Abstract

We theoretically and experimentally investigate the accuracy with which the magnitude and direction of the electric field (E-field) vector of electromagnetic waves can be determined using the crystal angle dependence of the electro-optic (EO) effect. The mathematical treatment in this paper is a large extension of our previous work to determine the E-field direction of terahertz electromagnetic waves by the spinning EO sensor method [Rev. Sci. Instrum. 83, 023104 (2012)]. Here we include misadjustments of the wave plate and polarizer in the experimental setup as well as the effect of the residual birefringence of the EO crystal due to uniform and local strains. The main results are as follows: (1) When there is no residual birefringence in the EO crystal, misadjustments of the wave plate and polarizer do not affect the experimentally determined direction of the E-field vector. This is true even when the term proportional to the square of the E-field magnitude of the EO signal becomes important. (2) The error due to residual birefringence can be effectively eliminated by a signal subtraction algorithm and it is roughly the product of the misadjustment angle of the wave plate and the degree of residual birefringence, which is very small. (3) The error does not depend on the magnitude of the E-field; thus, we can apply the technique when the E-field is weak and the polarization rotation of the probe pulse caused by the EO effect is much smaller than that induced by residual birefringence. These results give a mathematical basis for the accuracy and reliability of the spinning EO sensor method, which is robust, and will be useful for ultrabroadband E-field vector sensing at far-infrared to mid-infrared frequencies.

© 2013 Optical Society of America

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2013

S. Watanabe, N. Yasumatsu, K. Oguchi, M. Takeda, T. Suzuki, T. Tachizaki, “A real-time terahertz time-domain polarization analyzer with 80-MHz repetition-rate femtosecond laser pulses,” Sensors 13, 3299–3312 (2013).
[CrossRef]

2012

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

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

N. Yasumatsu, S. Watanabe, “Precise real-time polarization measurement of terahertz electromagnetic waves by a spinning electro-optic sensor,” Rev. Sci. Instrum. 83, 023104 (2012).
[CrossRef]

M. Neshat, N. P. Armitage, “Improved measurement of polarization state in terahertz polarization spectroscopy,” Opt. Lett. 37, 1811–1813 (2012).
[CrossRef]

C. M. Morris, R. V. Aguilar, A. V. Stier, N. P. Armitage, “Polarization modulation time-domain terahertz polarimetry,” Opt. Express 20, 12303–12317 (2012).
[CrossRef]

N. Yasumatsu, S. Watanabe, “T-ray topography by time-domain polarimetry,” Opt. Lett. 37, 2706–2708 (2012).
[CrossRef]

R. Imai, N. Kanda, T. Higuchi, Z. Zheng, K. Konishi, M. Kuwata-Gonokami, “Terahertz vector beam generation using segmented nonlinear optical crystals with threefold rotational symmetry,” Opt. Express 20, 21896–21904 (2012).
[CrossRef]

S. Katletz, M. Pfleger, H. Puhringer, M. Mikulics, N. Vieweg, O. Peters, B. Scherger, M. Scheller, M. Koch, K. Wiesauer, “Polarization sensitive terahertz imaging: detection of birefringence and optical axis,” Opt. Express 20, 23025–23035 (2012).
[CrossRef]

2011

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “THz wave polarization-controlled spectroscopic imaging for anisotropic materials,” Opt. Commun. 284, 4356–4359 (2011).
[CrossRef]

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002–17006 (2011).
[CrossRef]

2010

G. S. Jenkins, D. C. Schmadel, 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]

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
[CrossRef]

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “Terahertz polarization imaging with birefringent materials,” Opt. Commun. 283, 4993–4995 (2010).
[CrossRef]

2008

2007

H. Makabe, Y. Hirota, M. Tani, M. Hangyo, “Polarization state measurement of terahertz electromagnetic radiation by three-contact photoconductive antenna,” Opt. Express 15, 11650–11657 (2007).
[CrossRef]

T. Kampfrath, J. Notzold, M. Wolf, “Sampling of broadband terahertz pulses with thick electro-optic crystals,” Appl. Phys. Lett. 90, 231113 (2007).
[CrossRef]

2005

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

N. C. J. van der Valk, W. A. M. van der Marel, P. C. M. Planken, “Terahertz polarization imaging,” Opt. Lett. 30, 2802–2804 (2005).
[CrossRef]

2004

2003

M. Usami, R. Fukasawa, M. Tani, M. Watanabe, K. Sakai, “Calibration free terahertz imaging based on 2D electro-optic sampling technique,” Electron. Lett. 39, 1746–1747 (2003).
[CrossRef]

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, J. Allen, 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]

2001

T. Nagashima, M. Hangyo, “Measurement of complex optical constants of a highly doped Si wafer using terahertz ellipsometry,” Appl. Phys. Lett. 79, 3917–3919 (2001).
[CrossRef]

P. C. M. Planken, H.-K. Nienhuys, H. J. Bakker, T. Wenckebach, “Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe,” J. Opt. Soc. Am. B 18, 313–317 (2001).
[CrossRef]

2000

1999

Z. Jiang, F. G. Sun, Q. Chen, X. C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

1997

Q. Wu, X. C. Zhang, “Free-space electro-optics sampling of mid-infrared pulses,” Appl. Phys. Lett. 71, 1285–1286 (1997).
[CrossRef]

D. M. Mittleman, J. Cunningham, M. C. Nuss, M. Geva, “Noncontact semiconductor wafer characterization with the terahertz Hall effect,” Appl. Phys. Lett. 71, 16–18 (1997).
[CrossRef]

1995

1969

Aguilar, R. V.

C. M. Morris, R. V. Aguilar, A. V. Stier, N. P. Armitage, “Polarization modulation time-domain terahertz polarimetry,” Opt. Express 20, 12303–12317 (2012).
[CrossRef]

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

Allen, J.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, J. Allen, 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]

Andrews, S. R.

Aoki, H.

Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
[CrossRef]

Armitage, N. P.

M. Neshat, N. P. Armitage, “Improved measurement of polarization state in terahertz polarization spectroscopy,” Opt. Lett. 37, 1811–1813 (2012).
[CrossRef]

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

C. M. Morris, R. V. Aguilar, A. V. Stier, N. P. Armitage, “Polarization modulation time-domain terahertz polarimetry,” Opt. Express 20, 12303–12317 (2012).
[CrossRef]

Bakker, H. J.

Bansal, N.

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

Bilbro, L. S.

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

Birge, R. R.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, J. Allen, 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]

Brunken, M.

M. Brunken, H. Genz, P. Göttlicher, C. Hessler, M. Hüning, H. Loos, A. Richter, H. Schlarb, “Electro-optic sampling at the TESLA test accelerator: experimental setup and first results,” , pp. 1–24 (2003).

Castro-Camus, E.

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

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

Cerne, J.

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

Chen, Q.

Z. Jiang, F. G. Sun, Q. Chen, X. C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

Cui, Y.

Cunningham, J.

D. M. Mittleman, J. Cunningham, M. C. Nuss, M. Geva, “Noncontact semiconductor wafer characterization with the terahertz Hall effect,” Appl. Phys. Lett. 71, 16–18 (1997).
[CrossRef]

Deng, C.

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “THz wave polarization-controlled spectroscopic imaging for anisotropic materials,” Opt. Commun. 284, 4356–4359 (2011).
[CrossRef]

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “Terahertz polarization imaging with birefringent materials,” Opt. Commun. 283, 4993–4995 (2010).
[CrossRef]

Drew, H. D.

G. S. Jenkins, D. C. Schmadel, 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]

Eggimann, T.

Fraser, M. D.

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

Fukasawa, R.

M. Usami, R. Fukasawa, M. Tani, M. Watanabe, K. Sakai, “Calibration free terahertz imaging based on 2D electro-optic sampling technique,” Electron. Lett. 39, 1746–1747 (2003).
[CrossRef]

Galan, J. F.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, J. Allen, 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]

Genz, H.

M. Brunken, H. Genz, P. Göttlicher, C. Hessler, M. Hüning, H. Loos, A. Richter, H. Schlarb, “Electro-optic sampling at the TESLA test accelerator: experimental setup and first results,” , pp. 1–24 (2003).

George, D. K.

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

Geva, M.

D. M. Mittleman, J. Cunningham, M. C. Nuss, M. Geva, “Noncontact semiconductor wafer characterization with the terahertz Hall effect,” Appl. Phys. Lett. 71, 16–18 (1997).
[CrossRef]

Göttlicher, P.

M. Brunken, H. Genz, P. Göttlicher, C. Hessler, M. Hüning, H. Loos, A. Richter, H. Schlarb, “Electro-optic sampling at the TESLA test accelerator: experimental setup and first results,” , pp. 1–24 (2003).

Hangyo, M.

H. Makabe, Y. Hirota, M. Tani, M. Hangyo, “Polarization state measurement of terahertz electromagnetic radiation by three-contact photoconductive antenna,” Opt. Express 15, 11650–11657 (2007).
[CrossRef]

T. Nagashima, M. Hangyo, “Measurement of complex optical constants of a highly doped Si wafer using terahertz ellipsometry,” Appl. Phys. Lett. 79, 3917–3919 (2001).
[CrossRef]

Hessler, C.

M. Brunken, H. Genz, P. Göttlicher, C. Hessler, M. Hüning, H. Loos, A. Richter, H. Schlarb, “Electro-optic sampling at the TESLA test accelerator: experimental setup and first results,” , pp. 1–24 (2003).

Higuchi, T.

Hirota, Y.

Huber, R.

Hüning, M.

M. Brunken, H. Genz, P. Göttlicher, C. Hessler, M. Hüning, H. Loos, A. Richter, H. Schlarb, “Electro-optic sampling at the TESLA test accelerator: experimental setup and first results,” , pp. 1–24 (2003).

Hussain, A.

Ikebe, Y.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002–17006 (2011).
[CrossRef]

Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
[CrossRef]

Imai, R.

Jagadish, C.

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

Jansen, C.

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Jenkins, G. S.

G. S. Jenkins, D. C. Schmadel, 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]

Jiang, Z.

Z. Jiang, F. G. Sun, Q. Chen, X. C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

Jiang, Z. P.

Johnston, M. B.

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

Jordens, C.

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Kampfrath, T.

T. Kampfrath, J. Notzold, M. Wolf, “Sampling of broadband terahertz pulses with thick electro-optic crystals,” Appl. Phys. Lett. 90, 231113 (2007).
[CrossRef]

Kanda, N.

Katletz, S.

Kawasaki, M.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002–17006 (2011).
[CrossRef]

Koch, M.

S. Katletz, M. Pfleger, H. Puhringer, M. Mikulics, N. Vieweg, O. Peters, B. Scherger, M. Scheller, M. Koch, K. Wiesauer, “Polarization sensitive terahertz imaging: detection of birefringence and optical axis,” Opt. Express 20, 23025–23035 (2012).
[CrossRef]

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Konishi, K.

Kuwata-Gonokami, M.

Lee, R. W.

Leitenstorfer, A.

Liu, W.

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

Lloyd-Hughes, J.

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

Loos, H.

M. Brunken, H. Genz, P. Göttlicher, C. Hessler, M. Hüning, H. Loos, A. Richter, H. Schlarb, “Electro-optic sampling at the TESLA test accelerator: experimental setup and first results,” , pp. 1–24 (2003).

Makabe, H.

Markelz, A. G.

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

Masutomi, R.

Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
[CrossRef]

Mikulics, M.

Mittleman, D. M.

D. M. Mittleman, J. Cunningham, M. C. Nuss, M. Geva, “Noncontact semiconductor wafer characterization with the terahertz Hall effect,” Appl. Phys. Lett. 71, 16–18 (1997).
[CrossRef]

Morimoto, T.

Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
[CrossRef]

Morris, C. M.

Nagaosa, N.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002–17006 (2011).
[CrossRef]

Nagashima, T.

T. Nagashima, M. Hangyo, “Measurement of complex optical constants of a highly doped Si wafer using terahertz ellipsometry,” Appl. Phys. Lett. 79, 3917–3919 (2001).
[CrossRef]

Neshat, M.

Nienhuys, H.-K.

Notzold, J.

T. Kampfrath, J. Notzold, M. Wolf, “Sampling of broadband terahertz pulses with thick electro-optic crystals,” Appl. Phys. Lett. 90, 231113 (2007).
[CrossRef]

Nuss, M. C.

D. M. Mittleman, J. Cunningham, M. C. Nuss, M. Geva, “Noncontact semiconductor wafer characterization with the terahertz Hall effect,” Appl. Phys. Lett. 71, 16–18 (1997).
[CrossRef]

Oguchi, K.

S. Watanabe, N. Yasumatsu, K. Oguchi, M. Takeda, T. Suzuki, T. Tachizaki, “A real-time terahertz time-domain polarization analyzer with 80-MHz repetition-rate femtosecond laser pulses,” Sensors 13, 3299–3312 (2013).
[CrossRef]

Oh, S.

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

Okamoto, T.

Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
[CrossRef]

Peters, O.

Pfleger, M.

Planken, P. C. M.

Plaxco, K. W.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, J. Allen, 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]

Puhringer, H.

Ramian, G. J.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, J. Allen, 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]

Reisecker, V.

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Richter, A.

M. Brunken, H. Genz, P. Göttlicher, C. Hessler, M. Hüning, H. Loos, A. Richter, H. Schlarb, “Electro-optic sampling at the TESLA test accelerator: experimental setup and first results,” , pp. 1–24 (2003).

Romeike, D.

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Sakai, K.

M. Usami, R. Fukasawa, M. Tani, M. Watanabe, K. Sakai, “Calibration free terahertz imaging based on 2D electro-optic sampling technique,” Electron. Lett. 39, 1746–1747 (2003).
[CrossRef]

Savvidis, P. G.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, J. Allen, 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]

Scheller, M.

S. Katletz, M. Pfleger, H. Puhringer, M. Mikulics, N. Vieweg, O. Peters, B. Scherger, M. Scheller, M. Koch, K. Wiesauer, “Polarization sensitive terahertz imaging: detection of birefringence and optical axis,” Opt. Express 20, 23025–23035 (2012).
[CrossRef]

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Scherger, B.

Schlarb, H.

M. Brunken, H. Genz, P. Göttlicher, C. Hessler, M. Hüning, H. Loos, A. Richter, H. Schlarb, “Electro-optic sampling at the TESLA test accelerator: experimental setup and first results,” , pp. 1–24 (2003).

Schmadel, D. C.

G. S. Jenkins, D. C. Schmadel, 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]

Scopatz, A. M.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, J. Allen, 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]

Sell, A.

Shimano, R.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002–17006 (2011).
[CrossRef]

Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
[CrossRef]

Stier, A. V.

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

C. M. Morris, R. V. Aguilar, A. V. Stier, N. P. Armitage, “Polarization modulation time-domain terahertz polarimetry,” Opt. Express 20, 12303–12317 (2012).
[CrossRef]

Sun, F. G.

Z. Jiang, F. G. Sun, Q. Chen, X. C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

Sun, W. F.

Suzuki, T.

S. Watanabe, N. Yasumatsu, K. Oguchi, M. Takeda, T. Suzuki, T. Tachizaki, “A real-time terahertz time-domain polarization analyzer with 80-MHz repetition-rate femtosecond laser pulses,” Sensors 13, 3299–3312 (2013).
[CrossRef]

Tachizaki, T.

S. Watanabe, N. Yasumatsu, K. Oguchi, M. Takeda, T. Suzuki, T. Tachizaki, “A real-time terahertz time-domain polarization analyzer with 80-MHz repetition-rate femtosecond laser pulses,” Sensors 13, 3299–3312 (2013).
[CrossRef]

Takahashi, K. S.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002–17006 (2011).
[CrossRef]

Takeda, M.

S. Watanabe, N. Yasumatsu, K. Oguchi, M. Takeda, T. Suzuki, T. Tachizaki, “A real-time terahertz time-domain polarization analyzer with 80-MHz repetition-rate femtosecond laser pulses,” Sensors 13, 3299–3312 (2013).
[CrossRef]

Tan, H. H.

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

Tani, M.

H. Makabe, Y. Hirota, M. Tani, M. Hangyo, “Polarization state measurement of terahertz electromagnetic radiation by three-contact photoconductive antenna,” Opt. Express 15, 11650–11657 (2007).
[CrossRef]

M. Usami, R. Fukasawa, M. Tani, M. Watanabe, K. Sakai, “Calibration free terahertz imaging based on 2D electro-optic sampling technique,” Electron. Lett. 39, 1746–1747 (2003).
[CrossRef]

Tokura, Y.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002–17006 (2011).
[CrossRef]

Tsankov, D.

Usami, M.

M. Usami, R. Fukasawa, M. Tani, M. Watanabe, K. Sakai, “Calibration free terahertz imaging based on 2D electro-optic sampling technique,” Electron. Lett. 39, 1746–1747 (2003).
[CrossRef]

van der Marel, W. A. M.

van der Valk, N. C. J.

Vieweg, N.

Watanabe, M.

M. Usami, R. Fukasawa, M. Tani, M. Watanabe, K. Sakai, “Calibration free terahertz imaging based on 2D electro-optic sampling technique,” Electron. Lett. 39, 1746–1747 (2003).
[CrossRef]

Watanabe, S.

