Abstract

This manuscript introduces a self-calibrating technique for terahertz (THz) time-domain material characterization in transmission mode. The self-calibrating technique utilizes the relations between the multiple transmissions of a transient plane wave through a sample under test in order to extract its material properties. The method presented in this paper is particularly useful in reducing the time required for simultaneous imaging and spectroscopy scans, since existing characterization methods require the knowledge of the waveform of both the plane wave transmitted through a reference sample and the waveform of the plane wave transmitted through the sample under test. The technique presented also ameliorates the problem caused by differences between sample and reference signals due to power and time drifts inherent in a THz system, hence reducing artifacts in the characterization results.

© 2011 Optical Society of America

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References

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  1. B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
    [CrossRef]
  2. W. Chan, J. Deibel, and D. Mittleman, “Imaging with Terahertz Radiation,” Rep. Prog. Phys. 70, 1325–1379 (2007).
    [CrossRef]
  3. K. Kawase, Y. Ogawa, and Y. Watanabe, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express 11, 2549–2554 (2003).
    [CrossRef] [PubMed]
  4. D. Zimdars, J. White, G. Stuk, A. Chernovsky, G. Fichter, and S. Williamson, “Large area terahertz imaging and nondestructive evaluation applications,” Insight—Non-Destructive Testing and Condition Monitoring: J. Sol-Gel Sci. Technol. 48, 537–539(2006).
    [CrossRef]
  5. T. D. Dorney, R. G. Baraniuk, and D. M. Mittleman, “Material parameter estimation with terahertz time-domain spectroscopy,” J. Opt. Soc. Am. A 18, 1562–1571 (2001).
    [CrossRef]
  6. M. Naftaly and R. E. Miles, “Terahertz time-domain spectroscopy for material characterization,” Proc. IEEE 95, 1658–1665(2007).
    [CrossRef]
  7. J. A. Hejase, V. Melapudi, E. J. Rothwell, and P. Chahal, “A self-calibrating technique for THz time domain material parameter extraction,” presented at the 2010 IEEE International Symposium on Antennas and Propagation and CNC/USNC/URSI Radio Science Meeting, Toronto, Canada, 11–17 July 2010.
  8. E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2001).
    [CrossRef]
  9. L. Duvillaret, F. Garet, and J. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2, 739–746(1996).
    [CrossRef]
  10. A. Ravindran, K. M. Ragsdell, and G. V. Reklaitis, Engineering Optimization: Methods and Applications (Wiley, 2006).
  11. J. 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]
  12. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1965).
  13. P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
    [CrossRef]
  14. M. V. Exter and D. Grischowsky, “Optical and electronic properties of doped silicon between 0.1 to 2 THz,” Appl. Phys. Lett. 56, 1694–1696 (1990).
    [CrossRef]
  15. J. Li and J. Li, “Dielectric properties of silicon in terahertz wave region,” Microw. Opt. Technol. Lett. 50, 1143–1146 (2008).
    [CrossRef]
  16. J. A. Hejase, P. R. Paladhi, and P. Chahal, “THz characterization of dielectric substrates for component design and non-destructive evaluation of packages,” IEEE Trans. Adv. Packaging (in print).

2008

J. Li and J. Li, “Dielectric properties of silicon in terahertz wave region,” Microw. Opt. Technol. Lett. 50, 1143–1146 (2008).
[CrossRef]

2007

W. Chan, J. Deibel, and D. Mittleman, “Imaging with Terahertz Radiation,” Rep. Prog. Phys. 70, 1325–1379 (2007).
[CrossRef]

M. Naftaly and R. E. Miles, “Terahertz time-domain spectroscopy for material characterization,” Proc. IEEE 95, 1658–1665(2007).
[CrossRef]

2006

D. Zimdars, J. White, G. Stuk, A. Chernovsky, G. Fichter, and S. Williamson, “Large area terahertz imaging and nondestructive evaluation applications,” Insight—Non-Destructive Testing and Condition Monitoring: J. Sol-Gel Sci. Technol. 48, 537–539(2006).
[CrossRef]

2003

P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
[CrossRef]

K. Kawase, Y. Ogawa, and Y. Watanabe, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express 11, 2549–2554 (2003).
[CrossRef] [PubMed]

2002

B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
[CrossRef]

2001

1998

J. 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]

1996

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

1990

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

1965

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1965).

Baraniuk, R. G.

Bolivar, P. H.

P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
[CrossRef]

Brucherseifer, M.

P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
[CrossRef]

Chahal, P.

