Abstract

We present a set of equations describing the electro-optic detection of terahertz electric fields that is easy to use and that is applicable to all optically isotropic detection crystals. This set of equations can be used for every possible propagation and polarization direction of the probe beam and for arbitrary direction of the terahertz electric field. Due to its wide applicability, our results are very suitable to describe complex detection geometries, as are, for instance, found in terahertz microscopy. Our results are in excellent agreement with previously published experimental results.

© 2004 Optical Society of America

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References

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  1. Q. Wu and X.-C. Zhang, “Free-space electro-optic sampling of terahertz beam,” Appl. Phys. Lett. 67, 3523–3525 (1995).
    [CrossRef]
  2. P. Uhd Jepsen, C. Winnewisser, M. Shall, V. Schya, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, 3052–3054 (1997).
    [CrossRef]
  3. A. Nahata, A. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2322 (1996).
    [CrossRef]
  4. M. Li, G. C. Cho, T.-M. Lu, and X.-C. Zhang, “Time-domain dielectric constant measurement of thin film in GHz-THz frequency range near the Brewster angle,” Appl. Phys. Lett. 74, 2113–2115 (1999).
    [CrossRef]
  5. K. Wynne and D. A. Jaroszynski, “Superluminal terahertz pulses,” Opt. Lett. 24, 25–27 (1999).
    [CrossRef]
  6. B. Ferguson, S. Wang, D. Gray, D. Abbot, and X.-C. Zhang, “T-ray computed tomography,” Opt. Lett. 27, 1312–1314 (2002).
    [CrossRef]
  7. Q. Chen and X.-C. Zhang, “Semiconductor dynamic aperture for near-field terahertz wave imaging,” IEEE J. Sel. Top. Quantum Electron. 7, 608–614 (2001).
    [CrossRef]
  8. N. C. J. van der Valk and P. C. M. Planken, “Electro-optic detection of subwavelength terahertz spot sizes in the near field of a metal tip,” Appl. Phys. Lett. 81, 1558–1560 (2002).
    [CrossRef]
  9. It is also possible to perform electro-optic detection by measuring the change in the phase of the probe beam instead of the change in the polarization. However, we consider only the method that uses polarization modulation because this is by far the most widely used approach.
  10. S. Namba, “Electro-optical effect of zincblende,” J. Opt. Soc. Am. 51, 76–79 (1961).
    [CrossRef]
  11. D. F. Nelson, “General solution for the electro-optic effect,” J. Opt. Soc. Am. 65, 1144–1151 (1975).
    [CrossRef]
  12. M. J. Gunning and R. E. Raab, “Algebraic determination of the principal refractive indices and axes in the electro-optic effect,” Appl. Opt. 37, 8438–8447 (1998).
    [CrossRef]
  13. W. L. She and W. K. Lee, “Wave coupling theory of linear electro-optic effect,” Opt. Commun. 195, 303–311 (2001).
    [CrossRef]
  14. P. C. M. Planken, H.-K. Nienhuys, H. J. Bakker, and T. Wenckebach, “Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe,” J. Opt. Soc. Am. B 18, 313–317 (2001).
    [CrossRef]
  15. Q. Chen, M. Tani, Zhiping Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B 18, 823–831 (2001).
    [CrossRef]
  16. L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. II. Choice of the crystals and complete optimization of their orientation,” J. Opt. Soc. Am. B 19, 2704–2715 (2002).
    [CrossRef]
  17. A. Yariv, Quantum Electronics (Wiley, New York, 1975).
  18. This can be verified by substitution of D0i≃εrE0i in Eq. (1).
  19. R. W. Boyd, Nonlinear Optics (Academic, New York, 1992).

