Abstract

The far-infrared absorption and index of refraction of high-resistivity, float-zone, crystalline silicon has been measured by terahertz time-domain spectroscopy. The measured new upper limit for the absorption of this most transparent dielectric material in the far infrared shows unprecedented transparency over the range from 0.5 to 2.5 THz and a well-resolved absorption feature at 3.6 THz. The index of refraction shows remarkably little dispersion, changing by only 0.0001 over the range from 0.5 to 4.5 THz.

© 2004 Optical Society of America

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  1. C. M. Randall and R. D. Rawcliffe, “Refractive indices of germanium, silicon, and fused quartz in the far-infrared,” Appl. Opt. 6, 1889–1894 (1967).
    [CrossRef] [PubMed]
  2. E. V. Loewenstein, D. R. Smith, and R. L. Morgan, “Optical constants of far infrared materials. Crystalline solids,” Appl. Opt. 12, 398–406 (1973).
    [CrossRef] [PubMed]
  3. W. R. Passchier, D. D. Honijk, M. Mandel, and M. N. Afsar, “A new method for the determination of complex refractive index spectra of transparent solids in the far-infrared spectral region; results of pure silicon and crystals quartz,” J. Phys. D 10, 509–517 (1977).
    [CrossRef]
  4. J. R. Birch, “The absolute determination of complex reflectivity,” Infrared Phys. 18, 613–620 (1978).
    [CrossRef]
  5. A. K. Wan Abdullah, K. A. Maslin, and T. J. Parker, “Observation of two-phonon difference bands in the FIR transmission spectrum of Si,” Infrared Phys. 24, 185–188 (1984).
    [CrossRef]
  6. M. N. Afsar, “Dielectric measurements of millimeter-wave materials,” IEEE Trans. Microwave Theory Tech. MTT-32, 1598–1609 (1984).
    [CrossRef]
  7. J. M. Dutta, C. R. Jones, and H. Dave, “Complex dielectric constants for selected near-millimeter-wave materials at 245 GHz,” IEEE Trans. Microwave Theory Tech. MTT-34, 932–936 (1986).
    [CrossRef]
  8. T. Ohba and S. Ikawa, “Far-infrared absorption of silicon crystals,” J. Appl. Phys. 64, 4141–4143 (1988).
    [CrossRef]
  9. K. Seeger, “Microwave dielectric constants of silicon, gallium arsenide, and quartz,” J. Appl. Phys. 63, 5439–5443 (1988).
    [CrossRef]
  10. 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]
  11. M. van Exter and D. Grischkowsky, “Carrier dynamics of electrons and holes in moderately-doped silicon,” Phys. Rev. B 41, 12140–12149 (1990).
    [CrossRef]
  12. M. N. Asfar and H. Chi, “Millimeter wave complex refractive index, complex dielectric permittivity and loss tangent of extra high purity and compensated silicon,” Int. J. Infrared Millim. Waves 15, 1181–1188 (1994).
    [CrossRef]
  13. Tae-In Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78, 1106–1109 (1997).
    [CrossRef]
  14. Tae-In Jeon and D. Grischkowsky, “Observation of a Cole–Davidson type complex conductivity in the limit of very low carrier densities in doped silicon,” Appl. Phys. Lett. 72, 2259–2261 (1998).
    [CrossRef]
  15. J. Lesurf, Millimeter-Wave Optics, Devices and Systems (Hilger, Bristol, UK, 1990).
  16. M. T. Reiten, S. A. Harmon, and R. A. Cheville, “Terahertz beam propagation measured through three-dimensional amplitude profile determination,” J. Opt. Soc. Am. B 20, 2215–2225 (2003).
    [CrossRef]
  17. L. Duvillaret, F. Garet, and J.-L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. 38, 409–415 (1999).
    [CrossRef]
  18. S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76, 1987–1989 (2000).
    [CrossRef]
  19. R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000).
    [CrossRef]

2003 (1)

2000 (2)

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76, 1987–1989 (2000).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000).
[CrossRef]

1999 (1)

1998 (1)

Tae-In Jeon and D. Grischkowsky, “Observation of a Cole–Davidson type complex conductivity in the limit of very low carrier densities in doped silicon,” Appl. Phys. Lett. 72, 2259–2261 (1998).
[CrossRef]

1997 (1)

Tae-In Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78, 1106–1109 (1997).
[CrossRef]

1994 (1)

M. N. Asfar and H. Chi, “Millimeter wave complex refractive index, complex dielectric permittivity and loss tangent of extra high purity and compensated silicon,” Int. J. Infrared Millim. Waves 15, 1181–1188 (1994).
[CrossRef]

1990 (2)

1988 (2)

