Abstract

Terahertz time domain spectroscopy allows for characterization of dielectrics even in cases where the samples thickness is unknown. However, a parameter extraction over a broad frequency range with simultaneous thickness determination is time consuming using conventional algorithms due to the large number of optimization steps. In this paper we present a novel method to extract the data. By employing a three dimensional optimization algorithm the calculation effort is significantly reduced while preserving the same accuracy level as conventional approaches. The presented method is even fast enough to be used in imaging applications.

© 2011 OSA

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  1. S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559–561 (1997).
    [CrossRef]
  2. S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. 73(26), 3824–3826 (1998).
    [CrossRef]
  3. K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
    [CrossRef]
  4. M. Scheller, J. M. Yarborough, J. V. Moloney, M. Fallahi, M. Koch, and S. W. Koch, “Room temperature continuous wave milliwatt terahertz source,” Opt. Express 18(26), 27112–27117 (2010).
    [CrossRef]
  5. X. Wang, L. Hou, and Y. Zhang, “Continuous-wave terahertz interferometry with multiwavelength phase unwrapping,” Appl. Opt. 49(27), 5095–5102 (2010).
    [CrossRef] [PubMed]
  6. M. Scheller, K. Baaske, and M. Koch, “Multifrequency continuous wave terahertz spectroscopy for absolute thickness determination,” Appl. Phys. Lett. 96(15), 151112 (2010).
    [CrossRef]
  7. D. H. Auston and K. P. Cheung, “Coherent time-domain far-infrared spectroscopy,” J. Opt. Soc. Am. B 2(4), 606–612 (1985).
    [CrossRef]
  8. B. Sartorius, H. Roehle, H. Künzel, J. Böttcher, M. Schlak, D. Stanze, H. Venghaus, and M. Schell, “All-fiber terahertz time-domain spectrometer operating at 1.5 μm telecom wavelengths,” Opt. Express 16(13), 9565–9570 (2008).
    [CrossRef] [PubMed]
  9. M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. 397(3), 1009–1017 (2010).
    [CrossRef] [PubMed]
  10. P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
    [CrossRef]
  11. 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]
  12. T. D. Dorney, R. G. Baraniuk, and D. M. Mittleman, “Material parameter estimation with terahertz time-domain spectroscopy,” J. Opt. Soc. Am. A 18(7), 1562–1571 (2001).
    [CrossRef]
  13. M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
    [CrossRef]
  14. I. Duling and D. Zimdars, “Terahertz imaging: revealing hidden defects,” Nat. Photonics 3(11), 630–632 (2009).
    [CrossRef]
  15. P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30(1), 29–31 (2005).
    [CrossRef] [PubMed]
  16. 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(1), 112–147 (1998).
    [CrossRef]
  17. M. Scheller and M. Koch, ““Fast and accurate thickness determination of unknown materials using terahertz time domain spectroscopy,” J. Infrared Milli. Terahz. Waves 30(7), 762–769 (2009).
    [CrossRef]

2011 (1)

P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
[CrossRef]

2010 (4)

M. Scheller, K. Baaske, and M. Koch, “Multifrequency continuous wave terahertz spectroscopy for absolute thickness determination,” Appl. Phys. Lett. 96(15), 151112 (2010).
[CrossRef]

M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. 397(3), 1009–1017 (2010).
[CrossRef] [PubMed]

X. Wang, L. Hou, and Y. Zhang, “Continuous-wave terahertz interferometry with multiwavelength phase unwrapping,” Appl. Opt. 49(27), 5095–5102 (2010).
[CrossRef] [PubMed]

M. Scheller, J. M. Yarborough, J. V. Moloney, M. Fallahi, M. Koch, and S. W. Koch, “Room temperature continuous wave milliwatt terahertz source,” Opt. Express 18(26), 27112–27117 (2010).
[CrossRef]

2009 (3)

M. Scheller and M. Koch, ““Fast and accurate thickness determination of unknown materials using terahertz time domain spectroscopy,” J. Infrared Milli. Terahz. Waves 30(7), 762–769 (2009).
[CrossRef]

M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
[CrossRef]

I. Duling and D. Zimdars, “Terahertz imaging: revealing hidden defects,” Nat. Photonics 3(11), 630–632 (2009).
[CrossRef]

2008 (1)

2005 (1)

2002 (1)

K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
[CrossRef]

2001 (1)

1999 (1)

1998 (2)

S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. 73(26), 3824–3826 (1998).
[CrossRef]

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(1), 112–147 (1998).
[CrossRef]

1997 (1)

S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559–561 (1997).
[CrossRef]

1985 (1)

Auston, D. H.

