Abstract

Two plates of different birefringence material can be combined to obtain an achromatic wave retarder. In this work, we achieve a correction for the overall retardation of the system that extends the relation to any azimuth. Current techniques for the design of achromatic wave retarders do not present a parameter that characterizes its achromatism on a range of wavelengths. Thus, an achromatic degree has been introduced, in order to determine the optimal achromatic design composed with retarder plates for a spectrum of incident light. In particular, we have optimized a quarter retarder using two wave plates for the visible spectrum. Our technique has been compared to previous results, showing significant improvement.

© 2013 Optical Society of America

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References

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  1. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).
  2. D. Goldstein, Polarized Light, 3rd ed. (CRC, 2003).
  3. S. Pancharatnam, “Achromatic combinations of birefringent plates. Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. 41A, 137–144 (1955).
    [CrossRef]
  4. P. Hariharan, “Achromatic retarders using quartz and mica,” Meas. Sci. Technol. 6, 1078–1079 (1995).
    [CrossRef]
  5. J. J. Gil and E. Bernabeu, “Diseño de rotores, compensadores y moduladores de retardo a partir de retardadores comerciales,” Opt. Pura Apl. 15, 39–43 (1982).
  6. P. Hariharan and D. Malacara, “A simple achromatic half-wave retarder,” J. Mod. Opt. 41, 15–18 (1994).
    [CrossRef]
  7. B. Boulbry, B. Bousquet, B. Le Jeune, Y. Guern, and J. Lotrian, “Polarization errors associated with zero-order achromatic quarter-wave plates in the whole visible spectral range,” Opt. Express 9, 225–235 (2001).
    [CrossRef]
  8. A. Saha, K. Bhattacharya, and A. K. Chakraborty, “Achromatic quarter-wave plate using crystalline quartz,” Appl. Opt. 51, 1976–1980 (2012).
    [CrossRef]
  9. J. B. Masson and G. Gallot, “Terahertz achromatic quarter-wave plate,” Opt. Lett. 31, 265–267 (2006).
    [CrossRef]
  10. R. Pan, C. Lai, C. Lin, C. Hsieh, and C. Pan, “Achromatic liquid crystal phase plate for short laser pulses, molecular crystals and liquid crystals,” Mol. Cryst. Liq. Cryst. 527, 65/[221]–71/[227] (2010).
    [CrossRef]
  11. G. Kang, Q. Tan, X. Wang, and G. Jin, “Achromatic phase retarder applied to MWIR & LWIR dual-band,” Opt. Express 18, 1695–1703 (2010).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).
  13. S. Chandrasekhat, “The dispersion and thermo-optic behavior of vitreous silica,” Proc. Indian Acad. Sci. 34A, 275–282 (1951).
    [CrossRef]
  14. G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. 163, 95–102 (1999).
    [CrossRef]
  15. J. M. Beckers, “Achromatic linear retarders,” Appl. Opt. 10, 973–975 (1971).
    [CrossRef]
  16. M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009).
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    [CrossRef]

2012 (1)

2010 (2)

R. Pan, C. Lai, C. Lin, C. Hsieh, and C. Pan, “Achromatic liquid crystal phase plate for short laser pulses, molecular crystals and liquid crystals,” Mol. Cryst. Liq. Cryst. 527, 65/[221]–71/[227] (2010).
[CrossRef]

G. Kang, Q. Tan, X. Wang, and G. Jin, “Achromatic phase retarder applied to MWIR & LWIR dual-band,” Opt. Express 18, 1695–1703 (2010).
[CrossRef]

2006 (1)

2001 (1)

1999 (1)

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

1995 (1)

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

1994 (1)

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

1988 (1)

1984 (1)

1982 (1)

J. J. Gil and E. Bernabeu, “Diseño de rotores, compensadores y moduladores de retardo a partir de retardadores comerciales,” Opt. Pura Apl. 15, 39–43 (1982).

