Abstract

The lensing behavior of sinusoidal phase gratings is studied in the limit of large phase amplitudes Φ. The resulting asymptotic expressions are compared with the exact results and are shown to represent excellent approximations even for moderate values of Φ. They are well suited to a quantitative description of focusing imperfections caused by the strong spherical aberrations encountered in this type of microlens array.

© 2004 Optical Society of America

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References

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  1. O. Nomoto, “Über eine neue Sichtbarmachungsmethode stehender Ultraschallwellen in Flüssigkeiten,” Proc. Phys. Math. Soc. Jpn. 18, 402–424 (1936).
  2. N. S. Nagendra Nath, “The visibility of ultrasonic waves and its periodic variations,” Proc. Indian Acad. Sci., Sect. A 4, 262–274 (1936).
  3. E. Hiedemann, E. Schreuer, “Zur Periodizität der Abbildung von Ultraschallwellen,” Z. Phys. 107, 463–473 (1937).
    [CrossRef]
  4. K. G. H. Baldwin, “Experiments in atom optics,” Aust. J. Phys. 49, 855–897 (1996).
    [CrossRef]
  5. S. Gupta, A. E. Leanhardt, A. D. Cronin, D. E. Pritchard, “Coherent manipulation of atoms with standing light waves,” C. R. Acad. Sci. Paris 2(IV), 479–495 (2001).
  6. D. Meschede, H. Metcalf, “Atomic nanofabrication: atomic deposition and lithography by laser and magnetic forces,” J. Phys. D 36, R17–R38 (2003).
    [CrossRef]
  7. B. Rohwedder, “Atom optical elements based on near-field grating sequences,” Fortschr. Phys. 47, 883–911 (1999).
    [CrossRef]
  8. M. Leibscher, I. Sh. Averbukh, “Squeezing of atoms in a pulsed optical lattice,” Phys. Rev. A 65, 053816 (2002).
    [CrossRef]
  9. D. V. Strekalov, A. Turlapov, A. Kumarakrishnan, T. Sleator, “Periodic structures generated in a cloud of cold atoms,” Phys. Rev. A 66, 023601 (2002).
    [CrossRef]
  10. P. Barberis, B. Rohwedder, “Spherical correction lens array for atom nanofabrication,” Phys. Rev. A 67, 033604 (2003).
    [CrossRef]
  11. J. S. Darlin, P. Senthilkumaran, S. Bhattacharya, M. P. Kothiyal, R. S. Sirohi, “Fabrication of an array illuminator using tandem Michelson interferometers,” Opt. Commun. 123, 1–4 (1996).
    [CrossRef]
  12. K. Patorski, “Optical testing of ultrasonic phase gratings using a Fresnel diffraction method,” Ultrasonics 19, 169–172 (1981).
    [CrossRef]
  13. S. P. Trainoff, D. S. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
    [CrossRef]
  14. M. V. Berry, S. Klein, “Integer, fractional, and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
    [CrossRef]
  15. A. Reichelt, E. Storck, U. Wolff, “Near field diffraction pattern behind a sinusoidal phase grating,” Opt. Commun. 3, 169–172 (1971).
    [CrossRef]
  16. N. G. De Bruijn, Asymptotic Methods in Analysis (North-Holland, Amsterdam, 1958), Chap. 4 (“Laplace Method”) and 5 (“Saddle-Point Method”).
  17. Strictly speaking, the exponentially small contribution of these “distant” saddles must be consistently included to obtain a convergent asymptotic series. For the details, see M. V. Berry, C. J. Howls, “Hyperasymptotics for integrals with saddles,” Proc. R. Soc. London Ser. A 434, 657–675 (1999).
    [CrossRef]
  18. M. V. Berry, E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
    [CrossRef]
  19. T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).
  20. J. L. Cohen, B. Dubetsky, P. R. Berman, “Atom focusing by far-detuned and resonant standing wave fields: thin-lens regime,” Phys. Rev. A 60, 4886–4901 (1999).
    [CrossRef]
  21. J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81–92 (2003).
    [CrossRef]
  22. K. K. Berggren, M. Prentiss, G. L. Timp, R. E. Behringer, “Calculation of atomic positions in nanometer-scale direct-write optical lithography with an optical standing wave,” J. Opt. Soc. Am. B 11, 1166–1176 (1994).
    [CrossRef]
  23. J. J. McClelland, “Atom-optical properties of a standing-wave light field,” J. Opt. Soc. Am. B 12, 1761–1767 (1995).
    [CrossRef]
  24. K. Okamoto, Y. Inouye, S. Kawata, “Atomic-beam propagation in a two-dimensional standing wave of light: a numerical analysis based on a particle-optics,” Jpn. J. Appl. Phys., Part 1 40, 609–613 (2001).
    [CrossRef]
  25. C. J. Lee, “Quantum-mechanical analysis of atom lithography,” Phys. Rev. A 61, 063604 (2000).
    [CrossRef]
  26. S. J. H. Petra, K. A. H. van Leeuwen, L. Feenstra, W. Hogervorst, W. Vassen, “Numerical simulations on the motion of atoms travelling through a standing-wave light field,” Eur. Phys. J. D 27, 83–91 (2003).
    [CrossRef]

