Abstract

A terahertz time-domain spectrometer is employed to study different birefringent samples. We develop a method based on the temporal waveform and the impulse response of a sample to map the anisotropy of their inner structure. To validate our algorithm, we study the polarization-affecting structure of various classes of materials such as crystals, plastics, and natural products. Among all samples we observe the largest birefringence for a rutile crystal with Δn=3.3 at 1THz.

© 2009 Optical Society of America

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References

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  1. D. Grischkowsky, S. Keiding, M. van Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with THz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006-2015 (1990).
    [CrossRef]
  2. N. B. Singh, T. B. Norris, T. Buma, R. N. Singh, M. Gottlieb, D. Suhre, and J. J. Hawkins, “Properties of nonlinear optical crystals in the terahertz wavelength region,” Opt. Eng. 45, 094002 (2006).
    [CrossRef]
  3. G. Luessem, D. Turchinovich, P. Knobloch, and M. Koch, “THz time-domain spectroscopy on 4-(trans-4'-pentylcyclohexyl)-benzonitril,” Proc. SPIE 4463, 65 (2001).
    [CrossRef]
  4. T. Tsai, C. Chen, C. Pan, R. Pan, and X.-C. Zhang, “Terahertz time-domain spectroscopy studies of the optical constants of the nematic liquid crystal 5CB,” Appl. Opt. 42, 2372-2376 (2003).
    [CrossRef] [PubMed]
  5. R. Pan, T. Tsai, C. Chen, C. Wang, and C. Pan, “The refractive indices of nematic liquid crystal 4'-n-pentyl-4- cyanobiphenyl in the THz frequency range,” Mol. Cryst. Liq. Cryst. 409, 137-144 (2004).
    [CrossRef]
  6. F. Rutz, T. Hasek, M. Koch, H. Richter, and U. Ewert, “Terahertz birefringence of liquid crystal polymers,” Appl. Phys. Lett. 89, 221911 (2006).
    [CrossRef]
  7. M. Reid and R. Fedosejevs, “Terahertz birefringence and attenuation properties of wood and paper,” Appl. Opt. 45, 2766-2772 (2006).
    [CrossRef] [PubMed]
  8. T. Tsai, C. Chen, R. Pan, C. Pan, and X.-C. Zhang, “Electrically controlled room temperature terahertz phase shifter with liquid crystal,” IEEE Microw. Wirel. Componen. Lett. 14, 77-79 (2004).
    [CrossRef]
  9. C. Hsieh, R. Pan, T. Tang, H. Chen, and C. Pan, “Voltage-controlled liquid-crystal terahertz phase shifter and quarter-wave plate,” Opt. Lett. 31, 1112-1114 (2006).
    [CrossRef] [PubMed]
  10. C. Hsieh, Y. Lai, R. Pan, and C. Pan, “Polarizing terahertz waves with nematic liquid crystals,” Opt. Lett. 33, 1174-1176 (2008).
    [CrossRef] [PubMed]
  11. N. C. J. Van Der Valk, W. A. M. Van Der Marel, and P. C. M. Planken, “Terahertz polarization imaging,” Opt. Lett. 30, 2802-2804 (2005).
    [CrossRef] [PubMed]
  12. J. Masson and G. Gallot, “Terahertz polarimetry,” in Conference on Lasers and Electro-Optics (CLEO) (OSA, 2005), Vol. 3, pp. 2111-2113.
    [CrossRef]
  13. S. Hunsche, D. M. Mittleman, M. Koch, and M. C. Nuss, “New dimensions in T-ray imaging,” IEICE Trans. Electron. E81-C, 269-275 (1998).
  14. I. Pupeza, R. Wilk, and M. Koch, “Highly accurate optical material parameter determination with THz time-domain spectroscopy,” Opt. Express 15, 4335-4350 (2007).
    [CrossRef] [PubMed]
  15. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
    [CrossRef]

2008

2007

2006

C. Hsieh, R. Pan, T. Tang, H. Chen, and C. Pan, “Voltage-controlled liquid-crystal terahertz phase shifter and quarter-wave plate,” Opt. Lett. 31, 1112-1114 (2006).
[CrossRef] [PubMed]

M. Reid and R. Fedosejevs, “Terahertz birefringence and attenuation properties of wood and paper,” Appl. Opt. 45, 2766-2772 (2006).
[CrossRef] [PubMed]

