Abstract

Time-domain spectroscopy allows fast and broadband measurement of the optical constants of materials in the terahertz domain. We present a method that improves the determination of the optical constants through simultaneous determination of the sample thickness. This method could be applied to any material with moderate absorption and requires only two measurements of the temporal profile of the terahertz pulses: a reference one without the sample and one transmitted through the sample.

© 1999 Optical Society of America

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References

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  1. See, for example, the special issue on terahertz electromagnetic pulse generation, physics, and applications, J. Opt. Soc. Am. B 11(12), (1994).
  2. L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in THz time-domain spectroscopy,” IEEE J. Select. Topics Quantum Electron. 2, 739–746 (1996).
    [CrossRef]
  3. S. Labbé-Lavigne, S. Barret, F. Garet, L. Duvillaret, J.-L. Coutaz, “Far-infrared dielectric constant of porous silicon layers measured by THz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
    [CrossRef]
  4. D. Grischkowsky, S. Keiding, M. van Exter, Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
    [CrossRef]
  5. M. van Exter, D. Grischkowsky, “Optical and electronic properties of doped silicon from 0.1 to 2 THz,” Appl. Phys. Lett. 56, 1694–1696 (1990).
    [CrossRef]
  6. J. E. Pedersen, S. R. Keiding, “THz time-domain spectroscopy of nonpolar liquids,” IEEE J. Quantum Electron. 28, 2518–2522 (1992).
    [CrossRef]
  7. J. F. Whitaker, F. Gao, Y. Liu, “THz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, N. Peyghambarian, R. C. Eckhardt, D. D. Lowenthal, eds., SPIE2145, 168–177 (1994).
  8. P. U. Jepsen, S. R. Keiding, “Radiation patterns from lens-coupled THz antennas,” Opt. Lett. 20, 807–809 (1995).
    [CrossRef] [PubMed]
  9. We showed in Ref. 2 that the Fabry–Perot effect can be treated as a perturbation for the extraction of the complex refractive index of the sample in all practical cases.

1998 (1)

S. Labbé-Lavigne, S. Barret, F. Garet, L. Duvillaret, J.-L. Coutaz, “Far-infrared dielectric constant of porous silicon layers measured by THz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

1996 (1)

L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in THz time-domain spectroscopy,” IEEE J. Select. Topics Quantum Electron. 2, 739–746 (1996).
[CrossRef]

1995 (1)

1994 (1)

See, for example, the special issue on terahertz electromagnetic pulse generation, physics, and applications, J. Opt. Soc. Am. B 11(12), (1994).

1992 (1)

J. E. Pedersen, S. R. Keiding, “THz time-domain spectroscopy of nonpolar liquids,” IEEE J. Quantum Electron. 28, 2518–2522 (1992).
[CrossRef]

1990 (2)

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

M. van Exter, D. Grischkowsky, “Optical and electronic properties of doped silicon from 0.1 to 2 THz,” Appl. Phys. Lett. 56, 1694–1696 (1990).
[CrossRef]

Barret, S.

S. Labbé-Lavigne, S. Barret, F. Garet, L. Duvillaret, J.-L. Coutaz, “Far-infrared dielectric constant of porous silicon layers measured by THz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

Coutaz, J.-L.

S. Labbé-Lavigne, S. Barret, F. Garet, L. Duvillaret, J.-L. Coutaz, “Far-infrared dielectric constant of porous silicon layers measured by THz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in THz time-domain spectroscopy,” IEEE J. Select. Topics Quantum Electron. 2, 739–746 (1996).
[CrossRef]

Duvillaret, L.

S. Labbé-Lavigne, S. Barret, F. Garet, L. Duvillaret, J.-L. Coutaz, “Far-infrared dielectric constant of porous silicon layers measured by THz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in THz time-domain spectroscopy,” IEEE J. Select. Topics Quantum Electron. 2, 739–746 (1996).
[CrossRef]

Fattinger, Ch.

