Abstract

Laser interferometry, as applied in cutting-edge length and displacement metrology, requires detailed analysis of systematic effects due to diffraction, which may affect the measurement uncertainty. When the measurements aim at subnanometer accuracy levels, it is possible that the description of interferometer operation by paraxial and scalar approximations is not sufficient. Therefore, in this paper, we place emphasis on models based on nonparaxial vector beams. We address this challenge by proposing a method that uses the Huygens integral to propagate the electromagnetic fields and ray tracing to achieve numerical computability. Toy models are used to test the method’s accuracy. Finally, we recalculate the diffraction correction for an interferometer, which was recently investigated by paraxial methods.

© 2015 Optical Society of America

Full Article  |  PDF Article

Corrections

Birk Andreas, Giovanni Mana, and Carlo Palmisano, "Vectorial ray-based diffraction integral: erratum," J. Opt. Soc. Am. A 33, 559-560 (2016)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-33-4-559

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References

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2014 (1)

2013 (5)

M. A. Alonso, “Ray-based diffraction calculations using stable aggregate of flexible elements,” J. Opt. Soc. Am. A 30, 1223–1235 (2013).
[Crossref]

A. V. Gitin, “Huygens-Feynman-Fresnel principle as the basis of applied optics,” Appl. Opt. 52, 7419–7434 (2013).
[Crossref]

C. Y. Kee and C. F. Wang, “Efficient GPU implementation of the high-frequency SBR-PO method,” IEEE Antennas Wireless Propagat. Lett. 12, 941–944 (2013).
[Crossref]

B. Andreas, G. Mana, E. Massa, and C. Palmisano, “Modeling laser interferometers for the measurement of the Avogadro constant,” Proc. SPIE 8789, 87890W (2013).

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 1–17 (2013).

2012 (2)

2011 (3)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011).
[Crossref]

N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurement of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

2010 (3)

T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[Crossref]

E. Byckling and J. Simola, “Wave optical calculation of diffraction effects in optical systems,” Opt. Acta 27, 337–344 (2010).
[Crossref]

Y. Tao, H. Lin, and H. Bao, “GPU-based shooting and bouncing ray method for fast RCS prediction,” IEEE Trans. Antennas Propagat. 58, 494–502 (2010).

2009 (1)

2007 (1)

2006 (2)

G. Cavagnero, G. Mana, and E. Massa, “Aberration effects in two-beam laser interferometers,” J. Opt. Soc. Am. A 23, 1951–1959 (2006).
[Crossref]

A. Drezet, J. C. Woehl, and S. Huant, “Diffraction of light by a planar aperture in a metallic screen,” J. Math. Phys. 47, 072901 (2006).
[Crossref]

2002 (3)

G. W. Forbes and M. A. Alonso, “The Holy Grail of ray-based optical modeling,” Proc. SPIE 4832, 186–197 (2002).

M. Harrigan, “General beam propagation through non-orthogonal optical systems,” Proc. SPIE 4832, 379–389 (2002).

C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002).
[Crossref]

1999 (1)

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

1997 (2)

R. Masui, “Effect of diffraction in a Saunders-type optical interferometer,” Metrologia 34, 125–131 (1997).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instr. Meas. 46, 196–200 (1997).
[Crossref]

1995 (1)

1994 (1)

A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
[Crossref]

1991 (1)

J. Baldauf, S. W. Lee, L. Lin, S. K. Jeng, S. M. Scarborough, and C. L. Yu, “High frequency scattering from trihedral corner reflectors and other benchmark targets: SBR versus experiment,” IEEE Trans. Antennas Propagat. 39, 1345–1351 (1991).

1989 (2)

H. Ling, R. C. Chou, and S. W. Lee, “Shooting and bouncing rays: calculating the RCS of an arbitrarily shaped cavity,” IEEE Trans. Antennas Propagat. 37, 194–205 (1989).

G. Mana, “Diffraction effects in optical interferometers illuminated by laser sources,” Metrologia 26, 87–93 (1989).
[Crossref]

1988 (1)

S. W. Lee, H. Ling, and R. C. Chou, “Ray-tube integration in shooting and bouncing ray method,” Microwave Opt. Technol. Lett. 1, 286–289 (1988).
[Crossref]

1986 (1)

S. W. Lee and P. T. Lam, “Correction to diffraction by an arbitrary subreflector: GTD solution,” IEEE Trans. Antennas Propagat. AP-34, 272 (1986).

1985 (2)

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).

J. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. 24, 538–543 (1985).
[Crossref]

1983 (2)

1982 (1)

S. W. Lee, M. S. Sheshadri, V. Jamnejad, and R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Trans. MTT-30, 12–19 (1982).

1981 (1)

1980 (1)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

1979 (2)

S. W. Lee, P. Cramer, K. Woo, and Y. Rahmat-Samii, “Diffraction by an arbitrary subreflector: GTD solution,” IEEE Trans. Antennas Propagat. AP-27, 305–316 (1979).

R. Mittra and A. Rushdi, “An efficient approach for computing the geometrical optics field reflected from a numerically specified surface,” IEEE Trans. Antennas Propagat. 27, 871–877 (1979).

1976 (1)

K. Dorenwendt and G. Bönsch, “Über den Einfluß der Beugung auf die interferentielle Längenmessung,” Metrologia 12, 57–60 (1976).
[Crossref]

1972 (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[Crossref]

1970 (2)

J. N. Dukes and G. B. Gordon, “A two-hundred-foot yardstick with graduations every microinch,” Hewlett-Packard J. 21, 2–8 (1970).

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Lett. 60, 1168–1177 (1970).

1966 (1)

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. A 253, 349–357 (1959).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

1948 (1)

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[Crossref]

1947 (1)

W. R. Smythe, “The double current sheet in diffraction,” Phys. Rev. 72, 1066–1070 (1947).
[Crossref]

1939 (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Alonso, M. A.

M. A. Alonso, “Ray-based diffraction calculations using stable aggregate of flexible elements,” J. Opt. Soc. Am. A 30, 1223–1235 (2013).
[Crossref]

G. W. Forbes and M. A. Alonso, “The Holy Grail of ray-based optical modeling,” Proc. SPIE 4832, 186–197 (2002).

Andreas, B.

