Abstract

The applicability of different approximate methods proposed to determine the paraxial properties of the gradient-index (GRIN) distribution resembling that of the human lens, by means of the system ABCD matrix, is tested. Thus, the parabolic-ray-path approximation has been extended to provide the ABCD matrix of a slab lens comprised of a rotationally GRIN medium. The results show that this method has good numerical stability, and it is also the easiest one in determining the Gaussian constants of the human lens GRIN profile.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics, 8th ed. (Cambridge U. Press, 2002).
  2. E. W. Marchand, Gradient Index Optics (Academic, 1978).
  3. E. W. Marchand, "Ray tracing in gradient-index media," J. Opt. Soc. Am. 60, 1-8 (1970).
    [CrossRef]
  4. S. Doric and N. Renaud, "Analytical expressions for the paraxial parameters of a single lens with a spherical distribution of refractive index," Appl. Opt. 131, 5197-5200 (1992).
    [CrossRef]
  5. D. T. Moore, "Design of singlets with continuously varying indices of refraction," J. Opt. Soc. Am. 61, 886-894 (1971).
    [CrossRef]
  6. S. Doric, "Paraxial raytrace for rotationally symmetric homogeneous and inhomogeneous media," J. Opt. Soc. Am. A 1, 818-821 (1984).
    [CrossRef]
  7. A. Sharma, D. V. Kumar, and A. K. Ghatak, "Tracing rays through graded-index media: a new method," Appl. Opt. 21, 984-987 (1982).
    [CrossRef] [PubMed]
  8. R. K. Luneburg, Mathematical Theory of Optics (Univ. of California Press, 1964).
  9. P. J. Sands, "Inhomogeneous lens, III. Paraxial optics," J. Opt. Soc. Am. 61, 879-885 (1971).
    [CrossRef]
  10. D. A. Atchison and G. Smith, "Continuous gradient index and shell models of the human lens," Vision Res. 35, 2529-2538 (1995).
    [CrossRef] [PubMed]
  11. G. Smith and D. A. Atchison, "Equivalent power of the crystalline lens of the human eye: comparison of methods of calculation," J. Opt. Soc. Am. A 14, 2537-2546 (1997).
    [CrossRef]
  12. L. Matthiessen, "Untersuchungen über den aplanatismus und die periscopie der krystalllinsen in den augen der fische," Pfluegers Arch. Gesamte Physiol. Menschen Tiere 231, 287-307 (1880).
    [CrossRef]
  13. L. Matthiessen, "Untersuchungen über den Aplanatismus und die Periscopie der Krystalllinsen in den Augen der Fische," Pfluegers Arch. Gesamte Physiol. Menschen Tiere 27, 510-523 (1882).
    [CrossRef]
  14. A. Gullstrand, Hemholtz's Handbuch der Physiologischen Optik, 3rd ed., Vol. 1, Appendix II, pp. 301-358 (English translation edited by J. P. Southall, Optical Society of America, 1924).
  15. J. W. Blaker, "Toward an adptative model of the human eye," J. Opt. Soc. Am. 70, 220-283 (1980).
    [CrossRef] [PubMed]
  16. H. Liou and N. A. Brennan, "Anatomically accurate, finite model eye for optical modeling," J. Opt. Soc. Am. A 14, 1684-1695 (1997).
    [CrossRef] [PubMed]
  17. Y. Huang and D. T. Moore, "Human eye modeling using a single equation of gradient index crystalline lens for relaxed and accommodated states," Proc. SPIE 6342, 63420D (2006).
    [CrossRef]
  18. J. A. Díaz, C. Pizarro, and J. Arasa, "A single dispersive GRIN profile for the aging human eye," J. Opt. Soc. Am. A 25, 250-261 (2008).
    [CrossRef] [PubMed]
  19. M. Dublemann and G. van der Heijde, "The shape of the human lens: curvature, equivalent refractive index and the lens paradox," Vision Res. 41, 1867-1877 (2001).
    [CrossRef]