2002 (3)

2001 (4)

W. L. She and W. K. Lee, “Wave coupling theory of linear electro-optic effect,” Opt. Commun. 195, 303–311 (2001).
[CrossRef]

P. C. M. Planken, H.-K. Nienhuys, H. J. Bakker, and T. Wenckebach, “Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe,” J. Opt. Soc. Am. B 18, 313–317 (2001).
[CrossRef]

Q. Chen, M. Tani, Zhiping Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B 18, 823–831 (2001).
[CrossRef]

Q. Chen and X.-C. Zhang, “Semiconductor dynamic aperture for near-field terahertz wave imaging,” IEEE J. Sel. Top. Quantum Electron. 7, 608–614 (2001).
[CrossRef]

1999 (2)

M. Li, G. C. Cho, T.-M. Lu, and X.-C. Zhang, “Time-domain dielectric constant measurement of thin film in GHz-THz frequency range near the Brewster angle,” Appl. Phys. Lett. 74, 2113–2115 (1999).
[CrossRef]

K. Wynne and D. A. Jaroszynski, “Superluminal terahertz pulses,” Opt. Lett. 24, 25–27 (1999).
[CrossRef]

1998 (1)

1997 (1)

P. Uhd Jepsen, C. Winnewisser, M. Shall, V. Schya, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, 3052–3054 (1997).
[CrossRef]

1996 (1)

A. Nahata, A. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2322 (1996).
[CrossRef]

1995 (1)

Q. Wu and X.-C. Zhang, “Free-space electro-optic sampling of terahertz beam,” Appl. Phys. Lett. 67, 3523–3525 (1995).
[CrossRef]

1975 (1)

1961 (1)

Abbot, D.

Bakker, H. J.

Chen, Q.

Q. Chen and X.-C. Zhang, “Semiconductor dynamic aperture for near-field terahertz wave imaging,” IEEE J. Sel. Top. Quantum Electron. 7, 608–614 (2001).
[CrossRef]

Q. Chen, M. Tani, Zhiping Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B 18, 823–831 (2001).
[CrossRef]

Cho, G. C.

M. Li, G. C. Cho, T.-M. Lu, and X.-C. Zhang, “Time-domain dielectric constant measurement of thin film in GHz-THz frequency range near the Brewster angle,” Appl. Phys. Lett. 74, 2113–2115 (1999).
[CrossRef]

Coutaz, J.-L.

Duvillaret, L.

Ferguson, B.

Gray, D.

Gunning, M. J.

Heinz, T. F.

A. Nahata, A. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2322 (1996).
[CrossRef]

Helm, H.

P. Uhd Jepsen, C. Winnewisser, M. Shall, V. Schya, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, 3052–3054 (1997).
[CrossRef]

Jaroszynski, D. A.

Jiang, Zhiping

Keiding, S. R.

P. Uhd Jepsen, C. Winnewisser, M. Shall, V. Schya, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, 3052–3054 (1997).
[CrossRef]

Lee, W. K.

W. L. She and W. K. Lee, “Wave coupling theory of linear electro-optic effect,” Opt. Commun. 195, 303–311 (2001).
[CrossRef]

Li, M.

M. Li, G. C. Cho, T.-M. Lu, and X.-C. Zhang, “Time-domain dielectric constant measurement of thin film in GHz-THz frequency range near the Brewster angle,” Appl. Phys. Lett. 74, 2113–2115 (1999).
[CrossRef]

Lu, T.-M.

M. Li, G. C. Cho, T.-M. Lu, and X.-C. Zhang, “Time-domain dielectric constant measurement of thin film in GHz-THz frequency range near the Brewster angle,” Appl. Phys. Lett. 74, 2113–2115 (1999).
[CrossRef]

Nahata, A.

A. Nahata, A. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2322 (1996).
[CrossRef]

Namba, S.

Nelson, D. F.

Nienhuys, H.-K.

Planken, P. C. M.

N. C. J. van der Valk and P. C. M. Planken, “Electro-optic detection of subwavelength terahertz spot sizes in the near field of a metal tip,” Appl. Phys. Lett. 81, 1558–1560 (2002).
[CrossRef]

P. C. M. Planken, H.-K. Nienhuys, H. J. Bakker, and T. Wenckebach, “Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe,” J. Opt. Soc. Am. B 18, 313–317 (2001).
[CrossRef]

Raab, R. E.

Rialland, S.

Schya, V.

