Abstract

We demonstrate large scale polarization contrast optical diffraction tomography (ODT). In cross-polarized sample arm detection configuration we determine, from the amplitude of the optical wavefield, a relative measure of the birefringence projection. In parallel-polarized sample arm detection configuration we image the conventional phase projection. For off-axis sample placement we observe for polarization contrast ODT, similar as for phase contrast ODT, a strongly reduced noise contribution. In the limit of small birefringence phase shift δ we demonstrate tomographic reconstruction of polarization contrast images into a full 3D image of an optically cleared zebrafish. The polarization contrast ODT reconstruction shows muscular zebrafish tissue, which cannot be visualized in conventional phase contrast ODT. Polarization contrast ODT images of the zebrafish show a much higher signal to noise ratio (SNR) than the corresponding phase contrast images, SNR=73 and SNR=15, respectively.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

2017 (1)

2016 (1)

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

2015 (2)

A. d’Esposito, D. Nikitichev, A. Desjardins, S. Walker-Samuel, and M. F. Lythgoea, “Quantification of light attenuation in optically cleared mouse brains,” J. Biomed. Opt. 20(8), 080503 (2015).
[Crossref]

M. Menzel, K. Michielsen, H. D. Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc., Interface 12(111), 20150734 (2015).
[Crossref]

2014 (1)

2013 (1)

2012 (1)

A. Limaye, “Drishti: a volume exploration and presentation tool,” Proc. SPIE 8506, 85060X (2012).
[Crossref]

2009 (1)

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

2008 (2)

2007 (1)

R. J. Bryson-Richardson, S. Berger, T. F. Schilling, T. E. Hall, N. J. Cole, A. J. Gibson, J. Sharpe, and P. D. Currie, “Fishnet: an online database of zebrafish anatomy,” BMC Biol. 5(1), 34 (2007).
[Crossref]

2003 (1)

F. Massoumia, R. Juškaitis, M. A. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref]

2002 (1)

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205(2), 165–176 (2002).
[Crossref]

1998 (1)

1997 (1)

1996 (1)

R. Oldenbourg, “A new view on polarization microscopy,” Nature 381(6585), 811–812 (1996).
[Crossref]

1994 (1)

Amunts, K.

M. Menzel, K. Michielsen, H. D. Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc., Interface 12(111), 20150734 (2015).
[Crossref]

Arranz, A.

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

Axer, M.

M. Menzel, K. Michielsen, H. D. Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc., Interface 12(111), 20150734 (2015).
[Crossref]

Badizadegan, K.

Berger, J.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

Berger, S.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

R. J. Bryson-Richardson, S. Berger, T. F. Schilling, T. E. Hall, N. J. Cole, A. J. Gibson, J. Sharpe, and P. D. Currie, “Fishnet: an online database of zebrafish anatomy,” BMC Biol. 5(1), 34 (2007).
[Crossref]

Bryson-Richardson, R. J.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

R. J. Bryson-Richardson, S. Berger, T. F. Schilling, T. E. Hall, N. J. Cole, A. J. Gibson, J. Sharpe, and P. D. Currie, “Fishnet: an online database of zebrafish anatomy,” BMC Biol. 5(1), 34 (2007).
[Crossref]

Busch-Nentwich, E.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

Chen, D.

Choi, W.

Cole, N. J.

R. J. Bryson-Richardson, S. Berger, T. F. Schilling, T. E. Hall, N. J. Cole, A. J. Gibson, J. Sharpe, and P. D. Currie, “Fishnet: an online database of zebrafish anatomy,” BMC Biol. 5(1), 34 (2007).
[Crossref]

Colston, B. W.

Currie, P. D.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

R. J. Bryson-Richardson, S. Berger, T. F. Schilling, T. E. Hall, N. J. Cole, A. J. Gibson, J. Sharpe, and P. D. Currie, “Fishnet: an online database of zebrafish anatomy,” BMC Biol. 5(1), 34 (2007).
[Crossref]

d’Esposito, A.

A. d’Esposito, D. Nikitichev, A. Desjardins, S. Walker-Samuel, and M. F. Lythgoea, “Quantification of light attenuation in optically cleared mouse brains,” J. Biomed. Opt. 20(8), 080503 (2015).
[Crossref]

Dasari, R. R.

Desjardins, A.

