Abstract

The optical properties of one-dimensional gradient-refractive-index lens arrays based on liquid crystals are studied. We find that it is quite possible, using theoretical methods, to predict angular distributions of the light emanating from such arrays when they are illuminated with collimated monochromatic light. We compare four theoretical methods in relation to experiments. The experimental data and the model, based on a combination of eikonal methods and diffraction, are in close correspondence. Features such as maximal beam width and number of extrema in the angular light distribution are discussed and explained theoretically. We also studied dispersion effects, both experimentally and theoretically, with good agreement between the two.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. S. T. Kowel, D. S. Cleverly, and P. G. Kornreich, "Focusing by electrical modulation of refraction in a liquid crystal cell," Appl. Opt. 23, 278-289 (1984).
    [CrossRef] [PubMed]
  2. S. T. Kowel, P. Kornreich, and A. Nouhi, "Adaptive spherical lens," Appl. Opt. 23, 2774-2777 (1984).
    [CrossRef] [PubMed]
  3. P. F. Brinkley, S. T. Kowel, and C. Chu, "Liquid crystal adaptive lens: beam translation and field meshing," Appl. Opt. 27, 4578-4586 (1988).
    [CrossRef] [PubMed]
  4. W. Chan, L. Ning, S. T. Kowel, and P. F. Brinkley, "The liquid crystal adaptive lens: aberration correction," Proc. SPIE 1773, 468-475 (1992).
    [CrossRef]
  5. N. A. Riza and M. C. DeJule, "Three terminal adaptive nematic liquid-crystal lens device," Opt. Lett. 19, 1013-1015 (1994).
    [CrossRef] [PubMed]
  6. S. Masuda, S. Takahashi, T. Nose, S. Sato, and H. Ito, "Liquid-crystal microlens with a beam-steering function," Appl. Opt. 36, 4772-4778 (1997).
    [CrossRef] [PubMed]
  7. W. W. Chan and S. T. Kowel, "Imaging performance of the liquid-crystal adaptive lens with conductive ladder meshing," Appl. Opt. 36, 8958-8969 (1997).
    [CrossRef]
  8. V. Laude, "Twisted-nematic liquid-crystal pixelated active lens," Opt. Commun. 153, 134-152 (1998).
    [CrossRef]
  9. A. F. Naumov, G. D. Love, M. Y. Loktev, and F. L. Vladimirov, "Control optimization of spherical modal liquid crystal lenses," Opt. Express 4, 344-352 (1999).
    [CrossRef]
  10. O. A. Zayakin, M. Y. Loktev, G. Love, and A. Naumov, "Cylindrical adaptive lenses," Proc. SPIE 3983, 112-117 (1999).
    [CrossRef]
  11. W. Liu, J. Kelly, and J. Chen, "Electro-optical performance of a self-compensating vertically-aligned liquid crystal display mode," Jpn. J. Appl. Phys., Part 1 38, 2779-2784 (1999).
    [CrossRef]
  12. W. Liu and J. Kelly, "Optical properties of a switchable diffraction grating," Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 358, 199-208 (2001).
    [CrossRef]
  13. H. Ren, Y.-H. Fan, S. Gauza, and S.-T. Wu, "Tunable-focus cylindrical liquid crystal lens," Jpn. J. Appl. Phys., Part 1 43, 652-653 (2004).
    [CrossRef]
  14. Y.-H. Lin, H. Ren, K.-H. Fan-Chiang, W.-K. Choi, S. Gauza, X. Zhu, and S.-T. Wu, "Tunable-focus cylindical liquid crystal lenses," Jpn. J. Appl. Phys., Part 1 44, 243-244 (2005).
    [CrossRef]
  15. M. Ye, B. Wang, and S. Sato, "Liquid-crystal lens with a focal length that is variable in a wide range," Appl. Opt. 43, 6407-6412 (2004).
    [CrossRef] [PubMed]
  16. B. Apter, E. Bahat-Treidel, and U. Efron, "Continuously controllable, wide-angle liquid crystal beam deflector based on the transversal field effect in a three-electrode cell," Opt. Eng. 44, 054001 (2005).
