Abstract

We examine the free time evolution of a rectangular one dimensional Schrödinger wave packet of constant phase during the early stage which in the paraxial wave approximation is identical to the diffraction of a scalar field from a single slit. Our analysis, based on numerics and the Cornu spiral reveals considerable intricate detail behavior in the density and phase of the wave. We also point out a concentration of the intensity that occurs on axis and propose a new measure of width that expresses this concentration.

© 2012 OSA

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  1. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, London, 1964).
  2. G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys.155, 427–474 (1959).
    [CrossRef]
  3. C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys.42, 4–11 (1974).
    [CrossRef]
  4. J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys.37, 905–912 (1969).
    [CrossRef]
  5. A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988).
    [CrossRef]
  6. O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys.71, 319–325 (2003).
    [CrossRef]
  7. In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev.88, 625–631 (1952).
    [CrossRef]
  8. J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis.VI.3, 105–115 (1957).
  9. M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch.56a, 173–177 (2001).
  10. M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter15, R233–R255 (2003).
    [CrossRef]
  11. W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009).
    [CrossRef] [PubMed]
  12. T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992).
    [CrossRef]
  13. W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express17(23), 20966–20974 (2009).
    [CrossRef] [PubMed]
  14. A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A71, 043612 (2005).
    [CrossRef]
  15. See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002).
    [CrossRef] [PubMed]
  16. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000).
    [CrossRef] [PubMed]
  17. Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett.105, 183601 (2010).
    [CrossRef]
  18. I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002).
    [CrossRef] [PubMed]
  19. M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen.36, 7113–7128 (2003).
    [CrossRef]
  20. K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010).
    [CrossRef]
  21. R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt.57, 1437–1444 (2010).
    [CrossRef]
  22. T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
    [CrossRef]
  23. L. Novotny, “The history of near-field optics” in Progress in Optics vol. 50, E. Wolf, ed. (Elsevier, Amsterdam, 2007) pp. 137–184.
    [CrossRef]
  24. D. Courjon, Near-field Microscopy and Near-Field Optics (World Scientific Publishing, Singapore, 2003).
    [CrossRef]
  25. T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
    [CrossRef]
  26. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  27. The paraxial approximation is expected to hold as long as the wavelength is much less than the slit width.
  28. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
  29. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  30. R. W. Wood, Physical Optics (Optical Society of America, Washington, 1988).
  31. M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen.37, 9549–9556 (2004).
    [CrossRef]
  32. E. Sadurní, W. B. Case, and W. P. Schleich, in preparation.
  33. M. Gonçalves (personal communication, 2011).
  34. A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010).
    [CrossRef]
  35. T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004).
    [CrossRef]

2010

Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett.105, 183601 (2010).
[CrossRef]

K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010).
[CrossRef]

R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt.57, 1437–1444 (2010).
[CrossRef]

A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010).
[CrossRef]

2009

W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express17(23), 20966–20974 (2009).
[CrossRef] [PubMed]

T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
[CrossRef]

W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009).
[CrossRef] [PubMed]

2005

A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A71, 043612 (2005).
[CrossRef]

2004

M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen.37, 9549–9556 (2004).
[CrossRef]

T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004).
[CrossRef]

2003

M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter15, R233–R255 (2003).
[CrossRef]

M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen.36, 7113–7128 (2003).
[CrossRef]

O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys.71, 319–325 (2003).
[CrossRef]

2002

I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002).
[CrossRef] [PubMed]

See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002).
[CrossRef] [PubMed]

2001

M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch.56a, 173–177 (2001).

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
[CrossRef]

2000

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000).
[CrossRef] [PubMed]

1992

T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992).
[CrossRef]

1988

A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988).
[CrossRef]

1987

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
[CrossRef] [PubMed]

1974

C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys.42, 4–11 (1974).
[CrossRef]

1969

J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys.37, 905–912 (1969).
[CrossRef]

1959

G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys.155, 427–474 (1959).
[CrossRef]

1957

J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis.VI.3, 105–115 (1957).

1952

In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev.88, 625–631 (1952).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000).
[CrossRef] [PubMed]

Al-Amri, M.

Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett.105, 183601 (2010).
[CrossRef]

Andreata, M.

M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen.36, 7113–7128 (2003).
[CrossRef]

Arndt, M.

Arsenovic, D.

M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch.56a, 173–177 (2001).

Balykin, V.

