Abstract

We present a method based on a Fabry–Perot model to efficiently and accurately estimate optical constants of wafer samples in transmission-only measurements performed by a vector network analyzer (VNA). The method is demonstrated on two separate wafer samples: one of silicon and the other of polymethylmethacrylate. Results show that the method can not only acquire optical constants accurately and simply over a broad frequency domain but also overcome the limitations of calculation for dispersive and lossy materials to which existing methods are susceptible, such as those based on VNA-driven quasi-optical transmissometers and terahertz time-domain spectrometry.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2013 (1)

A. I. McIntosh, B. Yang, S. M. Goldup, M. Watkinson, and R. S. Donnan, Chem. Phys. Lett. 558, 104 (2013).
[CrossRef]

2012 (2)

2011 (3)

R. Singh, I. A. I. Al-Naib, M. Koch, and W. Zhang, Opt. Express 19, 6312 (2011).
[CrossRef]

U. C. Hasar, IEEE Geosci. Remote Sens. Lett. 8, 562 (2011).
[CrossRef]

D. F. Williams, IEEE Trans. Terahertz Sci. Technol. 1, 364 (2011).
[CrossRef]

2010 (1)

B. Yang, R. J. Wylde, D. H. Martin, P. Goy, R. S. Donnan, and S. Caroopen, IEEE Trans. Terahertz Sci. Technol. 58, 3587 (2010).

2009 (1)

U. C. Hasar, IEEE Trans. Microwave Theory Tech. 57, 1595 (2009).
[CrossRef]

2008 (2)

J. S. Melinger, N. Laman, and D. Grischkowsky, Appl. Phys. Lett. 93, 011102 (2008).
[CrossRef]

J. Liu, X. Guo, J. Dai, and X.-C. Zhang, Appl. Phys. Lett. 93, 171102 (2008).
[CrossRef]

2007 (1)

2006 (1)

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, J. Korean Phys. Soc. 49, 513 (2006).

2003 (1)

T. C. Williams, M. A. Stuchly, and P. Saville, IEEE Trans. Microwave Theory Tech. 51, 1560 (2003).
[CrossRef]

2001 (1)

2000 (2)

M. Brucherseifer, M. Nagel, P. Haring Bolivar, H. Kurz, A. Bosserhoff, and R. Büttner, Appl. Phys. Lett. 77, 4049 (2000).
[CrossRef]

A. G. Markelz, A. Roitberg, and E. J. Heilweil, Chem. Phys. Lett. 320, 42 (2000).
[CrossRef]

1999 (1)

1995 (1)

1963 (1)

D. W. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963).
[CrossRef]

1944 (1)

K. Levenberg, Q. Appl. Math. 2, 164 (1944).

Alexander, P.

Al-Naib, I. A. I.

Baraniuk, R. G.

Bosserhoff, A.

M. Brucherseifer, M. Nagel, P. Haring Bolivar, H. Kurz, A. Bosserhoff, and R. Büttner, Appl. Phys. Lett. 77, 4049 (2000).
[CrossRef]

Brucherseifer, M.

M. Brucherseifer, M. Nagel, P. Haring Bolivar, H. Kurz, A. Bosserhoff, and R. Büttner, Appl. Phys. Lett. 77, 4049 (2000).
[CrossRef]

Büttner, R.

M. Brucherseifer, M. Nagel, P. Haring Bolivar, H. Kurz, A. Bosserhoff, and R. Büttner, Appl. Phys. Lett. 77, 4049 (2000).
[CrossRef]

Caroopen, S.

B. Yang, R. J. Wylde, D. H. Martin, P. Goy, R. S. Donnan, and S. Caroopen, IEEE Trans. Terahertz Sci. Technol. 58, 3587 (2010).

Coutaz, J.-L.

Dai, J.

J. Liu, X. Guo, J. Dai, and X.-C. Zhang, Appl. Phys. Lett. 93, 171102 (2008).
[CrossRef]

Donnan, R. S.

A. I. McIntosh, B. Yang, S. M. Goldup, M. Watkinson, and R. S. Donnan, Chem. Phys. Lett. 558, 104 (2013).
[CrossRef]

B. Yang, R. J. Wylde, D. H. Martin, P. Goy, R. S. Donnan, and S. Caroopen, IEEE Trans. Terahertz Sci. Technol. 58, 3587 (2010).

Dorney, T. D.

Duvillaret, L.

Garet, F.

Goldup, S. M.

