Abstract

We present high precision measurements of polarization rotations in the frequency range from 0.1 to 2.5 THz using a polarization modulation technique. A motorized stage rotates a polarizer at ∼ 80 Hz, and the resulting modulation of the polarization is measured by a lock-in technique. We achieve an accuracy of 0.050° (900 μrad) and a precision of 0.02° (350 μrad) for small rotation angles. A detailed mathematical description of the technique is presented, showing its ability to fully characterize elliptical polarizations from 0.1 to 2.5 THz.

© 2012 OSA

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    [CrossRef] [PubMed]
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2012 (4)

E. Castro-Camus, “Polarization-resolved terahertz time-domain spectroscopy,” Int. J. Infrared Millim. Waves 33, 418–430 (2012).

R. Nandkishore, L. S. Levitov, and A. V. Chubukov, “Chiral superconductivity from repulsive interactions in doped graphene,” Nat. Phys. 8, 158–163 (2012).
[CrossRef]

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

L. Ren, C. L. Pint, T. Arikawa, K. Takeya, I. Kawayama, M. Tonouchi, R. H. Hauge, and J. Kono, “Broadband terahertz polarizers with ideal performance based on aligned carbon nanotube stacks,” Nano Lett. 12, 787–790 (2012).
[CrossRef] [PubMed]

2011 (3)

W.-K. Tse and A. H. MacDonald, “Magneto-optical Faraday and Kerr effects in topological insulator films and in other layered quantized Hall systems,” Phys. Rev. B 84, 205327 (2011).
[CrossRef]

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002 (2011).
[CrossRef]

L. S. Bilbro, R. Valdés Aguilar, G. Logvenov, O. Pelleg, I. Bozovic, and N. P. Armitage, “Temporal correlations of superconductivity above the transition temperature in La2–xSrxCuO4 probed by terahertz spectroscopy,” Nat. Phys. 7, 298–302 (2011).
[CrossRef]

2010 (3)

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[CrossRef]

C. Huang, S. Zhao, H. Chen, and Z. Liao, “Attenuation characterization of multiple combinations of imperfect polarizers,” J. Opt. Soc. Am. A 27, 1060–1068 (2010).
[CrossRef]

G. S. Jenkins, D. C. Schmadel, and H. D. Drew, “Simultaneous measurement of circular dichroism and Faraday rotation at terahertz frequencies utilizing electric field sensitive detection via polarization modulation,” Rev. Sci. Instrum. 81, 083903 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (2)

A Hussain and S. R. Andrews, “Ultrabroadband polarization analysis of terahertz pulses,” Opt. Express 16, 7251–7257 (2008).
[CrossRef] [PubMed]

E. Castro-Camus and M. B. Johnson, “Conformational changes of photoactive yellow protein monitored by terahertz spectroscopy,” Chem. Phys. Lett. 455, 289–292 (2008).
[CrossRef]

2007 (2)

2006 (2)

J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, “High resolution polar Kerr effect measurements of Sr2RuO4: evidence for broken time-reversal symmetry in the superconducting state,” Phys. Rev. Lett. 97, 167002 (2006).
[CrossRef] [PubMed]

J.-B. Masson and G. Gallot, “Terahertz achromatic quarter-wave plate,” Opt. Lett. 31, 265–267 (2006).
[CrossRef] [PubMed]

2005 (1)

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]

2004 (1)

C. Bernhard, J. Humlicek, and B. Keimer, “Far-infrared ellipsometry using a synchrotron light source-the dielectric response of the cuprate high Tc superconductors,” Thin Solid Films 455, 143–149 (2004).
[CrossRef]

2003 (2)

R. A. Kaindl, M. A. Carnahan, D. Hägele, R. Lövenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

S. Wang, B. Ferguson, D. Abbott, and X.-C. Zhang, “T-ray imaging and tomography,” J. Biol. Phys. Chem. 29, 247–256 (2003).

2002 (3)

M. Nagel, P. H. Bolivar, M. Brucherseifer, H. Kurz, A. Bosserhoff, and R. Büttner, “Integrated planar terahertz resonators for femtomolar sensitivity label-free detection of DNA hybridization,” Appl. Opt. 41, 2074–2078 (2002).
[CrossRef] [PubMed]

X-C Zhang, “Terahertz wave imaging: horizons and hurdles,” Phys. Med. Biol. 47, 3667 (2002).
[CrossRef] [PubMed]

M. Grayson, L. B. Rigal, D. C. Schmadel, H. D. Drew, and P.-J. Kung, “Spectral measurement of the Hall angle response in normal state cuprate superconductors,” Phys. Rev. Lett. 89, 037003 (2002).
[CrossRef] [PubMed]

2001 (1)

J. L. Johnson, T. D. Dorney, and D. M. Mittleman, “Enhanced depth resolution in terahertz imaging using phase-shift interferometry,” Appl. Phys. Lett. 78, 835–837 (2001).
[CrossRef]

2000 (1)

A.J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yagvesson, “An anti-reflection coating for silicon optics at terahertz frequencies,” IEEE Microw. Guid. Wave Lett. 10, 264–266 (2000).
[CrossRef]

1999 (1)

J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic, “Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+δ,” Nature 398, 221–223 (1999).
[CrossRef]

1998 (1)

J. N. Heyman, R. Kersting, and K. Unterrainer, “Time-domain measurement of intersubband oscillations in a quantum well,” Appl. Phys. Lett. 72, 644–646 (1998).
[CrossRef]

1995 (1)

1994 (1)

A. Kapitulnik, J. S. Dodge, and M. M. Fejer, “High-resolution magneto-optic measurements with a Sagnac interferometer,” J. Appl. Phys. 75, 6872–6877 (1994).
[CrossRef]

1954 (1)

X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Topological field theory of time-reversal invariant insulators” Phys. Rev. B 78, 195424 (2008).

