Abstract

Conventional imaging of the human cornea with optical coherence tomography (OCT) relies on telecentric scanning optics with sampling beams that are parallel to the optical axis of the eye. Because of the shape of the cornea, the beams have in some areas considerable inclination to the corneal surface which is accompanied by low signal intensities in these areas and thus an inhomogeneous appearance of corneal structures. In addition, alterations in the polarization state of the probing light depend on the angle between the imaging beam and the birefringent axis of the sample. Therefore, changes in the polarization state observed with polarization-sensitive (PS-) OCT originate mainly from the shape of the cornea. In order to minimize the effects of the corneal shape on intensity and polarization-sensitive based data, we developed a conical scanning optics design. This design provides imaging beams that are essentially orthogonal to the corneal surface. Thus, high signal intensity throughout the entire imaged volume is obtained and the influence of the corneal shape on polarization-sensitive data is greatly reduced. We demonstrate the benefit of the concept by comparing PS-OCT imaging results of the human cornea in healthy volunteers using both scanning schemes.

© 2017 Optical Society of America

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References

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2015 (4)

2014 (1)

2013 (2)

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus Diagnosis Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 54(2), 1384–1391 (2013).
[Crossref] [PubMed]

R. Poddar, D. E. Cortés, J. S. Werner, M. J. Mannis, and R. J. Zawadzki, “Three-dimensional anterior segment imaging in patients with type 1 Boston Keratoprosthesis with switchable full depth range swept source optical coherence tomography,” J. Biomed. Opt. 18(8), 86002 (2013).
[Crossref] [PubMed]

2011 (5)

M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Prog. Retin. Eye Res. 30(6), 431–451 (2011).
[Crossref] [PubMed]

A. N. Tan, L. D. C. Sauren, J. de Brabander, T. T. J. M. Berendschot, V. Lima Passos, C. A. B. Webers, R. M. M. A. Nuijts, and H. J. M. Beckers, “Reproducibility of Anterior Chamber Angle Measurements with Anterior Segment Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 52(5), 2095–2099 (2011).
[Crossref]

D. W. DelMonte and T. Kim, “Anatomy and physiology of the cornea,” J. Cataract Refract. Surg. 37(1), 588–598 (2011).
[Crossref] [PubMed]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2(8), 2392–2402 (2011).
[Crossref] [PubMed]

T. Klein, W. Wieser, C. M. Eigenwillig, B. R. Biedermann, and R. Huber, “Megahertz OCT for ultrawide-field retinal imaging with a 1050nm Fourier domain mode-locked laser,” Opt. Express 19(4), 3044–3062 (2011).
[Crossref] [PubMed]

2010 (3)

2009 (1)

K. M. Meek and C. Boote, “The use of X-ray scattering techniques to quantify the orientation and distribution of collagen in the corneal stroma,” Prog. Retin. Eye Res. 28(5), 369–392 (2009).
[Crossref] [PubMed]

2008 (2)

2007 (2)

M. Pircher, E. Götzinger, B. Baumann, and C. K. Hitzenberger, “Corneal birefringence compensation for polarization sensitive optical coherence tomography of the human retina,” J. Biomed. Opt. 12(4), 041210 (2007).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, I. Dejaco-Ruhswurm, S. Kaminski, C. Skorpik, and C. K. Hitzenberger, “Imaging of Birefringent Properties of Keratoconus Corneas by Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 48(8), 3551–3558 (2007).
[Crossref] [PubMed]

2006 (1)

E. Borasio, J. S. Mehta, and V. Maurino, “Surgically induced astigmatism after phacoemulsification in eyes with mild to moderate corneal astigmatism,” J. Cataract Refract. Surg. 32(4), 565–572 (2006).
[Crossref] [PubMed]

2005 (2)

M. Pircher, E. Goetzinger, R. A. Leitgeb, H. Sattmann, and C. K. Hitzenberger, “Ultrahigh resolution polarization sensitive optical coherence tomography,” SPIE Proc. 5690, 257–262 (2005).
[Crossref]

E. Götzinger, M. Pircher, and C. K. Hitzenberger, “High speed spectral domain polarization sensitive optical coherence tomography of the human retina,” Opt. Express 13(25), 10217–10229 (2005).
[Crossref] [PubMed]

2004 (3)

E. Götzinger, M. Pircher, M. Sticker, A. Fercher, and C. K. Hitzenberger, “Measurement and imaging of birefringent properties of the human cornea with phase-resolved, polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 9(1), 94–102 (2004).
[Crossref] [PubMed]

M. Pircher, E. Götzinger, R. Leitgeb, and C. K. Hitzenberger, “Transversal phase resolved polarization sensitive optical coherence tomography,” Phys. Med. Biol. 49(7), 1257 (2004).
[Crossref] [PubMed]

H. Aghamohammadzadeh, R. H. Newton, and K. M. Meek, “X-Ray Scattering Used to Map the Preferred Collagen Orientation in the Human Cornea and Limbus,” Structure 12(2), 249–256 (2004).
[Crossref] [PubMed]

2003 (1)

S. Woo and J. Lee, “Effect of central corneal thickness on surgically induced astigmatism in cataract surgery,” J. Cataract Refract. Surg. 29(12), 2401–2406 (2003).
[Crossref]