F. Rutz, T. Hasek, M. Koch, H. Richter, and U. Ewert, “Terahertz birefringence of liquid crystal polymers,” Appl. Phys. Lett. 89, 221911 (2006).
[CrossRef]

N. B. Singh, T. B. Norris, T. Buma, R. N. Singh, M. Gottlieb, D. Suhre, and J. J. Hawkins, “Properties of nonlinear optical crystals in the terahertz wavelength region,” Opt. Eng. 45, 094002 (2006).
[CrossRef]

2005

2004

R. Pan, T. Tsai, C. Chen, C. Wang, and C. Pan, “The refractive indices of nematic liquid crystal 4'-n-pentyl-4- cyanobiphenyl in the THz frequency range,” Mol. Cryst. Liq. Cryst. 409, 137-144 (2004).
[CrossRef]

T. Tsai, C. Chen, R. Pan, C. Pan, and X.-C. Zhang, “Electrically controlled room temperature terahertz phase shifter with liquid crystal,” IEEE Microw. Wirel. Componen. Lett. 14, 77-79 (2004).
[CrossRef]

2003

2001

G. Luessem, D. Turchinovich, P. Knobloch, and M. Koch, “THz time-domain spectroscopy on 4-(trans-4'-pentylcyclohexyl)-benzonitril,” Proc. SPIE 4463, 65 (2001).
[CrossRef]

1998

S. Hunsche, D. M. Mittleman, M. Koch, and M. C. Nuss, “New dimensions in T-ray imaging,” IEICE Trans. Electron. E81-C, 269-275 (1998).

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

1990

Buma, T.

N. B. Singh, T. B. Norris, T. Buma, R. N. Singh, M. Gottlieb, D. Suhre, and J. J. Hawkins, “Properties of nonlinear optical crystals in the terahertz wavelength region,” Opt. Eng. 45, 094002 (2006).
[CrossRef]

Chen, C.

T. Tsai, C. Chen, R. Pan, C. Pan, and X.-C. Zhang, “Electrically controlled room temperature terahertz phase shifter with liquid crystal,” IEEE Microw. Wirel. Componen. Lett. 14, 77-79 (2004).
[CrossRef]

R. Pan, T. Tsai, C. Chen, C. Wang, and C. Pan, “The refractive indices of nematic liquid crystal 4'-n-pentyl-4- cyanobiphenyl in the THz frequency range,” Mol. Cryst. Liq. Cryst. 409, 137-144 (2004).
[CrossRef]

T. Tsai, C. Chen, C. Pan, R. Pan, and X.-C. Zhang, “Terahertz time-domain spectroscopy studies of the optical constants of the nematic liquid crystal 5CB,” Appl. Opt. 42, 2372-2376 (2003).
[CrossRef] [PubMed]

Chen, H.

Ewert, U.

F. Rutz, T. Hasek, M. Koch, H. Richter, and U. Ewert, “Terahertz birefringence of liquid crystal polymers,” Appl. Phys. Lett. 89, 221911 (2006).
[CrossRef]

Fattinger, C.

Fedosejevs, R.

Gallot, G.

J. Masson and G. Gallot, “Terahertz polarimetry,” in Conference on Lasers and Electro-Optics (CLEO) (OSA, 2005), Vol. 3, pp. 2111-2113.
[CrossRef]

Gottlieb, M.

N. B. Singh, T. B. Norris, T. Buma, R. N. Singh, M. Gottlieb, D. Suhre, and J. J. Hawkins, “Properties of nonlinear optical crystals in the terahertz wavelength region,” Opt. Eng. 45, 094002 (2006).
[CrossRef]

Grischkowsky, D.

Hasek, T.

F. Rutz, T. Hasek, M. Koch, H. Richter, and U. Ewert, “Terahertz birefringence of liquid crystal polymers,” Appl. Phys. Lett. 89, 221911 (2006).
[CrossRef]

Hawkins, J. J.

N. B. Singh, T. B. Norris, T. Buma, R. N. Singh, M. Gottlieb, D. Suhre, and J. J. Hawkins, “Properties of nonlinear optical crystals in the terahertz wavelength region,” Opt. Eng. 45, 094002 (2006).
[CrossRef]

Hsieh, C.

Hunsche, S.

S. Hunsche, D. M. Mittleman, M. Koch, and M. C. Nuss, “New dimensions in T-ray imaging,” IEICE Trans. Electron. E81-C, 269-275 (1998).