Abbott, D.

S. Wang, B. Ferguson, D. Abbott, and X.-C. Zhang, “T-ray imaging and tomography,” J. Biol. Phys. Chem. 29, 247–256 (2003).

Andrews, S. R.

Arikawa, T.

L. Ren, C. L. Pint, T. Arikawa, K. Takeya, I. Kawayama, M. Tonouchi, R. H. Hauge, and J. Kono, “Broadband terahertz polarizers with ideal performance based on aligned carbon nanotube stacks,” Nano Lett. 12, 787–790 (2012).
[CrossRef] [PubMed]

Armitage, N. P.

L. S. Bilbro, R. Valdés Aguilar, G. Logvenov, O. Pelleg, I. Bozovic, and N. P. Armitage, “Temporal correlations of superconductivity above the transition temperature in La2–xSrxCuO4 probed by terahertz spectroscopy,” Nat. Phys. 7, 298–302 (2011).
[CrossRef]

M. Neshat and N. P. Armitage, “Improved measurement of polarization state in terahertz polarization spectroscopy,” (submitted) Opt. Lett. (2012).

Armitager, N. P.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

Bansal, N.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

Bernhard, C.

C. Bernhard, J. Humlicek, and B. Keimer, “Far-infrared ellipsometry using a synchrotron light source-the dielectric response of the cuprate high Tc superconductors,” Thin Solid Films 455, 143–149 (2004).
[CrossRef]

Beyersdorf, P. T.

J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, “High resolution polar Kerr effect measurements of Sr2RuO4: evidence for broken time-reversal symmetry in the superconducting state,” Phys. Rev. Lett. 97, 167002 (2006).
[CrossRef] [PubMed]

Bilbro, L. S.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

L. S. Bilbro, R. Valdés Aguilar, G. Logvenov, O. Pelleg, I. Bozovic, and N. P. Armitage, “Temporal correlations of superconductivity above the transition temperature in La2–xSrxCuO4 probed by terahertz spectroscopy,” Nat. Phys. 7, 298–302 (2011).
[CrossRef]

Bolivar, P. H.

Born, M.

M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, 1997).

Bosserhoff, A.

Bozovic, I.

L. S. Bilbro, R. Valdés Aguilar, G. Logvenov, O. Pelleg, I. Bozovic, and N. P. Armitage, “Temporal correlations of superconductivity above the transition temperature in La2–xSrxCuO4 probed by terahertz spectroscopy,” Nat. Phys. 7, 298–302 (2011).
[CrossRef]

J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic, “Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+δ,” Nature 398, 221–223 (1999).
[CrossRef]

Brucherseifer, M.

Büttner, R.

Carnahan, M. A.

R. A. Kaindl, M. A. Carnahan, D. Hägele, R. Lövenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

Castro-Camus, E.

E. Castro-Camus, “Polarization-resolved terahertz time-domain spectroscopy,” Int. J. Infrared Millim. Waves 33, 418–430 (2012).

E. Castro-Camus and M. B. Johnson, “Conformational changes of photoactive yellow protein monitored by terahertz spectroscopy,” Chem. Phys. Lett. 455, 289–292 (2008).
[CrossRef]

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]

Cerne, J.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

D. K. George, A. V. Stier, C. T. Ellis, B. D. McCombe, J. Černe, and A. G. Markelz, “Terahertz magneto optical polarization modulation spectroscopy,” (submitted) J. Opt. Soc. Am. B (2012).

Chemla, D. S.

R. A. Kaindl, M. A. Carnahan, D. Hägele, R. Lövenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

Chen, H.

Chubukov, A. V.

R. Nandkishore, L. S. Levitov, and A. V. Chubukov, “Chiral superconductivity from repulsive interactions in doped graphene,” Nat. Phys. 8, 158–163 (2012).
[CrossRef]

Corson, J.

J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic, “Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+δ,” Nature 398, 221–223 (1999).
[CrossRef]

Dadap, J. I.

Dodge, J. S.

A. Kapitulnik, J. S. Dodge, and M. M. Fejer, “High-resolution magneto-optic measurements with a Sagnac interferometer,” J. Appl. Phys. 75, 6872–6877 (1994).
[CrossRef]

Dorney, T. D.

J. L. Johnson, T. D. Dorney, and D. M. Mittleman, “Enhanced depth resolution in terahertz imaging using phase-shift interferometry,” Appl. Phys. Lett. 78, 835–837 (2001).
[CrossRef]

Drew, H. D.

