Abstract

A general mathematical model based on Mueller-matrix calculation is presented to describe the optical behavior of a dual-crystal electro-optic modulator. The two crystals inside the modulator are oriented at ± 45° with respect to the horizontal, thereby cancelling natural birefringence and temperature-induced birefringence. We describe the behavior of the modulator as a function of the ellipticity of the crystals, the rotation angles of the crystals and the applied voltage. By fitting the measured data with a Mueller-matrix model that uses values for the ellipticity and orientation angles of the crystals, the simulated data and the experimental measurements could be matched. This Mueller-matrix includes physical properties of the thermally compensated electro optic modulator, and the matrix can be used in simulations where these device-specific properties are important, for instance in the modeling of a polarization-sensitive optical coherence tomography system.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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  22. W. Sima, T. Liu, Q. Yang, R. Han, and S. Sun, “Temperature characteristics of Pockels electro-optic voltage sensor with double crystal compensation,” AIP Adv. 6(5), 055109 (2016).
    [Crossref]
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2016 (4)

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016).
[Crossref] [PubMed]

T. Ferreira da Silva, C. S. Nobre, and G. P. Temporão, “Polarization-dependent loss characterization method based on optical frequency beat,” Appl. Opt. 55(8), 1838–1843 (2016).
[Crossref] [PubMed]

M. R. Fernández-Ruiz, J. Huh, and J. Azaña, “Time-domain Vander-Lugt filters for in-fiber complex (amplitude and phase) optical pulse shaping,” Opt. Lett. 41(9), 2121–2124 (2016).
[Crossref] [PubMed]

W. Sima, T. Liu, Q. Yang, R. Han, and S. Sun, “Temperature characteristics of Pockels electro-optic voltage sensor with double crystal compensation,” AIP Adv. 6(5), 055109 (2016).
[Crossref]

2013 (1)

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A. 87, 062313 (2013).

2012 (1)

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

2010 (1)

2009 (1)

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

2008 (2)

2007 (1)

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

2006 (1)

R. K. Choubey, P. Sen, P. K. Sen, R. Bhatt, S. Kar, V. Shukla, and K. S. Bartwal, “Optical properties of MgO doped LiNbO3 single crystals,” Opt. Mater. 28(5), 467–472 (2006).
[Crossref]

2002 (1)

2000 (1)

1999 (1)

J. R. Mackey, K. K. Das, S. L. Anna, and G. H. McKinley, “A compact dual-crystal modulated birefringence-measurement system for microgravity applications,” Meas. Sci. Technol. 10(10), 946–955 (1999).
[Crossref]

Andersen, U. L.

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

Anna, S. L.

J. R. Mackey, K. K. Das, S. L. Anna, and G. H. McKinley, “A compact dual-crystal modulated birefringence-measurement system for microgravity applications,” Meas. Sci. Technol. 10(10), 946–955 (1999).
[Crossref]

Azaña, J.

Bartwal, K. S.

R. K. Choubey, P. Sen, P. K. Sen, R. Bhatt, S. Kar, V. Shukla, and K. S. Bartwal, “Optical properties of MgO doped LiNbO3 single crystals,” Opt. Mater. 28(5), 467–472 (2006).
[Crossref]

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Bhatt, R.

R. K. Choubey, P. Sen, P. K. Sen, R. Bhatt, S. Kar, V. Shukla, and K. S. Bartwal, “Optical properties of MgO doped LiNbO3 single crystals,” Opt. Mater. 28(5), 467–472 (2006).
[Crossref]

Bloch, M.

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Burns, S. A.

Cense, B.

Cerf, N. J.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Chen, T. C.

Chen, Z.

Choubey, R. K.

R. K. Choubey, P. Sen, P. K. Sen, R. Bhatt, S. Kar, V. Shukla, and K. S. Bartwal, “Optical properties of MgO doped LiNbO3 single crystals,” Opt. Mater. 28(5), 467–472 (2006).
[Crossref]

Chui, Y. T.

Das, K. K.

J. R. Mackey, K. K. Das, S. L. Anna, and G. H. McKinley, “A compact dual-crystal modulated birefringence-measurement system for microgravity applications,” Meas. Sci. Technol. 10(10), 946–955 (1999).
[Crossref]

de Boer, J. F.

Debuisschert, T.

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Diamanti, E.

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A. 87, 062313 (2013).

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Dusek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Fernández-Ruiz, M. R.

Ferreira da Silva, T.

Filip, R.

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

R. Filip, “Continuous-variable quantum key distribution with noisy coherent states,” Phys. Rev. A 77(2), 022310 (2008).
[Crossref]

Fossier, S.