Keiding, S.

Knobloch, P.

G. Luessem, D. Turchinovich, P. Knobloch, and M. Koch, “THz time-domain spectroscopy on 4-(trans-4'-pentylcyclohexyl)-benzonitril,” Proc. SPIE 4463, 65 (2001).
[CrossRef]

Koch, M.

I. Pupeza, R. Wilk, and M. Koch, “Highly accurate optical material parameter determination with THz time-domain spectroscopy,” Opt. Express 15, 4335-4350 (2007).
[CrossRef] [PubMed]

F. Rutz, T. Hasek, M. Koch, H. Richter, and U. Ewert, “Terahertz birefringence of liquid crystal polymers,” Appl. Phys. Lett. 89, 221911 (2006).
[CrossRef]

G. Luessem, D. Turchinovich, P. Knobloch, and M. Koch, “THz time-domain spectroscopy on 4-(trans-4'-pentylcyclohexyl)-benzonitril,” Proc. SPIE 4463, 65 (2001).
[CrossRef]

S. Hunsche, D. M. Mittleman, M. Koch, and M. C. Nuss, “New dimensions in T-ray imaging,” IEICE Trans. Electron. E81-C, 269-275 (1998).

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Lai, Y.

Luessem, G.

G. Luessem, D. Turchinovich, P. Knobloch, and M. Koch, “THz time-domain spectroscopy on 4-(trans-4'-pentylcyclohexyl)-benzonitril,” Proc. SPIE 4463, 65 (2001).
[CrossRef]

Masson, J.

J. Masson and G. Gallot, “Terahertz polarimetry,” in Conference on Lasers and Electro-Optics (CLEO) (OSA, 2005), Vol. 3, pp. 2111-2113.
[CrossRef]

Mittleman, D. M.

S. Hunsche, D. M. Mittleman, M. Koch, and M. C. Nuss, “New dimensions in T-ray imaging,” IEICE Trans. Electron. E81-C, 269-275 (1998).

Norris, T. B.

N. B. Singh, T. B. Norris, T. Buma, R. N. Singh, M. Gottlieb, D. Suhre, and J. J. Hawkins, “Properties of nonlinear optical crystals in the terahertz wavelength region,” Opt. Eng. 45, 094002 (2006).
[CrossRef]

Nuss, M. C.

S. Hunsche, D. M. Mittleman, M. Koch, and M. C. Nuss, “New dimensions in T-ray imaging,” IEICE Trans. Electron. E81-C, 269-275 (1998).

Pan, C.

C. Hsieh, Y. Lai, R. Pan, and C. Pan, “Polarizing terahertz waves with nematic liquid crystals,” Opt. Lett. 33, 1174-1176 (2008).
[CrossRef] [PubMed]

C. Hsieh, R. Pan, T. Tang, H. Chen, and C. Pan, “Voltage-controlled liquid-crystal terahertz phase shifter and quarter-wave plate,” Opt. Lett. 31, 1112-1114 (2006).
[CrossRef] [PubMed]

R. Pan, T. Tsai, C. Chen, C. Wang, and C. Pan, “The refractive indices of nematic liquid crystal 4'-n-pentyl-4- cyanobiphenyl in the THz frequency range,” Mol. Cryst. Liq. Cryst. 409, 137-144 (2004).
[CrossRef]

T. Tsai, C. Chen, R. Pan, C. Pan, and X.-C. Zhang, “Electrically controlled room temperature terahertz phase shifter with liquid crystal,” IEEE Microw. Wirel. Componen. Lett. 14, 77-79 (2004).
[CrossRef]

T. Tsai, C. Chen, C. Pan, R. Pan, and X.-C. Zhang, “Terahertz time-domain spectroscopy studies of the optical constants of the nematic liquid crystal 5CB,” Appl. Opt. 42, 2372-2376 (2003).
[CrossRef] [PubMed]

Pan, R.

