Abstract

We obtain the absolute phase in Fresnel diffraction for Gaussian beams by using a modified polarization Sagnac interferometer in which counterpropagating paths are spatially separated and labeled according to polarization. By erasing the polarization “which-path” information with an analyzing polarizer situated after the modified interferometer, we are able to regain interference and to precisely control the relative intensities of the diffracted and the reference beams. The resulting optimized visibility allows for a precise phase determination. This setup is very stable, requiring no active elements.

© 2007 Optical Society of America

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References

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  1. M. Mansuripur, The Physical Principles of Magneto-Optical Recording, 1st ed. (Cambridge U. Press. 1998).
  2. W. Treimer, M. Strobl, and A. Hilger, "Observation of edge diffraction with a double crystal diffractometer," Cryst. Res. Technol. 37, 727-733 (2002).
    [CrossRef]
  3. Z. Ping, G. Wei-Jian, and Y. Jian-Ping, "Diffracted field distribution from a knife-edge truncated semi-Gaussian beam as an atomic (molecular) mirror," Chin. Phys. 15, 116-125 (2006).
    [CrossRef]
  4. R. W. Boyd and D. T. Moore, "Interferometric measurement of the optical phase distribution for Fresnel diffraction by a straightedge," Appl. Opt. 18, 2013-2016 (1979).
    [CrossRef] [PubMed]
  5. T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeiliriger, "Complementarity and the quantum eraser," Phys. Rev. Lett. 75, 3034-3037 (1995).
    [CrossRef] [PubMed]
  6. J. E. Pearson, T. C. McGill, S. Kurtin, and A. Yariv, "Diffraction of Gaussian laser beams by a semi-infinite plane," J. Opt. Soc. Am. 59, 1440-1445 (1969).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  8. R. Sambasivan, "Diffraction of focused laser beams," Opt. Acta 21, 323-328 (1974).
    [CrossRef]
  9. J. A. Murphy and A. Egan, "Examples of Fresnel diffraction using Gaussian modes," Eur. J. Phys. 14, 121-127 (1993).
    [CrossRef]
  10. L. E. R. Petersson and G. S. Smith, "Three-dimensional electromagnetic diffraction of a Gaussian beam by a perfectly conducting plane," J. Opt. Soc. Am. A 19, 2265-2280 (2002).
    [CrossRef]
  11. P. W. Milonni and J. H. Eberly, Lasers, 1st ed. (Wiley, 1988), Chap. 14.5.
  12. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 4.2.
  13. D. A. Jackson, A. D. Kersey, and A. C. Lewin, "Fibre gyroscope with passive quadrature detection," Electron. Lett. 20, 399-401 (1984).
    [CrossRef]
  14. M. A. Novikov, "Polarization ring interferometer-ellipsometer," Opt. Spektrosk. 61, 424-427 (1986).
  15. K.-X. Sun, M. M. Fejer. E. Gustafson, and R. L. Byer, "Sagnac interferometer for gravitational-wave detection," Phys. Rev. Lett. 76, 3053-3056 (1996).
    [CrossRef] [PubMed]
  16. P. T. Beyersdorf, M. M. Fejer, and R. L. Byer, "Polarization Sagnac interferometer with a common-path local oscillator for heterodyne detection," J. Opt. Soc. Am. B 16, 1354-1358 (1999).
    [CrossRef]
  17. T. Shirai, T. H. Barnies, and T. G. Haskel, "Surface-profile measurement by means of a polarization Sagnac interferometer with parallel optical feedback," Opt. Lett. 24, 297-299 (1999).
    [CrossRef]
  18. J. Xia, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, "Modified Sagriac interferometer for high-sensitivity rnagneto-optic measurements at cryogenic temperatures," Appl. Phys. Lett. 89, 062508 (2006).
    [CrossRef]
  19. T. Kim, M. Fiorentino, and F. N. C. Wong, "Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer," Phys. Rev. A 73, 012316 (2006).
    [CrossRef]
  20. P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, "Grover's search algorithm: an optical approach," J. Mod. Opt. 47, 257-266 (2000).
    [CrossRef]