2003 (4)

D. Meschede, H. Metcalf, “Atomic nanofabrication: atomic deposition and lithography by laser and magnetic forces,” J. Phys. D 36, R17–R38 (2003).
[CrossRef]

P. Barberis, B. Rohwedder, “Spherical correction lens array for atom nanofabrication,” Phys. Rev. A 67, 033604 (2003).
[CrossRef]

J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81–92 (2003).
[CrossRef]

S. J. H. Petra, K. A. H. van Leeuwen, L. Feenstra, W. Hogervorst, W. Vassen, “Numerical simulations on the motion of atoms travelling through a standing-wave light field,” Eur. Phys. J. D 27, 83–91 (2003).
[CrossRef]

2002 (3)

S. P. Trainoff, D. S. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[CrossRef]

M. Leibscher, I. Sh. Averbukh, “Squeezing of atoms in a pulsed optical lattice,” Phys. Rev. A 65, 053816 (2002).
[CrossRef]

D. V. Strekalov, A. Turlapov, A. Kumarakrishnan, T. Sleator, “Periodic structures generated in a cloud of cold atoms,” Phys. Rev. A 66, 023601 (2002).
[CrossRef]

2001 (2)

S. Gupta, A. E. Leanhardt, A. D. Cronin, D. E. Pritchard, “Coherent manipulation of atoms with standing light waves,” C. R. Acad. Sci. Paris 2(IV), 479–495 (2001).

K. Okamoto, Y. Inouye, S. Kawata, “Atomic-beam propagation in a two-dimensional standing wave of light: a numerical analysis based on a particle-optics,” Jpn. J. Appl. Phys., Part 1 40, 609–613 (2001).
[CrossRef]

2000 (1)

C. J. Lee, “Quantum-mechanical analysis of atom lithography,” Phys. Rev. A 61, 063604 (2000).
[CrossRef]

1999 (4)

J. L. Cohen, B. Dubetsky, P. R. Berman, “Atom focusing by far-detuned and resonant standing wave fields: thin-lens regime,” Phys. Rev. A 60, 4886–4901 (1999).
[CrossRef]

Strictly speaking, the exponentially small contribution of these “distant” saddles must be consistently included to obtain a convergent asymptotic series. For the details, see M. V. Berry, C. J. Howls, “Hyperasymptotics for integrals with saddles,” Proc. R. Soc. London Ser. A 434, 657–675 (1999).
[CrossRef]

M. V. Berry, E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

B. Rohwedder, “Atom optical elements based on near-field grating sequences,” Fortschr. Phys. 47, 883–911 (1999).
[CrossRef]

1996 (3)

K. G. H. Baldwin, “Experiments in atom optics,” Aust. J. Phys. 49, 855–897 (1996).
[CrossRef]

M. V. Berry, S. Klein, “Integer, fractional, and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

J. S. Darlin, P. Senthilkumaran, S. Bhattacharya, M. P. Kothiyal, R. S. Sirohi, “Fabrication of an array illuminator using tandem Michelson interferometers,” Opt. Commun. 123, 1–4 (1996).
[CrossRef]