S. Watanabe, N. Yasumatsu, K. Oguchi, M. Takeda, T. Suzuki, T. Tachizaki, “A real-time terahertz time-domain polarization analyzer with 80-MHz repetition-rate femtosecond laser pulses,” Sensors 13, 3299–3312 (2013).
[CrossRef]

N. Yasumatsu, S. Watanabe, “Precise real-time polarization measurement of terahertz electromagnetic waves by a spinning electro-optic sensor,” Rev. Sci. Instrum. 83, 023104 (2012).
[CrossRef]

N. Yasumatsu, S. Watanabe, “T-ray topography by time-domain polarimetry,” Opt. Lett. 37, 2706–2708 (2012).
[CrossRef]

Wenckebach, T.

Wiesauer, K.

S. Katletz, M. Pfleger, H. Puhringer, M. Mikulics, N. Vieweg, O. Peters, B. Scherger, M. Scheller, M. Koch, K. Wiesauer, “Polarization sensitive terahertz imaging: detection of birefringence and optical axis,” Opt. Express 20, 23025–23035 (2012).
[CrossRef]

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Wieser, H.

Wietzke, S.

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Wolf, M.

T. Kampfrath, J. Notzold, M. Wolf, “Sampling of broadband terahertz pulses with thick electro-optic crystals,” Appl. Phys. Lett. 90, 231113 (2007).
[CrossRef]

Wu, L.

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

Wu, Q.

Q. Wu, X. C. Zhang, “Free-space electro-optics sampling of mid-infrared pulses,” Appl. Phys. Lett. 71, 1285–1286 (1997).
[CrossRef]

Xu, J.

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, J. Allen, 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]

Xu, X. G.

Yasumatsu, N.

S. Watanabe, N. Yasumatsu, K. Oguchi, M. Takeda, T. Suzuki, T. Tachizaki, “A real-time terahertz time-domain polarization analyzer with 80-MHz repetition-rate femtosecond laser pulses,” Sensors 13, 3299–3312 (2013).
[CrossRef]

N. Yasumatsu, S. Watanabe, “Precise real-time polarization measurement of terahertz electromagnetic waves by a spinning electro-optic sensor,” Rev. Sci. Instrum. 83, 023104 (2012).
[CrossRef]

N. Yasumatsu, S. Watanabe, “T-ray topography by time-domain polarimetry,” Opt. Lett. 37, 2706–2708 (2012).
[CrossRef]

Zentgraf, T.

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Zhang, C. L.

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “THz wave polarization-controlled spectroscopic imaging for anisotropic materials,” Opt. Commun. 284, 4356–4359 (2011).
[CrossRef]

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “Terahertz polarization imaging with birefringent materials,” Opt. Commun. 283, 4993–4995 (2010).
[CrossRef]

Zhang, L. L.

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “THz wave polarization-controlled spectroscopic imaging for anisotropic materials,” Opt. Commun. 284, 4356–4359 (2011).
[CrossRef]

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “Terahertz polarization imaging with birefringent materials,” Opt. Commun. 283, 4993–4995 (2010).
[CrossRef]

Zhang, R.

Zhang, X. C.

Z. P. Jiang, X. G. Xu, X. C. Zhang, “Improvement of terahertz imaging with a dynamic subtraction technique,” Appl. Opt. 39, 2982–2987 (2000).
[CrossRef]

Z. Jiang, F. G. Sun, Q. Chen, X. C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

Q. Wu, X. C. Zhang, “Free-space electro-optics sampling of mid-infrared pulses,” Appl. Phys. Lett. 71, 1285–1286 (1997).
[CrossRef]

Zhang, Y.

Zhao, Y. J.

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “THz wave polarization-controlled spectroscopic imaging for anisotropic materials,” Opt. Commun. 284, 4356–4359 (2011).
[CrossRef]

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “Terahertz polarization imaging with birefringent materials,” Opt. Commun. 283, 4993–4995 (2010).
[CrossRef]

Zheng, Z.

Zhong, H.

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “THz wave polarization-controlled spectroscopic imaging for anisotropic materials,” Opt. Commun. 284, 4356–4359 (2011).
[CrossRef]

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “Terahertz polarization imaging with birefringent materials,” Opt. Commun. 283, 4993–4995 (2010).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. M. Mittleman, J. Cunningham, M. C. Nuss, M. Geva, “Noncontact semiconductor wafer characterization with the terahertz Hall effect,” Appl. Phys. Lett. 71, 16–18 (1997).
[CrossRef]

T. Nagashima, M. Hangyo, “Measurement of complex optical constants of a highly doped Si wafer using terahertz ellipsometry,” Appl. Phys. Lett. 79, 3917–3919 (2001).
[CrossRef]

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

Q. Wu, X. C. Zhang, “Free-space electro-optics sampling of mid-infrared pulses,” Appl. Phys. Lett. 71, 1285–1286 (1997).
[CrossRef]

T. Kampfrath, J. Notzold, M. Wolf, “Sampling of broadband terahertz pulses with thick electro-optic crystals,” Appl. Phys. Lett. 90, 231113 (2007).
[CrossRef]

Z. Jiang, F. G. Sun, Q. Chen, X. C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[CrossRef]

Appl. Spectrosc.

Astrobiology

J. Xu, G. J. Ramian, J. F. Galan, P. G. Savvidis, A. M. Scopatz, R. R. Birge, J. Allen, 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]

Compos. Sci. Technol.

C. Jordens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, V. Reisecker, M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70, 472–477 (2010).
[CrossRef]

Electron. Lett.

M. Usami, R. Fukasawa, M. Tani, M. Watanabe, K. Sakai, “Calibration free terahertz imaging based on 2D electro-optic sampling technique,” Electron. Lett. 39, 1746–1747 (2003).
[CrossRef]

Europhys. Lett.

R. Shimano, Y. Ikebe, K. S. Takahashi, M. Kawasaki, N. Nagaosa, Y. Tokura, “Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3,” Europhys. Lett. 95, 17002–17006 (2011).
[CrossRef]

J. Infrared Millim. Terahertz Waves

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Commun.

