Abstract

A method for measuring birefringence by use of thermal-light polarization-sensitive optical coherence tomography is presented. The use of thermal light brings to polarization-sensitive optical coherence tomography a resolution in the micrometer range in three dimensions. The instrument is based on a Linnik interference microscope and makes use of achromatic quarter-wave plates. A mathematical representation of the instrument is presented here, and the detection scheme is described, together with a discussion of the validity domain of the equations used to evaluate the birefringence in the presence of white-light illumination.

© 2003 Optical Society of America

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  1. J. M. Mackowski, L. Pinard, L. Dognin, P. Ganau, B. Lagrange, C. Michel, M. Morgue, “Different approaches to improve the wavefront of low-loss mirrors used in the VIRGO gravitational wave antenna,” Appl. Surf. Sci. 151, 86–90 (1999).
    [CrossRef]
  2. J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, E. Tourni, “Scattered light noise in gravitational wave interferometric detectors: a statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
    [CrossRef]
  3. L. Vabre, V. Loriette, A. Dubois, J. Moreau, A. C. Boccara, “Imagery of local defects in multilayer components by short coherence length interferometry,” Opt. Lett. 27, 1899–1901 (2002).
    [CrossRef]
  4. J. F. de Boer, T. E. Milner, M. J. C. van Gemert, J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997).
    [CrossRef] [PubMed]
  5. J. F. de Boer, S. M. Srinivas, A. Malekafzali, Z. Chen, J. S. Nelson, “Imaging thermally damaged tissue by polarization sensitive optical coherence tomography,” Opt. Express 3, 212–218 (1998), http://www.opticsexpress.org .
    [CrossRef]
  6. M. J. Everett, K. Schoenenberger, B. W. Colston, L. B. Da Silva, “Birefringence characterization of biological tissue by use of optical coherence tomography,” Opt. Lett. 23, 228–230 (1998).
    [CrossRef]
  7. K. Schoenenberger, B. W. Colston, D. J. Maitland, L. B. Da Silva, M. J. Everett, “Mapping of birefringence and thermal damage in tissue by use of polarization-sensitive optical coherence tomography,” Appl. Opt. 37, 6026–6036 (1998).
    [CrossRef]
  8. J. F. de Boer, T. E. Milnerand, J. S. Nelson, “Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography,” Opt. Lett. 24, 300–302 (1999).
    [CrossRef]
  9. G. Yao, L. V. Wang, “Two-dimensional depth-resolved Mueller matrix characterization of biological tissue by optical coherence tomography,” Opt. Lett. 24, 537–539 (1999).
    [CrossRef]
  10. C. E. Saxer, J. F. de Boer, B. H. Park, Y. Zhao, Z. Chen, J. S. Nelson, “High-speed fiber-based polarization-sensitive optical coherence tomography of in vivo human skin,” Opt. Lett. 25, 1355–1357 (2000).
    [CrossRef]
  11. C. K. Hitzenberger, E. Götzinger, M. Sticker, M. Pircher, A. F. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express 9, 780–790 (2001), http://www.opticsexpress.org .
  12. J. E. Roth, J. A. Kozak, S. Yazdanfar, A. M. Rollins, J. A. Izatt, “Simplified method for polarization-sensitive optical coherence tomography,” Opt. Lett. 26, 1069–1071 (2001).
    [CrossRef]
  13. W. Drexler, U. Morgner, F. X. Kärtner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, J. G. Fujimoto, “In vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. 24, 1221–1223 (1999).
    [CrossRef]
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    [CrossRef]
  15. L. Vabre, A. Dubois, A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. 27, 530–532 (2002).
    [CrossRef]
  16. P. de Groot, L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
    [CrossRef]
  17. E. Beaurepaire, A. C. Boccara, M. Lebec, L. Blanchot, H. Saint-Jalmes, “Full-field optical coherence microscopy,” Opt. Lett. 23, 244–246 (1998).
    [CrossRef]
  18. A. Dubois, L. Vabre, A.-C. Boccara, E. Beaurepaire, “High-resolution full-field optical coherence tomography with a Linnick microscope,” Appl. Opt. 41, 805–812 (2002).
    [CrossRef] [PubMed]
  19. A. Dubois, L. Vabre, A. C. Boccara, “Sinusoidally phase-modulated interference microscope for high-speed high-resolution topographic imagery,” Opt. Lett. 26, 1873–1875 (2001).
    [CrossRef]
  20. A. Dubois, “Phase-map measurements by interferometry with sinusoidal phase modulation and four integrating buckets,” J. Opt. Soc. Am. A 18, 1972–1979 (2001).
    [CrossRef]
  21. L. Vabre, A. Dubois, M. C. Potier, J. L. Stehlé, A. C. Boccara, “DNA microarray inspection by interference microscopy,” Rev. Sci. Instrum. 72, 2834–2836 (2001).
    [CrossRef]
  22. A. Dubois, J. Selb, L. Vabre, A.-C. Boccara, “Phase measurements with wide-aperture interferometers,” Appl. Opt. 39, 2326–2331 (2000).
    [CrossRef]
  23. E. Collett, Polarized Light: Fundamental and Applications, Vol. 36 of Optical Engineering Series (Marcel Dekker, New York, 1993).
  24. J. F. Mosiño, A. Starodumov, O. Barbosa-García, V. N. Filippov, “Propagation of partially polarized light in dichroic and birefringent media,” J. Opt. B 3, 159–165 (2001).
    [CrossRef]
  25. P. Hariharan, M. Roy, “White-light phase stepping interferometry: measurement of the fractional interference order,” J. Mod. Opt. 42, 2357–2360 (1995).
    [CrossRef]
  26. A. Harasaki, J. Schmit, J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. 39, 2107–2115 (2000).
    [CrossRef]
  27. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
    [CrossRef]
  28. P. Sandoz, “An algorithm for profilometry by white-light phase-shifting interferometry,” J. Mod. Opt. 43, 1545–1554 (1996).