  20. M. Dublemann, G. van der Heijde, and H. A. Weeber, "Change in the shape of the aging human crystalline lens with accommodation," Vision Res. 45, 117-132 (2005).
    [CrossRef] [PubMed]
  21. C. Jones, D. Atchison, R. Meder, and J. Pope, "Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI)," Vision Res. 45, 2352-2366 (2005).
    [CrossRef] [PubMed]
  22. G. Smith, "Paraxial raytracing in gradient-index media: problems of convergence," J. Opt. Soc. Am. A 9, 331-333 (1992).
    [CrossRef] [PubMed]
  23. K. Halbach, "Matrix representation of Gaussian optics," Am. J. Phys. 3, 90-108 (1964).
    [CrossRef] [PubMed]
  24. A. Gerrard and J. Burch, Introduction to Matrix Methods in Optics (Dover, 1975).
    [PubMed]
  25. H. Arsenault, "Generalization of the principal plane concept in matrix optics," Am. J. Phys. 66, 397-399 (1980).
    [CrossRef] [PubMed]
  26. H. Arsenault and B. Macukow, "Factorization of the transfer matrix for symmetrical optical systems," J. Opt. Soc. Am. 73, 1350-1359 (1983).
    [CrossRef]
  27. W. M. Rosenblum, J. W. Blaker, and M. G. Block, "Matrix methods for the evaluation of lens systems with radial gradient-index elements," Am. J. Optom. Physiol. Opt. 65, 661-665 (1988).
    [PubMed]
  28. D. S. Goodman, "Geometrical Optics" in Handbook of Optics, Vol. I (Optical Society of America, 1995), pp. 1.1-1.80.
  29. W. T. Weltford, Aberrations of Optical Systems (Hilger, 1986).
  30. M. V. Pérez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gómez-Reino, "Description of gradient-index crystalline lens by a first order optical system," J. Opt. A, Pure Appl. Opt. 7, 103-110 (2005).
    [CrossRef]
  31. M. V. Pérez, C. Bao, M. T. Flores-Arias, M. Rama, and C. Gómez-Reino, "Gradient parameter and axial and field rays in the gradient-index crystalline lens model," J. Opt. A, Pure Appl. Opt. 5, S293-S297 (2003).
    [CrossRef] [PubMed]
  32. V. Lakshminarayanan and M. Calvo, "The optical transmission function and point spread function of the human eye treated as a cascade linear system in the Fresnel regime," in Selected Topics in Mathematical Physics, K.S. R. R.Sridhar and V.Lakshminarayanan, eds. (Allied, 1995).
    [PubMed]
  33. M. T. Flores-Arias, M. V. Pérez, C. Bao, A. Castelo, and C. Gómez-Reino, "Gradient-index human lens as a quadratic phase transformer," J. Mod. Opt. 4, 495-506 (2006).
    [CrossRef] [PubMed]
  34. P. J. Sands, "Third-order aberrations of inhomogeneous lenses," J. Opt. Soc. Am. 60, 1436-1443 (1970).
    [CrossRef]
  35. H. A. Buchdahl, Optical Aberrations Coefficients (Dover, 1968).
    [PubMed]
  36. K. Tanaka, "Paraxial theory of rotationally distributed-index media by means of Gaussian constants," Appl. Opt. 23, 1700-1706 (1984).
    [CrossRef] [PubMed]
  37. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Wiley, 2001).
    [PubMed]
  38. E. D. Rainville, Intermediate Differential Equations, 2nd ed. (Macmillan, 1964).
  39. C. Gomez-Reino, M. V. Pérez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).
  40. C. Palma and V. Bagini, "Extension of the fresnel transform to ABCD systems," J. Opt. Soc. Am. A 14, 1774-1779 (1997).
    [CrossRef]
  41. S. Wolfram, The Mathematica Book, 5th ed. (Wolfram Media, 2003).
  42. G. Smith, D. A. Atchison, and B. K. Pierscionek, "Modeling the power of the aging human eye," J. Opt. Soc. Am. A 9, 2111-2117 (1992).
    [CrossRef] [PubMed]
  43. B. A. Moffat, D. A. Atchison, and J. Pope, "Explanation of the lens paradox," Optom. Vision Sci. 79, 148-150 (2002).
    [CrossRef]

