Abstract

Terahertz spectroscopic measurements are usually performed in focused beam geometry while the standard routine for the retrieval of the sample refractive index assumes plane-wave approximation. In this paper we propose a model for the transmission function which accounts for spatially limited Gaussian terahertz beams. We demonstrate experimentally its validity and applicability for an accurate extraction of the refractive index from experimental data.

© 2010 OSA

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References

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  1. 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(10), 2006 (1990).
    [CrossRef]
  2. M. C. Nuss, K. W. Goossen, P. M. Mankiewich, and M. L. O'Malley, “Terahertz surface impedance of thin YBa2Cu3O7 superconducting films,” Appl. Phys. Lett. 58(22), 2561 (1991).
    [CrossRef]
  3. M. Misra, K. Kotani, I. Kawayama, H. Murakami, and M. Tonouchi, “Observation of TO1 soft mode in SrTiO3 films by terahertz time domain spectroscopy,” Appl. Phys. Lett. 87(18), 182909 (2005).
    [CrossRef]
  4. C. Kadlec, F. Kadlec, H. Němec, P. Kužel, J. Schubert, and G. Panaitov, “High tunability of the soft mode in strained SrTiO3/DyScO3 multilayers,” J. Phys. Condens. Matter 21(11), 115902 (2009).
    [CrossRef] [PubMed]
  5. 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(2), 409–415 (1999).
    [CrossRef]
  6. L. Duvillaret, F. Garet, and J.-L. Coutaz, “A Reliable Method for Extraction of Material Parameters in Terahertz Time-Domain Spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2(3), 739–746 (1996).
    [CrossRef]
  7. I. Pupeza, R. Wilk, and M. Koch, “Highly accurate optical material parameter determination with THz time-domain spectroscopy,” Opt. Express 15(7), 4335–4350 (2007).
    [CrossRef] [PubMed]
  8. S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23(5), 385–387 (1998).
    [CrossRef]
  9. P. Kužel, M. A. Khazan, and J. Kroupa, “Spatio-temporal transformations of ultrashort terahertz pulses,” J. Opt. Soc. Am. B 16(10), 1795–1800 (1999).
    [CrossRef]
  10. 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(10), 2215 (2003).
    [CrossRef]
  11. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4(12), 1562 (1965).
    [CrossRef]
  12. A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, “High-intensity terahertz radiation from a microstructured large-area photoconductor,” Appl. Phys. Lett. 86(12), 121114 (2005).
    [CrossRef]
  13. J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Am. B 21(7), 1379–1386 (2004).
    [CrossRef]
  14. J. Petzelt, P. Kužel, I. Rychetský, A. Pashkin, and T. Ostapchuk, “Dielectric response of soft modes in ferroelectric thin films,” Ferroelectrics 288(1), 169–185 (2003).
    [CrossRef]
  15. C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
    [CrossRef]

2009 (2)

C. Kadlec, F. Kadlec, H. Němec, P. Kužel, J. Schubert, and G. Panaitov, “High tunability of the soft mode in strained SrTiO3/DyScO3 multilayers,” J. Phys. Condens. Matter 21(11), 115902 (2009).
[CrossRef] [PubMed]

C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
[CrossRef]

2007 (1)

2005 (2)

A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, “High-intensity terahertz radiation from a microstructured large-area photoconductor,” Appl. Phys. Lett. 86(12), 121114 (2005).
[CrossRef]

M. Misra, K. Kotani, I. Kawayama, H. Murakami, and M. Tonouchi, “Observation of TO1 soft mode in SrTiO3 films by terahertz time domain spectroscopy,” Appl. Phys. Lett. 87(18), 182909 (2005).
[CrossRef]

2004 (1)

2003 (2)

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(10), 2215 (2003).
[CrossRef]

J. Petzelt, P. Kužel, I. Rychetský, A. Pashkin, and T. Ostapchuk, “Dielectric response of soft modes in ferroelectric thin films,” Ferroelectrics 288(1), 169–185 (2003).
[CrossRef]

1999 (2)

1998 (1)

1996 (1)

L. Duvillaret, F. Garet, and J.-L. Coutaz, “A Reliable Method for Extraction of Material Parameters in Terahertz Time-Domain Spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2(3), 739–746 (1996).
[CrossRef]

1991 (1)

M. C. Nuss, K. W. Goossen, P. M. Mankiewich, and M. L. O'Malley, “Terahertz surface impedance of thin YBa2Cu3O7 superconducting films,” Appl. Phys. Lett. 58(22), 2561 (1991).
[CrossRef]

1990 (1)

1965 (1)

Cheville, R. A.

Coutaz, J.-L.