2002 (2)

Q. Zhou and R. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Vis. Sci. 43(7), 2221–2228 (2002).
[PubMed]

J. M. Bueno and F. Vargas-Martín, “Measurements of the corneal birefringence with a liquid-crystal imaging polariscope,” Appl. Opt. 41(1), 116–124 (2002).
[Crossref] [PubMed]

2001 (2)

1998 (1)

1997 (2)

A. Daxer and P. Fratzl, “Collagen fibril orientation in the human corneal stroma and its implication in keratoconus,” Invest. Ophthalmol. Vis. Sci. 38(1), 121–129 (1997).
[PubMed]

J. F. de Boer, T. E. Millner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22(12), 934–936 (1997).
[Crossref] [PubMed]

1995 (1)

T. Kohnen, B. Dick, and K. W. Jacobi, “Comparison of the induced astigmatism after temporal clear corneal tunnel incisions of different sizes,” J. Cataract Refract. Surg. 21(4), 417–424 (1995).
[Crossref] [PubMed]

1992 (1)

1991 (1)

Y. Komai and T. Ushiki, “The three-dimensional organisation of collagen fibrils in the human cornea and sclera,” Invest. Ophthalmol. Vis. Sci. 32(8), 2244–2258 (1991).
[PubMed]

1987 (1)

Aghamohammadzadeh, H.

H. Aghamohammadzadeh, R. H. Newton, and K. M. Meek, “X-Ray Scattering Used to Map the Preferred Collagen Orientation in the Human Cornea and Limbus,” Structure 12(2), 249–256 (2004).
[Crossref] [PubMed]

Ahlers, C.

Asrani, S.

S. Asrani, M. Sarunic, C. Santiago, and J. Izatt, “Detailed Visualization of the Anterior Segment Using Fourier-Domain Optical Coherence Tomography,” Arch Ophthalmol. 126(6), 765–771 (2008).
[Crossref] [PubMed]

Baumann, B.

Beckers, H. J. M.

A. N. Tan, L. D. C. Sauren, J. de Brabander, T. T. J. M. Berendschot, V. Lima Passos, C. A. B. Webers, R. M. M. A. Nuijts, and H. J. M. Beckers, “Reproducibility of Anterior Chamber Angle Measurements with Anterior Segment Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 52(5), 2095–2099 (2011).
[Crossref]

Beheregaray, S.

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus Diagnosis Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 54(2), 1384–1391 (2013).
[Crossref] [PubMed]

Berendschot, T. T. J. M.

A. N. Tan, L. D. C. Sauren, J. de Brabander, T. T. J. M. Berendschot, V. Lima Passos, C. A. B. Webers, R. M. M. A. Nuijts, and H. J. M. Beckers, “Reproducibility of Anterior Chamber Angle Measurements with Anterior Segment Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 52(5), 2095–2099 (2011).
[Crossref]

Biedermann, B. R.

Boote, C.

K. M. Meek and C. Boote, “The use of X-ray scattering techniques to quantify the orientation and distribution of collagen in the corneal stroma,” Prog. Retin. Eye Res. 28(5), 369–392 (2009).
[Crossref] [PubMed]

Borasio, E.

E. Borasio, J. S. Mehta, and V. Maurino, “Surgically induced astigmatism after phacoemulsification in eyes with mild to moderate corneal astigmatism,” J. Cataract Refract. Surg. 32(4), 565–572 (2006).
[Crossref] [PubMed]

Bueno, J. M.

Challa, P.

Colston, B.W.

Cortés, D. E.

R. Poddar, D. E. Cortés, J. S. Werner, M. J. Mannis, and R. J. Zawadzki, “Three-dimensional anterior segment imaging in patients with type 1 Boston Keratoprosthesis with switchable full depth range swept source optical coherence tomography,” J. Biomed. Opt. 18(8), 86002 (2013).
[Crossref] [PubMed]

Daxer, A.

A. Daxer and P. Fratzl, “Collagen fibril orientation in the human corneal stroma and its implication in keratoconus,” Invest. Ophthalmol. Vis. Sci. 38(1), 121–129 (1997).
[PubMed]

de Boer, J. F.

de Brabander, J.

A. N. Tan, L. D. C. Sauren, J. de Brabander, T. T. J. M. Berendschot, V. Lima Passos, C. A. B. Webers, R. M. M. A. Nuijts, and H. J. M. Beckers, “Reproducibility of Anterior Chamber Angle Measurements with Anterior Segment Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 52(5), 2095–2099 (2011).
[Crossref]

Dejaco-Ruhswurm, I.

E. Götzinger, M. Pircher, I. Dejaco-Ruhswurm, S. Kaminski, C. Skorpik, and C. K. Hitzenberger, “Imaging of Birefringent Properties of Keratoconus Corneas by Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 48(8), 3551–3558 (2007).
[Crossref] [PubMed]

DelMonte, D. W.

D. W. DelMonte and T. Kim, “Anatomy and physiology of the cornea,” J. Cataract Refract. Surg. 37(1), 588–598 (2011).
[Crossref] [PubMed]

Dick, B.