B. Andreas, G. Mana, E. Massa, and C. Palmisano, “Modeling laser interferometers for the measurement of the Avogadro constant,” Proc. SPIE 8789, 87890W (2013).

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

Antos, R.

Arnaud, J.

Baldauf, J.

J. Baldauf, S. W. Lee, L. Lin, S. K. Jeng, S. M. Scarborough, and C. L. Yu, “High frequency scattering from trihedral corner reflectors and other benchmark targets: SBR versus experiment,” IEEE Trans. Antennas Propagat. 39, 1345–1351 (1991).

Bao, H.

Y. Tao, H. Lin, and H. Bao, “GPU-based shooting and bouncing ray method for fast RCS prediction,” IEEE Trans. Antennas Propagat. 58, 494–502 (2010).

Bastiaans, M. J.

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

Bekshaev, A. Y.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 1–17 (2013).

Bergamin, A.

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instr. Meas. 46, 196–200 (1997).
[Crossref]

A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 1–17 (2013).

Bönsch, G.

G. Bönsch, “Wavelength ratio of stabilized laser radiation at 3.39  μm and 0.633  μm,” Appl. Opt. 22, 3414–3420 (1983).
[Crossref]

K. Dorenwendt and G. Bönsch, “Über den Einfluß der Beugung auf die interferentielle Längenmessung,” Metrologia 12, 57–60 (1976).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

Burch, J. M.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

Byckling, E.

E. Byckling and J. Simola, “Wave optical calculation of diffraction effects in optical systems,” Opt. Acta 27, 337–344 (2010).
[Crossref]

Cavagnero, G.

G. Cavagnero, G. Mana, and E. Massa, “Aberration effects in two-beam laser interferometers,” J. Opt. Soc. Am. A 23, 1951–1959 (2006).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instr. Meas. 46, 196–200 (1997).
[Crossref]

A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
[Crossref]

Chen, C. G.

Chou, R. C.

H. Ling, R. C. Chou, and S. W. Lee, “Shooting and bouncing rays: calculating the RCS of an arbitrarily shaped cavity,” IEEE Trans. Antennas Propagat. 37, 194–205 (1989).

S. W. Lee, H. Ling, and R. C. Chou, “Ray-tube integration in shooting and bouncing ray method,” Microwave Opt. Technol. Lett. 1, 286–289 (1988).
[Crossref]

Chu, L. J.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Collins, S. A.

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Lett. 60, 1168–1177 (1970).

Cordiali, L.

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instr. Meas. 46, 196–200 (1997).
[Crossref]

Cramer, P.

S. W. Lee, P. Cramer, K. Woo, and Y. Rahmat-Samii, “Diffraction by an arbitrary subreflector: GTD solution,” IEEE Trans. Antennas Propagat. AP-27, 305–316 (1979).

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[Crossref]

Dorenwendt, K.

K. Dorenwendt and G. Bönsch, “Über den Einfluß der Beugung auf die interferentielle Längenmessung,” Metrologia 12, 57–60 (1976).
[Crossref]

Douglas, N. G.

Drezet, A.

A. Drezet, J. C. Woehl, and S. Huant, “Diffraction of light by a planar aperture in a metallic screen,” J. Math. Phys. 47, 072901 (2006).
[Crossref]

Dukes, J. N.

J. N. Dukes and G. B. Gordon, “A two-hundred-foot yardstick with graduations every microinch,” Hewlett-Packard J. 21, 2–8 (1970).

Dürr, F.

Ferrera, J.

Ferroglio, L.

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

Feynman, R. P.

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[Crossref]

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (California Institute of Technology, 2010), http://www.feynmanlectures.caltech.edu .

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Forbes, G. W.

G. W. Forbes and M. A. Alonso, “The Holy Grail of ray-based optical modeling,” Proc. SPIE 4832, 186–197 (2002).

Fujii, K.

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurement of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

Gerrard, A.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

Ghorpade, S. R.

S. R. Ghorpade and B. V. Limaye, A Course in Multivariable Calculus and Analysis (Springer, 2010).

Gillen, G. D.

Gitin, A. V.

Glassner, A. S.

A. S. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gordon, G. B.

J. N. Dukes and G. B. Gordon, “A two-hundred-foot yardstick with graduations every microinch,” Hewlett-Packard J. 21, 2–8 (1970).

Greynolds, A. W.

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).

Guha, S.

Harrigan, M.

M. Harrigan, “General beam propagation through non-orthogonal optical systems,” Proc. SPIE 4832, 379–389 (2002).

Heilmann, R. K.

Herloski, R.

Hillenbrand, M.

Huant, S.

A. Drezet, J. C. Woehl, and S. Huant, “Diffraction of light by a planar aperture in a metallic screen,” J. Math. Phys. 47, 072901 (2006).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Jamnejad, V.

S. W. Lee, M. S. Sheshadri, V. Jamnejad, and R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Trans. MTT-30, 12–19 (1982).

Javan, A.

Jeng, S. K.

J. Baldauf, S. W. Lee, L. Lin, S. K. Jeng, S. M. Scarborough, and C. L. Yu, “High frequency scattering from trihedral corner reflectors and other benchmark targets: SBR versus experiment,” IEEE Trans. Antennas Propagat. 39, 1345–1351 (1991).

Jones, A. R.

Kee, C. Y.

C. Y. Kee and C. F. Wang, “Efficient GPU implementation of the high-frequency SBR-PO method,” IEEE Antennas Wireless Propagat. Lett. 12, 941–944 (2013).
[Crossref]

Kelly, D. P.

Kelly, M. J.

Kofman, A. G.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 1–17 (2013).

Kogelnik, H.

Konkola, P. T.

Kuhn, M.

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011).
[Crossref]

Kuramoto, N.

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurement of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

Kurnit, N. A.

Lam, P. T.

S. W. Lee and P. T. Lam, “Correction to diffraction by an arbitrary subreflector: GTD solution,” IEEE Trans. Antennas Propagat. AP-34, 272 (1986).

Lee, K. S.

Lee, P. H.

Lee, S. W.

J. Baldauf, S. W. Lee, L. Lin, S. K. Jeng, S. M. Scarborough, and C. L. Yu, “High frequency scattering from trihedral corner reflectors and other benchmark targets: SBR versus experiment,” IEEE Trans. Antennas Propagat. 39, 1345–1351 (1991).