Gao, F.

J. F. Whitaker, F. Gao, Y. Liu, “THz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, N. Peyghambarian, R. C. Eckhardt, D. D. Lowenthal, eds., SPIE2145, 168–177 (1994).

Garet, F.

S. Labbé-Lavigne, S. Barret, F. Garet, L. Duvillaret, J.-L. Coutaz, “Far-infrared dielectric constant of porous silicon layers measured by THz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in THz time-domain spectroscopy,” IEEE J. Select. Topics Quantum Electron. 2, 739–746 (1996).
[CrossRef]

Grischkowsky, D.

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

M. van Exter, D. Grischkowsky, “Optical and electronic properties of doped silicon from 0.1 to 2 THz,” Appl. Phys. Lett. 56, 1694–1696 (1990).
[CrossRef]

Jepsen, P. U.

Keiding, S.

Keiding, S. R.

P. U. Jepsen, S. R. Keiding, “Radiation patterns from lens-coupled THz antennas,” Opt. Lett. 20, 807–809 (1995).
[CrossRef] [PubMed]

J. E. Pedersen, S. R. Keiding, “THz time-domain spectroscopy of nonpolar liquids,” IEEE J. Quantum Electron. 28, 2518–2522 (1992).
[CrossRef]

Labbé-Lavigne, S.

S. Labbé-Lavigne, S. Barret, F. Garet, L. Duvillaret, J.-L. Coutaz, “Far-infrared dielectric constant of porous silicon layers measured by THz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

Liu, Y.

J. F. Whitaker, F. Gao, Y. Liu, “THz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, N. Peyghambarian, R. C. Eckhardt, D. D. Lowenthal, eds., SPIE2145, 168–177 (1994).

Pedersen, J. E.

J. E. Pedersen, S. R. Keiding, “THz time-domain spectroscopy of nonpolar liquids,” IEEE J. Quantum Electron. 28, 2518–2522 (1992).
[CrossRef]

van Exter, M.

M. van Exter, D. Grischkowsky, “Optical and electronic properties of doped silicon from 0.1 to 2 THz,” Appl. Phys. Lett. 56, 1694–1696 (1990).
[CrossRef]

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

Whitaker, J. F.

J. F. Whitaker, F. Gao, Y. Liu, “THz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, N. Peyghambarian, R. C. Eckhardt, D. D. Lowenthal, eds., SPIE2145, 168–177 (1994).

Appl. Phys. Lett. (1)

M. van Exter, D. Grischkowsky, “Optical and electronic properties of doped silicon from 0.1 to 2 THz,” Appl. Phys. Lett. 56, 1694–1696 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. E. Pedersen, S. R. Keiding, “THz time-domain spectroscopy of nonpolar liquids,” IEEE J. Quantum Electron. 28, 2518–2522 (1992).
[CrossRef]

IEEE J. Select. Topics Quantum Electron. (1)

L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in THz time-domain spectroscopy,” IEEE J. Select. Topics Quantum Electron. 2, 739–746 (1996).
[CrossRef]

J. Appl. Phys. (1)

S. Labbé-Lavigne, S. Barret, F. Garet, L. Duvillaret, J.-L. Coutaz, “Far-infrared dielectric constant of porous silicon layers measured by THz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

J. Opt. Soc. Am. B (2)

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

See, for example, the special issue on terahertz electromagnetic pulse generation, physics, and applications, J. Opt. Soc. Am. B 11(12), (1994).

Opt. Lett. (1)

Other (2)

We showed in Ref. 2 that the Fabry–Perot effect can be treated as a perturbation for the extraction of the complex refractive index of the sample in all practical cases.

J. F. Whitaker, F. Gao, Y. Liu, “THz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, N. Peyghambarian, R. C. Eckhardt, D. D. Lowenthal, eds., SPIE2145, 168–177 (1994).