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “Terahertz polarization imaging with birefringent materials,” Opt. Commun. 283, 4993–4995 (2010).
[CrossRef]

L. L. Zhang, H. Zhong, C. Deng, C. L. Zhang, Y. J. Zhao, “THz wave polarization-controlled spectroscopic imaging for anisotropic materials,” Opt. Commun. 284, 4356–4359 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
[CrossRef]

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

Rev. Sci. Instrum.

G. S. Jenkins, D. C. Schmadel, 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]

N. Yasumatsu, S. Watanabe, “Precise real-time polarization measurement of terahertz electromagnetic waves by a spinning electro-optic sensor,” Rev. Sci. Instrum. 83, 023104 (2012).
[CrossRef]

Sensors

S. Watanabe, N. Yasumatsu, K. Oguchi, M. Takeda, T. Suzuki, T. Tachizaki, “A real-time terahertz time-domain polarization analyzer with 80-MHz repetition-rate femtosecond laser pulses,” Sensors 13, 3299–3312 (2013).
[CrossRef]

Other

M. Brunken, H. Genz, P. Göttlicher, C. Hessler, M. Hüning, H. Loos, A. Richter, H. Schlarb, “Electro-optic sampling at the TESLA test accelerator: experimental setup and first results,” , pp. 1–24 (2003).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of general experimental setup of the spinning EO sensor method. A probe pulse for terahertz pulse detection, which is linearly polarized by passing through a Glan–Thompson (GT) prism, is incident on the 110 -oriented ZnTe EO crystal. The ZnTe crystal is placed on a hollow-shaft motor that continuously rotates with angular frequency ω during the measurements. After passing the spinning ZnTe crystal, the probe pulse travels through a quarter-wave plate (QWP) and is then split into two pulses with mutually orthogonal polarization components via a Wollaston prism (WP). We detect the intensity difference between the two probe beams by using a balanced photodetector. In Section 4, we additionally put a QWP between the GT prism and the ZnTe crystal as a compensator, to compensate for the undesired ellipticity of the probe pulse in the path after the GT prism and make it linearly polarized on the ZnTe crystal. The X ^ axis is defined as the direction parallel to the initial probe beam polarization E ^ p . In addition, s ^ is the propagation direction of the probe beam, and Y ^ = X ^ × s ^ .

Fig. 2.
Fig. 2.

Definitions of angles used in this paper. As in [27] we use a hat ( ^ ) to indicate a unit vector, and we define variables α , β , and γ as the angles between the polarization direction of the probe beam ( E ^ p ) and the direction of one of propagation modes of the detection crystal ( r ^ c 1 ), between E ^ p and the direction of one of propagation modes of the quarter-wave plate ( r ^ l 1 ), and between r ^ l 1 and one of the polarization directions of the two beams after the Wollaston prism ( r ^ w 1 ), respectively. In addition, we also define the variables θ and φ as the angles between E ^ p and the direction of the terahertz E-field ( E ^ T ), and between E ^ p and the [ 1 1 ¯ 0 ] axis ( z ^ ) of the 110 -oriented zincblende crystal, respectively. The variables α ( t ) and φ ( t ) are time-dependent in our experimental configuration because we mechanically rotate the detection crystal.

Fig. 3.
Fig. 3.

Intensity difference signal Δ I res / I tot as a function of ϕ without applying the terahertz E-field, when (a)  γ = 40 ° , (b)  γ = 45 ° (optimum condition), and (c)  γ = 50 ° .

Fig. 4.
Fig. 4.

Amplitudes of the dc ( n = 0 ), fundamental ( n = 1 ), and n th harmonic ( n = 2 10 ) Fourier components of Δ I res / I tot at γ = 45 ° in Fig. 3(b).

Fig. 5.
Fig. 5.

Experimentally measured phase of the 3 ω Fourier component in the signal as a function of Δ γ (circle points). For each point with a different angle Δ γ , we repeated the measurement 1000 times and plotted the average value. The length of the line at each point is twice the standard error of the mean of all 1000 measurements. A solid line shows the numerical estimation.

Equations (65)