A. I. McIntosh, B. Yang, S. M. Goldup, M. Watkinson, and R. S. Donnan, Chem. Phys. Lett. 558, 104 (2013).
[CrossRef]

Goy, P.

B. Yang, R. J. Wylde, D. H. Martin, P. Goy, R. S. Donnan, and S. Caroopen, IEEE Trans. Terahertz Sci. Technol. 58, 3587 (2010).

Grischkowsky, D.

J. S. Melinger, N. Laman, and D. Grischkowsky, Appl. Phys. Lett. 93, 011102 (2008).
[CrossRef]

Guo, X.

J. Liu, X. Guo, J. Dai, and X.-C. Zhang, Appl. Phys. Lett. 93, 171102 (2008).
[CrossRef]

Haring Bolivar, P.

M. Brucherseifer, M. Nagel, P. Haring Bolivar, H. Kurz, A. Bosserhoff, and R. Büttner, Appl. Phys. Lett. 77, 4049 (2000).
[CrossRef]

Hasar, U. C.

U. C. Hasar, IEEE Geosci. Remote Sens. Lett. 8, 562 (2011).
[CrossRef]

U. C. Hasar, IEEE Trans. Microwave Theory Tech. 57, 1595 (2009).
[CrossRef]

Heilweil, E. J.

A. G. Markelz, A. Roitberg, and E. J. Heilweil, Chem. Phys. Lett. 320, 42 (2000).
[CrossRef]

Hirsch, O.

Hu, B. B.

Jeon, S.-G.

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, J. Korean Phys. Soc. 49, 513 (2006).

Jin, Y.-S.

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, J. Korean Phys. Soc. 49, 513 (2006).

Kim, G.-J.

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, J. Korean Phys. Soc. 49, 513 (2006).

Koch, M.

Kurz, H.

M. Brucherseifer, M. Nagel, P. Haring Bolivar, H. Kurz, A. Bosserhoff, and R. Büttner, Appl. Phys. Lett. 77, 4049 (2000).
[CrossRef]

Laman, N.

J. S. Melinger, N. Laman, and D. Grischkowsky, Appl. Phys. Lett. 93, 011102 (2008).
[CrossRef]

Levenberg, K.

K. Levenberg, Q. Appl. Math. 2, 164 (1944).

Liu, J.

J. Liu, X. Guo, J. Dai, and X.-C. Zhang, Appl. Phys. Lett. 93, 171102 (2008).
[CrossRef]

Markelz, A. G.

A. G. Markelz, A. Roitberg, and E. J. Heilweil, Chem. Phys. Lett. 320, 42 (2000).
[CrossRef]

Marquardt, D. W.

D. W. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963).
[CrossRef]

Martin, D. H.

B. Yang, R. J. Wylde, D. H. Martin, P. Goy, R. S. Donnan, and S. Caroopen, IEEE Trans. Terahertz Sci. Technol. 58, 3587 (2010).

McIntosh, A. I.

A. I. McIntosh, B. Yang, S. M. Goldup, M. Watkinson, and R. S. Donnan, Chem. Phys. Lett. 558, 104 (2013).
[CrossRef]

Melinger, J. S.

J. S. Melinger, N. Laman, and D. Grischkowsky, Appl. Phys. Lett. 93, 011102 (2008).
[CrossRef]

Mittleman, D. M.

Nagel, M.

M. Brucherseifer, M. Nagel, P. Haring Bolivar, H. Kurz, A. Bosserhoff, and R. Büttner, Appl. Phys. Lett. 77, 4049 (2000).
[CrossRef]

Nuss, M. C.

Roitberg, A.

A. G. Markelz, A. Roitberg, and E. J. Heilweil, Chem. Phys. Lett. 320, 42 (2000).
[CrossRef]

Saville, P.

T. C. Williams, M. A. Stuchly, and P. Saville, IEEE Trans. Microwave Theory Tech. 51, 1560 (2003).
[CrossRef]

Singh, R.

Stuchly, M. A.

T. C. Williams, M. A. Stuchly, and P. Saville, IEEE Trans. Microwave Theory Tech. 51, 1560 (2003).
[CrossRef]

Sun, W.

Wang, X.

Watkinson, M.

A. I. McIntosh, B. Yang, S. M. Goldup, M. Watkinson, and R. S. Donnan, Chem. Phys. Lett. 558, 104 (2013).
[CrossRef]

Williams, D. F.

D. F. Williams, IEEE Trans. Terahertz Sci. Technol. 1, 364 (2011).
[CrossRef]

Williams, T. C.