2006

Z. Ping, G. Wei-Jian, and Y. Jian-Ping, "Diffracted field distribution from a knife-edge truncated semi-Gaussian beam as an atomic (molecular) mirror," Chin. Phys. 15, 116-125 (2006).
[CrossRef]

J. Xia, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, "Modified Sagriac interferometer for high-sensitivity rnagneto-optic measurements at cryogenic temperatures," Appl. Phys. Lett. 89, 062508 (2006).
[CrossRef]

T. Kim, M. Fiorentino, and F. N. C. Wong, "Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer," Phys. Rev. A 73, 012316 (2006).
[CrossRef]

2002

W. Treimer, M. Strobl, and A. Hilger, "Observation of edge diffraction with a double crystal diffractometer," Cryst. Res. Technol. 37, 727-733 (2002).
[CrossRef]

L. E. R. Petersson and G. S. Smith, "Three-dimensional electromagnetic diffraction of a Gaussian beam by a perfectly conducting plane," J. Opt. Soc. Am. A 19, 2265-2280 (2002).
[CrossRef]

2000

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, "Grover's search algorithm: an optical approach," J. Mod. Opt. 47, 257-266 (2000).
[CrossRef]

1999

1998

M. Mansuripur, The Physical Principles of Magneto-Optical Recording, 1st ed. (Cambridge U. Press. 1998).

1996

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 4.2.

K.-X. Sun, M. M. Fejer. E. Gustafson, and R. L. Byer, "Sagnac interferometer for gravitational-wave detection," Phys. Rev. Lett. 76, 3053-3056 (1996).
[CrossRef] [PubMed]

1995

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeiliriger, "Complementarity and the quantum eraser," Phys. Rev. Lett. 75, 3034-3037 (1995).
[CrossRef] [PubMed]

1993

J. A. Murphy and A. Egan, "Examples of Fresnel diffraction using Gaussian modes," Eur. J. Phys. 14, 121-127 (1993).
[CrossRef]

1988

P. W. Milonni and J. H. Eberly, Lasers, 1st ed. (Wiley, 1988), Chap. 14.5.

1986

M. A. Novikov, "Polarization ring interferometer-ellipsometer," Opt. Spektrosk. 61, 424-427 (1986).

1984

D. A. Jackson, A. D. Kersey, and A. C. Lewin, "Fibre gyroscope with passive quadrature detection," Electron. Lett. 20, 399-401 (1984).
[CrossRef]

1979

1974

R. Sambasivan, "Diffraction of focused laser beams," Opt. Acta 21, 323-328 (1974).
[CrossRef]

1969

Barnies, T. H.

Beyersdorf, P. T.

J. Xia, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, "Modified Sagriac interferometer for high-sensitivity rnagneto-optic measurements at cryogenic temperatures," Appl. Phys. Lett. 89, 062508 (2006).
[CrossRef]

P. T. Beyersdorf, M. M. Fejer, and R. L. Byer, "Polarization Sagnac interferometer with a common-path local oscillator for heterodyne detection," J. Opt. Soc. Am. B 16, 1354-1358 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Boyd, R. W.

Byer, R. L.

P. T. Beyersdorf, M. M. Fejer, and R. L. Byer, "Polarization Sagnac interferometer with a common-path local oscillator for heterodyne detection," J. Opt. Soc. Am. B 16, 1354-1358 (1999).
[CrossRef]

K.-X. Sun, M. M. Fejer. E. Gustafson, and R. L. Byer, "Sagnac interferometer for gravitational-wave detection," Phys. Rev. Lett. 76, 3053-3056 (1996).
[CrossRef] [PubMed]

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers, 1st ed. (Wiley, 1988), Chap. 14.5.

Egan, A.

J. A. Murphy and A. Egan, "Examples of Fresnel diffraction using Gaussian modes," Eur. J. Phys. 14, 121-127 (1993).
[CrossRef]

Fejer, M. M.