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[CrossRef]

G. S. Jenkins, D. C. Schmadel, and H. D. Drew, “Simultaneous measurement of circular dichroism and Faraday rotation at terahertz frequencies utilizing electric field sensitive detection via polarization modulation,” Rev. Sci. Instrum. 81, 083903 (2010).
[CrossRef] [PubMed]

M. Grayson, L. B. Rigal, D. C. Schmadel, H. D. Drew, and P.-J. Kung, “Spectral measurement of the Hall angle response in normal state cuprate superconductors,” Phys. Rev. Lett. 89, 037003 (2002).
[CrossRef] [PubMed]

Eckstein, J. N.

J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic, “Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+δ,” Nature 398, 221–223 (1999).
[CrossRef]

Ellis, C. T.

D. K. George, A. V. Stier, C. T. Ellis, B. D. McCombe, J. Černe, and A. G. Markelz, “Terahertz magneto optical polarization modulation spectroscopy,” (submitted) J. Opt. Soc. Am. B (2012).

Fejer, M. M.

J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, “High resolution polar Kerr effect measurements of Sr2RuO4: evidence for broken time-reversal symmetry in the superconducting state,” Phys. Rev. Lett. 97, 167002 (2006).
[CrossRef] [PubMed]

A. Kapitulnik, J. S. Dodge, and M. M. Fejer, “High-resolution magneto-optic measurements with a Sagnac interferometer,” J. Appl. Phys. 75, 6872–6877 (1994).
[CrossRef]

Ferguson, B.

S. Wang, B. Ferguson, D. Abbott, and X.-C. Zhang, “T-ray imaging and tomography,” J. Biol. Phys. Chem. 29, 247–256 (2003).

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]

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, (John Wiley & Sons, Ltd., 2006).

Gallot, G.

Gatesman, A.J.

A.J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yagvesson, “An anti-reflection coating for silicon optics at terahertz frequencies,” IEEE Microw. Guid. Wave Lett. 10, 264–266 (2000).
[CrossRef]

George, D. K.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

D. K. George, A. V. Stier, C. T. Ellis, B. D. McCombe, J. Černe, and A. G. Markelz, “Terahertz magneto optical polarization modulation spectroscopy,” (submitted) J. Opt. Soc. Am. B (2012).

Grayson, M.

M. Grayson, L. B. Rigal, D. C. Schmadel, H. D. Drew, and P.-J. Kung, “Spectral measurement of the Hall angle response in normal state cuprate superconductors,” Phys. Rev. Lett. 89, 037003 (2002).
[CrossRef] [PubMed]

Hägele, D.

R. A. Kaindl, M. A. Carnahan, D. Hägele, R. Lövenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

Hangyo, M.

Hauge, R. H.

L. Ren, C. L. Pint, T. Arikawa, K. Takeya, I. Kawayama, M. Tonouchi, R. H. Hauge, and J. Kono, “Broadband terahertz polarizers with ideal performance based on aligned carbon nanotube stacks,” Nano Lett. 12, 787–790 (2012).
[CrossRef] [PubMed]

Heinz, T. F.

Heyman, J. N.

J. N. Heyman, R. Kersting, and K. Unterrainer, “Time-domain measurement of intersubband oscillations in a quantum well,” Appl. Phys. Lett. 72, 644–646 (1998).
[CrossRef]

Hirota, Y.

Hu, B. B.

Huang, C.

Hughes, T. L.

X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Topological field theory of time-reversal invariant insulators” Phys. Rev. B 78, 195424 (2008).

Humlicek, J.

C. Bernhard, J. Humlicek, and B. Keimer, “Far-infrared ellipsometry using a synchrotron light source-the dielectric response of the cuprate high Tc superconductors,” Thin Solid Films 455, 143–149 (2004).
[CrossRef]

Hussain, A

Ikebe, Y.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002 (2011).
[CrossRef]

Jagadish, C.

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]

Jenkins, G. S.

G. S. Jenkins, D. C. Schmadel, and H. D. Drew, “Simultaneous measurement of circular dichroism and Faraday rotation at terahertz frequencies utilizing electric field sensitive detection via polarization modulation,” Rev. Sci. Instrum. 81, 083903 (2010).
[CrossRef] [PubMed]

Ji, M.

A.J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yagvesson, “An anti-reflection coating for silicon optics at terahertz frequencies,” IEEE Microw. Guid. Wave Lett. 10, 264–266 (2000).
[CrossRef]

Johnson, J. L.

J. L. Johnson, T. D. Dorney, and D. M. Mittleman, “Enhanced depth resolution in terahertz imaging using phase-shift interferometry,” Appl. Phys. Lett. 78, 835–837 (2001).
[CrossRef]

Johnson, M. B.

E. Castro-Camus and M. B. Johnson, “Conformational changes of photoactive yellow protein monitored by terahertz spectroscopy,” Chem. Phys. Lett. 455, 289–292 (2008).
[CrossRef]

Johnston, M. B.

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]

Kaindl, R. A.

R. A. Kaindl, M. A. Carnahan, D. Hägele, R. Lövenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

Kapitulnik, A.

J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, “High resolution polar Kerr effect measurements of Sr2RuO4: evidence for broken time-reversal symmetry in the superconducting state,” Phys. Rev. Lett. 97, 167002 (2006).
[CrossRef] [PubMed]

A. Kapitulnik, J. S. Dodge, and M. M. Fejer, “High-resolution magneto-optic measurements with a Sagnac interferometer,” J. Appl. Phys. 75, 6872–6877 (1994).
[CrossRef]

Kawasaki, M.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002 (2011).
[CrossRef]

Kawayama, I.