T. Ohba and S. Ikawa, “Far-infrared absorption of silicon crystals,” J. Appl. Phys. 64, 4141–4143 (1988).
[CrossRef]

K. Seeger, “Microwave dielectric constants of silicon, gallium arsenide, and quartz,” J. Appl. Phys. 63, 5439–5443 (1988).
[CrossRef]

1986 (1)

J. M. Dutta, C. R. Jones, and H. Dave, “Complex dielectric constants for selected near-millimeter-wave materials at 245 GHz,” IEEE Trans. Microwave Theory Tech. MTT-34, 932–936 (1986).
[CrossRef]

1984 (2)

A. K. Wan Abdullah, K. A. Maslin, and T. J. Parker, “Observation of two-phonon difference bands in the FIR transmission spectrum of Si,” Infrared Phys. 24, 185–188 (1984).
[CrossRef]

M. N. Afsar, “Dielectric measurements of millimeter-wave materials,” IEEE Trans. Microwave Theory Tech. MTT-32, 1598–1609 (1984).
[CrossRef]

1978 (1)

J. R. Birch, “The absolute determination of complex reflectivity,” Infrared Phys. 18, 613–620 (1978).
[CrossRef]

1977 (1)

W. R. Passchier, D. D. Honijk, M. Mandel, and M. N. Afsar, “A new method for the determination of complex refractive index spectra of transparent solids in the far-infrared spectral region; results of pure silicon and crystals quartz,” J. Phys. D 10, 509–517 (1977).
[CrossRef]

1973 (1)

1967 (1)

Afsar, M. N.

M. N. Afsar, “Dielectric measurements of millimeter-wave materials,” IEEE Trans. Microwave Theory Tech. MTT-32, 1598–1609 (1984).
[CrossRef]

W. R. Passchier, D. D. Honijk, M. Mandel, and M. N. Afsar, “A new method for the determination of complex refractive index spectra of transparent solids in the far-infrared spectral region; results of pure silicon and crystals quartz,” J. Phys. D 10, 509–517 (1977).
[CrossRef]

Asfar, M. N.

M. N. Asfar and H. Chi, “Millimeter wave complex refractive index, complex dielectric permittivity and loss tangent of extra high purity and compensated silicon,” Int. J. Infrared Millim. Waves 15, 1181–1188 (1994).
[CrossRef]

Birch, J. R.

J. R. Birch, “The absolute determination of complex reflectivity,” Infrared Phys. 18, 613–620 (1978).
[CrossRef]

Cheville, R. A.

Chi, H.

M. N. Asfar and H. Chi, “Millimeter wave complex refractive index, complex dielectric permittivity and loss tangent of extra high purity and compensated silicon,” Int. J. Infrared Millim. Waves 15, 1181–1188 (1994).
[CrossRef]

Coutaz, J.-L.

Dave, H.

J. M. Dutta, C. R. Jones, and H. Dave, “Complex dielectric constants for selected near-millimeter-wave materials at 245 GHz,” IEEE Trans. Microwave Theory Tech. MTT-34, 932–936 (1986).
[CrossRef]

Dutta, J. M.

J. M. Dutta, C. R. Jones, and H. Dave, “Complex dielectric constants for selected near-millimeter-wave materials at 245 GHz,” IEEE Trans. Microwave Theory Tech. MTT-34, 932–936 (1986).
[CrossRef]

Duvillaret, L.

Fattinger, Ch.

Garet, F.

Grischkowsky, D.

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76, 1987–1989 (2000).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000).
[CrossRef]

Tae-In Jeon and D. Grischkowsky, “Observation of a Cole–Davidson type complex conductivity in the limit of very low carrier densities in doped silicon,” Appl. Phys. Lett. 72, 2259–2261 (1998).
[CrossRef]

Tae-In Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78, 1106–1109 (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]

M. van Exter and D. Grischkowsky, “Carrier dynamics of electrons and holes in moderately-doped silicon,” Phys. Rev. B 41, 12140–12149 (1990).
[CrossRef]

Harmon, S. A.

Honijk, D. D.

W. R. Passchier, D. D. Honijk, M. Mandel, and M. N. Afsar, “A new method for the determination of complex refractive index spectra of transparent solids in the far-infrared spectral region; results of pure silicon and crystals quartz,” J. Phys. D 10, 509–517 (1977).
[CrossRef]

Ikawa, S.

T. Ohba and S. Ikawa, “Far-infrared absorption of silicon crystals,” J. Appl. Phys. 64, 4141–4143 (1988).
[CrossRef]

Jamison, S. P.

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76, 1987–1989 (2000).
[CrossRef]

Jeon, Tae-In

Tae-In Jeon and D. Grischkowsky, “Observation of a Cole–Davidson type complex conductivity in the limit of very low carrier densities in doped silicon,” Appl. Phys. Lett. 72, 2259–2261 (1998).
[CrossRef]

Tae-In Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78, 1106–1109 (1997).
[CrossRef]

Jones, C. R.