2008 (1)

2006 (2)

Y. Huang and D. T. Moore, "Human eye modeling using a single equation of gradient index crystalline lens for relaxed and accommodated states," Proc. SPIE 6342, 63420D (2006).
[CrossRef]

M. T. Flores-Arias, M. V. Pérez, C. Bao, A. Castelo, and C. Gómez-Reino, "Gradient-index human lens as a quadratic phase transformer," J. Mod. Opt. 4, 495-506 (2006).
[CrossRef] [PubMed]

2005 (3)

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gómez-Reino, "Description of gradient-index crystalline lens by a first order optical system," J. Opt. A, Pure Appl. Opt. 7, 103-110 (2005).
[CrossRef]

M. Dublemann, G. van der Heijde, and H. A. Weeber, "Change in the shape of the aging human crystalline lens with accommodation," Vision Res. 45, 117-132 (2005).
[CrossRef] [PubMed]

C. Jones, D. Atchison, R. Meder, and J. Pope, "Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI)," Vision Res. 45, 2352-2366 (2005).
[CrossRef] [PubMed]

2003 (1)

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. Rama, and C. Gómez-Reino, "Gradient parameter and axial and field rays in the gradient-index crystalline lens model," J. Opt. A, Pure Appl. Opt. 5, S293-S297 (2003).
[CrossRef] [PubMed]

2002 (1)

B. A. Moffat, D. A. Atchison, and J. Pope, "Explanation of the lens paradox," Optom. Vision Sci. 79, 148-150 (2002).
[CrossRef]

2001 (1)

M. Dublemann and G. van der Heijde, "The shape of the human lens: curvature, equivalent refractive index and the lens paradox," Vision Res. 41, 1867-1877 (2001).
[CrossRef]

1997 (3)

1995 (1)

D. A. Atchison and G. Smith, "Continuous gradient index and shell models of the human lens," Vision Res. 35, 2529-2538 (1995).
[CrossRef] [PubMed]

1992 (3)

1988 (1)

W. M. Rosenblum, J. W. Blaker, and M. G. Block, "Matrix methods for the evaluation of lens systems with radial gradient-index elements," Am. J. Optom. Physiol. Opt. 65, 661-665 (1988).
[PubMed]

1984 (2)

1983 (1)

1982 (1)

1980 (2)

H. Arsenault, "Generalization of the principal plane concept in matrix optics," Am. J. Phys. 66, 397-399 (1980).
[CrossRef] [PubMed]

J. W. Blaker, "Toward an adptative model of the human eye," J. Opt. Soc. Am. 70, 220-283 (1980).
[CrossRef] [PubMed]

1971 (2)

1970 (2)

1964 (1)

K. Halbach, "Matrix representation of Gaussian optics," Am. J. Phys. 3, 90-108 (1964).
[CrossRef] [PubMed]

1882 (1)

L. Matthiessen, "Untersuchungen über den Aplanatismus und die Periscopie der Krystalllinsen in den Augen der Fische," Pfluegers Arch. Gesamte Physiol. Menschen Tiere 27, 510-523 (1882).
[CrossRef]

1880 (1)

L. Matthiessen, "Untersuchungen über den aplanatismus und die periscopie der krystalllinsen in den augen der fische," Pfluegers Arch. Gesamte Physiol. Menschen Tiere 231, 287-307 (1880).
[CrossRef]

Am. J. Optom. Physiol. Opt. (1)

W. M. Rosenblum, J. W. Blaker, and M. G. Block, "Matrix methods for the evaluation of lens systems with radial gradient-index elements," Am. J. Optom. Physiol. Opt. 65, 661-665 (1988).
[PubMed]

Am. J. Phys. (2)

H. Arsenault, "Generalization of the principal plane concept in matrix optics," Am. J. Phys. 66, 397-399 (1980).
[CrossRef] [PubMed]

K. Halbach, "Matrix representation of Gaussian optics," Am. J. Phys. 3, 90-108 (1964).
[CrossRef] [PubMed]

Appl. Opt. (3)