J. A. Hejase, V. Melapudi, E. J. Rothwell, and P. Chahal, “A self-calibrating technique for THz time domain material parameter extraction,” presented at the 2010 IEEE International Symposium on Antennas and Propagation and CNC/USNC/URSI Radio Science Meeting, Toronto, Canada, 11–17 July 2010.

J. A. Hejase, P. R. Paladhi, and P. Chahal, “THz characterization of dielectric substrates for component design and non-destructive evaluation of packages,” IEEE Trans. Adv. Packaging (in print).

Chan, W.

W. Chan, J. Deibel, and D. Mittleman, “Imaging with Terahertz Radiation,” Rep. Prog. Phys. 70, 1325–1379 (2007).
[CrossRef]

Chernovsky, A.

D. Zimdars, J. White, G. Stuk, A. Chernovsky, G. Fichter, and S. Williamson, “Large area terahertz imaging and nondestructive evaluation applications,” Insight—Non-Destructive Testing and Condition Monitoring: J. Sol-Gel Sci. Technol. 48, 537–539(2006).
[CrossRef]

Cloud, M. J.

E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2001).
[CrossRef]

Coutaz, J.

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

de Maagt, P.

P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
[CrossRef]

Deibel, J.

W. Chan, J. Deibel, and D. Mittleman, “Imaging with Terahertz Radiation,” Rep. Prog. Phys. 70, 1325–1379 (2007).
[CrossRef]

Dorney, T. D.

Duvillaret, L.

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

Eddera, I.

P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
[CrossRef]

Exter, M. V.

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

Ferguson, B.

B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
[CrossRef]

Fichter, G.

D. Zimdars, J. White, G. Stuk, A. Chernovsky, G. Fichter, and S. Williamson, “Large area terahertz imaging and nondestructive evaluation applications,” Insight—Non-Destructive Testing and Condition Monitoring: J. Sol-Gel Sci. Technol. 48, 537–539(2006).
[CrossRef]

Garet, F.

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

Gonzalo, R.

P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
[CrossRef]

Grischowsky, D.

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

Hejase, J. A.

J. A. Hejase, P. R. Paladhi, and P. Chahal, “THz characterization of dielectric substrates for component design and non-destructive evaluation of packages,” IEEE Trans. Adv. Packaging (in print).

J. A. Hejase, V. Melapudi, E. J. Rothwell, and P. Chahal, “A self-calibrating technique for THz time domain material parameter extraction,” presented at the 2010 IEEE International Symposium on Antennas and Propagation and CNC/USNC/URSI Radio Science Meeting, Toronto, Canada, 11–17 July 2010.

Holker, M.

P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
[CrossRef]

Kawase, K.

Lagarias, J.

J. 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]

Li, J.

J. Li and J. Li, “Dielectric properties of silicon in terahertz wave region,” Microw. Opt. Technol. Lett. 50, 1143–1146 (2008).
[CrossRef]

J. Li and J. Li, “Dielectric properties of silicon in terahertz wave region,” Microw. Opt. Technol. Lett. 50, 1143–1146 (2008).
[CrossRef]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1965).

Melapudi, V.

J. A. Hejase, V. Melapudi, E. J. Rothwell, and P. Chahal, “A self-calibrating technique for THz time domain material parameter extraction,” presented at the 2010 IEEE International Symposium on Antennas and Propagation and CNC/USNC/URSI Radio Science Meeting, Toronto, Canada, 11–17 July 2010.

Miles, R. E.

M. Naftaly and R. E. Miles, “Terahertz time-domain spectroscopy for material characterization,” Proc. IEEE 95, 1658–1665(2007).
[CrossRef]

Mittleman, D.

W. Chan, J. Deibel, and D. Mittleman, “Imaging with Terahertz Radiation,” Rep. Prog. Phys. 70, 1325–1379 (2007).
[CrossRef]

Mittleman, D. M.

Naftaly, M.

M. Naftaly and R. E. Miles, “Terahertz time-domain spectroscopy for material characterization,” Proc. IEEE 95, 1658–1665(2007).
[CrossRef]

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1965).

Ogawa, Y.

Paladhi, P. R.

J. A. Hejase, P. R. Paladhi, and P. Chahal, “THz characterization of dielectric substrates for component design and non-destructive evaluation of packages,” IEEE Trans. Adv. Packaging (in print).

Ragsdell, K. M.

A. Ravindran, K. M. Ragsdell, and G. V. Reklaitis, Engineering Optimization: Methods and Applications (Wiley, 2006).

Ravindran, A.