1995 (1)

1994 (1)

1981 (1)

K. Patorski, “Optical testing of ultrasonic phase gratings using a Fresnel diffraction method,” Ultrasonics 19, 169–172 (1981).
[CrossRef]

1971 (1)

A. Reichelt, E. Storck, U. Wolff, “Near field diffraction pattern behind a sinusoidal phase grating,” Opt. Commun. 3, 169–172 (1971).
[CrossRef]

1946 (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

1937 (1)

E. Hiedemann, E. Schreuer, “Zur Periodizität der Abbildung von Ultraschallwellen,” Z. Phys. 107, 463–473 (1937).
[CrossRef]

1936 (2)

O. Nomoto, “Über eine neue Sichtbarmachungsmethode stehender Ultraschallwellen in Flüssigkeiten,” Proc. Phys. Math. Soc. Jpn. 18, 402–424 (1936).

N. S. Nagendra Nath, “The visibility of ultrasonic waves and its periodic variations,” Proc. Indian Acad. Sci., Sect. A 4, 262–274 (1936).

Averbukh, I. Sh.

M. Leibscher, I. Sh. Averbukh, “Squeezing of atoms in a pulsed optical lattice,” Phys. Rev. A 65, 053816 (2002).
[CrossRef]

Baldwin, K. G. H.

K. G. H. Baldwin, “Experiments in atom optics,” Aust. J. Phys. 49, 855–897 (1996).
[CrossRef]

Barberis, P.

P. Barberis, B. Rohwedder, “Spherical correction lens array for atom nanofabrication,” Phys. Rev. A 67, 033604 (2003).
[CrossRef]

Behringer, R. E.

Berggren, K. K.

Berman, P. R.

J. L. Cohen, B. Dubetsky, P. R. Berman, “Atom focusing by far-detuned and resonant standing wave fields: thin-lens regime,” Phys. Rev. A 60, 4886–4901 (1999).
[CrossRef]

Berry, M. V.

Strictly speaking, the exponentially small contribution of these “distant” saddles must be consistently included to obtain a convergent asymptotic series. For the details, see M. V. Berry, C. J. Howls, “Hyperasymptotics for integrals with saddles,” Proc. R. Soc. London Ser. A 434, 657–675 (1999).
[CrossRef]

M. V. Berry, E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

M. V. Berry, S. Klein, “Integer, fractional, and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Bhattacharya, S.

J. S. Darlin, P. Senthilkumaran, S. Bhattacharya, M. P. Kothiyal, R. S. Sirohi, “Fabrication of an array illuminator using tandem Michelson interferometers,” Opt. Commun. 123, 1–4 (1996).
[CrossRef]

Bodenschatz, E.

M. V. Berry, E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

Cannell, D. S.

S. P. Trainoff, D. S. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[CrossRef]

Cohen, J. L.

J. L. Cohen, B. Dubetsky, P. R. Berman, “Atom focusing by far-detuned and resonant standing wave fields: thin-lens regime,” Phys. Rev. A 60, 4886–4901 (1999).
[CrossRef]

Cronin, A. D.

S. Gupta, A. E. Leanhardt, A. D. Cronin, D. E. Pritchard, “Coherent manipulation of atoms with standing light waves,” C. R. Acad. Sci. Paris 2(IV), 479–495 (2001).

Darlin, J. S.

J. S. Darlin, P. Senthilkumaran, S. Bhattacharya, M. P. Kothiyal, R. S. Sirohi, “Fabrication of an array illuminator using tandem Michelson interferometers,” Opt. Commun. 123, 1–4 (1996).
[CrossRef]

De Bruijn, N. G.

N. G. De Bruijn, Asymptotic Methods in Analysis (North-Holland, Amsterdam, 1958), Chap. 4 (“Laplace Method”) and 5 (“Saddle-Point Method”).

Dubetsky, B.