T. Kohnen, B. Dick, and K. W. Jacobi, “Comparison of the induced astigmatism after temporal clear corneal tunnel incisions of different sizes,” J. Cataract Refract. Surg. 21(4), 417–424 (1995).
[Crossref] [PubMed]

Eigenwillig, C. M.

Everett, M. J.

Fanjul-Vélez, F.

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, and C. K. Hitzenberger, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15(5), 056004 (2010).
[Crossref] [PubMed]

Fercher, A.

E. Götzinger, M. Pircher, M. Sticker, A. Fercher, and C. K. Hitzenberger, “Measurement and imaging of birefringent properties of the human cornea with phase-resolved, polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 9(1), 94–102 (2004).
[Crossref] [PubMed]

C. K. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express 9(13), 780–790 (2001).
[Crossref] [PubMed]

Fratzl, P.

A. Daxer and P. Fratzl, “Collagen fibril orientation in the human corneal stroma and its implication in keratoconus,” Invest. Ophthalmol. Vis. Sci. 38(1), 121–129 (1997).
[PubMed]

Fujimoto, J. G.

Fukuda, S.

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus Diagnosis Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 54(2), 1384–1391 (2013).
[Crossref] [PubMed]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2(8), 2392–2402 (2011).
[Crossref] [PubMed]

Fullwood, N. J.

K. M. Meek and N. J. Fullwood, “Corneal and scleral collagens - A microscopist’s perspective,” Micron 32(3), 261–272 (2001).
[Crossref]

Geitzenauer, W.

Goetzinger, E.

M. Pircher, E. Goetzinger, R. A. Leitgeb, H. Sattmann, and C. K. Hitzenberger, “Ultrahigh resolution polarization sensitive optical coherence tomography,” SPIE Proc. 5690, 257–262 (2005).
[Crossref]

C. K. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express 9(13), 780–790 (2001).
[Crossref] [PubMed]

Götzinger, E.

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, and C. K. Hitzenberger, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15(5), 056004 (2010).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, W. Geitzenauer, C. Ahlers, B. Baumann, S. Michels, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Retinal pigment epithelium segmentation by polarization sensitive optical coherence tomography,” Opt. Express 16(21), 16410–16422 (2008).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, I. Dejaco-Ruhswurm, S. Kaminski, C. Skorpik, and C. K. Hitzenberger, “Imaging of Birefringent Properties of Keratoconus Corneas by Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 48(8), 3551–3558 (2007).
[Crossref] [PubMed]

M. Pircher, E. Götzinger, B. Baumann, and C. K. Hitzenberger, “Corneal birefringence compensation for polarization sensitive optical coherence tomography of the human retina,” J. Biomed. Opt. 12(4), 041210 (2007).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, and C. K. Hitzenberger, “High speed spectral domain polarization sensitive optical coherence tomography of the human retina,” Opt. Express 13(25), 10217–10229 (2005).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, M. Sticker, A. Fercher, and C. K. Hitzenberger, “Measurement and imaging of birefringent properties of the human cornea with phase-resolved, polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 9(1), 94–102 (2004).
[Crossref] [PubMed]

M. Pircher, E. Götzinger, R. Leitgeb, and C. K. Hitzenberger, “Transversal phase resolved polarization sensitive optical coherence tomography,” Phys. Med. Biol. 49(7), 1257 (2004).
[Crossref] [PubMed]

Grulkowski, I.

Haindl, R.

Hee, M. R.

Himori, N.

Hitzenberger, C. K.

M. Sugita, M. Pircher, S. Zotter, B. Baumann, P. Roberts, T. Makihira, N. Tomatsu, M. Sato, C. Vass, and C. K. Hitzenberger, “Retinal nerve fiber bundle tracing and analysis in human eye by polarization sensitive OCT,” Biomed. Opt. Express 6(3), 1030–1054 (2015).
[Crossref] [PubMed]

W. Trasischker, S. Zotter, T. Torzicky, B. Baumann, R. Haindl, M. Pircher, and C. K. Hitzenberger, “Single input state polarization sensitive swept source optical coherence tomography based on an all single mode fiber interferometer,” Biomed. Opt. Express 5(8), 2798–2809 (2014).
[Crossref] [PubMed]

M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Prog. Retin. Eye Res. 30(6), 431–451 (2011).
[Crossref] [PubMed]

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, and C. K. Hitzenberger, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15(5), 056004 (2010).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, W. Geitzenauer, C. Ahlers, B. Baumann, S. Michels, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Retinal pigment epithelium segmentation by polarization sensitive optical coherence tomography,” Opt. Express 16(21), 16410–16422 (2008).
[Crossref] [PubMed]

M. Pircher, E. Götzinger, B. Baumann, and C. K. Hitzenberger, “Corneal birefringence compensation for polarization sensitive optical coherence tomography of the human retina,” J. Biomed. Opt. 12(4), 041210 (2007).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, I. Dejaco-Ruhswurm, S. Kaminski, C. Skorpik, and C. K. Hitzenberger, “Imaging of Birefringent Properties of Keratoconus Corneas by Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 48(8), 3551–3558 (2007).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, and C. K. Hitzenberger, “High speed spectral domain polarization sensitive optical coherence tomography of the human retina,” Opt. Express 13(25), 10217–10229 (2005).
[Crossref] [PubMed]

M. Pircher, E. Goetzinger, R. A. Leitgeb, H. Sattmann, and C. K. Hitzenberger, “Ultrahigh resolution polarization sensitive optical coherence tomography,” SPIE Proc. 5690, 257–262 (2005).
[Crossref]

M. Pircher, E. Götzinger, R. Leitgeb, and C. K. Hitzenberger, “Transversal phase resolved polarization sensitive optical coherence tomography,” Phys. Med. Biol. 49(7), 1257 (2004).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, M. Sticker, A. Fercher, and C. K. Hitzenberger, “Measurement and imaging of birefringent properties of the human cornea with phase-resolved, polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 9(1), 94–102 (2004).
[Crossref] [PubMed]

C. K. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express 9(13), 780–790 (2001).
[Crossref] [PubMed]

Hoshi, S.