A. d’Esposito, D. Nikitichev, A. Desjardins, S. Walker-Samuel, and M. F. Lythgoea, “Quantification of light attenuation in optically cleared mouse brains,” J. Biomed. Opt. 20(8), 080503 (2015).
[Crossref]

Dong, D.

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

Dudek, M.

Everett, M. J.

Fang, M.

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

Fang-Yen, C.

Feld, M. S.

Geisler, R.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

Ghiglia, D. C.

Gibson, A. J.

R. J. Bryson-Richardson, S. Berger, T. F. Schilling, T. E. Hall, N. J. Cole, A. J. Gibson, J. Sharpe, and P. D. Currie, “Fishnet: an online database of zebrafish anatomy,” BMC Biol. 5(1), 34 (2007).
[Crossref]

Gillette, M. U.

Hall, T. E.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

R. J. Bryson-Richardson, S. Berger, T. F. Schilling, T. E. Hall, N. J. Cole, A. J. Gibson, J. Sharpe, and P. D. Currie, “Fishnet: an online database of zebrafish anatomy,” BMC Biol. 5(1), 34 (2007).
[Crossref]

He, H.

Hui, H.

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

Jacoby, A. S.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

Juškaitis, R.

F. Massoumia, R. Juškaitis, M. A. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 2001).

Kalkman, J.

Kim, K.

Kim, K. S.

Kostencka, J.

Kozacki, T.

Kujawinska, M.

Lauer, V.

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205(2), 165–176 (2002).
[Crossref]

Liang, X.

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

Limaye, A.

A. Limaye, “Drishti: a volume exploration and presentation tool,” Proc. SPIE 8506, 85060X (2012).
[Crossref]

Lythgoea, M. F.

A. d’Esposito, D. Nikitichev, A. Desjardins, S. Walker-Samuel, and M. F. Lythgoea, “Quantification of light attenuation in optically cleared mouse brains,” J. Biomed. Opt. 20(8), 080503 (2015).
[Crossref]

Ma, H.

Massoumia, F.

F. Massoumia, R. Juškaitis, M. A. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref]

Menzel, M.

M. Menzel, K. Michielsen, H. D. Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc., Interface 12(111), 20150734 (2015).
[Crossref]

Michielsen, K.

M. Menzel, K. Michielsen, H. D. Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc., Interface 12(111), 20150734 (2015).
[Crossref]

Millet, L. J.

Neil, M. A. A.

F. Massoumia, R. Juškaitis, M. A. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref]

Nikitichev, D.

A. d’Esposito, D. Nikitichev, A. Desjardins, S. Walker-Samuel, and M. F. Lythgoea, “Quantification of light attenuation in optically cleared mouse brains,” J. Biomed. Opt. 20(8), 080503 (2015).
[Crossref]

Oldenbourg, R.

R. Oldenbourg, “A new view on polarization microscopy,” Nature 381(6585), 811–812 (1996).
[Crossref]

Park, H.

Park, Y.

Popescu, G.

Raedt, H. D.

M. Menzel, K. Michielsen, H. D. Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc., Interface 12(111), 20150734 (2015).
[Crossref]

Ravanfar, M.

Reckfort, J.

M. Menzel, K. Michielsen, H. D. Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc., Interface 12(111), 20150734 (2015).
[Crossref]

Ripoll, J.

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

Romero, L. A.

Sachs, C.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

Schilling, T. F.

R. J. Bryson-Richardson, S. Berger, T. F. Schilling, T. E. Hall, N. J. Cole, A. J. Gibson, J. Sharpe, and P. D. Currie, “Fishnet: an online database of zebrafish anatomy,” BMC Biol. 5(1), 34 (2007).
[Crossref]

Schoenenberger, K.

Sharpe, J.

R. J. Bryson-Richardson, S. Berger, T. F. Schilling, T. E. Hall, N. J. Cole, A. J. Gibson, J. Sharpe, and P. D. Currie, “Fishnet: an online database of zebrafish anatomy,” BMC Biol. 5(1), 34 (2007).
[Crossref]

Silva, L. B. D.

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 2001).

Sonntag, C.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

Stemple, D. L.

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

Tian, J.

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

Tuchin, V. V.

van Rooij, J.

Walker-Samuel, S.

