Abstract

We propose a method for high-sensitivity optical rotatory dispersion (ORD) measurement of optically active samples that takes advantage of the nonlinear behavior of the geometric phase (GP). To measure the GP as a function of wavelength, we use a multichannel Fourier transform spectrometer (MC-FTS) that is based on Savart plate birefringent-polarization interference, into which we newly insert a zeroth-order quarter-wave plate (QWP). The modified MC-FTS allows us to measure the wavelength dependence of the GP and thus that of the optical rotation angle due to the sample. In this paper, we describe the proposed approach and demonstrate proof-of-principle experiments.

© 2017 Optical Society of America

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References

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    [Crossref]
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2011 (4)

2009 (2)

S. Tamate, H. Kobayashi, T. Nakanishi, S. Sugiyama, and M. Kitano, “Geometrical aspects of weak measurements and quantum erasers,” New J. Phys. 11(9), 093025 (2009).
[Crossref]

J.-F. Lin, C.-C. Chang, C.-D. Syu, Y.-L. Lo, and S.-Y. Lee, “A new electro-optic modulated circular heterodyne interferometer for measuring the rotation angle in a chiral medium,” Opt. Lasers Eng. 47(17), 39–44 (2009).
[Crossref]

2007 (1)

2006 (4)

K. Oka and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508 (2006).
[Crossref]

T. Wakayama, Y. Otani, and N. Umeda, “Birefringence dispersion measurement based on achromatic four points of geometric phases,” Opt. Eng. 45(8), 083603 (2006).
[Crossref]

Y. L. Lo and T. C. Yu, “A polarimetric glucose sensor using a liquid-crystal polarization modulator driven by a sinusoidal signal,” Opt. Commun. 259(1), 40–48 (2006).
[Crossref]

Y. Hori, T. Yasui, and T. Araki, “Optical glucose monitoring based on femtosecond two-color pulse interferometry,” Opt. Rev. 13(1), 29–33 (2006).
[Crossref]

2004 (3)

J. Y. Lin, K. H. Chen, and D. C. Su, “Improved method for measuring small optical rotation angle of chiral medium,” Opt. Commun. 238(1–3), 113–118 (2004).
[Crossref]

C. Chou, W. C. Kuo, T. S. Hsieh, and H. K. Teng, “A phase sensitive optical rotation measurement in a scattered chiral medium using a Zeeman laser,” Opt. Commun. 230(4–6), 259–266 (2004).
[Crossref]

N. Murakami, K. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237(1–3), 9–15 (2004).
[Crossref]

2003 (1)

1999 (1)

J. Courtial, “Wave plates and the Pancharatnam phase,” Opt. Commun. 171(4–6), 179–183 (1999).
[Crossref]

1998 (1)

R. M. Ribeiro, A. B. A. Fiasca, and P. A. M. dos Santos, “Automatic optical activity measurement system,” Opt. Laser Technol. 30(2), 121–124 (1998).
[Crossref]

1997 (1)

1996 (2)

1994 (1)

R. Bhandari, “Geometric phase in interference experiments,” Curr. Sci. 67(4), 224–230 (1994).

1993 (1)

H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71(10), 1530–1533 (1993).
[Crossref] [PubMed]

1992 (3)

1986 (1)

1985 (2)

1984 (2)

T. Okamoto, S. Kawata, and S. Minami, “Fourier transform spectrometer with a self-scanning photodiode array,” Appl. Opt. 23(2), 269–273 (1984).
[Crossref] [PubMed]

M. V. Berry, “Quantum phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 45–57 (1984).
[Crossref]

1956 (2)

S. Pancharatnam, “Generalized theory of interference and its applications: Part I. Coherent pencils,” Proc. Ind. Acad. Sci. A 44(5), 247–251 (1956).

S. Pancharatnam, “Generalized theory of interference and its applications: Part II. Partially coherent pencils,” Proc. Ind. Acad. Sci. A 44(6), 398–412 (1956).

1951 (1)

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

Araki, T.

Y. Hori, T. Yasui, and T. Araki, “Optical glucose monitoring based on femtosecond two-color pulse interferometry,” Opt. Rev. 13(1), 29–33 (2006).
[Crossref]

Aravind, P. K.