L. Ren, C. L. Pint, T. Arikawa, K. Takeya, I. Kawayama, M. Tonouchi, R. H. Hauge, and J. Kono, “Broadband terahertz polarizers with ideal performance based on aligned carbon nanotube stacks,” Nano Lett. 12, 787–790 (2012).
[CrossRef] [PubMed]

Keimer, B.

C. Bernhard, J. Humlicek, and B. Keimer, “Far-infrared ellipsometry using a synchrotron light source-the dielectric response of the cuprate high Tc superconductors,” Thin Solid Films 455, 143–149 (2004).
[CrossRef]

Kersting, R.

J. N. Heyman, R. Kersting, and K. Unterrainer, “Time-domain measurement of intersubband oscillations in a quantum well,” Appl. Phys. Lett. 72, 644–646 (1998).
[CrossRef]

Kono, J.

L. Ren, C. L. Pint, T. Arikawa, K. Takeya, I. Kawayama, M. Tonouchi, R. H. Hauge, and J. Kono, “Broadband terahertz polarizers with ideal performance based on aligned carbon nanotube stacks,” Nano Lett. 12, 787–790 (2012).
[CrossRef] [PubMed]

Kung, P.-J.

M. Grayson, L. B. Rigal, D. C. Schmadel, H. D. Drew, and P.-J. Kung, “Spectral measurement of the Hall angle response in normal state cuprate superconductors,” Phys. Rev. Lett. 89, 037003 (2002).
[CrossRef] [PubMed]

Kurz, H.

Levitov, L. S.

R. Nandkishore, L. S. Levitov, and A. V. Chubukov, “Chiral superconductivity from repulsive interactions in doped graphene,” Nat. Phys. 8, 158–163 (2012).
[CrossRef]

Liao, Z.

Liu, W.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

Lloyd-Hughes, J.

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]

Logvenov, G.

L. S. Bilbro, R. Valdés Aguilar, G. Logvenov, O. Pelleg, I. Bozovic, and N. P. Armitage, “Temporal correlations of superconductivity above the transition temperature in La2–xSrxCuO4 probed by terahertz spectroscopy,” Nat. Phys. 7, 298–302 (2011).
[CrossRef]

Lövenich, R.

R. A. Kaindl, M. A. Carnahan, D. Hägele, R. Lövenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

MacDonald, A. H.

W.-K. Tse and A. H. MacDonald, “Magneto-optical Faraday and Kerr effects in topological insulator films and in other layered quantized Hall systems,” Phys. Rev. B 84, 205327 (2011).
[CrossRef]

Maciejko, J.

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[CrossRef]

Maeno, Y.

J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, “High resolution polar Kerr effect measurements of Sr2RuO4: evidence for broken time-reversal symmetry in the superconducting state,” Phys. Rev. Lett. 97, 167002 (2006).
[CrossRef] [PubMed]

Makabe, H.

Mallozzi, R.

J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic, “Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+δ,” Nature 398, 221–223 (1999).
[CrossRef]

Markelz, A. G.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

D. K. George, A. V. Stier, C. T. Ellis, B. D. McCombe, J. Černe, and A. G. Markelz, “Terahertz magneto optical polarization modulation spectroscopy,” (submitted) J. Opt. Soc. Am. B (2012).

Masson, J.-B.

McCombe, B. D.

D. K. George, A. V. Stier, C. T. Ellis, B. D. McCombe, J. Černe, and A. G. Markelz, “Terahertz magneto optical polarization modulation spectroscopy,” (submitted) J. Opt. Soc. Am. B (2012).

Mittleman, D. M.

J. L. Johnson, T. D. Dorney, and D. M. Mittleman, “Enhanced depth resolution in terahertz imaging using phase-shift interferometry,” Appl. Phys. Lett. 78, 835–837 (2001).
[CrossRef]

Musante, C.

A.J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yagvesson, “An anti-reflection coating for silicon optics at terahertz frequencies,” IEEE Microw. Guid. Wave Lett. 10, 264–266 (2000).
[CrossRef]

Nagaosa, N.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002 (2011).
[CrossRef]

Nagel, M.

Nandkishore, R.

R. Nandkishore, L. S. Levitov, and A. V. Chubukov, “Chiral superconductivity from repulsive interactions in doped graphene,” Nat. Phys. 8, 158–163 (2012).
[CrossRef]

Neshat, M.

M. Neshat and N. P. Armitage, “Improved measurement of polarization state in terahertz polarization spectroscopy,” (submitted) Opt. Lett. (2012).

Nuss, M.

M. Nuss and J. Orenstein, “Terahertz time-domain spectroscopy” in Millimeter and Submillimeter Wave Spectroscopy of Solids, George Grüner, ed. (Springer, 1998), Topics in Applied Physics 74, pp. 7–50.
[CrossRef]

Nuss, M. C.

Oh, S.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

Orenstein, J.

J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic, “Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+δ,” Nature 398, 221–223 (1999).
[CrossRef]

M. Nuss and J. Orenstein, “Terahertz time-domain spectroscopy” in Millimeter and Submillimeter Wave Spectroscopy of Solids, George Grüner, ed. (Springer, 1998), Topics in Applied Physics 74, pp. 7–50.
[CrossRef]

Pelleg, O.

