Abstract

The generation of circularly polarized light with a high circularity degree and low azimuthal error sensitivity was analyzed using a system composed by two waveplates. It is shown how the high circularity degree is achieved using a combination of a half- (λ/2) and a quarter- (λ/4) waveplate λ/2+λ/4 configuration. However, the lowest azimuthal sensitivity under small variations in the azimuths of the waveplates is obtained by employing a λ/4+λ/2 configuration. Analytical calculus particularized for quartz and MgF2 waveplates is presented.

© 2014 Optical Society of America

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References

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  1. C. J. Yu, C. E. Lin, L. P. Yu, and C. Chou, “Paired circularly polarized heterodyne ellipsometer,” Appl. Opt. 48, 758–764 (2009).
    [CrossRef]
  2. V. Samkaran, “Comparison of polarized-light propagation in biological tissue and phantoms,” Opt. Lett. 24, 1044–1046 (1999).
  3. C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
    [CrossRef]
  4. J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006).
    [CrossRef]
  5. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).
  6. S. Pancharatnam, “Achromatic combinations of birefringent plates, Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. 41A, 137–144 (1955).
  7. P. Violino, “Polariseur circulaire réglable sur une large domaine de longueurs d’onde,” Rev. Optique 44, 109–114 (1965).
  8. R. Corbalan and E. Bernabeu, “On the obtaining circular polarization for each of the optical doublet Caesium lines with available commercial components,” Opt. Pura Apl. 5, 80–84 (1972).
  9. E. Bernabeu and J. Aporta, “On obtaining circularly polarized light,” Atti della Fondazione Giorgio Ronchi 2, 351–355 (1975).
  10. P. Hariharan, “Achromatic retarders using quartz and mica,” Meas. Sci. Technol. 6, 1078–1079 (1995).
    [CrossRef]
  11. P. Hariharan and D. Malacara, “A simple achromatic half-wave retarder,” J. Mod. Opt. 41, 15–18 (1994).
    [CrossRef]
  12. A. Saha, K. Bhattacharya, and A. K. Chakraborty, “Achromatic quarter-wave plate using crystalline quartz,” Appl. Opt. 51, 1976–1980 (2012).
    [CrossRef]
  13. J. L. Vilas, L. M. Sanchez-Brea, and E. Bernabeu, “Optimal achromatic wave retarders using two birefringent wave plates,” Appl. Opt. 52, 7078–7080 (2013).
    [CrossRef]
  14. X. Zhang, “Optimal achromatic wave retarders using two birefringent wave plates: comment,” Appl. Opt. 52, 7078–7080 (2013).
    [CrossRef]
  15. J. L. Vilas, L. M. Sanchez-Brea, and E. Bernabeu, “Optimal achromatic wave retarders using two birefringent wave plates: reply,” Appl. Opt. 52, 7081–7082 (2013).
    [CrossRef]
  16. X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C.-Z. Peng, “Experimental quantum error detection,” Sci. Rep. 2, 626 (2012).
    [CrossRef]
  17. K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996).
    [CrossRef]
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    [CrossRef]
  19. J. B. Masson and G. Gallot, “Terahertz achromatic quarter-wave plate,” Opt. Lett. 31, 265–267 (2006).
    [CrossRef]
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    [CrossRef]
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  22. G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. 163, 95–102 (1999).
    [CrossRef]
  23. S. Chandrasekhar, “The dispersion and thermo-optic behaviour of vitreous silica,” Proc. Indian Acad. Sci. 34A, 275–282 (1951).

2013

2012

X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C.-Z. Peng, “Experimental quantum error detection,” Sci. Rep. 2, 626 (2012).
[CrossRef]

A. Saha, K. Bhattacharya, and A. K. Chakraborty, “Achromatic quarter-wave plate using crystalline quartz,” Appl. Opt. 51, 1976–1980 (2012).
[CrossRef]

2010

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

2009

2006

J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006).
[CrossRef]

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

1999

V. Samkaran, “Comparison of polarized-light propagation in biological tissue and phantoms,” Opt. Lett. 24, 1044–1046 (1999).