P. Uhd Jepsen, C. Winnewisser, M. Shall, V. Schya, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, 3052–3054 (1997).
[CrossRef]

Shall, M.

P. Uhd Jepsen, C. Winnewisser, M. Shall, V. Schya, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, 3052–3054 (1997).
[CrossRef]

She, W. L.

W. L. She and W. K. Lee, “Wave coupling theory of linear electro-optic effect,” Opt. Commun. 195, 303–311 (2001).
[CrossRef]

Tani, M.

Uhd Jepsen, P.

P. Uhd Jepsen, C. Winnewisser, M. Shall, V. Schya, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, 3052–3054 (1997).
[CrossRef]

van der Valk, N. C. J.

N. C. J. van der Valk and P. C. M. Planken, “Electro-optic detection of subwavelength terahertz spot sizes in the near field of a metal tip,” Appl. Phys. Lett. 81, 1558–1560 (2002).
[CrossRef]

Wang, S.

Weling, A.

A. Nahata, A. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2322 (1996).
[CrossRef]

Wenckebach, T.

Winnewisser, C.

P. Uhd Jepsen, C. Winnewisser, M. Shall, V. Schya, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, 3052–3054 (1997).
[CrossRef]

Wu, Q.

Q. Wu and X.-C. Zhang, “Free-space electro-optic sampling of terahertz beam,” Appl. Phys. Lett. 67, 3523–3525 (1995).
[CrossRef]

Wynne, K.

Zhang, X.-C.

B. Ferguson, S. Wang, D. Gray, D. Abbot, and X.-C. Zhang, “T-ray computed tomography,” Opt. Lett. 27, 1312–1314 (2002).
[CrossRef]

Q. Chen and X.-C. Zhang, “Semiconductor dynamic aperture for near-field terahertz wave imaging,” IEEE J. Sel. Top. Quantum Electron. 7, 608–614 (2001).
[CrossRef]

Q. Chen, M. Tani, Zhiping Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B 18, 823–831 (2001).
[CrossRef]

M. Li, G. C. Cho, T.-M. Lu, and X.-C. Zhang, “Time-domain dielectric constant measurement of thin film in GHz-THz frequency range near the Brewster angle,” Appl. Phys. Lett. 74, 2113–2115 (1999).
[CrossRef]

Q. Wu and X.-C. Zhang, “Free-space electro-optic sampling of terahertz beam,” Appl. Phys. Lett. 67, 3523–3525 (1995).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

A. Nahata, A. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2322 (1996).
[CrossRef]

M. Li, G. C. Cho, T.-M. Lu, and X.-C. Zhang, “Time-domain dielectric constant measurement of thin film in GHz-THz frequency range near the Brewster angle,” Appl. Phys. Lett. 74, 2113–2115 (1999).
[CrossRef]

Q. Wu and X.-C. Zhang, “Free-space electro-optic sampling of terahertz beam,” Appl. Phys. Lett. 67, 3523–3525 (1995).
[CrossRef]

N. C. J. van der Valk and P. C. M. Planken, “Electro-optic detection of subwavelength terahertz spot sizes in the near field of a metal tip,” Appl. Phys. Lett. 81, 1558–1560 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

Q. Chen and X.-C. Zhang, “Semiconductor dynamic aperture for near-field terahertz wave imaging,” IEEE J. Sel. Top. Quantum Electron. 7, 608–614 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

W. L. She and W. K. Lee, “Wave coupling theory of linear electro-optic effect,” Opt. Commun. 195, 303–311 (2001).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. E (1)

P. Uhd Jepsen, C. Winnewisser, M. Shall, V. Schya, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, 3052–3054 (1997).
[CrossRef]

Other (4)

It is also possible to perform electro-optic detection by measuring the change in the phase of the probe beam instead of the change in the polarization. However, we consider only the method that uses polarization modulation because this is by far the most widely used approach.

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

This can be verified by substitution of D0i≃εrE0i in Eq. (1).

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992).

