Abstract

The generalized Jones matrix (GJM) is a recently introduced tool to describe linear transformations of three-dimensional light fields. Based on this framework, a specific method for obtaining the GJM of uniaxial anisotropic media was recently presented. However, the GJM of biaxial media had not been tackled so far, as the previous method made use of a simplified rotation matrix that lacks a degree of freedom in the three-dimensional rotation, thus being not suitable for calculating the GJM of biaxial media. In this work we propose a general method to derive the GJM of arbitrarily-oriented homogeneous biaxial media. It is based on the differential generalized Jones matrix (dGJM), which is the three-dimensional counterpart of the conventional differential Jones matrix. We show that the dGJM provides a simple and elegant way to describe uniaxial and biaxial media, with the capacity to model multiple simultaneous optical effects. The practical usefulness of this method is illustrated by the GJM modeling of the polarimetric properties of a negative uniaxial KDP crystal and a biaxial KTP crystal for any three-dimensional sample orientation. The results show that this method constitutes an advantageous and straightforward way to model biaxial media, which show a growing relevance for many interesting applications.

© 2015 Optical Society of America

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References

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2014 (2)

2013 (1)

2011 (1)

2010 (2)

A. Voss, M. Abdou-Ahmed, and T. Graf, “Application of the extended Jones matrix formalism for higher-order transverse modes to laser resonators,” Opt. Express 18, 21540–21550 (2010).
[Crossref] [PubMed]

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, C. K. Hitzenberger, and J. L. Arce-Diego, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15, 056004 (2010).
[Crossref] [PubMed]

2009 (1)

2007 (1)

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[Crossref]

2000 (1)

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[Crossref]

1999 (1)

1994 (2)

1993 (1)

1982 (1)

1972 (2)

1948 (1)

Abdou-Ahmed, M.

Arce-Diego, J. L.

Azzam, R. M. A.

Bashara, N. M.

Baumann, B.

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, C. K. Hitzenberger, and J. L. Arce-Diego, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15, 056004 (2010).
[Crossref] [PubMed]

Bergman, J.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[Crossref]

Berreman, D. W.

Booso, B.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
[Crossref]

Carozzi, T.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[Crossref]

Chipman, R. A.

Fanjul-Vélez, F.

N. Ortega-Quijano, F. Fanjul-Vélez, and J. L. Arce-Diego, “Polarimetric study of birefringent turbid media with three-dimensional optic axis orientation,” Biomed. Opt. Express 5, 287–292 (2014).
[Crossref] [PubMed]

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, C. K. Hitzenberger, and J. L. Arce-Diego, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15, 056004 (2010).
[Crossref] [PubMed]

Gil, J. J.

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[Crossref]

Goldstein, H.

H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics (Addison Wesley, 2002).

Götzinger, E.

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, C. K. Hitzenberger, and J. L. Arce-Diego, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15, 056004 (2010).
[Crossref] [PubMed]

Graf, T.

Gu, C.

Hitzenberger, C. K.

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, C. K. Hitzenberger, and J. L. Arce-Diego, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15, 056004 (2010).
[Crossref] [PubMed]

Hofmann, T.

Huard, S.

S. Huard, Polarization of Light (Wiley, 1997).

Irene, E. A.

H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew Publishing, Springer, 2005).
[Crossref]

Jones, R. C.

Karlsson, R.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[Crossref]

Kwok, H. S.

Lu, S.-Y.

Ortega-Quijano, N.

Pircher, M.

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, C. K. Hitzenberger, and J. L. Arce-Diego, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15, 056004 (2010).
[Crossref] [PubMed]

Poole, C. P.

H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics (Addison Wesley, 2002).

Safko, J. L.

H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics (Addison Wesley, 2002).

Sarangan, A.

Scharf, T.

T. Scharf, Polarized Light in Liquid Crystals and Polymers (Wiley, 2007).

Schmidt, D.

Schubert, E.

Schubert, M.

Sheppard, C. J. R.

C. J. R. Sheppard, “Jones and Stokes parameters for polarization in three dimensions,” Phys. Rev. A 90, 023809 (2014).
[Crossref]

Tompkins, H. G.

H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew Publishing, Springer, 2005).
[Crossref]

Tudor, T.

Voss, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
[Crossref]

Yeh, P.

Yu, F. H.

