crossref]
A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074,106 (2007).
[Crossref]
A. Olivo and R. Speller, “Phase contrast imaging.”, International Patent WO/2008/029107, (2008).
A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94(4) (2009).
[Crossref]
A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52(22), 6555–6573 (2007).
[Crossref] [PubMed]
T. Gureyev and S. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15(3), 579–585 (1998).
[Crossref]
F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]
M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90(22), 224101 (pages 3) (2007).
[Crossref]
M. Engelhardt, C. Kottler, o. Bunk, C. David, C. Schroer, J. Baumann, m. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232, 145–157 (2008).
[Crossref] [PubMed]
A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 242(Part 2, No. 7B), L866–L868 (2003).
[Crossref]
A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32(12), 3617–3627 (2005).
[Crossref]
J. B. Keller, “Geometrical Theory of Diffraction,” J. Opt. Soc. Am. A 52(2), 116–130 (1962).
[Crossref]
G. James, Geometrical theory of diffraction for electromagnetic waves (Peter Peregrinus Ltd., 1976).
J. Arsac, Fourier transforms and the theory of distributions (Prentice-Hall, 1966).
A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system,” Phys. Med. Biol. 51(12), 3015–3030 (2006).
[Crossref] [PubMed]
A. Erdélyi, ed., Tables of integral transforms : based, in part, on notes left by Harry Bateman and compiled by the staff of the Bateman Manuscript Project, vol. I (New York; London: McGraw-Hill, 1954).
F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Val-lazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998).
[Crossref] [PubMed]
G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic Press, Boston, 1985).
A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53(22), 6461–6474 (2008).
[Crossref] [PubMed]
M. Born and E. Wolf, Principles of Optics, seventh ed. (Cambridge University Press, Cambridge, 1999).
J. Murray, Asymptotic analysis (Springer Verlag, 1984).
[Crossref]
R. Buchal and J. Keller, “Boundary layer problems in diffraction theory,” Commun. Pur. Appl. Math. 13, 85–114 (1960).
[Crossref]

2009 (1)

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94(4) (2009).
[Crossref]

2008 (2)

A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53(22), 6461–6474 (2008).
[Crossref] [PubMed]

M. Engelhardt, C. Kottler, o. Bunk, C. David, C. Schroer, J. Baumann, m. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232, 145–157 (2008).
[Crossref] [PubMed]

2007 (4)

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90(22), 224101 (pages 3) (2007).
[Crossref]

E. Castelli, F. Arfelli, D. Dreossi, R. Longo, T. Rokvic, M. Cova, E. Quaia, M. Tonutti, F. Zanconati, A. Abrami, V. Chenda, R. Menk, E. Quai, G. Tromba, P. Bregant, and F. de Guarrini, “Clinical mammography at the SYRMEP beam line,” Nucl. Instrum. Meth. A 572(1), 237–240 (2007).
[Crossref]

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074,106 (2007).
[Crossref]

A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52(22), 6555–6573 (2007).
[Crossref] [PubMed]

2006 (2)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system,” Phys. Med. Biol. 51(12), 3015–3030 (2006).
[Crossref] [PubMed]

2005 (1)

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32(12), 3617–3627 (2005).
[Crossref]

2004 (1)

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004).
[Crossref] [PubMed]

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 242(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

1998 (2)

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Val-lazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998).
[Crossref] [PubMed]

T. Gureyev and S. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15(3), 579–585 (1998).
[Crossref]

1962 (1)

J. B. Keller, “Geometrical Theory of Diffraction,” J. Opt. Soc. Am. A 52(2), 116–130 (1962).
[Crossref]

1960 (1)

R. Buchal and J. Keller, “Boundary layer problems in diffraction theory,” Commun. Pur. Appl. Math. 13, 85–114 (1960).
[Crossref]

Abrami, A.

Arfelli, F.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic Press, Boston, 1985).

Arsac, J.

