Hybrid continuous wave terahertz spectroscopy

Maik Scheller; Matthias Stecher; Marina Gerhard; Martin Koch

doi:10.1364/OE.18.015887

Hybrid continuous wave terahertz spectroscopy

Maik Scheller, Matthias Stecher, Marina Gerhard, and Martin Koch

Optics Express
Vol. 18,
Issue 15,
pp. 15887-15892
(2010)
•https://doi.org/10.1364/OE.18.015887

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Maik Scheller, Matthias Stecher, Marina Gerhard, and Martin Koch, "Hybrid continuous wave terahertz spectroscopy," Opt. Express 18, 15887-15892 (2010)

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Related Topics
Table of Contents Category
- Spectroscopy
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Interferometry (120.3180)
- Industrial optical metrology (150.3045)
- Spectroscopy, teraherz (300.6495)

About this Article
History
- Original Manuscript: May 27, 2010
- Revised Manuscript: July 2, 2010
- Manuscript Accepted: July 2, 2010
- Published: July 12, 2010

Accessible

Open Access

Abstract

We propose a hybrid architecture for continuous wave terahertz spectroscopy employing a conventional two color photomixing system combined with a quasi time domain spectrometer, driven by a multimode laser diode. This approach fuses high spectral intensity with broadband frequency information and overcomes the ambiguity of standard continuous wave thickness measurements.

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References

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Publication

K. Fukunaga, Y. Ogawa, S. Hayashi, and I. Hosako, “Terahertz spectroscopy for art conservation,” IEICE Electron. Express 4(8), 258–263 (2007).
[Crossref]
J. B. Jackson, M. Mourou, J. F. Whitaker, I. N. Duling, S. L. Williamson, M. Menu, and G. A. Mourou, “Terahertz imaging for non-destructive evaluation of mural paintings,” Opt. Commun. 281(4), 527–532 (2008).
[Crossref]
K. Fukunaga, N. Sekine, I. Hosako, N. Oda, H. Yoneyama, and T. Sudoh, “Real-time terahertz imaging for art conservation science,” J. Eur. Opt. Soc. Rapid Publ. 3, 08027 (2008).
[Crossref]
K. Yamamoto, M. Yamaguchi, M. Tani, M. Hangyo, S. Teramura, T. Isu, and N. Tomita, “Degradation diagnosis of ultrahigh-molecular weight polyethylene with terahertz-time-domain spectroscopy,” Appl. Phys. Lett. 85(22), 5194 (2004).
[Crossref]
C. D. Stoik, M. J. Bohn, and J. L. Blackshire, “Nondestructive evaluation of aircraft composites using transmissive terahertz time domain spectroscopy,” Opt. Express 16(21), 17039–17051 (2008).
[Crossref] [PubMed]
N. Krumbholz, T. Hochrein, N. Vieweg, T. Hasek, K. Kretschmer, M. Bastian, M. Mikulics, and M. Koch, “Monitoring polymeric compounding processes in line with THz time-domain spectroscopy,” Polym. Test. 28(1), 30–35 (2009).
[Crossref]
M. Nagai, H. Yada, T. Arikawa, and K. Tanaka, “Terahertz time-domain attenuated total reflection spectroscopy in water and biological solution,” Int. J. Infrared Millim. Waves 27, 1572–9559 (2006).
C. Jördens, M. Scheller, B. Breitenstein, D. Selmar, and M. Koch, “Evaluation of leaf water status by means of permittivity at terahertz frequencies,” J. Biol. Phys. 35(3), 255–264 (2009).
[Crossref] [PubMed]
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]
E. Knoesel, M. Bonn, J. Shan, and T. F. Heinz, “Charge transport and carrier dynamics in liquids probed by THz time-domain spectroscopy,” Phys. Rev. Lett. 86(2), 340–343 (2001).
[Crossref] [PubMed]
G. Segschneider, F. Jacob, T. Löffler, H. G. Roskos, S. Tautz, P. Kiesel, and G. Döhler, “Free-carrier dynamics in low-temperature-grown GaAs at high excitation densities investigated by time-domain terahertz spectroscopy,” Phys. Rev. B 65(12), 125205 (2002).
[Crossref]
H. B. Liu, G. Plopper, S. Earley, Y.-J. Chen, B. S. Ferguson, and X. C. Zhang, “Sensing minute changes in biological cell monolayers with THz differential time-domain spectroscopy,” Biosens. Bioelectron. 22(6), 1075–1080 (2007).
[Crossref]
W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70(8), 1325–1379 (2007).
[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 (1997).
[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]
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]
R. Wilk, F. Breitfeld, M. Mikulics, and M. Koch, “Continuous wave terahertz spectrometer as a noncontact thickness measuring device,” Appl. Opt. 47(16), 3023–3026 (2008).
[Crossref] [PubMed]
M. Scheller and M. Koch, “Terahertz quasi time domain spectroscopy,” Opt. Express 17(20), 17723–17733 (2009).
[Crossref] [PubMed]
T. Yasui, T. Yasuda, K. Sawanaka, and T. Araki, “Terahertz paintmeter for noncontact monitoring of thickness and drying progress in paint film,” Appl. Opt. 44(32), 6849–6856 (2005).
[Crossref] [PubMed]
M. Scheller and M. Koch, “Fast and accurate thickness determination of unknown materials using terahertz time domain spectroscopy,” J. Infrared, Millim, Terahertz Waves 30(7), 762–769 (2009).
[Crossref]
E. R. Brown, F. W. Smith, and K. A. McIntosh, “Coherent millimeter-wave generation by heterodyne conversion in low-temperature-grown GaAs photoconductors,” J. Appl. Phys. 73(3), 1480–1484 (1993).
[Crossref]
N. Vieweg, M. Mikulics, M. Scheller, K. Ezdi, R. Wilk, H. W. Hübers, and M. Koch, “Impact of the contact metallization on the performance of photoconductive THz antennas,” Opt. Express 16(24), 19695–19705 (2008).
[Crossref] [PubMed]
A. J. Deninger, T. Göbel, D. Schönherr, T. Kinder, A. Roggenbuck, M. Köberle, F. Lison, T. Müller-Wirts, and P. Meissner, “Precisely tunable continuous-wave terahertz source with interferometric frequency control,” Rev. Sci. Instrum. 79(4), 044702 (2008).
[Crossref] [PubMed]

