Abstract

The reflective frequency-domain optical delay line employing a diffraction grating, a lens, and a tiltable mirror has emerged as a device particularly suitable for interferometry and optical coherence tomography. The device is comprehensively described, both theoretically and experimentally, in the context of interferometry. The variations of phase and group delay produced by the device as well as its dispersive properties are described and demonstrated experimentally.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  2. A. M. Rollins, M. D. Kulkarni, S. Yazdanafar, R. Ung-arunyawee, J. A. Izatt, “In vivo video rate optical coherence tomography,” Opt. Express 3, 219–229 (1998); http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  3. G. J. Tearney, B. E. Bouma, S. A. Boppart, B. Golubovic, E. A. Swanson, J. G. Fujimoto, “Rapid acquisition of in vivo biological images by use of optical coherence tomography,” Opt. Lett. 21, 1408–1410 (1996).
    [CrossRef] [PubMed]
  4. R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
    [CrossRef]
  5. K. F. Kwong, D. Yankelevich, K. C. Chu, J. P. Heritage, A. Dienes, “400-Hz mechanical scanning optical delay line,” Opt. Lett. 18, 558–560 (1993).
    [CrossRef] [PubMed]
  6. G. J. Tearney, B. E. Bouma, J. G. Fujimoto, “High-speed phase- and group-delay scanning with a grating-based phase control delay line,” Opt. Lett. 22, 1811–1813 (1997).
    [CrossRef]
  7. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
    [CrossRef]
  8. O. E. Martinez, J. P. Gordon, R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A 1, 1003–1006 (1984).
    [CrossRef]
  9. A. M. Weiner, J. P. Heritage, E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1988).
    [CrossRef]
  10. J. X. Tull, M. A. Dugan, W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
    [CrossRef]
  11. A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. 15, 326–328 (1990).
    [CrossRef] [PubMed]
  12. K. K. M. B. D. Silva, A. V. Zvyagin, D. D. Sampson, “Extended range, rapid scanning optical delay line for biomedical interferometric imaging,” Electron. Lett. 35, 1404–1406 (1999).
    [CrossRef]
  13. Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. de Boer, J. S. Nelson, “Phase-resolved optical coherence tomography and Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25, 114–116 (2000).
    [CrossRef]
  14. W. G. Yang, D. Keusters, D. Goswami, W. S. Warren, “Rapid ultrafine-tunable optical delay line at the 1.55-mm wavelength,” Opt. Lett. 23, 1843–1845 (1998).
    [CrossRef]
  15. G. D. Love, “The unbounded nature of geometrical and dynamical phases in polarization optics,” Opt. Commun. 131, 237–240 (1996).
    [CrossRef]
  16. A. V. Zvyagin, D. D. Sampson, “Achromatic optical phase shifter–modulator,” Opt. Lett. 26, 187–189 (2001).
    [CrossRef]
  17. M. Stern, J. P. Heritage, E. W. Chase, “Grating compensation of third-order fiber dispersion,” IEEE J. Quantum Electron. 28, 2742–2748 (1992).
    [CrossRef]
  18. L. Deck, P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 (1994).
    [CrossRef] [PubMed]
  19. E. D. J. Smith, A. V. Zvyagin, D. D. Sampson, “Real-time dispersion compensation in scanning interferometry,” Opt. Lett. 27, 1998–2000 (2002).
    [CrossRef]
  20. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [CrossRef]
  21. G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
    [CrossRef]

2002 (1)

2001 (1)

2000 (1)

1999 (1)

K. K. M. B. D. Silva, A. V. Zvyagin, D. D. Sampson, “Extended range, rapid scanning optical delay line for biomedical interferometric imaging,” Electron. Lett. 35, 1404–1406 (1999).
[CrossRef]

1998 (2)

1997 (2)

G. J. Tearney, B. E. Bouma, J. G. Fujimoto, “High-speed phase- and group-delay scanning with a grating-based phase control delay line,” Opt. Lett. 22, 1811–1813 (1997).
[CrossRef]

