Abstract

We present three methods for the distortion-free enhancement of THz signals measured by electro-optic sampling in zinc blende-type detector crystals, e.g., ZnTe or GaP. A technique commonly used in optically heterodyne-detected optical Kerr effect spectroscopy is introduced, which is based on two measurements at opposite optical biases near the zero transmission point in a crossed polarizer detection geometry. In contrast to other techniques for an undistorted THz signal enhancement, it also works in a balanced detection scheme and does not require an elaborate procedure for the reconstruction of the true signal as the two measured waveforms are simply subtracted to remove distortions. We study three different approaches for setting an optical bias using the Jones matrix formalism and discuss them also in the framework of optical heterodyne detection. We show that there is an optimal bias point in realistic situations where a small fraction of the probe light is scattered by optical components. The experimental demonstration will be given in the second part of this two-paper series [J. Opt. Soc. Am. B, doc. ID 204877 (2014, posted online)].

© 2014 Optical Society of America

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References

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  1. Q. Wu and X.-C. Zhang, “Ultrafast electro-optic field sensors,” Appl. Phys. Lett. 68, 1604–1606 (1996).
    [CrossRef]
  2. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
    [CrossRef]
  3. 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]
  4. Q. Chen, M. Tani, Z. Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B 18, 823–831 (2001).
    [CrossRef]
  5. Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
    [CrossRef]
  6. A. Schneider and P. Günter, “Measurement of the terahertz-induced phase shift in electro-optic sampling for an arbitrary biasing phase,” Appl. Opt. 45, 6598–6601 (2006).
    [CrossRef]
  7. S. Ahmed, J. Savolainen, and P. Hamm, “Detectivity enhancement in THz electrooptical sampling,” Rev. Sci. Instrum. 85, 013114 (2014).
    [CrossRef]
  8. R. Torre, Time-Resolved Spectroscopy in Complex Liquids: An Experimental Perspective (Springer, 2007).
  9. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2001).
  10. J. A. Johnson, F. D. J. Brunner, S. Grübel, A. Ferrer, S. L. Johnson, and T. Feurer, “Distortion-free enhancement of THz signals measured by electro-optic sampling. II. Experiments,” J. Opt. Soc. Am. B31 (to be published).
  11. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  12. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

2014 (1)

S. Ahmed, J. Savolainen, and P. Hamm, “Detectivity enhancement in THz electrooptical sampling,” Rev. Sci. Instrum. 85, 013114 (2014).
[CrossRef]

2007 (1)

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[CrossRef]

2006 (1)

2002 (1)

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]

2001 (1)

1999 (1)

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

1996 (1)

Q. Wu and X.-C. Zhang, “Ultrafast electro-optic field sensors,” Appl. Phys. Lett. 68, 1604–1606 (1996).
[CrossRef]

1941 (1)

Ahmed, S.

S. Ahmed, J. Savolainen, and P. Hamm, “Detectivity enhancement in THz electrooptical sampling,” Rev. Sci. Instrum. 85, 013114 (2014).
[CrossRef]

Brunner, F. D. J.

J. A. Johnson, F. D. J. Brunner, S. Grübel, A. Ferrer, S. L. Johnson, and T. Feurer, “Distortion-free enhancement of THz signals measured by electro-optic sampling. II. Experiments,” J. Opt. Soc. Am. B31 (to be published).

Chen, Q.

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

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

Ferrer, A.

J. A. Johnson, F. D. J. Brunner, S. Grübel, A. Ferrer, S. L. Johnson, and T. Feurer, “Distortion-free enhancement of THz signals measured by electro-optic sampling. II. Experiments,” J. Opt. Soc. Am. B31 (to be published).

Feurer, T.

J. A. Johnson, F. D. J. Brunner, S. Grübel, A. Ferrer, S. L. Johnson, and T. Feurer, “Distortion-free enhancement of THz signals measured by electro-optic sampling. II. Experiments,” J. Opt. Soc. Am. B31 (to be published).