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Figures (3)

Fig. 1
Fig. 1

General setup used in electro-optic detection. A probe pulse, which is originally polarized linearly, travels through a detection crystal and a λ/4 plate, and is then split by a Wollaston prism (WP). The difference in the intensities of the two beams is measured with a differential detector, which, apart from the electronics, consists of two photodiodes (D1 and D2).

Fig. 2
Fig. 2

Relative orientation of the initial probe polarization Eˆp, the polarization of the two propagation modes of the detection crystal rˆc1 and rˆc2, the polarization of the two propagation modes of the λ/4 plate rˆl1 and rˆl2, and the polarization directions of the two beams after the Wollaston prism rˆw1 and rˆw2. The angle between Eˆp and rˆc1 is defined as α, the angle between Eˆp and rˆl1 as β, and the angle between rˆl1 and rˆw1 as γ. All vectors, except sˆ, are in the plane of the paper. The propagation direction of the probe sˆ is perpendicular to the plane of the paper, and points into the paper.

Fig. 3
Fig. 3

(a) Values that can be obtained for the lengths of V and V as a function of the probe propagation direction. (b) The path followed for the propagation direction displayed on the horizontal axis of graph (a).

Tables (1)

Tables Icon

Table 1 Calculation Results for Different Propagation Directions of the Probe Beama

Equations (51)