2010 (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]

2009 (4)

N. Krumbholz, T. Hochrein, N. Vieweg, T. Hasek, K. Kretschmer, M. Bastian, M. Mikulics, and M. Koch, “Monitoring polymeric compounding processes in line with THz time-domain spectroscopy,” Polym. Test. 28(1), 30–35 (2009).
[Crossref]

C. Jördens, M. Scheller, B. Breitenstein, D. Selmar, and M. Koch, “Evaluation of leaf water status by means of permittivity at terahertz frequencies,” J. Biol. Phys. 35(3), 255–264 (2009).
[Crossref] [PubMed]

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

M. Scheller and M. Koch, “Terahertz quasi time domain spectroscopy,” Opt. Express 17(20), 17723–17733 (2009).
[Crossref] [PubMed]

2008 (6)

R. Wilk, F. Breitfeld, M. Mikulics, and M. Koch, “Continuous wave terahertz spectrometer as a noncontact thickness measuring device,” Appl. Opt. 47(16), 3023–3026 (2008).
[Crossref] [PubMed]

C. D. Stoik, M. J. Bohn, and J. L. Blackshire, “Nondestructive evaluation of aircraft composites using transmissive terahertz time domain spectroscopy,” Opt. Express 16(21), 17039–17051 (2008).
[Crossref] [PubMed]

N. Vieweg, M. Mikulics, M. Scheller, K. Ezdi, R. Wilk, H. W. Hübers, and M. Koch, “Impact of the contact metallization on the performance of photoconductive THz antennas,” Opt. Express 16(24), 19695–19705 (2008).
[Crossref] [PubMed]

A. J. Deninger, T. Göbel, D. Schönherr, T. Kinder, A. Roggenbuck, M. Köberle, F. Lison, T. Müller-Wirts, and P. Meissner, “Precisely tunable continuous-wave terahertz source with interferometric frequency control,” Rev. Sci. Instrum. 79(4), 044702 (2008).
[Crossref] [PubMed]

J. B. Jackson, M. Mourou, J. F. Whitaker, I. N. Duling, S. L. Williamson, M. Menu, and G. A. Mourou, “Terahertz imaging for non-destructive evaluation of mural paintings,” Opt. Commun. 281(4), 527–532 (2008).
[Crossref]

K. Fukunaga, N. Sekine, I. Hosako, N. Oda, H. Yoneyama, and T. Sudoh, “Real-time terahertz imaging for art conservation science,” J. Eur. Opt. Soc. Rapid Publ. 3, 08027 (2008).
[Crossref]

2007 (3)

H. B. Liu, G. Plopper, S. Earley, Y.-J. Chen, B. S. Ferguson, and X. C. Zhang, “Sensing minute changes in biological cell monolayers with THz differential time-domain spectroscopy,” Biosens. Bioelectron. 22(6), 1075–1080 (2007).
[Crossref]

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70(8), 1325–1379 (2007).
[Crossref]

K. Fukunaga, Y. Ogawa, S. Hayashi, and I. Hosako, “Terahertz spectroscopy for art conservation,” IEICE Electron. Express 4(8), 258–263 (2007).
[Crossref]

2006 (1)

M. Nagai, H. Yada, T. Arikawa, and K. Tanaka, “Terahertz time-domain attenuated total reflection spectroscopy in water and biological solution,” Int. J. Infrared Millim. Waves 27, 1572–9559 (2006).

2005 (1)

T. Yasui, T. Yasuda, K. Sawanaka, and T. Araki, “Terahertz paintmeter for noncontact monitoring of thickness and drying progress in paint film,” Appl. Opt. 44(32), 6849–6856 (2005).
[Crossref] [PubMed]

2004 (1)

K. Yamamoto, M. Yamaguchi, M. Tani, M. Hangyo, S. Teramura, T. Isu, and N. Tomita, “Degradation diagnosis of ultrahigh-molecular weight polyethylene with terahertz-time-domain spectroscopy,” Appl. Phys. Lett. 85(22), 5194 (2004).
[Crossref]

2002 (2)

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]

G. Segschneider, F. Jacob, T. Löffler, H. G. Roskos, S. Tautz, P. Kiesel, and G. Döhler, “Free-carrier dynamics in low-temperature-grown GaAs at high excitation densities investigated by time-domain terahertz spectroscopy,” Phys. Rev. B 65(12), 125205 (2002).
[Crossref]

2001 (1)

E. Knoesel, M. Bonn, J. Shan, and T. F. Heinz, “Charge transport and carrier dynamics in liquids probed by THz time-domain spectroscopy,” Phys. Rev. Lett. 86(2), 340–343 (2001).
[Crossref] [PubMed]

1998 (1)

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]

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 (1997).
[Crossref]

1993 (1)

E. R. Brown, F. W. Smith, and K. A. McIntosh, “Coherent millimeter-wave generation by heterodyne conversion in low-temperature-grown GaAs photoconductors,” J. Appl. Phys. 73(3), 1480–1484 (1993).
[Crossref]

Araki, T.

Arikawa, T.

Bastian, M.

Bauer, T.

Bitzer, A.

Blackshire, J. L.

Bohn, M. J.

Bonn, M.

Breitenstein, B.

Breitfeld, F.

R. Wilk, F. Breitfeld, M. Mikulics, and M. Koch, “Continuous wave terahertz spectrometer as a noncontact thickness measuring device,” Appl. Opt. 47(16), 3023–3026 (2008).
[Crossref] [PubMed]

Brown, E. R.

Calawa, S.

Chan, W. L.

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70(8), 1325–1379 (2007).
[Crossref]

Chen, Y.-J.

Czasch, S.

Deibel, J.

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70(8), 1325–1379 (2007).
[Crossref]

Deninger, A. J.

Dinatale, W. F.

Döhler, G.

Duerr, E. K.

Duling, I. N.

Earley, S.

Ezdi, K.

Ferguson, B. S.

Fischer, B. M.

Fukunaga, K.

K. Fukunaga, N. Sekine, I. Hosako, N. Oda, H. Yoneyama, and T. Sudoh, “Real-time terahertz imaging for art conservation science,” J. Eur. Opt. Soc. Rapid Publ. 3, 08027 (2008).
[Crossref]

K. Fukunaga, Y. Ogawa, S. Hayashi, and I. Hosako, “Terahertz spectroscopy for art conservation,” IEICE Electron. Express 4(8), 258–263 (2007).
[Crossref]

Göbel, T.

Hangyo, M.

Hasek, T.

Hayashi, S.