Grübel, S.

J. A. Johnson, F. D. J. Brunner, S. Grübel, A. Ferrer, S. L. Johnson, and T. Feurer, “Distortion-free enhancement of THz signals measured by electro-optic sampling. II. Experiments,” J. Opt. Soc. Am. B31 (to be published).

Günter, P.

Hamm, P.

S. Ahmed, J. Savolainen, and P. Hamm, “Detectivity enhancement in THz electrooptical sampling,” Rev. Sci. Instrum. 85, 013114 (2014).
[CrossRef]

Jiang, Z.

Jiang, Z. P.

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

Johnson, J. A.

J. A. Johnson, F. D. J. Brunner, S. Grübel, A. Ferrer, S. L. Johnson, and T. Feurer, “Distortion-free enhancement of THz signals measured by electro-optic sampling. II. Experiments,” J. Opt. Soc. Am. B31 (to be published).

Johnson, S. L.

J. A. Johnson, F. D. J. Brunner, S. Grübel, A. Ferrer, S. L. Johnson, and T. Feurer, “Distortion-free enhancement of THz signals measured by electro-optic sampling. II. Experiments,” J. Opt. Soc. Am. B31 (to be published).

Jones, R. C.

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]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2001).

Savolainen, J.

S. Ahmed, J. Savolainen, and P. Hamm, “Detectivity enhancement in THz electrooptical sampling,” Rev. Sci. Instrum. 85, 013114 (2014).
[CrossRef]

Schneider, A.

Sun, F. G.

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

Tani, M.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2001).

Tonouchi, M.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[CrossRef]

Torre, R.

R. Torre, Time-Resolved Spectroscopy in Complex Liquids: An Experimental Perspective (Springer, 2007).

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]

Wu, Q.

Q. Wu and X.-C. Zhang, “Ultrafast electro-optic field sensors,” Appl. Phys. Lett. 68, 1604–1606 (1996).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

Zhang, X.-C.

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

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

Q. Wu and X.-C. Zhang, “Ultrafast electro-optic field sensors,” Appl. Phys. Lett. 68, 1604–1606 (1996).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

Q. Wu and X.-C. Zhang, “Ultrafast electro-optic field sensors,” Appl. Phys. Lett. 68, 1604–1606 (1996).
[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]

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nat. Photonics (1)

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[CrossRef]

Rev. Sci. Instrum. (1)

S. Ahmed, J. Savolainen, and P. Hamm, “Detectivity enhancement in THz electrooptical sampling,” Rev. Sci. Instrum. 85, 013114 (2014).
[CrossRef]

Other (4)

R. Torre, Time-Resolved Spectroscopy in Complex Liquids: An Experimental Perspective (Springer, 2007).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2001).

J. A. Johnson, F. D. J. Brunner, S. Grübel, A. Ferrer, S. L. Johnson, and T. Feurer, “Distortion-free enhancement of THz signals measured by electro-optic sampling. II. Experiments,” J. Opt. Soc. Am. B31 (to be published).

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

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Figures (4)

Fig. 1.
Fig. 1.

Detection of THz pulses in electro-optic crystals with zinc blende structure. The polarization ellipses of the probe beam are depicted at the given positions. The dotted lines indicate the polarization states after the detector crystal when the THz electric field E⃗THz is zero. (a) In the typical configuration, E⃗THz modulates the polarization state of the probe pulse, a quarter waveplate or a compensator adds an optical bias, and a polarizing beamsplitter converts the polarization state modulation into an intensity modulation. The transmission axis of the polarizer and the main axes of the waveplate and of the detector crystal in the presence of E⃗THz are indicated as dashed lines. Alternative configurations: (b) an optical bias is added first. (c) The polarizer is rotated instead of the quarter waveplate.

Fig. 2.
Fig. 2.