T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992).
[CrossRef]

Bialynicki-Birula, I.

I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002).
[CrossRef] [PubMed]

Bills, F. A.

J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys.37, 905–912 (1969).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, London, 1964).

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000).
[CrossRef] [PubMed]

Bozic, M.

M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch.56a, 173–177 (2001).

Bracco, G.

T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
[CrossRef]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000).
[CrossRef] [PubMed]

Buse, K.

See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002).
[CrossRef] [PubMed]

Case, W. B.

Charron, E.

A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010).
[CrossRef]

Chin, C.

T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004).
[CrossRef]

Cirone, M. A.

I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002).
[CrossRef] [PubMed]

Courjon, D.

D. Courjon, Near-field Microscopy and Near-Field Optics (World Scientific Publishing, Singapore, 2003).
[CrossRef]

Dahl, J. P.

I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002).
[CrossRef] [PubMed]

Deachapunya, S.

Dodonov, D.

M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen.36, 7113–7128 (2003).
[CrossRef]

Dowling, J. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000).
[CrossRef] [PubMed]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
[CrossRef] [PubMed]

Engel, E.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
[CrossRef]

Ernst, W. E.

T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
[CrossRef]

Fedorov, M.

I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002).
[CrossRef] [PubMed]

Feynman, R. P.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Fickler, R.

W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009).
[CrossRef] [PubMed]

Fladischer, K.

T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
[CrossRef]

Flaspöhler, M.

See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002).
[CrossRef] [PubMed]

Gaaloul, N.

A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010).
[CrossRef]

Gähler, R.

A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988).
[CrossRef]

Gleisberg, F.

K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010).
[CrossRef]

Goldemberg, J.

J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis.VI.3, 105–115 (1957).

Gonçalves, M.

M. Gonçalves (personal communication, 2011).

Grimm, R.

T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004).
[CrossRef]

Hall, M. J. W.

M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen.37, 9549–9556 (2004).
[CrossRef]

Harshman, N. L.

K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010).
[CrossRef]

Haubrich, D.

See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002).
[CrossRef] [PubMed]

Hell, S. W.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
[CrossRef]

Herbig, J.

T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004).
[CrossRef]

Hibbs, A. R.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Holst, B.

T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
[CrossRef]

Jaouadi, A.

A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010).
[CrossRef]

Jönsson, C.

C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys.42, 4–11 (1974).
[CrossRef]

G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys.155, 427–474 (1959).
[CrossRef]

Kazemi, P.

K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010).
[CrossRef]

Klar, T. A.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
[CrossRef]

Kok, P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000).
[CrossRef] [PubMed]

Kraemer, T.

T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004).
[CrossRef]

Leavitt, J. A.

J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys.37, 905–912 (1969).
[CrossRef]

Liao, Z.

Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett.105, 183601 (2010).
[CrossRef]

Linke, N. M.

W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009).
[CrossRef] [PubMed]

Mack, R.

K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010).
[CrossRef]

R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt.57, 1437–1444 (2010).
[CrossRef]

Mampe, W.

A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988).
[CrossRef]

Mark, M.

T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004).
[CrossRef]

Meijer, J.

W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009).
[CrossRef] [PubMed]

Meschede, D.

See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002).
[CrossRef] [PubMed]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
[CrossRef] [PubMed]

Mlynek, J.

T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992).
[CrossRef]

Möllenstedt, G.

G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys.155, 427–474 (1959).
[CrossRef]

Moshinsky, M.

In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev.88, 625–631 (1952).
[CrossRef]

Mützel, M.

See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002).
[CrossRef] [PubMed]

Nägerl, H. C.

T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004).
[CrossRef]

Nairz, O.

O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys.71, 319–325 (2003).
[CrossRef]

Novotny, L.

L. Novotny, “The history of near-field optics” in Progress in Optics vol. 50, E. Wolf, ed. (Elsevier, Amsterdam, 2007) pp. 137–184.
[CrossRef]

Nussenzveig, H. M.

J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis.VI.3, 105–115 (1957).

Oberthaler, M.

M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter15, R233–R255 (2003).
[CrossRef]

Patel, A.

T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
[CrossRef]

Peithmann, K.

See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002).
[CrossRef] [PubMed]

Pfau, T.

M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter15, R233–R255 (2003).
[CrossRef]

T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992).
[CrossRef]

Plimak, L.

K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010).
[CrossRef]

Pruvost, L.

A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010).
[CrossRef]

Reingruber, H.