K. Fukunaga, Y. Ogawa, S. Hayashi, and I. Hosako, “Terahertz spectroscopy for art conservation,” IEICE Electron. Express 4(8), 258–263 (2007).
[Crossref]

Heinz, T. F.

Helm, H.

Hochrein, T.

Hosako, I.

K. Fukunaga, N. Sekine, I. Hosako, N. Oda, H. Yoneyama, and T. Sudoh, “Real-time terahertz imaging for art conservation science,” J. Eur. Opt. Soc. Rapid Publ. 3, 08027 (2008).
[Crossref]

K. Fukunaga, Y. Ogawa, S. Hayashi, and I. Hosako, “Terahertz spectroscopy for art conservation,” IEICE Electron. Express 4(8), 258–263 (2007).
[Crossref]

Hübers, H. W.

Isu, T.

Jackson, J. B.

Jacob, F.

Jördens, C.

Kiesel, P.

Kinder, T.

Knoesel, E.

Köberle, M.

Koch, M.

M. Scheller and M. Koch, “Terahertz quasi time domain spectroscopy,” Opt. Express 17(20), 17723–17733 (2009).
[Crossref] [PubMed]

R. Wilk, F. Breitfeld, M. Mikulics, and M. Koch, “Continuous wave terahertz spectrometer as a noncontact thickness measuring device,” Appl. Opt. 47(16), 3023–3026 (2008).
[Crossref] [PubMed]

Kretschmer, K.

Krumbholz, N.

Leonhardt, R.

Lison, F.

Liu, H. B.

Löffler, T.

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 (1997).
[Crossref]

McIntosh, K. A.

Meissner, P.

Menu, M.

Mikulics, M.

R. Wilk, F. Breitfeld, M. Mikulics, and M. Koch, “Continuous wave terahertz spectrometer as a noncontact thickness measuring device,” Appl. Opt. 47(16), 3023–3026 (2008).
[Crossref] [PubMed]

Mittleman, D. M.

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70(8), 1325–1379 (2007).
[Crossref]

Molvar, K. A.

Mourou, G. A.

Mourou, M.

Müller-Wirts, T.

Nagai, M.

Oda, N.

K. Fukunaga, N. Sekine, I. Hosako, N. Oda, H. Yoneyama, and T. Sudoh, “Real-time terahertz imaging for art conservation science,” J. Eur. Opt. Soc. Rapid Publ. 3, 08027 (2008).
[Crossref]

Ogawa, Y.

K. Fukunaga, Y. Ogawa, S. Hayashi, and I. Hosako, “Terahertz spectroscopy for art conservation,” IEICE Electron. Express 4(8), 258–263 (2007).
[Crossref]

Ortner, A.

Plopper, G.

Quast, H.

Roggenbuck, A.

Roskos, H. G.

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 (1997).
[Crossref]

Sawanaka, K.

Scheller, M.

M. Scheller and M. Koch, “Terahertz quasi time domain spectroscopy,” Opt. Express 17(20), 17723–17733 (2009).
[Crossref] [PubMed]

Schönherr, D.

Segschneider, G.

Sekine, N.

K. Fukunaga, N. Sekine, I. Hosako, N. Oda, H. Yoneyama, and T. Sudoh, “Real-time terahertz imaging for art conservation science,” J. Eur. Opt. Soc. Rapid Publ. 3, 08027 (2008).
[Crossref]

Selmar, D.

Shan, J.

Siebert, K. J.

Smith, F. W.

Stoik, C. D.

Sudoh, T.

K. Fukunaga, N. Sekine, I. Hosako, N. Oda, H. Yoneyama, and T. Sudoh, “Real-time terahertz imaging for art conservation science,” J. Eur. Opt. Soc. Rapid Publ. 3, 08027 (2008).
[Crossref]

Tanaka, K.

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 (1997).
[Crossref]

Tautz, S.

Teramura, S.

Thoman, A.

Thomson, M.

Tomita, N.

Verghese, S.

Vieweg, N.

Walther, M.

Whitaker, J. F.

Wilk, R.

R. Wilk, F. Breitfeld, M. Mikulics, and M. Koch, “Continuous wave terahertz spectrometer as a noncontact thickness measuring device,” Appl. Opt. 47(16), 3023–3026 (2008).
[Crossref] [PubMed]

Williamson, S. L.

Yada, H.

Yamaguchi, M.

Yamamoto, K.