H. Ling, R. C. Chou, and S. W. Lee, “Shooting and bouncing rays: calculating the RCS of an arbitrarily shaped cavity,” IEEE Trans. Antennas Propagat. 37, 194–205 (1989).

S. W. Lee, H. Ling, and R. C. Chou, “Ray-tube integration in shooting and bouncing ray method,” Microwave Opt. Technol. Lett. 1, 286–289 (1988).
[Crossref]

S. W. Lee and P. T. Lam, “Correction to diffraction by an arbitrary subreflector: GTD solution,” IEEE Trans. Antennas Propagat. AP-34, 272 (1986).

S. W. Lee, M. S. Sheshadri, V. Jamnejad, and R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Trans. MTT-30, 12–19 (1982).

S. W. Lee, P. Cramer, K. Woo, and Y. Rahmat-Samii, “Diffraction by an arbitrary subreflector: GTD solution,” IEEE Trans. Antennas Propagat. AP-27, 305–316 (1979).

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (California Institute of Technology, 2010), http://www.feynmanlectures.caltech.edu .

Lemmer, U.

Li, T.

Limaye, B. V.

S. R. Ghorpade and B. V. Limaye, A Course in Multivariable Calculus and Analysis (Springer, 2010).

Lin, H.

Y. Tao, H. Lin, and H. Bao, “GPU-based shooting and bouncing ray method for fast RCS prediction,” IEEE Trans. Antennas Propagat. 58, 494–502 (2010).

Lin, L.

J. Baldauf, S. W. Lee, L. Lin, S. K. Jeng, S. M. Scarborough, and C. L. Yu, “High frequency scattering from trihedral corner reflectors and other benchmark targets: SBR versus experiment,” IEEE Trans. Antennas Propagat. 39, 1345–1351 (1991).

Ling, H.

H. Ling, R. C. Chou, and S. W. Lee, “Shooting and bouncing rays: calculating the RCS of an arbitrarily shaped cavity,” IEEE Trans. Antennas Propagat. 37, 194–205 (1989).

S. W. Lee, H. Ling, and R. C. Chou, “Ray-tube integration in shooting and bouncing ray method,” Microwave Opt. Technol. Lett. 1, 286–289 (1988).
[Crossref]

Mana, G.

B. Andreas, G. Mana, E. Massa, and C. Palmisano, “Modeling laser interferometers for the measurement of the Avogadro constant,” Proc. SPIE 8789, 87890W (2013).

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

G. Cavagnero, G. Mana, and E. Massa, “Aberration effects in two-beam laser interferometers,” J. Opt. Soc. Am. A 23, 1951–1959 (2006).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instr. Meas. 46, 196–200 (1997).
[Crossref]

A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
[Crossref]

G. Mana, “Diffraction effects in optical interferometers illuminated by laser sources,” Metrologia 26, 87–93 (1989).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 2008).

Marshall, S.

Massa, E.

B. Andreas, G. Mana, E. Massa, and C. Palmisano, “Modeling laser interferometers for the measurement of the Avogadro constant,” Proc. SPIE 8789, 87890W (2013).

G. Cavagnero, G. Mana, and E. Massa, “Aberration effects in two-beam laser interferometers,” J. Opt. Soc. Am. A 23, 1951–1959 (2006).
[Crossref]

Masui, R.

R. Masui, “Effect of diffraction in a Saunders-type optical interferometer,” Metrologia 34, 125–131 (1997).
[Crossref]

Mittra, R.

S. W. Lee, M. S. Sheshadri, V. Jamnejad, and R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Trans. MTT-30, 12–19 (1982).

R. Mittra and A. Rushdi, “An efficient approach for computing the geometrical optics field reflected from a numerically specified surface,” IEEE Trans. Antennas Propagat. 27, 871–877 (1979).

Monchalin, J.-P.

Nori, F.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 1–17 (2013).

Palmisano, C.

B. Andreas, G. Mana, E. Massa, and C. Palmisano, “Modeling laser interferometers for the measurement of the Avogadro constant,” Proc. SPIE 8789, 87890W (2013).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Rahmat-Samii, Y.

S. W. Lee, P. Cramer, K. Woo, and Y. Rahmat-Samii, “Diffraction by an arbitrary subreflector: GTD solution,” IEEE Trans. Antennas Propagat. AP-27, 305–316 (1979).

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Riechert, F.

Rohlfing, U.

Rolland, J. P.

Rushdi, A.

R. Mittra and A. Rushdi, “An efficient approach for computing the geometrical optics field reflected from a numerically specified surface,” IEEE Trans. Antennas Propagat. 27, 871–877 (1979).

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (California Institute of Technology, 2010), http://www.feynmanlectures.caltech.edu .

Scarborough, S. M.

J. Baldauf, S. W. Lee, L. Lin, S. K. Jeng, S. M. Scarborough, and C. L. Yu, “High frequency scattering from trihedral corner reflectors and other benchmark targets: SBR versus experiment,” IEEE Trans. Antennas Propagat. 39, 1345–1351 (1991).

Schattenburg, M. L.

Schmid, T.

Sheshadri, M. S.

S. W. Lee, M. S. Sheshadri, V. Jamnejad, and R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Trans. MTT-30, 12–19 (1982).

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Simola, J.

E. Byckling and J. Simola, “Wave optical calculation of diffraction effects in optical systems,” Opt. Acta 27, 337–344 (2010).
[Crossref]

Sinzinger, S.

Smith, G. S.

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge University, 1997).

Smythe, W. R.

W. R. Smythe, “The double current sheet in diffraction,” Phys. Rev. 72, 1066–1070 (1947).
[Crossref]

Stratton, J. A.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Szöke, A.

Tamkin, J.

Tao, Y.

Y. Tao, H. Lin, and H. Bao, “GPU-based shooting and bouncing ray method for fast RCS prediction,” IEEE Trans. Antennas Propagat. 58, 494–502 (2010).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Thomas, J. E.

Thompson, K. P.

van Hoesel, F. J.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Visser, T. D.

T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[Crossref]

Walther, A.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, 1995).

Wang, C. F.