L. S. Bilbro, R. Valdés Aguilar, G. Logvenov, O. Pelleg, I. Bozovic, and N. P. Armitage, “Temporal correlations of superconductivity above the transition temperature in La2–xSrxCuO4 probed by terahertz spectroscopy,” Nat. Phys. 7, 298–302 (2011).
[CrossRef]

Pint, C. L.

L. Ren, C. L. Pint, T. Arikawa, K. Takeya, I. Kawayama, M. Tonouchi, R. H. Hauge, and J. Kono, “Broadband terahertz polarizers with ideal performance based on aligned carbon nanotube stacks,” Nano Lett. 12, 787–790 (2012).
[CrossRef] [PubMed]

Qi, X.-L.

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[CrossRef]

X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Topological field theory of time-reversal invariant insulators” Phys. Rev. B 78, 195424 (2008).

Ren, L.

L. Ren, C. L. Pint, T. Arikawa, K. Takeya, I. Kawayama, M. Tonouchi, R. H. Hauge, and J. Kono, “Broadband terahertz polarizers with ideal performance based on aligned carbon nanotube stacks,” Nano Lett. 12, 787–790 (2012).
[CrossRef] [PubMed]

Rigal, L. B.

M. Grayson, L. B. Rigal, D. C. Schmadel, H. D. Drew, and P.-J. Kung, “Spectral measurement of the Hall angle response in normal state cuprate superconductors,” Phys. Rev. Lett. 89, 037003 (2002).
[CrossRef] [PubMed]

Schmadel, D. C.

G. S. Jenkins, D. C. Schmadel, and H. D. Drew, “Simultaneous measurement of circular dichroism and Faraday rotation at terahertz frequencies utilizing electric field sensitive detection via polarization modulation,” Rev. Sci. Instrum. 81, 083903 (2010).
[CrossRef] [PubMed]

M. Grayson, L. B. Rigal, D. C. Schmadel, H. D. Drew, and P.-J. Kung, “Spectral measurement of the Hall angle response in normal state cuprate superconductors,” Phys. Rev. Lett. 89, 037003 (2002).
[CrossRef] [PubMed]

Shan, J.

Shimano, R.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002 (2011).
[CrossRef]

Stier, A. V.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

D. K. George, A. V. Stier, C. T. Ellis, B. D. McCombe, J. Černe, and A. G. Markelz, “Terahertz magneto optical polarization modulation spectroscopy,” (submitted) J. Opt. Soc. Am. B (2012).

Takahashi, K. S.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002 (2011).
[CrossRef]

Takeya, K.

L. Ren, C. L. Pint, T. Arikawa, K. Takeya, I. Kawayama, M. Tonouchi, R. H. Hauge, and J. Kono, “Broadband terahertz polarizers with ideal performance based on aligned carbon nanotube stacks,” Nano Lett. 12, 787–790 (2012).
[CrossRef] [PubMed]

Tan, H. H.

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.

Tokura, Y.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002 (2011).
[CrossRef]

Tonouchi, M.

L. Ren, C. L. Pint, T. Arikawa, K. Takeya, I. Kawayama, M. Tonouchi, R. H. Hauge, and J. Kono, “Broadband terahertz polarizers with ideal performance based on aligned carbon nanotube stacks,” Nano Lett. 12, 787–790 (2012).
[CrossRef] [PubMed]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[CrossRef]

Tse, W.-K.

W.-K. Tse and A. H. MacDonald, “Magneto-optical Faraday and Kerr effects in topological insulator films and in other layered quantized Hall systems,” Phys. Rev. B 84, 205327 (2011).
[CrossRef]

Unterrainer, K.

J. N. Heyman, R. Kersting, and K. Unterrainer, “Time-domain measurement of intersubband oscillations in a quantum well,” Appl. Phys. Lett. 72, 644–646 (1998).
[CrossRef]

Valdés Aguilar, R.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

L. S. Bilbro, R. Valdés Aguilar, G. Logvenov, O. Pelleg, I. Bozovic, and N. P. Armitage, “Temporal correlations of superconductivity above the transition temperature in La2–xSrxCuO4 probed by terahertz spectroscopy,” Nat. Phys. 7, 298–302 (2011).
[CrossRef]

Waldman, J.

A.J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yagvesson, “An anti-reflection coating for silicon optics at terahertz frequencies,” IEEE Microw. Guid. Wave Lett. 10, 264–266 (2000).
[CrossRef]

Wang, S.

S. Wang, B. Ferguson, D. Abbott, and X.-C. Zhang, “T-ray imaging and tomography,” J. Biol. Phys. Chem. 29, 247–256 (2003).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, 1997).

Wu, L.

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

Xia, J.

J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, “High resolution polar Kerr effect measurements of Sr2RuO4: evidence for broken time-reversal symmetry in the superconducting state,” Phys. Rev. Lett. 97, 167002 (2006).
[CrossRef] [PubMed]

Yagvesson, S.

A.J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yagvesson, “An anti-reflection coating for silicon optics at terahertz frequencies,” IEEE Microw. Guid. Wave Lett. 10, 264–266 (2000).
[CrossRef]

Zhang, S.-C.

