Abstract

Composite waveplates (CWs) consisting of multiple single waveplates are basic polarization elements and widely used to manipulate the polarized light in optical systems, and their performances affect the final accuracy and precision significantly. This research proposes a method for the comprehensive characterization of an arbitrary CW based on spectroscopic Mueller matrix polarimetry. An analytical model is established to describe a general CW by extending Jones’ equivalent theorem with Mueller matrix calculus. In this model, an arbitrary CW is optically equivalent to a cascaded system consisting of a linear retarder with slight diattenuation followed by an optical rotator, and its polarization properties are completely described by four polarization parameters, including the retardance, the fast axis azimuth, the rotation angle, and the diattenuation angle. Analytical relations between the polarization properties, the structure, and the Mueller matrix of the CW are then derived from the established model. By the proposed method, the polarization parameters of an arbitrary CW can be comprehensively characterized over an ultra-wide spectral range via only one measurement. Moreover, the actual structure of the CW, including the thicknesses and fast axis azimuths of the single waveplates, as well as the axis alignment errors, can be completely reconstructed from the polarization spectra. Experiments performed with a house-developed broadband Mueller matrix polarimeter on three typical CWs including a compound zero-order waveplate, an achromatic waveplate and a specially designed biplate have demonstrated the capability of the proposed method.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (3)

2017 (3)

X. Tu, L. Jiang, M. Ibn-Elhaj, and S. Pau, “Design, fabrication and testing of achromatic elliptical polarizer,” Opt. Express 25(9), 10355–10367 (2017).
[Crossref] [PubMed]

J. L. Vilas and A. Lazarova-Lazarova, “A simple analytical method to obtain achromatic waveplate retarders,” J. Opt. 19(4), 045701 (2017).
[Crossref]

Z. Han and Q. Zheng, “Self-spectral calibration of a compound zero-order waveplate at blind rotation angles,” Opt. Lasers Eng. 91, 257–260 (2017).
[Crossref]

2016 (4)

H. Gu, X. Chen, H. Jiang, C. Zhang, and S. Liu, “Optimal broadband Mueller matrix ellipsometer using multi-waveplates with flexibly oriented axes,” J. Opt. 18(2), 025702 (2016).
[Crossref]

Q. Zheng, Z. Han, and L. Chen, “Determination of the misalignment error of a compound zero-order waveplate using the spectroscopic phase shifting method,” Opt. Commun. 374, 18–23 (2016).
[Crossref]

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

C. C. Chou, S. Y. Lu, T. Lin, S. H. Lu, and R. J. Jeng, “Environment-noise-free optical heterodyne retardation measurement using a double-pass acousto-optic frequency shifter,” Opt. Lett. 41(22), 5138–5141 (2016).
[Crossref] [PubMed]

2015 (6)

2014 (2)

X. Chen, S. Liu, H. Gu, and C. Zhang, “Formulation of error propagation and estimation in grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter,” Thin Solid Films 571, 653–659 (2014).
[Crossref]

L. Liu, A. Zeng, L. Zhu, and H. Huang, “Lateral shearing interferometer with variable shearing for measurement of a small beam,” Opt. Lett. 39(7), 1992–1995 (2014).
[Crossref] [PubMed]

2013 (4)

W. Chen, S. Zhang, and X. Long, “Optic axis determination based on polarization flipping effect induced by optical feedback,” Opt. Lett. 38(7), 1080–1082 (2013).
[Crossref] [PubMed]

C. C. Liao and Y. L. Lo, “Extraction of anisotropic parameters of turbid media using hybrid model comprising differential- and decomposition-based Mueller matrices,” Opt. Express 21(14), 16831–16853 (2013).
[Crossref] [PubMed]

X. Chen, C. Zhang, and S. Liu, “Depolarization effects from nanoimprinted grating structures as measured by Mueller matrix polarimetry,” Appl. Phys. Lett. 103(15), 151605 (2013).
[Crossref]

P. Zhang, Y. Tan, W. Liu, and W. Chen, “Methods for optical phase retardation measurement: A review,” Sci. China Technol. Sci. 56(5), 1155–1163 (2013).
[Crossref]

2012 (2)

W. Chen, H. Li, S. Zhang, and X. Long, “Measurement of phase retardation of waveplate online based on laser feedback,” Rev. Sci. Instrum. 83(1), 013101 (2012).
[Crossref] [PubMed]

H. Dong, M. Tang, and Y. Gong, “Measurement errors induced by deformation of optical axes of achromatic waveplate retarders in RRFP Stokes polarimeters,” Opt. Express 20(24), 26649–26666 (2012).
[Crossref] [PubMed]

2011 (2)

2009 (3)

2008 (2)

2007 (1)

2006 (2)

2004 (2)

2001 (3)

2000 (1)

D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, “Entangled state quantum cryptography: eavesdropping on the ekert protocol,” Phys. Rev. Lett. 84(20), 4733–4736 (2000).
[Crossref] [PubMed]

1997 (2)

1996 (1)

1995 (1)

E. A. West and M. H. Smith, “Polarization errors associated with birefringent waveplates,” Opt. Eng. 34(6), 1574–1580 (1995).
[Crossref]

1993 (1)

1988 (1)

1984 (1)

1971 (1)

1951 (1)

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

1941 (1)

Aitken, D. K.

