Abstract

The calculation of phase shift and optical path difference in birefringent media is related to a wide range of applications and devices. We obtain an explicit formula for the phase shift introduced by an anisotropic uniaxial plane-parallel plate with arbitrary orientation of the optical axis when the incident wave has an arbitrary direction. This allows us to calculate the phase shift introduced by waveplates when considering oblique incidence as well as optical axis misalignments. The expressions were obtained by using Maxwell’s equations and boundary conditions without any approximation. They can be applied both to single plane wave and space-limited beams.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. W. Q. Zhang, “New phase shift formulas and stability of waveplate in oblique incident beam,” Opt. Commun. 176, 9–15 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. C. C. Tsai, H. C. Wei, C. H. Hsieh, J. S. Wu, C. E. Lin, and C. Chou, “Linear birefringence parameters determination of a multi-order wave plate via phase detection at large oblique incidence angles,” Opt. Commun. 281, 3036–3041 (2008).
    [CrossRef]
  8. M. C. Simon and K. V. Gottschalk, “Optical path in birefringent media and Fermat’s principle,” Pure Appl. Opt. 7, 1403–1410 (1998).
    [CrossRef]
  9. L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik (Jena) 111, 297–306 (2000).
  10. Meadowlark Optics Inc., “Sources of error in retarders and waveplates,” www.meadowlarkoptics.com/applicationNotes.
  11. Alphalas, www.alphalas.com/images/stories/products/polarization.
  12. E. Kubacki, CVI Laser (Melles Griot), “Waveplates offer precise control of polarization,” www.cvilaser.com/Common/PDFs/OLEreprintMar2005CVI.pdf.
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  14. M. C. Simon and R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
    [CrossRef] [PubMed]
  15. L. I. Perez, M. T. Garea, and R. M. Echarri, “Isotropic-uniaxial crystal interfaces: negative refraction and backward wave phenomena,” Opt. Commun. 254, 10–18 (2005).
    [CrossRef]
  16. M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
    [CrossRef] [PubMed]
  17. O. N. Stavroudis, “Ray-tracing formulas for uniaxial crystals,” J. Opt. Soc. Am. 52, 187–189 (1962).
    [CrossRef]
  18. M. Avendaño-Alejo and O. N. Stavroudis, “Huygens’s principle and rays in uniaxial anisotropic media. II. Crystal axis orientation arbitrary,” J. Opt. Soc. Am. A 19, 1674–1679 (2002).
    [CrossRef]
  19. M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik (Jena) 118, 457–470 (2007).
  20. M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008), .
    [CrossRef]
  21. F. E. Veiras and L. I. Perez, “Phase shift formulas for waveplates in oblique incidence,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), paper PDPA2.

2008

C. C. Tsai, H. C. Wei, C. H. Hsieh, J. S. Wu, C. E. Lin, and C. Chou, “Linear birefringence parameters determination of a multi-order wave plate via phase detection at large oblique incidence angles,” Opt. Commun. 281, 3036–3041 (2008).
[CrossRef]

M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008), .
[CrossRef]

2007

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik (Jena) 118, 457–470 (2007).

2006

2005

L. I. Perez, M. T. Garea, and R. M. Echarri, “Isotropic-uniaxial crystal interfaces: negative refraction and backward wave phenomena,” Opt. Commun. 254, 10–18 (2005).
[CrossRef]

2004

D. Clarke, “Interference effects in single wave plates,” J. Opt. A: Pure Appl. Opt. 6, 1036–1040 (2004).
[CrossRef]

2002

2000

W. Q. Zhang, “New phase shift formulas and stability of waveplate in oblique incident beam,” Opt. Commun. 176, 9–15 (2000).
[CrossRef]

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik (Jena) 111, 297–306 (2000).

1999

S. Prunet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983–990 (1999).
[CrossRef]

1998

M. C. Simon and K. V. Gottschalk, “Optical path in birefringent media and Fermat’s principle,” Pure Appl. Opt. 7, 1403–1410 (1998).
[CrossRef]

1994

1988

1986

1983

1962

Avendaño-Alejo, M.

Born, M.

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

Chou, C.