    [CrossRef]
  17. H. Ren, Y.-H. Lin, and S.-T. Wu, "Adaptive lens using liquid crystal concentration redistribution," Appl. Phys. Lett. 88, 191116 (2006).
    [CrossRef]
  18. M.Born and E.Wolf, eds., Principles of Optics (Pergamon, 1959).
  19. M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, 1965).
  20. Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
    [CrossRef]
  21. H. L. Ong, "Optical properties of general twisted nematic liquid crystals," Appl. Phys. Lett. 51, 1398-1400 (1987).
    [CrossRef]
  22. J. A. Kosmopoulos and H. M. Zenginoglou, "Geometrical optics approach to the nematic liquid crystal grating: numerical results," Appl. Opt. 26, 1714-1721 (1987).
    [CrossRef] [PubMed]
  23. H. M. Zenginoglou and J. A. Kosmopoulos, "Geometrical optics approach to the nematic liquid crystal grating: leading term formulas," Appl. Opt. 28, 3516-3519 (1989).
    [CrossRef] [PubMed]
  24. A. L. Rivera, S. M. Chumakov, and K. B. Wolf, "Hamiltonian foundation of geometrical anisotropic optics," J. Opt. Soc. Am. A 12, 1380-1389 (1995).
    [CrossRef]
  25. W. Shen, J. Zhang, S. Wang, and S. Zhu, "Fermat's principle, the general eikonal equation, and space geometry in a static anisotropic medium," J. Opt. Soc. Am. A 14, 2850-2854 (1997).
    [CrossRef]
  26. C. G. Parazzoli, B. E. C. Koltenbah, R. B. Greegor, T. A. Lam, and M. H. Tanielan, "Eikonal equation for a general anisotropic or chiral medium: application to a negative-graded index-of-refraction lens with an anisotropic material," J. Opt. Soc. Am. B 23, 439-450 (2006).
    [CrossRef]
  27. S. C. McClain, L. W. Hillman, and R. A. Chipman, "Polarization ray tracing in anisotropic optically active media. I. Algorithms," J. Opt. Soc. Am. A 10, 2371-2382 (1993).
    [CrossRef]
  28. S. C. McClain, L. W. Hillman, and R. A. Chipman, "Polarization ray tracing in anisotropic optically active media. II. Theory and physics," J. Opt. Soc. Am. A 10, 2383-2393 (1993).
    [CrossRef]
  29. G. Panasyuk, J. Kelly, E. C. Gartland, and D. W. Allender, "Geometrical optics approach in liquid crystal films with three-dimensional director variations," Phys. Rev. E 67, 041702 (2003).
    [CrossRef]
  30. O. Gielkens, Polymer Vision Ltd., High Tech Campus 48, 5656 AE Eindhoven, The Netherlands (personal communication, 2001).
  31. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 1999).
  32. Mathematica website, http://www.wolfram.com.
  33. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).
  34. ELDIM website, http://www.eldim.fr.
  35. R. A. M. Hikmet, T. van Bommel, and T. C. Kraan, "Study of light distribution through lc field induced lens arrays," submitted to J. Appl. Phys..
  36. Shintech website, http://www.shintech.jp.
  37. J. F. Strömer, E. P. Raynes, and C. V. Brown, "Study of elastic constant ratios in nematic liquid crystals," Appl. Phys. Lett. 88, 051915 (2006).
    [CrossRef]
  38. B. A. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]
  39. E. Hecht, Optics, 2nd ed. (Addison Wesley, 1989).
  40. K. Neyts, S. Vermeirsch, S. Vermael, H. de Vleeschouwer, F. Bougrioua, S. Rozanski, D. de Boer, J. van Haaren, and S. Day, "Simulation of refraction, retardation and transmission in liquid crystal displays with slow variations," in Proceedings of the 20th International Display Research Conference (Society for Information Display, 2000), pp. 225-228.

