Abstract

A new method for calculating electromagnetic scattering from an arbitrarily shaped, inhomogeneous, dielectric object is developed. The method is based on an invariant imbedding procedure for computing the T matrix that was originally developed to solve quantum mechanical scattering problems. The final outcome of this approach is a two-term recurrence relation which can be solved numerically for the T matrix. The limiting form of this recurrence relation is a first-order nonlinear differential equation that is identical in form to the quantum mechanical Calogero equation. The results of several test calculations are also presented.

© 1988 Optical Society of America

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References

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  1. B. R. Johnson, D. Secrest, “Quantum Mechanical Calculations of the Inelastic Cross Sections for Rotational Excitation of Para and Ortho H2 upon Collision with He,” J. Chem. Phys. 48, 4682 (1968).
    [CrossRef]
  2. C. T. Tai, Dyadic Green’s Function in Electromagnetic Theory (Intext Educational Publishers, Scranton, PA, 1971).
  3. R. Newton, Scattering Theory of Particles and Waves (McGraw-Hill, New York, 1966).
  4. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  5. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  6. A. D. Yaghjian, “Electric Dyadic Green’s Function in the Source Region,” Proc. IEEE 68, 248 (1980).
    [CrossRef]
  7. C. T. Tai, “Comments on Electric Dyadic Green’s Functions in the Source Region,” Proc. IEEE 69, 282 (1981).
    [CrossRef]
  8. J. Van Bladel, “Some Remarks on Green’s Dyadic for Infinite Space,” IRE Trans. Antennas Propag. AP-9, 563 (1961).
    [CrossRef]
  9. C. T. Tai, “On the Eigenfunction Expansion of Dyadic Green’s Functions,” Proc. IEEE 61, 480 (1973).
    [CrossRef]
  10. R. E. Collin, “On the Incompleteness of E and H Modes in Wave Guides,” Can. J. Phys. 51, 1135 (1973).
    [CrossRef]
  11. A. Q. Howard, “On the Longitudinal Component of the Green’s Function Dyadic,” Proc. IEEE 62, 1704 (1974).
    [CrossRef]
  12. Y. Rahmit-Sami, “On the Question of Computation of the Dyadic Green’s Function at the Source Region in Waveguides and Cavities,” IEEE Trans. Microwave Theory Tech. MTT-23, 762 (1975).
    [CrossRef]
  13. K. M. Chen, “A Simple Physical Picture of Tensor Green’s Function in Source Region,” Proc. IEEE 65, 1202 (1977).
    [CrossRef]
  14. R. Bellman, G. M. Wing, An Introduction to Invariant Imbedding (Wiley, New York, 1975).
  15. S. Flugge, Practical Quantum Mechanics (Springer-Verlag, New York, 1974), problem 97, Calogero’s equation.
  16. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  17. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  18. S. R. Aragon, M. Elwenspoek, “Mie Scattering from Thin Spherical Bubbles,” J. Chem. Phys. 77, 3406 (1982).
    [CrossRef]
  19. S. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
    [CrossRef]
  20. B. Carnahan, H. Luther, J. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).
  21. R. K. Luneburg, The Mathemtical Theory of Optics (U. California Press, Berkeley, 1964).
  22. C. T. Tai, “The Electromagnetic Theory of the Spherical Luneberg Lens,” Appl. Scient. Res. Sect. B 7, 113 (1958).
    [CrossRef]
  23. P. J. Wyatt, “Scattering of Electromagnetic Plane Waves from Inhomogeneous Spherically Symmetric Objects,” Phys. Rev. 127, 1837 (1962); Errata Phys. Rev. 134, AB1 (1964).
    [CrossRef]
  24. S. Asano, G. Yamamoto, “Light Scattering by a Spheroidal Particle,” Appl. Opt. 14, 29 (1975).
    [PubMed]

1982 (1)

S. R. Aragon, M. Elwenspoek, “Mie Scattering from Thin Spherical Bubbles,” J. Chem. Phys. 77, 3406 (1982).
[CrossRef]

1981 (1)

C. T. Tai, “Comments on Electric Dyadic Green’s Functions in the Source Region,” Proc. IEEE 69, 282 (1981).
[CrossRef]

1980 (1)

A. D. Yaghjian, “Electric Dyadic Green’s Function in the Source Region,” Proc. IEEE 68, 248 (1980).
[CrossRef]

1977 (1)

K. M. Chen, “A Simple Physical Picture of Tensor Green’s Function in Source Region,” Proc. IEEE 65, 1202 (1977).
[CrossRef]

1975 (2)

S. Asano, G. Yamamoto, “Light Scattering by a Spheroidal Particle,” Appl. Opt. 14, 29 (1975).
[PubMed]

Y. Rahmit-Sami, “On the Question of Computation of the Dyadic Green’s Function at the Source Region in Waveguides and Cavities,” IEEE Trans. Microwave Theory Tech. MTT-23, 762 (1975).
[CrossRef]

1974 (1)

A. Q. Howard, “On the Longitudinal Component of the Green’s Function Dyadic,” Proc. IEEE 62, 1704 (1974).
[CrossRef]

1973 (2)

C. T. Tai, “On the Eigenfunction Expansion of Dyadic Green’s Functions,” Proc. IEEE 61, 480 (1973).
[CrossRef]

R. E. Collin, “On the Incompleteness of E and H Modes in Wave Guides,” Can. J. Phys. 51, 1135 (1973).
[CrossRef]

1968 (1)

B. R. Johnson, D. Secrest, “Quantum Mechanical Calculations of the Inelastic Cross Sections for Rotational Excitation of Para and Ortho H2 upon Collision with He,” J. Chem. Phys. 48, 4682 (1968).
[CrossRef]

1962 (1)

P. J. Wyatt, “Scattering of Electromagnetic Plane Waves from Inhomogeneous Spherically Symmetric Objects,” Phys. