Abstract

We present a complete frequency-domain description of electro-optic (EO) detection of terahertz (THz) electromagnetic radiation, including a description of the ellipsometry technique employed. These frequency-domain results show the effect of EO detection of a pulse of THz radiation as the product of three spectral filters acting on the complex amplitude spectrum of the THz pulse that is entering the EO crystal. For the usual experimental situation in which the optical bandwidth of the interrogating light pulse is small compared with the optical carrier frequency, we obtain an important simplification of our general result for the detected EO signal. When this simplified result is rewritten in the time domain, a more general description of the previous time-domain picture of EO detection is obtained.

© 1999 Optical Society of America

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  1. D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
    [CrossRef]
  2. D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
    [CrossRef]
  3. Q. Wu, T. D. Hewitt, and X.-C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69, 1026–1028 (1996).
    [CrossRef]
  4. R. A. Cheville, R. W. McGowan, and D. Grischkowsky, “Late-time target response measured with terahertz impulse ranging,” IEEE Trans. Antennas Propag. 45, 1518–1524 (1997).
    [CrossRef]
  5. Q. Wu and X.-C. Zhang, “Free-space electro-optics sampling of mid-infrared pulses,” Appl. Phys. Lett. 71, 1285–1286 (1997).
    [CrossRef]
  6. Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
    [CrossRef]
  7. S.-G. Park, M. R. Melloch, and A. M. Weiner, “Comparison of terahertz waveforms measured by electro-optic and photoconductive sampling,” Appl. Phys. Lett. 73, 3184–3186 (1998).
    [CrossRef]
  8. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  9. 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]
  10. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1987).
  11. Clearly the frequency spectrum ATHz(Ω) of the THz pulse that is entering the EO crystal is equal to the incident spectrum AIn(Ω) multiplied by the complex frequency-dependent Fresnel transmission coefficient t12(Ω) that describes the EO crystal; i.e., ATHz(Ω)=t12(Ω)AIn(Ω). See Section 7.
  12. D. H. Auston and M. C. Nuss, “Electrooptic generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24, 184–197 (1988).
    [CrossRef]
  13. 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]
  14. Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70, 1784–1786 (1997).
    [CrossRef]
  15. G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7, 5345–5359 (1973).
    [CrossRef]
  16. T. Hattori, Y. Homma, A. Mitsushi, and M. Tacke, “Indices of refraction of ZnS, ZnSe, ZnTe, CdS, and CdTe in the far infrared,” Opt. Commun. 7, 229–232 (1973).
    [CrossRef]

1998

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

S.-G. Park, M. R. Melloch, and A. M. Weiner, “Comparison of terahertz waveforms measured by electro-optic and photoconductive sampling,” Appl. Phys. Lett. 73, 3184–3186 (1998).
[CrossRef]

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]

1997

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, “Late-time target response measured with terahertz impulse ranging,” IEEE Trans. Antennas Propag. 45, 1518–1524 (1997).
[CrossRef]

Q. Wu and X.-C. Zhang, “Free-space electro-optics sampling of mid-infrared pulses,” Appl. Phys. Lett. 71, 1285–1286 (1997).
[CrossRef]

Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70, 1784–1786 (1997).
[CrossRef]

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]

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

Q. Wu, T. D. Hewitt, and X.-C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69, 1026–1028 (1996).
[CrossRef]

1990

1988

D. H. Auston and M. C. Nuss, “Electrooptic generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24, 184–197 (1988).
[CrossRef]

1973

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7, 5345–5359 (1973).
[CrossRef]

T. Hattori, Y. Homma, A. Mitsushi, and M. Tacke, “Indices of refraction of ZnS, ZnSe, ZnTe, CdS, and CdTe in the far infrared,” Opt. Commun. 7, 229–232 (1973).
[CrossRef]

Auston, D. H.

D. H. Auston and M. C. Nuss, “Electrooptic generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24, 184–197 (1988).
[CrossRef]

Bakker, H. J.

Boyd, G. D.

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7, 5345–5359 (1973).
[CrossRef]

Brener, I.

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

Cai, Y.

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

Cheville, R. A.

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, “Late-time target response measured with terahertz impulse ranging,” IEEE Trans. Antennas Propag. 45, 1518–1524 (1997).
[CrossRef]

Cho, G. C.

Fattinger, Ch.

Federici, J. F.

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

Grischkowsky, D.

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, “Late-time target response measured with terahertz impulse ranging,” IEEE Trans. Antennas Propag. 45, 1518–1524 (1997).
[CrossRef]

D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

Hattori, T.

T. Hattori, Y. Homma, A. Mitsushi, and M. Tacke, “Indices of refraction of ZnS, ZnSe, ZnTe, CdS, and CdTe in the far infrared,” Opt. Commun. 7, 229–232 (1973).
[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]

Hewitt, T. D.

Q. Wu, T. D. Hewitt, and X.-C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69, 1026–1028 (1996).
[CrossRef]

Homma, Y.

T. Hattori, Y. Homma, A. Mitsushi, and M. Tacke, “Indices of refraction of ZnS, ZnSe, ZnTe, CdS, and CdTe in the far infrared,” Opt. Commun. 7, 229–232 (1973).
[CrossRef]

Jacobsen, R. H.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

Keiding, S.

Kurz, H.

Lopata, J.

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

McGowan, R. W.

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, “Late-time target response measured with terahertz impulse ranging,” IEEE Trans. Antennas Propag. 45, 1518–1524 (1997).
[CrossRef]

Melloch, M. R.

S.-G. Park, M. R. Melloch, and A. M. Weiner, “Comparison of terahertz waveforms measured by electro-optic and photoconductive sampling,” Appl. Phys. Lett. 73, 3184–3186 (1998).
[CrossRef]

Mitsushi, A.

T. Hattori, Y. Homma, A. Mitsushi, and M. Tacke, “Indices of refraction of ZnS, ZnSe, ZnTe, CdS, and CdTe in the far infrared,” Opt. Commun. 7, 229–232 (1973).
[CrossRef]

Mittleman, D. M.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

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]

Nuss, M. C.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

D. H. Auston and M. C. Nuss, “Electrooptic generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24, 184–197 (1988).
[CrossRef]

Park, S.-G.

S.-G. Park, M. R. Melloch, and A. M. Weiner, “Comparison of terahertz waveforms measured by electro-optic and photoconductive sampling,” Appl. Phys. Lett. 73, 3184–3186 (1998).
[CrossRef]

Pfeiffer, L.

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

Pollack, M. A.

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7, 5345–5359 (1973).
[CrossRef]

Stark, J. B.

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

Tacke, M.

T. Hattori, Y. Homma, A. Mitsushi, and M. Tacke, “Indices of refraction of ZnS, ZnSe, ZnTe, CdS, and CdTe in the far infrared,” Opt. Commun. 7, 229–232 (1973).
[CrossRef]

van Exter, M.

Weiner, A. M.

S.-G. Park, M. R. Melloch, and A. M. Weiner, “Comparison of terahertz waveforms measured by electro-optic and photoconductive sampling,” Appl. Phys. Lett. 73, 3184–3186 (1998).
[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]

Wu, Q.

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]

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70, 1784–1786 (1997).
[CrossRef]

Q. Wu and X.-C. Zhang, “Free-space electro-optics sampling of mid-infrared pulses,” Appl. Phys. Lett. 71, 1285–1286 (1997).
[CrossRef]

Q. Wu, T. D. Hewitt, and X.-C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69, 1026–1028 (1996).
[CrossRef]

Wynn, J.

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

Xhang, X. C.

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

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]

Q. Wu and X.-C. Zhang, “Free-space electro-optics sampling of mid-infrared pulses,” Appl. Phys. Lett. 71, 1285–1286 (1997).
[CrossRef]

Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70, 1784–1786 (1997).
[CrossRef]

Q. Wu, T. D. Hewitt, and X.-C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69, 1026–1028 (1996).
[CrossRef]

Appl. Phys. Lett.

