Abstract

From the minimal-action principle follow the Hamilton equations of evolution for geometrical-optics rays in anisotropic media. In such media the direction of the ray and the canonical momentum are not generally parallel but differ by an anisotropy vector. The refractive index of this version of geometrical optics may have, in principle, any dependence on ray direction. The tangential component of momentum is conserved at surfaces of index discontinuity. It is shown that the factorization theorem of refraction holds for interfaces between two anisotropic media. We find the Lie–Seidel coefficients for axisymmetric interfaces between homogeneous aligned uniaxial anisotropic media to third aberration order.

© 1995 Optical Society of America

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References

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  1. S. C. McClain, L. W. Hillman, R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993); “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
    [CrossRef]
  2. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1963).
  3. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958); R. J. Regis, “The modern development of Hamiltonian optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. I, pp. 1–29; J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1973), Vol. XI, pp. 247–304; R. D. Guenther, Modern Optics (Wiley, New York, 1990); E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1979); K. D. Möller, Optics (University Science, Mill Valley, Calif., 1988); C. S. Hastings, New Methods in Geometrical Optics (Macmillan, New York, 1927); J. W. Blaker, Geometric Optics (Dekker, New York, 1971).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1984).
  5. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970).
  6. P. W. Hawkes, “Lie methods in optics: an assessment,” in Lie Methods in Optics, K. B. Wolf, ed., Vol. 352 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1989), pp. 1–17.
    [CrossRef]
  7. A. Mercier, Variational Principles of Physics (Dover, New York, 1963), p. 222.
  8. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Los Angeles, 1944).
  9. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).
  10. L. D. Landau, E. M. Lifshitz, Field Theory (Nauka, Moscow, 1972).
  11. A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, K. B. Wolf, ed., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 4, pp. 105–158.
    [CrossRef]
  12. R. Rashed, “A pioneer in anaclastics—Ibn Sahl on burning mirrors and lenses,” Isis 81, 464–491 (1990); Géométrie et Dioptrique au Xesiècle: Ibn Sahl, al-Qûhî, et Ibn al-Hayatham, Collection Sciences et Philosophie Arabes, Textes et Etudes (Les Belles Lettres, Paris, 1993).
    [CrossRef]
  13. R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974).
  14. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976).
  15. K. B. Wolf, “Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988).
    [CrossRef]
  16. A. J. Dragt, “Lie-algebraic theory of geometrical optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982).
    [CrossRef]
  17. M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986).
    [CrossRef]
  18. J. Delgado, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Iztapalapa, Mexico (personal communication, 1994).
  19. K. B. Wolf, “Nonlinearity in aberration optics,” in Symmetries and Nonlinear Phenomena, D. Levi, P. Winternitz, eds., Proceedings of the International School on Applied Mathematics, Centro Internacional de Física (CIF), Paipa, Colombia, February 22–26, 1988, Vol. 9 of CIF Series (World Scientific, Singapore, 1989), pp. 376–429.
  20. K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (1991).
    [CrossRef]
  21. P. Debye, Polar Molecules (Chemical Catalog, New York, 1929).

1993 (1)

1991 (1)

K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (1991).
[CrossRef]

1990 (1)

R. Rashed, “A pioneer in anaclastics—Ibn Sahl on burning mirrors and lenses,” Isis 81, 464–491 (1990); Géométrie et Dioptrique au Xesiècle: Ibn Sahl, al-Qûhî, et Ibn al-Hayatham, Collection Sciences et Philosophie Arabes, Textes et Etudes (Les Belles Lettres, Paris, 1993).
[CrossRef]

1988 (1)

1986 (1)

1982 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1984).

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970).

Chipman, R. A.

Debye, P.

P. Debye, Polar Molecules (Chemical Catalog, New York, 1929).

Delgado, J.

J. Delgado, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Iztapalapa, Mexico (personal communication, 1994).

Dragt, A. J.