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus Diagnosis Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 54(2), 1384–1391 (2013).
[Crossref] [PubMed]

Huang, D.

Huber, R.

Izatt, J.

S. Asrani, M. Sarunic, C. Santiago, and J. Izatt, “Detailed Visualization of the Anterior Segment Using Fourier-Domain Optical Coherence Tomography,” Arch Ophthalmol. 126(6), 765–771 (2008).
[Crossref] [PubMed]

Izatt, J. A.

Jacobi, K. W.

T. Kohnen, B. Dick, and K. W. Jacobi, “Comparison of the induced astigmatism after temporal clear corneal tunnel incisions of different sizes,” J. Cataract Refract. Surg. 21(4), 417–424 (1995).
[Crossref] [PubMed]

Kaji, Y.

Kaminski, S.

E. Götzinger, M. Pircher, I. Dejaco-Ruhswurm, S. Kaminski, C. Skorpik, and C. K. Hitzenberger, “Imaging of Birefringent Properties of Keratoconus Corneas by Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 48(8), 3551–3558 (2007).
[Crossref] [PubMed]

Kim, T.

D. W. DelMonte and T. Kim, “Anatomy and physiology of the cornea,” J. Cataract Refract. Surg. 37(1), 588–598 (2011).
[Crossref] [PubMed]

Kiuchi, T.

Klein, T.

Knupp, C.

K. M. Meek and C. Knupp, “Corneal structure and transparency,” Prog. Retin. Eye Res. 49, 1–16 (2015).
[Crossref] [PubMed]

Kohnen, T.

T. Kohnen, B. Dick, and K. W. Jacobi, “Comparison of the induced astigmatism after temporal clear corneal tunnel incisions of different sizes,” J. Cataract Refract. Surg. 21(4), 417–424 (1995).
[Crossref] [PubMed]

Kokubun, T.

Komai, Y.

Y. Komai and T. Ushiki, “The three-dimensional organisation of collagen fibrils in the human cornea and sclera,” Invest. Ophthalmol. Vis. Sci. 32(8), 2244–2258 (1991).
[PubMed]

Kunikata, H.

Kunimatsu-Sanuki, S.

Kuo, A. N.

Lee, J.

S. Woo and J. Lee, “Effect of central corneal thickness on surgically induced astigmatism in cataract surgery,” J. Cataract Refract. Surg. 29(12), 2401–2406 (2003).
[Crossref]

Leitgeb, R.

M. Pircher, E. Götzinger, R. Leitgeb, and C. K. Hitzenberger, “Transversal phase resolved polarization sensitive optical coherence tomography,” Phys. Med. Biol. 49(7), 1257 (2004).
[Crossref] [PubMed]

Leitgeb, R. A.

M. Pircher, E. Goetzinger, R. A. Leitgeb, H. Sattmann, and C. K. Hitzenberger, “Ultrahigh resolution polarization sensitive optical coherence tomography,” SPIE Proc. 5690, 257–262 (2005).
[Crossref]

Lim, Y.

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus Diagnosis Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 54(2), 1384–1391 (2013).
[Crossref] [PubMed]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2(8), 2392–2402 (2011).
[Crossref] [PubMed]

Lima Passos, V.

A. N. Tan, L. D. C. Sauren, J. de Brabander, T. T. J. M. Berendschot, V. Lima Passos, C. A. B. Webers, R. M. M. A. Nuijts, and H. J. M. Beckers, “Reproducibility of Anterior Chamber Angle Measurements with Anterior Segment Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 52(5), 2095–2099 (2011).
[Crossref]

Maitland, D. J.

Makihira, T.

Makita, S.

Mannis, M. J.

R. Poddar, D. E. Cortés, J. S. Werner, M. J. Mannis, and R. J. Zawadzki, “Three-dimensional anterior segment imaging in patients with type 1 Boston Keratoprosthesis with switchable full depth range swept source optical coherence tomography,” J. Biomed. Opt. 18(8), 86002 (2013).
[Crossref] [PubMed]

Marcos, S.

Maruyama, K.

Maurino, V.

E. Borasio, J. S. Mehta, and V. Maurino, “Surgically induced astigmatism after phacoemulsification in eyes with mild to moderate corneal astigmatism,” J. Cataract Refract. Surg. 32(4), 565–572 (2006).
[Crossref] [PubMed]

McNabb, R. P.

Meek, K. M.