P. K. Aravind, “A simple proof of Pancharatnam’s theory,” Opt. Commun. 94(4), 191–196 (1992).
[Crossref]

Aryamanya-Mugisha, H.

Baba, N.

N. Murakami, K. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237(1–3), 9–15 (2004).
[Crossref]

Barnes, T. H.

Barton, J. K.

Berry, M. V.

M. V. Berry, “Quantum phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 45–57 (1984).
[Crossref]

Bhandari, R.

R. Bhandari, “Geometric phase in interference experiments,” Curr. Sci. 67(4), 224–230 (1994).

Bonnema, G. T.

Chandrasekhar, S.

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

Chang, C.-C.

J.-F. Lin, C.-C. Chang, C.-D. Syu, Y.-L. Lo, and S.-Y. Lee, “A new electro-optic modulated circular heterodyne interferometer for measuring the rotation angle in a chiral medium,” Opt. Lasers Eng. 47(17), 39–44 (2009).
[Crossref]

Chang, M.

Chen, K. H.

J. Y. Lin, K. H. Chen, and D. C. Su, “Improved method for measuring small optical rotation angle of chiral medium,” Opt. Commun. 238(1–3), 113–118 (2004).
[Crossref]

Chou, C.

C. Chou, W. C. Kuo, T. S. Hsieh, and H. K. Teng, “A phase sensitive optical rotation measurement in a scattered chiral medium using a Zeeman laser,” Opt. Commun. 230(4–6), 259–266 (2004).
[Crossref]

C. Chou, Y. C. Huang, and M. Chang, “Precise optical activity measurement of quartz plate by using a true phase-sensitive technique,” Appl. Opt. 36(16), 3604–3609 (1997).
[Crossref] [PubMed]

Courtial, J.

dos Santos, P. A. M.

R. M. Ribeiro, A. B. A. Fiasca, and P. A. M. dos Santos, “Automatic optical activity measurement system,” Opt. Laser Technol. 30(2), 121–124 (1998).
[Crossref]

Dultz, W.

H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71(10), 1530–1533 (1993).
[Crossref] [PubMed]

Fiasca, A. B. A.

R. M. Ribeiro, A. B. A. Fiasca, and P. A. M. dos Santos, “Automatic optical activity measurement system,” Opt. Laser Technol. 30(2), 121–124 (1998).
[Crossref]

Gutiérrez-Vega, J. C.

Harvey, A. R.

Hashimoto, M.

Hori, Y.

Y. Hori, T. Yasui, and T. Araki, “Optical glucose monitoring based on femtosecond two-color pulse interferometry,” Opt. Rev. 13(1), 29–33 (2006).
[Crossref]

Hsieh, T. S.

C. Chou, W. C. Kuo, T. S. Hsieh, and H. K. Teng, “A phase sensitive optical rotation measurement in a scattered chiral medium using a Zeeman laser,” Opt. Commun. 230(4–6), 259–266 (2004).
[Crossref]

Huang, Y. C.

Ishigaki, T.

N. Murakami, K. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237(1–3), 9–15 (2004).
[Crossref]

Jonasz, M.

Junttila, M.-L.

Kaneko, T.

Kato, K.

N. Murakami, K. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237(1–3), 9–15 (2004).
[Crossref]

Kawata, S.

Kitamura, R.

Kitano, M.

H. Kobayashi, S. Tamate, T. Nakanishi, K. Sugiyama, and M. Kitano, “Observation of geometric phases in quantum erasers,” J. Phys. Soc. Jpn. 80(3), 034401 (2011).
[Crossref]

S. Tamate, H. Kobayashi, T. Nakanishi, S. Sugiyama, and M. Kitano, “Geometrical aspects of weak measurements and quantum erasers,” New J. Phys. 11(9), 093025 (2009).
[Crossref]

Klein, S.

H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71(10), 1530–1533 (1993).
[Crossref] [PubMed]

Kobayashi, H.