C. Hsieh, Y. Lai, R. Pan, and C. Pan, “Polarizing terahertz waves with nematic liquid crystals,” Opt. Lett. 33, 1174-1176 (2008).
[CrossRef] [PubMed]

C. Hsieh, R. Pan, T. Tang, H. Chen, and C. Pan, “Voltage-controlled liquid-crystal terahertz phase shifter and quarter-wave plate,” Opt. Lett. 31, 1112-1114 (2006).
[CrossRef] [PubMed]

R. Pan, T. Tsai, C. Chen, C. Wang, and C. Pan, “The refractive indices of nematic liquid crystal 4'-n-pentyl-4- cyanobiphenyl in the THz frequency range,” Mol. Cryst. Liq. Cryst. 409, 137-144 (2004).
[CrossRef]

T. Tsai, C. Chen, R. Pan, C. Pan, and X.-C. Zhang, “Electrically controlled room temperature terahertz phase shifter with liquid crystal,” IEEE Microw. Wirel. Componen. Lett. 14, 77-79 (2004).
[CrossRef]

T. Tsai, C. Chen, C. Pan, R. Pan, and X.-C. Zhang, “Terahertz time-domain spectroscopy studies of the optical constants of the nematic liquid crystal 5CB,” Appl. Opt. 42, 2372-2376 (2003).
[CrossRef] [PubMed]

Planken, P. C. M.

Pupeza, I.

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Reid, M.

Richter, H.

F. Rutz, T. Hasek, M. Koch, H. Richter, and U. Ewert, “Terahertz birefringence of liquid crystal polymers,” Appl. Phys. Lett. 89, 221911 (2006).
[CrossRef]

Rutz, F.

F. Rutz, T. Hasek, M. Koch, H. Richter, and U. Ewert, “Terahertz birefringence of liquid crystal polymers,” Appl. Phys. Lett. 89, 221911 (2006).
[CrossRef]

Singh, N. B.

N. B. Singh, T. B. Norris, T. Buma, R. N. Singh, M. Gottlieb, D. Suhre, and J. J. Hawkins, “Properties of nonlinear optical crystals in the terahertz wavelength region,” Opt. Eng. 45, 094002 (2006).
[CrossRef]

Singh, R. N.

N. B. Singh, T. B. Norris, T. Buma, R. N. Singh, M. Gottlieb, D. Suhre, and J. J. Hawkins, “Properties of nonlinear optical crystals in the terahertz wavelength region,” Opt. Eng. 45, 094002 (2006).
[CrossRef]

Suhre, D.

N. B. Singh, T. B. Norris, T. Buma, R. N. Singh, M. Gottlieb, D. Suhre, and J. J. Hawkins, “Properties of nonlinear optical crystals in the terahertz wavelength region,” Opt. Eng. 45, 094002 (2006).
[CrossRef]

Tang, T.

Tsai, T.

R. Pan, T. Tsai, C. Chen, C. Wang, and C. Pan, “The refractive indices of nematic liquid crystal 4'-n-pentyl-4- cyanobiphenyl in the THz frequency range,” Mol. Cryst. Liq. Cryst. 409, 137-144 (2004).
[CrossRef]

T. Tsai, C. Chen, R. Pan, C. Pan, and X.-C. Zhang, “Electrically controlled room temperature terahertz phase shifter with liquid crystal,” IEEE Microw. Wirel. Componen. Lett. 14, 77-79 (2004).
[CrossRef]

T. Tsai, C. Chen, C. Pan, R. Pan, and X.-C. Zhang, “Terahertz time-domain spectroscopy studies of the optical constants of the nematic liquid crystal 5CB,” Appl. Opt. 42, 2372-2376 (2003).
[CrossRef] [PubMed]

Turchinovich, D.

G. Luessem, D. Turchinovich, P. Knobloch, and M. Koch, “THz time-domain spectroscopy on 4-(trans-4'-pentylcyclohexyl)-benzonitril,” Proc. SPIE 4463, 65 (2001).
[CrossRef]

Van Der Marel, W. A. M.

Van Der Valk, N. C. J.

van Exter, M.

Wang, C.

R. Pan, T. Tsai, C. Chen, C. Wang, and C. Pan, “The refractive indices of nematic liquid crystal 4'-n-pentyl-4- cyanobiphenyl in the THz frequency range,” Mol. Cryst. Liq. Cryst. 409, 137-144 (2004).
[CrossRef]

Wilk, R.

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Zhang, X.-C.