2006 (3)

H. Ren, Y.-H. Lin, and S.-T. Wu, "Adaptive lens using liquid crystal concentration redistribution," Appl. Phys. Lett. 88, 191116 (2006).
[CrossRef]

J. F. Strömer, E. P. Raynes, and C. V. Brown, "Study of elastic constant ratios in nematic liquid crystals," Appl. Phys. Lett. 88, 051915 (2006).
[CrossRef]

C. G. Parazzoli, B. E. C. Koltenbah, R. B. Greegor, T. A. Lam, and M. H. Tanielan, "Eikonal equation for a general anisotropic or chiral medium: application to a negative-graded index-of-refraction lens with an anisotropic material," J. Opt. Soc. Am. B 23, 439-450 (2006).
[CrossRef]

2005 (2)

Y.-H. Lin, H. Ren, K.-H. Fan-Chiang, W.-K. Choi, S. Gauza, X. Zhu, and S.-T. Wu, "Tunable-focus cylindical liquid crystal lenses," Jpn. J. Appl. Phys., Part 1 44, 243-244 (2005).
[CrossRef]

B. Apter, E. Bahat-Treidel, and U. Efron, "Continuously controllable, wide-angle liquid crystal beam deflector based on the transversal field effect in a three-electrode cell," Opt. Eng. 44, 054001 (2005).
[CrossRef]

2004 (2)

H. Ren, Y.-H. Fan, S. Gauza, and S.-T. Wu, "Tunable-focus cylindrical liquid crystal lens," Jpn. J. Appl. Phys., Part 1 43, 652-653 (2004).
[CrossRef]

M. Ye, B. Wang, and S. Sato, "Liquid-crystal lens with a focal length that is variable in a wide range," Appl. Opt. 43, 6407-6412 (2004).
[CrossRef] [PubMed]

2003 (1)

G. Panasyuk, J. Kelly, E. C. Gartland, and D. W. Allender, "Geometrical optics approach in liquid crystal films with three-dimensional director variations," Phys. Rev. E 67, 041702 (2003).
[CrossRef]

2001 (1)

W. Liu and J. Kelly, "Optical properties of a switchable diffraction grating," Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 358, 199-208 (2001).
[CrossRef]

1999 (3)

O. A. Zayakin, M. Y. Loktev, G. Love, and A. Naumov, "Cylindrical adaptive lenses," Proc. SPIE 3983, 112-117 (1999).
[CrossRef]

W. Liu, J. Kelly, and J. Chen, "Electro-optical performance of a self-compensating vertically-aligned liquid crystal display mode," Jpn. J. Appl. Phys., Part 1 38, 2779-2784 (1999).
[CrossRef]

A. F. Naumov, G. D. Love, M. Y. Loktev, and F. L. Vladimirov, "Control optimization of spherical modal liquid crystal lenses," Opt. Express 4, 344-352 (1999).
[CrossRef]

1998 (1)

V. Laude, "Twisted-nematic liquid-crystal pixelated active lens," Opt. Commun. 153, 134-152 (1998).
[CrossRef]

1997 (3)

1995 (1)

1994 (1)

1993 (2)

1992 (1)

W. Chan, L. Ning, S. T. Kowel, and P. F. Brinkley, "The liquid crystal adaptive lens: aberration correction," Proc. SPIE 1773, 468-475 (1992).
[CrossRef]

1989 (1)

1988 (1)

1987 (2)

1984 (2)

Appl. Opt. (8)

Appl. Phys. Lett. (3)

J. F. Strömer, E. P. Raynes, and C. V. Brown, "Study of elastic constant ratios in nematic liquid crystals," Appl. Phys. Lett. 88, 051915 (2006).
[CrossRef]

H. Ren, Y.-H. Lin, and S.-T. Wu, "Adaptive lens using liquid crystal concentration redistribution," Appl. Phys. Lett. 88, 191116 (2006).
[CrossRef]

H. L. Ong, "Optical properties of general twisted nematic liquid crystals," Appl. Phys. Lett. 51, 1398-1400 (1987).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

Jpn. J. Appl. Phys., Part 1 (3)

H. Ren, Y.-H. Fan, S. Gauza, and S.-T. Wu, "Tunable-focus cylindrical liquid crystal lens," Jpn. J. Appl. Phys., Part 1 43, 652-653 (2004).
[CrossRef]

