Abstract

A formalism based on plane-wave decomposition is applied to the linear propagation of terahertz pulses in experimental geometries. The approach is general and is not restricted to any particular polarization (or current) source. Near- and far-field expressions easily amenable to numerical computation are obtained for the temporal profiles and spectra of terahertz pulses in layered structures, as often encountered in experiments. The effects of polarization and angle-dependent multiple reflection and transmission, as well as of material dispersion, are included. Examples of optical rectification in GaAs and ZnTe are presented to illustrate the simplicity of the method and are compared with experiments. The numerical evaluation of the expressions for the terahertz electric field in practical experimental geometries is straightforward.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 9.
  2. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.
  3. R. Ell, U. Morgner, F. X. Kärtner, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, M. J. Lederer, and A. B. B. Luther-Davies, “Generation of 5-fs pulses and octave-spanning spectra directly from a Ti:sapphire laser,” Opt. Lett. 26, 373–375 (2001).
    [CrossRef]
  4. C. Fattinger and D. Grischkowsky, “Point source terahertz optics,” Appl. Phys. Lett. 53, 1480–1482 (1988).
    [CrossRef]
  5. J. T. Darrow, X.-C. Zhang, and D. H. Auston, “Power scaling of large-aperture photoconducting antennas,” Appl. Phys. Lett. 58, 25–27 (1991).
    [CrossRef]
  6. A. Bonvalet, M. Joffre, J. L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
    [CrossRef]
  7. O. Dietrich, F. Krausz, and P. B. Corkum, “Determining the absolute carrier phase of a few-cycle laser pulse,” Opt. Lett. 25, 16–18 (2000).
    [CrossRef]
  8. S.-G. Park, M. R. Melloch, and A. M. Weiner, “Analysis of terahertz waveforms measured by photoconductive and electrooptic sampling,” IEEE J. Quantum Electron. 35, 810–819 (1999).
    [CrossRef]
  9. E. Budiharto, N.-W. Pu, S. Jeong, and J. Bokor, “Near-field propagation of terahertz pulses from large-aperture antenna,” Opt. Lett. 23, 213–215 (1998).
    [CrossRef]
  10. A. E. Kaplan, “Diffraction-induced transformation of near-cycle and subcycle pulses,” J. Opt. Soc. Am. B 15, 951–956 (1998).
    [CrossRef]
  11. S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998); errata 23, 1141 (1998).
    [CrossRef]
  12. S. Hunsche, S. Feng, H. G. Winful, A. Leitenstorfer, M. C. Nuss, and E. P. Ippen, “Spatiotemporal focusing of single-cycle light pulses,” J. Opt. Soc. Am. A 16, 2025–2028 (1999).
    [CrossRef]
  13. S. Nekkanti, D. Sullivan, and D. S. Citrin, “Simulation of spatiotemporal terahertz pulse shaping in 3-D using conductive apertures of finite thickness,” IEEE J. Quantum Electron. 37, 1226–1231 (2001).
    [CrossRef]
  14. D. Co⁁té, N. Laman, and H. M. van Driel, “Rectification and shift currents in GaAs,” Appl. Phys. Lett. 80, 905–907 (2002).
    [CrossRef]
  15. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1985), Chap. 1, p. 51.
  17. A. E. Siegman, Lasers (University Science, California, 1986), Chap. 20.
  18. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 3, p. 131.
  19. J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
    [CrossRef]
  20. A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
    [CrossRef]
  21. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).
  22. A. E. Siegman, Lasers (University Science, California, 1986), p. 637.
  23. P. Y. Han and X.-C. Zhang, “Coherent, broadband midinfrared terahertz beam sensors,” Appl. Phys. Lett. 73, 3049–3051 (1998).
    [CrossRef]
  24. G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16, 1204–1212 (1999).
    [CrossRef]
  25. A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: experiment and theory,” Appl. Phys. Lett. 74, 1516–1518 (1999).
    [CrossRef]
  26. H. J. Bakker, G. C. Cho, H. Kurz, Q. Wu, and X.-C. Zhang, “Distortion of terahertz pulses in electro-optic sampling,” J. Opt. Soc. Am. B 15, 1795–1801 (1998).
    [CrossRef]
  27. A computer implementation of the calculations in C/C++ can be obtained at http://www.novajo.ca/thzcode.html.