T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
[CrossRef]

Reisinger, T.

T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
[CrossRef]

Sadurní, E.

E. Sadurní, W. B. Case, and W. P. Schleich, in preparation.

Schleich, W. P.

R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt.57, 1437–1444 (2010).
[CrossRef]

K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010).
[CrossRef]

I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002).
[CrossRef] [PubMed]

E. Sadurní, W. B. Case, and W. P. Schleich, in preparation.

Schmidt-Kaler, F.

W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009).
[CrossRef] [PubMed]

Schnitzler, W.

W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009).
[CrossRef] [PubMed]

Shull, C. G.

A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988).
[CrossRef]

Singer, K.

W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009).
[CrossRef] [PubMed]

Sleator, T.

A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A71, 043612 (2005).
[CrossRef]

T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992).
[CrossRef]

Smith, H. I.

T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Tandler, S.

See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002).
[CrossRef] [PubMed]

Telmini, M.

A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010).
[CrossRef]

Tomandl, M.

Tonyushkin, A.

A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A71, 043612 (2005).
[CrossRef]

Treimer, W.

A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988).
[CrossRef]

Turlapov, A.

A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A71, 043612 (2005).
[CrossRef]

Viaris de Lesegno, B.

A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010).
[CrossRef]

Vogel, K.

K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010).
[CrossRef]

Vuskovic, L.

M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch.56a, 173–177 (2001).

Weber, T.

T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004).
[CrossRef]

Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, London, 1964).

Wood, R. W.

R. W. Wood, Physical Optics (Optical Society of America, Washington, 1988).

Yakovlev, V. P.

R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt.57, 1437–1444 (2010).
[CrossRef]

Zeilinger, A.

O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys.71, 319–325 (2003).
[CrossRef]

A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988).
[CrossRef]

Zubairy, M.S.

Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett.105, 183601 (2010).
[CrossRef]

Am. J. Phys.

C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys.42, 4–11 (1974).
[CrossRef]

J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys.37, 905–912 (1969).
[CrossRef]

O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys.71, 319–325 (2003).
[CrossRef]

App. Phys. B

T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004).
[CrossRef]

Appl. Phys. B.

T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992).
[CrossRef]

Chemical Physics

K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010).
[CrossRef]

J. Mod. Opt.

R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt.57, 1437–1444 (2010).
[CrossRef]

J. Phys. A: Math. Gen.

M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen.37, 9549–9556 (2004).
[CrossRef]

M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen.36, 7113–7128 (2003).
[CrossRef]

J. Phys.: Condens. Matter

M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter15, R233–R255 (2003).
[CrossRef]

Opt. Express

Phys. Rev.

In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev.88, 625–631 (1952).
[CrossRef]

Phys. Rev. A

A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010).
[CrossRef]

T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009).
[CrossRef]

A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A71, 043612 (2005).
[CrossRef]

Phys. Rev. E

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
[CrossRef]

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
[CrossRef] [PubMed]

See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002).
[CrossRef] [PubMed]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000).
[CrossRef] [PubMed]

Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett.105, 183601 (2010).
[CrossRef]

I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002).
[CrossRef] [PubMed]

W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009).
[CrossRef] [PubMed]

Rev. Mex. Fis.

J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis.VI.3, 105–115 (1957).

Rev. Mod. Phys.

A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988).
[CrossRef]

Z. f. Phys.

G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys.155, 427–474 (1959).
[CrossRef]

Z. Naturforsch.

M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch.56a, 173–177 (2001).

Other

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, London, 1964).

L. Novotny, “The history of near-field optics” in Progress in Optics vol. 50, E. Wolf, ed. (Elsevier, Amsterdam, 2007) pp. 137–184.
[CrossRef]

D. Courjon, Near-field Microscopy and Near-Field Optics (World Scientific Publishing, Singapore, 2003).
[CrossRef]

The paraxial approximation is expected to hold as long as the wavelength is much less than the slit width.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

R. W. Wood, Physical Optics (Optical Society of America, Washington, 1988).

E. Sadurní, W. B. Case, and W. P. Schleich, in preparation.

M. Gonçalves (personal communication, 2011).

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Figures (4)