A. J. Dragt, “Lie-algebraic theory of geometrical optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982).
[CrossRef]

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, K. B. Wolf, ed., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 4, pp. 105–158.
[CrossRef]

Feynman, R. P.

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1963).

Forest, E.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, K. B. Wolf, ed., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 4, pp. 105–158.
[CrossRef]

Gilmore, R.

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974).

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

Hawkes, P. W.

P. W. Hawkes, “Lie methods in optics: an assessment,” in Lie Methods in Optics, K. B. Wolf, ed., Vol. 352 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1989), pp. 1–17.
[CrossRef]

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958); R. J. Regis, “The modern development of Hamiltonian optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. I, pp. 1–29; J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1973), Vol. XI, pp. 247–304; R. D. Guenther, Modern Optics (Wiley, New York, 1990); E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1979); K. D. Möller, Optics (University Science, Mill Valley, Calif., 1988); C. S. Hastings, New Methods in Geometrical Optics (Macmillan, New York, 1927); J. W. Blaker, Geometric Optics (Dekker, New York, 1971).
[CrossRef]

Hillman, L. W.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976).

Krötzsch, G.

K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (1991).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Field Theory (Nauka, Moscow, 1972).

Leighton, R. B.

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1963).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Field Theory (Nauka, Moscow, 1972).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Los Angeles, 1944).

McClain, S. C.

Mercier, A.

A. Mercier, Variational Principles of Physics (Dover, New York, 1963), p. 222.

Navarro-Saad, M.

Rashed, R.

R. Rashed, “A pioneer in anaclastics—Ibn Sahl on burning mirrors and lenses,” Isis 81, 464–491 (1990); Géométrie et Dioptrique au Xesiècle: Ibn Sahl, al-Qûhî, et Ibn al-Hayatham, Collection Sciences et Philosophie Arabes, Textes et Etudes (Les Belles Lettres, Paris, 1993).
[CrossRef]

Sands, M.

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1963).

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1984).

Wolf, K. B.

K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (1991).
[CrossRef]

K. B. Wolf, “Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986).
[CrossRef]

K. B. Wolf, “Nonlinearity in aberration optics,” in Symmetries and Nonlinear Phenomena, D. Levi, P. Winternitz, eds., Proceedings of the International School on Applied Mathematics, Centro Internacional de Física (CIF), Paipa, Colombia, February 22–26, 1988, Vol. 9 of CIF Series (World Scientific, Singapore, 1989), pp. 376–429.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, K. B. Wolf, ed., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 4, pp. 105–158.
[CrossRef]

Isis (1)

R. Rashed, “A pioneer in anaclastics—Ibn Sahl on burning mirrors and lenses,” Isis 81, 464–491 (1990); Géométrie et Dioptrique au Xesiècle: Ibn Sahl, al-Qûhî, et Ibn al-Hayatham, Collection Sciences et Philosophie Arabes, Textes et Etudes (Les Belles Lettres, Paris, 1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Symb. Comput. (1)

K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (1991).
[CrossRef]

Other (15)

P. Debye, Polar Molecules (Chemical Catalog, New York, 1929).

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974).

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976).

J. Delgado, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Iztapalapa, Mexico (personal communication, 1994).

K. B. Wolf, “Nonlinearity in aberration optics,” in Symmetries and Nonlinear Phenomena, D. Levi, P. Winternitz, eds., Proceedings of the International School on Applied Mathematics, Centro Internacional de Física (CIF), Paipa, Colombia, February 22–26, 1988, Vol. 9 of CIF Series (World Scientific, Singapore, 1989), pp. 376–429.

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1963).

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958); R. J. Regis, “The modern development of Hamiltonian optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. I, pp. 1–29; J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1973), Vol. XI, pp. 247–304; R. D. Guenther, Modern Optics (Wiley, New York, 1990); E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1979); K. D. Möller, Optics (University Science, Mill Valley, Calif., 1988); C. S. Hastings, New Methods in Geometrical Optics (Macmillan, New York, 1927); J. W. Blaker, Geometric Optics (Dekker, New York, 1971).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1984).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970).