2002 (4)

2001 (6)

2000 (3)

1999 (4)

1998 (4)

1997 (2)

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, E. Tourni, “Scattered light noise in gravitational wave interferometric detectors: a statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[CrossRef]

J. F. de Boer, T. E. Milner, M. J. C. van Gemert, J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997).
[CrossRef] [PubMed]

1996 (2)

P. Sandoz, “An algorithm for profilometry by white-light phase-shifting interferometry,” J. Mod. Opt. 43, 1545–1554 (1996).

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
[CrossRef]

1995 (2)

P. de Groot, L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

P. Hariharan, M. Roy, “White-light phase stepping interferometry: measurement of the fractional interference order,” J. Mod. Opt. 42, 2357–2360 (1995).
[CrossRef]

Barbosa-García, O.

J. F. Mosiño, A. Starodumov, O. Barbosa-García, V. N. Filippov, “Propagation of partially polarized light in dichroic and birefringent media,” J. Opt. B 3, 159–165 (2001).
[CrossRef]

Beaurepaire, E.

Blanchot, L.

Boccara, A. C.

Boccara, A.-C.

Bondu, F.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, E. Tourni, “Scattered light noise in gravitational wave interferometric detectors: a statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[CrossRef]

Boppart, S. A.

Braccini, S.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, E. Tourni, “Scattered light noise in gravitational wave interferometric detectors: a statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[CrossRef]

Brisson, V.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, E. Tourni, “Scattered light noise in gravitational wave interferometric detectors: a statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[CrossRef]

Chen, Z.

Collett, E.

E. Collett, Polarized Light: Fundamental and Applications, Vol. 36 of Optical Engineering Series (Marcel Dekker, New York, 1993).

Colston, B. W.

Da Silva, L. B.

de Boer, J. F.

de Groot, P.

P. de Groot, L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

Deck, L.