H. Kobayashi, S. Tamate, T. Nakanishi, K. Sugiyama, and M. Kitano, “Observation of geometric phases in quantum erasers,” J. Phys. Soc. Jpn. 80(3), 034401 (2011).
[Crossref]

S. Tamate, H. Kobayashi, T. Nakanishi, S. Sugiyama, and M. Kitano, “Geometrical aspects of weak measurements and quantum erasers,” New J. Phys. 11(9), 093025 (2009).
[Crossref]

Kuo, W. C.

C. Chou, W. C. Kuo, T. S. Hsieh, and H. K. Teng, “A phase sensitive optical rotation measurement in a scattered chiral medium using a Zeeman laser,” Opt. Commun. 230(4–6), 259–266 (2004).
[Crossref]

Kurzynowski, P.

Lee, S.-Y.

J.-F. Lin, C.-C. Chang, C.-D. Syu, Y.-L. Lo, and S.-Y. Lee, “A new electro-optic modulated circular heterodyne interferometer for measuring the rotation angle in a chiral medium,” Opt. Lasers Eng. 47(17), 39–44 (2009).
[Crossref]

Lin, J. Y.

J. Y. Lin, K. H. Chen, and D. C. Su, “Improved method for measuring small optical rotation angle of chiral medium,” Opt. Commun. 238(1–3), 113–118 (2004).
[Crossref]

Lin, J.-F.

J.-F. Lin, C.-C. Chang, C.-D. Syu, Y.-L. Lo, and S.-Y. Lee, “A new electro-optic modulated circular heterodyne interferometer for measuring the rotation angle in a chiral medium,” Opt. Lasers Eng. 47(17), 39–44 (2009).
[Crossref]

Lo, Y. L.

Y. L. Lo and T. C. Yu, “A polarimetric glucose sensor using a liquid-crystal polarization modulator driven by a sinusoidal signal,” Opt. Commun. 259(1), 40–48 (2006).
[Crossref]

Lo, Y.-L.

J.-F. Lin, C.-C. Chang, C.-D. Syu, Y.-L. Lo, and S.-Y. Lee, “A new electro-optic modulated circular heterodyne interferometer for measuring the rotation angle in a chiral medium,” Opt. Lasers Eng. 47(17), 39–44 (2009).
[Crossref]

Love, G. D.

G. D. Love, “The unbounded nature of geometrical and dynamical phases in polarization optics,” Opt. Commun. 131(4–6), 236–240 (1996).
[Crossref]

Minami, S.

Murakami, N.

N. Murakami, K. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237(1–3), 9–15 (2004).
[Crossref]

Nakanishi, T.

H. Kobayashi, S. Tamate, T. Nakanishi, K. Sugiyama, and M. Kitano, “Observation of geometric phases in quantum erasers,” J. Phys. Soc. Jpn. 80(3), 034401 (2011).
[Crossref]

S. Tamate, H. Kobayashi, T. Nakanishi, S. Sugiyama, and M. Kitano, “Geometrical aspects of weak measurements and quantum erasers,” New J. Phys. 11(9), 093025 (2009).
[Crossref]

Oka, K.

K. Oka and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508 (2006).
[Crossref]

K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express 11(13), 1510–1519 (2003).
[Crossref] [PubMed]

Okamoto, T.

Otani, Y.

T. Wakayama, Y. Otani, and N. Umeda, “Birefringence dispersion measurement based on achromatic four points of geometric phases,” Opt. Eng. 45(8), 083603 (2006).
[Crossref]

Padgett, M. J.

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference and its applications: Part II. Partially coherent pencils,” Proc. Ind. Acad. Sci. A 44(6), 398–412 (1956).

S. Pancharatnam, “Generalized theory of interference and its applications: Part I. Coherent pencils,” Proc. Ind. Acad. Sci. A 44(5), 247–251 (1956).

Patterson, B. A.

Pilon, L.

Ribeiro, R. M.

R. M. Ribeiro, A. B. A. Fiasca, and P. A. M. dos Santos, “Automatic optical activity measurement system,” Opt. Laser Technol. 30(2), 121–124 (1998).
[Crossref]

Saito, N.

K. Oka and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508 (2006).
[Crossref]

Schmitzer, H.

H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71(10), 1530–1533 (1993).
[Crossref] [PubMed]

Sibbett, W.

Su, D. C.