J. Xia, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, "Modified Sagriac interferometer for high-sensitivity rnagneto-optic measurements at cryogenic temperatures," Appl. Phys. Lett. 89, 062508 (2006).
[CrossRef]

P. T. Beyersdorf, M. M. Fejer, and R. L. Byer, "Polarization Sagnac interferometer with a common-path local oscillator for heterodyne detection," J. Opt. Soc. Am. B 16, 1354-1358 (1999).
[CrossRef]

K.-X. Sun, M. M. Fejer. E. Gustafson, and R. L. Byer, "Sagnac interferometer for gravitational-wave detection," Phys. Rev. Lett. 76, 3053-3056 (1996).
[CrossRef] [PubMed]

Fiorentino, M.

T. Kim, M. Fiorentino, and F. N. C. Wong, "Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer," Phys. Rev. A 73, 012316 (2006).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 4.2.

Gustafson, E.

K.-X. Sun, M. M. Fejer. E. Gustafson, and R. L. Byer, "Sagnac interferometer for gravitational-wave detection," Phys. Rev. Lett. 76, 3053-3056 (1996).
[CrossRef] [PubMed]

Haskel, T. G.

Herzog, T. J.

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeiliriger, "Complementarity and the quantum eraser," Phys. Rev. Lett. 75, 3034-3037 (1995).
[CrossRef] [PubMed]

Hilger, A.

W. Treimer, M. Strobl, and A. Hilger, "Observation of edge diffraction with a double crystal diffractometer," Cryst. Res. Technol. 37, 727-733 (2002).
[CrossRef]

Jackson, D. A.

D. A. Jackson, A. D. Kersey, and A. C. Lewin, "Fibre gyroscope with passive quadrature detection," Electron. Lett. 20, 399-401 (1984).
[CrossRef]

Jian-Ping, Y.

Z. Ping, G. Wei-Jian, and Y. Jian-Ping, "Diffracted field distribution from a knife-edge truncated semi-Gaussian beam as an atomic (molecular) mirror," Chin. Phys. 15, 116-125 (2006).
[CrossRef]

Kapitulnik, A.

J. Xia, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, "Modified Sagriac interferometer for high-sensitivity rnagneto-optic measurements at cryogenic temperatures," Appl. Phys. Lett. 89, 062508 (2006).
[CrossRef]

Kersey, A. D.

D. A. Jackson, A. D. Kersey, and A. C. Lewin, "Fibre gyroscope with passive quadrature detection," Electron. Lett. 20, 399-401 (1984).
[CrossRef]

Kim, T.

T. Kim, M. Fiorentino, and F. N. C. Wong, "Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer," Phys. Rev. A 73, 012316 (2006).
[CrossRef]

Kurtin, S.

Kwiat, P. G.

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, "Grover's search algorithm: an optical approach," J. Mod. Opt. 47, 257-266 (2000).
[CrossRef]

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeiliriger, "Complementarity and the quantum eraser," Phys. Rev. Lett. 75, 3034-3037 (1995).
[CrossRef] [PubMed]

Lewin, A. C.

D. A. Jackson, A. D. Kersey, and A. C. Lewin, "Fibre gyroscope with passive quadrature detection," Electron. Lett. 20, 399-401 (1984).
[CrossRef]

Mansuripur, M.

M. Mansuripur, The Physical Principles of Magneto-Optical Recording, 1st ed. (Cambridge U. Press. 1998).

McGill, T. C.

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers, 1st ed. (Wiley, 1988), Chap. 14.5.

Mitchell, J. R.

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, "Grover's search algorithm: an optical approach," J. Mod. Opt. 47, 257-266 (2000).
[CrossRef]

Moore, D. T.

Murphy, J. A.

J. A. Murphy and A. Egan, "Examples of Fresnel diffraction using Gaussian modes," Eur. J. Phys. 14, 121-127 (1993).
[CrossRef]

Novikov, M. A.