G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. 163, 95–102 (1999).
[CrossRef]

1996

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996).
[CrossRef]

1995

P. Hariharan, “Achromatic retarders using quartz and mica,” Meas. Sci. Technol. 6, 1078–1079 (1995).
[CrossRef]

1994

P. Hariharan and D. Malacara, “A simple achromatic half-wave retarder,” J. Mod. Opt. 41, 15–18 (1994).
[CrossRef]

1988

1984

1975

E. Bernabeu and J. Aporta, “On obtaining circularly polarized light,” Atti della Fondazione Giorgio Ronchi 2, 351–355 (1975).

1972

R. Corbalan and E. Bernabeu, “On the obtaining circular polarization for each of the optical doublet Caesium lines with available commercial components,” Opt. Pura Apl. 5, 80–84 (1972).

1965

P. Violino, “Polariseur circulaire réglable sur une large domaine de longueurs d’onde,” Rev. Optique 44, 109–114 (1965).

1955

S. Pancharatnam, “Achromatic combinations of birefringent plates, Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. 41A, 137–144 (1955).

1951

S. Chandrasekhar, “The dispersion and thermo-optic behaviour of vitreous silica,” Proc. Indian Acad. Sci. 34A, 275–282 (1951).

Aporta, J.

E. Bernabeu and J. Aporta, “On obtaining circularly polarized light,” Atti della Fondazione Giorgio Ronchi 2, 351–355 (1975).

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).

Bao, X. H.

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).

Bass, M.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, in Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 4.

Bernabeu, E.

J. L. Vilas, L. M. Sanchez-Brea, and E. Bernabeu, “Optimal achromatic wave retarders using two birefringent wave plates,” Appl. Opt. 52, 7078–7080 (2013).
[CrossRef]

J. L. Vilas, L. M. Sanchez-Brea, and E. Bernabeu, “Optimal achromatic wave retarders using two birefringent wave plates: reply,” Appl. Opt. 52, 7081–7082 (2013).
[CrossRef]

E. Bernabeu and J. Aporta, “On obtaining circularly polarized light,” Atti della Fondazione Giorgio Ronchi 2, 351–355 (1975).

R. Corbalan and E. Bernabeu, “On the obtaining circular polarization for each of the optical doublet Caesium lines with available commercial components,” Opt. Pura Apl. 5, 80–84 (1972).

Bhattacharya, K.

Chakraborty, A. K.

Chandrasekhar, S.

S. Chandrasekhar, “The dispersion and thermo-optic behaviour of vitreous silica,” Proc. Indian Acad. Sci. 34A, 275–282 (1951).

Chen, K.

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

Chen, Y. A.

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

Chou, C.

Cirac, I.

J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006).
[CrossRef]

Corbalan, R.

R. Corbalan and E. Bernabeu, “On the obtaining circular polarization for each of the optical doublet Caesium lines with available commercial components,” Opt. Pura Apl. 5, 80–84 (1972).

Day, G. W.

DeCusatis, C.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, in Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 4.

Dodge, M. J.

Enoch, J.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, in Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 4.

Gallot, G.

Ghosh, G.

G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. 163, 95–102 (1999).
[CrossRef]

Goebel, A.

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

Hale, P. D.

Hammerer, K.

J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006).
[CrossRef]

Hariharan, P.

P. Hariharan, “Achromatic retarders using quartz and mica,” Meas. Sci. Technol. 6, 1078–1079 (1995).
[CrossRef]

P. Hariharan and D. Malacara, “A simple achromatic half-wave retarder,” J. Mod. Opt. 41, 15–18 (1994).
[CrossRef]

Jin, X. M.

X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C.-Z. Peng, “Experimental quantum error detection,” Sci. Rep. 2, 626 (2012).
[CrossRef]

Julsgaard, B.