J. Y. Lin, K. H. Chen, and D. C. Su, “Improved method for measuring small optical rotation angle of chiral medium,” Opt. Commun. 238(1–3), 113–118 (2004).
[Crossref]

Sugiyama, K.

H. Kobayashi, S. Tamate, T. Nakanishi, K. Sugiyama, and M. Kitano, “Observation of geometric phases in quantum erasers,” J. Phys. Soc. Jpn. 80(3), 034401 (2011).
[Crossref]

Sugiyama, S.

S. Tamate, H. Kobayashi, T. Nakanishi, S. Sugiyama, and M. Kitano, “Geometrical aspects of weak measurements and quantum erasers,” New J. Phys. 11(9), 093025 (2009).
[Crossref]

Syu, C.-D.

J.-F. Lin, C.-C. Chang, C.-D. Syu, Y.-L. Lo, and S.-Y. Lee, “A new electro-optic modulated circular heterodyne interferometer for measuring the rotation angle in a chiral medium,” Opt. Lasers Eng. 47(17), 39–44 (2009).
[Crossref]

Szarycz, M.

Tamate, S.

H. Kobayashi, S. Tamate, T. Nakanishi, K. Sugiyama, and M. Kitano, “Observation of geometric phases in quantum erasers,” J. Phys. Soc. Jpn. 80(3), 034401 (2011).
[Crossref]

S. Tamate, H. Kobayashi, T. Nakanishi, S. Sugiyama, and M. Kitano, “Geometrical aspects of weak measurements and quantum erasers,” New J. Phys. 11(9), 093025 (2009).
[Crossref]

Teng, H. K.

C. Chou, W. C. Kuo, T. S. Hsieh, and H. K. Teng, “A phase sensitive optical rotation measurement in a scattered chiral medium using a Zeeman laser,” Opt. Commun. 230(4–6), 259–266 (2004).
[Crossref]

Umeda, N.

T. Wakayama, Y. Otani, and N. Umeda, “Birefringence dispersion measurement based on achromatic four points of geometric phases,” Opt. Eng. 45(8), 083603 (2006).
[Crossref]

Wakayama, T.

T. Wakayama, Y. Otani, and N. Umeda, “Birefringence dispersion measurement based on achromatic four points of geometric phases,” Opt. Eng. 45(8), 083603 (2006).
[Crossref]

Williams, R. R.

Winkler, A. M.

Wozniak, W. A.

Yasui, T.

Y. Hori, T. Yasui, and T. Araki, “Optical glucose monitoring based on femtosecond two-color pulse interferometry,” Opt. Rev. 13(1), 29–33 (2006).
[Crossref]

Yu, T. C.

Y. L. Lo and T. C. Yu, “A polarimetric glucose sensor using a liquid-crystal polarization modulator driven by a sinusoidal signal,” Opt. Commun. 259(1), 40–48 (2006).
[Crossref]

Appl. Opt. (8)

Appl. Spectrosc. (2)

Curr. Sci. (1)

R. Bhandari, “Geometric phase in interference experiments,” Curr. Sci. 67(4), 224–230 (1994).

J. Opt. Soc. Am. A (1)

J. Phys. Soc. Jpn. (1)

H. Kobayashi, S. Tamate, T. Nakanishi, K. Sugiyama, and M. Kitano, “Observation of geometric phases in quantum erasers,” J. Phys. Soc. Jpn. 80(3), 034401 (2011).
[Crossref]

New J. Phys. (1)

S. Tamate, H. Kobayashi, T. Nakanishi, S. Sugiyama, and M. Kitano, “Geometrical aspects of weak measurements and quantum erasers,” New J. Phys. 11(9), 093025 (2009).
[Crossref]

Opt. Commun. (7)

J. Y. Lin, K. H. Chen, and D. C. Su, “Improved method for measuring small optical rotation angle of chiral medium,” Opt. Commun. 238(1–3), 113–118 (2004).
[Crossref]

C. Chou, W. C. Kuo, T. S. Hsieh, and H. K. Teng, “A phase sensitive optical rotation measurement in a scattered chiral medium using a Zeeman laser,” Opt. Commun. 230(4–6), 259–266 (2004).
[Crossref]