C. Y. Kee and C. F. Wang, “Efficient GPU implementation of the high-frequency SBR-PO method,” IEEE Antennas Wireless Propagat. Lett. 12, 941–944 (2013).
[Crossref]

Woehl, J. C.

A. Drezet, J. C. Woehl, and S. Huant, “Diffraction of light by a planar aperture in a metallic screen,” J. Math. Phys. 47, 072901 (2006).
[Crossref]

Wolf, E.

J. P. Rolland, K. P. Thompson, K. S. Lee, and J. Tamkin, T. Schmid and E. Wolf, “Observation of the Gouy phase anomaly in astigmatic beams,” Appl. Opt. 51, 2902–2908 (2012).
[Crossref]

T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[Crossref]

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. A 253, 349–357 (1959).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 2008).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

Woo, K.

S. W. Lee, P. Cramer, K. Woo, and Y. Rahmat-Samii, “Diffraction by an arbitrary subreflector: GTD solution,” IEEE Trans. Antennas Propagat. AP-27, 305–316 (1979).

Wyrowski, F.

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011).
[Crossref]

Yamazawa, K.

N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurement of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
[Crossref]

Yu, C. L.

J. Baldauf, S. W. Lee, L. Lin, S. K. Jeng, S. M. Scarborough, and C. L. Yu, “High frequency scattering from trihedral corner reflectors and other benchmark targets: SBR versus experiment,” IEEE Trans. Antennas Propagat. 39, 1345–1351 (1991).

Zernike, F.

Appl. Opt. (8)

Eur. Phys. J. D (1)

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

Hewlett-Packard J. (1)

J. N. Dukes and G. B. Gordon, “A two-hundred-foot yardstick with graduations every microinch,” Hewlett-Packard J. 21, 2–8 (1970).

IEEE Antennas Wireless Propagat. Lett. (1)

C. Y. Kee and C. F. Wang, “Efficient GPU implementation of the high-frequency SBR-PO method,” IEEE Antennas Wireless Propagat. Lett. 12, 941–944 (2013).
[Crossref]

IEEE Trans. Antennas Propagat. (6)

J. Baldauf, S. W. Lee, L. Lin, S. K. Jeng, S. M. Scarborough, and C. L. Yu, “High frequency scattering from trihedral corner reflectors and other benchmark targets: SBR versus experiment,” IEEE Trans. Antennas Propagat. 39, 1345–1351 (1991).

Y. Tao, H. Lin, and H. Bao, “GPU-based shooting and bouncing ray method for fast RCS prediction,” IEEE Trans. Antennas Propagat. 58, 494–502 (2010).

S. W. Lee, P. Cramer, K. Woo, and Y. Rahmat-Samii, “Diffraction by an arbitrary subreflector: GTD solution,” IEEE Trans. Antennas Propagat. AP-27, 305–316 (1979).

R. Mittra and A. Rushdi, “An efficient approach for computing the geometrical optics field reflected from a numerically specified surface,” IEEE Trans. Antennas Propagat. 27, 871–877 (1979).

S. W. Lee and P. T. Lam, “Correction to diffraction by an arbitrary subreflector: GTD solution,” IEEE Trans. Antennas Propagat. AP-34, 272 (1986).

H. Ling, R. C. Chou, and S. W. Lee, “Shooting and bouncing rays: calculating the RCS of an arbitrarily shaped cavity,” IEEE Trans. Antennas Propagat. 37, 194–205 (1989).

IEEE Trans. Instr. Meas. (1)

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instr. Meas. 46, 196–200 (1997).
[Crossref]

IEEE Trans. Microwave Theory Trans. (1)

S. W. Lee, M. S. Sheshadri, V. Jamnejad, and R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Trans. MTT-30, 12–19 (1982).

J. Math. Phys. (1)

A. Drezet, J. C. Woehl, and S. Huant, “Diffraction of light by a planar aperture in a metallic screen,” J. Math. Phys. 47, 072901 (2006).
[Crossref]

J. Mod. Opt. (1)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011).
[Crossref]

J. Opt. Lett. (1)

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Lett. 60, 1168–1177 (1970).

J. Opt. Soc. Am. A (5)

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

Metrologia (6)

K. Dorenwendt and G. Bönsch, “Über den Einfluß der Beugung auf die interferentielle Längenmessung,” Metrologia 12, 57–60 (1976).
[Crossref]

G. Mana, “Diffraction effects in optical interferometers illuminated by laser sources,” Metrologia 26, 87–93 (1989).
[Crossref]

R. Masui, “Effect of diffraction in a Saunders-type optical interferometer,” Metrologia 34, 125–131 (1997).
[Crossref]

N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurement of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

Microwave Opt. Technol. Lett. (1)

S. W. Lee, H. Ling, and R. C. Chou, “Ray-tube integration in shooting and bouncing ray method,” Microwave Opt. Technol. Lett. 1, 286–289 (1988).
[Crossref]

New J. Phys. (1)

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 1–17 (2013).

Opt. Acta (1)

E. Byckling and J. Simola, “Wave optical calculation of diffraction effects in optical systems,” Opt. Acta 27, 337–344 (2010).
[Crossref]

Opt. Commun. (1)

T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[Crossref]

Optik (1)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

Phys. Rev. (2)

W. R. Smythe, “The double current sheet in diffraction,” Phys. Rev. 72, 1066–1070 (1947).
[Crossref]

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Phys. Rev. A (1)

A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
[Crossref]

Proc. IEEE (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[Crossref]

Proc. R. Soc. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. A 253, 349–357 (1959).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Proc. SPIE (4)

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).

M. Harrigan, “General beam propagation through non-orthogonal optical systems,” Proc. SPIE 4832, 379–389 (2002).

B. Andreas, G. Mana, E. Massa, and C. Palmisano, “Modeling laser interferometers for the measurement of the Avogadro constant,” Proc. SPIE 8789, 87890W (2013).

G. W. Forbes and M. A. Alonso, “The Holy Grail of ray-based optical modeling,” Proc. SPIE 4832, 186–197 (2002).

Rev. Mod. Phys. (1)

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[Crossref]

Other (14)

R. Mittra, ed., Computer Techniques for Electromagnetics (Pergamon, 1973).

A. S. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).

A. E. Siegman, Lasers (University Science Books, 1986).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge University, 1997).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 2008).