J. Mod. Opt. (1)

M. T. Flores-Arias, M. V. Pérez, C. Bao, A. Castelo, and C. Gómez-Reino, "Gradient-index human lens as a quadratic phase transformer," J. Mod. Opt. 4, 495-506 (2006).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt. (2)

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gómez-Reino, "Description of gradient-index crystalline lens by a first order optical system," J. Opt. A, Pure Appl. Opt. 7, 103-110 (2005).
[CrossRef]

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. Rama, and C. Gómez-Reino, "Gradient parameter and axial and field rays in the gradient-index crystalline lens model," J. Opt. A, Pure Appl. Opt. 5, S293-S297 (2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (7)

Optom. Vision Sci. (1)

B. A. Moffat, D. A. Atchison, and J. Pope, "Explanation of the lens paradox," Optom. Vision Sci. 79, 148-150 (2002).
[CrossRef]

Pfluegers Arch. Gesamte Physiol. Menschen Tiere (2)

L. Matthiessen, "Untersuchungen über den aplanatismus und die periscopie der krystalllinsen in den augen der fische," Pfluegers Arch. Gesamte Physiol. Menschen Tiere 231, 287-307 (1880).
[CrossRef]

L. Matthiessen, "Untersuchungen über den Aplanatismus und die Periscopie der Krystalllinsen in den Augen der Fische," Pfluegers Arch. Gesamte Physiol. Menschen Tiere 27, 510-523 (1882).
[CrossRef]

Proc. SPIE (1)

Y. Huang and D. T. Moore, "Human eye modeling using a single equation of gradient index crystalline lens for relaxed and accommodated states," Proc. SPIE 6342, 63420D (2006).
[CrossRef]

Vision Res. (4)

M. Dublemann and G. van der Heijde, "The shape of the human lens: curvature, equivalent refractive index and the lens paradox," Vision Res. 41, 1867-1877 (2001).
[CrossRef]

M. Dublemann, G. van der Heijde, and H. A. Weeber, "Change in the shape of the aging human crystalline lens with accommodation," Vision Res. 45, 117-132 (2005).
[CrossRef] [PubMed]

C. Jones, D. Atchison, R. Meder, and J. Pope, "Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI)," Vision Res. 45, 2352-2366 (2005).
[CrossRef] [PubMed]

D. A. Atchison and G. Smith, "Continuous gradient index and shell models of the human lens," Vision Res. 35, 2529-2538 (1995).
[CrossRef] [PubMed]

Other (13)

D. S. Goodman, "Geometrical Optics" in Handbook of Optics, Vol. I (Optical Society of America, 1995), pp. 1.1-1.80.

W. T. Weltford, Aberrations of Optical Systems (Hilger, 1986).

A. Gullstrand, Hemholtz's Handbuch der Physiologischen Optik, 3rd ed., Vol. 1, Appendix II, pp. 301-358 (English translation edited by J. P. Southall, Optical Society of America, 1924).

A. Gerrard and J. Burch, Introduction to Matrix Methods in Optics (Dover, 1975).
[PubMed]

R. K. Luneburg, Mathematical Theory of Optics (Univ. of California Press, 1964).

M. Born and E. Wolf, Principles of Optics, 8th ed. (Cambridge U. Press, 2002).

E. W. Marchand, Gradient Index Optics (Academic, 1978).

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Wiley, 2001).
[PubMed]

E. D. Rainville, Intermediate Differential Equations, 2nd ed. (Macmillan, 1964).

C. Gomez-Reino, M. V. Pérez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).

H. A. Buchdahl, Optical Aberrations Coefficients (Dover, 1968).
[PubMed]

S. Wolfram, The Mathematica Book, 5th ed. (Wolfram Media, 2003).

V. Lakshminarayanan and M. Calvo, "The optical transmission function and point spread function of the human eye treated as a cascade linear system in the Fresnel regime," in Selected Topics in Mathematical Physics, K.S. R. R.Sridhar and V.Lakshminarayanan, eds. (Allied, 1995).
[PubMed]

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 (7)

Fig. 1
Fig. 1

Ray tracing parameters for axial and field rays in a slab lens of a GRIN medium with rotational symmetry.