Y.-H. Lin, H. Ren, K.-H. Fan-Chiang, W.-K. Choi, S. Gauza, X. Zhu, and S.-T. Wu, "Tunable-focus cylindical liquid crystal lenses," Jpn. J. Appl. Phys., Part 1 44, 243-244 (2005).
[CrossRef]

W. Liu, J. Kelly, and J. Chen, "Electro-optical performance of a self-compensating vertically-aligned liquid crystal display mode," Jpn. J. Appl. Phys., Part 1 38, 2779-2784 (1999).
[CrossRef]

Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A (1)

W. Liu and J. Kelly, "Optical properties of a switchable diffraction grating," Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 358, 199-208 (2001).
[CrossRef]

Opt. Commun. (1)

V. Laude, "Twisted-nematic liquid-crystal pixelated active lens," Opt. Commun. 153, 134-152 (1998).
[CrossRef]

Opt. Eng. (1)

B. Apter, E. Bahat-Treidel, and U. Efron, "Continuously controllable, wide-angle liquid crystal beam deflector based on the transversal field effect in a three-electrode cell," Opt. Eng. 44, 054001 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

G. Panasyuk, J. Kelly, E. C. Gartland, and D. W. Allender, "Geometrical optics approach in liquid crystal films with three-dimensional director variations," Phys. Rev. E 67, 041702 (2003).
[CrossRef]

Proc. SPIE (2)

W. Chan, L. Ning, S. T. Kowel, and P. F. Brinkley, "The liquid crystal adaptive lens: aberration correction," Proc. SPIE 1773, 468-475 (1992).
[CrossRef]

O. A. Zayakin, M. Y. Loktev, G. Love, and A. Naumov, "Cylindrical adaptive lenses," Proc. SPIE 3983, 112-117 (1999).
[CrossRef]

Other (13)

M.Born and E.Wolf, eds., Principles of Optics (Pergamon, 1959).

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, 1965).

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[CrossRef]

O. Gielkens, Polymer Vision Ltd., High Tech Campus 48, 5656 AE Eindhoven, The Netherlands (personal communication, 2001).

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 1999).

Mathematica website, http://www.wolfram.com.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

ELDIM website, http://www.eldim.fr.

R. A. M. Hikmet, T. van Bommel, and T. C. Kraan, "Study of light distribution through lc field induced lens arrays," submitted to J. Appl. Phys..

Shintech website, http://www.shintech.jp.

B. A. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

E. Hecht, Optics, 2nd ed. (Addison Wesley, 1989).

K. Neyts, S. Vermeirsch, S. Vermael, H. de Vleeschouwer, F. Bougrioua, S. Rozanski, D. de Boer, J. van Haaren, and S. Day, "Simulation of refraction, retardation and transmission in liquid crystal displays with slow variations," in Proceedings of the 20th International Display Research Conference (Society for Information Display, 2000), pp. 225-228.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Layout of the system considered and definition of the coordinates.

Fig. 2
Fig. 2

Director d ̂ in the co-ordinate frame and the director polar angle θ.

Fig. 3
Fig. 3

LC layer with the relevant variables and Huygens spheres for the derivation of the extension of Snell’s law to GRIN materials.

Fig. 4
Fig. 4

LC lens array system. The light impinges at the bottom side of the figure and is focused at the top.

Fig. 5
Fig. 5

Director profile as computed by Shintech. The light is incident from the bottom of the LC layer.

Fig. 6
Fig. 6

Radiance of monochromatic light with a wavelength of 550 nm as a function of the refraction angle for a cell with a LC orientation perpendicular to the linear electrodes and d = 34 μ m , electrode width 4 μ m , h = 18 μ m at 4 V rms. Comparison of theoretical models with the measurement. We present the measured radiance and (a) the radiance according to Huygens spheres method, (b) the radiance according to the Kline-Kay method, (c) the radiance according to the Fraunhofer integral, and (d) the radiance according to the diffraction integral employing the eikonal of the Kline–Kay method.

Fig. 7
Fig. 7

Average effective index of refraction n eff computed from the Shintech data of the director profile.