P. W. Hawkes, “Lie methods in optics: an assessment,” in Lie Methods in Optics, K. B. Wolf, ed., Vol. 352 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1989), pp. 1–17.
[CrossRef]

A. Mercier, Variational Principles of Physics (Dover, New York, 1963), p. 222.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Los Angeles, 1944).

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

L. D. Landau, E. M. Lifshitz, Field Theory (Nauka, Moscow, 1972).

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, K. B. Wolf, ed., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 4, pp. 105–158.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

In isotropic but inhomogeneous media the Descartes sphere guides the ray by changing its radius in response to the local refractive index. The component of p normal to the gradient of the refractive index is conserved.

Fig. 2
Fig. 2

In anisotropic media the unit ray-direction vector q ˙ is multiplied by n(q, q ˙) and summed to the anisotropy vector A(q, q ˙) to yield the optical momentum three-vector p. While q ˙ ranges over the unit sphere, n q ˙ draws out the ray surface (dotted curve) and p sweeps over the Descartes ovoid (solid curve).

Fig. 3
Fig. 3

Quadrupole medium: n0 q ˙ ranges over the sphere (circle in the two dimensions of the figure) of radius n0, n q ˙ ranges over the peanut-shaped surface, and the momentum (two-) vector p draws a Descartes oval (ovoid in two dimensions). The lines joining points on the circle to those on the oval relate the direction of the ray with the corresponding direction of the momentum vector. In the two dimensions of the figure the quadrupole matrix is Q ^ = diag(Q, −Q).

Fig. 4
Fig. 4

Maxwell electromagnetic vectors in an anisotropic medium, where B = μH points into the page. The electric and displacement vectors, E and D, are not parallel. The Poynting vector S ||E × H is the direction of flow of energy and corresponds to the direction of propagation of the ray q ˙ in geometrical optics. The wave-front normals ∊Φ are parallel to the optical momentum p and orthogonal to the displacement vector D.

Fig. 5
Fig. 5

Behavior of spherical aberration coefficients of free propagation [Eq. (6.6)] for k = 0, 1, 2, 3 as a function of the anisotropy parameter ν in an uniaxial crystal.

Fig. 6
Fig. 6

Descartes diagram for construction of refraction angles between two anisotropic media n and n′: (1) Draw two circles and the corresponding Descartes ovoids for the two media n and n′, joined at a line parallel to the refracting interface at the point of incidence. (2) Draw the direction q ˙ of the incoming ray and the corresponding momentum vector p. (3) Project the momentum on the interface; this quantity is conserved. (4) Draw the conserved segment for the second ovoid and its corresponding p′. (5) The associated q ˙ yields the direction of the refracted ray.

Fig. 7
Fig. 7

Refraction at a surface is a map between phase-space points (q, p) and (q′, p′). This transformation visibly factors into transformations back and forth from the point of impact q ¯ on the surface z ¯ = ζ ( q ¯ ).

Equations (82)