C. C. Tsai, H. C. Wei, C. H. Hsieh, J. S. Wu, C. E. Lin, and C. Chou, “Linear birefringence parameters determination of a multi-order wave plate via phase detection at large oblique incidence angles,” Opt. Commun. 281, 3036–3041 (2008).
[CrossRef]

Clarke, D.

D. Clarke, “Interference effects in single wave plates,” J. Opt. A: Pure Appl. Opt. 6, 1036–1040 (2004).
[CrossRef]

Day, G. W.

Echarri, R. M.

L. I. Perez, M. T. Garea, and R. M. Echarri, “Isotropic-uniaxial crystal interfaces: negative refraction and backward wave phenomena,” Opt. Commun. 254, 10–18 (2005).
[CrossRef]

M. C. Simon and R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
[CrossRef] [PubMed]

Fortunato, G.

S. Prunet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983–990 (1999).
[CrossRef]

Garea, M. T.

L. I. Perez, M. T. Garea, and R. M. Echarri, “Isotropic-uniaxial crystal interfaces: negative refraction and backward wave phenomena,” Opt. Commun. 254, 10–18 (2005).
[CrossRef]

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik (Jena) 111, 297–306 (2000).

Gottschalk, K. V.

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik (Jena) 118, 457–470 (2007).

M. C. Simon and K. V. Gottschalk, “Optical path in birefringent media and Fermat’s principle,” Pure Appl. Opt. 7, 1403–1410 (1998).
[CrossRef]

Hale, P. D.

Hsieh, C. H.

C. C. Tsai, H. C. Wei, C. H. Hsieh, J. S. Wu, C. E. Lin, and C. Chou, “Linear birefringence parameters determination of a multi-order wave plate via phase detection at large oblique incidence angles,” Opt. Commun. 281, 3036–3041 (2008).
[CrossRef]

Journet, B.

S. Prunet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983–990 (1999).
[CrossRef]

Kubacki, E.

E. Kubacki, CVI Laser (Melles Griot), “Waveplates offer precise control of polarization,” www.cvilaser.com/Common/PDFs/OLEreprintMar2005CVI.pdf.

Lin, C. E.

C. C. Tsai, H. C. Wei, C. H. Hsieh, J. S. Wu, C. E. Lin, and C. Chou, “Linear birefringence parameters determination of a multi-order wave plate via phase detection at large oblique incidence angles,” Opt. Commun. 281, 3036–3041 (2008).
[CrossRef]

Perez, L. I.

M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008), .
[CrossRef]

L. I. Perez, M. T. Garea, and R. M. Echarri, “Isotropic-uniaxial crystal interfaces: negative refraction and backward wave phenomena,” Opt. Commun. 254, 10–18 (2005).
[CrossRef]

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik (Jena) 111, 297–306 (2000).

F. E. Veiras and L. I. Perez, “Phase shift formulas for waveplates in oblique incidence,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), paper PDPA2.

Prunet, S.

S. Prunet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983–990 (1999).
[CrossRef]

Rosete-Aguilar, M.

Simon, M. C.

M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008), .
[CrossRef]

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik (Jena) 118, 457–470 (2007).

M. C. Simon and K. V. Gottschalk, “Optical path in birefringent media and Fermat’s principle,” Pure Appl. Opt. 7, 1403–1410 (1998).
[CrossRef]

M. C. Simon and R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
[CrossRef] [PubMed]

M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
[CrossRef] [PubMed]

Stavroudis, O. N.

Tsai, C. C.

C. C. Tsai, H. C. Wei, C. H. Hsieh, J. S. Wu, C. E. Lin, and C. Chou, “Linear birefringence parameters determination of a multi-order wave plate via phase detection at large oblique incidence angles,” Opt. Commun. 281, 3036–3041 (2008).
[CrossRef]

Veiras, F. E.

M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008), .
[CrossRef]

F. E. Veiras and L. I. Perez, “Phase shift formulas for waveplates in oblique incidence,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), paper PDPA2.

Wei, H. C.