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

García-Patrón, R.

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Grangier, P.

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Han, R.

W. Sima, T. Liu, Q. Yang, R. Han, and S. Sun, “Temperature characteristics of Pockels electro-optic voltage sensor with double crystal compensation,” AIP Adv. 6(5), 055109 (2016).
[Crossref]

Huang, D.

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016).
[Crossref] [PubMed]

Huang, P.

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016).
[Crossref] [PubMed]

Huh, J.

Jouguet, P.

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A. 87, 062313 (2013).

Kar, S.

R. K. Choubey, P. Sen, P. K. Sen, R. Bhatt, S. Kar, V. Shukla, and K. S. Bartwal, “Optical properties of MgO doped LiNbO3 single crystals,” Opt. Mater. 28(5), 467–472 (2006).
[Crossref]

Karpov, E.

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Kunz-Jacques, S.

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A. 87, 062313 (2013).

Lassen, M.

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

Lin, D.

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016).
[Crossref] [PubMed]

Liu, T.

W. Sima, T. Liu, Q. Yang, R. Han, and S. Sun, “Temperature characteristics of Pockels electro-optic voltage sensor with double crystal compensation,” AIP Adv. 6(5), 055109 (2016).
[Crossref]

Lodewyck, J.

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Lütkenhaus, N.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Mackey, J. R.

J. R. Mackey, K. K. Das, S. L. Anna, and G. H. McKinley, “A compact dual-crystal modulated birefringence-measurement system for microgravity applications,” Meas. Sci. Technol. 10(10), 946–955 (1999).
[Crossref]

Madsen, L. S.

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

McKinley, G. H.

J. R. Mackey, K. K. Das, S. L. Anna, and G. H. McKinley, “A compact dual-crystal modulated birefringence-measurement system for microgravity applications,” Meas. Sci. Technol. 10(10), 946–955 (1999).
[Crossref]

McLaughlin, S. W.

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Nelson, J. S.

Nobre, C. S.

Park, B. H.

Peev, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Pierce, M. C.

Qi, X.

Saxer, C. E.

Scarani, V.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Sen, P.

R. K. Choubey, P. Sen, P. K. Sen, R. Bhatt, S. Kar, V. Shukla, and K. S. Bartwal, “Optical properties of MgO doped LiNbO3 single crystals,” Opt. Mater. 28(5), 467–472 (2006).
[Crossref]

Sen, P. K.

R. K. Choubey, P. Sen, P. K. Sen, R. Bhatt, S. Kar, V. Shukla, and K. S. Bartwal, “Optical properties of MgO doped LiNbO3 single crystals,” Opt. Mater. 28(5), 467–472 (2006).
[Crossref]

Shukla, V.

R. K. Choubey, P. Sen, P. K. Sen, R. Bhatt, S. Kar, V. Shukla, and K. S. Bartwal, “Optical properties of MgO doped LiNbO3 single crystals,” Opt. Mater. 28(5), 467–472 (2006).
[Crossref]

Sima, W.

W. Sima, T. Liu, Q. Yang, R. Han, and S. Sun, “Temperature characteristics of Pockels electro-optic voltage sensor with double crystal compensation,” AIP Adv. 6(5), 055109 (2016).
[Crossref]

Song, H.

Sun, S.

W. Sima, T. Liu, Q. Yang, R. Han, and S. Sun, “Temperature characteristics of Pockels electro-optic voltage sensor with double crystal compensation,” AIP Adv. 6(5), 055109 (2016).
[Crossref]

Temporão, G. P.

Tualle-Brouri, R.

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Usenko, V. C.

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

Yang, Q.

W. Sima, T. Liu, Q. Yang, R. Han, and S. Sun, “Temperature characteristics of Pockels electro-optic voltage sensor with double crystal compensation,” AIP Adv. 6(5), 055109 (2016).
[Crossref]

Zeng, G.

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016).
[Crossref] [PubMed]

Zhao, Y.

Zhong, Z.

Zou, W.

AIP Adv. (1)

W. Sima, T. Liu, Q. Yang, R. Han, and S. Sun, “Temperature characteristics of Pockels electro-optic voltage sensor with double crystal compensation,” AIP Adv. 6(5), 055109 (2016).
[Crossref]

Appl. Opt. (1)

Meas. Sci. Technol. (1)

J. R. Mackey, K. K. Das, S. L. Anna, and G. H. McKinley, “A compact dual-crystal modulated birefringence-measurement system for microgravity applications,” Meas. Sci. Technol. 10(10), 946–955 (1999).
[Crossref]

Nat. Commun. (1)

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (4)

Opt. Mater. (1)

R. K. Choubey, P. Sen, P. K. Sen, R. Bhatt, S. Kar, V. Shukla, and K. S. Bartwal, “Optical properties of MgO doped LiNbO3 single crystals,” Opt. Mater. 28(5), 467–472 (2006).
[Crossref]

Phys. Rev. A (2)

R. Filip, “Continuous-variable quantum key distribution with noisy coherent states,” Phys. Rev. A 77(2), 022310 (2008).
[Crossref]

J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007).
[Crossref]

Phys. Rev. A. (1)

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A. 87, 062313 (2013).