A. d’Esposito, D. Nikitichev, A. Desjardins, S. Walker-Samuel, and M. F. Lythgoea, “Quantification of light attenuation in optically cleared mouse brains,” J. Biomed. Opt. 20(8), 080503 (2015).
[Crossref]

Wang, Z.

Wilson, T.

F. Massoumia, R. Juškaitis, M. A. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref]

Xie, Q.

Yamaguchi, I.

Yang, X.

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

Yao, G.

Ye, J. C.

Zeng, C.

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

Zeng, N.

Zhang, T.

Appl. Opt. (1)

Biomed. Opt. Express (2)

BMC Biol. (1)

R. J. Bryson-Richardson, S. Berger, T. F. Schilling, T. E. Hall, N. J. Cole, A. J. Gibson, J. Sharpe, and P. D. Currie, “Fishnet: an online database of zebrafish anatomy,” BMC Biol. 5(1), 34 (2007).
[Crossref]

Development (1)

A. S. Jacoby, E. Busch-Nentwich, R. J. Bryson-Richardson, T. E. Hall, J. Berger, S. Berger, C. Sonntag, C. Sachs, R. Geisler, D. L. Stemple, and P. D. Currie, “The zebrafish dystrophic mutant softy maintains muscle fibre viability despite basement membrane rupture and muscle detachment,” Development 136(19), 3367–3376 (2009).
[Crossref]

J. Biomed. Opt. (1)

A. d’Esposito, D. Nikitichev, A. Desjardins, S. Walker-Samuel, and M. F. Lythgoea, “Quantification of light attenuation in optically cleared mouse brains,” J. Biomed. Opt. 20(8), 080503 (2015).
[Crossref]

J. Microsc. (2)

F. Massoumia, R. Juškaitis, M. A. A. Neil, and T. Wilson, “Quantitative polarized light microscopy,” J. Microsc. 209(1), 13–22 (2003).
[Crossref]

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205(2), 165–176 (2002).
[Crossref]

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

J. R. Soc., Interface (1)

M. Menzel, K. Michielsen, H. D. Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc., Interface 12(111), 20150734 (2015).
[Crossref]

Nature (1)

R. Oldenbourg, “A new view on polarization microscopy,” Nature 381(6585), 811–812 (1996).
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Proc. SPIE (1)

A. Limaye, “Drishti: a volume exploration and presentation tool,” Proc. SPIE 8506, 85060X (2012).
[Crossref]

Sci. Rep. (1)

M. Fang, D. Dong, C. Zeng, X. Liang, X. Yang, A. Arranz, J. Ripoll, H. Hui, and J. Tian, “Polarization-sensitive optical projection tomography for muscle fiber imaging,” Sci. Rep. 6(1), 19241 (2016).
[Crossref]

Other (1)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 2001).

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Figures (7)