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δ A B d t = 0.
δ A B d s n [ q ( s ) , q ˙ ( s ) ] = 0 .
q = ( x y z ) = ( q z ) ,             q = ( x y ) ,
q ˙ = d q d s = ( x ˙ y ˙ z ˙ ) = ( q ˙ σ 1 - q ˙ 2 ) = ( sin θ cos ϕ sin θ sin ϕ cos θ ) .
v = d q d z = ( tan θ cos ϕ tan θ sin ϕ ) ,
d s = d z 1 - q ˙ 2 = d z 1 + v 2 = d z sec θ
δ z A z B d z L [ q ( z ) , z ; v ( z ) ] = 0 ,
L ( q , z ; v ) = 1 + v 2 n ( q , z ; v ) .
d d z p = L q ,
p = ( p x p y ) = L v = n v 1 + v 2 + 1 + v 2 n v .
d q d z = H p ,             d p d z = - H q ,
H ( q , z ; p ) = p · v - 1 + v 2 n ( q , z ; v ) .
h = { f , g } = f q · g p - f p · g q ,
d q = - { H , p } d z ,             d p = - { H , q } d z .
d w = - { H , w } d z w ( z ) = τ lim N m = 1 N [ 1 - z N { H ( w , z m ) , } ] w ( 0 ) ,
w ( z ) = exp ( - z { H , } ) w = w - z { H , w } + ½ z 2 { H , { H , w } } +
{ H ( p ) , q } = - H p ,             { H ( p ) , H p } = 0 ,             { H ( p ) , p } = 0.
v = q ˙ 1 - q ˙ 2 ,             q ˙ = v 1 + v 2 .
v = ( / v x / v y ) = ( cos 2 θ cos ϕ θ - cot θ sin ϕ ϕ cos 2 θ sin ϕ θ + cot θ cos ϕ ϕ ) = 1 - q ˙ 2 [ 1 - x ˙ 2 - x ˙ y ˙ - x ˙ y ˙ 1 - y ˙ 2 ] ( / x ˙ / y ˙ ) = 1 - q ˙ 2 ( 1 - q ˙ q ˙ T ) q ˙ ,
p = n q ˙ + A ( q , z , q ˙ ) ,
A = 1 + v 2 n v = ( 1 - q ˙ q ˙ T ) n ( q , z , q ˙ ) q ˙ .
H ( q , z ; p ) = p · q ˙ - n ( q , q ˙ ) 1 - q ˙ 2 = - ( n 2 - p - A 2 ) 1 / 2 + ( p - A ) · A ( n 2 - p - A 2 ) 1 / 2 .
n z ˙ = n 1 - q ˙ 2 = ( n 2 - p - A 2 ) 1 / 2
p z = - H = ( n 2 - p - A 2 ) 1 / 2 + A z ,
A z = - n q ˙ · A 1 - q ˙ 2 = - ( p - A ) · A ( n 2 - p - A 2 ) 1 / 2 .
p = n q ˙ + A ( q , q ˙ ) ,             p z = - H ,
q ˙ = 1 ,             q ˙ · A = 0 ,
p - A = n ( q , q ˙ ) ;
A = n q ˙ | q ˙ = 1 = ( 1 - q ˙ q ˙ T ) n q ˙ | q ˙ = 1 = e ^ θ n θ + e ^ ϕ sin θ n ϕ .
d q d z = H p = p - A p z - A z ,
d p d z = - H q = n p z - A z n q .
d p × n q = 0.
n ( q , q ˙ ) = n 0 ( q ) + Q ( q , q ˙ ) ,
Q ( q , q ˙ ) j , k = x , y , z Q j , k q ˙ j q ˙ k = q ˙ T Q ^ q ˙ .