J. L. Cohen, B. Dubetsky, P. R. Berman, “Atom focusing by far-detuned and resonant standing wave fields: thin-lens regime,” Phys. Rev. A 60, 4886–4901 (1999).
[CrossRef]

Feenstra, L.

S. J. H. Petra, K. A. H. van Leeuwen, L. Feenstra, W. Hogervorst, W. Vassen, “Numerical simulations on the motion of atoms travelling through a standing-wave light field,” Eur. Phys. J. D 27, 83–91 (2003).
[CrossRef]

Gupta, S.

S. Gupta, A. E. Leanhardt, A. D. Cronin, D. E. Pritchard, “Coherent manipulation of atoms with standing light waves,” C. R. Acad. Sci. Paris 2(IV), 479–495 (2001).

Hiedemann, E.

E. Hiedemann, E. Schreuer, “Zur Periodizität der Abbildung von Ultraschallwellen,” Z. Phys. 107, 463–473 (1937).
[CrossRef]

Hogervorst, W.

S. J. H. Petra, K. A. H. van Leeuwen, L. Feenstra, W. Hogervorst, W. Vassen, “Numerical simulations on the motion of atoms travelling through a standing-wave light field,” Eur. Phys. J. D 27, 83–91 (2003).
[CrossRef]

Howls, C. J.

Strictly speaking, the exponentially small contribution of these “distant” saddles must be consistently included to obtain a convergent asymptotic series. For the details, see M. V. Berry, C. J. Howls, “Hyperasymptotics for integrals with saddles,” Proc. R. Soc. London Ser. A 434, 657–675 (1999).
[CrossRef]

Inouye, Y.

K. Okamoto, Y. Inouye, S. Kawata, “Atomic-beam propagation in a two-dimensional standing wave of light: a numerical analysis based on a particle-optics,” Jpn. J. Appl. Phys., Part 1 40, 609–613 (2001).
[CrossRef]

Kawata, S.

K. Okamoto, Y. Inouye, S. Kawata, “Atomic-beam propagation in a two-dimensional standing wave of light: a numerical analysis based on a particle-optics,” Jpn. J. Appl. Phys., Part 1 40, 609–613 (2001).
[CrossRef]

Klein, S.

M. V. Berry, S. Klein, “Integer, fractional, and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Kothiyal, M. P.

J. S. Darlin, P. Senthilkumaran, S. Bhattacharya, M. P. Kothiyal, R. S. Sirohi, “Fabrication of an array illuminator using tandem Michelson interferometers,” Opt. Commun. 123, 1–4 (1996).
[CrossRef]

Kumarakrishnan, A.

D. V. Strekalov, A. Turlapov, A. Kumarakrishnan, T. Sleator, “Periodic structures generated in a cloud of cold atoms,” Phys. Rev. A 66, 023601 (2002).
[CrossRef]

Leanhardt, A. E.

S. Gupta, A. E. Leanhardt, A. D. Cronin, D. E. Pritchard, “Coherent manipulation of atoms with standing light waves,” C. R. Acad. Sci. Paris 2(IV), 479–495 (2001).

Lee, C. J.

C. J. Lee, “Quantum-mechanical analysis of atom lithography,” Phys. Rev. A 61, 063604 (2000).
[CrossRef]

Leibscher, M.

M. Leibscher, I. Sh. Averbukh, “Squeezing of atoms in a pulsed optical lattice,” Phys. Rev. A 65, 053816 (2002).
[CrossRef]

McClelland, J. J.

Meschede, D.

D. Meschede, H. Metcalf, “Atomic nanofabrication: atomic deposition and lithography by laser and magnetic forces,” J. Phys. D 36, R17–R38 (2003).
[CrossRef]

Metcalf, H.

D. Meschede, H. Metcalf, “Atomic nanofabrication: atomic deposition and lithography by laser and magnetic forces,” J. Phys. D 36, R17–R38 (2003).
[CrossRef]

Nagendra Nath, N. S.

N. S. Nagendra Nath, “The visibility of ultrasonic waves and its periodic variations,” Proc. Indian Acad. Sci., Sect. A 4, 262–274 (1936).

Nomoto, O.