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E ^ p c = exp ( i Ω n 1 L / c ) ( r ^ c 1 · E ^ p ) r ^ c 1 + exp ( i Ω n 2 L / c ) ( r ^ c 2 · E ^ p ) r ^ c 2 = exp ( i Ω n 1 L / c ) cos ( α ) r ^ c 1 exp ( i Ω n 2 L / c ) sin ( α ) r ^ c 2 = exp ( i Ω n 1 L / c ) [ cos ( α ) r ^ c 1 exp ( i C ) sin ( α ) r ^ c 2 ] .
E ^ p l = r ^ l 1 ( r ^ l 1 · E ^ p c ) + i r ^ l 2 ( r ^ l 2 · E ^ p c ) ,
E ^ p l = exp ( i Ω n 1 L / c ) [ cos ( α ) cos ( β α ) exp ( i C ) sin ( α ) sin ( β α ) ] r ^ l 1 i exp ( i Ω n 1 L / c ) [ cos ( α ) sin ( β α ) + exp ( i C ) sin ( α ) cos ( β α ) ] r ^ l 2 .
exp ( i C ) 1 + i C C 2 / 2 .
I 1 / I tot = | E ^ p l · r ^ w 1 | 2 1 2 [ 1 + cos ( 2 β ) cos ( 2 γ ) + C sin ( 2 γ ) sin ( 2 α ) + C 2 2 cos ( 2 γ ) sin ( 2 β 2 α ) sin ( 2 α ) ] ,
I 2 / I tot = | E ^ p l · r ^ w 2 | 2 1 2 [ 1 cos ( 2 β ) cos ( 2 γ ) C sin ( 2 γ ) sin ( 2 α ) C 2 2 cos ( 2 γ ) sin ( 2 β 2 α ) sin ( 2 α ) ] ,
Δ I / I tot cos ( 2 β ) cos ( 2 γ ) + C sin ( 2 γ ) sin ( 2 α ) + C 2 2 cos ( 2 γ ) sin ( 2 β 2 α ) sin ( 2 α ) .
s ^ = s x x ^ + s y y ^ + s z z ^ = 1 2 x ^ 1 2 y ^ ,
E T E T x x ^ + E T y y ^ + E T z z ^ .
x ^ = s ^ = 1 2 x ^ 1 2 y ^ ,
y ^ = z ^ ,
z ^ = 1 2 x ^ 1 2 y ^ ,
X ^ = cos ( φ ( t ) ) z ^ sin ( φ ( t ) ) y ^ ,
Y ^ = sin ( φ ( t ) ) z ^ + cos ( φ ( t ) ) y ^ ,
Z ^ = x ^ = s ^ ,
E ^ p = X ^ .
E T = | E T | cos ( θ ) X ^ + | E T | sin ( θ ) Y ^ .
E T x = 1 2 | E T | cos ( φ ( t ) θ ) ,
E T y = 1 2 | E T | cos ( φ ( t ) θ ) ,
E T z = | E T | sin ( φ ( t ) θ ) .
Δ n 1 = χ ( 2 ) | E T | 4 ε r ( sin ( φ ( t ) θ ) 1 + 3 cos 2 ( φ ( t ) θ ) ) ,
Δ n 2 = χ ( 2 ) | E T | 4 ε r ( sin ( φ ( t ) θ ) + 1 + 3 cos 2 ( φ ( t ) θ ) ) ,
C = Ω ( n 2 n 1 ) L / c Ω ( Δ n 2 Δ n 1 ) L / c = Ω L χ ( 2 ) | E T | 2 c ε r 1 + 3 cos 2 ( φ ( t ) θ ) η | E T | 1 + 3 cos 2 ( φ ( t ) θ ) ,
E 01 = | E T | cos ( φ ( t ) θ ) y ^ | E T | 2 ( 1 + 3 cos 2 ( φ ( t ) θ ) sin ( φ ( t ) θ ) ) z ^ ,
r ^ c 1 E 01 | E 01 | = 1 2 cos ( φ ( t ) θ ) | cos ( φ ( t ) θ ) | 1 + sin ( φ ( t ) θ ) 1 + 3 cos 2 ( φ ( t ) θ ) y ^ 1 2 2 1 + 3 cos 2 ( φ ( t ) θ ) sin ( φ ( t ) θ ) | cos ( φ ( t ) θ ) | 1 + sin ( φ ( t ) θ ) 1 + 3 cos 2 ( φ ( t ) θ ) z ^ ,
cos ( α φ ( t ) ) = z ^ · r ^ c 1 = 1 2 2 1 + 3 cos 2 ( φ ( t ) θ ) sin ( φ ( t ) θ ) | cos ( φ ( t ) θ ) | 1 + sin ( φ ( t ) θ ) 1 + 3 cos 2 ( φ ( t ) θ ) ,
sin ( α φ ( t ) ) = s ^ · ( z ^ × r ^ c 1 ) = 1 2 cos ( φ ( t ) θ ) | cos ( φ ( t ) θ ) | 1 + sin ( φ ( t ) θ ) 1 + 3 cos 2 ( φ ( t ) θ ) ,
sin ( 2 α 2 φ ( t ) ) = 2 sin ( α φ ( t ) ) cos ( α φ ( t ) ) = 2 cos ( φ ( t ) θ ) 1 + 3 cos 2 ( φ ( t ) θ ) ,
cos ( 2 α 2 φ ( t ) ) = 1 2 sin 2 ( α φ ( t ) ) = sin ( φ ( t ) θ ) 1 + 3 cos 2 ( φ ( t ) θ ) .
C sin ( 2 α 2 φ ( t ) ) = 2 η | E T | cos ( φ ( t ) θ ) ,
C cos ( 2 α 2 φ ( t ) ) = η | E T | sin ( φ ( t ) θ ) .
C sin ( 2 α ) = C sin { ( 2 α 2 φ ( t ) ) + 2 φ ( t ) } = C [ sin ( 2 α 2 φ ( t ) ) cos ( 2 φ ( t ) ) + cos ( 2 α 2 φ ( t ) ) sin ( 2 φ ( t ) ) ] = η | E T | [ 2 cos ( φ ( t ) θ ) cos ( 2 φ ( t ) ) sin ( φ ( t ) θ ) sin ( 2 φ ( t ) ) ] = η | E T | [ 1 2 cos ( φ ( t ) + θ ) + 3 2 cos ( 3 φ ( t ) θ ) ] .