Fig. 2
Fig. 2

Cardinal elements in a slab lens having a GRIN medium with rotational symmetry.

Fig. 3
Fig. 3

GRIN profiles for the examples analyzed in this study. The values for the isoindicial contours for the Gullstrand model are 1.386, 1.396, 1.398, 1.4, and 1.401; for the lens resembling that of a 45 year old subject (see [21]) are 1.371, 1.381, 1.391 and 1.401; and those for the Huang and Moore model are 1.368, 1.375, 1.38, 1.39, and 1.401.

Fig. 4
Fig. 4

(Color online) Height and slope corresponding to the axial and field ray traced for the Gullstrand model profile. The continuous curves are the results achieved by the series solution to Eq. (4), the dashed curve by the exact solution (coincident with the series solution), and the short dashed curve by the parabolic-ray-path approximation.

Fig. 5
Fig. 5

Weak inhomogeneity condition value [Eq. (11)] for the lens examples of this study as a function of the lens thickness. For example 2, only the first part of the lens is shown in order to demonstrate that this condition is not fulfilled.

Fig. 6
Fig. 6

(Color online) Height and slope corresponding to the axial and field ray traced for the lens profile of a 45 year old subject. The continuous curve is the results achieved by the series solution to Eq. (4), the dashed curve by the exact solution, and the short dashed curve by the parabolic-ray-path approximation.

Fig. 7
Fig. 7

(Color online) Height and slope corresponding to the axial and field ray traced for the Huang and Moore model. The continuous curve is the results achieved by the series solution to Eq. (4), the dashed curve by the exact solution, the long dashed curve by the weak inhomogeneity condition, and the short dashed curve by the parabolic-ray-path approximation (almost coincident with the exact solution).

Tables (1)

Tables Icon

Table 1 Cardinal Elements Determined by Means of the ABCD Matrix Provided by the Different Calculation Methods Analyzed a

Equations (46)