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[CrossRef]

X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Topological field theory of time-reversal invariant insulators” Phys. Rev. B 78, 195424 (2008).

Zhang, X.-C.

S. Wang, B. Ferguson, D. Abbott, and X.-C. Zhang, “T-ray imaging and tomography,” J. Biol. Phys. Chem. 29, 247–256 (2003).

Zhang, X-C

X-C Zhang, “Terahertz wave imaging: horizons and hurdles,” Phys. Med. Biol. 47, 3667 (2002).
[CrossRef] [PubMed]

Zhao, S.

Appl. Opt. (1)

Appl. Phys. Lett. (3)

J. L. Johnson, T. D. Dorney, and D. M. Mittleman, “Enhanced depth resolution in terahertz imaging using phase-shift interferometry,” Appl. Phys. Lett. 78, 835–837 (2001).
[CrossRef]

J. N. Heyman, R. Kersting, and K. Unterrainer, “Time-domain measurement of intersubband oscillations in a quantum well,” Appl. Phys. Lett. 72, 644–646 (1998).
[CrossRef]

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]

Chem. Phys. Lett. (1)

E. Castro-Camus and M. B. Johnson, “Conformational changes of photoactive yellow protein monitored by terahertz spectroscopy,” Chem. Phys. Lett. 455, 289–292 (2008).
[CrossRef]

Europhys. Lett. (1)

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002 (2011).
[CrossRef]

IEEE Microw. Guid. Wave Lett. (1)

A.J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yagvesson, “An anti-reflection coating for silicon optics at terahertz frequencies,” IEEE Microw. Guid. Wave Lett. 10, 264–266 (2000).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

E. Castro-Camus, “Polarization-resolved terahertz time-domain spectroscopy,” Int. J. Infrared Millim. Waves 33, 418–430 (2012).

J. Appl. Phys. (1)

A. Kapitulnik, J. S. Dodge, and M. M. Fejer, “High-resolution magneto-optic measurements with a Sagnac interferometer,” J. Appl. Phys. 75, 6872–6877 (1994).
[CrossRef]

J. Biol. Phys. Chem. (1)

S. Wang, B. Ferguson, D. Abbott, and X.-C. Zhang, “T-ray imaging and tomography,” J. Biol. Phys. Chem. 29, 247–256 (2003).

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

Nano Lett. (1)

L. Ren, C. L. Pint, T. Arikawa, K. Takeya, I. Kawayama, M. Tonouchi, R. H. Hauge, and J. Kono, “Broadband terahertz polarizers with ideal performance based on aligned carbon nanotube stacks,” Nano Lett. 12, 787–790 (2012).
[CrossRef] [PubMed]

Nat. Photonics (1)

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[CrossRef]

Nat. Phys. (2)

L. S. Bilbro, R. Valdés Aguilar, G. Logvenov, O. Pelleg, I. Bozovic, and N. P. Armitage, “Temporal correlations of superconductivity above the transition temperature in La2–xSrxCuO4 probed by terahertz spectroscopy,” Nat. Phys. 7, 298–302 (2011).
[CrossRef]

R. Nandkishore, L. S. Levitov, and A. V. Chubukov, “Chiral superconductivity from repulsive interactions in doped graphene,” Nat. Phys. 8, 158–163 (2012).
[CrossRef]

Nature (2)

J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic, “Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+δ,” Nature 398, 221–223 (1999).
[CrossRef]

R. A. Kaindl, M. A. Carnahan, D. Hägele, R. Lövenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (2)

Phys. Med. Biol. (1)

X-C Zhang, “Terahertz wave imaging: horizons and hurdles,” Phys. Med. Biol. 47, 3667 (2002).
[CrossRef] [PubMed]

Phys. Rev. B (2)

W.-K. Tse and A. H. MacDonald, “Magneto-optical Faraday and Kerr effects in topological insulator films and in other layered quantized Hall systems,” Phys. Rev. B 84, 205327 (2011).
[CrossRef]

X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Topological field theory of time-reversal invariant insulators” Phys. Rev. B 78, 195424 (2008).

Phys. Rev. Lett. (4)

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[CrossRef]

M. Grayson, L. B. Rigal, D. C. Schmadel, H. D. Drew, and P.-J. Kung, “Spectral measurement of the Hall angle response in normal state cuprate superconductors,” Phys. Rev. Lett. 89, 037003 (2002).
[CrossRef] [PubMed]

R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitager, “Thz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[CrossRef] [PubMed]

J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, “High resolution polar Kerr effect measurements of Sr2RuO4: evidence for broken time-reversal symmetry in the superconducting state,” Phys. Rev. Lett. 97, 167002 (2006).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

G. S. Jenkins, D. C. Schmadel, and H. D. Drew, “Simultaneous measurement of circular dichroism and Faraday rotation at terahertz frequencies utilizing electric field sensitive detection via polarization modulation,” Rev. Sci. Instrum. 81, 083903 (2010).
[CrossRef] [PubMed]

Thin Solid Films (1)

C. Bernhard, J. Humlicek, and B. Keimer, “Far-infrared ellipsometry using a synchrotron light source-the dielectric response of the cuprate high Tc superconductors,” Thin Solid Films 455, 143–149 (2004).
[CrossRef]

Other (6)

M. Neshat and N. P. Armitage, “Improved measurement of polarization state in terahertz polarization spectroscopy,” (submitted) Opt. Lett. (2012).