A. Ravindran, K. M. Ragsdell, and G. V. Reklaitis, Engineering Optimization: Methods and Applications (Wiley, 2006).

Reeds, J. A.

J. 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]

Reklaitis, G. V.

A. Ravindran, K. M. Ragsdell, and G. V. Reklaitis, Engineering Optimization: Methods and Applications (Wiley, 2006).

Reynolds, A. L.

P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
[CrossRef]

Rivas, J. G.

P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
[CrossRef]

Rothwell, E. J.

J. A. Hejase, V. Melapudi, E. J. Rothwell, and P. Chahal, “A self-calibrating technique for THz time domain material parameter extraction,” presented at the 2010 IEEE International Symposium on Antennas and Propagation and CNC/USNC/URSI Radio Science Meeting, Toronto, Canada, 11–17 July 2010.

E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2001).
[CrossRef]

Stuk, G.

D. Zimdars, J. White, G. Stuk, A. Chernovsky, G. Fichter, and S. Williamson, “Large area terahertz imaging and nondestructive evaluation applications,” Insight—Non-Destructive Testing and Condition Monitoring: J. Sol-Gel Sci. Technol. 48, 537–539(2006).
[CrossRef]

Watanabe, Y.

White, J.

D. Zimdars, J. White, G. Stuk, A. Chernovsky, G. Fichter, and S. Williamson, “Large area terahertz imaging and nondestructive evaluation applications,” Insight—Non-Destructive Testing and Condition Monitoring: J. Sol-Gel Sci. Technol. 48, 537–539(2006).
[CrossRef]

Williamson, S.

D. Zimdars, J. White, G. Stuk, A. Chernovsky, G. Fichter, and S. Williamson, “Large area terahertz imaging and nondestructive evaluation applications,” Insight—Non-Destructive Testing and Condition Monitoring: J. Sol-Gel Sci. Technol. 48, 537–539(2006).
[CrossRef]

Wright, M. H.

J. 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. 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.

B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
[CrossRef]

Zimdars, D.

D. Zimdars, J. White, G. Stuk, A. Chernovsky, G. Fichter, and S. Williamson, “Large area terahertz imaging and nondestructive evaluation applications,” Insight—Non-Destructive Testing and Condition Monitoring: J. Sol-Gel Sci. Technol. 48, 537–539(2006).
[CrossRef]

Appl. Phys. Lett.

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

Comput. J. (UK)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1965).

IEEE J. Sel. Top. Quantum Electron.

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

IEEE Trans. Microwave Theory Tech.

P. H. Bolivar, M. Brucherseifer, J. G. Rivas, R. Gonzalo, I. Eddera, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062–1066 (2003).
[CrossRef]

Insight—Non-Destructive Testing and Condition Monitoring: J. Sol-Gel Sci. Technol.

D. Zimdars, J. White, G. Stuk, A. Chernovsky, G. Fichter, and S. Williamson, “Large area terahertz imaging and nondestructive evaluation applications,” Insight—Non-Destructive Testing and Condition Monitoring: J. Sol-Gel Sci. Technol. 48, 537–539(2006).
[CrossRef]

J. Opt. Soc. Am. A

Microw. Opt. Technol. Lett.

J. Li and J. Li, “Dielectric properties of silicon in terahertz wave region,” Microw. Opt. Technol. Lett. 50, 1143–1146 (2008).
[CrossRef]

Nat. Mater.

B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
[CrossRef]

Opt. Express

Proc. IEEE

M. Naftaly and R. E. Miles, “Terahertz time-domain spectroscopy for material characterization,” Proc. IEEE 95, 1658–1665(2007).
[CrossRef]

Rep. Prog. Phys.

W. Chan, J. Deibel, and D. Mittleman, “Imaging with Terahertz Radiation,” Rep. Prog. Phys. 70, 1325–1379 (2007).
[CrossRef]

SIAM J. Optim.

J. 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. A. Hejase, V. Melapudi, E. J. Rothwell, and P. Chahal, “A self-calibrating technique for THz time domain material parameter extraction,” presented at the 2010 IEEE International Symposium on Antennas and Propagation and CNC/USNC/URSI Radio Science Meeting, Toronto, Canada, 11–17 July 2010.

E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2001).
[CrossRef]

A. Ravindran, K. M. Ragsdell, and G. V. Reklaitis, Engineering Optimization: Methods and Applications (Wiley, 2006).

J. A. Hejase, P. R. Paladhi, and P. Chahal, “THz characterization of dielectric substrates for component design and non-destructive evaluation of packages,” IEEE Trans. Adv. Packaging (in print).