Yasuda, T.

Yasui, T.

Yoneyama, H.

K. Fukunaga, N. Sekine, I. Hosako, N. Oda, H. Yoneyama, and T. Sudoh, “Real-time terahertz imaging for art conservation science,” J. Eur. Opt. Soc. Rapid Publ. 3, 08027 (2008).
[Crossref]

Zhang, X. C.

Anal. Bioanal. Chem. (1)

Appl. Opt. (2)

R. Wilk, F. Breitfeld, M. Mikulics, and M. Koch, “Continuous wave terahertz spectrometer as a noncontact thickness measuring device,” Appl. Opt. 47(16), 3023–3026 (2008).
[Crossref] [PubMed]

Appl. Phys. Lett. (4)

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

Biosens. Bioelectron. (1)

IEICE Electron. Express (1)

K. Fukunaga, Y. Ogawa, S. Hayashi, and I. Hosako, “Terahertz spectroscopy for art conservation,” IEICE Electron. Express 4(8), 258–263 (2007).
[Crossref]

Int. J. Infrared Millim. Waves (1)

J. Appl. Phys. (1)

J. Biol. Phys. (1)

J. Eur. Opt. Soc. Rapid Publ. (1)

K. Fukunaga, N. Sekine, I. Hosako, N. Oda, H. Yoneyama, and T. Sudoh, “Real-time terahertz imaging for art conservation science,” J. Eur. Opt. Soc. Rapid Publ. 3, 08027 (2008).
[Crossref]

J. Infrared, Millim, Terahertz Waves (1)

Opt. Commun. (1)

Opt. Express (3)

M. Scheller and M. Koch, “Terahertz quasi time domain spectroscopy,” Opt. Express 17(20), 17723–17733 (2009).
[Crossref] [PubMed]

Phys. Rev. B (1)

Phys. Rev. Lett. (1)

Polym. Test. (1)

Rep. Prog. Phys. (1)

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70(8), 1325–1379 (2007).
[Crossref]

Rev. Sci. Instrum. (1)

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Fig. 1 Simulated THz signals (a) and the corresponding spectra (b) of the proposed hybrid system approach. In the case of a two color laser system excitation, a pure sinusoidal signal is generated, inhibiting a high spectral brightness. The signal of the MM laser diode driven spectrometer consists of a pulse train with a THz frequency like spectrum. The hybrid system combines both characteristics.

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Fig. 2 Schematic of the hybrid CW THz System. A two color laser system and a MM laser diode are superimposed in a dichroic mirror. The resulting laser beam is divided into two parts by a beam splitter (BS). One of the beams is focused onto an emitter PCA, the other one is guided over a delay stage and then focused onto a detector PCA.

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Fig. 3 Measured THz signals of the hybrid system for different difference frequency of the two color laser system ((a): waveforms, (b): spectra).

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Fig. 4 (a). THz signal through a Si wafer and silica glass compared with the reference measurement. The difference frequency of the two color laser system was set to 500 GHz. (b): The spectral amplitude of the signals. (c): The phase shift between reference and sample measurements.

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Equations (6)

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

I_{D} (Δ X) = A (Δ f) P_{1} P_{2} \cos (\frac{Δ X}{c_{0}} 2 π Δ f),

I_{D} (Δ X) = \sum_{m = 1}^{M - 1} \frac{(M - m)}{M^{2}} (2 π m δ f) A (m δ f) P^{2} \cos (\frac{Δ X}{c_{0}} 2 π m δ f) .

\begin{array}{l} I_{D} (Δ X) = A (Δ f) P_{1} P_{2} \cos (\frac{Δ X}{c_{0}} 2 π Δ f) + \\ \sum_{m = 1}^{M - 1} \frac{(M - m)}{M^{2}} (2 π m δ f) A (m δ f) P^{2} \cos (\frac{Δ X}{c_{0}} 2 π m δ f) . \end{array}

H (Δ f) = \frac{1}{\sqrt{1 + 2 π Δ f τ}} \cdot (1 - exp(-i 2 π \frac{Δ f}{f_{0}})),

Δ ϕ_{c w, c o r} = Δ ϕ_{c w} - 2 π \cdot R D {Δ t_{p} f_{c w}},

T = \frac{| Δ ϕ_{c w, c o r} | c_{0}}{2 π f_{c w} (n_{c w} - 1)} .