Abstract

We have fabricated a three-contact photoconductive antenna for the polarization-sensitive detection of terahertz electromagnetic radiation. Taking into account all three photoconductive signal current components, this three-contact photoconductive antenna can measure the polarization state of pulsed THz radiation at an accuracy comparable to that achieved using the conventional method which employs a set of wire-grid polarizers. The three-contact photoconductive receiver may be useful for polarization-sensitive spectroscopy such as vibrational circular dichroism spectroscopy and ellipsometry in the THz frequency region.

©2007 Optical Society of America

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References

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  1. J. H. W. G. den Boer, G. M. W. Kroesen, W. de Zeeuw, and F. J. de Hoog, “Improved polarizer in the infrared: two wire-grid polarizers in tandem,” Opt. Lett. 20, 800–802 (1995).
    [Crossref]
  2. N. V. Cohan and H. F. Hameka, “Isotope effects in optical rotation,” J. Am. Chem. Soc. 88, 2136–2142 (1966).
    [Crossref]
  3. L. A. Nafie, “Infrared and raman vibrational optical activity: theoretical and experimental aspects,” Annu. Rev. Phys. Chem. 48, 357–386 (1997).
    [Crossref] [PubMed]
  4. P. L. Polavarapu and Z. Deng, “Measurement of vibrational circular dichroism below 600 cm-1: progress towards meeting the challenge,” Appl. Spectrosc. 50, 686–692 (1996).
    [Crossref]
  5. J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
    [Crossref] [PubMed]
  6. E. Castro-Camus, J. Lloyd-Hughes, M. B. Johnston, M. D. Fraser, H. H. Tan, and C. Jagadish, “Polarization-sensitive terahertz detection by multicontact photoconductive receivers,” Appl. Phys. Lett. 86, 254102 (2005).
    [Crossref]
  7. E. Castro-Camus, J. Lloyd-Hughes, L. Fu, H. H. Tan, C. Jagadish, and M. B. Johnston, “An ion-implanted InP receiver for polarization resolved terahertz spectroscopy,” Opt. Express 15, 7047–7057 (2007).
    [Crossref] [PubMed]
  8. Y. Hirota, R. Hattori, M. Tani, and M. Hangyo, “Polarization modulation of terahertz electromagnetic radiation by four-contact photoconductive antenna,” Opt. Express 14, 4486–4493 (2006).
    [Crossref] [PubMed]
  9. Q. Chen and X.-C. Zhang, “Polarization modulation in optoelectronic generation and detection of terahertz beams,” Appl. Phys. Lett. 74, 3435–3437 (1999).
    [Crossref]
  10. R. Shimano, H. Nishimura, and T. Sato, “Frequency tunable circular polarization control of terahertz radiation,” Jpn. J. Appl. Phys. 44, L676–L678 (2005).
    [Crossref]
  11. D. C. Look, “Molecular beam epitaxial GaAs grown at low temperatures,” Thin Solid Films 231, 61–73 (1993).
    [Crossref]
  12. M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
    [Crossref]
  13. C. L. Mok, W. G. Chambers, T. J. Parker, and A. E. Costley, “The far-infrared performance and application of free-standing grids wound from 5µm diameter tungsten wire,” Infrared Phys. 19, 437–442 (1979).
    [Crossref]
  14. A. Filin, M. Stowe, and R. Kersting, “Time-domain differentiation of terahertz pulses,” Opt. Lett. 26, 2008–2010 (2001).
    [Crossref]

2007 (1)

2006 (1)

2005 (2)

E. Castro-Camus, J. Lloyd-Hughes, M. B. Johnston, M. D. Fraser, H. H. Tan, and C. Jagadish, “Polarization-sensitive terahertz detection by multicontact photoconductive receivers,” Appl. Phys. Lett. 86, 254102 (2005).
[Crossref]

R. Shimano, H. Nishimura, and T. Sato, “Frequency tunable circular polarization control of terahertz radiation,” Jpn. J. Appl. Phys. 44, L676–L678 (2005).
[Crossref]

2003 (1)

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
[Crossref] [PubMed]

2001 (1)

1999 (1)

Q. Chen and X.-C. Zhang, “Polarization modulation in optoelectronic generation and detection of terahertz beams,” Appl. Phys. Lett. 74, 3435–3437 (1999).
[Crossref]

1997 (1)

L. A. Nafie, “Infrared and raman vibrational optical activity: theoretical and experimental aspects,” Annu. Rev. Phys. Chem. 48, 357–386 (1997).
[Crossref] [PubMed]