S. R. Ghorpade and B. V. Limaye, A Course in Multivariable Calculus and Analysis (Springer, 2010).

A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, 1995).

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (California Institute of Technology, 2010), http://www.feynmanlectures.caltech.edu .

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

http://en.wikipedia.org/wiki/Rodrigues'_rotation_formula .

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Figures (25)

Fig. 1.
Fig. 1. Propagation through interface. The versors for reflection r ^ r and refraction r ^ t are calculated for each versor r ^ .
Fig. 2.
Fig. 2. Toy models 1 (top) and 2 (bottom) for the numerical test of power conservation for reflection, refraction, and propagation of electromagnetic fields by vectorial diffraction integrals. An interface separates two transparent, isotropic, homogeneous, and nonmagnetic media with refractive indices n 1 and n 2 . The fixed parameters are listed in Table 1; the variable parameters are given in Table 2.
Fig. 3.
Fig. 3. Directly from S 0 to S 2 [Fig. 2 (top), Test 1 in Tables 1 and 2] propagated electric field components of a linearly x -polarized Gaussian beam by use of vectorial diffraction integrals. The viewing angle is perpendicular to S 2 . The corresponding magnetic components can be found in Appendix C.
Fig. 4.
Fig. 4. Relative deviations of directly [ S 0 to S 2 , Fig. 2 (top), Test 1 in Tables 1 and 2] propagated electric field components of a linearly x -polarized Gaussian beam by use of vectorial diffraction integrals from indirectly propagated ones ( S 0 via S 1 to S 2 ). The viewing angle is perpendicular to S 2 . The large relative deviation for the y -component results from the fact that the y -component of the directly propagated field, i.e., E 2 , y , is exactly zero. The corresponding magnetic components can be found in Appendix C.
Fig. 5.
Fig. 5. Irradiance (left) and its relative deviation of the directly [ S 0 to S 2 , Fig. 2 (top), Test 1 in Tables 1 and 2] propagated linearly x -polarized Gaussian beam by use of vectorial diffraction integrals from the indirectly propagated one ( S 0 via S 1 to S 2 ). The viewing angle is perpendicular to S 2 .
Fig. 6.
Fig. 6. Relative deviations of directly [ S 0 to S 2 , Fig. 2 (bottom), Test 2 in Tables 1 and 2] propagated electric field components of a linearly x -polarized Gaussian beam by use of vectorial diffraction integrals from indirectly propagated ones ( S 0 via S 1 to S 2 ). The viewing angle is perpendicular to S 2 . The large relative deviation for the y -component results from the fact that the y component of the directly propagated field, i.e., E 2 , y , is exactly zero. The corresponding magnetic components can be found in Appendix C.
Fig. 7.
Fig. 7. Left: from S 0 via S 1 to S 2 [Fig. 2 (bottom), Test 2 in Tables 1 and 2] propagated y component of the electric field of a linearly x -polarized Gaussian beam by use of vectorial diffraction integrals. Right: relative deviation of the irradiance between the directly from S 0 to S 2 and the indirectly ( S 0 via S 1 to S 2 ) propagated beam. The viewing angle is perpendicular to S 2 .
Fig. 8.
Fig. 8. Types of used input field decompositions in the ray picture. Only one component is shown, respectively. Initially, a grid on the aperture plane defines the ray spacing. Top: decomposition into plane waves: each component is represented by parallel rays starting from the wave front, which intersects the origin of the input plane, i.e., the plane where the decomposition is done by Eqs. (1) and (2). Bottom: decomposition into spherical waves: each component is represented by divergent rays starting from one of the sampling points where the input field is given.
Fig. 9.
Fig. 9. Flow chart of the VRBDI algorithm.
Fig. 10.
Fig. 10. Comparison between different propagation methods by toy model 2 shown in Fig. 2 with the parameters from Table 7. Left: electric field components propagated by a vectorial ray-based diffraction integral where a linearly x -polarized Gaussian beam is decomposed by spherical waves (VRBDI-SW). Middle: relative deviation from a stepwise propagation by vectorial diffraction integrals. Right: relative deviation of the electric field components propagated by a VRBDI with plane wave decomposition (VRBDI-PW) from the ones obtained by the stepwise method. The viewing angle is perpendicular to S 2 . The corresponding magnetic components can be found in Appendix C.
Fig. 11.
Fig. 11. Comparison between different propagation methods by toy model 2 shown in Fig. 2 with the parameters from Table 7. Left: irradiance obtained by a vectorial ray-based diffraction integral where a linearly x -polarized Gaussian beam is decomposed by spherical waves (VRBDI-SW). Middle: relative deviation from a stepwise propagation by vectorial diffraction integrals. Right: relative deviation of the irradiance obtained by a vectorial diffraction integral with plane wave decomposition (VRBDI-PW) from the one obtained by VRBDI-SW. The viewing angle is perpendicular to S 2 .
Fig. 12.
Fig. 12. Comparison between different propagation methods by toy model 2 shown in Fig. 2 with the parameters from Table 7: relative deviation of the electric field components propagated by a vectorial diffraction integral with spherical wave decomposition (VRBDI-SW) of a linearly x -polarized Gaussian beam from the ones propagated by a VRBDI with plane wave decomposition (VRBDI-PW). The viewing angle is perpendicular to S 2 . The corresponding magnetic components can be found in Appendix C.
Fig. 13.
Fig. 13. Phase bias for empty and loaded cavity in the ideal configuration of the interferometer described in [1113] but calculated by the vectorial ray-based diffraction integral with spherical wave decomposition (VRBDI-SW).
Fig. 14.
Fig. 14. Difference of the phase bias for empty (top) and loaded cavity in the ideal configuration of the interferometer described in [1113] between the calculation by a vectorial ray-based diffraction integral with spherical wave decomposition (VRBDI-SW) and a vectorial general astigmatic Gaussian beam-tracing algorithm (VGAGB) [13].
Fig. 15.
Fig. 15. Left: distribution of chosen source points on the input plane for the test of reachability of the detector sampling grid positions of an optical system. The numbering corresponds to the respective source intensity. Right: example of an optical system under test (only marginal rays from a single source point are shown).
Fig. 16.
Fig. 16. Test of overlap and ray aiming for a simple imaging system with a 4 mm aperture. Left: additive sampling of the rays from the nine input source points yields nine separated spots on the detector grid. The plotted quantity is arbitrary and corresponds to the ray density times the respective source intensity. The depicted values are b x , y , total point spread width including all sources; b ¯ x , y , mean of the point spread widths taken from the individual source points; b , largest width of the point spread of the central source point, i.e., source point 9 in Fig. 15, which is also the calculation window size of the right picture; Δ z , distance from the last lens surface apex to the detector plane. Right: the ray aiming error according to Eq. (A7) for the central source point. The viewing angle is perpendicular to the detector plane.
Fig. 17.
Fig. 17. Test of overlap and ray aiming for a simple imaging system with a 30 mm aperture. Left: additive sampling of the rays from the nine input source points on the detector grid. The plotted quantity is arbitrary and corresponds to the ray density times the respective source intensity. The depicted values are b x , y , total point spread width including all sources; b ¯ x , y , mean of the point spread widths taken from the individual source points; b , largest width of the point spread of the central source point, i.e., source point 9 in Fig. 15, which is also the calculation window size of the right picture; Δ z , distance from the last lens surface apex to the detector plane. Right: the ray aiming error according to Eq. (A7) for the central source point. The viewing angle is perpendicular to the detector plane.
Fig. 18.
Fig. 18. Test of overlap and ray aiming for a telescope with a 30 mm aperture. Left: additive sampling of the rays from the nine input source points on the detector grid. The plotted quantity is arbitrary and corresponds to the ray density times the respective source intensity. The depicted values are b x , y , total point spread width including all sources; b ¯ x , y , mean of the point spread widths taken from the individual source points; b , largest width of the point spread of the central source point, i.e., source point 9 in Fig. 15, which is also the calculation window size of the right picture; Δ z , distance from the last lens surface apex to the detector plane. Right: the ray aiming error according to Eq. (A7) for the central source point. The viewing angle is perpendicular to the detector plane.
Fig. 19.
Fig. 19. Situation after ray tracing: the points Q ± x , y on the plane normal to r ^ have to be found from known intersections P and P ± x , y and known versors r ^ and r ^ ± x , y .
Fig. 20.
Fig. 20. Situation after ray tracing. Left: the intersection P of the base ray with the detector plane defines the orthogonal base plane, which does, in general, not contain the nearby sample-grid point P sgp . Further propagation by Δ z along the base ray is necessary. Right: the four possible ±- combinations of ray vectors are decided by the position of P sgp .
Fig. 21.
Fig. 21. Magnetic field components corresponding to Fig. 3.
Fig. 22.
Fig. 22. Magnetic field components corresponding to Fig. 4.
Fig. 23.
Fig. 23. Magnetic field components corresponding to Fig. 6.
Fig. 24.
Fig. 24. Magnetic field components corresponding to Fig. 10.
Fig. 25.
Fig. 25. Magnetic field components corresponding to Fig. 12.