Biomed. Opt. Express (1)

Eur. Phys. J. Appl. Phys. (1)

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[Crossref]

J. Biomed. Opt. (1)

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, C. K. Hitzenberger, and J. L. Arce-Diego, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15, 056004 (2010).
[Crossref] [PubMed]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (1)

C. J. R. Sheppard, “Jones and Stokes parameters for polarization in three dimensions,” Phys. Rev. A 90, 023809 (2014).
[Crossref]

Phys. Rev. E (1)

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[Crossref]

Other (7)

H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew Publishing, Springer, 2005).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
[Crossref]

S. Huard, Polarization of Light (Wiley, 1997).

T. Scharf, Polarized Light in Liquid Crystals and Polymers (Wiley, 2007).

H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics (Addison Wesley, 2002).

United Crystals Inc, “High damage threshold KTP single crystal,” http://www.unitedcrystals.com .

Eksma Optics, “KTP crystals,” http://eksmaoptics.com .

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Figures (6)

Fig. 1
Fig. 1 (a) Euler angles α, β, and γ that specify the orientation of the local reference system xyz′ relative to the laboratory reference system xyz. (b) Index ellipsoid of the biaxial medium, whose principal axes are aligned with the local reference system. The principal refractive indices are nx, ny, and nz. The light beam propagates along z.
Fig. 2
Fig. 2 Anisotropic crystal sample of thickness d with its principal axes aligned with the local xyz′ reference system. The beam propagates along z in the laboratory system xyz.
Fig. 3
Fig. 3 Real part (left) and imaginary part (right) of the nine generalized Jones matrix (GJM) elements for a 3 mm section of KDP at λ = 1064 nm.
Fig. 4
Fig. 4 Real part (left) and imaginary part (right) of the GJM elements of the KTP sample at λ = 1064 nm for: (a) γ = 0, and (b) γ = π/3.
Fig. 5
Fig. 5 Absolute value of the GJM elements of a 3 mm sample of: (a) KDP, (b) KTP for γ = 0, (c) KTP for γ = π/3, and (d) phase of the latter.
Fig. 6
Fig. 6 Absolute value (a) and phase (b) of the GJM elements of the KTP sample for α =π/4.

Tables (1)

Tables Icon

Table 1 Differential generalized Jones matrix parameters.

Equations (19)

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d D d l z = g D ,
G = exp ( g l z ) ,
g = 1 2 [ 2 I + L x y + L z 3 L x y + C x y * L x z + C x z * L x y C x y * 2 I L x y + L z 3 L y z + C y z * L x z C x z * L y z C y z * 2 I 2 L z 3 ] ,
g uniaxial x y z = 1 2 [ L z / 3 C B i C D C B i C D C B i C D L z / 3 C B i C D C B i C D C B i C D 2 L z / 3 ] .
G uniaxial xyz = C G uniaxial x y z C 1 ,
C = [ sin ϕ cos θ cos ϕ sin θ cos ϕ cos ϕ cos θ sin ϕ sin θ sin ϕ 0 sin θ cos θ ] ,
g biaxial x y z = 1 2 [ L x y + L z / 3 C B i C D C B i C D C B i C D L x y + L z / 3 C B i C D C B i C D C B i C D 2 L z / 3 ] .
G biaxial xyz = T G biaxial x y z T 1 ,
T = R α R β R γ ,
R α = [ cos α sin α 0 sin α cos α 0 0 0 1 ] ,
R β = [ 1 0 0 0 cos β sin β 0 sin β cos β ] ,
R γ = [ cos γ sin γ 0 sin γ cos γ 0 0 0 1 ] .
g 1 x y z = 1 2 [ L z / 3 0 0 0 L z / 3 0 0 0 2 L z / 3 ] ,
g 1 x y z = 10 5 [ 1.3346 i 0 0 0 1.3346 i 0 0 0 2.6692 i ] .
G 1 xyz ( α , β , γ ) = T exp ( g 1 x y z d ) T 1 .
G 1 xyz ( 0 , 0 , 0 ) = G 1 x y z = [ 0.176 + 0.984 i 0 0 0 0.176 + 0.984 i 0 0 0 0.938 + 0.347 i ] .
g 2 x y z = 1 2 [ L x y + L z / 3 0 0 0 L x y + L z / 3 0 0 0 2 L z / 3 ] ,
g 2 x y z = 10 5 [ 3.1426 i 0 0 0 3.5855 i 0 0 0 6.7280 i ] .
G 2 xyz ( 0 , 0 , 0 ) = G 2 x y z = [ 0.957 + 0.291 i 0 0 0 0.347 + 0.938 i 0 0 0 0.059 0.998 i ] .

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