A ( 2 ) = 2 ( 1 - q ˙ q ˙ T ) Q ^ q ˙ = 2 [ Q ^ q ˙ - Q ( q , q ˙ ) q ˙ ] .
p = [ n ( q , q ˙ ) + 2 ( 1 - q ˙ q ˙ T ) Q ^ ] q ˙ = ( n 0 + 2 Q ^ - q ˙ T Q ^ q ˙ ) q ˙ .
n ( q , q ˙ ) = m 0 N ( m ) ( q , q ˙ ) ,
N ( m ) ( q , q ˙ ) i N i 1 , i 2 , , i m ( m ) q ˙ i 1 q ˙ i 2 q ˙ i m q ˙ T · N ( m ) .
A ( m ) = m ( 1 - q ˙ q ˙ T ) N ( m ) = m [ N ( m ) - N ( m ) ( q , q ˙ ) q ˙ ] .
p = n q ˙ + m 1 A ( m ) = [ n 0 - m 2 ( m - 1 ) N ( m ) ] q ˙ + m 1 m N ( m ) .
n o = μ ,
n e = μ / ( 1 + - z z sin 2 θ ) 1 / 2 ,
n ( q ˙ ) = n 0 + ( x ˙ , y ˙ , z ˙ ) [ ν 0 0 0 ν 0 0 0 - 2 ν ] ( x ˙ y ˙ z ˙ ) = n 0 + ν ( x ˙ 2 + y ˙ 2 ) - 2 ν z ˙ 2 = ( n 0 - 2 ν ) + 3 ν q ˙ 2 = n 0 - 2 ν + 3 ν sin 2 θ .
μ z = n 0 + ν ,             n o = μ = n 0 - 2 ν ,
n e ( n 0 - 2 ν ) ( 1 + 3 ν n 0 sin 2 θ ) n 0 - 2 ν + 3 ν sin 2 θ
p = ( n 0 + 4 ν ) q ˙ - 3 ν q ˙ 2 q ˙ = ( 1 + v 2 ) - 3 / 2 [ ( n 0 + 4 ν ) + ( n 0 + ν ) v 2 ] v = [ ( n 0 + 4 ν ) sin θ - 3 ν sin 3 θ ] ( cos ϕ sin ϕ ) .
p = n 0 v 1 + v 2 ,             p = n 0 sin θ             ( ν = 0 ) .
d q d z = v q ( z ) = q ( 0 ) + z v ,
d p d z = 0 p ( z ) = p ( 0 ) .
v = p ( n 0 ) 2 - p 2 = p p z ,             v = n 0 tan θ             ( ν = 0 ) .
v ( p ) = 1 n 0 + 4 ν p + ½ n 0 + 5 ν ( n 0 + 4 ν ) 4 p 2 p + ³ / ( n 0 ) 2 + ¹⁵ / n 0 ν + 51 ν 2 ( n 0 + 4 ν ) 7 ( p 2 ) 2 p + .
H ( p ) = p · v - ( n 0 - 2 ν ) 1 + v 2 - 3 ν v 2 1 + v 2 = - ( n 0 - 2 ν ) + 1 2 n e p 2 + n 0 + 10 ν 8 ( n e ) 4 ( p 2 ) 2 + 3 ( n 0 ) 2 + 60 n 0 ν + 408 ν 2 16 ( n e ) 7 ( p 2 ) 3 + .
p = p .
sin θ = n e n e sin θ + 3 n e [ ( n e n e ) 3 ν - ν ] sin 3 θ + 27 ( n e ) 2 ( n e n e ) 2 ν [ ( n e n e ) 3 ν - ν ] sin 5 θ + .
q ( z ¯ ) = q + ζ ( q ¯ ) v = q ¯ = q + ζ ( q ¯ ) v = q ( z ¯ ) .
p × S ( q ¯ ) = p ¯ × k ^ = p × S ( q ¯ ) .
p - H ( p ) Σ ( q ¯ ) = p ¯ = p - H ( p ) Σ ( q ¯ ) .
R n ; ζ :             q q ¯ = q + ζ ( q ¯ ) v ( p ) ,
R n ; ζ :             p p ¯ = p - H ( p ) Σ ( q ¯ ) ,
S n , n ; ζ :             { q q p p
S n , n ; ζ = R n ; ζ R n ; ζ - 1 .