C 2 sin ( 2 β 2 α ) sin ( 2 α ) = C sin { ( 2 β 2 φ ( t ) ) ( 2 α 2 φ ( t ) ) } × C sin { ( 2 α 2 φ ( t ) ) + 2 φ ( t ) } = η | E T | [ sin ( φ ( t ) θ ) sin ( 2 β 2 φ ( t ) ) 2 cos ( φ ( t ) θ ) cos ( 2 β 2 φ ( t ) ) ] × η | E T | [ 2 cos ( φ ( t ) θ ) cos ( 2 φ ( t ) ) sin ( φ ( t ) θ ) sin ( 2 φ ( t ) ) ] = 1 4 η 2 | E T | 2 [ cos ( φ ( t ) 2 β + θ ) + 3 cos ( 3 φ ( t ) 2 β θ ) ] × [ cos ( φ ( t ) + θ ) + 3 cos ( 3 φ ( t ) θ ) ] .
C 2 sin ( 2 β 2 α ) sin ( 2 α ) = 1 4 η 2 | E T | 2 × [ 5 cos ( 2 β ) + 3 2 cos ( 2 φ ( t ) 2 β 2 θ ) + 1 2 cos ( 2 φ ( t ) 2 β + 2 θ ) + 3 2 cos ( 2 φ ( t ) + 2 β 2 θ ) + 3 cos ( 4 φ ( t ) 2 β ) + 9 2 cos ( 6 φ ( t ) 2 β 2 θ ) ] .
I 1 / I tot 1 2 + 1 2 cos ( 2 β ) cos ( 2 γ ) + η 2 sin ( 2 γ ) | E T | [ 1 2 cos ( φ ( t ) + θ ) + 3 2 cos ( 3 φ ( t ) θ ) ] η 2 16 | E T | 2 cos ( 2 γ ) [ 5 cos ( 2 β ) + 3 2 cos ( 2 φ ( t ) 2 β 2 θ ) + 1 2 cos ( 2 φ ( t ) 2 β + 2 θ ) + 3 2 cos ( 2 φ ( t ) + 2 β 2 θ ) + 3 cos ( 4 φ ( t ) 2 β ) + 9 2 cos ( 6 φ ( t ) 2 β 2 θ ) ] ,
I 2 / I tot 1 2 1 2 cos ( 2 β ) cos ( 2 γ ) η 2 sin ( 2 γ ) | E T | [ 1 2 cos ( φ ( t ) + θ ) + 3 2 cos ( 3 φ ( t ) θ ) ] + η 2 16 | E T | 2 cos ( 2 γ ) [ 5 cos ( 2 β ) + 3 2 cos ( 2 φ ( t ) 2 β 2 θ ) + 1 2 cos ( 2 φ ( t ) 2 β + 2 θ ) + 3 2 cos ( 2 φ ( t ) + 2 β 2 θ ) + 3 cos ( 4 φ ( t ) 2 β ) + 9 2 cos ( 6 φ ( t ) 2 β 2 θ ) ] ,
Δ I / I tot cos ( 2 β ) cos ( 2 γ ) + η sin ( 2 γ ) | E T | [ 1 2 cos ( φ ( t ) + θ ) + 3 2 cos ( 3 φ ( t ) θ ) ] η 2 8 | E T | 2 cos ( 2 γ ) [ 5 cos ( 2 β ) + 3 2 cos ( 2 φ ( t ) 2 β 2 θ ) + 1 2 cos ( 2 φ ( t ) 2 β + 2 θ ) + 3 2 cos ( 2 φ ( t ) + 2 β 2 θ ) + 3 cos ( 4 φ ( t ) 2 β ) + 9 2 cos ( 6 φ ( t ) 2 β 2 θ ) ] ,
E ω = sin ( 2 γ ) η 2 | E T | ,
Φ ω = φ 0 + θ + 2 n 1 π ,
E 3 ω = sin ( 2 γ ) 3 η 2 | E T | ,
Φ 3 ω = 3 φ 0 θ + 2 n 2 π ,
C = C + C res ( φ ( t ) ) ,
Δ I obs / I tot cos ( 2 β ) cos ( 2 γ ) + { C + C res ( φ ( t ) ) } sin ( 2 γ ) sin ( 2 α ) + { C + C res ( φ ( t ) ) } 2 2 cos ( 2 γ ) sin ( 2 β 2 α ) sin ( 2 α ) .
Δ I res / I tot cos ( 2 β ) cos ( 2 γ ) + C res ( φ ( t ) ) sin ( 2 γ ) sin ( 2 α ) + C res 2 ( φ ( t ) ) 2 cos ( 2 γ ) sin ( 2 β 2 α ) sin ( 2 α ) .
S = Δ I obs / I tot Δ I res / I tot C sin ( 2 γ ) sin ( 2 α ) + C 2 2 cos ( 2 γ ) sin ( 2 β 2 α ) sin ( 2 α ) + C C res ( φ ( t ) ) cos ( 2 γ ) sin ( 2 β 2 α ) sin ( 2 α ) .
S res C C res ( φ ( t ) ) cos ( 2 γ ) sin ( 2 β 2 α ) sin ( 2 α ) ,
S res = C res ( φ ( t ) ) sin ( 2 α ) cos ( 2 γ ) · C sin [ 2 β 2 φ ( t ) ( 2 α 2 φ ( t ) ) ] = C res ( φ ( t ) ) sin ( 2 α ) cos ( 2 γ ) · ( η | E T | ) [ 1 2 cos ( φ ( t ) 2 β + θ ) + 3 2 cos ( 3 φ ( t ) 2 β θ ) ] = C res ( φ ( t ) ) sin ( 2 α ) cos ( 2 γ ) · ( η | E T | ) [ 1 2 cos ( ω t + φ 0 2 β + θ ) + 3 2 cos ( 3 ω t + 3 φ 0 2 β θ ) ] .
C res ( φ ( t ) ) sin ( 2 α ) = n = 0 A n cos ( n ω t + ϕ n ) .
sin ( 2 γ ) · η | E T | [ 1 2 cos ( ω t + φ 0 + θ ) + 3 2 cos ( 3 ω t + 3 φ 0 θ ) ] cos ( 2 γ ) · η | E T | n = 0 A n cos ( n ω t + ϕ n ) × [ 1 2 cos ( ω t + φ 0 2 β + θ ) + 3 2 cos ( 3 ω t + 3 φ 0 2 β θ ) ] ,