D. K. George, A. V. Stier, C. T. Ellis, B. D. McCombe, J. Černe, and A. G. Markelz, “Terahertz magneto optical polarization modulation spectroscopy,” (submitted) J. Opt. Soc. Am. B (2012).

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, (John Wiley & Sons, Ltd., 2006).

M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, 1997).

M. Nuss and J. Orenstein, “Terahertz time-domain spectroscopy” in Millimeter and Submillimeter Wave Spectroscopy of Solids, George Grüner, ed. (Springer, 1998), Topics in Applied Physics 74, pp. 7–50.
[CrossRef]

M. S. Sherwin, C. A. Schmuttenmaer, and P. H. Bucksbaumeds., “Opportunities in THz science,” DOE-NSF-NIH Workshop, Feb. 12–14, 2004, http://science.energy.gov/~/media/bes/pdf/reports/files/thz_rpt.pdf .

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Figures (3)

Fig. 1
Fig. 1

a) The rotator with the QMC polarizer mounted in the rotating cylinder. b) Experimental layout: a linearly polarized terahertz waveform passes through a sample, which introduces ellipticity. The rotating polarizer modulates the polarization, and the modulation is detected by the lock-in amplifier.

Fig. 2
Fig. 2

System resolution characterization. a) Measurement of test polarizer angles from 20° to 80°. b) Comparison of the accuracy and precision for small and large test polarizer angles. c) Ellipticity angle ε for two polarizer angles. d) A test polarizer angle of ∼ 6°, showing 0.05° accuracy and 0.02° precision up to 0.8 THz. The shaded area in the top panel indicates the error level shown in the bottom panel.

Fig. 3
Fig. 3

Birefringence of sapphire. a) Single measurement of the birefringence of sapphire with the electric field incident on the sample at a 45° angle to the two principle axes. b) Calculation of the birefringence (neno) for the ordinary and extraordinary axes for a static polarizer measurement (requiring 2 measurements plus a reference) and a polarization modulation measurement (requiring 1 measurement).

Equations (36)