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

L = P o P n d s ,
d d s ( n ( r ) d r d s ) = n ,
n ( z , y ) = N o ( z ) + N 1 ( z ) y 2 + N 2 ( z ) y 4 + N 3 ( z ) y 6 + ,
N o ( z ) d 2 y d z 2 + N ˙ o ( z ) d y d z 2 N 1 ( z ) y = 0 ,
y ( z ) = j = 0 A j z j ,    d y ( z ) d z = u ( z ) = j = 1 j A j z j 1 .
N o ( z ) = j = 0 N o , j z j ,    N 1 ( z ) = j = 0 N 1 , j z j ,   …   ,
y ( 0 ) = A o ,    u ( 0 ) = A 1 .
A m = 1 m ( m 1 ) N 0 , 0 n = 2 m 1 [ 2 A n 2 N 1 , m n n ( n 1 ) A n N 0 , m n ( m n + 1 ) ( n 1 ) A n 1 N 0 , m n + 1 ] ( m 1 ) A m 1 N 0 , 1 + 2 A m 2 N 1 ,0 .
[ y ( z ) u ( z ) ] = [ y f ( z ) y a ( z ) u f ( z ) u a ( z ) ] [ y ( 0 ) u ( 0 ) ] .
n ( z , y ) = n o ( z ) [ 1 g 2 ( z ) 2 y 2 ] .
| g ˙ ( z ) | g 2 ( z ) 1 ,
y ( z ) = y ( 0 ) ( 1 + β z 2 ) , with   | β | 1 .
Φ = N o ( d ) 2 β d ,    Φ b = N o ( d ) 2 β d 1 + β d 2 ,
β = 0 d N 1 ( z ) d z d N o ( d ) 2 0 d N 1 ( z ) z 2 d z .
[ A B C D ] = [ 1 0 C / A 1 ] [ A 0 0 1 / A ] [ 1 B / A 0 1 ] .
V F ¯ = C / A = 2 β d 1 + β d 2 .
H F ¯ = 1 / C = 1 2 β d A = 1 + β d 2 .
[ A B C D ] = [ 1 B / D 0 1 ] [ 1 / D 0 0 D / A ] [ 1 0 C / D 1 ] ,
N o , i * ( z ) = j = 0 N j , i d j ,   N 1 , i * ( z ) = j = 1 N j , i j ! ( j 1 ) ! d j 1 ,   …   .
V H ¯ = 1 C ( D N o ( 0 ) N o ( d ) ) = V F ¯ + F H ¯ = 1 + β * d 2 2 β * d + 1 2 β * d ,
D = N o ( 0 ) N o ( d ) + β d 2 ,
A D B C = N o ( 0 ) N o ( d ) B = d 2 ( 1 + N o ( 0 ) N o ( d ) + β d 2 ) .
[ A B C D ] = [ 1 + β d 2 d 2 ( 1 + N o ( 0 ) N o ( d ) + β d 2 ) 2 β d N o ( 0 ) N o ( d ) + β d 2 ] ,
n ( z , y ) = 1.406 0.0062685 ( z z o ) 2 + 0.0003834 ( z z o ) 3 [ 0.00052375 + 0.00005735 ( z z o ) + 0.00027875 ( z z o ) 2 ] y 2 0.000066717 y 4 .
[ 0.99292 3.55801 0.0043165 0.99167 ] .
N o ( z ) = 1.386 + 0.026637 z 0.00822384 z 2 + 0.0038344 z 3 ,
N 1 ( z ) = 0.0012318425 + 0.0008904 z 0.00027875 z 2 .
[ 0.99292 3.55801 0.00431678 0.99167 ] .
[ 0.99226 3.58606 0.0043023 0.99226 ] .
n ( z , y ) = j = 0 c j r 2 j ( z , y ) ,
r 2 ( z , y ) = ( z a 1 ) 2 a 1 2 + y 2 b 2 ,
r 2 ( z , y ) = ( z a 1 ) 2 a 2 2 + y 2 b 2 .
[ 0.98802 3.86496 0.007367 0.98331 ] .
N o ( z ) = 1.371 + 0.33749 z 1.03996 z 2 + 1.83277 z 3 2.06925 z 4 + 1.57783 z 5 0.82819 z 6 + 0.29680 z 7 0.06970 z 8 + 0.0097 z 9 0.00061 z 10 ,
N 1 ( z ) = 0.013306 + 0.07367 z 7064 z 2 + 0.21947 z 3 0.17360 z 4 + 0.08720 z 5 0.027305 z 6 + 0.00489 z 7 0.00038 z 8 .
[ 0.98543 3.96292 0.007299 0.98543 ] ,
n ( z , y ) = 1.3678 0.001978 y 2 + 0.03455 z + 0.001103 z 2 0.015657 z 3 + 0.006855 z 4 0.001065 z 5 + 9.9 × 10 6 z 7 0.001938 ( z 2 ) 1 / 5 + 0.00978 ( z 2 ) 1 / 3 ,
[ 0.97715 3.91366 0.0115278 0.97668 ] .
N o ( z ) = 1.368 + 0.03455 z 0.001103 z 2 0.015657 z 3 + 0.006855 z 4 0.001065 z 5 0.001938 p = 0 | d p z 2 / 5 d z p | z = 0 z p 0.00978 q = 0 | d q z 2 / 3 d z q | z = 0 z q
N 1 ( z ) = 0.001978 ,
[ 0.998217 3.95288 0.0115371 0.956101 ] .
[ 0.976701 3.97317 0.0115916 0.976701 ] ,
y ¨ ( z ) + g 2 ( z ) y = 0 .
y f ( z ) u a ( z ) y a ( z ) u f ( z ) = N o ( 0 ) N o ( d ) ,
Φ L = Φ s a + Φ GRIN + Φ s b t 1 Φ GRIN Φ s a N o ( 0 ) t 2 Φ GRIN Φ s b N o ( d ) Φ s a Φ s b ( t 1 N o ( 0 ) + t 2 N o ( d ) ) + t 1 t 2 Φ s a Φ GRIN Φ s b N o ( d ) ,
Δ Φ L = | Φ L t 1 | Δ t 1 + | Φ L t 2 | Δ t 2 .

Metrics