Abstract

We are presenting an extension for the method of obtaining the geometric phase introduced by the birefringent medium. This extension takes into account the possible dichroism of the examined medium. Its influence has been described formally in two kinds of interferometric experiments using the Jones formalism. The mathematical formulas have been visualized with the specific-triangles construction on the Poincaré sphere. This medium’s dichroism can affect the final formulas for the obtained interferometric pattern in two ways, depending on the type of experiment. Dichroism can change the geometrical phase in the setup with a Mach–Zehnder interferometer; however, only the contrast of possible interference fringes can be changed in a polariscopic setup.

© 2011 Optical Society of America

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References

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  1. S. Pancharatnam, “Generalized theory of interference and its applications. Part I. Coherent pencils,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956).
  2. M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
    [CrossRef]
  3. J. Courtial, “Wave plates and the Pancharatnam phase,” Opt. Commun. 171, 179–183 (1999).
    [CrossRef]
  4. P. Kurzynowski, W. A. Woźniak, and M. Szarycz, “Geometric phase: two triangles on the Poincaré sphere,” J. Opt. Soc. Am. A 28, 475–482 (2011).
    [CrossRef]
  5. P. Yeh and C. Gu, “Chapter 4.1. Jones matrix formulation,” in Optics of Liquid Crystal Displays, P.Yeh and C.Gu, eds. (Wiley, 2010), pp. 173–198.
  6. P. Kurzynowski, W. A. Woźniak, and F. Ratajczyk, “Coherent superposition of polarized light—intensity vector with continuously changed relative amplitudes of component waves,” Optik 111, 201–203 (2000).
  7. J. C. Polking, “The area of a spherical triangle. Girard’s Theorem.,” http://math.rice.edu/~pcmi/sphere/gos4.html#1.
  8. I. Todhunter, “Spherical trigonometry,” http://www.gutenberg.org/files/19770/19770-pdf.pdf, p. 26.
  9. P. Kurzynowski and W. A. Woźniak, “Phase difference superposition rule for dichroic media,” Optik 103, 66–68(1996).

2011

2000

P. Kurzynowski, W. A. Woźniak, and F. Ratajczyk, “Coherent superposition of polarized light—intensity vector with continuously changed relative amplitudes of component waves,” Optik 111, 201–203 (2000).

1999

J. Courtial, “Wave plates and the Pancharatnam phase,” Opt. Commun. 171, 179–183 (1999).
[CrossRef]

1996

P. Kurzynowski and W. A. Woźniak, “Phase difference superposition rule for dichroic media,” Optik 103, 66–68(1996).

1987

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

1956

S. Pancharatnam, “Generalized theory of interference and its applications. Part I. Coherent pencils,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956).

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

Courtial, J.

J. Courtial, “Wave plates and the Pancharatnam phase,” Opt. Commun. 171, 179–183 (1999).
[CrossRef]

Gu, C.

P. Yeh and C. Gu, “Chapter 4.1. Jones matrix formulation,” in Optics of Liquid Crystal Displays, P.Yeh and C.Gu, eds. (Wiley, 2010), pp. 173–198.

Kurzynowski, P.

P. Kurzynowski, W. A. Woźniak, and M. Szarycz, “Geometric phase: two triangles on the Poincaré sphere,” J. Opt. Soc. Am. A 28, 475–482 (2011).
[CrossRef]

P. Kurzynowski, W. A. Woźniak, and F. Ratajczyk, “Coherent superposition of polarized light—intensity vector with continuously changed relative amplitudes of component waves,” Optik 111, 201–203 (2000).

P. Kurzynowski and W. A. Woźniak, “Phase difference superposition rule for dichroic media,” Optik 103, 66–68(1996).

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference and its applications. Part I. Coherent pencils,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956).

Polking, J. C.

J. C. Polking, “The area of a spherical triangle. Girard’s Theorem.,” http://math.rice.edu/~pcmi/sphere/gos4.html#1.

Ratajczyk, F.

P. Kurzynowski, W. A. Woźniak, and F. Ratajczyk, “Coherent superposition of polarized light—intensity vector with continuously changed relative amplitudes of component waves,” Optik 111, 201–203 (2000).

Szarycz, M.

Todhunter, I.