1971 (1)

1955 (1)

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

1951 (1)

S. Chandrasekhat, “The dispersion and thermo-optic behavior of vitreous silica,” Proc. Indian Acad. Sci. 34A, 275–282 (1951).
[CrossRef]

Bass, M.

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

Beckers, J. M.

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Diseño de rotores, compensadores y moduladores de retardo a partir de retardadores comerciales,” Opt. Pura Apl. 15, 39–43 (1982).

Bhattacharya, K.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Boulbry, B.

Bousquet, B.

Chakraborty, A. K.

Chandrasekhat, S.

S. Chandrasekhat, “The dispersion and thermo-optic behavior of vitreous silica,” Proc. Indian Acad. Sci. 34A, 275–282 (1951).
[CrossRef]

Day, G. W.

DeCusatis, C.

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

Dodge, M. J.

Enoch, J.

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

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]

Gil, J. J.

J. J. Gil and E. Bernabeu, “Diseño de rotores, compensadores y moduladores de retardo a partir de retardadores comerciales,” Opt. Pura Apl. 15, 39–43 (1982).

Goldstein, D.

D. Goldstein, Polarized Light, 3rd ed. (CRC, 2003).

Guern, Y.

Hale, P. D.

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]

Hsieh, C.

R. Pan, C. Lai, C. Lin, C. Hsieh, and C. Pan, “Achromatic liquid crystal phase plate for short laser pulses, molecular crystals and liquid crystals,” Mol. Cryst. Liq. Cryst. 527, 65/[221]–71/[227] (2010).
[CrossRef]

Jeune, B. Le

Jin, G.

Kang, G.

Kliger, D. S.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Lai, C.

R. Pan, C. Lai, C. Lin, C. Hsieh, and C. Pan, “Achromatic liquid crystal phase plate for short laser pulses, molecular crystals and liquid crystals,” Mol. Cryst. Liq. Cryst. 527, 65/[221]–71/[227] (2010).
[CrossRef]

Lakshminarayanan, V.

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

Lewis, J. W.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Li, G.

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

Lin, C.

R. Pan, C. Lai, C. Lin, C. Hsieh, and C. Pan, “Achromatic liquid crystal phase plate for short laser pulses, molecular crystals and liquid crystals,” Mol. Cryst. Liq. Cryst. 527, 65/[221]–71/[227] (2010).
[CrossRef]

Lotrian, J.

MacDonald, C.

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

Mahajan, V.

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

Malacara, D.

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

Masson, J. B.

Pan, C.

R. Pan, C. Lai, C. Lin, C. Hsieh, and C. Pan, “Achromatic liquid crystal phase plate for short laser pulses, molecular crystals and liquid crystals,” Mol. Cryst. Liq. Cryst. 527, 65/[221]–71/[227] (2010).
[CrossRef]

Pan, R.

R. Pan, C. Lai, C. Lin, C. Hsieh, and C. Pan, “Achromatic liquid crystal phase plate for short laser pulses, molecular crystals and liquid crystals,” Mol. Cryst. Liq. Cryst. 527, 65/[221]–71/[227] (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).
[CrossRef]

Randall, C. E.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Saha, A.

Tan, Q.

Van Stryland, E.

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

Wang, X.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Appl. Opt. (4)

J. Mod. Opt. (1)

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

Meas. Sci. Technol. (1)

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

Mol. Cryst. Liq. Cryst. (1)

R. Pan, C. Lai, C. Lin, C. Hsieh, and C. Pan, “Achromatic liquid crystal phase plate for short laser pulses, molecular crystals and liquid crystals,” Mol. Cryst. Liq. Cryst. 527, 65/[221]–71/[227] (2010).
[CrossRef]

Opt. Commun. (1)

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

Opt. Express (2)

Opt. Lett. (1)

Opt. Pura Apl. (1)

J. J. Gil and E. Bernabeu, “Diseño de rotores, compensadores y moduladores de retardo a partir de retardadores comerciales,” Opt. Pura Apl. 15, 39–43 (1982).