2002

D. Co⁁té, N. Laman, and H. M. van Driel, “Rectification and shift currents in GaAs,” Appl. Phys. Lett. 80, 905–907 (2002).
[CrossRef]

2001

S. Nekkanti, D. Sullivan, and D. S. Citrin, “Simulation of spatiotemporal terahertz pulse shaping in 3-D using conductive apertures of finite thickness,” IEEE J. Quantum Electron. 37, 1226–1231 (2001).
[CrossRef]

R. Ell, U. Morgner, F. X. Kärtner, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, M. J. Lederer, and A. B. B. Luther-Davies, “Generation of 5-fs pulses and octave-spanning spectra directly from a Ti:sapphire laser,” Opt. Lett. 26, 373–375 (2001).
[CrossRef]

2000

1999

S.-G. Park, M. R. Melloch, and A. M. Weiner, “Analysis of terahertz waveforms measured by photoconductive and electrooptic sampling,” IEEE J. Quantum Electron. 35, 810–819 (1999).
[CrossRef]

S. Hunsche, S. Feng, H. G. Winful, A. Leitenstorfer, M. C. Nuss, and E. P. Ippen, “Spatiotemporal focusing of single-cycle light pulses,” J. Opt. Soc. Am. A 16, 2025–2028 (1999).
[CrossRef]

G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16, 1204–1212 (1999).
[CrossRef]

A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: experiment and theory,” Appl. Phys. Lett. 74, 1516–1518 (1999).
[CrossRef]

1998

1996

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
[CrossRef]

1995

A. Bonvalet, M. Joffre, J. L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[CrossRef]

1991

J. T. Darrow, X.-C. Zhang, and D. H. Auston, “Power scaling of large-aperture photoconducting antennas,” Appl. Phys. Lett. 58, 25–27 (1991).
[CrossRef]

1988

C. Fattinger and D. Grischkowsky, “Point source terahertz optics,” Appl. Phys. Lett. 53, 1480–1482 (1988).
[CrossRef]

1987

1981

J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
[CrossRef]

Angelow, G.

Auston, D. H.

J. T. Darrow, X.-C. Zhang, and D. H. Auston, “Power scaling of large-aperture photoconducting antennas,” Appl. Phys. Lett. 58, 25–27 (1991).
[CrossRef]

Bakker, H. J.

Bokor, J.

Bonvalet, A.

A. Bonvalet, M. Joffre, J. L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[CrossRef]

Budiharto, E.

Cho, G. C.

Citrin, D. S.

S. Nekkanti, D. Sullivan, and D. S. Citrin, “Simulation of spatiotemporal terahertz pulse shaping in 3-D using conductive apertures of finite thickness,” IEEE J. Quantum Electron. 37, 1226–1231 (2001).
[CrossRef]

Co?té, D.

D. Co⁁té, N. Laman, and H. M. van Driel, “Rectification and shift currents in GaAs,” Appl. Phys. Lett. 80, 905–907 (2002).
[CrossRef]

Corkum, P. B.

Darrow, J. T.

J. T. Darrow, X.-C. Zhang, and D. H. Auston, “Power scaling of large-aperture photoconducting antennas,” Appl. Phys. Lett. 58, 25–27 (1991).
[CrossRef]

Dietrich, O.

Ell, R.

Fattinger, C.

C. Fattinger and D. Grischkowsky, “Point source terahertz optics,” Appl. Phys. Lett. 53, 1480–1482 (1988).
[CrossRef]

Feng, S.

Fujimoto, J. G.

Gallot, G.

Grischkowsky, D.

G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16, 1204–1212 (1999).
[CrossRef]

C. Fattinger and D. Grischkowsky, “Point source terahertz optics,” Appl. Phys. Lett. 53, 1480–1482 (1988).
[CrossRef]

Han, P. Y.

P. Y. Han and X.-C. Zhang, “Coherent, broadband midinfrared terahertz beam sensors,” Appl. Phys. Lett. 73, 3049–3051 (1998).
[CrossRef]

Heinz, T. F.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
[CrossRef]

Hunsche, S.