P. de Groot, L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

Dognin, L.

J. M. Mackowski, L. Pinard, L. Dognin, P. Ganau, B. Lagrange, C. Michel, M. Morgue, “Different approaches to improve the wavefront of low-loss mirrors used in the VIRGO gravitational wave antenna,” Appl. Surf. Sci. 151, 86–90 (1999).
[CrossRef]

Drexler, W.

Dubois, A.

Everett, M. J.

Fercher, A. F.

Ferrante, I.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, E. Tourni, “Scattered light noise in gravitational wave interferometric detectors: a statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[CrossRef]

Filippov, V. N.

J. F. Mosiño, A. Starodumov, O. Barbosa-García, V. N. Filippov, “Propagation of partially polarized light in dichroic and birefringent media,” J. Opt. B 3, 159–165 (2001).
[CrossRef]

Fujimoto, J. G.

Ganau, P.

J. M. Mackowski, L. Pinard, L. Dognin, P. Ganau, B. Lagrange, C. Michel, M. Morgue, “Different approaches to improve the wavefront of low-loss mirrors used in the VIRGO gravitational wave antenna,” Appl. Surf. Sci. 151, 86–90 (1999).
[CrossRef]

Götzinger, E.

Harasaki, A.

Hariharan, P.

P. Hariharan, M. Roy, “White-light phase stepping interferometry: measurement of the fractional interference order,” J. Mod. Opt. 42, 2357–2360 (1995).
[CrossRef]

Hartl, I.

Hitzenberger, C. K.

Ippen, E. P.

Izatt, J. A.

Kärtner, F. X.

Ko, T.

Kowalevicz, A. M.

Kozak, J. A.

Lagrange, B.

J. M. Mackowski, L. Pinard, L. Dognin, P. Ganau, B. Lagrange, C. Michel, M. Morgue, “Different approaches to improve the wavefront of low-loss mirrors used in the VIRGO gravitational wave antenna,” Appl. Surf. Sci. 151, 86–90 (1999).
[CrossRef]

Larkin, K. G.

Lebec, M.

Li, X. D.

Loriette, V.

Mackowski, J. M.

J. M. Mackowski, L. Pinard, L. Dognin, P. Ganau, B. Lagrange, C. Michel, M. Morgue, “Different approaches to improve the wavefront of low-loss mirrors used in the VIRGO gravitational wave antenna,” Appl. Surf. Sci. 151, 86–90 (1999).
[CrossRef]

Maitland, D. J.

Malekafzali, A.

Michel, C.

J. M. Mackowski, L. Pinard, L. Dognin, P. Ganau, B. Lagrange, C. Michel, M. Morgue, “Different approaches to improve the wavefront of low-loss mirrors used in the VIRGO gravitational wave antenna,” Appl. Surf. Sci. 151, 86–90 (1999).
[CrossRef]

Milner, T. E.

Milnerand, T. E.

Moreau, J.

Morgner, U.

Morgue, M.

J. M. Mackowski, L. Pinard, L. Dognin, P. Ganau, B. Lagrange, C. Michel, M. Morgue, “Different approaches to improve the wavefront of low-loss mirrors used in the VIRGO gravitational wave antenna,” Appl. Surf. Sci. 151, 86–90 (1999).
[CrossRef]

Mosiño, J. F.

J. F. Mosiño, A. Starodumov, O. Barbosa-García, V. N. Filippov, “Propagation of partially polarized light in dichroic and birefringent media,” J. Opt. B 3, 159–165 (2001).
[CrossRef]

Nelson, J. S.

Park, B. H.

Pinard, L.

J. M. Mackowski, L. Pinard, L. Dognin, P. Ganau, B. Lagrange, C. Michel, M. Morgue, “Different approaches to improve the wavefront of low-loss mirrors used in the VIRGO gravitational wave antenna,” Appl. Surf. Sci. 151, 86–90 (1999).
[CrossRef]

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, E. Tourni, “Scattered light noise in gravitational wave interferometric detectors: a statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[CrossRef]

Pircher, M.