Tables (8)

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Table 1. Fixed Simulation Parameters for the Toy Models Shown in Fig. 2

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Table 2. Variable Simulation Parameters for Toy Models Shown in Fig. 2 a

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Table 3. Numerical Test of Power Conservation for Propagation through a Real Interface between Two Transparent, Isotropic, Homogeneous, and Nonmagnetic Media a

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Table 4. Numerical Test of Power Conservation for Different Sampling Resolutions for Propagation through an Interface between Two Transparent, Isotropic, Homogeneous, and Nonmagnetic Media a

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Table 5. Numerical Test of Power Conservation for Different Radii of Curvature of an Interface between Two Transparent, Isotropic, Homogeneous, and Nonmagnetic Media a

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Table 6. Numerical Test of Power Conservation for Different Sampling Resolutions for Propagation in a Focusing Toy Model with Stronger Refractive Index Contrast n 2 / n 1 3 a

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Table 7. Simulation Parameters for the Comparison Between the Vectorial Ray-Based Diffraction Integral and Stepwise Method Based on Vectorial Diffraction Integrals a

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Table 8. Measurement Bias for Empty Gap Δ D E , Loaded Gap Δ D L , and Resulting Sphere Diameter Δ D a

Equations (72)

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E x , y ( k x , k y ) = + d x 0 d y 0 E x , y ( x 0 , y 0 ) exp [ i ( k x x 0 + k y y 0 ) ] ,
E z = E x k x + E y k y k z ,
H ( k x , k y ) = n ϵ 0 μ 0 k ^ × E ( k x , k y ) ,
E ( x , y , z ) = 1 4 π 2 + d k x d k y · E ( k x , k y ) exp [ i ( k x x + k y y + k z z ) ] .
E = 1 4 π 2 + d k x d k y [ E x x ^ + E y y ^ ( E x k x k z + E y k y k z ) z ^ ] exp [ i ( k x x + k y y + k z z ) ] ,
E ( x , y , z ) = 2 × + d x 0 d y 0 z ^ × E ( x 0 , y 0 ) { i 8 π 2 + d k x d k y · exp [ i ( k x ( x x 0 ) + k y ( y y 0 ) + k z z ) ] k z } .
exp ( i k r ) 4 π r = i 8 π 2 + d k x d k y · exp [ i ( k x ( x x 0 ) + k y ( y y 0 ) + k z z ) ] k z ,
E ( x , y , z ) = 1 2 π × + d x 0 d y 0 z ^ × E ( x 0 , y 0 ) exp ( i k r ) r .
E ( x , y , z ) = 1 2 π × S 0 d x 0 d y 0 z ^ × E ( x 0 , y 0 ) exp ( i k r ) r .
E ( P 1 ) = 1 2 π × S 0 d A 0 N ^ 0 × E ( P 0 ) exp ( i k r ) r ,
E 1 = i λ S 0 d A 0 exp ( i k r ) r ( 1 i k r ) ( N ^ 0 × E 0 ) × r ^ + 1 2 π S 0 d A 0 exp ( i k r ) r × ( N ^ 0 × E 0 ) ,
E 1 = i λ S 0 d A 0 exp ( i k r ) r ( 1 i k r ) ( N ^ 0 × E 0 ) × r ^ ,
H 1 = i λ S 0 d A 0 exp ( i k r ) r ( 1 i k r ) ( N ^ 0 × H 0 ) × r ^ .
S = 1 2 Re ( E × H * ) ,
d S = d A 0 2 λ r 2 ( 1 + 1 k 2 r 2 ) Re { r ^ [ r ^ N ^ ( N ^ ( E 0 × H 0 ) ) ] } r ^