S. Hunsche, S. Feng, H. G. Winful, A. Leitenstorfer, M. C. Nuss, and E. P. Ippen, “Spatiotemporal focusing of single-cycle light pulses,” J. Opt. Soc. Am. A 16, 2025–2028 (1999).
[CrossRef]

A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: experiment and theory,” Appl. Phys. Lett. 74, 1516–1518 (1999).
[CrossRef]

Ippen, E. P.

Jeong, S.

Joffre, M.

A. Bonvalet, M. Joffre, J. L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[CrossRef]

Kaplan, A. E.

Kärtner, F. X.

Knox, W. H.

A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: experiment and theory,” Appl. Phys. Lett. 74, 1516–1518 (1999).
[CrossRef]

Krausz, F.

Kurz, H.

Laman, N.

D. Co⁁té, N. Laman, and H. M. van Driel, “Rectification and shift currents in GaAs,” Appl. Phys. Lett. 80, 905–907 (2002).
[CrossRef]

Lederer, M. J.

Leitenstorfer, A.

S. Hunsche, S. Feng, H. G. Winful, A. Leitenstorfer, M. C. Nuss, and E. P. Ippen, “Spatiotemporal focusing of single-cycle light pulses,” J. Opt. Soc. Am. A 16, 2025–2028 (1999).
[CrossRef]

A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: experiment and theory,” Appl. Phys. Lett. 74, 1516–1518 (1999).
[CrossRef]

Luther-Davies, A. B. B.

Martin, J. L.

A. Bonvalet, M. Joffre, J. L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[CrossRef]

Melloch, M. R.

S.-G. Park, M. R. Melloch, and A. M. Weiner, “Analysis of terahertz waveforms measured by photoconductive and electrooptic sampling,” IEEE J. Quantum Electron. 35, 810–819 (1999).
[CrossRef]

Migus, A.

A. Bonvalet, M. Joffre, J. L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[CrossRef]

Morgner, U.

Nahata, A.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
[CrossRef]

Nekkanti, S.

S. Nekkanti, D. Sullivan, and D. S. Citrin, “Simulation of spatiotemporal terahertz pulse shaping in 3-D using conductive apertures of finite thickness,” IEEE J. Quantum Electron. 37, 1226–1231 (2001).
[CrossRef]

Nuss, M. C.

A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: experiment and theory,” Appl. Phys. Lett. 74, 1516–1518 (1999).
[CrossRef]

S. Hunsche, S. Feng, H. G. Winful, A. Leitenstorfer, M. C. Nuss, and E. P. Ippen, “Spatiotemporal focusing of single-cycle light pulses,” J. Opt. Soc. Am. A 16, 2025–2028 (1999).
[CrossRef]

Park, S.-G.

S.-G. Park, M. R. Melloch, and A. M. Weiner, “Analysis of terahertz waveforms measured by photoconductive and electrooptic sampling,” IEEE J. Quantum Electron. 35, 810–819 (1999).
[CrossRef]

Pu, N.-W.

Scheuer, V.

Shah, J.

A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: experiment and theory,” Appl. Phys. Lett. 74, 1516–1518 (1999).
[CrossRef]

Sipe, J. E.

J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
[CrossRef]

J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
[CrossRef]

Sullivan, D.

S. Nekkanti, D. Sullivan, and D. S. Citrin, “Simulation of spatiotemporal terahertz pulse shaping in 3-D using conductive apertures of finite thickness,” IEEE J. Quantum Electron. 37, 1226–1231 (2001).
[CrossRef]

Tschudi, T.

van Driel, H. M.

D. Co⁁té, N. Laman, and H. M. van Driel, “Rectification and shift currents in GaAs,” Appl. Phys. Lett. 80, 905–907 (2002).
[CrossRef]

Weiner, A. M.

S.-G. Park, M. R. Melloch, and A. M. Weiner, “Analysis of terahertz waveforms measured by photoconductive and electrooptic sampling,” IEEE J. Quantum Electron. 35, 810–819 (1999).
[CrossRef]

Weling, A. S.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
[CrossRef]

Winful, H. G.