J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006).
[CrossRef]

Krauter, H.

J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006).
[CrossRef]

Kwiat, P. G.

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996).
[CrossRef]

Lakshminarayanan, V.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, in Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 4.

Li, C. M.

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

Li, G.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, in Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 4.

Lin, C. E.

MacDonald, C.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, in Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 4.

Mahajan, V.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, in Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 4.

Malacara, D.

P. Hariharan and D. Malacara, “A simple achromatic half-wave retarder,” J. Mod. Opt. 41, 15–18 (1994).
[CrossRef]

Masson, J. B.

Mattle, K.

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996).
[CrossRef]

Olsson, R. K.

J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006).
[CrossRef]

Pan, J. W.

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “Achromatic combinations of birefringent plates, Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. 41A, 137–144 (1955).

Peng, C.-Z.

X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C.-Z. Peng, “Experimental quantum error detection,” Sci. Rep. 2, 626 (2012).
[CrossRef]

Polzik, E. S.

J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006).
[CrossRef]

Reingruber, A.

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

Saha, A.

Samkaran, V.

Sanchez-Brea, L. M.

Sherson, J. F.

J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006).
[CrossRef]

Van Stryland, E.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, in Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 4.

Vilas, J. L.

Violino, P.

P. Violino, “Polariseur circulaire réglable sur une large domaine de longueurs d’onde,” Rev. Optique 44, 109–114 (1965).

Wagenknecht, C.

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

Weinfurter, H.

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996).
[CrossRef]

Yang, B.

X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C.-Z. Peng, “Experimental quantum error detection,” Sci. Rep. 2, 626 (2012).
[CrossRef]

Yang, T.

X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C.-Z. Peng, “Experimental quantum error detection,” Sci. Rep. 2, 626 (2012).
[CrossRef]

Yi, Z. H.

X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C.-Z. Peng, “Experimental quantum error detection,” Sci. Rep. 2, 626 (2012).
[CrossRef]

Yu, C. J.

Yu, L. P.

Zeilinger, A.

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996).
[CrossRef]

Zhang, Q.

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

Zhang, X.

Zhou, F.

X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C.-Z. Peng, “Experimental quantum error detection,” Sci. Rep. 2, 626 (2012).
[CrossRef]

Appl. Opt.

Atti della Fondazione Giorgio Ronchi

E. Bernabeu and J. Aporta, “On obtaining circularly polarized light,” Atti della Fondazione Giorgio Ronchi 2, 351–355 (1975).

J. Mod. Opt.

P. Hariharan and D. Malacara, “A simple achromatic half-wave retarder,” J. Mod. Opt. 41, 15–18 (1994).
[CrossRef]

Meas. Sci. Technol.

P. Hariharan, “Achromatic retarders using quartz and mica,” Meas. Sci. Technol. 6, 1078–1079 (1995).
[CrossRef]

Nat. Photonics

C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010).
[CrossRef]

Nature

J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006).
[CrossRef]

Opt. Commun.

G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. 163, 95–102 (1999).
[CrossRef]

Opt. Lett.

Opt. Pura Apl.

R. Corbalan and E. Bernabeu, “On the obtaining circular polarization for each of the optical doublet Caesium lines with available commercial components,” Opt. Pura Apl. 5, 80–84 (1972).

Phys. Rev. Lett.

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996).
[CrossRef]

Proc. Indian Acad. Sci.

S. Chandrasekhar, “The dispersion and thermo-optic behaviour of vitreous silica,” Proc. Indian Acad. Sci. 34A, 275–282 (1951).

S. Pancharatnam, “Achromatic combinations of birefringent plates, Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. 41A, 137–144 (1955).

Rev. Optique

P. Violino, “Polariseur circulaire réglable sur une large domaine de longueurs d’onde,” Rev. Optique 44, 109–114 (1965).

Sci. Rep.

X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C.-Z. Peng, “Experimental quantum error detection,” Sci. Rep. 2, 626 (2012).
[CrossRef]

Other

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, in Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 4.