Y. L. Lo and T. C. Yu, “A polarimetric glucose sensor using a liquid-crystal polarization modulator driven by a sinusoidal signal,” Opt. Commun. 259(1), 40–48 (2006).
[Crossref]

G. D. Love, “The unbounded nature of geometrical and dynamical phases in polarization optics,” Opt. Commun. 131(4–6), 236–240 (1996).
[Crossref]

J. Courtial, “Wave plates and the Pancharatnam phase,” Opt. Commun. 171(4–6), 179–183 (1999).
[Crossref]

N. Murakami, K. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237(1–3), 9–15 (2004).
[Crossref]

P. K. Aravind, “A simple proof of Pancharatnam’s theory,” Opt. Commun. 94(4), 191–196 (1992).
[Crossref]

Opt. Eng. (1)

T. Wakayama, Y. Otani, and N. Umeda, “Birefringence dispersion measurement based on achromatic four points of geometric phases,” Opt. Eng. 45(8), 083603 (2006).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

R. M. Ribeiro, A. B. A. Fiasca, and P. A. M. dos Santos, “Automatic optical activity measurement system,” Opt. Laser Technol. 30(2), 121–124 (1998).
[Crossref]

Opt. Lasers Eng. (1)

J.-F. Lin, C.-C. Chang, C.-D. Syu, Y.-L. Lo, and S.-Y. Lee, “A new electro-optic modulated circular heterodyne interferometer for measuring the rotation angle in a chiral medium,” Opt. Lasers Eng. 47(17), 39–44 (2009).
[Crossref]

Opt. Lett. (1)

Opt. Rev. (1)

Y. Hori, T. Yasui, and T. Araki, “Optical glucose monitoring based on femtosecond two-color pulse interferometry,” Opt. Rev. 13(1), 29–33 (2006).
[Crossref]

Phys. Rev. Lett. (1)

H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71(10), 1530–1533 (1993).
[Crossref] [PubMed]

Proc. Ind. Acad. Sci. A (2)

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Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

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Proc. SPIE (1)

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[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Optical setup of modified MC-FTS based on Savart plate (SP) birefringent interference, in which a zeroth-order quarter-wave plate (QWP) is inserted. W-LED: white LED; P: linear polarizer; A: linear analyzer; L: Fourier transform lens; PDA: photodiode array; ϕ: orientation angle of the fast axis (f) of the QWP; θ: orientation angle of P; α: optical rotation angle of the sample. All angles are defined with respect to the horizontal axis (x) and their signs are positive for counterclockwise rotation in the x-y coordinate system, as shown in the figure.
Fig. 2
Fig. 2 Poincaré sphere diagrams used to explain the geometric phases, γ, which are obtained from the MC-FTS with the zeroth-order QWP (the fast axis of which is represented by f), where α stands for the rotatory angle provided by the sample, θ is the orientation angle of the analyzer, ϕ is the orientation angle of the fast axis of the QWP, and δ(λ) is the retardation angle introduced by the QWP depending on the wavelength λ; (a) φ=π/4, α=0,δ(λ)=π/2 , (b) φ=π/4 ,α0,δ(λ)=π/2 , (c) φπ/4, α0,δ(λ)=π/2 , (d) φ=π/4, α=0,δ(λ)π/2 , (e) φ=π/4 ,α0,δ(λ)π/2 , and (f) φπ/4, α0,δ(λ)π/2 .
Fig. 3
Fig. 3 Flow chart for estimation of ORD, α(λ).
Fig. 4
Fig. 4 Plots of Γ as a function of θ, where: (a) shows Γ(θ, φ=π/4, δ=π/2, α=0) , (b) shows Γ(θ, φ=π/4, δ=π/2, α=π/18) , (c) shows Γ(θ, φ=17π/36, δ=π/2, α=π/18) , (d) shows Γ(θ, φ=π/4, δ=7π/18, α=0) , (e) shows Γ(θ, φ=π/4, δ=7π/18, α=π/18) , and (f) shows Γ(θ, φ=17π/18, δ=7π/18, α=π/18) .
Fig. 5
Fig. 5 Plots of Γ/α as a function of θ for α= 0.1 with φ= 45 , 6 0 , 80 , and 8 5 : (a) δ= 150.3 (λ=400 nm) , (b) δ= 116.5 (λ=500 nm) , (c) δ= 95.4 (λ=600 nm) , and (d) δ= 80.8 (λ=700 nm) .
Fig. 6
Fig. 6 (a) Plots of Γ/α as a function of α with φ= 85 and θ= 80 at λ = 400 nm, 500 nm, 600 nm, and 700 nm, and (b) the same plots as in (a) again, but with φ= 6 0 and θ= 5 5 .
Fig. 7
Fig. 7 α(λ) of dextro-rotatory quartz plates with thicknesses of l = 2.1 mm and l = 0.5 mm. Solid lines denote numerically calculated values based on Eq. (20).