M. A. Novikov, "Polarization ring interferometer-ellipsometer," Opt. Spektrosk. 61, 424-427 (1986).

Pearson, J. E.

Petersson, L. E. R.

Ping, Z.

Z. Ping, G. Wei-Jian, and Y. Jian-Ping, "Diffracted field distribution from a knife-edge truncated semi-Gaussian beam as an atomic (molecular) mirror," Chin. Phys. 15, 116-125 (2006).
[CrossRef]

Sambasivan, R.

R. Sambasivan, "Diffraction of focused laser beams," Opt. Acta 21, 323-328 (1974).
[CrossRef]

Schwindt, P. D. D.

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, "Grover's search algorithm: an optical approach," J. Mod. Opt. 47, 257-266 (2000).
[CrossRef]

Shirai, T.

Smith, G. S.

Strobl, M.

W. Treimer, M. Strobl, and A. Hilger, "Observation of edge diffraction with a double crystal diffractometer," Cryst. Res. Technol. 37, 727-733 (2002).
[CrossRef]

Sun, K.-X.

K.-X. Sun, M. M. Fejer. E. Gustafson, and R. L. Byer, "Sagnac interferometer for gravitational-wave detection," Phys. Rev. Lett. 76, 3053-3056 (1996).
[CrossRef] [PubMed]

Treimer, W.

W. Treimer, M. Strobl, and A. Hilger, "Observation of edge diffraction with a double crystal diffractometer," Cryst. Res. Technol. 37, 727-733 (2002).
[CrossRef]

Wei-Jian, G.

Z. Ping, G. Wei-Jian, and Y. Jian-Ping, "Diffracted field distribution from a knife-edge truncated semi-Gaussian beam as an atomic (molecular) mirror," Chin. Phys. 15, 116-125 (2006).
[CrossRef]

Weinfurter, H.

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeiliriger, "Complementarity and the quantum eraser," Phys. Rev. Lett. 75, 3034-3037 (1995).
[CrossRef] [PubMed]

White, A. G.

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, "Grover's search algorithm: an optical approach," J. Mod. Opt. 47, 257-266 (2000).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Wong, F. N. C.

T. Kim, M. Fiorentino, and F. N. C. Wong, "Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer," Phys. Rev. A 73, 012316 (2006).
[CrossRef]

Xia, J.

J. Xia, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, "Modified Sagriac interferometer for high-sensitivity rnagneto-optic measurements at cryogenic temperatures," Appl. Phys. Lett. 89, 062508 (2006).
[CrossRef]

Yariv, A.

Zeiliriger, A.

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeiliriger, "Complementarity and the quantum eraser," Phys. Rev. Lett. 75, 3034-3037 (1995).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

J. Xia, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, "Modified Sagriac interferometer for high-sensitivity rnagneto-optic measurements at cryogenic temperatures," Appl. Phys. Lett. 89, 062508 (2006).
[CrossRef]

Chin. Phys.

Z. Ping, G. Wei-Jian, and Y. Jian-Ping, "Diffracted field distribution from a knife-edge truncated semi-Gaussian beam as an atomic (molecular) mirror," Chin. Phys. 15, 116-125 (2006).
[CrossRef]

Cryst. Res. Technol.

W. Treimer, M. Strobl, and A. Hilger, "Observation of edge diffraction with a double crystal diffractometer," Cryst. Res. Technol. 37, 727-733 (2002).
[CrossRef]

Electron. Lett.

D. A. Jackson, A. D. Kersey, and A. C. Lewin, "Fibre gyroscope with passive quadrature detection," Electron. Lett. 20, 399-401 (1984).
[CrossRef]

Eur. J. Phys.

J. A. Murphy and A. Egan, "Examples of Fresnel diffraction using Gaussian modes," Eur. J. Phys. 14, 121-127 (1993).
[CrossRef]

J. Mod. Opt.

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, "Grover's search algorithm: an optical approach," J. Mod. Opt. 47, 257-266 (2000).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Acta

R. Sambasivan, "Diffraction of focused laser beams," Opt. Acta 21, 323-328 (1974).
[CrossRef]

Opt. Lett.