Pitris, C.

Pollnau, M.

Potier, M. C.

L. Vabre, A. Dubois, M. C. Potier, J. L. Stehlé, A. C. Boccara, “DNA microarray inspection by interference microscopy,” Rev. Sci. Instrum. 72, 2834–2836 (2001).
[CrossRef]

Rollins, A. M.

Roth, J. E.

Roy, M.

P. Hariharan, M. Roy, “White-light phase stepping interferometry: measurement of the fractional interference order,” J. Mod. Opt. 42, 2357–2360 (1995).
[CrossRef]

Saint-Jalmes, H.

Salathé, R. P.

Sandoz, P.

P. Sandoz, “An algorithm for profilometry by white-light phase-shifting interferometry,” J. Mod. Opt. 43, 1545–1554 (1996).

Saxer, C. E.

Schmit, J.

Schoenenberger, K.

Selb, J.

Srinivas, S. M.

Starodumov, A.

J. F. Mosiño, A. Starodumov, O. Barbosa-García, V. N. Filippov, “Propagation of partially polarized light in dichroic and birefringent media,” J. Opt. B 3, 159–165 (2001).
[CrossRef]

Stehlé, J. L.

L. Vabre, A. Dubois, M. C. Potier, J. L. Stehlé, A. C. Boccara, “DNA microarray inspection by interference microscopy,” Rev. Sci. Instrum. 72, 2834–2836 (2001).
[CrossRef]

Sticker, M.

Tourni, E.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, E. Tourni, “Scattered light noise in gravitational wave interferometric detectors: a statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[CrossRef]

Vabre, L.

van Gemert, M. J. C.

Vinet, J.-Y.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, E. Tourni, “Scattered light noise in gravitational wave interferometric detectors: a statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[CrossRef]

Wang, L. V.

Wyant, J. C.

Yao, G.

Yazdanfar, S.

Zhao, Y.

Appl. Opt. (4)

Appl. Surf. Sci. (1)

J. M. Mackowski, L. Pinard, L. Dognin, P. Ganau, B. Lagrange, C. Michel, M. Morgue, “Different approaches to improve the wavefront of low-loss mirrors used in the VIRGO gravitational wave antenna,” Appl. Surf. Sci. 151, 86–90 (1999).
[CrossRef]

J. Mod. Opt. (3)

P. de Groot, L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

P. Hariharan, M. Roy, “White-light phase stepping interferometry: measurement of the fractional interference order,” J. Mod. Opt. 42, 2357–2360 (1995).
[CrossRef]

P. Sandoz, “An algorithm for profilometry by white-light phase-shifting interferometry,” J. Mod. Opt. 43, 1545–1554 (1996).

J. Opt. B (1)

J. F. Mosiño, A. Starodumov, O. Barbosa-García, V. N. Filippov, “Propagation of partially polarized light in dichroic and birefringent media,” J. Opt. B 3, 159–165 (2001).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Express (3)

Opt. Lett. (11)

L. Vabre, V. Loriette, A. Dubois, J. Moreau, A. C. Boccara, “Imagery of local defects in multilayer components by short coherence length interferometry,” Opt. Lett. 27, 1899–1901 (2002).
[CrossRef]

L. Vabre, A. Dubois, A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. 27, 530–532 (2002).
[CrossRef]

J. E. Roth, J. A. Kozak, S. Yazdanfar, A. M. Rollins, J. A. Izatt, “Simplified method for polarization-sensitive optical coherence tomography,” Opt. Lett. 26, 1069–1071 (2001).
[CrossRef]

A. Dubois, L. Vabre, A. C. Boccara, “Sinusoidally phase-modulated interference microscope for high-speed high-resolution topographic imagery,” Opt. Lett. 26, 1873–1875 (2001).
[CrossRef]