K. M. Meek and C. Knupp, “Corneal structure and transparency,” Prog. Retin. Eye Res. 49, 1–16 (2015).
[Crossref] [PubMed]

K. M. Meek and C. Boote, “The use of X-ray scattering techniques to quantify the orientation and distribution of collagen in the corneal stroma,” Prog. Retin. Eye Res. 28(5), 369–392 (2009).
[Crossref] [PubMed]

H. Aghamohammadzadeh, R. H. Newton, and K. M. Meek, “X-Ray Scattering Used to Map the Preferred Collagen Orientation in the Human Cornea and Limbus,” Structure 12(2), 249–256 (2004).
[Crossref] [PubMed]

K. M. Meek and N. J. Fullwood, “Corneal and scleral collagens - A microscopist’s perspective,” Micron 32(3), 261–272 (2001).
[Crossref]

Mehta, J. S.

E. Borasio, J. S. Mehta, and V. Maurino, “Surgically induced astigmatism after phacoemulsification in eyes with mild to moderate corneal astigmatism,” J. Cataract Refract. Surg. 32(4), 565–572 (2006).
[Crossref] [PubMed]

Michels, S.

Millner, T. E.

Miura, M.

Nakazawa, T.

Nelson, J. S.

Newton, R. H.

H. Aghamohammadzadeh, R. H. Newton, and K. M. Meek, “X-Ray Scattering Used to Map the Preferred Collagen Orientation in the Human Cornea and Limbus,” Structure 12(2), 249–256 (2004).
[Crossref] [PubMed]

Nuijts, R. M. M. A.

A. N. Tan, L. D. C. Sauren, J. de Brabander, T. T. J. M. Berendschot, V. Lima Passos, C. A. B. Webers, R. M. M. A. Nuijts, and H. J. M. Beckers, “Reproducibility of Anterior Chamber Angle Measurements with Anterior Segment Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 52(5), 2095–2099 (2011).
[Crossref]

Omodaka, K.

Ortiz, S.

Oshika, T.

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus Diagnosis Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 54(2), 1384–1391 (2013).
[Crossref] [PubMed]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2(8), 2392–2402 (2011).
[Crossref] [PubMed]

Pascual, D.

Pircher, M.

M. Sugita, M. Pircher, S. Zotter, B. Baumann, P. Roberts, T. Makihira, N. Tomatsu, M. Sato, C. Vass, and C. K. Hitzenberger, “Retinal nerve fiber bundle tracing and analysis in human eye by polarization sensitive OCT,” Biomed. Opt. Express 6(3), 1030–1054 (2015).
[Crossref] [PubMed]

W. Trasischker, S. Zotter, T. Torzicky, B. Baumann, R. Haindl, M. Pircher, and C. K. Hitzenberger, “Single input state polarization sensitive swept source optical coherence tomography based on an all single mode fiber interferometer,” Biomed. Opt. Express 5(8), 2798–2809 (2014).
[Crossref] [PubMed]

M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Prog. Retin. Eye Res. 30(6), 431–451 (2011).
[Crossref] [PubMed]

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, and C. K. Hitzenberger, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15(5), 056004 (2010).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, W. Geitzenauer, C. Ahlers, B. Baumann, S. Michels, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Retinal pigment epithelium segmentation by polarization sensitive optical coherence tomography,” Opt. Express 16(21), 16410–16422 (2008).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, I. Dejaco-Ruhswurm, S. Kaminski, C. Skorpik, and C. K. Hitzenberger, “Imaging of Birefringent Properties of Keratoconus Corneas by Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 48(8), 3551–3558 (2007).
[Crossref] [PubMed]

M. Pircher, E. Götzinger, B. Baumann, and C. K. Hitzenberger, “Corneal birefringence compensation for polarization sensitive optical coherence tomography of the human retina,” J. Biomed. Opt. 12(4), 041210 (2007).
[Crossref] [PubMed]

M. Pircher, E. Goetzinger, R. A. Leitgeb, H. Sattmann, and C. K. Hitzenberger, “Ultrahigh resolution polarization sensitive optical coherence tomography,” SPIE Proc. 5690, 257–262 (2005).
[Crossref]

E. Götzinger, M. Pircher, and C. K. Hitzenberger, “High speed spectral domain polarization sensitive optical coherence tomography of the human retina,” Opt. Express 13(25), 10217–10229 (2005).
[Crossref] [PubMed]

M. Pircher, E. Götzinger, R. Leitgeb, and C. K. Hitzenberger, “Transversal phase resolved polarization sensitive optical coherence tomography,” Phys. Med. Biol. 49(7), 1257 (2004).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, M. Sticker, A. Fercher, and C. K. Hitzenberger, “Measurement and imaging of birefringent properties of the human cornea with phase-resolved, polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 9(1), 94–102 (2004).
[Crossref] [PubMed]

C. K. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express 9(13), 780–790 (2001).
[Crossref] [PubMed]

Poddar, R.

R. Poddar, D. E. Cortés, J. S. Werner, M. J. Mannis, and R. J. Zawadzki, “Three-dimensional anterior segment imaging in patients with type 1 Boston Keratoprosthesis with switchable full depth range swept source optical coherence tomography,” J. Biomed. Opt. 18(8), 86002 (2013).
[Crossref] [PubMed]

Remon, L.

Roberts, P.

Ryu, M.

Santiago, C.