J. Arsac, Fourier transforms and the theory of distributions (Prentice-Hall, 1966).

Assante, M.

Baumann, J.

Bohndiek, S. E.

Bonvicini, V.

Bravin, A.

Bregant, P.

Buchal, R.

R. Buchal and J. Keller, “Boundary layer problems in diffraction theory,” Commun. Pur. Appl. Math. 13, 85–114 (1960).
[Crossref]

Bunk, o.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Cantatore, G.

Castelli, E.

Chenda, V.

Cova, M.

Dalla Palma, L.

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Dreossi, D.

Engelhardt, M.

Griffiths, J. A.

Guarrini, F. de

Gureyev, T.

T. Gureyev and S. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15(3), 579–585 (1998).
[Crossref]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 242(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

James, G.

G. James, Geometrical theory of diffraction for electromagnetic waves (Peter Peregrinus Ltd., 1976).

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 242(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Keller, J.

R. Buchal and J. Keller, “Boundary layer problems in diffraction theory,” Commun. Pur. Appl. Math. 13, 85–114 (1960).
[Crossref]

Keller, J. B.

J. B. Keller, “Geometrical Theory of Diffraction,” J. Opt. Soc. Am. A 52(2), 116–130 (1962).
[Crossref]

Konstantinidis, A.

Kottler, C.

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 242(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Lewis, R.

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004).
[Crossref] [PubMed]

Longo, R.

Menk, R.

Michiel, M. Di

Momose, A.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 242(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Murray, J.

J. Murray, Asymptotic analysis (Springer Verlag, 1984).
[Crossref]

Olivo, A.

A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53(22), 6461–6474 (2008).
[Crossref] [PubMed]

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074,106 (2007).
[Crossref]

A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52(22), 6555–6573 (2007).
[Crossref] [PubMed]

A. Olivo and R. Speller, “Phase contrast imaging.”, International Patent WO/2008/029107, (2008).

Pani, S.

Peterzol, A.

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Pontoni, D.

Poropat, P.

Prest, M.

Quai, E.

Quaia, E.

Rashevsky, A.

Rigon, L.

Rokvic, T.

Schroer, C.

Schuster, m.

Speller, R.

A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53(22), 6461–6474 (2008).
[Crossref] [PubMed]

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074,106 (2007).
[Crossref]

A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52(22), 6555–6573 (2007).
[Crossref] [PubMed]

A. Olivo and R. Speller, “Phase contrast imaging.”, International Patent WO/2008/029107, (2008).

Speller, R. D.

Suzuki, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 242(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 242(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Tonutti, M.

Tromba, G.

Vacchi, A.

Val-lazza, E.

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Wilkins, S.

T. Gureyev and S. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15(3), 579–585 (1998).
[Crossref]

Zanconati, F.

Appl. Phys. Lett. (3)

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074,106 (2007).
[Crossref]

Commun. Pur. Appl. Math. (1)

R. Buchal and J. Keller, “Boundary layer problems in diffraction theory,” Commun. Pur. Appl. Math. 13, 85–114 (1960).
[Crossref]

J. Microsc. (1)

J. Opt. Soc. Am. A (2)

J. B. Keller, “Geometrical Theory of Diffraction,” J. Opt. Soc. Am. A 52(2), 116–130 (1962).
[Crossref]

T. Gureyev and S. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15(3), 579–585 (1998).
[Crossref]

Jpn. J. Appl. Phys. (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 242(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Med. Phys. (1)

Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Nucl. Instrum. Meth. A (1)

Phys. Med. Biol. (5)

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004).
[Crossref] [PubMed]

A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52(22), 6555–6573 (2007).
[Crossref] [PubMed]

A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53(22), 6461–6474 (2008).
[Crossref] [PubMed]

Other (7)

M. Born and E. Wolf, Principles of Optics, seventh ed. (Cambridge University Press, Cambridge, 1999).

J. Murray, Asymptotic analysis (Springer Verlag, 1984).
[Crossref]

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic Press, Boston, 1985).