T. C. Williams, M. A. Stuchly, and P. Saville, IEEE Trans. Microwave Theory Tech. 51, 1560 (2003).
[CrossRef]

Wu, T. Y.

T. Y. Wu, Meas. Sci. Technol. 23, 085904 (2012).
[CrossRef]

Wylde, R. J.

B. Yang, R. J. Wylde, D. H. Martin, P. Goy, R. S. Donnan, and S. Caroopen, IEEE Trans. Terahertz Sci. Technol. 58, 3587 (2010).

Xiong, W.

Yang, B.

A. I. McIntosh, B. Yang, S. M. Goldup, M. Watkinson, and R. S. Donnan, Chem. Phys. Lett. 558, 104 (2013).
[CrossRef]

B. Yang, R. J. Wylde, D. H. Martin, P. Goy, R. S. Donnan, and S. Caroopen, IEEE Trans. Terahertz Sci. Technol. 58, 3587 (2010).

Zhang, W.

Zhang, X.-C.

J. Liu, X. Guo, J. Dai, and X.-C. Zhang, Appl. Phys. Lett. 93, 171102 (2008).
[CrossRef]

Zhang, Y.

Appl. Opt. (1)

Appl. Phys. Lett. (3)

J. S. Melinger, N. Laman, and D. Grischkowsky, Appl. Phys. Lett. 93, 011102 (2008).
[CrossRef]

J. Liu, X. Guo, J. Dai, and X.-C. Zhang, Appl. Phys. Lett. 93, 171102 (2008).
[CrossRef]

M. Brucherseifer, M. Nagel, P. Haring Bolivar, H. Kurz, A. Bosserhoff, and R. Büttner, Appl. Phys. Lett. 77, 4049 (2000).
[CrossRef]

Chem. Phys. Lett. (2)

A. G. Markelz, A. Roitberg, and E. J. Heilweil, Chem. Phys. Lett. 320, 42 (2000).
[CrossRef]

A. I. McIntosh, B. Yang, S. M. Goldup, M. Watkinson, and R. S. Donnan, Chem. Phys. Lett. 558, 104 (2013).
[CrossRef]

IEEE Geosci. Remote Sens. Lett. (1)

U. C. Hasar, IEEE Geosci. Remote Sens. Lett. 8, 562 (2011).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

U. C. Hasar, IEEE Trans. Microwave Theory Tech. 57, 1595 (2009).
[CrossRef]

T. C. Williams, M. A. Stuchly, and P. Saville, IEEE Trans. Microwave Theory Tech. 51, 1560 (2003).
[CrossRef]

IEEE Trans. Terahertz Sci. Technol. (2)

B. Yang, R. J. Wylde, D. H. Martin, P. Goy, R. S. Donnan, and S. Caroopen, IEEE Trans. Terahertz Sci. Technol. 58, 3587 (2010).

D. F. Williams, IEEE Trans. Terahertz Sci. Technol. 1, 364 (2011).
[CrossRef]

J. Korean Phys. Soc. (1)

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, J. Korean Phys. Soc. 49, 513 (2006).

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

J. Soc. Ind. Appl. Math. (1)

D. W. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963).
[CrossRef]

Meas. Sci. Technol. (1)

T. Y. Wu, Meas. Sci. Technol. 23, 085904 (2012).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Q. Appl. Math. (1)

K. Levenberg, Q. Appl. Math. 2, 164 (1944).

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Figures (4)

Fig. 1.
Fig. 1.

Sketch of THz wave propagation in the wafer performed by a VNA.

Fig. 2.
Fig. 2.

Measured transmittance and phase difference of the silicon wafer.

Fig. 3.
Fig. 3.

Refractive index of the silicon wafer obtained with the VNA and by TDS by the proposed method and a published method.

Fig. 4.
Fig. 4.

Measured transmittance and phase difference of (a) the PMMA wafer and (b) its complex refractive index n and k.

Equations (10)

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

H˙=m=0EtmETHz=(1R)exp(iω(n˜21)L/c)1Rexp(iωn˜2L/c),
f(x)=12j=1mrj2(x),
f(x)=j=1mrj(x)rj(x)=J(x)Tr(x),
2f(x)=J(x)T+j=1mrj(x)2rj(x).
f(x)=f(x0)+(xx0)T2f(x0)+ht,
xi+1=xi(2f(x))1f(xi).
xi+1=xi(H+λI)1f(xi),
ITI0=11+Fsin2δ,
(IT)max=I0,δ=pπ,pan integer
(IT)min=I01+F,δ=(p+12π).

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