J. F. de Boer, T. E. Milner, M. J. C. van Gemert, J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997).
[CrossRef] [PubMed]

M. J. Everett, K. Schoenenberger, B. W. Colston, L. B. Da Silva, “Birefringence characterization of biological tissue by use of optical coherence tomography,” Opt. Lett. 23, 228–230 (1998).
[CrossRef]

E. Beaurepaire, A. C. Boccara, M. Lebec, L. Blanchot, H. Saint-Jalmes, “Full-field optical coherence microscopy,” Opt. Lett. 23, 244–246 (1998).
[CrossRef]

J. F. de Boer, T. E. Milnerand, J. S. Nelson, “Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography,” Opt. Lett. 24, 300–302 (1999).
[CrossRef]

G. Yao, L. V. Wang, “Two-dimensional depth-resolved Mueller matrix characterization of biological tissue by optical coherence tomography,” Opt. Lett. 24, 537–539 (1999).
[CrossRef]

W. Drexler, U. Morgner, F. X. Kärtner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, J. G. Fujimoto, “In vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. 24, 1221–1223 (1999).
[CrossRef]

C. E. Saxer, J. F. de Boer, B. H. Park, Y. Zhao, Z. Chen, J. S. Nelson, “High-speed fiber-based polarization-sensitive optical coherence tomography of in vivo human skin,” Opt. Lett. 25, 1355–1357 (2000).
[CrossRef]

Phys. Rev. D (1)

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, E. Tourni, “Scattered light noise in gravitational wave interferometric detectors: a statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[CrossRef]

Rev. Sci. Instrum. (1)

L. Vabre, A. Dubois, M. C. Potier, J. L. Stehlé, A. C. Boccara, “DNA microarray inspection by interference microscopy,” Rev. Sci. Instrum. 72, 2834–2836 (2001).
[CrossRef]

Other (1)

E. Collett, Polarized Light: Fundamental and Applications, Vol. 36 of Optical Engineering Series (Marcel Dekker, New York, 1993).

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Figures (8)

Fig. 1
Fig. 1

Schematic of the instrument: NPBS, nonpolarizing beam-splitter cube; AQWPs, achromatic quarter-wave plates; PZT, piezoactuated translation stage.

Fig. 2
Fig. 2

Typical interferometric signal I det recorded by a CCD pixel as a function of time. This signal is integrated over four consecutive quarters of the modulation period to give four images, S 1S 4.

Fig. 3
Fig. 3

Values of Γ A and Γ B as a function of wavelength for δ0 = δ0opt.

Fig. 4
Fig. 4

Value of Eq. (76) as a function of the reference mirror’s modulation amplitude: circle, point for calculating Fig. 5(b); triangles, points used in Figs. 5(a) and 5(c).

Fig. 5
Fig. 5

Simulation of the procedure proposed for fixing the value of δ0: interferograms with micrometer times micrometer dimensions: (a) δ0 = 0.85 × δ0opt, (b) δ0 = δ0opt, and (c) δ0 = 1.15 × δ0opt.

Fig. 6
Fig. 6

Estimated versus true topography δ z for a source spectrum that is flat from 650 to 950 nm.

Fig. 7
Fig. 7

Magnitudes of estimated versus true birefringence δ B for various values of topography δ z ranging from 0 (dashed curves) to 0.75 × λ0 (solid curves) for the flat source spectrum: (a) β = π/4, (b) β = 0.

Fig. 8
Fig. 8

Values of Ω as a function of δ z expressed in units of λ0 for φ B = 0 (solid curve) and φ B = π (dashed curve).

Equations (82)

Equations on this page are rendered with MathJax. Learn more.