J. M. Dutta, C. R. Jones, and H. Dave, “Complex dielectric constants for selected near-millimeter-wave materials at 245 GHz,” IEEE Trans. Microwave Theory Tech. MTT-34, 932–936 (1986).
[CrossRef]

Keiding, S.

Loewenstein, E. V.

Mandel, M.

W. R. Passchier, D. D. Honijk, M. Mandel, and M. N. Afsar, “A new method for the determination of complex refractive index spectra of transparent solids in the far-infrared spectral region; results of pure silicon and crystals quartz,” J. Phys. D 10, 509–517 (1977).
[CrossRef]

Maslin, K. A.

A. K. Wan Abdullah, K. A. Maslin, and T. J. Parker, “Observation of two-phonon difference bands in the FIR transmission spectrum of Si,” Infrared Phys. 24, 185–188 (1984).
[CrossRef]

McGowan, R. W.

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76, 1987–1989 (2000).
[CrossRef]

Mendis, R.

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000).
[CrossRef]

Morgan, R. L.

Ohba, T.

T. Ohba and S. Ikawa, “Far-infrared absorption of silicon crystals,” J. Appl. Phys. 64, 4141–4143 (1988).
[CrossRef]

Parker, T. J.

A. K. Wan Abdullah, K. A. Maslin, and T. J. Parker, “Observation of two-phonon difference bands in the FIR transmission spectrum of Si,” Infrared Phys. 24, 185–188 (1984).
[CrossRef]

Passchier, W. R.

W. R. Passchier, D. D. Honijk, M. Mandel, and M. N. Afsar, “A new method for the determination of complex refractive index spectra of transparent solids in the far-infrared spectral region; results of pure silicon and crystals quartz,” J. Phys. D 10, 509–517 (1977).
[CrossRef]

Randall, C. M.

Rawcliffe, R. D.

Reiten, M. T.

Seeger, K.

K. Seeger, “Microwave dielectric constants of silicon, gallium arsenide, and quartz,” J. Appl. Phys. 63, 5439–5443 (1988).
[CrossRef]

Smith, D. R.

van Exter, M.

Wan Abdullah, A. K.

A. K. Wan Abdullah, K. A. Maslin, and T. J. Parker, “Observation of two-phonon difference bands in the FIR transmission spectrum of Si,” Infrared Phys. 24, 185–188 (1984).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76, 1987–1989 (2000).
[CrossRef]

Tae-In Jeon and D. Grischkowsky, “Observation of a Cole–Davidson type complex conductivity in the limit of very low carrier densities in doped silicon,” Appl. Phys. Lett. 72, 2259–2261 (1998).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

M. N. Afsar, “Dielectric measurements of millimeter-wave materials,” IEEE Trans. Microwave Theory Tech. MTT-32, 1598–1609 (1984).
[CrossRef]

J. M. Dutta, C. R. Jones, and H. Dave, “Complex dielectric constants for selected near-millimeter-wave materials at 245 GHz,” IEEE Trans. Microwave Theory Tech. MTT-34, 932–936 (1986).
[CrossRef]

Infrared Phys. (2)

J. R. Birch, “The absolute determination of complex reflectivity,” Infrared Phys. 18, 613–620 (1978).
[CrossRef]

A. K. Wan Abdullah, K. A. Maslin, and T. J. Parker, “Observation of two-phonon difference bands in the FIR transmission spectrum of Si,” Infrared Phys. 24, 185–188 (1984).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

M. N. Asfar and H. Chi, “Millimeter wave complex refractive index, complex dielectric permittivity and loss tangent of extra high purity and compensated silicon,” Int. J. Infrared Millim. Waves 15, 1181–1188 (1994).
[CrossRef]

J. Appl. Phys. (3)

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000).
[CrossRef]

T. Ohba and S. Ikawa, “Far-infrared absorption of silicon crystals,” J. Appl. Phys. 64, 4141–4143 (1988).
[CrossRef]

K. Seeger, “Microwave dielectric constants of silicon, gallium arsenide, and quartz,” J. Appl. Phys. 63, 5439–5443 (1988).
[CrossRef]

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

J. Phys. D (1)

W. R. Passchier, D. D. Honijk, M. Mandel, and M. N. Afsar, “A new method for the determination of complex refractive index spectra of transparent solids in the far-infrared spectral region; results of pure silicon and crystals quartz,” J. Phys. D 10, 509–517 (1977).
[CrossRef]

Phys. Rev. B (1)

M. van Exter and D. Grischkowsky, “Carrier dynamics of electrons and holes in moderately-doped silicon,” Phys. Rev. B 41, 12140–12149 (1990).
[CrossRef]

Phys. Rev. Lett. (1)

Tae-In Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78, 1106–1109 (1997).
[CrossRef]

Other (1)

J. Lesurf, Millimeter-Wave Optics, Devices and Systems (Hilger, Bristol, UK, 1990).

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Figures (6)

Fig. 1
Fig. 1

The optoelectronic THz-TDS confocal quasi-optic system.