D. K. Aitken and J. H. Hough, “Spectral modulation, or ripple, in retardation plates for linear and circular polarization,” Publ. Astron. Soc. Pac. 113(788), 1300–1305 (2001).
[Crossref]

An, I.

Beckers, J. M.

Berglund, A. J.

D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, “Entangled state quantum cryptography: eavesdropping on the ekert protocol,” Phys. Rev. Lett. 84(20), 4733–4736 (2000).
[Crossref] [PubMed]

Bernabeu, E.

Booth, M. J.

Boulbry, B.

Bousquet, B.

Broch, L.

Cai, Y.

Chandrasekhar, S.

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

Chang, M.

Chen, H.

Chen, L.

Q. Zheng, Z. Han, and L. Chen, “Determination of the misalignment error of a compound zero-order waveplate using the spectroscopic phase shifting method,” Opt. Commun. 374, 18–23 (2016).
[Crossref]

Chen, P. C.

Chen, W.

W. Chen, S. Zhang, and X. Long, “Optic axis determination based on polarization flipping effect induced by optical feedback,” Opt. Lett. 38(7), 1080–1082 (2013).
[Crossref] [PubMed]

P. Zhang, Y. Tan, W. Liu, and W. Chen, “Methods for optical phase retardation measurement: A review,” Sci. China Technol. Sci. 56(5), 1155–1163 (2013).
[Crossref]

W. Chen, H. Li, S. Zhang, and X. Long, “Measurement of phase retardation of waveplate online based on laser feedback,” Rev. Sci. Instrum. 83(1), 013101 (2012).
[Crossref] [PubMed]

Chen, X.

H. Gu, X. Chen, H. Jiang, C. Zhang, and S. Liu, “Optimal broadband Mueller matrix ellipsometer using multi-waveplates with flexibly oriented axes,” J. Opt. 18(2), 025702 (2016).
[Crossref]

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

H. Gu, S. Liu, X. Chen, and C. Zhang, “Calibration of misalignment errors in composite waveplates using Mueller matrix ellipsometry,” Appl. Opt. 54(4), 684–693 (2015).
[Crossref] [PubMed]

S. Liu, X. Chen, and C. Zhang, “Development of a broadband Mueller matrix ellipsometer as a powerful tool for nanostructure metrology,” Thin Solid Films 584, 176–185 (2015).
[Crossref]

X. Chen, S. Liu, H. Gu, and C. Zhang, “Formulation of error propagation and estimation in grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter,” Thin Solid Films 571, 653–659 (2014).
[Crossref]

X. Chen, C. Zhang, and S. Liu, “Depolarization effects from nanoimprinted grating structures as measured by Mueller matrix polarimetry,” Appl. Phys. Lett. 103(15), 151605 (2013).
[Crossref]

X. Chen, L. Yan, and X. S. Yao, “Waveplate analyzer using binary magneto-optic rotators,” Opt. Express 15(20), 12989–12994 (2007).
[Crossref] [PubMed]

Chenault, D. B.

Chipman, R. A.

Chou, C.

Chou, C. C.

Chou, L. D.

Clarke, D.

D. Clarke, “Interference effects in compound and achromatic wave plates,” J. Opt. A, Pure Appl. Opt. 6(11), 1041–1046 (2004).
[Crossref]

Collins, R. W.

Danilishin, S. L.

S. L. Danilishin, E. Knyazev, N. V. Voronchev, F. Y. Khalili, C. Graf, S. Steinlechner, J. S. Hennig, and S. Hild, “A new quantum speed-meter interferometer: measuring speed to search for intermediate mass black holes,” Light Sci. Appl. 7(1), 11 (2018).
[Crossref]

Dodge, M. J.

Dong, H.

Elston, S. J.

En Naciri, A.

Fells, J. A. J.

Gallot, G.

Gong, Y.

Graf, C.