Wu, Q.

Zhang, X.-C.

H. J. Bakker, G. C. Cho, H. Kurz, Q. Wu, and X.-C. Zhang, “Distortion of terahertz pulses in electro-optic sampling,” J. Opt. Soc. Am. B 15, 1795–1801 (1998).
[CrossRef]

P. Y. Han and X.-C. Zhang, “Coherent, broadband midinfrared terahertz beam sensors,” Appl. Phys. Lett. 73, 3049–3051 (1998).
[CrossRef]

J. T. Darrow, X.-C. Zhang, and D. H. Auston, “Power scaling of large-aperture photoconducting antennas,” Appl. Phys. Lett. 58, 25–27 (1991).
[CrossRef]

Appl. Phys. Lett.

C. Fattinger and D. Grischkowsky, “Point source terahertz optics,” Appl. Phys. Lett. 53, 1480–1482 (1988).
[CrossRef]

J. T. Darrow, X.-C. Zhang, and D. H. Auston, “Power scaling of large-aperture photoconducting antennas,” Appl. Phys. Lett. 58, 25–27 (1991).
[CrossRef]

A. Bonvalet, M. Joffre, J. L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[CrossRef]

D. Co⁁té, N. Laman, and H. M. van Driel, “Rectification and shift currents in GaAs,” Appl. Phys. Lett. 80, 905–907 (2002).
[CrossRef]

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
[CrossRef]

P. Y. Han and X.-C. Zhang, “Coherent, broadband midinfrared terahertz beam sensors,” Appl. Phys. Lett. 73, 3049–3051 (1998).
[CrossRef]

A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: experiment and theory,” Appl. Phys. Lett. 74, 1516–1518 (1999).
[CrossRef]

IEEE J. Quantum Electron.

S. Nekkanti, D. Sullivan, and D. S. Citrin, “Simulation of spatiotemporal terahertz pulse shaping in 3-D using conductive apertures of finite thickness,” IEEE J. Quantum Electron. 37, 1226–1231 (2001).
[CrossRef]

S.-G. Park, M. R. Melloch, and A. M. Weiner, “Analysis of terahertz waveforms measured by photoconductive and electrooptic sampling,” IEEE J. Quantum Electron. 35, 810–819 (1999).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Surf. Sci.

J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
[CrossRef]

Other

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).

A. E. Siegman, Lasers (University Science, California, 1986), p. 637.

A computer implementation of the calculations in C/C++ can be obtained at http://www.novajo.ca/thzcode.html.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 9.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998); errata 23, 1141 (1998).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1985), Chap. 1, p. 51.

A. E. Siegman, Lasers (University Science, California, 1986), Chap. 20.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 3, p. 131.

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 (11)

Fig. 1
Fig. 1

Coordinate systems used in the calculations, sketched for sˆ, p^±, and ν± real. (a) Axial direction zˆ, wave-vector component K, and polarization sˆ. (b) Polarization vectors sˆ and p^± for plane-wave directions ν^±. (c) Vector decomposition of plane-wave vector ν± into its transverse (K) and axial (±wzˆ) components.

Fig. 2
Fig. 2

Lens geometry used in the calculation. The upward-propagating beam is incident from z<do.

Fig. 3
Fig. 3

Three-layer geometry with polarization source in the medium with index n3.

Fig. 4
Fig. 4

Temporal profile showing the far field at z=50 mm for 2πσΩ-1=200 fs and 2πσK-1=1 mm in a medium with n12=10+i10-4, similar to dispersionless GaAs. The field follows the second time derivative of the polarization envelope function.

Fig. 5
Fig. 5

Temporal profiles showing the transitions from near to far field for common THz parameters with 2πσΩ-1=200 fs and 2πσK-1=1 mm in a medium with a dielectric constant n12=10+i10-4 similar to dispersionless GaAs. (a) z=0 mm, (b) z=8 mm, (c) z=50 mm. The arbitrary units are the same for all three figures and are the same as those used in Fig. 4.