O. Nomoto, “Über eine neue Sichtbarmachungsmethode stehender Ultraschallwellen in Flüssigkeiten,” Proc. Phys. Math. Soc. Jpn. 18, 402–424 (1936).

Okamoto, K.

K. Okamoto, Y. Inouye, S. Kawata, “Atomic-beam propagation in a two-dimensional standing wave of light: a numerical analysis based on a particle-optics,” Jpn. J. Appl. Phys., Part 1 40, 609–613 (2001).
[CrossRef]

Patorski, K.

K. Patorski, “Optical testing of ultrasonic phase gratings using a Fresnel diffraction method,” Ultrasonics 19, 169–172 (1981).
[CrossRef]

Pearcey, T.

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Petra, S. J. H.

S. J. H. Petra, K. A. H. van Leeuwen, L. Feenstra, W. Hogervorst, W. Vassen, “Numerical simulations on the motion of atoms travelling through a standing-wave light field,” Eur. Phys. J. D 27, 83–91 (2003).
[CrossRef]

Prentiss, M.

Primot, J.

J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81–92 (2003).
[CrossRef]

Pritchard, D. E.

S. Gupta, A. E. Leanhardt, A. D. Cronin, D. E. Pritchard, “Coherent manipulation of atoms with standing light waves,” C. R. Acad. Sci. Paris 2(IV), 479–495 (2001).

Reichelt, A.

A. Reichelt, E. Storck, U. Wolff, “Near field diffraction pattern behind a sinusoidal phase grating,” Opt. Commun. 3, 169–172 (1971).
[CrossRef]

Rohwedder, B.

P. Barberis, B. Rohwedder, “Spherical correction lens array for atom nanofabrication,” Phys. Rev. A 67, 033604 (2003).
[CrossRef]

B. Rohwedder, “Atom optical elements based on near-field grating sequences,” Fortschr. Phys. 47, 883–911 (1999).
[CrossRef]

Schreuer, E.

E. Hiedemann, E. Schreuer, “Zur Periodizität der Abbildung von Ultraschallwellen,” Z. Phys. 107, 463–473 (1937).
[CrossRef]

Senthilkumaran, P.

J. S. Darlin, P. Senthilkumaran, S. Bhattacharya, M. P. Kothiyal, R. S. Sirohi, “Fabrication of an array illuminator using tandem Michelson interferometers,” Opt. Commun. 123, 1–4 (1996).
[CrossRef]

Sirohi, R. S.

J. S. Darlin, P. Senthilkumaran, S. Bhattacharya, M. P. Kothiyal, R. S. Sirohi, “Fabrication of an array illuminator using tandem Michelson interferometers,” Opt. Commun. 123, 1–4 (1996).
[CrossRef]

Sleator, T.

D. V. Strekalov, A. Turlapov, A. Kumarakrishnan, T. Sleator, “Periodic structures generated in a cloud of cold atoms,” Phys. Rev. A 66, 023601 (2002).
[CrossRef]

Storck, E.

A. Reichelt, E. Storck, U. Wolff, “Near field diffraction pattern behind a sinusoidal phase grating,” Opt. Commun. 3, 169–172 (1971).
[CrossRef]

Strekalov, D. V.

D. V. Strekalov, A. Turlapov, A. Kumarakrishnan, T. Sleator, “Periodic structures generated in a cloud of cold atoms,” Phys. Rev. A 66, 023601 (2002).
[CrossRef]

Timp, G. L.

Trainoff, S. P.

S. P. Trainoff, D. S. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[CrossRef]

Turlapov, A.

D. V. Strekalov, A. Turlapov, A. Kumarakrishnan, T. Sleator, “Periodic structures generated in a cloud of cold atoms,” Phys. Rev. A 66, 023601 (2002).
[CrossRef]

van Leeuwen, K. A. H.

S. J. H. Petra, K. A. H. van Leeuwen, L. Feenstra, W. Hogervorst, W. Vassen, “Numerical simulations on the motion of atoms travelling through a standing-wave light field,” Eur. Phys. J. D 27, 83–91 (2003).
[CrossRef]

Vassen, W.