S. L. Danilishin, E. Knyazev, N. V. Voronchev, F. Y. Khalili, C. Graf, S. Steinlechner, J. S. Hennig, and S. Hild, “A new quantum speed-meter interferometer: measuring speed to search for intermediate mass black holes,” Light Sci. Appl. 7(1), 11 (2018).
[Crossref]

Gu, H.

H. Gu, X. Chen, H. Jiang, C. Zhang, and S. Liu, “Optimal broadband Mueller matrix ellipsometer using multi-waveplates with flexibly oriented axes,” J. Opt. 18(2), 025702 (2016).
[Crossref]

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

H. Gu, S. Liu, X. Chen, and C. Zhang, “Calibration of misalignment errors in composite waveplates using Mueller matrix ellipsometry,” Appl. Opt. 54(4), 684–693 (2015).
[Crossref] [PubMed]

X. Chen, S. Liu, H. Gu, and C. Zhang, “Formulation of error propagation and estimation in grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter,” Thin Solid Films 571, 653–659 (2014).
[Crossref]

Guern, Y.

Han, Z.

Z. Han and Q. Zheng, “Self-spectral calibration of a compound zero-order waveplate at blind rotation angles,” Opt. Lasers Eng. 91, 257–260 (2017).
[Crossref]

Q. Zheng, Z. Han, and L. Chen, “Determination of the misalignment error of a compound zero-order waveplate using the spectroscopic phase shifting method,” Opt. Commun. 374, 18–23 (2016).
[Crossref]

Hennig, J. S.

S. L. Danilishin, E. Knyazev, N. V. Voronchev, F. Y. Khalili, C. Graf, S. Steinlechner, J. S. Hennig, and S. Hild, “A new quantum speed-meter interferometer: measuring speed to search for intermediate mass black holes,” Light Sci. Appl. 7(1), 11 (2018).
[Crossref]

Herrera-Fernandez, J. M.

Hild, S.

S. L. Danilishin, E. Knyazev, N. V. Voronchev, F. Y. Khalili, C. Graf, S. Steinlechner, J. S. Hennig, and S. Hild, “A new quantum speed-meter interferometer: measuring speed to search for intermediate mass black holes,” Light Sci. Appl. 7(1), 11 (2018).
[Crossref]

Hough, J. H.

D. K. Aitken and J. H. Hough, “Spectral modulation, or ripple, in retardation plates for linear and circular polarization,” Publ. Astron. Soc. Pac. 113(788), 1300–1305 (2001).
[Crossref]

Hsu, P. F.

Hu, P.

Huang, H.

Huang, Y. C.

Hurwitz, H.

Ibn-Elhaj, M.

Jeng, R. J.

Jeng, Y. T.

Jiang, H.

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

H. Gu, X. Chen, H. Jiang, C. Zhang, and S. Liu, “Optimal broadband Mueller matrix ellipsometer using multi-waveplates with flexibly oriented axes,” J. Opt. 18(2), 025702 (2016).
[Crossref]

Jiang, L.

Johann, L.

Jones, R. C.

Khalili, F. Y.

S. L. Danilishin, E. Knyazev, N. V. Voronchev, F. Y. Khalili, C. Graf, S. Steinlechner, J. S. Hennig, and S. Hild, “A new quantum speed-meter interferometer: measuring speed to search for intermediate mass black holes,” Light Sci. Appl. 7(1), 11 (2018).
[Crossref]

Knyazev, E.

S. L. Danilishin, E. Knyazev, N. V. Voronchev, F. Y. Khalili, C. Graf, S. Steinlechner, J. S. Hennig, and S. Hild, “A new quantum speed-meter interferometer: measuring speed to search for intermediate mass black holes,” Light Sci. Appl. 7(1), 11 (2018).
[Crossref]

Kwiat, P. G.

D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, “Entangled state quantum cryptography: eavesdropping on the ekert protocol,” Phys. Rev. Lett. 84(20), 4733–4736 (2000).
[Crossref] [PubMed]

Kyoseva, E.

A. A. Rangelov and E. Kyoseva, “Broadband composite polarization rotator,” Opt. Commun. 338, 574–577 (2015).
[Crossref]

Lai, C. H.

Lazarova-Lazarova, A.

J. L. Vilas and A. Lazarova-Lazarova, “A simple analytical method to obtain achromatic waveplate retarders,” J. Opt. 19(4), 045701 (2017).
[Crossref]

Le Jeune, B.

Lee, C. C.

Lee, J.

Li, H.

W. Chen, H. Li, S. Zhang, and X. Long, “Measurement of phase retardation of waveplate online based on laser feedback,” Rev. Sci. Instrum. 83(1), 013101 (2012).
[Crossref] [PubMed]

Li, J.

Li, L.

Li, Q.

Li, W.