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E x ( t ) = 1 2 π d ω e i ω t E x ( ω )
( E x f ( ω ) E y f ( ω ) ) = ( M ˜ x x ( ω ) M ˜ x y ( ω ) M ˜ y x ( ω ) M ˜ y y ( ω ) ) ( E x 0 ( ω ) E y 0 ( ω ) )
P Θ = ( cos 2 ( Θ ) cos ( Θ ) sin ( Θ ) cos ( Θ ) sin ( Θ ) sin 2 ( Θ ) )
T ( ω ) = ( t ˜ x x ( ω ) t ˜ x y ( ω ) t ˜ y x ( ω ) t ˜ y y ( ω ) )
( E x f E y f ) = P Ω t T ( E x 0 E y 0 ) = ( E x 0 [ t ˜ x x cos 2 ( Ω t r ) + t ˜ y x cos ( Ω t r ) sin ( Ω t r ) ] E x 0 [ t ˜ x x cos ( Ω t r ) sin ( Ω t r ) + t ˜ y x sin 2 ( Ω t r ) ] + E y 0 [ t ˜ x y cos 2 ( Ω t r ) + t ˜ y y cos ( Ω t r ) sin ( Ω t r ) ] + E y 0 [ t ˜ x y cos ( Ω t r ) sin ( Ω t r ) + t ˜ y y sin 2 ( Ω t r ) ] )
( E x f E y f ) = P x P Ω t T P x ( E x 0 E y 0 ) = ( E x 0 ( t ˜ x x cos 2 ( Ω t r ) + t ˜ y x cos ( Ω t r ) sin ( Ω t r ) ) 0 ) = ( E x 0 2 ( t ˜ x x ( 1 + cos ( 2 Ω t r ) ) + t ˜ y x sin ( 2 Ω t r ) ) 0 )
( E x f E y f ) = P y P Ω t T P y ( E x 0 E y 0 ) = ( 0 E y 0 2 ( t ˜ x y sin ( 2 Ω t r ) + t ˜ y y ( 1 cos ( 2 Ω t r ) ) ) )
E x f ( t p , t r ) = 1 2 π d ω e i ω t p E x f ( ω , t r )
S X ( t p ) = 1 τ 0 τ d t r cos ( 2 Ω t r ) E x f ( t p , t r ) = 1 τ 2 π 0 τ d t r cos ( 2 Ω t r ) d ω e i ω t p E x f ( ω , t r )
S X ( ω ) = 1 2 π τ 0 τ d t r cos ( 2 Ω t r ) d t p e i ω t p d ω e i ω t p E x f ( ω , t r ) = 1 τ 0 τ d t r cos ( 2 Ω t r ) E x f ( ω , t r )
S X ( ω ) = R 0 ( ω ) τ 0 τ d t r ( E x 0 ( ω ) t ˜ x x ( ω ) 2 ( 1 + cos ( 2 Ω t r ) ) + E x 0 ( ω ) t ˜ y x ( ω ) 2 ( sin ( 2 Ω t r ) ) ) cos ( 2 Ω t r ) = R 0 ( ω ) τ ( E x 0 ( ω ) t ˜ x x ( ω ) 4 0 τ d t r ( 2 cos ( 2 Ω t r ) + cos ( 4 Ω t r ) + 1 ) + E x 0 ( ω ) t ˜ y x ( ω ) 4 0 τ d t r sin ( 4 Ω t r ) ) = 1 4 R 0 ( ω ) E x 0 ( ω ) t ˜ x x ( ω )
S Y ( ω ) = 1 4 R 0 ( ω ) E x 0 ( ω ) t ˜ y x ( ω )
( E x f E y f ) = T P x ( E x 0 E y 0 ) = ( t ˜ x x E x 0 t ˜ y x E x 0 )
( cos ( ϕ 34 ) sin ( ϕ 34 ) sin ( ϕ 34 ) cos ( ϕ 34 ) )
S X = R 0 E x 0 4 ( t ˜ x x cos ( ϕ 34 ) t ˜ y x sin ( ϕ 34 ) ) S Y = R 0 E x 0 4 ( t ˜ y x cos ( ϕ 34 ) + t ˜ x x sin ( ϕ 34 ) )
S X = R 0 E x 0 4 t ˜ x x cos ( ϕ 34 ) , S Y = R 0 E x 0 4 t ˜ y x cos ( ϕ 34 )
P = ( cos 2 ( θ ) + η sin 2 ( θ ) ( 1 η ) cos ( θ ) sin ( θ ) ( 1 η ) cos ( θ ) sin ( θ ) η cos 2 ( θ ) + sin 2 ( θ ) )
S X = R 0 E x 0 4 t ˜ x x cos ( ϕ 34 ) ( 1 η r ) , S Y = R 0 E x 0 4 t ˜ y x cos ( ϕ 34 ) ( 1 η r )
( 1 0 0 0 ) ( 1 0 0 η )
S Y ( ω ) S X ( ω ) = E x 0 ( ω ) t ˜ y x ( ω ) E x 0 ( ω ) t ˜ x x ( ω ) = t ˜ y x ( ω ) t ˜ x x ( ω )
t ˜ y x ( ω ) t ˜ x x ( ω ) = sin [ ϕ ( ω ) ] cos [ ϕ ( ω ) ]
arctan ( sin [ ϕ ( ω ) ] cos [ ϕ ( ω ) ] ) = ϕ ( ω )
arctan ( S Y ( ω ) S X ( ω ) ) = ϕ ( ω ) ( E x 0 E y 0 + E x 0 cot ( ϕ 12 ) + E y 0 E x 0 + E y 0 tan ( ϕ 12 ) + tan ( ϕ 34 ) ) η
ρ = E y ( ω ) E x ( ω ) = tan [ ψ ( ω ) ] e i Δ ( ω )
tan ( 2 θ ) = 2 tan ψ cos Δ 1 tan 2 ψ
sin ( 2 ε ) = 2 tan ψ sin Δ 1 + tan 2 ψ
arctan ρ = 1 2 arg ( 1 + tan ψ e i Δ ) 1 2 arg ( 1 tan ψ e i Δ ) + i 1 4 log ( tan 2 ψ cos 2 Δ + ( 1 + tan ψ sin Δ ) 2 tan 2 ψ cos 2 Δ + ( 1 tan ψ sin Δ ) 2 )
Re [ arctan ρ ] = 1 2 arg ( 1 + tan ψ e i Δ ) 1 2 arg ( 1 tan ψ e i Δ ) = 1 2 [ arctan ( tan ψ cos Δ 1 tan ψ sin Δ ) + arctan ( tan ψ cos Δ 1 + tan ψ sin Δ ) ] = 1 2 arctan ( 2 tan ψ cos Δ 1 tan 2 ψ )
Re [ arctan ρ ] = Re [ arctan ( E y ( ω ) E x ( ω ) ) ] = 1 2 arctan ( tan 2 θ ) = θ
Im [ arctan ρ ] = 1 4 log ( tan 2 ψ cos Δ 2 + ( 1 + tan ψ sin Δ ) 2 tan 2 ψ cos Δ 2 + ( 1 tan ψ sin Δ ) 2 ) = 1 4 log [ 1 + 2 tan ψ sin Δ 1 + tan 2 ψ 1 2 tan ψ sin Δ 1 + tan 2 ψ ] = 1 2 arctanh ( 2 tan ψ sin Δ 1 + tan 2 ψ )
Im [ arctan ρ ] = 1 2 arctanh ( sin ( 2 ε ) )
( e i φ x cos 2 α + e i φ y sin 2 α ( e i φ x e i φ y ) cos α sin α ( e i φ x e i φ y ) cos α sin α e i φ x sin 2 α + e i φ y cos 2 α )
1 2 ( e i φ x + e i φ y e i φ x e i φ y e i φ x e i φ y e i φ x + e i φ y )
S Y ( ω ) S X ( ω ) = e i φ x e i φ y e i φ x + e i φ y = i sin ( Δ φ / 2 ) cos ( Δ φ / 2 )
Δ φ = 2 arctan ( Im [ S Y ( ω ) S X ( ω ) ] )
S 45 ° = e i φ x e i φ y S + 45 ° = e i φ x + e i φ y

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