Fig. 8
Fig. 8

Refraction angle α air as a function of the lateral position x along the lens cell as computed by using the Kline–Kay method (method 2).

Fig. 9
Fig. 9

Lowest-order quadratic approximation of the effective index of refraction in a cylinder GRIN lens cell with Eq. (33) used for theoretical considerations.

Fig. 10
Fig. 10

Geometry indicating the focal distance, maximum refraction angle α max , and the half-period of the cell.

Fig. 11
Fig. 11

Discretized trajectory in the discretized Huygens spheres method, a refinement of method 1.

Fig. 12
Fig. 12

Scheme for the discretized Huygens spheres method.

Fig. 13
Fig. 13

Measured birefringence Δ n of the LC material BL009 as a function of the wavelength λ.

Fig. 14
Fig. 14

Radiance of monochromatic light with a wavelength of (a) 450 nm and (b) 691 nm as a function of the refraction angle for a cell with a LC orientation perpendicular to the linear electrodes and d = 34 μ m , electrode width 4 μ m , h = 18 μ m at 4 V rms. Comparison of theoretical models with the measurement. We present the measured radiance and the radiance according to the diffraction integral employing the eikonal of the Kline–Kay method.

Equations (61)

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

d ̂ = ( sin θ , 0 , cos θ ) .
n eff = n o n e n o 2 sin 2 χ + n e 2 cos 2 χ .
n eff ( x ) = 1 h 0 h n eff ( x , z ) d z .
r i = ( t t 0 i τ i ) c n g = ( t t 0 i ) c n g h n i n g , i = 1 , 2 ,
c n g ( t 02 t 01 ) = ( x 2 x 1 ) sin α glass , in ,
r 1 r 2 = h n g ( n 1 + n 2 ) + ( x 2 x 1 ) sin α glass , in .
sin α glass , out = r 1 r 2 x 2 x 1 = h n g n 2 n 1 x 2 x 1 + sin α glass , in .
sin α air , out = sin α air , in + h d n eff ( x ) d x .
ψ ( x , y , z ) = c t .
E * ( x , y , z ) = E ( x , y , z , ψ c ) , H * ( x , y , z ) = H ( x , y , z , ψ c ) .
p = ψ ;
ϵ E * + p × H * = 0 ,
p × E * + μ H * = 0 .
H = det ϵ x x ϵ x y ϵ x z 0 p z p y ϵ y x ϵ y y ϵ y z p z 0 p x ϵ z x ϵ z y ϵ z z p y p x 0 0 p z p y μ x x μ x y μ x z p z 0 p x μ y x μ y y μ y z p y p x 0 μ z x μ z y μ z z ,
ϵ i j ( x , z ) = ϵ δ i j + Δ ϵ d ̂ i ( x , z ) d ̂ j ( x , z ) .
d ̂ ( x , z ) = ( sin θ ( x , z ) , 0 , cos θ ( x , z ) ) .
Δ ϵ ϵ ϵ .
ϵ = n o 2 , ϵ = n e 2 .
μ i j ( x , z ) = δ i j .
H = [ p x 2 + p z 2 ϵ ] [ ( p x 2 + p z 2 ϵ ) ϵ + Δ ϵ ( ( p z cos θ + p x sin θ ) 2 ϵ ) ] .
p x 2 + p z 2 ϵ = 0 ,
( p x 2 + p z 2 ϵ ) ϵ + Δ ϵ ( ( p z cos θ + p x sin θ ) 2 ϵ ) = 0 .
H red = ( p x 2 + p z 2 ϵ ) ϵ + Δ ϵ ( ( p z cos θ + p x sin θ ) 2 ϵ ) .