Li, Y. C.

Liang, R.

Liao, C. C.

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Figures (6)

Fig. 1
Fig. 1 Schematic of the structure of a general CW. F n and θn (n = 1, 2, …, N) refer to the fast axis of the n-th single-waveplate and its azimuth angle with respect to the x-axis, respectively.

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Fig. 2
Fig. 2 (a) Schematic of the MMP-based experimental set-up for the characterization of a CW; (b) Prototype of the experimental set-up based on a house-developed broadband MMP.

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Fig. 3
Fig. 3 Mueller matrix spectra: (a) the quartz compound zero-order waveplate; (b) the MgF2-MgF2-quartz achromatic waveplate; (c) the MgF2 biplate.

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Fig. 4
Fig. 4 Spectra for the equivalent polarization parameters over concerned wavelength ranges and their corresponding standard deviations (STD) for 30 repeated measurements: (a) the quartz compound zero-order waveplate; (b) the MgF2-MgF2-quartz achromatic waveplate; (c) the MgF2 biplate.

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Fig. 5
Fig. 5 Elliptical retardances and ellipticities of the tested CWs.

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Fig. 6
Fig. 6 Designed and measured retardance spectra for the tested CWs.

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Tables (1)

Tables Icon

Table 1 Structure parameters of the CWs tested in this paper.

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Equations (29)

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M ( δ , θ ) = R ( − θ ) M ( δ ) R ( θ ) .
M ( δ ) = [ 1 0 0 0 0 1 0 0 0 0 cos δ sin δ 0 0 − sin δ cos δ ] ,
R ( θ ) = [ 1 0 0 0 0 cos ( 2 θ ) sin ( 2 θ ) 0 0 − sin ( 2 θ ) cos ( 2 θ ) 0 0 0 0 1 ] ,
δ = 2 π λ d ⋅ Δ n ,
tan ψ = t f t s ,
M ( δ , ψ ) = [ 1 − cos ( 2 ψ ) 0 0 − cos ( 2 ψ ) 1 0 0 0 0 sin ( 2 ψ ) cos δ sin ( 2 ψ ) sin δ 0 0 − sin ( 2 ψ ) sin δ sin ( 2 ψ ) cos δ ] .
M = ∏ n = 1 N M ( δ n , θ n ) ≐ R ( ρ e ) R ( − θ e ) M ( δ e , ψ e ) R ( θ e ) = [ 1 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ] ,
m 12 = − C θ e C ψ e ,
m 13 = − S θ e C ψ e ,
m 14 = m 41 = 0 ,
m 21 = − ( C θ e C ρ e + S θ e S ρ e ) C ψ e ,
m 22 = ( C θ e 2 + S θ e 2 S ψ e cos δ e ) C ρ e + S θ e C θ e ( 1 − S ψ e cos δ e ) S ρ e ,
m 23 = S θ e C θ e ( 1 − S ψ e cos δ e ) C ρ e + ( S θ e 2 + C θ e 2 S ψ e cos δ e ) S ρ e ,
m 24 = ( C θ e S ρ e − S θ e C ρ e ) S ψ e sin δ e ,
m 31 = ( C θ e S ρ e − S θ e C ρ e ) C ψ e ,
m 32 = S θ e C θ e ( 1 − S ψ e cos δ e ) C ρ e − ( C θ e 2 + S θ e 2 S ψ e cos δ e ) S ρ e ,
m 33 = ( S θ e 2 + C θ e 2 S ψ e cos δ e ) C ρ e − S θ e C θ e ( 1 − S ψ e cos δ e ) S ρ e ,
m 34 = ( S θ e S ρ e + C θ e C ρ e ) S ψ e sin δ e ,
m 42 = S θ e S ψ e sin δ e ,
m 43 = − C θ e S ψ e sin δ e ,
m 44 = S ψ e cos δ e ,
S κ = sin ( 2 κ ) , C κ = cos ( 2 κ ) ,     ( κ = θ e , ψ e , ρ e ) .
δ e = tan − 1 m 42 2 + m 43 2 m 44 2 ,
θ e = 1 2 tan − 1 ( m 42 m 43 ) ,
ρ e = 1 2 tan − 1 ( m 23 − m 32 m 22 + m 33 ) ,
ψ e = 1 2 cos − 1 m 12 2 + m 13 2 = 1 2 cos − 1 m 21 2 + m 31 2 ,
γ = 2 cos − 1 ( cos δ e 2 cos ρ e ) ,
ε = 1 2 tan − 1 ( cot δ e 2 sin ρ e ) .
x ^ = arg min x ∈ Ω ∑ k [ y k c − y k m σ ( y k ) ] 2 ,

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