Equations (20)

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P( θ )=A(θ)=( cos 2 θ cosθsinθ cosθsinθ sin 2 θ ).
C( φ,δ )=( cos δ 2 +icos2φsin δ 2 isin2φsin δ 2 isin2φsin δ 2 cos δ 2 icos2φsin δ 2 ).
R( α )=( cosα sinα sinα cosα ).
J 1 (θ,φ,δ,α)=A( θ )C(φ,δ)R( α )P( 0 ) P in = 1 2 { cos δ 2 cos( θα )+isin δ 2 cos( θ2φ+α ) }( cosθ sinθ ).
J 2 (θ,φ,δ,α)=A( θ )C(φ,δ)R( α )P( π/2 ) P in = 1 2 { cos δ 2 sin( θα )isin δ 2 cos( θ2φ+α ) }( cosθ sinθ ).
γ 1 (θ,φ,δ,α)=arg{ cos δ 2 cos( θα )+isin δ 2 cos( θ2φ+α ) } = tan 1 { cos( θ2φ+α ) cos( θα ) tan δ 2 }.
γ 2 (θ,φ,δ,α)=arg{ cos δ 2 sin( θα )isin δ 2 sin( θ2φ+α ) } = tan 1 { sin( θ2φ+α ) sin( θα ) tan δ 2 }.
γ(θ,φ,δ,α)= γ 1 (θ,φ,δ,α) γ 2 (θ,φ,δ,α) = tan 1 { cos( θ2φ+α ) cos( θα ) tan δ 2 } tan 1 { sin( θ2φ+α ) sin( θα ) tan δ 2 }.
γ(θ,φ,δ, α=0)= tan 1 { cos( θ2φ ) cosθ tan δ 2 } tan 1 { sin( 2φθ ) sinθ tan δ 2 },
γ(θ, φ=π/4, δ=π/2, α=0)= γ 1 (θ, π/4, π/2, 0) γ 2 (θ, π/4, π/2, 0) = tan 1 ( tanθ ) tan 1 ( cotθ )=2θπ/2.
e= i=0 N { p( θ i ,φ,δ(λ),α(λ))γ( θ i ,φ,δ(λ),α(λ)) } 2 .
P(θ,φ,δ(λ),α(λ))=p(θ,φ,δ(λ),α(λ))p(0,φ,δ(λ),α(λ)).
Γ(θ,φ,δ(λ),α(λ))=γ(θ,φ,δ(λ),α(λ))γ(0,φ,δ(λ),α(λ)).
ΔP(θ,φ,δ(λ),α(λ))=P(θ,φ,δ(λ),α(λ))P(θ,φ,δ(λ),0).
ΔΓ(θ,φ,δ(λ),α(λ))=Γ(θ,φ,δ(λ),α(λ))Γ(θ,φ,δ(λ),0).
E= i=0 N { ΔP( θ i ,φ,δ(λ),α(λ))ΔΓ( θ i ,φ,δ(λ),α(λ)) } 2 .
Γ/α=γ/α [ γ/α ] θ=0 .
γ/α=X cos 2 γ 1 Y cos 2 γ 2 ,
X= sin( θ2φ+α ) cos( θα ) tan δ 2 cos( θ2φ+α ) cos 2 ( θα ) sin( θα )tan δ 2 Y= cos( θ2φ+α ) sin( θα ) tan δ 2 sin( θ2φ+α ) sin 2 ( θα ) cos( θα )tan δ 2 .
α λ =0.1963657+7.262667/ λ 2 +0.1171867/ λ 4 +0.0019554/ λ 6 ,

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