I. Todhunter, “Spherical trigonometry,” http://www.gutenberg.org/files/19770/19770-pdf.pdf, p. 26.

Wozniak, W.

Wozniak, W. A.

P. Kurzynowski, W. A. Woźniak, and F. Ratajczyk, “Coherent superposition of polarized light—intensity vector with continuously changed relative amplitudes of component waves,” Optik 111, 201–203 (2000).

P. Kurzynowski and W. A. Woźniak, “Phase difference superposition rule for dichroic media,” Optik 103, 66–68(1996).

Yeh, P.

P. Yeh and C. Gu, “Chapter 4.1. Jones matrix formulation,” in Optics of Liquid Crystal Displays, P.Yeh and C.Gu, eds. (Wiley, 2010), pp. 173–198.

J. Mod. Opt.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J. Courtial, “Wave plates and the Pancharatnam phase,” Opt. Commun. 171, 179–183 (1999).
[CrossRef]

Optik

P. Kurzynowski, W. A. Woźniak, and F. Ratajczyk, “Coherent superposition of polarized light—intensity vector with continuously changed relative amplitudes of component waves,” Optik 111, 201–203 (2000).

P. Kurzynowski and W. A. Woźniak, “Phase difference superposition rule for dichroic media,” Optik 103, 66–68(1996).

Proc. Ind. Acad. Sci. A

S. Pancharatnam, “Generalized theory of interference and its applications. Part I. Coherent pencils,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956).

Other

J. C. Polking, “The area of a spherical triangle. Girard’s Theorem.,” http://math.rice.edu/~pcmi/sphere/gos4.html#1.

I. Todhunter, “Spherical trigonometry,” http://www.gutenberg.org/files/19770/19770-pdf.pdf, p. 26.

P. Yeh and C. Gu, “Chapter 4.1. Jones matrix formulation,” in Optics of Liquid Crystal Displays, P.Yeh and C.Gu, eds. (Wiley, 2010), pp. 173–198.

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Figures (3)

Fig. 1
Fig. 1

(a) Birefringent medium in a Mach–Zehnder setup and (b) idea of construction of two triangles on the Poincaré sphere using the points representing both eigenwaves of the birefringent medium (see description in text).

Fig. 2
Fig. 2

Idea of construction of two specific triangles on the Poincaré sphere in the case of a dichroic medium using the new starting point A (see description in text).

Fig. 3
Fig. 3

Construction of the characteristic lune on the Poincaré sphere which explains the geometric phase calculating method for dichroic media in the case of the polariscopic setup.

Equations (18)

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φ D = δ ¯ ,
φ G = arctan ( tan γ 2 · cos 2 α ) ,
φ G = Ω S / 2 Ω F / 2 2 ,
[ B ] = [ M d ] · [ A ] ,
[ T F · cos α · exp ( i γ 2 ) T S · sin α · exp ( i γ 2 ) ] = [ T F · exp ( i γ 2 ) 0 0 T S · exp ( i γ 2 ) ] · [ cos α sin α ] .
[ B ] = [ M nd ] · [ A ] ,
[ T · cos β · exp ( i γ 2 ) T · sin β · exp ( i γ 2 ) ] = [ T · exp ( i γ 2 ) 0 0 T · exp ( i γ 2 ) ] · [ cos β sin β ] .
tan β = tan α · T S T F .
tan φ G = tan ( γ 2 ) · cos ( β + α ) cos ( β α ) .
| Ω F | = | F + A + B π | ,
cot [ 1 2 ( A + B ) ] = cos [ 1 2 ( a + b ) ] cos [ 1 2 ( a b ) ] · tan ( 1 2 F ) .
tan [ 1 2 ( A + B π ) ] = cos ( β + α ) cos ( β α ) · tan ( γ 2 ) .
φ G = arctan ( tan 2 α A · sin 2 ϑ F ) ,
tan μ = T S T F .
T M 2 = T F 2 + T S 2 2 .
I out = 1 2 I 0 T M 2 [ cos 2 ϑ f · cos ( 2 α A 2 μ ) + sin 2 μ · cos ( γ φ G ) ] .
φ G = arctan ( tan 2 α A · sin 2 ϑ F ) ,
I out = A + B · cos ( ϕ ) ,

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