S. Asrani, M. Sarunic, C. Santiago, and J. Izatt, “Detailed Visualization of the Anterior Segment Using Fourier-Domain Optical Coherence Tomography,” Arch Ophthalmol. 126(6), 765–771 (2008).
[Crossref] [PubMed]

Sarunic, M.

S. Asrani, M. Sarunic, C. Santiago, and J. Izatt, “Detailed Visualization of the Anterior Segment Using Fourier-Domain Optical Coherence Tomography,” Arch Ophthalmol. 126(6), 765–771 (2008).
[Crossref] [PubMed]

Sato, M.

Sattmann, H.

M. Pircher, E. Goetzinger, R. A. Leitgeb, H. Sattmann, and C. K. Hitzenberger, “Ultrahigh resolution polarization sensitive optical coherence tomography,” SPIE Proc. 5690, 257–262 (2005).
[Crossref]

Sauren, L. D. C.

A. N. Tan, L. D. C. Sauren, J. de Brabander, T. T. J. M. Berendschot, V. Lima Passos, C. A. B. Webers, R. M. M. A. Nuijts, and H. J. M. Beckers, “Reproducibility of Anterior Chamber Angle Measurements with Anterior Segment Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 52(5), 2095–2099 (2011).
[Crossref]

Schmidt-Erfurth, U.

Schoenenberger, K.

Shiga, Y.

Sielecki, D.

Silva, L. B. D.

Skorpik, C.

E. Götzinger, M. Pircher, I. Dejaco-Ruhswurm, S. Kaminski, C. Skorpik, and C. K. Hitzenberger, “Imaging of Birefringent Properties of Keratoconus Corneas by Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 48(8), 3551–3558 (2007).
[Crossref] [PubMed]

Sticker, M.

E. Götzinger, M. Pircher, M. Sticker, A. Fercher, and C. K. Hitzenberger, “Measurement and imaging of birefringent properties of the human cornea with phase-resolved, polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 9(1), 94–102 (2004).
[Crossref] [PubMed]

C. K. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express 9(13), 780–790 (2001).
[Crossref] [PubMed]

Sugita, M.

Swanson, E.

Takahashi, H.

Tan, A. N.

A. N. Tan, L. D. C. Sauren, J. de Brabander, T. T. J. M. Berendschot, V. Lima Passos, C. A. B. Webers, R. M. M. A. Nuijts, and H. J. M. Beckers, “Reproducibility of Anterior Chamber Angle Measurements with Anterior Segment Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 52(5), 2095–2099 (2011).
[Crossref]

Tomatsu, N.

Torzicky, T.

Trasischker, W.

Tsuda, S.

Ushiki, T.

Y. Komai and T. Ushiki, “The three-dimensional organisation of collagen fibrils in the human cornea and sclera,” Invest. Ophthalmol. Vis. Sci. 32(8), 2244–2258 (1991).
[PubMed]

Van Blokland, G. J.

van Gemert, M. J. C.

Vargas-Martín, F.

Vass, C.

Verhelst, S. C.

Webers, C. A. B.

A. N. Tan, L. D. C. Sauren, J. de Brabander, T. T. J. M. Berendschot, V. Lima Passos, C. A. B. Webers, R. M. M. A. Nuijts, and H. J. M. Beckers, “Reproducibility of Anterior Chamber Angle Measurements with Anterior Segment Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 52(5), 2095–2099 (2011).
[Crossref]

Weinreb, R.

Q. Zhou and R. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Vis. Sci. 43(7), 2221–2228 (2002).
[PubMed]

Werner, J. S.

R. Poddar, D. E. Cortés, J. S. Werner, M. J. Mannis, and R. J. Zawadzki, “Three-dimensional anterior segment imaging in patients with type 1 Boston Keratoprosthesis with switchable full depth range swept source optical coherence tomography,” J. Biomed. Opt. 18(8), 86002 (2013).
[Crossref] [PubMed]

Wieser, W.

Wojtkowski, M.

Woo, S.

S. Woo and J. Lee, “Effect of central corneal thickness on surgically induced astigmatism in cataract surgery,” J. Cataract Refract. Surg. 29(12), 2401–2406 (2003).
[Crossref]

Yamanari, M.

Yasuno, Y.

Yokoyama, Y.

Zawadzki, R. J.

R. Poddar, D. E. Cortés, J. S. Werner, M. J. Mannis, and R. J. Zawadzki, “Three-dimensional anterior segment imaging in patients with type 1 Boston Keratoprosthesis with switchable full depth range swept source optical coherence tomography,” J. Biomed. Opt. 18(8), 86002 (2013).
[Crossref] [PubMed]

Zhou, Q.

Q. Zhou and R. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Vis. Sci. 43(7), 2221–2228 (2002).
[PubMed]

Zotter, S.