J. X. Tull, M. A. Dugan, W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
[CrossRef]

1996 (2)

1994 (1)

1993 (1)

1992 (1)

M. Stern, J. P. Heritage, E. W. Chase, “Grating compensation of third-order fiber dispersion,” IEEE J. Quantum Electron. 28, 2742–2748 (1992).
[CrossRef]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1990 (1)

1988 (1)

1986 (1)

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

1984 (1)

1977 (2)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

1969 (1)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

Boppart, S. A.

Bouma, B. E.

Celli, V.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Chase, E. W.

M. Stern, J. P. Heritage, E. W. Chase, “Grating compensation of third-order fiber dispersion,” IEEE J. Quantum Electron. 28, 2742–2748 (1992).
[CrossRef]

Chen, Z.

Chu, K. C.

de Boer, J. F.

de Groot, P.

Deck, L.

Dienes, A.

Dugan, M. A.

J. X. Tull, M. A. Dugan, W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
[CrossRef]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fork, R. L.

Fujimoto, J. G.

Golubovic, B.

Gordon, J. P.

Goswami, D.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Heritage, J. P.

K. F. Kwong, D. Yankelevich, K. C. Chu, J. P. Heritage, A. Dienes, “400-Hz mechanical scanning optical delay line,” Opt. Lett. 18, 558–560 (1993).
[CrossRef] [PubMed]

M. Stern, J. P. Heritage, E. W. Chase, “Grating compensation of third-order fiber dispersion,” IEEE J. Quantum Electron. 28, 2742–2748 (1992).
[CrossRef]

A. M. Weiner, J. P. Heritage, E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1988).
[CrossRef]

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Hill, N. R.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Izatt, J. A.

Keusters, D.

Kirschner, E. M.

Kulkarni, M. D.

Kwong, K. F.

Leaird, D. E.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Love, G. D.

G. D. Love, “The unbounded nature of geometrical and dynamical phases in polarization optics,” Opt. Commun. 131, 237–240 (1996).
[CrossRef]

Martinez, O. E.

Marvin, A.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Nelson, J. S.

Patel, J. S.

Pulifito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Rollins, A. M.

Sampson, D. D.

Saxer, C.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Silva, K. K. M. B. D.

K. K. M. B. D. Silva, A. V. Zvyagin, D. D. Sampson, “Extended range, rapid scanning optical delay line for biomedical interferometric imaging,” Electron. Lett. 35, 1404–1406 (1999).
[CrossRef]

Smith, E. D. J.

Stern, M.

M. Stern, J. P. Heritage, E. W. Chase, “Grating compensation of third-order fiber dispersion,” IEEE J. Quantum Electron. 28, 2742–2748 (1992).
[CrossRef]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

G. J. Tearney, B. E. Bouma, S. A. Boppart, B. Golubovic, E. A. Swanson, J. G. Fujimoto, “Rapid acquisition of in vivo biological images by use of optical coherence tomography,” Opt. Lett. 21, 1408–1410 (1996).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tearney, G. J.

Thurston, R. N.

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Tomlinson, W. J.

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Treacy, E. B.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

Tull, J. X.

J. X. Tull, M. A. Dugan, W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
[CrossRef]

Ung-arunyawee, R.

Warren, W. S.

W. G. Yang, D. Keusters, D. Goswami, W. S. Warren, “Rapid ultrafine-tunable optical delay line at the 1.55-mm wavelength,” Opt. Lett. 23, 1843–1845 (1998).
[CrossRef]

J. X. Tull, M. A. Dugan, W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
[CrossRef]

Weiner, A. M.

Wullert, J. R.

Xiang, S.

Yang, W. G.

Yankelevich, D.

Yazdanafar, S.

Zhao, Y.

Zvyagin, A. V.