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(2), 409–415 (1999).
[CrossRef]

L. Duvillaret, F. Garet, and J.-L. Coutaz, “A Reliable Method for Extraction of Material Parameters in Terahertz Time-Domain Spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2(3), 739–746 (1996).
[CrossRef]

Dai, J.

Dekorsy, T.

A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, “High-intensity terahertz radiation from a microstructured large-area photoconductor,” Appl. Phys. Lett. 86(12), 121114 (2005).
[CrossRef]

Dreyhaupt, A.

A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, “High-intensity terahertz radiation from a microstructured large-area photoconductor,” Appl. Phys. Lett. 86(12), 121114 (2005).
[CrossRef]

Duvillaret, L.

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(2), 409–415 (1999).
[CrossRef]

L. Duvillaret, F. Garet, and J.-L. Coutaz, “A Reliable Method for Extraction of Material Parameters in Terahertz Time-Domain Spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2(3), 739–746 (1996).
[CrossRef]

Fattinger, Ch.

Feng, S.

Garet, F.

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(2), 409–415 (1999).
[CrossRef]

L. Duvillaret, F. Garet, and J.-L. Coutaz, “A Reliable Method for Extraction of Material Parameters in Terahertz Time-Domain Spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2(3), 739–746 (1996).
[CrossRef]

Goossen, K. W.

M. C. Nuss, K. W. Goossen, P. M. Mankiewich, and M. L. O'Malley, “Terahertz surface impedance of thin YBa2Cu3O7 superconducting films,” Appl. Phys. Lett. 58(22), 2561 (1991).
[CrossRef]

Grischkowsky, D.

Harmon, S. A.

Hellwarth, R. W.

Helm, M.

A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, “High-intensity terahertz radiation from a microstructured large-area photoconductor,” Appl. Phys. Lett. 86(12), 121114 (2005).
[CrossRef]

Hlinka, J.

C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
[CrossRef]

Kadlec, C.

C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
[CrossRef]

C. Kadlec, F. Kadlec, H. Němec, P. Kužel, J. Schubert, and G. Panaitov, “High tunability of the soft mode in strained SrTiO3/DyScO3 multilayers,” J. Phys. Condens. Matter 21(11), 115902 (2009).
[CrossRef] [PubMed]

Kadlec, F.

C. Kadlec, F. Kadlec, H. Němec, P. Kužel, J. Schubert, and G. Panaitov, “High tunability of the soft mode in strained SrTiO3/DyScO3 multilayers,” J. Phys. Condens. Matter 21(11), 115902 (2009).
[CrossRef] [PubMed]

C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
[CrossRef]

Kawayama, I.

M. Misra, K. Kotani, I. Kawayama, H. Murakami, and M. Tonouchi, “Observation of TO1 soft mode in SrTiO3 films by terahertz time domain spectroscopy,” Appl. Phys. Lett. 87(18), 182909 (2005).
[CrossRef]

Keiding, S.

Khazan, M. A.

Koch, M.

Kogelnik, H.

Kotani, K.

M. Misra, K. Kotani, I. Kawayama, H. Murakami, and M. Tonouchi, “Observation of TO1 soft mode in SrTiO3 films by terahertz time domain spectroscopy,” Appl. Phys. Lett. 87(18), 182909 (2005).
[CrossRef]

Kroupa, J.

Kužel, P.

C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
[CrossRef]

C. Kadlec, F. Kadlec, H. Němec, P. Kužel, J. Schubert, and G. Panaitov, “High tunability of the soft mode in strained SrTiO3/DyScO3 multilayers,” J. Phys. Condens. Matter 21(11), 115902 (2009).
[CrossRef] [PubMed]

J. Petzelt, P. Kužel, I. Rychetský, A. Pashkin, and T. Ostapchuk, “Dielectric response of soft modes in ferroelectric thin films,” Ferroelectrics 288(1), 169–185 (2003).
[CrossRef]

P. Kužel, M. A. Khazan, and J. Kroupa, “Spatio-temporal transformations of ultrashort terahertz pulses,” J. Opt. Soc. Am. B 16(10), 1795–1800 (1999).
[CrossRef]

Mankiewich, P. M.

M. C. Nuss, K. W. Goossen, P. M. Mankiewich, and M. L. O'Malley, “Terahertz surface impedance of thin YBa2Cu3O7 superconducting films,” Appl. Phys. Lett. 58(22), 2561 (1991).
[CrossRef]

Misra, M.

M. Misra, K. Kotani, I. Kawayama, H. Murakami, and M. Tonouchi, “Observation of TO1 soft mode in SrTiO3 films by terahertz time domain spectroscopy,” Appl. Phys. Lett. 87(18), 182909 (2005).
[CrossRef]

Murakami, H.