Rev. Mod. Phys. (1)

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Sci. Rep. (1)

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016).
[Crossref] [PubMed]

Other (10)

R. Hunsperger, “Integrated optical detectors,” in Integrated Optics: Theory and Technology, R. Hunsperger, ed. (Springer Series in Optical Sciences, 1984).

T. A. Maldonado, “Electro-optic modulators,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, 1995).

Newport, “Practical Uses and Applications of Electro-Optic Modulators,” www.newport.com/practical-uses-and-applications-of-electro-optic-modulators .

Thorlabs, “Free-Space EO Modulator Lab Facts,” www.thorlabs.com/images/TabImages/Free-Space_EO_Modulator_Lab_Facts.pdf .

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley Sons Inc, 1984).

P. R. Hobson, “Lasers and Electro-Optics: Fundamentals and Engineering (2nd Edition), by Christopher C. Davis,” Contemp. Phys. 1–1 (2014).

T. T. Ng, A Fast Polarization Independent Phase Shifter Of Light, (PhD thesis, National University of Singapore, 2007).

R. A. Chipman, “Polarimetry,” in Handbook of Optics II, M. Bass, ed. (McGraw Hill, 1995), pp 22.1–22.37.

D. Kliger and J. Lewis, Polarized Light in Optics and Spectroscopy (Academic, 2012).

D. H. Goldstein, Polarized Light (CRC, 2010).

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Figures (6)

Fig. 1
Fig. 1 Configuration of a transverse modulator crystal. The electric field is applied along the z direction and the light propagation is along the y direction. The length of the crystal is L and d is the crystal thickness. Electrodes are placed on the top and the bottom of the crystal.
Fig. 2
Fig. 2 Schematic representation of the beam propagation through the different components. Light from a light source with a wavelength λ is coupled into an optical fiber and collimated. Then the light passes through a linear polarizer oriented at an angle θ p with respect to the axis of polarization. After the polarizer, the light passes through the EOM which consists of two MgO:LiNbO3 crystals. The first crystal is oriented at an angle θ and the second one is rotated by 90° with respect to the first crystal’s fast axis, θ + 90 ° . The voltage applied to the crystals is the same. The arrows that are drawn onto the crystals represent the orientation of the electric field. After the modulator, the light travels towards an analyzer, which is a linear polarizer with an angle θ A with respect to the axis of polarization. Finally, the light power is collected by a photodetector.
Fig. 3
Fig. 3 Normalized amplitude obtained by means of Eq. (26) at different orientation angles of the analyzer. a) Amplitude plot for perfectly aligned crystals without ellipticity and rotation error. b) Amplitude plot for the ideal situation, with w = 0.001, Δθ1 = −0.5°, Δθ2 = −0.5° and θA = 0°, 30°, + 45°, −45°, 90°, 120°. Due to these non-ideal parameters, the plots do not cross at a normalized amplitude of 0.5. The most obvious difference for these non-ideal parameters with the ideal situation is the gap that develops between the + 45° and −45° measurements.
Fig. 4
Fig. 4 Intensity detected with a photodetector when the EOM modulator was placed between a polarizer at 90° and an analyzer with orientations of 0°, 30°, +/−45°, 90° and 120°, respectively. The black plot represents the input voltage, a triangle function, sent to the amplifier.
Fig. 5
Fig. 5 Graph of the recorded photodiode voltage when the analyzer was set at 0°, 30°, + 45°, −45°, 90° and 120° respectively, for voltages ranging from −3.8 V to 1.2 V, normalized with respect to the highest detected signal. The highest signal is obtained with the analyzer at 0° and 90°, while the analyzer at 45° provides almost no modulation in the detected signal. The data was fit with our model, a Vπ of 4.8V, w = −0.003 and orientation parameters of Δθ1 = −0.3° and Δθ2 = −0.6° related to a common and relative difference parameter of both crystals.
Fig. 6
Fig. 6 Measurements on the electro-optic modulator using a narrow bandwidth laser (Δλ = 1 nm (FWHM) at λc = 785 nm). Figure 6(a) shows the recorded photodiode voltage when the analyzer was set at 0°, 30°, + 45°, −45°, 90° and 120° respectively, for input voltages ranging from −0.36 V to 4.44 V, normalized with respect to the highest detected signal. The elliptical parameter after fitting was equal to w = −0.003 and the orientation parameters were Δθ1 = −0.03° and Δθ2 = −0.79°, the common and relative difference parameter of both crystals, respectively. Figure 6(b) shows the temperature-dependent variation of the elliptical parameter w, which is very small over this temperature range.