Adv. Magn. Opt. Reson. (1)

J. X. Tull, M. A. Dugan, W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

K. K. M. B. D. Silva, A. V. Zvyagin, D. D. Sampson, “Extended range, rapid scanning optical delay line for biomedical interferometric imaging,” Electron. Lett. 35, 1404–1406 (1999).
[CrossRef]

IEEE J. Quantum Electron. (3)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

M. Stern, J. P. Heritage, E. W. Chase, “Grating compensation of third-order fiber dispersion,” IEEE J. Quantum Electron. 28, 2742–2748 (1992).
[CrossRef]

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

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

Opt. Commun. (1)

G. D. Love, “The unbounded nature of geometrical and dynamical phases in polarization optics,” Opt. Commun. 131, 237–240 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (8)

G. J. Tearney, B. E. Bouma, S. A. Boppart, B. Golubovic, E. A. Swanson, J. G. Fujimoto, “Rapid acquisition of in vivo biological images by use of optical coherence tomography,” Opt. Lett. 21, 1408–1410 (1996).
[CrossRef] [PubMed]

K. F. Kwong, D. Yankelevich, K. C. Chu, J. P. Heritage, A. Dienes, “400-Hz mechanical scanning optical delay line,” Opt. Lett. 18, 558–560 (1993).
[CrossRef] [PubMed]

G. J. Tearney, B. E. Bouma, J. G. Fujimoto, “High-speed phase- and group-delay scanning with a grating-based phase control delay line,” Opt. Lett. 22, 1811–1813 (1997).
[CrossRef]

A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. 15, 326–328 (1990).
[CrossRef] [PubMed]

A. V. Zvyagin, D. D. Sampson, “Achromatic optical phase shifter–modulator,” Opt. Lett. 26, 187–189 (2001).
[CrossRef]

Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. de Boer, J. S. Nelson, “Phase-resolved optical coherence tomography and Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25, 114–116 (2000).
[CrossRef]

W. G. Yang, D. Keusters, D. Goswami, W. S. Warren, “Rapid ultrafine-tunable optical delay line at the 1.55-mm wavelength,” Opt. Lett. 23, 1843–1845 (1998).
[CrossRef]

E. D. J. Smith, A. V. Zvyagin, D. D. Sampson, “Real-time dispersion compensation in scanning interferometry,” Opt. Lett. 27, 1998–2000 (2002).
[CrossRef]

Phys. Rev. B (2)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Pulifito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Schematic diagrams of (a) scanning Michelson interferometer employing the frequency-domain optical delay line (FD-ODL) and (b) FD-ODL.

Fig. 2
Fig. 2

Ray representation of plane-wave diffraction from a diffraction grating (Gr).

Fig. 3
Fig. 3

Plot of phase-to-group-delay ratio ξ versus mirror offset x0. Points A, B, and C denote three regimes of interest.

Fig. 4
Fig. 4

Schematic diagram of FD-ODL in the case of a grating tilted from normal and offset from the focal plane: L, lens; MI, mirror.

Fig. 5
Fig. 5

Interferograms for zero pivot offset plotted versus mirror tilt angle and the corresponding axial scan distance. Solid curves represent experimental results, and the dashed curve is the theoretical envelope.

Fig. 6
Fig. 6

Interferograms for small pivot offset plotted versus axial scan distance. Solid and dashed curves represent experimental and theoretical results, respectively (with the use of 2δk0 values as shown, where x0=200 μm). The dotted curves represents the calculated interferogram envelope.

Fig. 7
Fig. 7

Interferograms versus mirror tilt angle measured at mirror offsets denoted by points A, B, and C in Fig. 3.

Fig. 8
Fig. 8

Interferograms for zero pivot offset and small dispersion plotted versus axial scan distance. Solid and dashed curves represent experimental and theoretical results, respectively (with the use of 2δk0 values as shown, where x0=0 and Dωρ2=-0.42).