M. Misra, K. Kotani, I. Kawayama, H. Murakami, and M. Tonouchi, “Observation of TO1 soft mode in SrTiO3 films by terahertz time domain spectroscopy,” Appl. Phys. Lett. 87(18), 182909 (2005).
[CrossRef]

Nemec, H.

C. Kadlec, F. Kadlec, H. Němec, P. Kužel, J. Schubert, and G. Panaitov, “High tunability of the soft mode in strained SrTiO3/DyScO3 multilayers,” J. Phys. Condens. Matter 21(11), 115902 (2009).
[CrossRef] [PubMed]

C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
[CrossRef]

Nuss, M. C.

M. C. Nuss, K. W. Goossen, P. M. Mankiewich, and M. L. O'Malley, “Terahertz surface impedance of thin YBa2Cu3O7 superconducting films,” Appl. Phys. Lett. 58(22), 2561 (1991).
[CrossRef]

O'Malley, M. L.

M. C. Nuss, K. W. Goossen, P. M. Mankiewich, and M. L. O'Malley, “Terahertz surface impedance of thin YBa2Cu3O7 superconducting films,” Appl. Phys. Lett. 58(22), 2561 (1991).
[CrossRef]

Ostapchuk, T.

J. Petzelt, P. Kužel, I. Rychetský, A. Pashkin, and T. Ostapchuk, “Dielectric response of soft modes in ferroelectric thin films,” Ferroelectrics 288(1), 169–185 (2003).
[CrossRef]

Panaitov, G.

C. Kadlec, F. Kadlec, H. Němec, P. Kužel, J. Schubert, and G. Panaitov, “High tunability of the soft mode in strained SrTiO3/DyScO3 multilayers,” J. Phys. Condens. Matter 21(11), 115902 (2009).
[CrossRef] [PubMed]

C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
[CrossRef]

Pashkin, A.

J. Petzelt, P. Kužel, I. Rychetský, A. Pashkin, and T. Ostapchuk, “Dielectric response of soft modes in ferroelectric thin films,” Ferroelectrics 288(1), 169–185 (2003).
[CrossRef]

Petzelt, J.

J. Petzelt, P. Kužel, I. Rychetský, A. Pashkin, and T. Ostapchuk, “Dielectric response of soft modes in ferroelectric thin films,” Ferroelectrics 288(1), 169–185 (2003).
[CrossRef]

Pupeza, I.

Reiten, M. T.

Rychetský, I.

J. Petzelt, P. Kužel, I. Rychetský, A. Pashkin, and T. Ostapchuk, “Dielectric response of soft modes in ferroelectric thin films,” Ferroelectrics 288(1), 169–185 (2003).
[CrossRef]

Schubert, J.

C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
[CrossRef]

C. Kadlec, F. Kadlec, H. Němec, P. Kužel, J. Schubert, and G. Panaitov, “High tunability of the soft mode in strained SrTiO3/DyScO3 multilayers,” J. Phys. Condens. Matter 21(11), 115902 (2009).
[CrossRef] [PubMed]

Skoromets, V.

C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
[CrossRef]

Tonouchi, M.

M. Misra, K. Kotani, I. Kawayama, H. Murakami, and M. Tonouchi, “Observation of TO1 soft mode in SrTiO3 films by terahertz time domain spectroscopy,” Appl. Phys. Lett. 87(18), 182909 (2005).
[CrossRef]

van Exter, M.

Wilk, R.

Winful, H. G.

Winnerl, S.

A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, “High-intensity terahertz radiation from a microstructured large-area photoconductor,” Appl. Phys. Lett. 86(12), 121114 (2005).
[CrossRef]

Zhang, J.

Zhang, W.

Appl. Opt. (2)

Appl. Phys. Lett. (3)

M. C. Nuss, K. W. Goossen, P. M. Mankiewich, and M. L. O'Malley, “Terahertz surface impedance of thin YBa2Cu3O7 superconducting films,” Appl. Phys. Lett. 58(22), 2561 (1991).
[CrossRef]

M. Misra, K. Kotani, I. Kawayama, H. Murakami, and M. Tonouchi, “Observation of TO1 soft mode in SrTiO3 films by terahertz time domain spectroscopy,” Appl. Phys. Lett. 87(18), 182909 (2005).
[CrossRef]

A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, “High-intensity terahertz radiation from a microstructured large-area photoconductor,” Appl. Phys. Lett. 86(12), 121114 (2005).
[CrossRef]

Ferroelectrics (1)