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D0i=ni2ε0[E0i-(sˆ·E0i)sˆ],
εij=ε0δij+χij(1)+kχijk(2)ETk,
χij(1)=(εr-1)δij,
ijk  χijk(2)=χ(2),
otherwise  χijk(2)=0.
ε¯¯=ε0εrχ(2)ETzχ(2)ETyχ(2)ETzεrχ(2)ETxχ(2)ETyχ(2)ETxεr,
εrχ(2)ETzχ(2)ETyχ(2)ETzεrχ(2)ETxχ(2)ETyχ(2)ETxεrE0i
=ni21-sx2-sxsy-sxsz-sxsy1-sy2-sysz-sxsz-sysz1-sz2E0i,
rˆciE0iE0i withi{1, 2}.
ni2=(εr+Δni)2εr+2Δniεr.
AETzETyETzAETxETyETxAE0i
A+2 Δniεrχ(2)1-sx2-sxsy-sxsz-sxsy1-sy2-sysz-sxsz-sysz1-sz2E0i,
xˆ=sxxˆ+syyˆ+szzˆ,
yˆ=(-szyˆ+syzˆ)/sy2+sz2,
zˆ=[(sy2+sz2)xˆ-sxsyyˆ-sxszzˆ]/sy2+sz2,
P¯¯=sx0sy2+sz2sy-sz/sy2+sz2-sxsy/sy2+sz2szsy/sy2+sz2-sxsz/sy2+sz2.
A+2ETxsysz+2ETysxsz+2ETzsysxETx(sy2-sz2)+ETysysx-ETzszsxsy2+sz2ETx(sy2-sz2)+ETysysx-ETzszsxsy2+sz2-2ETx syszsy2+sz2-2 Δniεrχ(2)-2ETxsxsysz+ETysz(-1+2sy2+2sz2)+2ETzsy(sz2+sy2)sy2+sz2ETxsx -sy2+sz2sy2+sz2+ETysy-ETzsz
-2ETxsxsysz+ETysz(-1+2sy2+2sz2)+2ETzsy(sz2+sy2)sy2+sz2ETxsx -sy2+sz2sy2+sz2+ETysy-ETzsz-2ETx sx2syszsy2+sz2-2ETysxsz-2ETzsxsy-2 Δniεrχ(2)E0i=0,
Aεr/χ(2)ET,
(A00)EixEiyEiz=0,
Eix=0.
-2ETx syszsy2+sz2-2 Δniεrχ(2)ETxsx -sy2+sz2sy2+sz2+ETysy-ETzszETxsx -sy2+sz2sy2+sz2+ETysy-ETzsz-2ETx sx2syszsy2+sz2-2ETysxsz-2ETzsxsy-2 Δniεrχ(2)EiyEiz=00.
Δn1=-χ(2)2εr(b-b2+c)
Δn2=-χ(2)2εr(b+b2+c),
b=ETxsysz+ETysxsz+ETzsysx,
c=ETx2sx2+ETy2sy2+ETz2sz2-2(ETxETysxsy+ETzETyszsy+ETxETzsxsz).
E01=0-ETxsx sz2-sy2sz2+sy2-ETysy+ETzsz-b2+c+b-2ETx syszsy2+sz2.
Eˆp=(rˆc1·Eˆp)rˆc1+(rˆc2·Eˆp)rˆc2.
Eˆpc=exp(iωn1L/c)rˆc1(rˆc1·Eˆp)+exp(iωn2L/c)rˆc2(rˆc2·Eˆp)=exp(iωn1L/c)cos(α)rˆc1-exp(iωn2L/c)sin(α)rˆc2=exp(iωn1L/c)[cos(α)rˆc1-exp(iωΔnL/c)sin(α)rˆc2],
Eˆpcexp(iωn1L/c)[cos(α)rˆc1-(1+iC)sin(α)rˆc2].
Eˆpl=rˆl1(rˆl1·Eˆpc)+irˆl2(rˆl2·Eˆpc)exp(iωn1L/c){rˆl1[cos(β-α)cos(α)-sin(β-α)×(1+iC)sin(α)]+irˆl2[-sin(β-α)cos(α)-cos(β-α)(1+iC)sin(α)]}=exp(iωn1L/c){[cos(β)rˆl1+C cos(β-α)sin(α)rˆl2]-i[sin(β)rˆl2+C sin(β-α)sin(α)rˆl1]},
I1=12ε0εrc|Eˆpl·rˆw1|2Itot{cos2(β)cos2(γ)+sin2(β)sin2(γ)+2C sin(α)sin(γ)cos(γ)[cos(β)cos(β-α)+sin(β)sin(β-α)]}=Itot[cos2(β)cos2(γ)+sin2(β)sin2(γ)+C sin(2α)sin(2γ)/2],
I2=12ε0εrc|Eˆpl·rˆw2|2Itot[cos2(β)sin2(γ)+sin2(β)cos2(γ)-C sin(2α)sin(2γ)/2],
ΔI=I1-I2Itot[cos(2β)cos(2γ)+C sin(2α)sin(2γ)].
ΔIItotsin(2α) ωΔnLc.
Eˆp0cos(δ)sin(δ),
Eˆp=1sy2+sz2 sin(δ)(sy2+sz2)-sz cos(δ)-sxsy sin(δ)sy cos(δ)-sxsz sin(δ).
sin(2δ)=2Epx(Epysz-Epzsy)sy2+sz2,
cos(2δ)=(Epysz-Epzsy)2-Epx2sy2+sz2,
ΔIItotsin(2α) ωΔnLc=2ωχ(2)Lcεr b2+c(Eˆp·rˆc1)[sˆ·(Eˆp×rˆc1)]=2ωχ(2)Lcεr b2+c(Eˆp·E01)×[sˆ·(Eˆp×E01)]/E012.
sin(α)=sˆ·(Eˆp×rˆc1),
cos(α)=Eˆp·rˆc1.
ΔIItot=-ωχ(2)Lcεr sin(2δ)ETxETyETz·sysz sz2+sy2-2sy2+sz2sxszsxsy+cos(2δ)ETxETyETz·sx -sz2+sy2sy2+sz2-sysz,
ΔIItot=ωn3r41Lc[sin(2δ)(ET·S1)+cos(2δ)(ET·S2)],
S1=sysz sz2+sy2-2sy2+sz2, sxsz, sxsy,
S2=sx -sz2+sy2sy2+sz2,-sy, sz,
ΔIItot=ωn3r41Lc(ET·V),
Vsin(2δ)S1+cos(2δ)S2,
tan(4δex)=2(S1·S2)S22-S12.
V=V-sˆ(sˆ·V)
=sin(2δ)[S1-sˆ(sˆ·S1)]+cos(2δ)[S2-sˆ(sˆ·S2)].

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