r ^ r = r ^ 2 ( r ^ N ^ 1 ) N ^ 1 ,
r ^ t = n 1 n 2 [ r ^ ( r ^ N ^ 1 ) ] + σ ( r ^ N ^ 1 ) 1 ( n 1 n 2 ) 2 [ 1 ( r ^ N ^ 1 ) 2 ] N ^ 1 ,
ξ ^ = r ^ × ( N ^ 1 × r ^ ) | r ^ × ( N ^ 1 × r ^ ) | , η ^ = r ^ × ξ ^ , ξ ^ r = η ^ × r ^ r , ξ ^ t = η ^ × r ^ t .
r TM = n 2 cos θ n 1 cos θ t n 2 cos θ + n 1 cos θ t , r TE = n 1 cos θ n 2 cos θ t n 1 cos θ + n 2 cos θ t , t TM = 2 n 1 cos θ n 2 cos θ + n 1 cos θ t , t TE = 2 n 1 cos θ n 1 cos θ + n 2 cos θ t ,
d E 1 = i n 1 λ 0 d A 0 exp ( i k 0 n 1 r ) r ( 1 i k 0 n 1 r ) ( N ^ 0 × E 0 ) × r ^ ;
d E 1 , r = r TM ( d E 1 ξ ^ ) ξ ^ r + r TE ( d E 1 η ^ ) η ^ ;
d E 1 , t = t TM ( d E 1 ξ ^ ) ξ ^ t + t TE ( d E 1 η ^ ) η ^ ;
d H 1 = i n 1 λ 0 d A 0 exp ( i k 0 n 1 r ) r ( 1 i k 0 n 1 r ) ( N ^ 0 × H 0 ) × r ^ ;
d H 1 , r = r TM ( d H 1 ξ ^ ) ξ ^ r + r TE ( d H 1 η ^ ) η ^ ;
d H 1 , t = n 2 n 1 [ t TM ( d H 1 ξ ^ ) ξ ^ t + t TE ( d H 1 η ^ ) η ^ ] .
z = f ( x , y ) = C z σ ( C z ) R 2 ( x C x ) 2 ( y C y ) 2 ,
d A sph ( x , y ) = R 2 R 2 ( x C x ) 2 ( y C y ) 2 d x d y .
I i = | S i N ^ i | ,
P i = S i d A i I i .
E 0 , x = exp [ ( x 2 + y 2 ) / w 0 2 ] V / m ,
l = ( T P 0 ) N ^ z r ^ N ^ z .
l 1,2 = ( C P 0 ) r ^ ± [ ( C P 0 ) r ^ ] 2 | C P 0 | 2 + R 2 .
d E 0 = d A 0 d k x d k y 4 π 2 E ( k x , k y ) ,
d E 0 = d A 0 ( N ^ 0 × E 0 ) × r ^ initial ,
d H m 1 = n m 1 ϵ 0 μ 0 r ^ × d E m 1 , r ^ = r ^ final .
OPL = i = 0 m 1 n i | P i + 1 P i | ,
( d E , d H ) sgp = AF PW exp ( i k 0 OPL ) exp ( i k 0 S PW ) ( d E , d H ) m 1 ,
( d E , d H ) sgp = i λ 0 AF SW exp ( i k 0 OPL ) ( 1 i k 0 OPL ) · exp ( i k 0 S SW ) ( d E , d H ) m 1 ,
( E , H ) sgp = input plane ( d E , d H ) sgp .
Δ x · Δ p x 2 sin Δ α x λ 4 π Δ x .
x = ( P i T ) N ^ x , y = ( P i T ) N ^ y ,
r ^ x = r ^ x ( x , y ) , r ^ y = r ^ y ( x , y ) .
ρ = ( x ) 2 + ( y ) 2 ρ max , φ = atan 2 ( y , x ) .
[ Z ] a = [ Z 0 0 ( ρ 1 , φ 1 ) Z 1 1 ( ρ 1 , φ 1 ) Z 0 0 ( ρ ν , φ ν ) Z 1 1 ( ρ ν , φ ν ) Z u u ( ρ 1 , φ 1 ) Z u u ( ρ ν , φ ν ) ] [ a 1 a ζ ] = [ F ( ρ 1 , φ 1 ) F ( ρ ν , φ ν ) ] = F ,
a = ( [ Z ] T [ Z ] ) 1 [ Z ] T F .
F sgp = [ Z ] sgp a ,
δ ra = ( x x sgp ) 2 + ( y y sgp ) 2 Δ x sgp 2 + Δ y sgp 2 ,
e ^ x = [ r ^ z 0 r ^ x ] 1 r ^ x 2 + r ^ z 2 , e ^ y = r ^ × e ^ x .
P 0 , ± x = P 0 ± D · e ^ x , P 0 , ± y = P 0 ± D · e ^ y ,
r ^ ± x = r ^ cos β ± e ^ x sin β , r ^ ± y = r ^ cos β ± e ^ y sin β .
[ x ˘ y ˘ x ˘ y ˘ ] 2 = [ A B C D ] [ x ˘ y ˘ x ˘ y ˘ ] 1 = [ A 11 A 12 B 11 B 12 A 21 A 22 B 21 B 22 C 11 C 12 D 11 D 12 C 21 C 22 D 21 D 22 ] [ x ˘ y ˘ x ˘ y ˘ ] 1 .
v = r ^ × r ^ r , t | r ^ × r ^ r , t | ,
θ = arc cos ( r ^ r ^ r , t ) .
M ( v ) = I + sin θ V × + ( 1 cos θ ) V × 2 ,
V × = [ 0 v z v y v z 0 v x v y v x 0 ] .
M ( v ) = [ 1 + ( 1 cos θ ) ( v y 2 v z 2 ) v z sin θ + ( 1 cos θ ) v x v y v y sin θ + ( 1 cos θ ) v x v z v z sin θ + ( 1 cos θ ) v x v y 1 + ( 1 cos θ ) ( v x 2 v z 2 ) v x sin θ + ( 1 cos θ ) v y v z v y sin θ + ( 1 cos θ ) v x v z v x sin θ + ( 1 cos θ ) v y v z 1 + ( 1 cos θ ) ( v x 2 v y 2 ) ] .