C. C. Tsai, H. C. Wei, C. H. Hsieh, J. S. Wu, C. E. Lin, and C. Chou, “Linear birefringence parameters determination of a multi-order wave plate via phase detection at large oblique incidence angles,” Opt. Commun. 281, 3036–3041 (2008).
[CrossRef]

Wolf, E.

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

Wu, J. S.

C. C. Tsai, H. C. Wei, C. H. Hsieh, J. S. Wu, C. E. Lin, and C. Chou, “Linear birefringence parameters determination of a multi-order wave plate via phase detection at large oblique incidence angles,” Opt. Commun. 281, 3036–3041 (2008).
[CrossRef]

Zhang, W. Q.

W. Q. Zhang, “New phase shift formulas and stability of waveplate in oblique incident beam,” Opt. Commun. 176, 9–15 (2000).
[CrossRef]

Zhu, X.

AIP Conf. Proc.

M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008), .
[CrossRef]

Appl. Opt.

J. Opt. A: Pure Appl. Opt.

D. Clarke, “Interference effects in single wave plates,” J. Opt. A: Pure Appl. Opt. 6, 1036–1040 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

C. C. Tsai, H. C. Wei, C. H. Hsieh, J. S. Wu, C. E. Lin, and C. Chou, “Linear birefringence parameters determination of a multi-order wave plate via phase detection at large oblique incidence angles,” Opt. Commun. 281, 3036–3041 (2008).
[CrossRef]

W. Q. Zhang, “New phase shift formulas and stability of waveplate in oblique incident beam,” Opt. Commun. 176, 9–15 (2000).
[CrossRef]

L. I. Perez, M. T. Garea, and R. M. Echarri, “Isotropic-uniaxial crystal interfaces: negative refraction and backward wave phenomena,” Opt. Commun. 254, 10–18 (2005).
[CrossRef]

Opt. Eng.

S. Prunet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983–990 (1999).
[CrossRef]

Optik (Jena)

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik (Jena) 118, 457–470 (2007).

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik (Jena) 111, 297–306 (2000).

Pure Appl. Opt.

M. C. Simon and K. V. Gottschalk, “Optical path in birefringent media and Fermat’s principle,” Pure Appl. Opt. 7, 1403–1410 (1998).
[CrossRef]

Other

F. E. Veiras and L. I. Perez, “Phase shift formulas for waveplates in oblique incidence,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), paper PDPA2.

Meadowlark Optics Inc., “Sources of error in retarders and waveplates,” www.meadowlarkoptics.com/applicationNotes.

Alphalas, www.alphalas.com/images/stories/products/polarization.

E. Kubacki, CVI Laser (Melles Griot), “Waveplates offer precise control of polarization,” www.cvilaser.com/Common/PDFs/OLEreprintMar2005CVI.pdf.

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

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Figures (9)

Fig. 1
Fig. 1

(a) Ordinary wave. (b) Extraordinary wave.

Fig. 2
Fig. 2

Equal phase planes in a uniaxial medium associated with an extraordinary plane wave.

Fig. 3
Fig. 3

(a) Equal phase planes in a system formed by a uniaxial plate immersed in an isotropic medium. (b) Detail of Fig. 3a.

Fig. 4
Fig. 4

Ordinary and extraordinary transmission through a uniaxial plane-parallel plate immersed in an isotropic medium. ( x , σ , t ) is the coordinate system. x , t is the plane of incidence. θ is the angle between the optical axis and the interface. δ is the angle between the plane of incidence and the optical axis projection on the interface. l t , l t , and l σ are the coordinates of the points of incidence on the second interface for the ordinary and extraordinary rays, respectively.

Fig. 5
Fig. 5

Ordinary (dashed line) and extraordinary (solid line) equal phase planes for a uniaxial plane-parallel plate immersed in an isotropic medium.

Fig. 6
Fig. 6

Diagram of the points of incidence on the second interface for a calcite plate: (a)  θ = 0 ° and (b)  θ = 45 ° , for | α | = 0 ° , 1 ° , and 2 ° , and δ = 0 ° , 45 ° , 90 ° , and 135 ° . L = 1 mm , n o = 1.66 , n e = 1.49 , and λ v = 632.8 nm .