R n ; ζ - 1 :             q ¯ q = q ¯ - ζ ( q ¯ ) v ( p ) ,
R n ; ζ - 1 :             p ¯ p = p ¯ + H ( p ) Σ ( q ¯ ) ,
q ( q ¯ , p ) = F ( q ¯ , p ) p ,             p ¯ ( q ¯ , p ) = F ( q ¯ , p ) q ¯ .
R ( q ¯ , p ) = q ¯ · p H , ζ = q ¯ · p + z ¯ p z = q ¯ . p - ζ ( q ¯ ) H ( p ) ,
ζ ( q ) = ζ 2 q 2 + ζ 4 ( q 2 ) 2 + ,
Σ ( q ) = 2 ζ 2 q + 4 ζ 4 q 2 q + .
q ¯ = q + ζ 2 n 0 + 4 ν q 2 p ,
p ¯ = p + 2 ζ 2 ( n 0 - 2 ν ) q - ζ 2 1 n 0 + 4 ν p 2 q + 2 ζ 2 2 n 0 - 2 ν n 0 + 4 ν q 2 p + 4 ζ 4 ( n 0 - 2 ν ) q 2 q .
R n ; ζ ( 3 ) = exp [ - ζ 2 / 2 n e p 2 q 2 + ζ 4 ( n 0 - 2 ν ) ( q 2 ) 2 , ] × exp [ - ζ 2 ( n 0 - 2 ν ) q 2 , ] .
q = q ¯ - ζ 2 1 n 0 + 4 ν q ¯ 2 p ¯ + 2 ζ 2 2 n 0 - 2 ν n 0 + 4 ν q ¯ 2 q ¯ ,
p = p ¯ - 2 ζ 2 ( n 0 - 2 ν ) q ¯ + ζ 2 1 n 0 + 4 ν p ¯ 2 q ¯ - 4 ζ 2 2 n 0 - 2 ν n 0 + 4 ν p ¯ · q ¯ q ¯ + 4 [ ζ 2 3 ( n 0 - 2 ν ) 2 n 0 + 4 ν - ζ 4 ( n 0 - 2 ν ) ] q ¯ 2 q ¯ .
q = q - ζ 2 ( 1 n 0 + 4 ν - 1 n 0 + 4 ν ) q 2 p + 2 ζ 2 2 ( n 0 - 2 ν ) - ( n 0 - 2 ν ) n 0 + 4 ν q 2 q ,
p = p - 2 ζ 2 [ ( n 0 - 2 ν ) - ( n 0 - 2 ν ) ] q + ζ 2 ( 1 n 0 + 4 ν - 1 n 0 + 4 ν ) p 2 q - 4 ζ 2 2 ( n 0 - 2 ν ) - ( n 0 - 2 ν ) n 0 + 4 ν p · qq - 2 ζ 2 2 ( n 0 - 2 ν ) - ( n 0 - 2 ν ) n 0 + 4 ν q 2 p + 4 { ζ 2 3 [ ( n 0 - 2 ν ) - ( n 0 - 2 ν ) ] 2 n 0 + 4 ν - ζ 4 [ ( n 0 - 2 ν ) - ( n 0 - 2 ν ) ] } q 2 q .
S n , n ; ζ ( 3 ) = exp [ A ( p 2 ) 2 + B p 2 p · q + C ( p · q ) 2 + D p 2 q 2 + E p · q q 2 + F ( q 2 ) 2 , ] × exp { - ζ 2 [ ( n 0 - 2 ν ) - ( n 0 - 2 ν ) ] q 2 , } ,
Spherical aberration :             A = 0 ,
Coma :             B = 0 ,
Curvature of field :             C = ζ 2 2 ( 1 n 0 + 4 ν - 1 n 0 + 4 ν ) ,
Astigmatism :             D = 0 ,
Distortion :             E = - 2 ζ 2 2 × ( n 0 - 2 ν ) - ( n 0 - 2 ν ) n 0 + 4 ν ,
Focus ( defocus ) :             F = 2 ζ 2 3 × [ ( n 0 - 2 ν ) - ( n 0 - 2 ν ) ] 2 n 0 + 4 ν - ζ 4 [ ( n 0 - 2 ν ) - ( n 0 - 2 ν ) ] .
E = m = 0 E m ( i k 0 ) m exp ( i k 0 ψ )

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