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Figures (4)

Fig. 1.
Fig. 1.

System under study. A linear polarizer P(θ) and two phase plates C1 and C2 with retardations δ1, δ2 and azimuths 0 and ϕ, respectively.

Fig. 2.
Fig. 2.

Eccentricity of the polarization ellipse for a system illuminated at λ=638.2nm and composed by (a) a linear polarizer and a configuration λ/4+λ/2 of quartz-MgF2, (b) a linear polarizer and a configuration λ/2+λ/4 of quartz-MgF2, and (c) a linear polarizer and a configuration λ/4+λ/4 of quartz-MgF2.

Fig. 3.
Fig. 3.

Curves of azimuthal error sensitivity for the configurations λ/2+λ/4, λ/4+λ/2, λ/4+λ/4, and λ/4 under illumination at λ=638.2nm: (a) e/ϕ versus ϕ; all the curves are essentially the same. (b) e/θ versus θ; the configuration λ/4+λ/2 exhibits better azimuthal error sensitivity.

Fig. 4.
Fig. 4.

Overall retardation for plane spectrum in the bandwidth [500, 700] nm: (continuous) configuration λ/2+λ/4 at 638.2 nm of quartz-MgF2 with ϕ=π/2, d1=369.87μm, and d2=298.12μm or equivalently the same curve is obtained with a combination λ/4+λ/2 at the same wavelength of MgF2-quartz with d1=298.12μm and d2=369.87μm. (Dashed) single quartz waveplate λ/4 with thickness d=369.87μm. λ/4+λ/2 at the same wavelength of MgF2-quartz with d1=298.12μm and d2=369.87μm, respectively.

Equations (19)

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

jo=C2C1ji(θ),
C(δ,ϕ)=(cosδ2+isinδ2cos2ϕisinδ2sin2ϕisinδ2sin2ϕcosδ2isinδ2cos2ϕ).
M=(ABB*A*),
tan2Δ2=Im(A)2+Im(B)2Re(A)2+Re(B)2.
tan2Δ2=sin2(δ12δ22)+12sinδ1sinδ2(1+cos2ϕ)cos2(δ12δ22)12sinδ1sinδ2(1+cos2ϕ).
Cr=R(ϕ)T2Ci(δ,0)T1R(ϕ),
T1=(2no+1002ne+1),T2=(2nono+1002nene+1),
R(ϕ)=(cosϕsinϕsinϕcosϕ),
Cr(δ,ϕ,tx,ty)=(txcos2ϕeiδ/2+tysin2ϕeiδ/212sin2ϕ(txeiδ/2tyeiδ/2)12sin2ϕ(txeiδ/2tyeiδ/2)txsin2ϕeiδ/2+tycos2ϕeiδ/2),
tx=4no(no+1)2,ty=4ne(ne+1)2.
x2Ex2+y2Ey22xyEx2Ey2cosδ=sin2δ,
e2=1b2a2,
1/a2=12[1sin2δ[(1Ex2+1Ey2)+4Ex2Ey2cos2δ+(1Ex21Ey2)2]],
1/b2=12[1sin2δ[(1Ex2+1Ey2)4Ex2Ey2cos2δ+(1Ex21Ey2)2]].
e2=2cos2γsin22α+cos22α1+cos2γsin22α+cos22α.
no21=0.48755108λ2λ20.043384082+0.39875031λ2λ20.094614422+2.3120353λ2λ223.7936042,
ne21=0.41344023λ2λ20.036842622+0.50497499λ2λ20.090761622+2.4904862λ2λ223.7719952,
no21=0.663044λλ20.06002+0.517852λ2λ20.10602+0.175912λ2λ20.11902+0.565380λ2λ28.8442+1.675299λ2λ220.7422,
ne21=0.665721λ.2λ20.06002+0.503511λ2λ20.10602+0.214792λ2λ20.11902+0.539173λ2λ28.7922+1.8076613λ2λ219.702.

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