MS=1000,
MP=0001;
MQ=expi π4100-i;
Mθ=cos θsin θ-sin θcos θ;
MQθ=M-θMQMθ=expi π4cos2 θ-i sin2 θcos θ sin θ1+icos θ sin θ1+isin2 θ-i cos2 θ;
MQ45=221ii1;
MR=r1001.
MB=rexp-iφB/200exp+iφB/2.
MBβ=rcos2 β exp-iφB/2+sin2 β exp+iφB/2-i sin 2βsinφB/2-i sin 2β sinφB/2cos2 β exp+iφB/2+sin2 β exp-iφB/2.
MD=rx00ry,
MDβ=rx cos2 β+ry sin2 βrx-rysin β cosβrx-rysin β cos βrx sin2 β+ry cos2 β.
MbsR=rbs1001,
MbsT=tbs1001.
EP=I01.
Mref=MbsTMQθMRMQθMbsR,
Mref=icos 2θsin 2θsin 2θ-cos 2θrbstbsrref expiϕref.
Mech=MbsRMQ45MSβMQ45MbsT,
Msam=12rx-ryexp2iβirx+ryirx+ry-rx-ryexp-2iβrbstbs expiϕsam.
Msam=i-sinφB/2exp2iβcosφB/2cosφB/2sinφB/2exp-2iβrbstbsr expiϕsam.
EdetP=MPMref+MsamEP.
EdetP=I2-rref cos 2θ expiϕref-rx-ry2exp-2iβexpiϕsam.
IdetP=I4rref cos 2θ2+|rx-ry|24+rref cos 2θRerx-rycosϕ-2β-Imrx-rysinϕ-2β
ϕx, y=ϕsam-ϕref¯+φzx, y.
EdetS=I2rref sin 2θ expiϕref+i rx+ry2expiϕsam
IdetS=I4rref sin 2θ2+|rx+ry|24-rref sin 2θRerx+rysinϕ+Imrx+rycosϕ.
IdetP=I4rref cos 2θ2+δr24+rrefδr cos 2θ cosϕ-2β,
IdetS=I4rref sin 2θ2+rsam2-2rrefrsam sin 2θ sinϕ.
IdetP=I4rref cos 2θ2+rsam sinφB22+2rrefrsam cos 2θ sinφB2sinφ-2β,
IdetS=I4rref sin 2θ2+rsam cosφB22-2rrefrsam sin 2θ cosφB2sinϕ.
IdetP=I0P+AP cosϕsam-ϕref¯+BP sinϕsam-ϕref¯,
IdetS=I0S+AS cosϕsam-ϕref¯+BS sinϕsam-ϕref¯,
I0P=14 Irref cos 2θ2+rsam sinφB22,
AP=+I2 rrefrsam cos 2θ sinφB2sinφz-2β,
BP=+I2 rrefrsam cos 2θ sinφB2cosφz-2β,
I0S=14 Irref sin 2θ2+rsam cosφB22,
AS=-I2 rrefrsam sin 2θ cosφB2sinφz,
BS=-I2 rrefrsam sin 2θ cosφB2cosφz.
tan2φB2=tan-22θAP2+BP2AS2+BS2;
tanφz=ASBS;
tan2β=ASBP-APBSBSBP+ASAP.
FWHM=0.6λ1-cosarcsin N.A.,
ϕsam-ϕref¯=ϕ0 sinωt+ψ,
ϕ0<2π
sinϕsam-ϕref¯=2 k=0 J2k+1ϕ0sin2k+1ωt+ψ,
cosϕsam-ϕref¯=J0ϕ0+2 k=1 J2kϕ0cos2kωt+ψ.