Baaske, K.

M. Scheller, K. Baaske, and M. Koch, “Multifrequency continuous wave terahertz spectroscopy for absolute thickness determination,” Appl. Phys. Lett. 96(15), 151112 (2010).
[CrossRef]

Baraniuk, R. G.

Bauer, T.

K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
[CrossRef]

Bitzer, A.

M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. 397(3), 1009–1017 (2010).
[CrossRef] [PubMed]

Böttcher, J.

Calawa, S.

S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. 73(26), 3824–3826 (1998).
[CrossRef]

Cheung, K. P.

Cooke, D. G.

P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
[CrossRef]

Coutaz, J. L.

Czasch, S.

K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
[CrossRef]

Dinatale, W. F.

S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. 73(26), 3824–3826 (1998).
[CrossRef]

Dorney, T. D.

Duerr, E. K.

S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. 73(26), 3824–3826 (1998).
[CrossRef]

Duling, I.

I. Duling and D. Zimdars, “Terahertz imaging: revealing hidden defects,” Nat. Photonics 3(11), 630–632 (2009).
[CrossRef]

Duvillaret, L.

Fallahi, M.

Fischer, B. M.

M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. 397(3), 1009–1017 (2010).
[CrossRef] [PubMed]

P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30(1), 29–31 (2005).
[CrossRef] [PubMed]

Garet, F.

Helm, H.

M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. 397(3), 1009–1017 (2010).
[CrossRef] [PubMed]

Hou, L.

Jansen, C.

M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
[CrossRef]

Jepsen, P.

P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
[CrossRef]

Jepsen, P. U.

Koch, M.

P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
[CrossRef]

M. Scheller, K. Baaske, and M. Koch, “Multifrequency continuous wave terahertz spectroscopy for absolute thickness determination,” Appl. Phys. Lett. 96(15), 151112 (2010).
[CrossRef]

M. Scheller, J. M. Yarborough, J. V. Moloney, M. Fallahi, M. Koch, and S. W. Koch, “Room temperature continuous wave milliwatt terahertz source,” Opt. Express 18(26), 27112–27117 (2010).
[CrossRef]

M. Scheller and M. Koch, ““Fast and accurate thickness determination of unknown materials using terahertz time domain spectroscopy,” J. Infrared Milli. Terahz. Waves 30(7), 762–769 (2009).
[CrossRef]

M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
[CrossRef]

Koch, S. W.

Künzel, H.

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(1), 112–147 (1998).
[CrossRef]

Leonhardt, R.

K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
[CrossRef]

Löffler, T.

K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
[CrossRef]

Matsuura, S.

S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559–561 (1997).
[CrossRef]

McIntosh, K. A.

S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. 73(26), 3824–3826 (1998).
[CrossRef]

Mittleman, D. M.

Moloney, J. V.

Molvar, K. A.

S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. 73(26), 3824–3826 (1998).
[CrossRef]

Ortner, A.

M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. 397(3), 1009–1017 (2010).
[CrossRef] [PubMed]

Quast, H.

K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
[CrossRef]

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(1), 112–147 (1998).
[CrossRef]

Roehle, H.

Roskos, H. G.

K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
[CrossRef]

Sakai, K.

S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559–561 (1997).
[CrossRef]

Sartorius, B.

Schell, M.

Scheller, M.

M. Scheller, J. M. Yarborough, J. V. Moloney, M. Fallahi, M. Koch, and S. W. Koch, “Room temperature continuous wave milliwatt terahertz source,” Opt. Express 18(26), 27112–27117 (2010).
[CrossRef]

M. Scheller, K. Baaske, and M. Koch, “Multifrequency continuous wave terahertz spectroscopy for absolute thickness determination,” Appl. Phys. Lett. 96(15), 151112 (2010).
[CrossRef]

M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
[CrossRef]

M. Scheller and M. Koch, ““Fast and accurate thickness determination of unknown materials using terahertz time domain spectroscopy,” J. Infrared Milli. Terahz. Waves 30(7), 762–769 (2009).
[CrossRef]

Schlak, M.

Siebert, K. J.