Q. Wu and X.-C. Zhang, “Free-space electro-optics sampling of mid-infrared pulses,” Appl. Phys. Lett. 71, 1285–1286 (1997).
[CrossRef]

Y. Cai, I. Brener, J. Lopata, J. Wynn, L. Pfeiffer, J. B. Stark, Q. Wu, X. C. Xhang, and J. F. Federici, “Coherent terahertz radiation detection: direct comparison between free-space electro-optic sampling and antenna detection,” Appl. Phys. Lett. 73, 444–446 (1998).
[CrossRef]

S.-G. Park, M. R. Melloch, and A. M. Weiner, “Comparison of terahertz waveforms measured by electro-optic and photoconductive sampling,” Appl. Phys. Lett. 73, 3184–3186 (1998).
[CrossRef]

Q. Wu, T. D. Hewitt, and X.-C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69, 1026–1028 (1996).
[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]

Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70, 1784–1786 (1997).
[CrossRef]

IEEE J. Quantum Electron.

D. H. Auston and M. C. Nuss, “Electrooptic generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24, 184–197 (1988).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

IEEE Trans. Antennas Propag.

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, “Late-time target response measured with terahertz impulse ranging,” IEEE Trans. Antennas Propag. 45, 1518–1524 (1997).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

T. Hattori, Y. Homma, A. Mitsushi, and M. Tacke, “Indices of refraction of ZnS, ZnSe, ZnTe, CdS, and CdTe in the far infrared,” Opt. Commun. 7, 229–232 (1973).
[CrossRef]

Phys. Rev. B

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7, 5345–5359 (1973).
[CrossRef]

Other

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1987).

Clearly the frequency spectrum ATHz(Ω) of the THz pulse that is entering the EO crystal is equal to the incident spectrum AIn(Ω) multiplied by the complex frequency-dependent Fresnel transmission coefficient t12(Ω) that describes the EO crystal; i.e., ATHz(Ω)=t12(Ω)AIn(Ω). See Section 7.

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Figures (4)

Fig. 1
Fig. 1

Ellipsometer with a λ/4 plate and a Wollaston polarizer.

Fig. 2
Fig. 2

Polarization ellipse (solid curve) for the electric field after the λ/4 plate, compared with a circle (dashed curve).

Fig. 3
Fig. 3

Propagation of optical intensity Iopt and EO pulse PEO in the nonlinear crystal.

Fig. 4
Fig. 4

Absolute value of the second-order susceptibility in ZnTe.

Equations (101)