Opt. Spektrosk.

M. A. Novikov, "Polarization ring interferometer-ellipsometer," Opt. Spektrosk. 61, 424-427 (1986).

Phys. Rev. A

T. Kim, M. Fiorentino, and F. N. C. Wong, "Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer," Phys. Rev. A 73, 012316 (2006).
[CrossRef]

Phys. Rev. Lett.

K.-X. Sun, M. M. Fejer. E. Gustafson, and R. L. Byer, "Sagnac interferometer for gravitational-wave detection," Phys. Rev. Lett. 76, 3053-3056 (1996).
[CrossRef] [PubMed]

T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeiliriger, "Complementarity and the quantum eraser," Phys. Rev. Lett. 75, 3034-3037 (1995).
[CrossRef] [PubMed]

Other

M. Mansuripur, The Physical Principles of Magneto-Optical Recording, 1st ed. (Cambridge U. Press. 1998).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

P. W. Milonni and J. H. Eberly, Lasers, 1st ed. (Wiley, 1988), Chap. 14.5.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 4.2.

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Figures (8)

Fig. 1
Fig. 1

Geometry of diffraction of a Gaussian beam by a straight edge.

Fig. 2
Fig. 2

(a) Cornu’s spiral and (b) intensity and phase profiles for the case of a straight edge blocking the semi-infinite plane ( x , y ) for x 0 ( w 0 ) , assuming an incident plane wave. (c) Typical intensity and phase profiles for a tightly focused Gaussian beam.

Fig. 3
Fig. 3

Intensity profiles from Eq. (7) for selected analyzer angles. For θ = 90 ° (the diffracted, V-polarized beam) the familiar straight-edge diffraction pattern appears. θ = 0 ° gives the intensity from the H-polarized reference beam. The plot at 45 ° shows the combined intensity when the analyzing polarizer has the same polarization as the beam entering the interferometer. The plot at 45 ° shows an apparent inversion of the diffracted beam with respect to w = 0 , although its intensity is smaller. The other analysis angles show the tuning of the minima at different values of w (positive, zero, and negative, see vertical arrows) for θ in the ( 90 ° , 45 ° ) interval, implying a perfect visibility with respect to the neighboring intensity maxima. The maximum achievable visibility decreases outside the ( 90 ° , 45 ° ) interval, as stressed in the 22.5 ° subplot.

Fig. 4
Fig. 4

Schematic view of our experimental setup. A coherent beam from a He Ne laser is split in a modified Sagnac interferometer into two paths labeled with orthogonal polarizations. We place a knife edge into the outermost path and tune the relative intensities from each contributing path, using a polarizer at θ, prior to recording the resulting image. PBS stands for polarizing beam splitter.

Fig. 5
Fig. 5

Intensity profiles for four analyzer settings. Black curves correspond to the measured data. Light gray (blue online) in all panels curves are theoretical predictions employing Cornu’s spiral and a best-fit Gaussian envelope. The knife edge blocks the right half of the beam. θ = ± 90 ° corresponds to vertically polarized light (path with knife edge), and θ = 0 ° corresponds to horizontally polarized light (path with no obstacle). For θ = 90 ° we also display the theoretical normalized intensity [upper (red online) curve in top panel] with no envelope function.

Fig. 6
Fig. 6

Intensity profiles used to determine the phase ϕ [from Eq. (7)] and the amount of noise in the experimental data. Note the reduced scale for panel 3 (destructive interference). Black curves correspond to experimental data, and the gray (green online) curves in panels 1, 2, and 4 are the best-fit Gaussian envelope function.

Fig. 7
Fig. 7

(a) “ cos ϕ ” function and the Fresnel phase (dots) as obtained from raw data with the polarization analyzer at 45 ° . Solid curve in the phase plot corresponds to the Fresnel phase as predicted using Cornu’s spiral. (b) From the symmetric data, taken with the analyzer at 63.5 ° , we were able to match local minima (triangles) and maxima (squares) from experiment to the plane-wave theory. The lowest panels depict a zoom-in of the dashed rectangle regions, for which w 0 (nonshadow region).