d x d τ = λ H red p x , d z d τ = λ H red p z ,
d p x d τ = λ H red x , d p z d τ = λ H red z .
d x d τ = H red p x , d z d τ = H red p z ,
d p x d τ = H red x , d p z d τ = H red z .
H red p x = 2 p x ϵ + 2 Δ ϵ sin θ ( p z cos θ + p x sin θ ) ,
H red p z = 2 p z ϵ + 2 Δ ϵ cos θ ( p z cos θ + p x sin θ ) ,
H red x = 2 Δ ϵ ( p z cos θ + p x sin θ ) ( p z sin θ + p x cos θ ) θ x ,
H red z = 2 Δ ϵ ( p z cos θ + p x sin θ ) ( p z sin θ + p x cos θ ) θ z .
n medium sin α = p x .
A ( α air ) P ( x ) exp { i [ Δ ϕ ( x ) 2 π x λ 0 sin α air ] } d x .
Δ ϕ ( x ) = 2 π λ 0 h n eff ( x ) .
I ( α air ) A ( α air ) 2 { P ( x ) cos [ ( Δ ϕ ( x ) 2 π x λ 0 sin α air ) ] d x } 2 + { P ( x ) sin [ ( Δ ϕ ( x ) 2 π x λ 0 sin α air ) ] d x } 2 .
Δ ψ ( x , y , z ) = τ d ψ d τ d τ .
d ψ d τ = ψ x d x d τ + ψ z d z d τ = p x H p x + p z H p z .
A eikonal ( α air ) P ( x ) exp [ i 2 π λ 0 ( Δ ψ ( x ) x sin α air ) ] d x = P ( x ) exp [ i 2 π λ 0 ( τ d ψ d τ d τ x sin α air ) ] d x = P ( x ) exp { i 2 π λ 0 [ τ ( p x H p x + p z H p z ) d τ x sin α air ] } d x .
n ( x ) = n 0 + ( Δ n ) swing [ 1 ( 2 x w ) 2 ] .
d n d x = 8 ( Δ n ) swing x w 2 .
sin α = h d n d x = 8 ( Δ n ) swing h x w 2 .
sin α max = h 4 ( Δ n ) swing w .
α max h 4 ( Δ n ) swing w .
tan α max = w 2 f ,
f = w 2 tan α max w 2 α max = w 2 8 ( Δ n ) swing h .
P ( x ) = { 1 for w 2 x w 2 0 elsewhere } ,
I ( α air ) = λ 0 w 2 16 Δ n h { [ C ( w sin α 2 Δ n h λ 0 + 2 Δ n h λ 0 ) C ( w sin α 2 Δ n h λ 0 2 Δ n h λ 0 ) ] 2 + [ S ( w sin α 2 Δ n h λ 0 + 2 Δ n h λ 0 ) S ( w sin α 2 Δ n h λ 0 2 Δ n h λ 0 ) ] 2 } .
C ( u ) = 0 u cos π u 2 2 d u , S ( u ) = 0 u sin π u 2 2 d u .
sin α max = ± 4 h Δ n w .
N min + N max 4 ( Δ n ) swing h λ ,
d Δ ϕ ( x ) d x = 2 π λ sin α air .
Δ ϕ = 2 π λ h n eff ,
sin α air = h d n eff d x .
Δ α = arcsin [ Δ s n eff , next ( cos α n eff x sin α n eff z ) ] ,
( cos α n eff x sin α n eff z ) .
Δ α Δ s = Δ ϵ sin ( θ α ) cos ( θ α ) n o 2 sin 2 ( θ α ) + n e 2 cos 2 ( θ α ) ( cos α θ x sin α θ z ) ,
Δ ψ = n eff Δ s ,
Δ α Δ ψ = 1 n eff Δ ϵ sin ( θ α ) cos ( θ α ) n o 2 sin 2 ( θ α ) + n e 2 cos 2 ( θ α ) ( cos α θ x sin α θ z ) .
d α d τ = d d τ arctan p x p z = 2 Δ ϵ n eff sin ( θ α ) cos ( θ α ) ( cos α θ x sin α θ z ) ,
d ψ d τ = 2 n eff 2 [ n o 2 sin 2 ( θ α ) + n e 2 cos 2 ( θ α ) ] ,
Δ α Δ ψ = Δ α Δ τ ( d ψ d τ ) 1 = 1 n eff Δ ϵ sin ( θ α ) cos ( θ α ) n o 2 sin 2 ( θ α ) + n e 2 cos 2 ( θ α ) ( cos α θ x sin α θ z ) ,

Metrics