Proc. Indian Acad. Sci. (2)

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

S. Chandrasekhat, “The dispersion and thermo-optic behavior of vitreous silica,” Proc. Indian Acad. Sci. 34A, 275–282 (1951).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

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

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

D. Goldstein, Polarized Light, 3rd ed. (CRC, 2003).

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Figures (3)

Fig. 1.
Fig. 1.

System to study. Two retarder plates L1 and L2 with retardations δ1 and δ2 and azimuths 0 and ϕ, respectively, are illuminated with a polychromatic light beam with spectrum g(λ).

Fig. 2.
Fig. 2.

Optimal overall retardation of the system for a Gaussian and plane spectrum (continuous curve) centered at λ0=600nm and a width δλ=100nm, with wavelength [500,700] nm. Both curves are coincident. A previous result of overall retardation is shown with a dashed curve [8].

Fig. 3.
Fig. 3.

(a) Overall retardation for plane or Gaussian spectra using several bandwidths: [500,700] nm (continuous), [525,675] nm (dashed), [550,650] nm (dash–dotted), and [575,625] nm (crosses for plane spectrum and dots for Gaussian spectrum). (b) Quadratic fit of maximum difference committed by the retarder as a function of the length interval for a Gaussian spectrum (dot points: continuous curve) and for a plane spectrum (cross points: dashed curve). Maximum difference, ϵ, in terms of the bandwidth Δλ for a plane spectrum (crosses) and a Gaussian spectrum (dots), and quadratic fit (continuous curve).

Equations (18)

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Li(δ,ϕ)=(cos(δ/2)+isin(δ/2)cos2ϕisin(δ/2)sin2ϕisin(δ/2)sin2ϕcos(δ/2)isin(δ/2)cos2ϕ),
σ0=(1001),σ1=(1001),σ2=(0110),σ3=(0ii0).
Li(δ,ϕ)=cosδ2·σ0+isinδ2cos2ϕ·σ1+isinδ2sin2ϕ·σ2,
M=L2L1=(cosδ2+isinδ2cos2ϕisinδ2sin2ϕisinδ2sin2ϕcosδ2isinδ2cos2ϕ)(eiδ1/200eiδ1/2)=(ABB*A*),
M=l0σ0+l1σ1+l2σ2+l3σ3,
l0=cosδ12cosδ22sinδ12sinδ22cos2ϕ,l1=i(cosδ12sinδ22cos2ϕ+sinδ12cosδ22),l2=icosδ12sinδ22sin2ϕ,l3=isinδ12sinδ22sin2ϕ.
tan2Δ2=|Im(A)|2+|Im(B)|2|Re(A)|2+|Re(B)|2,Ψ=0.
μ1=s0+s021,μ2=s0s021,
s0=cosδ12cosδ22sinδ12sinδ22cos2ϕ.
Tr(M)=2s0.
(μ100μ2)=(eiΔ/200eiΔ/2),
cosΔ2=TrM2=s0.
AcD=(|Δ(ϕ,d1,d2,λ)Δ0|2g(λ)dλ)1/2g(λ)dλ,
Δ=δ1+δ2ϕ=0,Δ=δ1δ2ϕ=π/2.
Δn1(λ1)d1±Δn2(λ1)d2=λ1/4,Δn1(λ2)d1±Δn2(λ2)d2=λ2/4.
no21=0.48755108λ2λ20.043384082+0.39875031λ2λ20.094614422+2.3120353λ2λ223.7936042,ne21=0.41344023λ2λ20.036842622+0.50497499λ2λ20.090761622+2.4904862λ2λ223.7719952,
no21=0.663044λ2λ2(0.0600)2+0.517852λ2λ20.10602+0.175912λ2λ20.11902+0.565380λ2λ28.8442+1.675299λ2λ220.7422,ne21=0.665721λ2λ20.06002+0.503511λ2λ20.10602+0.214792λ2λ20.11902+0.539173λ2λ28.7922+1.8076613λ2λ219.702.
ϵ=sup|ΔΔ0|,

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