S. J. H. Petra, K. A. H. van Leeuwen, L. Feenstra, W. Hogervorst, W. Vassen, “Numerical simulations on the motion of atoms travelling through a standing-wave light field,” Eur. Phys. J. D 27, 83–91 (2003).
[CrossRef]

Wolff, U.

A. Reichelt, E. Storck, U. Wolff, “Near field diffraction pattern behind a sinusoidal phase grating,” Opt. Commun. 3, 169–172 (1971).
[CrossRef]

Aust. J. Phys. (1)

K. G. H. Baldwin, “Experiments in atom optics,” Aust. J. Phys. 49, 855–897 (1996).
[CrossRef]

C. R. Acad. Sci. Paris (1)

S. Gupta, A. E. Leanhardt, A. D. Cronin, D. E. Pritchard, “Coherent manipulation of atoms with standing light waves,” C. R. Acad. Sci. Paris 2(IV), 479–495 (2001).

Eur. Phys. J. D (1)

S. J. H. Petra, K. A. H. van Leeuwen, L. Feenstra, W. Hogervorst, W. Vassen, “Numerical simulations on the motion of atoms travelling through a standing-wave light field,” Eur. Phys. J. D 27, 83–91 (2003).
[CrossRef]

Fortschr. Phys. (1)

B. Rohwedder, “Atom optical elements based on near-field grating sequences,” Fortschr. Phys. 47, 883–911 (1999).
[CrossRef]

J. Mod. Opt. (2)

M. V. Berry, S. Klein, “Integer, fractional, and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

M. V. Berry, E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

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

J. Phys. D (1)

D. Meschede, H. Metcalf, “Atomic nanofabrication: atomic deposition and lithography by laser and magnetic forces,” J. Phys. D 36, R17–R38 (2003).
[CrossRef]

Jpn. J. Appl. Phys., Part 1 (1)

K. Okamoto, Y. Inouye, S. Kawata, “Atomic-beam propagation in a two-dimensional standing wave of light: a numerical analysis based on a particle-optics,” Jpn. J. Appl. Phys., Part 1 40, 609–613 (2001).
[CrossRef]

Opt. Commun. (3)

J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81–92 (2003).
[CrossRef]

A. Reichelt, E. Storck, U. Wolff, “Near field diffraction pattern behind a sinusoidal phase grating,” Opt. Commun. 3, 169–172 (1971).
[CrossRef]

J. S. Darlin, P. Senthilkumaran, S. Bhattacharya, M. P. Kothiyal, R. S. Sirohi, “Fabrication of an array illuminator using tandem Michelson interferometers,” Opt. Commun. 123, 1–4 (1996).
[CrossRef]

Philos. Mag. (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Phys. Fluids (1)

S. P. Trainoff, D. S. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[CrossRef]

Phys. Rev. A (5)

J. L. Cohen, B. Dubetsky, P. R. Berman, “Atom focusing by far-detuned and resonant standing wave fields: thin-lens regime,” Phys. Rev. A 60, 4886–4901 (1999).
[CrossRef]

M. Leibscher, I. Sh. Averbukh, “Squeezing of atoms in a pulsed optical lattice,” Phys. Rev. A 65, 053816 (2002).
[CrossRef]

D. V. Strekalov, A. Turlapov, A. Kumarakrishnan, T. Sleator, “Periodic structures generated in a cloud of cold atoms,” Phys. Rev. A 66, 023601 (2002).
[CrossRef]

P. Barberis, B. Rohwedder, “Spherical correction lens array for atom nanofabrication,” Phys. Rev. A 67, 033604 (2003).
[CrossRef]

C. J. Lee, “Quantum-mechanical analysis of atom lithography,” Phys. Rev. A 61, 063604 (2000).
[CrossRef]

Proc. Indian Acad. Sci., Sect. A (1)

N. S. Nagendra Nath, “The visibility of ultrasonic waves and its periodic variations,” Proc. Indian Acad. Sci., Sect. A 4, 262–274 (1936).