Fig. 7
Fig. 7

Equal phase planes: (a) the arbitrary points on the second interface, r 2 o and r 2 e , respectively, coincide with the coordinates of P and P . (b)  r 2 = r 2 o = r 2 e .

Fig. 8
Fig. 8

Phase shift Δ ϕ ( α , δ ) for a quartz plate. (a) The phase shift was expressed in degrees (vertical axis), the angle of incidence α in the radial coordinate, and the angle that forms the plane of incidence with the projection of the optical axis on the interfaces δ as an azimuthal coordinate. (b) Phase shift contour lines: θ = 0 ° , L = 1.973 mm , n o = 1.54264 , n e = 1.5517 , and λ v = 632.8 nm .

Fig. 9
Fig. 9

Phase shift of the region of minimum variation. (a) Phase shift contour lines Δ ϕ ( α , δ ) for a quartz plate ( θ = 1 ° ). (b) Angle of incidence, α mv ( θ ) , and phase shift, Δ ϕ mv ( θ ) , associated with the center of the region of minimum variation of the phase shift as a function of the direction of the optical axis ( 2.5 ° < θ < 2.5 ° ). L = 1.973 mm , n o = 1.54264 , n e = 1.5517 , and λ v = 632.8 nm .

Equations (19)

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E ( r , t ) = E exp [ i ω ( r · u t ) ] ,
u = [ u e 2 + ( u o 2 u e 2 ) ( · 3 ) 2 ] 1 2 ,
e = 1 f e [ n o 2 e + ( n e 2 n o 2 ) ( e · 3 ) 3 ] ,
OPL e = c u O Q ¯ = c v O P 1 ¯ ,
v = u e · e .
OPL o = c u o O P ¯ = n o O P ¯ .
OPL o = L n o 2 [ n o 2 n 2 sin 2 α ] 1 2 .
OPL e = n L e · e e · ,
OPL e = L n o n e 2 { n e 2 ( n e 2 sin 2 θ + n o 2 cos 2 θ ) [ n e 2 ( n e 2 n o 2 ) cos 2 θ sin 2 δ ] n 2 sin 2 α } 1 2 .
Δ o e = ( OPL o + n P Q ¯ ) ( OPL e + n P Q ¯ ) .
n ( P Q ¯ ) n ( P Q ¯ ) = n ( l t l t ) sin α .
Δ ϕ = 2 π L λ v ( ( n o 2 n 2 sin 2 α ) 1 2 + n ( n o 2 n e 2 ) sin θ cos θ cos δ sin α n e 2 sin 2 θ + n o 2 cos 2 θ + n o { n e 2 ( n e 2 sin 2 θ + n o 2 cos 2 θ ) [ n e 2 ( n e 2 n o 2 ) cos 2 θ sin 2 δ ] n 2 sin 2 α } 1 2 n e 2 sin 2 θ + n o 2 cos 2 θ ) .
E II ( r , t ) = E o II exp [ i ω ( r · o u o t ) ] + E e II exp [ i ω ( r · e u t ) ] .
E III ( r , t ) = E o III exp ( i { ω [ ( r r 2 o ) · u t ] + ϕ o ( r 2 o ) } ) + E e III exp ( i { ω [ ( r r 2 e ) · u t ] + ϕ e ( r 2 e ) } ) .
ϕ o ( r 2 o ) = ω r 2 o · o u o ,
ϕ e ( r 2 e ) = ω r 2 e · e u .
E III ( r , t ) = E o III exp ( i { ω [ ( r r 2 ) · u t ] + ϕ o ( r 2 ) } ) + E e III exp ( i { ω [ ( r r 2 ) · u t ] + ϕ e ( r 2 ) } ) .
Δ ϕ | θ = 0 ° = 2 π L λ v { ( n o 2 n 2 sin 2 α ) 1 2 [ n o 2 n e 2 n 2 sin 2 α ( n o 2 sin 2 δ + n e 2 cos 2 δ ) ] 1 2 n o } .
Δ ϕ | α = 0 ° = 2 π L λ v n o [ 1 n e ( n e 2 sin 2 θ + n o 2 cos 2 θ ) 1 2 ] .

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