Appl. Opt. (2)

Arch Ophthalmol. (1)

S. Asrani, M. Sarunic, C. Santiago, and J. Izatt, “Detailed Visualization of the Anterior Segment Using Fourier-Domain Optical Coherence Tomography,” Arch Ophthalmol. 126(6), 765–771 (2008).
[Crossref] [PubMed]

Biomed. Opt. Express (5)

Invest. Ophthalmol. Vis. Sci. (6)

E. Götzinger, M. Pircher, I. Dejaco-Ruhswurm, S. Kaminski, C. Skorpik, and C. K. Hitzenberger, “Imaging of Birefringent Properties of Keratoconus Corneas by Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 48(8), 3551–3558 (2007).
[Crossref] [PubMed]

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus Diagnosis Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 54(2), 1384–1391 (2013).
[Crossref] [PubMed]

Y. Komai and T. Ushiki, “The three-dimensional organisation of collagen fibrils in the human cornea and sclera,” Invest. Ophthalmol. Vis. Sci. 32(8), 2244–2258 (1991).
[PubMed]

A. Daxer and P. Fratzl, “Collagen fibril orientation in the human corneal stroma and its implication in keratoconus,” Invest. Ophthalmol. Vis. Sci. 38(1), 121–129 (1997).
[PubMed]

A. N. Tan, L. D. C. Sauren, J. de Brabander, T. T. J. M. Berendschot, V. Lima Passos, C. A. B. Webers, R. M. M. A. Nuijts, and H. J. M. Beckers, “Reproducibility of Anterior Chamber Angle Measurements with Anterior Segment Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 52(5), 2095–2099 (2011).
[Crossref]

Q. Zhou and R. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Vis. Sci. 43(7), 2221–2228 (2002).
[PubMed]

J. Biomed. Opt. (4)

R. Poddar, D. E. Cortés, J. S. Werner, M. J. Mannis, and R. J. Zawadzki, “Three-dimensional anterior segment imaging in patients with type 1 Boston Keratoprosthesis with switchable full depth range swept source optical coherence tomography,” J. Biomed. Opt. 18(8), 86002 (2013).
[Crossref] [PubMed]

F. Fanjul-Vélez, M. Pircher, B. Baumann, E. Götzinger, and C. K. Hitzenberger, “Polarimetric analysis of the human cornea measured by polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 15(5), 056004 (2010).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, M. Sticker, A. Fercher, and C. K. Hitzenberger, “Measurement and imaging of birefringent properties of the human cornea with phase-resolved, polarization-sensitive optical coherence tomography,” J. Biomed. Opt. 9(1), 94–102 (2004).
[Crossref] [PubMed]

M. Pircher, E. Götzinger, B. Baumann, and C. K. Hitzenberger, “Corneal birefringence compensation for polarization sensitive optical coherence tomography of the human retina,” J. Biomed. Opt. 12(4), 041210 (2007).
[Crossref] [PubMed]

J. Cataract Refract. Surg. (4)

T. Kohnen, B. Dick, and K. W. Jacobi, “Comparison of the induced astigmatism after temporal clear corneal tunnel incisions of different sizes,” J. Cataract Refract. Surg. 21(4), 417–424 (1995).
[Crossref] [PubMed]

S. Woo and J. Lee, “Effect of central corneal thickness on surgically induced astigmatism in cataract surgery,” J. Cataract Refract. Surg. 29(12), 2401–2406 (2003).
[Crossref]

E. Borasio, J. S. Mehta, and V. Maurino, “Surgically induced astigmatism after phacoemulsification in eyes with mild to moderate corneal astigmatism,” J. Cataract Refract. Surg. 32(4), 565–572 (2006).
[Crossref] [PubMed]

D. W. DelMonte and T. Kim, “Anatomy and physiology of the cornea,” J. Cataract Refract. Surg. 37(1), 588–598 (2011).
[Crossref] [PubMed]

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

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

Micron (1)

K. M. Meek and N. J. Fullwood, “Corneal and scleral collagens - A microscopist’s perspective,” Micron 32(3), 261–272 (2001).
[Crossref]

Opt. Express (6)

S. Ortiz, D. Sielecki, I. Grulkowski, L. Remon, D. Pascual, M. Wojtkowski, and S. Marcos, “Optical distortion correction in optical coherence tomography for quantitative ocular anterior segment by three-dimensional imaging,” Opt. Express 18(3), 2782–2796 (2010).
[Crossref] [PubMed]

C. K. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express 9(13), 780–790 (2001).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, and C. K. Hitzenberger, “High speed spectral domain polarization sensitive optical coherence tomography of the human retina,” Opt. Express 13(25), 10217–10229 (2005).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, W. Geitzenauer, C. Ahlers, B. Baumann, S. Michels, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Retinal pigment epithelium segmentation by polarization sensitive optical coherence tomography,” Opt. Express 16(21), 16410–16422 (2008).
[Crossref] [PubMed]

T. Klein, W. Wieser, C. M. Eigenwillig, B. R. Biedermann, and R. Huber, “Megahertz OCT for ultrawide-field retinal imaging with a 1050nm Fourier domain mode-locked laser,” Opt. Express 19(4), 3044–3062 (2011).
[Crossref] [PubMed]

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[Crossref] [PubMed]

Opt. Lett. (1)

Phys. Med. Biol. (1)

M. Pircher, E. Götzinger, R. Leitgeb, and C. K. Hitzenberger, “Transversal phase resolved polarization sensitive optical coherence tomography,” Phys. Med. Biol. 49(7), 1257 (2004).
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M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Prog. Retin. Eye Res. 30(6), 431–451 (2011).
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[Crossref]

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of the all SM fiber based SS-PS-OCT setup: SS, swept source; PC, polarization control paddles; LP, linear polarizer; BS, beam splitter; UP, unused port; SA, sample arm; RA, reference arm; C, collimator; GS, galvo-scanner; L, scanning lens; M, mirror; PBS, polarizing beam splitter; BD, balanced detection unit.