Fig. 8
Fig. 8

(a) Theoretical phase from straight-edge diffraction for a focused beam, (b) Intensity profiles for two analyzer settings. Experimental curves are in black, theoretical curves in gray (blue online).

Equations (16)

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

U 0 ( ρ , z ) = i A Z 0 q ( z ) exp ( i k z ) exp [ i k 2 q ( z ) ρ 2 ] .
U d ( ρ , z ) = i k exp [ i k ( z z 1 ) ] 2 π ( z z 1 ) + d y x 1 + d x U 0 ( x , y , z 1 ) exp [ i k ( ρ ρ ) 2 2 ( z z 1 ) ] ,
U d ( ρ , z ) = U 0 ( ρ , z ) 2 [ 1 + ( i 1 ) ( C ( w ) + i S ( w ) ) ] U d ( ρ , z ) exp [ i ϕ ( w ) ] ,
C ( w ) 0 w d t cos ( π t 2 2 ) ,
S ( w ) 0 w d t sin ( π t 2 2 ) ,
w ( x , z ) = k ( z i Z 0 ) π ( z 1 i Z 0 ) ( z z 1 ) ( x 1 z 1 i Z 0 z i Z 0 x ) .
lim Z 0 z 1 w = k π ( z z 1 ) ( x 1 x ) .
I ( θ , w ) = 1 2 ( cos θ , sin θ ) ( U 0 U d e i ( ϕ ( w ) + δ ) ) 2 = 1 2 ( U 0 2 cos 2 θ + U d ( w ) 2 sin 2 θ + U 0 U d ( w ) sin ( 2 θ ) cos [ ϕ ( w ) + δ ] ) .
I ( w ) = 1 2 ( U 0 2 + U d ( w ) 2 + 2 U 0 U d ( w ) cos [ θ ( w ) + δ ] ) .
U d ( ρ , z ) = k A Z 0 e i k z 2 π ( z z 1 ) q ( z 1 ) + d y x 1 + d x exp [ i k 2 q ( z 1 ) ρ 2 ] exp [ i k ( ρ ρ ) 2 2 ( z z 1 ) ] .
i k 2 ( z z 1 ) [ α ( ρ ρ α ) 2 + ( 1 1 α ) ρ 2 ] ,
U d ( ρ , z ) = k A Z 0 e i k z 2 π ( z z 1 ) q ( z 1 ) exp [ i k 2 q ( z ) ρ 2 ] × + d y exp [ i k α ( y y α ) 2 2 ( z z 1 ) ] x 1 + d x exp [ i k α ( x x α ) 2 2 ( z z 1 ) ] .
x 1 + d x exp [ i k α ( x x α ) 2 2 ( z z 1 ) ] = 0 + d x exp [ i k α x 2 2 ( z z 1 ) ] 0 x 1 ( x α ) d x exp [ i k α x 2 2 ( z z 1 ) ] = 2 π ( z z 1 ) i k α { 1 2 i k α 2 π ( z z 1 ) 0 x 1 ( x α ) d x exp [ i k α x 2 2 ( z z 1 ) ] } ,
U d ( ρ , z ) = i A Z 0 e i k z q ( z ) exp [ i k 2 q ( z ) ρ 2 ] { 1 2 i k α 2 π ( z z 1 ) 0 x 1 ( x α ) d x exp [ i k α x 2 2 ( z z 1 ) ] } .
U 0 ( ρ , z ) = i A Z 0 e i k z q ( z ) exp [ i k 2 q ( x ) ρ 2 ]
U d ( ρ , z ) = U 0 ( ρ , z ) 2 { 1 + ( i 1 ) 0 w d t exp [ i π t 2 2 ] } = U 0 ( ρ , z ) 2 { 1 + ( i 1 ) [ C ( w ) + i S ( w ) ] } ,

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