Fig. 6
Fig. 6

Temporal profile of THz radiation from below bandgap optical rectification with σΩ/2π=3.8 THz at z=5 cm. Inset, spectrum amplitude.

Fig. 7
Fig. 7

Temporal profile of THz radiation from below bandgap optical rectification with σΩ/2π=1 THz near and far from the source.

Fig. 8
Fig. 8

Two lenses of focal length f used to collimate and refocus a point source located at the focus of one of the lenses are equivalent to a single lens with a focal length twice as short, f=f/2; do (di), distance between object (image) plane and lens.

Fig. 9
Fig. 9

Diffraction of a Gaussian beam originating from z=0 and transformation by a lens at z=do. The field at z=do+di is obtained. Inset, estimation of the cutoff frequency when the lens is far from a point source with tan θco=L/do|Kco|/Ω˜.

Fig. 10
Fig. 10

Magnitude of filter functions F full (solid curve) and F approx (dotted curve) for σK-1=100 μm, L=2.5 cm, f=2.5 cm, and do=di=5 cm.

Fig. 11
Fig. 11

Calculation in a 30-μm ZnTe crystal of the temporal profile and spectrum of THz radiation from below bandgap optical rectification detected in a 27-μm-thick ZnTe crystal. These calculations agree well with published results.23 EO, electro-optic.

Equations (88)

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

E(r, t)=0dΩ2πdK(2π)2E(Ω, K; z)exp(iK  R)×exp(-iΩt)+c.c.,
E(Ω, K; z)=E+(Ω, K; z)+E-(Ω, K; z),
E±(Ω, K; z)=E±(Ω, K)exp(±iwz),
E±(Ω, K)=sˆE±s(Ω, K)+p^±E±p±(Ω, K),
E+(Ω, K; z°)=E+Ω°,K°δ(Ω-Ω°)δ(K-K°),
ν±K±wzˆ,
w(Ω˜2n2-K2)1/2,
sˆKˆ×zˆ,
p^±ν-1(KzˆwKˆ),
ei(z)=E+(Ω, K)exp(iwiz)E-(Ω, K)exp(-iwiz),
Mi(z)=exp(iwiz)00exp(-iwiz),
Mij=1tij1rijrij1,
rijp=winj2-wjni2winj2+wjni2,rijs=wi-wjwi+wj,
tijp=2ninjwiwinj2+wjni2,tijs=2wiwi+wj.
E(R+[do+DL]zˆ, t)
=E+Ω°,K°exp[iwvacdvac(R)+iwlensdlens(R)+iK°  R-iΩ°t]+c.c.,
E(R+[do+DL]zˆ, t)
=0dΩ2πdK(2π)2E+(Ω, K; do+DL)
×exp(iK  R-iΩt)+c.c.,
E(R+[do+DL]zˆ, t)=E+Ω°,K°exp(-iΔwR2/2R˜+iwlens DL+iK°  R-iΩ°t)+c.c.R<LE+Ω°,K°exp(iwvac DL+iK°  R-iΩ°t)+c.c.R>L,
E+(Ω, K; do+DL)=E+Ω°,K°δ(Ω-Ω°)exp(iwlensDL)R<LdR×exp[i(K°-K)  R-iΔwR2/2R˜]+exp(iwvacDL)R>LdRexp[i(K°-K)R].
1R˜=1R°-iΔwL2
KL2=2ΔwR°-2iL2,
E+(Ω, K; do+DL)
E+Ω°,K°δ(Ω-Ω°)exp(iΔwDL)dR×exp[i(K°-K)  R]exp-i R2KL24
=4πiKL2E+Ω°,K°δ(Ω-Ω°)×expi |K°-K|2KL2+iΔwDL,
02πdϕ exp(imϕ)exp(ix cos ϕ)=2πimJm(x),
0RdR exp(iaR2)J0(bR)=i2aexp-ib24a.
E+(Ω, K; do+)=4πiKL2dK(2π)2E+(Ω, K; do-)×expi |K-K|2KL2,
P(r, t)=0dΩ2πdK(2π)2P(Ω, K; z)exp(iK  R)×exp(-iΩt)+c.c.,
E(Ω, K; z)=dzG(Ω, K; z-z)  P(Ω, K; z).
G(Ω, K; z)=iΩ˜22°w1 (sˆsˆ+p^1+p^1+)θ(z)exp(iw1z)+iΩ˜22°w1 (sˆsˆ+p^1-p^1-)θ(-z)×exp(-iw1z)-1n12° zˆzˆδ(z),
E+(Ω, K; z)=iΩ˜22°w3qˆCqexp(iw1z)×q^1+q^3+-0dzexp(-iw2z)×P(Ω, K;z)+r32qexp(2iw3D)q^1+q^3-
-D0dzexp(iw3z)P(Ω, K; z),
 