Fig. 2
Fig. 2

Sketch of the different scanning optics geometries: (a) Conical setup for perpendicular scanning, using an aspheric condenser lens (ACL) and (b) telecentric setup with sampling beams parallel to the optical axis, using an achromatic doublet lens (ADL). The position of the collimator lens (VC) is variable, to adjust the focal plane onto the surface of the cornea in the conical setup.

Fig. 3
Fig. 3

Conceptual representation of the image transformation: (a) shows a point P(px, py, pz) in the original data volume. (b) shows the same point P(pr, χ, ψ) with new dependencies in the transformed volume. The values used for the transformation are indicated.

Fig. 4
Fig. 4

(a) 3D model of the test object. (b) B-scan of the test target and (c) en-face projection imaged with the standard scanning optics as shown in Fig. 2(b). (d) B-scan of the same feature as in (b) and the en-face projection (e) of the test target imaged with the conical setup shown in Fig. 2(a). (f) and (g) show the corresponding B-scan and the en-face map of the surface of the test target after correcting for the distortions introduced by the scan pattern (but not for field distortions introduced by the lens, which are responsible for the remaining distortions at the periphery). The red dashed line represents the position of the corresponding B-scans.

Fig. 5
Fig. 5

En-face image of the axis orientation of a test target surface before (a) and after (b) the compensation for polarizing effects introduced by the scanning lens.

Fig. 6
Fig. 6

Surface maps of retardation and axis orientation of a flat test target behind different retarders using the conical scanning optics. (a) Retardation and (b) axis orientation when the test target was imaged behind a HWP. (c) Retardation and (d) axis orientation when the test target was imaged behind a QWP, respectively. The axis orientation offset was not compensated.

Fig. 7
Fig. 7

Comparison of representative B-scans of a human cornea using the conventional scanning optics (a–d), the conical scanning optics before (e–h), and also after the coordinate transformation (i–l). The reflectivity is shown in (a), (e) and (i), the retardation in (b), (f) and (j), the axis orientation in (c), (g) and (k) and the DOPU images in (d), (h) and (l). All polarization-sensitive images were intensity thresholded as described in the methods section (sub-threshold pixels were set to grey).

Fig. 8
Fig. 8

Comparison of regions of interest (ROI) of red rectangles in Fig. 7. The ROI of the image recorded with the standard scanning optics (a) shows less signal than the ROI of the image recorded with the conical scanning scheme (b). (c) shows the ROI of the distortion corrected image. For (a)–(c) the aspect ratio was changed to better visualize features of the corneal layer structure. (d) shows a different ROI as indicated in the red rectangles of Fig. 7(f–h) of retardation, (e) axis orientation and (f) DOPU.

Fig. 9
Fig. 9

En-face projections generated from a 3D data set recorded in a right (a, c and e) and left (b, d and f) eye with the conventional scanning optics (polarization data retrieved from the iris). (a) and (b) show the reflectivity projections of the total depth, (c) and (d) the retardation and (e) and (f) the axis orientation. The axis orientation was not corrected for the offset θ0.

Fig. 10
Fig. 10

Comparison of the incident angles on the corneal surface calculated for central B-scans recorded with both scanning schemes.

Fig. 11
Fig. 11

En-face projections generated from a 3D data set recorded in a right (a, c and e) and left (b, d and f) eye with the conical scanning optics design (retardation and offset compensated axis orientation retrieved from the posterior corneal surface). (a) and (b) show the reflectivity projections of the total depth, (c) and (d) the retardation and (e) and (f) the axis orientation.

Fig. 12
Fig. 12

Vector representation of axis orientation measured at the backside of the cornea in areas of high retardation. Red lines show the idealized, theoretical orientation of preferentially aligned, reinforcing collagen fibrils, which are superimposed on the in vivo measurement data. The red circle represents the limbus region and therefore the FoV.

Equations (10)

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R ( z ) ~ A 1 ( z ) 2 + A 2 ( z ) 2
δ ( z ) = arctan ( A 2 ( z ) A 1 ( z ) )
θ = 180 ° Δ ϕ 2 + θ 0
p x ¯ = p x p x , max and p y ¯ = p y p y , max
P ( p x , p y , p z ) P ( P r , χ , ψ ) P ( x , y , z ) ,
ψ ( p x ¯ , p y ¯ ) = arcsin ( p y ¯ p r ) with p r = p x ¯ 2 + p y ¯ 2
χ ( p x ¯ , p y ¯ ) = p r δ χ with δ χ = χ max p x , max .
x = ( p z + ζ ) sin ( χ ( p x ¯ , p y ¯ ) ) cos ( ψ ( p x ¯ , p y ¯ ) ) ,
y = ( p z + ζ ) sin ( χ ( p x ¯ , p y ¯ ) ) sin ( ψ ( p x ¯ , p y ¯ ) ) ,
z = ( p z + ζ ) cos ( ψ ( p x ¯ , p y ¯ ) ) .