1996 (1)

1995 (1)

1994 (1)

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

1993 (1)

D. C. Look, “Molecular beam epitaxial GaAs grown at low temperatures,” Thin Solid Films 231, 61–73 (1993).
[Crossref]

1979 (1)

C. L. Mok, W. G. Chambers, T. J. Parker, and A. E. Costley, “The far-infrared performance and application of free-standing grids wound from 5µm diameter tungsten wire,” Infrared Phys. 19, 437–442 (1979).
[Crossref]

1966 (1)

N. V. Cohan and H. F. Hameka, “Isotope effects in optical rotation,” J. Am. Chem. Soc. 88, 2136–2142 (1966).
[Crossref]

Abe, H.

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Abe, Y.

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Allen, S. J.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
[Crossref] [PubMed]

Birge, R. R.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
[Crossref] [PubMed]

Castro-Camus, E.

E. Castro-Camus, J. Lloyd-Hughes, L. Fu, H. H. Tan, C. Jagadish, and M. B. Johnston, “An ion-implanted InP receiver for polarization resolved terahertz spectroscopy,” Opt. Express 15, 7047–7057 (2007).
[Crossref] [PubMed]

E. Castro-Camus, J. Lloyd-Hughes, M. B. Johnston, M. D. Fraser, H. H. Tan, and C. Jagadish, “Polarization-sensitive terahertz detection by multicontact photoconductive receivers,” Appl. Phys. Lett. 86, 254102 (2005).
[Crossref]

Chambers, W. G.

C. L. Mok, W. G. Chambers, T. J. Parker, and A. E. Costley, “The far-infrared performance and application of free-standing grids wound from 5µm diameter tungsten wire,” Infrared Phys. 19, 437–442 (1979).
[Crossref]

Chen, Q.

Q. Chen and X.-C. Zhang, “Polarization modulation in optoelectronic generation and detection of terahertz beams,” Appl. Phys. Lett. 74, 3435–3437 (1999).
[Crossref]

Cohan, N. V.

N. V. Cohan and H. F. Hameka, “Isotope effects in optical rotation,” J. Am. Chem. Soc. 88, 2136–2142 (1966).
[Crossref]

Costley, A. E.

C. L. Mok, W. G. Chambers, T. J. Parker, and A. E. Costley, “The far-infrared performance and application of free-standing grids wound from 5µm diameter tungsten wire,” Infrared Phys. 19, 437–442 (1979).
[Crossref]

de Hoog, F. J.

de Zeeuw, W.

den Boer, J. H. W. G.

Deng, Z.

Filin, A.

Fraser, M. D.

E. Castro-Camus, J. Lloyd-Hughes, M. B. Johnston, M. D. Fraser, H. H. Tan, and C. Jagadish, “Polarization-sensitive terahertz detection by multicontact photoconductive receivers,” Appl. Phys. Lett. 86, 254102 (2005).
[Crossref]

Fu, L.

Galan, J. F.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
[Crossref] [PubMed]

Hameka, H. F.

N. V. Cohan and H. F. Hameka, “Isotope effects in optical rotation,” J. Am. Chem. Soc. 88, 2136–2142 (1966).
[Crossref]

Hangyo, M.

Y. Hirota, R. Hattori, M. Tani, and M. Hangyo, “Polarization modulation of terahertz electromagnetic radiation by four-contact photoconductive antenna,” Opt. Express 14, 4486–4493 (2006).
[Crossref] [PubMed]

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Harima, H.

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Hattori, R.

Hirota, Y.

Jagadish, C.

E. Castro-Camus, J. Lloyd-Hughes, L. Fu, H. H. Tan, C. Jagadish, and M. B. Johnston, “An ion-implanted InP receiver for polarization resolved terahertz spectroscopy,” Opt. Express 15, 7047–7057 (2007).
[Crossref] [PubMed]

E. Castro-Camus, J. Lloyd-Hughes, M. B. Johnston, M. D. Fraser, H. H. Tan, and C. Jagadish, “Polarization-sensitive terahertz detection by multicontact photoconductive receivers,” Appl. Phys. Lett. 86, 254102 (2005).
[Crossref]

Johnston, M. B.