p 1 , ± x = [ ± D 0 0 0 ] , p 1 , ± y = [ 0 ± D 0 0 ] , q 1 , ± x = [ 0 0 ± n 0 tan β 0 ] , q 1 , ± y = [ 0 0 0 ± n 0 tan β ] ,
Q ± x , y = ( P P ± x , y ) r ^ r ^ ± x , y r ^ r ^ ± x , y + P ± x , y .
x ˘ ± x , y = ( Q ± x , y P ) e ^ x , y ˘ ± x , y = ( Q ± x , y P ) e ^ y .
s ^ ± x , y = [ r ^ ± x , y e ^ x r ^ ± x , y e ^ y r ^ ± x , y r ^ ] .
p 2 , ± x , q 2 , ± x = [ x ˘ ± x y ˘ ± x n m 1 s ^ x ˘ , ± x / s ^ z ˘ , ± x n m 1 s ^ y ˘ , ± x / s ^ z ˘ , ± x ] , p 2 , ± y , q 2 , ± y = [ x ˘ ± y y ˘ ± y n m 1 s ^ x ˘ , ± y / s ^ z ˘ , ± y n m 1 s ^ y ˘ , ± y / s ^ z ˘ , ± y ] .
[ A B C D ] = [ p 2 , ± x 1 / 1 p 1 , ± x p 2 , ± y 1 / 2 p 1 , ± y p 2 , ± x 2 / 1 p 1 , ± x p 2 , ± y 2 / 2 p 1 , ± y p 2 , ± x 3 / 1 p 1 , ± x p 2 , ± y 3 / 2 p 1 , ± y p 2 , ± x 4 / 1 p 1 , ± x p 2 , ± y 4 / 2 p 1 , ± y q 2 , ± x 1 / 3 q 1 , ± x q 2 , ± y 1 / 4 q 1 , ± y q 2 , ± x 2 / 3 q 1 , ± x q 2 , ± y 2 / 4 q 1 , ± y q 2 , ± x 3 / 3 q 1 , ± x q 2 , ± y 3 / 4 q 1 , ± y q 2 , ± x 4 / 3 q 1 , ± x q 2 , ± y 4 / 4 q 1 , ± y ] .
( d E , d H ) bp = exp ( i k 0 OPL ) det ( A ) exp { i k 0 2 det ( A ) · [ x ˘ 2 ( C 11 A 22 C 12 A 21 ) + y ˘ 2 ( C 22 A 11 C 21 A 12 ) + x ˘ y ˘ ( C 12 A 11 C 11 A 12 + C 21 A 22 C 22 A 21 ) ] } · ( d E , d H ) m 1
( d E , d H ) bp = i λ 0 exp ( i k 0 OPL ) det ( B ) ( 1 i k 0 OPL ) exp { i k 0 2 det ( B ) · [ x ˘ 2 ( D 11 B 22 D 12 B 21 ) + y ˘ 2 ( D 22 B 11 D 21 B 12 ) + x ˘ y ˘ ( D 12 B 11 D 11 B 12 + D 21 B 22 D 22 B 21 ) ] } · ( d E , d H ) m 1
[ x ˘ L y ˘ L z ˘ L ] = [ ( P sgp P ) e ^ x ( P sgp P ) e ^ y ( P sgp P ) r ^ ] .
[ A ˜ B ˜ C D ] = [ I Δ z / n m 1 I 0 I ] [ A B C D ] .
A ˜ = A + Δ z / n m 1 C or B ˜ = B + Δ z / n m 1 D
1 det ( A ˜ ) σ ( À 11 ) À 11 σ ( À 22 ) À 22 , 1 det ( B ˜ ) σ ( 11 ) 11 σ ( 22 ) 22 ,
( d E , d H ) sgp = σ ( À 11 ) À 11 σ ( À 22 ) À 22 exp ( i k 0 OPL ) · exp { i k 0 2 det ( A ˜ ) [ x ˘ L 2 ( C 11 A ˜ 22 C 12 A ˜ 21 ) + y ˘ L 2 ( C 22 A ˜ 11 C 21 A ˜ 12 ) + x ˘ L y ˘ L ( C 12 A ˜ 11 C 11 A ˜ 12 + C 21 A ˜ 22 C 22 A ˜ 21 ) ] } · ( d E , d H ) m 1
( d E , d H ) sgp = i λ 0 σ ( 11 ) 11 σ ( 22 ) 22 exp ( i k 0 OPL ) · ( 1 i k 0 OPL ) exp { i k 0 2 det ( B ˜ ) · [ L 2 ( D 11 B ˜ 22 D 12 B ˜ 21 ) + L 2 ( D 22 B ˜ 11 D 21 B ˜ 12 ) + L L ( D 12 B ˜ 11 D 11 B ˜ 12 + D 21 B ˜ 22 D 22 B ˜ 21 ) ] } · ( d E , d H ) m 1
A F PW = σ ( À 11 ) À 11 σ ( À 22 ) À 22 , S PW = 1 2 det ( A ˜ ) [ x ˘ L 2 ( C 11 A ˜ 22 C 12 A ˜ 21 ) + y ˘ L 2 ( C 22 A ˜ 11 C 21 A ˜ 12 ) + x ˘ L y ˘ L ( C 12 A ˜ 11 C 11 A ˜ 12 + C 21 A ˜ 22 C 22 A ˜ 21 ) ]
A F SW = σ ( 11 ) 11 σ ( 22 ) 22 , S SW = 1 2 det ( B ˜ ) [ L 2 ( D 11 B ˜ 22 D 12 B ˜ 21 ) + L 2 ( D 22 B ˜ 11 D 21 B ˜ 12 ) + L L ( D 12 B ˜ 11 D 11 B ˜ 12 + D 21 B ˜ 22 D 22 B ˜ 21 ) ]

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