A. Erdélyi, ed., Tables of integral transforms : based, in part, on notes left by Harry Bateman and compiled by the staff of the Bateman Manuscript Project, vol. I (New York; London: McGraw-Hill, 1954).

G. James, Geometrical theory of diffraction for electromagnetic waves (Peter Peregrinus Ltd., 1976).

J. Arsac, Fourier transforms and the theory of distributions (Prentice-Hall, 1966).

A. Olivo and R. Speller, “Phase contrast imaging.”, International Patent WO/2008/029107, (2008).

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Fig. 1.

Schematic diagram of imaging system including reference frames used in the paper. Note that (x̄,ȳ,z), (ξ, ψ, z) and (x, y, z) all form right handed coordinate systems. The imaging system is assumed to have no y dependence. Note that dL is defined by the displacement between the detector apertures and the projection of the sample apertures onto the detector apertures.

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Fig. 2.

Diagram illustrating the phase function ϕ(ξ) of a phase object of extent z_ob in the z direction.

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Fig. 3.

Plot of K(x) for some values of source FWHM, given in the legend in μm. Values of M = 1.25 and L = 40μm were used.

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Fig. 4.

Three cases which must be considered in order to evaluate the integral in Eq. (8).

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Fig. 5.

Diagram illustrating the three regions which must be considered when analysing the field incident upon the detector apertures.

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Fig. 6.

Contours of Δx′ for a variety of values of R and δ. This diagram effectively shows the minimum separation required between adjacent sample/detector aperture pairs to ensure detector apertures principally detect photons originating from their associated sample aperture.

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Fig. 7.

Plot of the difference between intensities calculated using the wave optics (full expression evaluated numerically) and geometrical optics approximations. The sensitive region of the pixel is shaded. Simulation parameters were the same as in Fig. 8.

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Fig. 8.

Plot of the intensity of the field incident upon the detector apertures for the geometrical and wave optics (full expression evaluated numerically) solutions. Simulation parameters used were R = 5μm and ε = 10⁻⁷, all other parameters were as described in Sec. (6.3).

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Fig. 9.

Plot of the intensity of the field incident upon the detector apertures as calculated using the exact and approximate wave optics formulations. Simulation parameters were the same as in Fig. 8.

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Fig. 10.

Plot wave optics (WO, full expression evaluated numerically) and geometrical optics (GO) XPCi signal traces for a cylinder of radius 5μm and δ = 10⁻⁶, for three different values of source FWHM. The cylinder is scanned from ξ ₀ = −L/4 − R to ξ ₀ = 0. The signals have been normalised by the signal for the object free case.

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Fig. 11.

Plots of ε against δ ² R for three different values of δ and a point source (left) and a source of FWHM 50μm.

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Fig. 12.

Contour plot of the error between the normalised XPCi signals as calculated by geometrical and wave optics models. Source FWHM is on the vertical axis and δ ² R is on the horizontal axis.

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Tables (1)

Table 1. Summary of components contributing to the field and intensity in the different regions for the case of wave and geometrical optics solutions respectively. Note that objects illuminated by adjacent sample apertures may be modelled by replacing each term I _1,0 and I _1,0 ² with summations, Σ_i I ⁱ _1,0 and Σ_i(I ⁱ _1,0)², over all objects i respectively.