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

TerahertzfieldE1(z, ω1)=E1(z, ω1)uˆj,
E1(z, ω1)=A1(z, ω1)×exp[ik1(ω1)z]×exp(-iω1t),
E1(z, t)=-+E1(z, ω1)dω1.
OpticalbeamE2(z, ω2)=E2(z, ω2)uˆk,
E2(z, ω2)=A2(z, ω2-ω0)exp[ik2(ω2)z]exp[-i(ω2-ω0)t],
E2(z, t)=-+E2(z, ω2-ω0)dω2.
Sum-frequencyfieldE3(z, ω3)=E3(z, ω3)uˆi,
E3(z, ω3)=A3(z, ω3-ω0)exp[ik3(ω3)z]exp[-i(ω3-ω0)t],
E3(z, t)=-+E3(z, ω3-ω0)dω3.
Difference-frequencyfieldE4(z, ω4)=E4(z, ω4)uˆi,
E4(z, ω4)=A4(z, ω4-ω0)exp[ik4(ω4)z]exp[-i(ω4-ω0)t],
E4(z, t)=-+E4(z, ω4-ω0)dω4.
k=k+iβ,
P+(2)(ω3)=p+(2) exp[i(k1+k2)z-iω3t]uˆ+=χijk(2)(ω3)E1(ω1)E2(ω2)uˆ+,
P-(2)(ω4)=p-(2) exp[i(-k1+k2)z-iω4t]uˆ-=χijk(2)(ω4)E1*(ω1)E2(ω2)uˆ-,
z+β3(ω3)A3(z, ω3-ω0)
=i 2πω32c2k3(ω3)p+(2) exp{iΔk+z-[β1(ω1)+β2(ω2)]z},
z+β4(ω4)A4(z, ω4-ω0)
=i 2πω42c2k4(ω4)p-(2) exp{iΔk-z-[β1(ω1)+β2(ω2)]z},
Δk+=-k3(ω2+ω1)+k1(ω1)+k2(ω2),
Δk-=-k4(ω2-ω1)-k1(ω1)+k2(ω2).
A3(l, ω3-ω0)=i 2πω32c2k3(ω3)p+(2)exp{iΔk+l-[β1(ω1)+β2(ω2)]l}-exp[-β3(ω3)l]iΔk+-[β1(ω1)+β2(ω2)-β3(ω3)].
A3(l, ω3-ω0)=i 2πω32c2k3(ω3)χijk(2)(ω3) [exp(iΔk+l)-1]iΔk+×exp[-β3(ω3)l]A1(ω1)A2(ω2-ω0).
A4(l, ω4-ω0)=i 2πω42c2k4(ω4)p-(2)exp{iΔk-l-[β1(ω1)+β2(ω2)]l}-exp[-β4(ω4)l]iΔk--[β1(ω1)+β2(ω2)-β4(ω4)],
A4(l, ω4-ω0)=i 2πω42c2k4(ω4)χijk(2)(ω4) exp(iΔk-l)-1iΔk-×exp[-β4(ω4)l]A1*(ω1)A2(ω2-ω0),
Δk+=-k3(ω2+ω1)+k1(ω1)+k2(ω2),
Δk-=-k4(ω2-ω1)-k1(ω1)+k2(ω2)
An(ω)=An(z=0, ω).
ω1=Ωandω2=ω-Ω(sumfrequency),
ω1=Ωandω2=ω+Ω(differencefrequency),
k3(ω)=k4(ω).
E3(l, ω)=i-+ 2πω2c2k3(ω)χijk(2)(ω; Ω, ω-Ω)×exp(iΔk+l)-1iΔk+exp[ik3(ω)l]A1(Ω)×exp[-iΩ(t+τ)]A2(ω-Ω-ω0)×exp[-i(ω-Ω-ω0)t]dΩ,
E4(l, ω)=i-+ 2πω2c2k3(ω)χijk(2)(ω; Ω, ω+Ω)×exp(iΔk-l)-1iΔk-exp[ik3(ω)l]A1*(Ω)×exp[+iΩ(t+τ)]A2(ω+Ω-ω0)×exp[-i(ω+Ω-ω0)t]dΩ,