Enhancement and distortion of THz signals for an optical bias set by the phase retardation Γc of a compensator oriented at θ=45° [see Fig. 1(a) or Fig. 1(b)] or by the polarizer angle ϕ in combination with a quarter waveplate at θ=0° [see Fig. 1(c)] for different values of the background b. (a) Enhancement factor gS,c(Γ0,b) as a function of the effective optical bias Γ0, which is either Γc or Γp(ϕ). (b) Parameter for the first-order distortion of an individual measured signal dS,c(Γ0,b) as a function of gS,c(Γ0,b). (c) Parameter for the remaining first-order distortion of the subtracted signals qS,c(Γ0,b) as a function of gS,c(Γ0,b).

Fig. 3.
Fig. 3.

Enhancement and distortion of THz signals for an optical bias ΓQWP set by the quarter waveplate angle θ [see Fig. 1(a) or Fig. 1(b)] for different values of the background b. (a) Enhancement factor gS,QWP(θ,b) as a function of ΓQWP(θ). (b) Parameter for the first order distortion of an individual measured signal dS,QWP(θ,b) as a function of gS,QWP(θ,b). (c) Parameter for the remaining first order distortion of the subtracted signals qS,QWP(Γc,b) as a function of gS,QWP(θ,b).

Fig. 4.
Fig. 4.

Electro-optic sampling expressed in the terminology of optical heterodyne detection (see text for details): polarizer (P), waveplate (WP), detector crystal (D), polarizing beamsplitter (PBS), and photodiode (PD). (a) Direct detection of the signal intensity Isig.|Esig.|2 at zero optical bias (denoted by Γ0=0), (b) direct detection of the local oscillator intensity ILO|ELO|2 at the absence of the THz electric field, and (c) optical heterodyne detection of the superposition of the signal and local oscillator fields I|Esig.+ELO|2.

Equations (58)