Cq=t31q1-r32qr31qexp(i2w3D),
E(r, t)=0dΩ2πidK2πwe+(Ω, K)exp(iwz)×exp(iK  R-iΩt)+c.c.,
e+(Ω, K)exp(iwz)=-iw2πE+(Ω, K; z).
E(r, t)0dΩ2πe+(Ω,K¯) exp(iΩ˜nr-iΩt)r+c.c.,
P(Ω, K; z)=xˆP(Ω, K)δ(z)
xˆ  E(zzˆ, t)14π°z0dΩ2π Ω˜2P(Ω, 0)×exp(iΩ˜n1z-iΩt)+c.c.
P(Ω, K)=4π3/2σΩσK2 Poexp(-Ω2/σΩ2)exp(-K2/σK2),
aˆbˆ=mfm(aˆbˆ)exp(-imϕ),
xˆ  E+(Ω, K; z)=iΩ˜22ow1m[fm(sˆsˆ)+fm(p^1+p^1+)]exp(-imϕ) xˆ exp(iw1z)P(Ω, K).
xˆ  E+(Ω, K; z)=iΩ˜22ow112-12cos 2ϕ+w122ν12+w122ν12cos 2ϕexp(iw1z)P(Ω, K).
xˆ  E(zzˆ, t)=i2°0dΩ2π Ω˜2exp(-iΩt)0KdK2π×12+w122ν12w1-1exp(iw1z)P(Ω, K).
E+(ω, K)exp(-iω°D/c)
=4π3/2σω°σκ°2 Eω°exp[-(ω-ω°)2/σω°2]
×exp(-|K|2/σκ°2),
P(Ω, K; z)=xˆP(Ω, K)exp[(inω°gΩ˜-2αω°)(z+D)],
P(Ω, K)=2°χ2eff|Eω°|24π3/2σω°σκ°2exp(-Ω2/σΩ2)×exp(-K2/σK2).
e+(Ω, K)=Ω˜24π°qˆCqq^3+-D0dz×exp(-iw3z)P(Ω, K; z)+r32qexp(2iw3D)q^3--D0dz×exp(iw3z)P(Ω, K; z),
dzexp(iw3z)P(Ω, K; z)
=xˆP(Ω, K)-D0dzexp[(inω°gΩ˜-2αω°)
×(z+D)]exp(iw3z)
xˆP(Ω, K)L±(Ω, K),
L±(Ω, K)=exp(inω°gΩ˜D-2αω°D)-exp(±iw3D)iw3+inω°gΩ˜-2αω°,
xˆ  E(zzˆ, t)dΩ2πΩ˜24π°CsP(Ω, 0)[L+(Ω, 0)+r32sexp(2iw3D)L-(Ω, 0)]× exp(iΩ˜nz-iΩt)z+c.c.,
xˆ  E+(Ω, K; z)
=iΩ˜22°w3exp(iw1z)×qˆC q-D0dzexp(-iw3z)xˆ  q^1+q^3+  xˆP(Ω, K; z)+r32qexp(2iw3D)-D0dz×exp(iw3z)zˆq^1+q^3-  xˆP(Ω, K; z).
xˆ  E+(Ω, K; z)=iΩ˜22ow3exp(iw1z)P(Ω, K)L+(Ω, K)×12Cs-12 Cscos 2ϕ+w1w32ν1ν3Cp+w1w32ν1ν3Cpcos 2ϕ+exp(2iw3D)L-(Ω, K)12 r32sCs-12 r32sCscos 2ϕ-w1w32ν1ν3 r32pCp-w1w32ν1ν3 r32pCpcos 2ϕ.