View Table

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

U_{i} (P) = C \int \int_{𝓐} \exp (ik (ξ^{2} + ψ^{2}) \frac{z_{s_{o}} + z_{od}}{2 z_{so} z_{od}}) \exp (- ik (\frac{ξx′ + ψy}{z_{od}})) dξdψ

x′ = x + x_{s} z_{od} / z_{so}

C = - \frac{i}{λ z_{so} z_{od}} \exp (ik (z_{so} + z_{od})) \exp (ik (\frac{x_{s}^{2}}{2 z_{so}} + \frac{x^{2} + y^{2}}{2 z_{od}}))

k = 2 π / λ

Γ (y) = \int_{- \infty}^{\infty} \exp (ik (\frac{z_{s_{o}} + z_{od}}{2 z_{so} z_{od}} ψ^{2} - \frac{y}{z_{od}} ψ)) d ψ

I = \int_{- \infty}^{\infty} f (x) \exp (ikg (x)) dx

I_{0} \sim \sqrt{\frac{2 π}{k ∣ g'' (x_{0}) ∣}} f (x_{0}) \exp (i [kg (x_{0}) + \frac{π}{4} sgn (g'' (x_{0}))])

Γ (y) \sim \sqrt{\frac{i z_{so} z_{od} λ}{z_{so} + z_{od}}} \exp (- ik y^{2} \frac{z_{so}}{2 z_{od} (z_{so} + z_{od})})

U_{i} (P) = C Γ (y) \int_{- \infty}^{\infty} T (ξ) \exp (ik ξ^{2} \frac{z_{s_{o}} + z_{od}}{2 z_{so} z_{od}}) \exp (- ik ξ \frac{x′}{z_{od}}) d ξ

U (P) = U_{i} (P) + \int_{- ℬ} T (ξ) [\exp (iϕ (ξ)) - 1] \exp (ik ξ^{2} \frac{z_{s_{o}} + z_{od}}{2 z_{so} z_{od}}) \exp (- ik ξ \frac{x′}{z_{od}}) d ξ

T (ξ) = \sum_{n = - \infty}^{\infty} C_{n} \exp (i 2 πn \frac{ξ}{L})

U_{i} (P) = C Γ (y) \sqrt{\frac{i λ_{z_{so} z_{od}}}{z_{so} + z_{od}}} \exp (- ik \frac{{x′}^{2} z_{so}}{2 z_{od} (z_{so} + z_{od})}) \sum_{n = - \infty}^{\infty} \exp (- iπλ {(\frac{n}{L})}^{2} \frac{z_{so} z_{od}}{z_{so} + z_{od}}) C_{n} \exp (i 2 π \frac{n}{L} x′ \frac{z_{so}}{z_{so} + z_{od}})

G (f_{p}) = \frac{1}{2 N} \sum_{κ = 0}^{2 N - 1} g (t_{κ}) \exp (i 2 π f_{p} t_{κ}), t_{κ} = \frac{κT}{2 N}, p = 0,1, \dots, 2 N - 1

g (t_{κ}) = \sum_{p = 0}^{2 N - 1} G (f_{p}) \exp (- i 2 π f_{p} t_{κ}), f_{p} = \frac{p}{T}, κ = 0,1, \dots, 2 N - 1

w = [\begin{matrix} w_{0} & w_{1} & w_{2} & \dots & w_{N} & w_{- N + 1} & w_{- N + 2} & w_{- N + 1} \end{matrix}]

w_{n} = C_{n} \exp (- iπλ {(\frac{n}{L})}^{2} \frac{z_{so} z_{od}}{z_{so} + z_{od}})

n \frac{d r}{d s} = \nabla 𝓢

ϕ (ξ) = k \int_{- z_{ob} / 2}^{z_{ob} / 2} n (ξ, z) d z

α (ξ) = \frac{1}{k} \frac{dϕ}{d ξ}

x = z_{so} \tan (θ_{i}) + z_{od} \tan (θ_{i} + α (ξ))

∣ \frac{U (z = z_{od})}{U (z = 0)} ∣ = \sqrt{∣ \frac{d ξ}{d x} ∣} = \sqrt{\frac{1}{1 + \frac{z_{od}}{z_{so}} + z_{od} \sec^{2} (α) \frac{d α}{d ξ}}}

S (p) = \int_{(p - 1 / 4) LM + dL}^{(p + 1 / 4) LM + dL} [\int_{- \infty}^{\infty} P (\bar{x}) I (x + \bar{x} \frac{z_{od}}{z_{so}}) d \bar{x}] d x