E3(l, ω)=i-+ 2πω2c2k3(ω)χijk(2)(ω; Ω, ω-Ω)×exp(iΔk+l)-1iΔk+×A1(Ω)A2(ω-Ω-ω0)exp(-iΩτ)dΩ×exp[ik3(ω)l-i(ω-ω0)t],
E4(l, ω)=i-+ 2πω2c2k3(ω)χijk(2)(ω; Ω, ω+Ω)×exp(iΔk-l)-1iΔk-×A1*(Ω)A2(ω+Ω-ω0)exp(+iΩτ)dΩ×exp[ik3(ω)l-i(ω-ω0)t].
E3+E4
=i-+ 2πω2c2k3(ω)exp[ik3(ω)l]χijk(2)(ω; Ω, ω-Ω)×exp[iΔk+(Ω, ω)l]-1iΔk+(Ω, ω)A1(Ω)A2(ω-Ω-ω0)×exp(-iΩτ)+χijk(2)(ω; Ω, ω+Ω)×exp[iΔk-(Ω, ω)l]-1iΔk-(Ω, ω)×A1*(Ω)A2(ω+Ω-ω0)exp(iΩτ)×exp[-i(ω-ω0)t]dΩ.
Δk-(-Ω, ω)=Δk+(Ω, ω),
A1*(-Ω)=A1(Ω),
χijk(2)(ω;-Ω, ω+Ω)=χijk(2)*(ω; Ω, ω+Ω),
E3+E4=i-+ 2πω2c2k3(ω)exp[ik3(ω)l]×[Re χijk(2)(ω; Ω, ω-Ω)]×exp[iΔk+(Ω, ω)l]-1iΔk+(Ω, ω)×A1(Ω)A2(ω-Ω-ω0)×exp[-iΩτ-i(ω-ω0)t]dΩ.
E(ω)=E2(ω)+E3(ω)+E4(ω).
E2(ω)=E2xeˆx+E2yeˆy=A2x exp[ikx(ω)l-i(ω-ω0)t]eˆx+A2y exp[iky(ω)l-i(ω-ω0)t]eˆy,
Ex(ω)=E2x(ω)+E3x(ω)+E4x(ω),
Ey(ω)=E2y(ω)+E3y(ω)+E4y(ω),
E3x+E4x
=i-+ 2πω2c2k(ω)exp[ikx(ω)l]χeffx(ω; Ω, ω-Ω)×exp[iΔk+(Ω, ω)l]-1iΔk+(Ω, ω)×A1(Ω)A2(ω-Ω-ω0)×exp[-iΩτ-i(ω-ω0)t]dΩ,
E3y+E4y
=i-+ 2πω2c2k(ω)exp[iky(ω)l]χeffy(ω; Ω, ω-Ω)×exp[iΔk+(Ω, ω)l]-1iΔk+(Ω, ω)A1(Ω)A2(ω-Ω-ω0)×exp[-iΩτ-i(ω-ω0)t]dΩ.
Ex(ω)E2x(ω)[1+iφx(ω, τ)],
Ey(ω)E2y(ω)[1+iφy(ω, τ)],
φx(ω, τ)=E3x(l, ω)+E4x(l, ω)iE2x(ω),
φy(ω, τ)=E3y(l, ω)+E4y(l, ω)iE2y(ω).
Ex(ω)=E2x(ω)exp[iφx(ω, τ)],
Ey(ω)=E2y(ω)exp[iφy(ω, τ)].
Ex=a0(ω)cos[k(ω)z-ωt+φx(ω)],
Ey=a0(ω)cos[k(ω)z-ωt+φy(ω)],
Ex=a0(ω)cos[k(ω)z-ωt+φx(ω)],
Ey=a0(ω)cos[k(ω)z-ωt+φy(ω)+(π/2)].
a(ω)=a0(ω)[1+½φ(ω)],
b(ω)=a0(ω)[1-½φ(ω)].
Ia=½0c0+|a(ω)|2dω,
Ib=½0c0+|b(ω)|2dω.
20cIa=0+|a0(ω)|2dω+¼0+|a0(ω)φ(ω)|2dω+Re 0+|a0(ω)|2φ(ω)dω,
20cIb=0+|a0(ω)|2dω+¼0+|a0(ω)φ(ω)|2dω-Re 0+|a0(ω)|2φ(ω)dω.
S=Ia-Ib=0c Re 0+|a0(ω)|2φ(ω)dω.