SqP=q-1T/4qT/4 IdetPtdt, SqS=q-1T/4qT/4 IdetStdt.
SqP=T4I0P+J0ϕ0AP+TAPπk=1J2kϕ02ksin2kq π2+ψ-sin2kq-1π2+ψ-TBPπk=0J2k+1ϕ02k+1cos2k+1q π2+ψ-cos2k+1q-1π2+ψ,
ΣBP=-S1P+S2P+S3P-S4P,
ΣAP=-S1P+S2P-S3P+S4P.
ΣBP=4Tπ BPΓBϕ0, ψ,
ΣAP=4Tπ APΓAϕ0, ψ,
ΓBϕ0, ψ=k=0-1kJ2k+1ϕ02k+1sin2k+1ψ,
ΓAϕ0, ψ=k=11--1kJ2kϕ02ksin2kψ.
tan2φB2=tan-22θΓBϕ0, ψΣAP2+ΓAϕ0, ψΣBP2ΓBϕ0, ψΣAS2+ΓAϕ0, ψΣBS2.
S0P=T4I0P+J0ϕ0AP.
S1P=S0P+Tπ-APΓAϕ0, ψ-BPf2(ϕ0, ψ,
S2P=S0P+TπAPΓAϕ0, ψ-BPf3ϕ0, ψ,
S3P=S0P+Tπ-APΓAϕ0, ψ+BPf2ϕ0, ψ,
S4P=S0P+TπAPΓAϕ0, ψ+BPf3ϕ0, ψ,
f2ϕ0, ψ=k=0J2k+1ϕ02k+1cos2k+1ψ+-1k sin2k+1ψ,
f3ϕ0, ψ=k=0J2k+1ϕ02k+1cos2k+1ψ--1k sin2k+1ψ.
ΓBϕ0, ψ=½f2ϕ0, ψ-f3ϕ0, ψ.
ΣAP2+ΣBP2=4Tπ2APΓAϕ0, ψ2+BPΓBϕ0, ψ2.
ΣAP2+ΣBP2=4Tπ ΓAϕ0, ψI2 rrefrsam cos 2θ sinφB22,
tan2φB2=tan-22θΣAP2+ΣBP2ΣAS2+ΣBS2.
IdetPν=I4rref cos 2θ2+rsam sinφBν22+2rrefrsam cos 2θ sinφBν2sinϕν-2β,
IdetSν=I4rref sin 2θ2+rsam cosφBν22-2rrefrsam sin 2θ cosφBν2sinϕν.
S1P=S0P+Tπ-0 fνAPν, δBΓAν, δ0, ψdν-0 fνBPν, δBf2ν, δ0, ψdν,
S2P=S0P+Tπ0 fνAPν, δBΓAν, δ0, ψdν-0 fνBPν, δBf3ν, δ0, ψdν,
S3P=S0P+Tπ-0 fνAPν, δBΓAν, δ0, ψdν+0 fνBPν, δBf2ν, δ0, ψdν,
S4P=S0P+Tπ0 fνAPν, δBΓAν, δ0, ψdν+0 fνBPν, δBf3ν, δ0, ψdν.
ΣAS2+ΣBS2=4Tπ20 fνAS×ν, δBΓAν, δ0, ψdν2+0 fνBS×ν, δBΓBν, δ0, ψdν2,
ASν, δB=-I2 rrefrsam sin 2θ sin4πνδz/c,
BSν, δB=-I2 rrefrsam sin 2θ cos4πνδz/c.
ddδzΣAS2+ΣBS2δz0=0,
0 fνΓAν, δ0optνdν2-0 fνΓBν, δ0optdν0 fνΓBν, δ0optν2dν=0.
ΓA4π δ0optλ0, π4=ΓB4π δ0optλ0, π4.
ΘAδ0opt=0 fνΓAν, δ0opt, π4dν,ΘBδ0opt=0 fνΓBν, δ0opt, π4dν
tan2π δBλ0=tan-22θ×ΘBδ0optΣAP2+ΘAδ0optΣBP2ΘBδ0optΣAS2+ΘAδ0optΣBS2,
tan2π δzλ0=ΘBδ0optΣASΘAδ0optΣBS,
tan2β=ΘAδ0optΘBδ0optΣASΣBP-ΣAPΣBSΘA2δ0optΣBSΣBP+ΘB2δ0optΣASΣAP.
ΩδB, β, δz=ΣAS2+ΣAP2+ΣBS2+ΣBP2.

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