Fig. 1
Fig. 1

Near-field patterns (top) originating from the time evolution of a rectangular wave packet, and their explanation (bottom) with the help of the Cornu spiral. Here we depict the probability density (a) and the phase (b) of the wave function ψ given by Eqs. (4) and (5) in their dependence on the dimensionless time τ and coordinate χ. The white line located at τ = 1/3 indicates both the dominant peak of the probability density corresponding to the focusing of the wave packet and the main plateau of the phase of ψ. The Cornu spirals of (c) and (d) show the complex-valued function F(w) = FC(w) + iFS(w) where the arc length of the curve is parameterized by the argument w. Maxima of |F(w)| correspond to positions on the spiral where the separation from the origin assumes a local maximum. The blue arrows in (c) indicate the first two maxima with n = 0, 1. The corresponding maxima in terms of the space-time variables τ and χ are given by Eqs. (12) and (13). The red arrows indicate maxima of the phase of F(w), which coincide with maxima in the phase of ψ for χ = 0; again we only show the first two corresponding to k = 0, 1. The central arrow in (d) labeled 1 shows the approximate location of the intensity maximum at τ = 1/3 and χ = 0. Arrows 2 and 3 represent arguments of F for τ = 1/3 but with nonzero χ. Since the arguments of the two Fresnel integrals in Eq. (5) defining the wave function ψ are w = 2 / τ ( 1 / 2 ± χ ), we see that the two contributions are symmetrically placed around χ = 0 and thus ψ has a smaller amplitude but a similar phase compared to ψ at χ = 0.

Fig. 2
Fig. 2

Probability density |ψ|2 of a freely propagating rectangular wave packet as a function of the space-time variables τ and χ, similar to Fig. 1(a) but on a finer time scale close to the origin. Light and dark colors represent high and low densities, respectively. Superimposed are the parabolas of Eq. (12) and Eq. (13) indicated by blue and red lines, respectively. At the crossings of the parabolas maxima of the intensity pattern occur.

Fig. 3
Fig. 3

Family of normalized Gaussian widths δ���� (κ, τ)/��(κ, 0) of the freely-propagating initial rectangular wave packet as a function of the dimensionless time τht/(ma2) and the parametrization κa of the measure. For the optimal parameter κa ≈ 4.5 and τ ≈ 0.3 a global minimum, corresponding to the focused probability peak occurs, in complete agreement with the numerical evaluation of the time-dependent probability density shown in Fig. 1(a) and the analytical considerations of section 2.3. This choice of the parameter κ indicates a maximal shrinkage of the width and therefore represents the best way of capturing focusing.

Fig. 4
Fig. 4

Influence of sharp edges on the spatio-temporal probability density patterns generated by flat distributions with Gaussian edges. In each frame a different slope is achieved by varying the width Δχ of the Gaussian: Δχ = 1/100 (top left), Δχ = 1/10 (top right), Δχ = 1 (lower left), and Δχ = 10 (lower right). For small values of Δχ, that is for large slopes at the edges we recover the focusing peak and the intricacies of the pattern, while for soft edges, that is for increasing values of Δχ the details near the slit disappear and the focusing effect is mitigated.

Equations (16)

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

ψ ( χ , 0 ) Θ ( 1 2 | χ | ) ,
K ( χ χ , τ ) = 1 τ exp [ i π / 4 ] exp [ i π ( χ χ ) 2 / τ ]
ψ ( χ , τ ) = 1 τ exp [ i π / 4 ] 1 / 2 1 / 2 d χ exp [ i π ( χ χ ) 2 / τ ]
F ( w ) F C ( w ) + i F S ( w ) 0 w cos [ π y 2 / 2 ] d y + i 0 w sin [ π y 2 / 2 ] d y
ψ ( χ , τ ) = 1 2 exp [ i π / 4 ] { F [ 2 τ ( 1 2 χ ) ] + F [ 2 τ ( 1 2 + χ ) ] } .
d | F | d w = d d w ( F C ) 2 + ( F S ) 2 = 1 | F | [ F C ( F C ) + F S ( F S ) ] = F C | F | [ F C + F S ] = 0
cos ( π w l 2 2 ) + sin ( π w l 2 2 ) = 0 .
w l = 3 2 + 2 l .
w n 2 = 3 2 + 4 n
w m 2 = 7 2 + 2 m
w = 2 τ ( 1 2 ± χ ) .
τ = 4 3 + 8 n ( 1 2 + χ ) 2
τ = 4 3 + 8 n ( 1 2 χ ) 2 ,
𝒲 1 κ 2 ( 1 e ( κ x ) 2 )
τ = z λ a 2 1 3 .
z 1 3 × a ( a λ ) 0.691 × a

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