Equations (26)

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Δ ( 1 n 2 ) i = j = 1 3 r i j E j ,
r i j = ( 0 r 22 r 13 0 r 22 r 13 0 0 r 33 0 r 51 0 r 51 0 0 r 22 0 0 ) .
Δ ( 1 n 2 ) 1 = r 13 E z
Δ ( 1 n 2 ) 2 = r 13 E z
Δ ( 1 n 2 ) 3 = r 33 E z
n x = n 0 1 2 n 0 3 r 13 E z ,
n y = n 0 1 2 n 0 3 r 13 E z ,
n z = n e 1 2 n e 3 r 33 E z .
n z n x = ( n e n o ) 1 2 ( n e 3 r 33 n o 3 r 13 ) E z .
Γ = 2 π λ ( n e n o ) L π λ ( n e 3 r 33 n o 3 r 13 ) L d V ,
Γ 1 = 2 π λ ( n e n o ) L + π λ ( n e 3 r 33 n o 3 r 13 ) L d V = δ s + δ i ,
Γ 2 = 2 π λ ( n e n o ) L π λ ( n e 3 r 33 n o 3 r 13 ) L d V = δ s δ i .
P ( θ ) = 1 2 ( 1 cos 2 θ sin 2 θ 0 cos 2 θ cos 2 2 θ sin 2 θ cos 2 θ 0 sin 2 θ sin 2 θ cos 2 θ sin 2 2 θ 0 0 0 0 0 )
R ( θ , δ , w ) = ( 1 0 0 0 0 D 2 E 2 F 2 + G 2 2 ( D E + F G ) 2 ( D F + E G ) 0 2 ( D E F G ) D 2 + E 2 F 2 + G 2 2 ( D G E F ) 0 2 ( D F E G ) 2 ( D G + E F ) D 2 E 2 + F 2 + G 2 ) ,
D = cos ( 2 w ) cos ( 2 θ ) sin ( δ / 2 ) ,
E = cos ( 2 w ) sin ( 2 θ ) sin ( δ / 2 ) ,
F = sin ( 2 w ) sin ( δ / 2 ) ,
G = cos ( δ / 2 ) .
M E O M = R ( θ + Δ θ 1 90 º , δ s + δ i , w ) R ( θ + Δ θ 1 + Δ θ 2 , δ s δ i , w ) .
M E O M ( w , Δ θ 1 , Δ θ 2 , δ s , δ i ) = ( 1 0 0 0 0 cos 2 δ i M 2 , 3 sin 2 δ i 0 M 3 , 2 1 R 3 , 4 0 2 cos δ i sin δ i M 4 , 3 cos 2 δ i ) ,
M 3 , 2 = 4 Δ θ 1 cos 2 δ i 2 w [ 2 cos ( δ s + δ i ) sin 2 δ s ] 2 Δ θ 2 [ sin ( δ s + δ i ) + 1 ]
M 2 , 3 = 4 Δ θ 1 cos 2 δ i 2 Δ θ 2 [ cos 2 δ i + sin ( δ s δ i ) ] 2 w [ 2 cos ( δ s δ i ) sin 2 δ s ]
M 4 , 3 = 4 w sin 2 δ s + Δ θ 2 [ 2 cos ( δ s δ i ) 2 sin 2 δ i ] 2 Δ θ 1 sin 2 δ i
M 3 , 4 = 4 w sin 2 δ s 2 Δ θ 2 cos ( δ s + δ i ) + 2 Δ θ 1 sin 2 δ i
S o u t [ w , Δ θ 1 , Δ θ 2 , δ s , δ i ] = A [ θ A ] M E O M [ w , Δ θ 1 , Δ θ 2 , δ s , δ i ] P [ 90 º ] S i n .
I ( θ A , Δ θ 1 , Δ θ 2 , δ s , δ , w ) = 1 4 [ 1 + cos 2 θ A cos 2 δ i + 2 sin 2 θ A ( Δ θ 1 + Δ θ 2 + Δ θ 1 cos 2 δ i + 4 w cos ( δ i + δ s ) 2 w sin 2 δ s + 4 Δ θ 1 sin ( δ i + δ s ) ) ] .

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