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Figures (12)

Fig. 1
Fig. 1

Sketch of idealized measurement setup.

Fig. 2
Fig. 2

Received time-domain signal for a typical sample showing multiple transmissions under ideal conditions.

Fig. 3
Fig. 3

Example time-domain signals obtained from the measurement.

Fig. 4
Fig. 4

Sample signal along with the shifted archived time-domain signal.

Fig. 5
Fig. 5

Sample signal and first subtraction result.

Fig. 6
Fig. 6

Second subtraction result.

Fig. 7
Fig. 7

Measured and final sample signals.

Fig. 8
Fig. 8

Extracted dielectric properties of Al 2 O 3 .

Fig. 9
Fig. 9

Extracted dielectric properties of HDPE.

Fig. 10
Fig. 10

Extracted dielectric properties of quartz.

Fig. 11
Fig. 11

Extracted dielectric properties of InP.

Fig. 12
Fig. 12

Extracted dielectric properties of doped Si.

Equations (17)

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

E r ( t ) = E i ( t p ) * T ( t ) ,
E r ( ω ) = E i ( ω ) exp [ j ω n ˜ a ( d l ) / c ] T ( ω ) ,
T ( ω ) = T a s ( ω ) T s a ( ω ) exp ( j ω n ˜ s l / c ) 1 + Γ a s ( ω ) Γ s a ( ω ) exp ( 2 j ω n ˜ s l / c ) ,
Γ a s ( ω ) = n ˜ s n ˜ a n ˜ a + n ˜ s .
Γ s a ( ω ) = Γ a s ( ω ) ,
T a s ( ω ) = 1 Γ a s ( ω ) ,
T s a ( ω ) = 1 + Γ a s ( ω ) .
T ( ω ) = T a s ( ω ) T s a ( ω ) exp ( j ω n ˜ s l / c ) T a s ( ω ) T s a ( ω ) exp ( j ω n ˜ s l / c ) [ Γ a s ( ω ) Γ s a ( ω ) exp ( 2 j ω n ˜ s l / c ) ] + T a s ( ω ) T s a ( ω ) exp ( j ω n ˜ s l / c ) [ Γ a s ( ω ) Γ s a ( ω ) exp ( 2 j ω n ˜ s l / c ) ] 2 .
T ( t ) = [ T a s ( t ) * T s a ( t ) * f s ( t ) ] U ( t τ s ) [ T a s ( t ) * T s a ( t ) * Γ a s ( t ) * Γ s a ( t ) * f s ( t ) * 3 ] U ( t 3 τ s ) + [ T a s ( t ) * T s a ( t ) * Γ a s ( t ) * Γ s a ( t ) * Γ a s ( t ) * Γ s a ( t ) * f s ( t ) * 5 ] U ( t 5 τ s ) ,
E r ( t ) = [ E i ( t p ) * T a s ( t ) * T s a ( t ) * f s ( t ) ] U ( t τ s ) [ E i ( t p ) * T a s ( t ) * T s a ( t ) * Γ a s ( t ) * Γ s a ( t ) * f s ( t ) * 3 ] U ( t 3 τ s ) + [ E i ( t p ) * T a s ( t ) * T s a ( t ) * Γ a s ( t ) * Γ s a ( t ) * Γ a s ( t ) * Γ s a ( t ) * f s ( t ) * 5 ] U ( t 5 τ s ) .
E a ( t ) = [ E i ( t p ) * T a s ( t ) * T s a ( t ) * f s ( t ) ] U ( t τ s ) W ( t )
E a ( t ) E 1 ( t ) = [ E i ( t p ) * T a s ( t ) * T s a ( t ) * f s ( t ) ] U ( t τ s ) .
E 1 ( ω ) = E i ( ω ) exp [ j ω n ˜ a ( d l ) / c ] T a s ( ω ) T s a ( ω ) exp ( j ω n ˜ s l / c ) .
E 1 ( ω ) E r ( ω ) = 1 Γ a s 2 exp ( 2 j ω n ˜ s l / c ) .
M ^ ( ω ) = 1 Γ a s 2 exp ( 2 j ω n ˜ s l / c ) = 1 ( n j k 1 1 + n j k ) 2 exp [ 2 j ω ( n j k ) l / c ] .
f ( ω ) = C 1 | | M ( ω ) | | M ^ ( ω ) | | + C 2 | M ( ω ) M ^ ( ω ) | .
n initial = c Δ t 2 l ,

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