E. Castro-Camus, J. Lloyd-Hughes, L. Fu, H. H. Tan, C. Jagadish, and M. B. Johnston, “An ion-implanted InP receiver for polarization resolved terahertz spectroscopy,” Opt. Express 15, 7047–7057 (2007).
[Crossref] [PubMed]

E. Castro-Camus, J. Lloyd-Hughes, M. B. Johnston, M. D. Fraser, H. H. Tan, and C. Jagadish, “Polarization-sensitive terahertz detection by multicontact photoconductive receivers,” Appl. Phys. Lett. 86, 254102 (2005).
[Crossref]

Kanamoto, K.

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Kersting, R.

Kroesen, G. M. W.

Lloyd-Hughes, J.

E. Castro-Camus, J. Lloyd-Hughes, L. Fu, H. H. Tan, C. Jagadish, and M. B. Johnston, “An ion-implanted InP receiver for polarization resolved terahertz spectroscopy,” Opt. Express 15, 7047–7057 (2007).
[Crossref] [PubMed]

E. Castro-Camus, J. Lloyd-Hughes, M. B. Johnston, M. D. Fraser, H. H. Tan, and C. Jagadish, “Polarization-sensitive terahertz detection by multicontact photoconductive receivers,” Appl. Phys. Lett. 86, 254102 (2005).
[Crossref]

Look, D. C.

D. C. Look, “Molecular beam epitaxial GaAs grown at low temperatures,” Thin Solid Films 231, 61–73 (1993).
[Crossref]

Mok, C. L.

C. L. Mok, W. G. Chambers, T. J. Parker, and A. E. Costley, “The far-infrared performance and application of free-standing grids wound from 5µm diameter tungsten wire,” Infrared Phys. 19, 437–442 (1979).
[Crossref]

Nafie, L. A.

L. A. Nafie, “Infrared and raman vibrational optical activity: theoretical and experimental aspects,” Annu. Rev. Phys. Chem. 48, 357–386 (1997).
[Crossref] [PubMed]

Nakashima, S.

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Nishimura, H.

R. Shimano, H. Nishimura, and T. Sato, “Frequency tunable circular polarization control of terahertz radiation,” Jpn. J. Appl. Phys. 44, L676–L678 (2005).
[Crossref]

Parker, T. J.

C. L. Mok, W. G. Chambers, T. J. Parker, and A. E. Costley, “The far-infrared performance and application of free-standing grids wound from 5µm diameter tungsten wire,” Infrared Phys. 19, 437–442 (1979).
[Crossref]

Plaxco, K. W.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
[Crossref] [PubMed]

Polavarapu, P. L.

Ramian, G. J.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
[Crossref] [PubMed]

Sakai, K.

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Sato, T.

R. Shimano, H. Nishimura, and T. Sato, “Frequency tunable circular polarization control of terahertz radiation,” Jpn. J. Appl. Phys. 44, L676–L678 (2005).
[Crossref]

Savvidis, P. G.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
[Crossref] [PubMed]

Scopatz, A. M.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
[Crossref] [PubMed]

Shimano, R.

R. Shimano, H. Nishimura, and T. Sato, “Frequency tunable circular polarization control of terahertz radiation,” Jpn. J. Appl. Phys. 44, L676–L678 (2005).
[Crossref]

Stowe, M.

Tan, H. H.

E. Castro-Camus, J. Lloyd-Hughes, L. Fu, H. H. Tan, C. Jagadish, and M. B. Johnston, “An ion-implanted InP receiver for polarization resolved terahertz spectroscopy,” Opt. Express 15, 7047–7057 (2007).
[Crossref] [PubMed]

E. Castro-Camus, J. Lloyd-Hughes, M. B. Johnston, M. D. Fraser, H. H. Tan, and C. Jagadish, “Polarization-sensitive terahertz detection by multicontact photoconductive receivers,” Appl. Phys. Lett. 86, 254102 (2005).
[Crossref]

Tani, M.

Y. Hirota, R. Hattori, M. Tani, and M. Hangyo, “Polarization modulation of terahertz electromagnetic radiation by four-contact photoconductive antenna,” Opt. Express 14, 4486–4493 (2006).
[Crossref] [PubMed]

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Tokuda, Y.

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Tsukada, N.

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Xu, J.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
[Crossref] [PubMed]

Zhang, X.-C.