3 η | E T | sin ( 2 γ ) 2 · [ cos ( 3 ω t + 3 φ 0 θ ) cot ( 2 γ ) 6 A 2 cos ( 3 ω t + φ 0 + θ 2 β + ϕ 2 ) ] = 3 η | E T | sin ( 2 γ ) 2 · [ { 1 cot ( 2 γ ) 6 A 2 cos ( 2 φ 0 + 2 θ 2 β + ϕ 2 ) } cos ( 3 ω t + 3 φ 0 θ ) + cot ( 2 γ ) 6 A 2 sin ( 2 φ 0 + 2 θ 2 β + ϕ 2 ) · sin ( 3 ω t + 3 φ 0 θ ) ] .
3 η | E T | sin ( 2 γ ) 2 · [ { 1 + tan ( 2 Δ γ ) 6 A 2 cos ( 2 φ 0 + 2 θ 2 β + ϕ 2 ) } cos ( 3 ω t + 3 φ 0 θ ) tan ( 2 Δ γ ) 6 A 2 sin ( 2 φ 0 + 2 θ 2 β + ϕ 2 ) · sin ( 3 ω t + 3 φ 0 θ ) ] .
3 η | E T | sin ( 2 γ ) 2 · [ C 1 · cos ( 3 ω t + 3 φ 0 θ + δ θ ) ] ,
C 1 = 1 + 1 3 A 2 sin ( 2 φ 0 + 2 θ + ϕ 2 ) · Δ γ + O ( ( Δ γ ) 2 ) ,
δ θ = A 2 3 cos ( 2 φ 0 + 2 θ + ϕ 2 ) · Δ γ + O ( ( Δ γ ) 2 ) .
Φ 3 ω = 3 φ 0 θ A 2 3 cos ( 2 φ 0 + 2 θ + ϕ 2 ) · Δ γ + O ( ( Δ γ ) 2 ) ,
| E ^ p l · r ^ w 1 | = | [ cos ( β ) cos ( γ ) + C sin ( α ) cos ( β α ) sin ( γ ) + C 2 2 sin ( α ) sin ( β α ) cos ( γ ) ] i [ sin ( β ) sin ( γ ) + C sin ( α ) sin ( β α ) cos ( γ ) C 2 2 sin ( α ) cos ( β α ) sin ( γ ) ] | .
| E ^ p l · r ^ w 1 | 2 cos 2 ( β ) cos 2 ( γ ) + sin 2 ( β ) sin 2 ( γ ) + C sin ( 2 γ ) [ sin ( α ) cos ( β α ) cos ( β ) + sin ( α ) sin ( β α ) sin ( β ) ] + C 2 [ cos 2 ( γ ) sin ( α ) sin ( β α ) cos ( β ) sin 2 ( γ ) sin ( α ) cos ( β α ) sin ( β ) + sin 2 ( γ ) sin 2 ( α ) cos 2 ( β α ) + cos 2 ( γ ) sin 2 ( α ) sin 2 ( β α ) ] ,
cos 2 ( β ) cos 2 ( γ ) + sin 2 ( β ) sin 2 ( γ ) = 1 + cos ( 2 β ) 2 · 1 + cos ( 2 γ ) 2 + 1 cos ( 2 β ) 2 · 1 cos ( 2 γ ) 2 = 1 2 { 1 + cos ( 2 β ) cos ( 2 γ ) } ,
C sin ( 2 γ ) [ sin ( α ) cos ( β α ) cos ( β ) + sin ( α ) sin ( β α ) sin ( β ) ] = ( C / 2 ) sin ( 2 γ ) sin ( 2 α ) ,
C 2 2 { sin ( α ) sin ( β α ) cos ( β ) sin ( α ) cos ( β α ) sin ( β ) } + C 2 2 cos ( 2 γ ) { sin ( α ) sin ( β α ) cos ( β ) + sin ( α ) cos ( β α ) sin ( β ) } + C 2 2 { sin 2 ( α ) cos 2 ( β α ) + sin 2 ( α ) sin 2 ( β α ) } C 2 2 cos ( 2 γ ) { sin 2 ( α ) cos 2 ( β α ) sin 2 ( α ) sin 2 ( β α ) } ,
C 2 2 cos ( 2 γ ) sin ( α ) sin ( 2 β α ) C 2 2 cos ( 2 γ ) sin 2 ( α ) cos ( 2 β 2 α ) = C 2 2 cos ( 2 γ ) sin ( α ) { sin ( 2 β 2 α + α ) sin ( α ) cos ( 2 β 2 α ) } = C 2 2 cos ( 2 γ ) sin ( α ) { sin ( 2 β 2 α ) cos ( α ) + cos ( 2 β 2 α ) sin ( α ) sin ( α ) cos ( 2 β 2 α ) } = C 2 4 cos ( 2 γ ) sin ( 2 β 2 α ) sin ( 2 α ) ,
| E ^ p l · r ^ w 2 | = | [ cos ( β ) sin ( γ ) + C sin ( α ) cos ( β α ) cos ( γ ) C 2 2 sin ( α ) sin ( β α ) sin ( γ ) ] i [ sin ( β ) cos ( γ ) C sin ( α ) sin ( β α ) sin ( γ ) C 2 2 sin ( α ) cos ( β α ) cos ( γ ) ] | .
| E 01 | 2 = | E T | 2 cos 2 ( φ θ ) + | E T | 2 4 ( 1 + 3 cos 2 ( φ θ ) sin ( φ θ ) ) 2 = | E T | 2 4 { 1 + 7 cos 2 ( φ θ ) + sin 2 ( φ θ ) 2 sin ( φ θ ) 1 + 3 cos 2 ( φ θ ) } = | E T | 2 2 1 + 3 cos 2 ( φ θ ) ( 1 + 3 cos 2 ( φ θ ) sin ( φ θ ) ) .
1 | E 01 | 2 = 2 | E T | 2 1 1 + 3 cos 2 ( φ θ ) 1 + 3 cos 2 ( φ θ ) + sin ( φ θ ) 1 + 3 cos 2 ( φ θ ) sin 2 ( φ θ ) = 1 2 | E T | 2 cos 2 ( φ θ ) [ 1 + sin ( φ θ ) 1 + 3 cos 2 ( φ θ ) ] ,
1 | E 01 | = 1 | E 01 | 2 = 1 2 | E T | · | cos ( φ θ ) | 1 + sin ( φ θ ) 1 + 3 cos 2 ( φ θ ) .

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