Proc. Phys. Math. Soc. Jpn. (1)

O. Nomoto, “Über eine neue Sichtbarmachungsmethode stehender Ultraschallwellen in Flüssigkeiten,” Proc. Phys. Math. Soc. Jpn. 18, 402–424 (1936).

Proc. R. Soc. London Ser. A (1)

Strictly speaking, the exponentially small contribution of these “distant” saddles must be consistently included to obtain a convergent asymptotic series. For the details, see M. V. Berry, C. J. Howls, “Hyperasymptotics for integrals with saddles,” Proc. R. Soc. London Ser. A 434, 657–675 (1999).
[CrossRef]

Ultrasonics (1)

K. Patorski, “Optical testing of ultrasonic phase gratings using a Fresnel diffraction method,” Ultrasonics 19, 169–172 (1981).
[CrossRef]

Z. Phys. (1)

E. Hiedemann, E. Schreuer, “Zur Periodizität der Abbildung von Ultraschallwellen,” Z. Phys. 107, 463–473 (1937).
[CrossRef]

Other (1)

N. G. De Bruijn, Asymptotic Methods in Analysis (North-Holland, Amsterdam, 1958), Chap. 4 (“Laplace Method”) and 5 (“Saddle-Point Method”).

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Figures (11)

Fig. 1
Fig. 1

Essentials of an experimental setup for the determination of diffraction-related effects in the focusing produced by a periodic phase structure (grating constant λx) placed at the origin (z=0) of the optical axis. The figure shows how an (initially well collimated) beam of wavelength λzλx (light gray) becomes gradually focused behind a sinusoidal phase grating. In the focal plane, the resulting intensity pattern consists of equidistant spikes superimposed on an unfocused constant background (dark gray).

Fig. 2
Fig. 2

Φ dependence of the localization factor L and asymptotic behavior.

Fig. 3
Fig. 3

Saddle-point function σ0(ξ) at the focal plane and its lowest and second-lowest Taylor approximations.

Fig. 4
Fig. 4

Behavior of i[(ξ/2)2-sin2(ξ/2)] in the first quadrant of the complex plane, including saddle points (circles), curves to determine their approximate positions (gray bands), and the curves through ξ=0 on which the function’s real (dashed–dotted curves) and imaginary (solid and dashed curves) parts vanish. The solid (dashed) curve represents the path of steepest ascent (descent).

Fig. 5
Fig. 5

As soon as ξ0, the degenerate saddle point at the origin splits into three simple ones. The steepest paths from Fig. 4 become distorted in a way such that they approach the corresponding ξ=0 paths asymptotically. In contrast to these (dashed and solid) curves, the dotted curve possesses mixed slope characteristics. The situation shown in the figure corresponds to ξ=0.01.

Fig. 6
Fig. 6

Comparison of asymptotic (a) intensity |ψ|2=A02 and (b) phase ϕ0 with the corresponding lowest and second-lowest Taylor approximations given in the text.

Fig. 7
Fig. 7

Comparison of asymptotic (a) intensity |ψ|2=A02 and (b) phase ϕ0 with the exact expressions, evaluated for two representative values of Φ by using Eq. (6).

Fig. 8
Fig. 8

Φ dependence of the on-axis value of (a) |ψ| and (b) |ψ| at the focal distance ζ=1/Φ (left ordinates, dashed–dotted curves). In the same graphics the absolute difference between these and their corresponding asymptotic expressions is shown (right ordinates, solid curves).

Fig. 9
Fig. 9

ξ dependence of the saddle points σ defined by Eq. (26) for different values of the parameter a. At the focal plane (a=1, solid curve) and in front of it (a<1, dashed–dotted curve) only one saddle point exists. Behind the focal plane (a>1), additional solutions arise in a finite interval around the origin (ξ=0). Two examples (dashed and dotted curves) are shown in the figure.

Fig. 10
Fig. 10

Circumfocal density plot of |ψ|2 including interference terms. (a) Exact intensity distribution for the choice Φ=50/π, (b) corresponding (lowest-order) asymptotic approximation.

Fig. 11
Fig. 11

Graphic representation of Eq. (31) without interference terms. The focal waists and their corresponding caustics are clearly visible in both the grayscale [(a)] and the contourline [(b)] plots.The numbers in (b) indicate the value of J corresponding to each region.