K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
[CrossRef]

Stanze, D.

Tani, M.

S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559–561 (1997).
[CrossRef]

Thoman, A.

M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. 397(3), 1009–1017 (2010).
[CrossRef] [PubMed]

Thomson, M.

K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
[CrossRef]

Venghaus, H.

Verghese, S.

S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. 73(26), 3824–3826 (1998).
[CrossRef]

Walther, M.

M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. 397(3), 1009–1017 (2010).
[CrossRef] [PubMed]

Wang, X.

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(1), 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(1), 112–147 (1998).
[CrossRef]

Yarborough, J. M.

Zhang, Y.

Zimdars, D.

I. Duling and D. Zimdars, “Terahertz imaging: revealing hidden defects,” Nat. Photonics 3(11), 630–632 (2009).
[CrossRef]

Anal. Bioanal. Chem. (1)

M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. 397(3), 1009–1017 (2010).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

M. Scheller, K. Baaske, and M. Koch, “Multifrequency continuous wave terahertz spectroscopy for absolute thickness determination,” Appl. Phys. Lett. 96(15), 151112 (2010).
[CrossRef]

S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559–561 (1997).
[CrossRef]

S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. 73(26), 3824–3826 (1998).
[CrossRef]

K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002).
[CrossRef]

J. Infrared Milli. Terahz. Waves (1)

M. Scheller and M. Koch, ““Fast and accurate thickness determination of unknown materials using terahertz time domain spectroscopy,” J. Infrared Milli. Terahz. Waves 30(7), 762–769 (2009).
[CrossRef]

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

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

Laser Photonics Rev. (1)

P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011).
[CrossRef]

Nat. Photonics (1)

I. Duling and D. Zimdars, “Terahertz imaging: revealing hidden defects,” Nat. Photonics 3(11), 630–632 (2009).
[CrossRef]

Opt. Commun. (1)

M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

SIAM J. Optim. (1)

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(1), 112–147 (1998).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Initial parameters of the refractive index (a) and the absorption coefficient (b) for an arbitrary data set. The parameters were obtained using Eqs. (1) and (2) and exhibit FP oscillations before filtering.

Fig. 2
Fig. 2

Initial values and extracted material parameters of the silica wafer; (a): refractive index, (b): absorption coefficient. (c): FP oscillation strength as function of the thickness and optical derived thickness.

Fig. 3
Fig. 3

Error values for constant ξ (a), constant ψ (b), and constant ζ (c) for the analysis of the silica wafer.

Fig. 4
Fig. 4

Initial values and extracted material parameters of the PET sample; (a): refractive index, (b): absorption coefficient. (c): FP oscillation strength as function of the thickness and optical derived thickness.

Fig. 5
Fig. 5

(a) Initial values and extracted material parameters of the alpha-lactose-PP compound; (a): refractive index, (b): absorption coefficient. (c): FP oscillation strength as function of the thickness and optical derived thickness.

Fig. 6
Fig. 6

Converged thickness as a function of the initial thickness for the PET sample (a). (b): Remaining values of the error function.

Equations (9)

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

n 0 ( f ) = 1 + H ( f ) 2 π f L 0 c 0 ,
κ 0 ( f ) = c 0 2 π f L 0 ln ( | H ( f ) | ( n 0 + 1 ) 2 4 n 0 ) ,
f F P = c 0 2 n 0 L 0 .
H ( f , n , κ , L ) = ( 1 ( n + i κ 1 n + i κ + 1 ) 2 ) g = 0 M ( exp ( i 2 π f L ( ( 2 g + 1 ) [ n + i κ ] 1 ) c 0 ) ( n + i κ 1 n + i κ + 1 ) 2 g ) .
n ( f ) = ξ ( n 0 ( f ) 1 ) + 1 ,     κ ( f ) = ψ κ 0 ( f ) ,     L = ζ L 0 .
H ( f ) = H ( f , ξ ( n 0 ( f ) 1 ) + 1 , ψ κ 0 ( f ) , ζ L 0 )
Δ H = f | H t h e o r e t i c a l ( f ) H m e a s u r e d ( f ) | .
n o p t = ξ o p t ( n 0 1 ) + 1 ,     κ o p t = ψ o p t κ 0 ,     L o p t = ζ o p t L 0 .
Δ H j = | H t h e o r e t i c a l ( f j ) H m e a s u r e d ( f j ) |     .

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