a0(ω)=A2x(ω)exp[ik(ω)l]=A2y(ω)exp[ik(ω)l]Aopt(ω)exp[ik(ω)l],
A1(Ω)ATHz(Ω).
φ=φy-φx=-+ 2πω2c2|k(ω)|χeff(2)(ω; Ω, ω-Ω)×exp[iΔk+(Ω, ω)l]-1iΔk+(Ω, ω)×ATHz(Ω)Aopt(ω-Ω-ω0)Aopt(ω-ω0)exp(-iΩτ)dΩ,
χeff(2)=χeffy-χeffx.
S(τ)=0c2Re -+-+ 2πω2c2|k(ω)|exp[-2β(ω)l]×χeff(2)(ω; Ω, ω-Ω)exp[iΔk+(Ω, ω)l]-1iΔk+(Ω, ω)×ATHz(Ω)Aopt*(ω-ω0)Aopt(ω-Ω-ω0)×exp(-iΩτ)dΩdω,
S(τ)=π0cRe -+ATHz(Ω)f(Ω)exp(-iΩτ)dΩ,
f(Ω)=-+ ω2|k(ω)|exp[-2β(ω)l]χeff(2)(ω; Ω, ω-Ω)×exp[iΔk+(ω, Ω)l]-1iΔk+(ω, Ω)×Aopt*(ω-ω0)Aopt(ω-Ω-ω0)dω.
S(τ)=π0c-+ATHz(Ω)f(Ω)exp(-iΩτ)dΩ.
S(τ)=π0cω02k(ω0)exp[-2β(ω0)l]×-+χeff(2)(ω0; Ω, ω0-Ω)×exp[iΔk+(ω0, Ω)l]-1iΔk+(ω0, Ω)×ATHz(Ω)Copt(Ω)exp(-iΩτ)dΩ,
Copt(Ω)=-+Aopt*(ω-ω0)Aopt(ω-ω0-Ω)dω.
S(τ)-+ATHz(Ω)f(Ω)exp(-iΩτ)dΩ,
f(Ω)=Copt(Ω)χeff(2)(ω0;Ω, ω0-Ω)×exp[iΔk+(ω0,Ω)l]-1iΔk+(ω0, Ω).
k(ω0+Δω)k(ω0)+dkdωω0Δω,
dkdω=1vgngc.
Δk+=-k(ω0+Ω)+k(Ω)+k(ω0)-k(ω0)-Ωdkdωω0+k(Ω)+k(ω0)k(Ω)-kg(Ω),
kg(Ω)Ωdkdωω0=Ωvg=Ωcng.
S(τ)0ldz-+Copt(Ω)×exp[-ikg(Ω)z]χeff(2)(ω0; Ω,ω0-Ω)ATHz(Ω)×exp[ik(Ω)z]exp(-iΩτ)dΩ;
S(τ)0ldz-+Iopt(z, t-τ)PEO(z, t)dt,
PEO(z, t)-+χeff(2)(ω0; Ω, ω0-Ω)ATHz(Ω)×exp[ik(Ω)z]exp(-iΩt)dΩ.
Iopt(t)=|Aopt(t)|2,
Iopt(z, t)=exp(-2β0z)Iopt(z=0, t-z/vg).
Sref(Ω)=π0ct12(Ω)AIn(Ω)f(Ω),
Ssample(Ω)=π0cH(Ω)t12(Ω)AIn(Ω)f(Ω).
Ssample(Ω)Sref(Ω)=H(Ω).
f(Ω)-+ωAopt*(ω-ω0)Aopt(ω-ω0-Ω)dω.
Aopt(ω)=1πΔω2exp-ω2Δω2.
f(Ω)1πΔω2-+ω×exp-1Δω2[(ω-ω0)2+(ω-ω0-Ω)2]dω1πΔω2-+ωg(ω)dω,
g(ω)=exp-1Δω2[2ω2-2ω(2ω0+Ω)+ω02+(ω0+Ω)2].
dgdω=-1Δω2(4ω-4ω0-2Ω)g(ω),
-+ωg(ω)dω=ω0+Ω2-+g(ω)dω.
f(Ω)1Δω2πω0+Ω2exp-Ω22Δω2.
f(Ω)1Δω2πω0 exp-Ω22Δω2=ω0Copt(Ω).
χχe+χi=0-1,χen2-1,
χi=0-n2,
χijk(2)(Ω)=χie(ω0)χje(ω0)[δijkeχke(Ω)+δijkiχki(Ω)].

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