Q. Chen and X.-C. Zhang, “Polarization modulation in optoelectronic generation and detection of terahertz beams,” Appl. Phys. Lett. 74, 3435–3437 (1999).
[Crossref]

Annu. Rev. Phys. Chem. (1)

L. A. Nafie, “Infrared and raman vibrational optical activity: theoretical and experimental aspects,” Annu. Rev. Phys. Chem. 48, 357–386 (1997).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

E. Castro-Camus, J. Lloyd-Hughes, M. B. Johnston, M. D. Fraser, H. H. Tan, and C. Jagadish, “Polarization-sensitive terahertz detection by multicontact photoconductive receivers,” Appl. Phys. Lett. 86, 254102 (2005).
[Crossref]

Q. Chen and X.-C. Zhang, “Polarization modulation in optoelectronic generation and detection of terahertz beams,” Appl. Phys. Lett. 74, 3435–3437 (1999).
[Crossref]

Appl. Spectrosc. (1)

Astrobiology (1)

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, S. J. Allen, and K. W. Plaxco, “Terahertz circular dichroism spectroscopy: a potential approach to the in situ detection of life’s metabolic and genetic machinery,” Astrobiology 3, 489–504 (2003).
[Crossref] [PubMed]

Infrared Phys. (1)

C. L. Mok, W. G. Chambers, T. J. Parker, and A. E. Costley, “The far-infrared performance and application of free-standing grids wound from 5µm diameter tungsten wire,” Infrared Phys. 19, 437–442 (1979).
[Crossref]

J. Am. Chem. Soc. (1)

N. V. Cohan and H. F. Hameka, “Isotope effects in optical rotation,” J. Am. Chem. Soc. 88, 2136–2142 (1966).
[Crossref]

Jpn. J. Appl. Phys. (2)

R. Shimano, H. Nishimura, and T. Sato, “Frequency tunable circular polarization control of terahertz radiation,” Jpn. J. Appl. Phys. 44, L676–L678 (2005).
[Crossref]

M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, “Spectroscopic characterization of low-temperature grown GaAs epitaxial films,” Jpn. J. Appl. Phys. 33, 4807–4811 (1994).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Thin Solid Films (1)

D. C. Look, “Molecular beam epitaxial GaAs grown at low temperatures,” Thin Solid Films 231, 61–73 (1993).
[Crossref]

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Figures (7)
Fig.1. .
Fig.1. . a). (Color online) Microscopic view of the three-contact PC antenna. (b) Equivalent circuit for the three-contact PC antenna.

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Fig. 2.
Fig. 2. (Color online) Schematic diagram of the polarization detection system with the three-contact photoconductive receiver.

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Fig. 3.
Fig. 3. (Color online) (a) The signal waveforms Ii (τ) (i=1,2) and (b) their Fourier-transformed power spectra Ii (ω) (i=1,2) for THz radiation with a polarization angle of 45°.

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Fig. 4.
Fig. 4. (Color online) Experimentally determined normalized response matrix components (solid lines) and the analytical ones (dashed lines with the same color to the corresponding experimental ones).

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Fig. 5.
Fig. 5. (Color online) “Polarization trajectories” of THz radiations with measurement using the three-contact photoconductive receiver for the wire-grid polarizer angles -60°, -45°, -30°, 0°, 30°, 45°, 60°, and 90°.

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Fig. 6.
Fig. 6. (Color online) Relative changes of the THz polarization angle measured with the three-contact PC receiver (open circles) for rotations of the wire grid polarizer by 0.1° steps from 0 to 1° (indicated by a straight line).

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Fig. 7.
Fig. 7. (Color online) Ellipticity of THz radiations with measurement using the three-contact photoconductive receiver for the wire-grid polarizer angles -90°, -60°, -30°, 0°, 30°, 60°, and 90°.

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Equations (4)

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

E = RI , ( E x ( ω ) E y ( ω ) ) = ( R x 1 ( ω ) , R x 2 ( ω ) R y 1 ( ω ) , R y 2 ( ω ) ) ( I 1 ( ω ) I 2 ( ω ) ) .
R ( ω ) R 0 = ( − γ ( ω ) α ( ω ) β ( ω ) − γ ( ω ) , β ( ω ) α ( ω ) β ( ω ) − γ ( ω ) α ( ω ) α ( ω ) β ( ω ) − γ ( ω ) , − 1 α ( ω ) β ( ω ) − γ ( ω ) ) ≡ r ( ω ) = ( r x 1 ( ω ) , r x 2 ( ω ) r y 1 ( ω ) , r y 2 ( ω ) ) .
I H 1 ( ω ) : I H 2 ( ω ) : I V 1 ( ω ) : I V 2 ( ω ) = 1 : α ( ω ) : β ( ω ) : γ ( ω ) .
r = ( 3 , 3 − 1 , 1 )

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