Fig. 1.
Fig. 1. Schematic of the ODT sample arm geometry. (a) The orientation of the uniaxial sample is defined by the inclination angle $\alpha$ and the direction angle $\varphi $. The sample is rotated around the $x$-axis for tomographic measurement. The input polarization state is along the $x$-axis, after which the parallel polarized $x$-component (a) or the cross-polarized $y$-component (b) of the complex wave is measured for each projection angle. The tomographic angle $\beta$ is defined with respect to the fiber orientation in the $y$-$z$ plane. The angle of the polarizers $\rho$ is defined with respect to the $x$ axis.
Fig. 2.
Fig. 2. Phase shift $\delta$ between the two orthogonal polarizations in the case of a uniaxial birefringent cylinder with maximum projected phase shift $\delta =18$ radians as quantified with polarization contrast imaging.
Fig. 3.
Fig. 3. Experimental setup for acquiring the digital holograms. HeNe: Helium Neon laser, BE: Beam expander, BS: Beam splitter, IML: Index matching liquid, S: Sample rotated around the z-axis, MO: Microscope objective, M: Mirror, TL: Tube lens, PZT: Mirror mounted on piezo stage, C: Camera, P: Polarizer, WP: Half-wave plate.
Fig. 4.
Fig. 4. (a) Logarithm of the standard deviation $\sigma (n_e - n_o)$ of a single polarization contrast reconstructed slice. (b) Cross-section along the dashed line in figure (a) and the average standard deviation over all slices of the stack (red). (c) Logarithm of the standard deviation $\sigma (\Delta n_{avg})$ of the phase contrast reconstructed slice. (d) Cross-section along the dashed line in figure (c) and the average standard deviation over all slices of the stack (red).
Fig. 5.
Fig. 5. Reconstructed amplitude (a) and (b) and phase projections (d) and (e) from two different angles of a 3 day old optically cleared zebrafish larva, illustrating the different contrasts obtained through polarization and phase contrast respectively. In (c) and (f) the histograms of the full 3D data set are plotted for the polarization and phase contrasts, respectively. The background contribution is indicated in both histograms, and the myotome and interstitial tissue for the polarization and phase contrast respectively.
Fig. 6.
Fig. 6. 3D visualization of the phase (a) and polarization (b) contrast, and combined (c) ODT reconstructions of a 3 day old zebrafish larva tail. In phase contrast, the tail (in red) and the spinal cord (in purple) appear, but not the developing muscle tissue (myotome), which is birefringent. In the polarization contrast reconstruction the structure of the myotome can be clearly discerned. Insets show transverse cross sections in linear intensity scale taken at the dashed line. Scalebar for 3D reconstruction corresponds to 200 µm.
Fig. 7.
Fig. 7. Plot of the projection functions $|U_{y}(p,\beta )|$ along with the resulting tomographic reconstructions for tilt angles $\gamma =0^{\circ }$ (a-b) and $\gamma =54^{\circ }$ (c-d). The simulation parameters are cross-polarizer angle $\rho =27^{\circ }$, $\delta n=1 \cdot 10^{-5}$, $R=1$ mm and $\lambda =633 \cdot 10^{-9}$. For comparison, the case for $\gamma =54^{\circ }$ for a non-birefringent cylinder is shown (e-f).

Equations (16)

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δ = k Δ cos 2 ( α ( β ) ) , with Δ = [ n e ( s ) n o ( s ) ]   d s ,
α = γ sin β and φ = γ cos β
U = ( e 1 2 i ( δ 2 ϵ ) ( sin 2 ( ρ φ ) + e i δ cos 2 ( ρ φ ) ) i e i ϵ sin ( δ 2 ) sin ( 2 ρ 2 φ ) ) ,
ϵ = 2 π λ n e ( s ) + n o ( s ) 2   d s .
ϕ U x = tan 1 ( cot ( δ 2 ) sin ( ϵ ) sec ( 2 ρ 2 φ ) + cos ( ϵ ) cot ( δ 2 ) cos ( ϵ ) sec ( 2 ρ 2 φ ) sin ( ϵ ) ) .
ϕ U x δ = csc 2 ( δ 2 ) sec ( 2 ρ 2 φ ) 2 cot 2 ( δ 2 ) sec 2 ( 2 ρ 2 φ ) + 2 ,
ϕ U x tan 1 ( tan ( ϵ ) ) + 1 2 δ cos ( 2 ρ 2 φ ) .
| U y | = | sin ( δ 2 ) | | sin ( 2 ρ 2 φ ) | .
sin 1 ( | sin ( δ 2 ) | ) = { δ 2 m π if 0 δ 2 < π 2 mod π δ 2 + m π if π 2 δ 2 < π mod π ,
| U y | 1 2 δ | sin ( 2 ρ 2 φ ) | .
| U y | ( β ) 1 2 k Δ cos 2 ( γ sin ( β ) ) | sin ( 2 ρ 2 γ cos ( β ) ) | .
U y ( β ) = i e i ϵ sin ( 2 ρ 2 γ cos ( β ) ) sin ( 1 2 k Δ cos 2 ( γ sin ( β ) ) ) ,
| U y ( β ) | = | sin ( 2 ρ 2 γ cos ( β ) ) | | sin ( 1 2 k Δ cos 2 ( γ sin ( β ) ) ) | .
( f ) = { 2 R 2 sec ( γ ) A p 2 A p 2 A 0 otherwise
A = R 2 cos 2 ( β ) sec 2 ( γ ) + R 2 sin 2 ( β ) ,
| U y ( p , β ) | = { | sin ( 2 ρ 2 γ cos ( β ) ) sin ( R 2 δ n k sec ( γ ) cos 2 ( γ sin ( β ) ) A p 2 A ) | p 2 A 0 otherwise