Fig. 2
Fig. 2

(a) Measured transmitted pulses 1–3. (b) Corresponding amplitude spectra for pulses 1–3.

Fig. 3
Fig. 3

(a) Measured transmitted pulses: Pulse 1, (Pulse 2)/(t12t21), (Pulse 3)/(t12r21r21t21). (b) Amplitude spectra for Pulse 1, (Pulse 2)/(t12t21), (Pulse 3)/(t12r21r21t21).

Fig. 4
Fig. 4

Measured indices of refraction, n21 (open circles), n31 (dots), n32 (connected dots). These measurements are considered accurate between 0.5 THz and 4.5 THz.

Fig. 5
Fig. 5

Measured power absorption coefficients α21(ω) open circles, α31(ω) dots, and α32(ω) connected dots. These measurements are considered accurate between 0.5 THz and 4.5 THz.

Fig. 6
Fig. 6

(a) Upper limit α32(ω) to the power absorption coefficient α(ω) of float-zone, high-resistivity silicon. The estimated experimental error in α32(ω) is ±0.0015 cm-1 in the region from 0.5 to 2.7 THz, ±0.003 cm-1 in the region from 2.7 to 4 THz, and ±0.006 cm-1 in the region from 4 to 4.5 THz. These frequency regions are located between the vertical dashed lines together with the corresponding calibration points centered in the circles indicating the experimental error. (b) Measured index of refraction n32(ω). The estimated experimental error of ±0.00001 for the frequency dependence of the index of refraction n32(ω) is smaller than the data-point diameters shown in Fig. 6(b); although because of possible unresolved frequency-independent, systematic experimental effects, the entire curve shown in Fig. 6(b) could move up or down by ±0.0002. These measurements are considered accurate between 0.5 THz and 4.5 THz.

Equations (18)

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A1(ω)=|A1(ω)|exp[iϕ1(ω)]=Ao(ω)U1(ω)exp[iko(ω)L],
A2(ω)=|A2(ω)|exp[iϕ2(ω)]=Ao(ω)U2(ω)t12t21 exp[-α(ω)L/2]exp[ik(ω)L],
A3(ω)=|A3(ω)|exp[iϕ3(ω)]=Ao(ω)U3(ω)t12r21r21t21 exp[-3α(ω)L/2]×exp[i3k(ω)L],
A2(ω)/A1(ω)=[U2(ω)/U1(ω)]t12t21 exp[-α(ω)L/2]×exp[i(k(ω)-ko)L],
A3(ω)/A1(ω)=[U3(ω)/U1(ω)]t12r21r21t21×exp[-3α(ω)L/2]×exp[i(3k(ω)-ko)L],
A3(ω)/A2(ω)=[U3(ω)/U2(ω)]r21r21 exp[-α(ω)L]×exp[i2k(ω)L].
[ϕ2(ω)-ϕ1(ω)](2π/λo)[n21(ω)-nAir(THz)]L-ωΔT21,
[ϕ3(ω)-ϕ1(ω)](2π/λo)[3n31(ω)-nAir(THz)]L-ωΔT31,
[ϕ3(ω)-ϕ2(ω)](2π/λo)2n32(ω)L-ωΔT32,
α(ω)=(2/L)Ln|U2(ω)/U1(ω)|-(2/L)Ln[|A2(ω)/A1(ω)|/(t12t21)],
α(ω)=[2/(3L)]Ln|U3(ω)/U1(ω)|-[2/(3L)]Ln[|A3(ω)/A1(ω)|/(t12r21r21t21)],
α(ω)=(1/L)Ln|U3(ω)/U2(ω)|-(1/L)Ln[|A3(ω)/A2(ω)|/(r21r21)].
|A2(ω)/A1(ω)|t12t21 exp[-α21(ω)L/2],
|A3(ω)/A1(ω)|t12r21r21t21 exp[-3α31(ω)L/2],
|A3(ω)/A2(ω)|r21r21 exp[-α32(ω)L].
α21(ω)=-(2/L)Ln)[|A2(ω)/A1(ω)|/(t12t21)]=α(ω)-(2/L)Ln|U2(ω)/U1(ω)|,
α31(ω)=-[2/(3L)]Ln[|A3(ω)/A1(ω)|/(t12r21r21t21)]=α(ω)-[2/(3L)]Ln|U3(ω)/U1(ω)|,
α32(ω)=-(1/L)Ln[|A3(ω)/A2(ω)|/(r21r21)]=α(ω)-(1/L)Ln|U3(ω)/U2(ω)|.

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