Fig. 9
Fig. 9

Interferograms for zero pivot offset and large dispersion plotted versus axial scan distance. Solid and dashed curves represent experimental and theoretical results, respectively (with the use of x0=0 and Dωρ2=-7.5). The dotted curve represents the calculated interferogram envelope.

Equations (38)

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

Es=ER cos[ψs(k)-ωt],
Er=E cos[ψr(k)-ωt],
id=ϕR+14E2+ϕR2E2 cos[ψs(k)-ψr(k)],
ψcorr(xg)=kgmxg,
ψ(k, x)=ψdyn(k, x)+ψcorr(x),
ΔψD(k, θ)=ψD(k, θ)-ψD(k, 0).
ΔψD(k, θ)=[ψDdyn(k, θ)-ψDdyn(k, 0)]+[ψcorr(θ)-ψcorr(0)]=ΔψDdyn(k, θ)+ψcorr(θ),
ΔψDdyn(k, θ)=k(2MN+BC)=2kx0θ+kBC.
p(sin α+sin β)=mλ,
k(BC+AB)=-kgmOB,
ΔψD(k, θ)=2kx0θ-kAB,
ΔψD(k, θ)=2kx0θ-2kθf sin β.
p sin β=m(λ-λ0),
ΔψD(k)=2kx0θ+4πmθf(k-k0)pk0,
id(k)=ϕR2E2 cos[2k(Lr-Ls)+ΔψD(k)]=ϕR2E2 cos[ψ(k)],
ψ(k)=2kδ+2kx0θ+4πmθf(k-k0)pk0,
i=0id(k)G(k)dk=ϕR2E2 Re0G(k)exp[jψ(k)]dk,
ψ(k)=ψ(k0)+ψ(k0)(k-k0)+ψ(k0)(k-k0)22!+ψ(k0)(k-k0)33!+,
tp=1cψ(k0)k0=2δc+2θx0c,
tg=1cψ(k0)=Lgc=2δc+2θx0c+4πmθfcpk0,
ξtptgδ=0=1+mλ0fpx0-1.
x0=-mλ0fp.
i=ϕR2E2 Reexp(jψ0)-G(κ+k0)exp[j(Lgκ+Dωc2κ2/2+Dω(1)c3κ3/6)]dκ,
G(k)=G(κ+k0)=2ln 2Δkπexp-4κ2 ln 2Δk2,
i(θ)=ϕE2R2(1+Dω2ρ4)-1/4 exp-Lg2ρ22c2(1+Dω2ρ4)×cosψ(k0)-DωLg2ρ42c2(1+Dω2ρ4)+12arctan(Dωρ2),
i=ϕE2R2exp-Lg2ρ22c2cos(2δk0+2θx0k0).
ψOC=k(OPin+[O0MO0]+2MN-A0B0+APout).
ΔψD(k)=2kΔz cos β+2kx0θ-2kθf sin β=2kΔz-4kΔz sin2 β2+2kx0θ-2kθf sin β,
p[sin(β+θg)-sin θg]=m(λ-λ0),
Dω=-8π2m2Δzc2p2k03 cos2 θg,
Dω(1)=24π2m2Δzc3p2k04 cos2 θg1+2mπ sin θgpk0 cos2 θg,
ΔψD(k)=2kx0θ-2kθf sin×arcsin-2πm(k-k0)pkk0+sin θg-θg.
ΔψD(k)2kx0θ+4πθfm(k-k0)pk0 cos θg,
E(i)(x, z)=E0(i) exp[j(kx(i)x-kz(i)z-ωt)],
E(d)(x, z)=m=-+Em,0(d) exp[j(km,x(d)x+km,z(d)z-ωt)],
km,x(d)=km,x(i)+mkg.
Em(d)(x, z)=Em,0(d) exp(j{(km,x(i)+mkg)x+[k2-(km,x(i)+mkg)2]1/2z-ωt}).
ψcorr(xg)=kgmxg.

Metrics