J. Petzelt, P. Kužel, I. Rychetský, A. Pashkin, and T. Ostapchuk, “Dielectric response of soft modes in ferroelectric thin films,” Ferroelectrics 288(1), 169–185 (2003).
[CrossRef]

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

L. Duvillaret, F. Garet, and J.-L. Coutaz, “A Reliable Method for Extraction of Material Parameters in Terahertz Time-Domain Spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2(3), 739–746 (1996).
[CrossRef]

J. Opt. Soc. Am. B (4)

J. Phys. Condens. Matter (1)

C. Kadlec, F. Kadlec, H. Němec, P. Kužel, J. Schubert, and G. Panaitov, “High tunability of the soft mode in strained SrTiO3/DyScO3 multilayers,” J. Phys. Condens. Matter 21(11), 115902 (2009).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

C. Kadlec, V. Skoromets, F. Kadlec, H. Němec, J. Hlinka, J. Schubert, G. Panaitov, and P. Kužel, “Temperature and electric field tuning of the ferroelectric soft mode in a strained SrTiO3/DyScO3 heterostructure,” Phys. Rev. B 80(17), 174116 (2009).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Scheme of an experimental setup containing 3 focal planes (for an emitter, sample and sensor). The displacement Δ of an object plane yields a displacement Δ' of its image. The arrows indicate the shifts of foci.

Fig. 2
Fig. 2

Experimental setup. Emitter is either an interdigitated semiconductor photoswitch TeraSED (Setup I) or a 1 mm thick ZnTe crystal (Setup II). The focal length of ellipsoidal mirrors is f = 7.5 cm.

Fig. 3
Fig. 3

Experimentally determined β(ν) for our experimental setups by measuring the Gouy shift. Insets: waist size and confocal parameter.

Fig. 4
Fig. 4

Refractive index of BaF2 retrieved from experiments with both setups by using plane-wave approximation and Gaussian beam approximation.

Fig. 5
Fig. 5

Refractive index of a 1.93 mm thick Si wafer calculated from the direct pass (closed symbols) and from the first echo (open symbols). Retrieval by using (a) Gaussian beam approximation and sample thickness optimization (d = 1.9326 mm); (b) Plane wave approximation with a thickness of d = 1.9326 mm; (c) Plane-wave approximation with a thickness optimization (d = 1.926 mm).

Tables (1)

Tables Icon

Table 1 Summary of measured samples. The thickness of samples is determined by using a mechanical gauge.

Equations (24)

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

Δ = Δ ( a / f 1 ) 2
e ( ν ; z ) = e 0 ( ν ) exp ( i k z ) ,
t ( ν ) = 4 N ( N + 1 ) 2 A exp ( i φ ) m = M 1 M 2 ( N 1 N + 1 A ) 2 m A m exp ( i φ m )
φ = 2 π ν ( n 1 ) d / c , φ m = 4 π m ν n d / c , A = exp ( 2 π ν κ d / c ) , A m = 1.
e ( ν ; z ) = e 0 ( ν ) w 0 w ( z ) exp [ i k z + i ψ ( z ) ]
w 2 ( z ) = w 0 2 [ 1 + ( z z 0 ) 2 ]
ψ = arc tan ( z z 0 )
z 0 = π w 0 2 c ν .
ψ Δ z 0 = φ β n ,
β = c 2 π ν z 0 ( a / f 1 ) 2 .
φ G = φ ( 1 + β n ) .
Δ m = ( 2 m + 1 ) Δ 2 m d = Δ 2 m d n
ψ 1 ( a / f 1 ) 2 Δ m z 0
φ m G = φ m ( 1 β n 2 ) .
A m G = 1 1 + β 2 ( φ / n φ m / n 2 ) 2 .
t ( ν ) = 4 N ( N + 1 ) 2 A exp ( i φ G ) m = M 1 M 2 ( N 1 N + 1 A ) 2 m A m G exp ( i φ m G )
2 π ν c ( n 1 1 ) d = 2 π ν c ( n 1 ) d + ( n 1 ) n ( a / f 1 ) 2 d z 0 .
n n 1 β ( n 1 1 n 1 ) ;
n n 1 β ( n 1 ( 2 m + 1 ) n 1 ( 2 m + 1 ) ) .
D = ( n ¯ 1 ) d n ¯ ( a / f 1 ) 2 ,
2 π ν c ( n 2 1 ) d = 2 π ν c ( n 1 ) d + ( n 1 ) n ( a / f 1 ) 2 d z 0 2 D z 0 .
β = ( n 2 n 1 ) ( a / f 1 ) 2 d 2 D .
ν m = 2 c L π w E A ,
Δ n n s Δ d s d f + | d s d r | d f Δ n s ,

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