S (p) = σ \int_{- \infty}^{\infty} I (x) K (x) d x

K (x) = \frac{\sqrt{π}}{2} {\erf [\frac{z_{so}}{z_{od} σ} (x - (p - 1 / 4) LM - dL)] - \erf [\frac{z_{so}}{z_{od} σ} (x - (p + 1 / 4) LM - dL)]}

\erf (z) = \frac{2}{\sqrt{π}} \int_{0}^{z} \exp ({- t}^{2}) d t

ϕ (ξ) = - k \int_{cylinder} δ d z = - 2 kδ \sqrt{R^{2} - {(ξ - ξ_{0})}^{2}}

U (P) = U_{i} + C Γ (y) [U_{1} - U_{2}]

U_{1} = \int_{Ω} \exp [ik (\frac{ξ^{2} M}{2 z_{od}} - \frac{x′ ξ}{z_{od}} - 2 δ \sqrt{R^{2} - {(ξ - ξ_{0})}^{2}})] d ξ

U_{2} = \int_{Ω} \exp [ik (\frac{ξ^{2} M}{2 z_{od}} - \frac{x′ ξ}{z_{od}})] d ξ

I_{1} \sim {[\frac{f \exp (ikg)}{ikg′}]}_{a}^{b}

g_{1} = \frac{M ξ^{2}}{2 z_{od}} - \frac{x′ξ}{z_{od}} - 2 δ \sqrt{R^{2} - {(ξ - ξ_{0})}^{2}} g_{2} = \frac{M ξ^{2}}{2 z_{od}} - \frac{x′ξ}{z_{od}}

{g′}_{1} = - \frac{x′}{z_{od}} + \frac{Mξ}{z_{od}} + \frac{2 δ (ξ - ξ_{0})}{\sqrt{R^{2} - {(ξ - ξ_{0})}^{2}}} {g′}_{2} = \frac{x′}{z_{od}} + \frac{Mξ}{z_{od}}

{g″}_{1} = \frac{M}{z_{od}} + \frac{2 δ {(ξ - ξ_{0})}^{2}}{{(R^{2} - {(ξ - ξ_{0})}^{2})}^{3 / 2}} + \frac{2 δ}{\sqrt{R^{2} - {(ξ - ξ_{0})}^{2}}} {g″}_{2} = \frac{M}{z_{od}}

I_{1,0} \sim \sqrt{\frac{2 π}{k ∣ {g″}_{1} (ξ_{1,0}) ∣}} \exp (i [{kg}_{1} (ξ_{1,0}) + \frac{π}{4}])

I_{2,0} \sim \sqrt{iλ z_{od} / M} \exp (- ik \frac{{x′}^{2}}{2 M z_{od}})

I_{2,1} \sim {[\frac{\exp (i {kg}_{2})}{ik {g′}_{2}}]}_{a}^{b}

I_{2,1} \sim \frac{z_{od} \exp (ik {(Ma)}^{2} / (2 z_{od}))}{ik ({x′}^{2} - {(Ma)}^{2})} (i 2 x′ \sin (kx' Ma / z_{od}) - 2 Ma \cos (kx' Ma / z_{od}))

x′ \approx M ξ + z_{od} \frac{2 (ξ - ξ_{0}) δ}{\sqrt{R^{2} - {(ξ - ξ_{0})}^{2}}}

Δ {x′}^{3} = \frac{M}{z_{od}} (1 + 2 \frac{δ z_{od}}{RM}) (4 δ^{2} R z_{od}^{3})

	Region I	Region II	Region III
Geometrical optics, \|U\|²	I ² _1,0	I ² _1,0 + I ² _2,0	I ² _1,0
Wave optics, U	U_i + I _1,0 − I _2,0 − I _2,1	U_i + I _1,0 − I _2,1	U_i + I _1,0 − I _2,1