T. Tsai, C. Chen, R. Pan, C. Pan, and X.-C. Zhang, “Electrically controlled room temperature terahertz phase shifter with liquid crystal,” IEEE Microw. Wirel. Componen. Lett. 14, 77-79 (2004).
[CrossRef]

T. Tsai, C. Chen, C. Pan, R. Pan, and X.-C. Zhang, “Terahertz time-domain spectroscopy studies of the optical constants of the nematic liquid crystal 5CB,” Appl. Opt. 42, 2372-2376 (2003).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

F. Rutz, T. Hasek, M. Koch, H. Richter, and U. Ewert, “Terahertz birefringence of liquid crystal polymers,” Appl. Phys. Lett. 89, 221911 (2006).
[CrossRef]

IEEE Microw. Wirel. Componen. Lett.

T. Tsai, C. Chen, R. Pan, C. Pan, and X.-C. Zhang, “Electrically controlled room temperature terahertz phase shifter with liquid crystal,” IEEE Microw. Wirel. Componen. Lett. 14, 77-79 (2004).
[CrossRef]

IEICE Trans. Electron.

S. Hunsche, D. M. Mittleman, M. Koch, and M. C. Nuss, “New dimensions in T-ray imaging,” IEICE Trans. Electron. E81-C, 269-275 (1998).

J. Opt. Soc. Am. B

Mol. Cryst. Liq. Cryst.

R. Pan, T. Tsai, C. Chen, C. Wang, and C. Pan, “The refractive indices of nematic liquid crystal 4'-n-pentyl-4- cyanobiphenyl in the THz frequency range,” Mol. Cryst. Liq. Cryst. 409, 137-144 (2004).
[CrossRef]

Opt. Eng.

N. B. Singh, T. B. Norris, T. Buma, R. N. Singh, M. Gottlieb, D. Suhre, and J. J. Hawkins, “Properties of nonlinear optical crystals in the terahertz wavelength region,” Opt. Eng. 45, 094002 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

G. Luessem, D. Turchinovich, P. Knobloch, and M. Koch, “THz time-domain spectroscopy on 4-(trans-4'-pentylcyclohexyl)-benzonitril,” Proc. SPIE 4463, 65 (2001).
[CrossRef]

SIAM J. Optim.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Other

J. Masson and G. Gallot, “Terahertz polarimetry,” in Conference on Lasers and Electro-Optics (CLEO) (OSA, 2005), Vol. 3, pp. 2111-2113.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Decomposition of incoming terahertz field and composition of outgoing terahertz field.

Fig. 2
Fig. 2

Refractive index and absorption coefficient of (a), (b) rutile and (c), (d)  a yucca leaf.

Fig. 3
Fig. 3

Terahertz time-domain signals of (a)–(c) an orientated LCP test bar and (d) orientation analysis.

Fig. 4
Fig. 4

Terahertz time-domain signals of (a) an orientated yucca leaf and (c) a rutile crystal and (b), (d) orientation analysis for both samples.

Fig. 5
Fig. 5

HSB sample. (a) Angle-resolved measurement. (b) Photograph.

Fig. 6
Fig. 6

Accuracy of the advanced algorithm. Influence of noise on (a) the calculated angle and (b) the correlation coefficient for varying angles.

Equations (14)

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

e ( t ) = sin 2 ( φ ) · e OA ( t ) + cos 2 ( φ ) · e EA ( t ) .
φ = atan ( e ^ 1 e ^ 2 ) .
H ( ω ) = 4 n ˜ ( 1 + n ˜ ) 2 · exp ( i · ω ( n ˜ 1 ) c 0 · L ) · [ k = 0 M ( exp ( i · 2 · ω n ˜ c 0 · L ) · ( 1 n ˜ ) ( 1 + n ˜ ) · ( 1 n ˜ ) ( 1 + n ˜ ) ) k ] .
h ( t ) = F 1 { H ( ω ) } .
e OA ( t ) = e ref ( t ) * h OA ( t ) ,
e EA ( t ) = e ref ( t ) * h EA ( t ) ,
E OA ( ω ) = E ref ( ω ) · H OA ( ω ) ,
E EA ( ω ) = E ref ( ω ) · H EA ( ω ) .
ξ = sin 2 ( φ ) cos 2 ( φ ) = 1 ξ ,
e sim ( t ) = ξ · e OA ( t ) + ( 1 ξ ) · e EA ( t ) .
E sim ( ω ) = ξ · E OA ( ω ) + ( 1 ξ ) · E EA ( ω ) .
e error ( t ) e sim ( t ) e meas ( t ) .
sse ( ξ ) k ( e error ( t k ) ) 2 .
R 2 1 sse k ( e meas ( t k ) e ¯ meas ) 2 .

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