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ΓTHz(ETHz)=2πLλn3r41ETHz,
I(δ)=Isig.+ILO+2(Isig.ILO)1/2cosδ,
I(δ)I(180°δ)=4(Isig.ILO)1/2cosδ.
I(ETHz,Γc)=I01+b+b[sin2ΓTHz(ETHz)+Γc2+b],
I(ETHz,Γc)=I01+b+b[cos2ΓTHz(ETHz)+Γc2+b],
T=sin2Γc2+bcos2Γc2+b.
S(ETHz,Γc)=I(ETHz,Γc)TI(ETHz,Γc)I(0,Γc)+TI(0,Γc).
S(ETHz,Γc)=cosΓccos[ΓTHz(ETHz)+Γc]4(sin2Γc2+b)(cos2Γc2+b).
S(ETHz,Γc=90°)=sinΓTHz(ETHz).
S=gS,cΓTHz+dS,c(gS,cΓTHz)2+O(ΓTHz3)
gS,c(Γc,b)=sinΓc4(sin2Γc2+b)cos2Γc2
dS,c(Γc,b)=cotΓc2gS,c(Γc,b)
|SgS,cΓTHzS|dS,cgS,c|ΓTHz|.
gS,c(Γc,max(b),b)14b
Γc,max(b)2b,
ΓTHz(ETHz)=arcsinS(ETHz,Γc=90°),
S(ETHz,Γc)S(ETHz,Γc)=2gS,c(Γc,b)S(ETHz,Γc=90°).
ΓTHz(ETHz)=arcsinS(ETHz,Γc)S(ETHz,Γc)2gS,c(Γc,b).
rS,c(Γc,ΔΓc,b)qS,c(Γc,b)gS,c(Γc,b)ΓTHzΔΓc,
qS,c(Γc,b)=dS,c(Γc,b)gS,c(Γc,b)gS,c(Γc,b)Γc+12dS,c(Γc,b)Γc.
qS,c(Γc,max(b),b)14b
I(ETHz,Γc)I(0,Γc)I(ETHz,Γc)I(0,Γc)=sinΓcsin2Γc2+bsinΓTHz(ETHz),
I(ETHz,Γc)I(0,Γc)I(ETHz,Γc)I(0,Γc)=2tanΓc2sinΓTHz(ETHz).
I(ETHz,Γp)=I02(1+b+b)×[sin2ΓTHz(ETHz)+Γp(ϕ)2+b],
I(ETHz,Γp)=I02(1+b+b)×[cos2ΓTHz(ETHz)+Γp(ϕ)2+b],
Γp(ϕ)=2ϕ
I(ETHz,θ)=I02(1+b+b)×{1+sin(2θ)sin[ΓTHz(ETHz)+Γs]cos2(2θ)cos[ΓTHz(ETHz)+Γs]+b},
I(ETHz,θ)=I02(1+b+b)×{1sin(2θ)sin[ΓTHz(ETHz)+Γs]+cos2(2θ)cos[ΓTHz(ETHz)+Γs]+b}.
T=sin2(2θ)+b1+cos2(2θ).
S(ETHz,θ)=A(θ)cosΓQWP(θ)cos[ΓTHz(ETHz)+ΓQWP(θ)][sin2(2θ)+b][1+cos2(2θ)],
ΓQWP(θ)=arccotcos2(2θ)sin(2θ)
A(θ)=sin2(2θ)+cos4(2θ)
S(ETHz,θ=45°)=sin[ΓTHz(ETHz)+Γs]sinΓs.
gS,QWP(θ,b)=sin(2θ)[sin2(2θ)+b][1+cos2(2θ)]
|SgS,QWPΓTHzS|dS,QWPgS,QWP|ΓTHz|,
dS,QWP(θ,b)=12[1+cos2(2θ)][sin2(2θ)+b]cot2(2θ).
gS,QWP(θmax(b),b)14b
θmax(b)b2,
S(ETHz,θ)S(ETHz,θ)=2gS,QWP(θ,b)S(ETHz,θ=45°).
ΓTHz(ETHz)=arcsinS(ETHz,θ)S(ETHz,θ)2gS,QWP(θ,b).
rS,QWP(θ,Δθ,b)2ΔθqS,QWP(θ,b)gS,QWP(θ,b)ΓTHz,
qS,QWP(θ,b)=dS,QWP(θ,b)gS,QWP(θ,b)gS,QWP(θ,b)θ+12dS,QWP(θ,b)θ.
qS,QWP(θmax(b),b)1b
I(ETHz,θ)I(0,θ)I(ETHz,θ)I(0,θ)=2sin(2θ)1cos2(2θ)+bsinΓTHz(ETHz),
I(ETHz,θ)I(0,θ)I(ETHz,θ)I(0,θ)=2sin(2θ)1+cos2(2θ)sinΓTHz(ETHz),
Isig.=I(ETHz,0)=I0sin2ΓTHz(ETHz)2.
I(ETHz,Γc)=I0[sinΓTHz(ETHz)2cosΓc2+sinΓc2cosΓTHz(ETHz)2]2
I(ETHz,Γc)I0[sinΓTHz(ETHz)2+sinΓc2]2.
ILO=I(0,Γc)=I0sin2Γc2,
δ={0°ifΓcΓTHz(ETHz)>0,180°ifΓcΓTHz(ETHz)<0.
I(ETHz,θ)=I0{sin2[ΓTHz(ETHz)2]+12sin2(2θ)cosΓTHz(ETHz)+sin[ΓTHz(ETHz)2]sin(2θ)cos[ΓTHz(ETHz)2]}
I(ETHz,θ)I0{sin2[ΓTHz(ETHz)2]+12sin2(2θ)+sin[ΓTHz(ETHz)2]sin(2θ)},
ILO=I(0,θ)=12I0sin2(2θ),
δ={45°ifθΓTHz(ETHz)>0,135°ifθΓTHz(ETHz)<0.
M=gΓTHz+O(ΓTHz2)(asΓTHz0)
|MgΓTHzM|dg|ΓTHz|.
g=(I0ILO)1/2|cosδ|,
d=14cos2δ.

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