xˆ  E(zzˆ, t)=0dΩ2πexp(-iΩt) iΩ˜22°×0KdK2π w3-1exp(iw1z)P(Ω, K)×12Cs+w1w32ν1ν3CpL+(Ω, K)+exp(2iw3D)12Csr32s-w1w32ν1ν3Cpr32pL-(Ω, K)+c.c.
E+(Ω, K; do-)=E+(Ω, K)expiΩ˜do-i K2do2Ω˜.
E+(Ω, K; do-)=4πσK2E+(Ω)exp(iΩ˜do)exp-K2σl2(do),
1σo2(do)=1σK2+i do2Ω˜.
E+(Ω, K; do+)=4πσK2σo2(do)σo2(do)+iKL2E+(Ω)×exp(iΩ˜do)exp-K2σo2(do)+iKL2.
E+(Ω, K; do+di)=4πσK2σo2(do)σi2(do, 0)E+(Ω)exp[iΩ˜(do+di)]exp-K2σi2(do, di),
1σi2(do, di)=1σo2(do)+iKL2+idi2Ω˜.
F full(Ω, do, di)=xˆ  [dK/(2π)2]E+(Ω, K; do+di)xˆ  [dK/(2π)2]E+(Ω, K; 0)
=σo2(do)σo2(0)σi2(do, di)σi2(do, 0),
F approx(Ω)=erf 2Ω˜LfσK,
E+(Ω, K; do+di)=E+(Ω, K; do-)F approx(Ω)×expi 2 fK2Ω˜exp(iwdi).
Kˆ=xˆ cos ϕ+yˆ sin ϕ,
sˆ=Kˆ×zˆ=-yˆ cos ϕ+xˆ sin ϕ,
p^i±=1νi (KzˆwiKˆ)=Kνi zˆwiνi xˆ cos ϕwiνi yˆ sin ϕ.
f0(sˆsˆ)=12 (xˆxˆ+yˆyˆ),
f+2(sˆsˆ)=12-12 (xˆxˆ-yˆyˆ)+i2-12 (xˆyˆ+yˆxˆ),
f-2(sˆsˆ)=12-12 (xˆxˆ-yˆyˆ)-i2-12 (xˆyˆ+yˆxˆ).
f0(p^±ip^±j)=K2νiνj zˆzˆ+wiwj2νiνj (xˆxˆ+yˆyˆ),
f+1(p^±ip^±j)=12Kwjνiνj zˆxˆ+Kwiνiνj xˆzˆi2Kwjνiνj zˆyˆ+Kwiνiνj yˆzˆ
f-1(p^±ip^±j)=12Kwjνiνj zˆxˆ+Kwiνiνj xˆzˆ±i2Kwjνiνj zˆyˆ+Kwiνiνj yˆzˆ
f+2(p^±ip^±j)=12wiwj2νiνj (xˆxˆ-yˆyˆ)+i2wiwj2νiνj (xˆyˆ+yˆxˆ),
f-2(p^±ip^±j)=12wiwj2νiνj (xˆxˆ-yˆyˆ)-i2wiwj2νiνj (xˆyˆ+yˆxˆ).
f0(p^±ip^j)=K2νiνj zˆzˆ-wiwj2νiνj (xˆxˆ+yˆyˆ),
f+1(p^±ip^j)=12Kwjνiνj zˆxˆKwiνiνj xˆzˆ+i2Kwjνiνj zˆyˆKwiνiνj yˆzˆ,
f-1(p^±ip^j)=12Kwjνiνj zˆxˆKwiνiνj xˆzˆ-i2Kwjνiνj zˆyˆKwiνiνj yˆzˆ,
f+2(p^±ip^j)=12-wiwj2νiνj (xˆxˆ-yˆyˆ)+i2wiwj2νiνj (xˆyˆ+yˆxˆ),
f-2(p^±ip^j)=12-wiwj2νiνj (xˆxˆ-yˆyˆ)-i2wiwj2νiνj (xˆyˆ+yˆxˆ).

Metrics