Tables (2)

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Table 1 Coefficients c^μ,2ν-μ Required for Evaluation of the Asymptotic Expansion Factors d^ν of Fˆ by Using Eqs. (A5 ) and (A6 )

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Table 2 Coefficients cˇμ,2ν-μ Required for Evaluation of the Asymptotic Expansion Factors dˇν of Fˇ by Using Eqs. (A5 ) and (A6 )

Equations (40)

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Ψ(x, z>0, t)=exp(-iωt)exp(ikzz)kzi2πz-dx  
×expi kz(x-x)22zT(x),
ψ(ξ, ζ)exp(iωt)exp(-ikzz)Ψ(x, z, t),
ψ(ξ, ζ)=1iπ4ζ-dξexpi (ξ-ξ)24ζψ(ξ, 0).
ψ(ξ, 0)=exp[-iΦ sin2(ξ/2)],
12π(2π)dξ|ψ(ξ, ζ)|2=1.
ψξ, 2π mn=r=1nC(r, m, n)ψξ+2π rn, 0,
C(r, m, n)1ns=1nexp-i2π ms2+rsn
L(ζ)1-12π(2π)dξ cos ξ|ψ(ξ, ζ)|2,
ψξ, 1Φ=Φi4π-dξexpΦiξ22-sin2ξ+ξ2
ξ=sin(ξ+ξ).
σ0(π+[ξ-π])-12 [ξ-π]+116[ξ-π]33!.
-dξexpiΦξ24-sin2ξ2F(Φ)
ψ(ξ, 1/Φ)A0(ξ)exp[-iΦϕ0(ξ)],
A0(ξ)1{1-cos[σ0(ξ)+ξ]}1/2
ϕ0(ξ)sin2σ0(ξ)+ξ2-σ0(ξ)22,
1-cos[σ0(ξ)+ξ]=[σ0(ξ)+1]-1,
12π(2π)dξ|ψ(ξ, 1/Φ)|2=12π {[σ0(ξ)](2π)+2π}=1.
A0(π+[ξ-π])121+12!ξ-π42+174!ξ-π44
ϕ0(π+[ξ-π])1-12!ξ-π22+14!12ξ-π24,
limΦ(2π)dξ cos(ξ)|ψ(ξ, 1/Φ)|2=(2π)dξ sin(ξ)σ0(ξ),
α=[3/(4π)]1/2Γ(1/4)exp(-iπ/8).
Fˆ(Φ)-dξsin2 ξexpiΦξ24-sin2ξ2,
Fˇ(Φ)-dξcos ξexpiΦξ24-sin2ξ2,
ψ(0, 1/Φ)=-Φi4πΦ24 Fˆ(Φ)+i Φ2 Fˇ(Φ).
ψξ, aΦ=Φi4πa-dξexpΦi1aξ22-sin2ξ+ξ2
σj=a sin(σj+ξ).
Kn,±(a)2πn±[arccos(1/a)-a2-1],
limΦ ψξ, aΦ=j=-J+JAj(ξ)exp[-iΦϕj(ξ)],
Aj(ξ)=1{1-a cos[σj(ξ)+ξ]}1/2,
ϕj(ξ)sin2σj(ξ)+ξ2-1aσj(ξ)22,
limΦψξ, aΦ2=j=-J+J|Aj(ξ)|2+interferenceterms.
limΦψξ, aΦ2=11-a cos ξ,
F(t)=-dxg(x)exp[th(x)]
g(x)exp[tx5(a5+a6x+a7x2+)]
=m=0n=0cm,n(tx5)mxn.
F(t)m=0n=0cm,ntm-dx exp(ta4x4)x5m+n
F(t)t-1/42m=0n=0cm,nt-(m+n)/4Γ5m+n+14(-a4)(5m+n+1)/4,
F(t)ν=0dνt-1/4-ν/2,
dν12μ=02νcμ,2ν-μΓ[(4μ+2ν+1)/4](-a4)(4μ+2ν+1)/4.

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