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Figures (9)

Fig. 1
Fig. 1

Extracted ordinary index of refraction of LiNbO3 for different thicknesses (thick curves) together with the standard deviation calculated over 12 records (thin curves).

Fig. 2
Fig. 2

Terahertz pulses transmitted through a 1.1-mm-thick plate of LiNbO3 with polarization parallel to the slow optical axis (ordinary axis) of the crystal.

Fig. 3
Fig. 3

Ordinary index of refraction of LiNbO3 extracted from echo 0 (solid curves) and echo 1 (dashed curves) for different estimated thicknesses of the sample.

Fig. 4
Fig. 4

Difference of the indices of refraction (points with error bars) extracted from echoes 0 and 1, as plotted in Fig. 2, versus the estimated thickness of the sample. The best theoretical fit given by relation (5) and obtained for L 0 = 1.100 mm is represented by the solid curve.

Fig. 5
Fig. 5

Index of refraction of silicon for different estimated thicknesses of the sample.

Fig. 6
Fig. 6

Representation of the oscillating term θ = arg[(1 - N)/(1 - D)], which is responsible for the artificial oscillations of the refractive index seen in Fig. 4 in the complex plane.

Fig. 7
Fig. 7

Amplitude of the oscillations of the extracted refractive indices (error bars), as plotted in Fig. 4, versus the estimated thickness of the sample. The best theoretical fit from relation (15) and obtained for L 0 = 368 µm is represented by the solid curve.

Fig. 8
Fig. 8

Extracted index of refraction of silicon for an estimated thickness of the wafer of 540 µm, which is quite different from its actual value of 368 µm.

Fig. 9
Fig. 9

Magnitude of the oscillation of the index of refraction of silicon (points), as plotted in Fig. 7, versus the frequency. Three theoretical fits given by relation (14) for three different values of the thickness are also represented.

Equations (17)

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

Tpω=4ñ0ñ0-12pñ0+12p+2×exp-iñ0ωL0/c2p+1 expiωL0/c=ρpωexp-iϕpω,
ρpω=4n02+κ021/2n0-12+κ02pn0+12+κ02p+1×exp-2p+1κ0·ωL0/c),
ϕpω=2p+1n0-1ωL0c+2p arctan2κ0n02+κ02-1+arctanκ0n0n0+1+κ02.
ϕpω2p+1n0-1ωL0/c.
Δn12p+1ωL0/cΔϕω+n0-12p+1ΔLL0,
np-nq=n0+Δnp-n0+Δnq2q-p2p+12q+1ΔLL0.
ΔL=n0-n1n0-n1+2/3 L,
Tω; ñ0, L0=4ñ0ñ0+12exp-iñ0-1ωL0/c1-ñ0-1ñ0+12 exp-2iñ0ωL0/c,
Tω; ñ0, L0=Tω; ñ, L,
Δñ=1-ñ0ΔLL-cωLargñ0ñ0+12ññ+12-cωLarg1-ñ-1ñ+12 exp-2iñ ωLc1-ñ0-1ñ0+12 exp-2iñ0ωL0c.
Δn1-n0ΔLL-cωL θ, θ=arg1-N1-D, N=n0-1n0+1+2ΔL/L02×exp-2in0ωL0cexp-2i ωΔLc, D=n0-1n0+12 exp-2in0ωL0c,
FFP=c2n0L0,  Fmod=c2ΔL,
minΔn=1-n0ΔLL-cωL θmax1-n0ΔLL-2cωLarctanR|sinωΔL/c|1-R,
maxΔn=1-n0ΔLL-cωL θmin1-n0ΔLL+2cωLarctanR|sinωΔL/c|1+R,
R=|N|+|D|2=n0-1n0+1+ΔL/L02n-1n+1+ΔL/L2.
magΔn=maxΔn-minΔn2cωLarctan2R1-R2sinω ΔLc.
magΔn4R1-R2|ΔL|L.

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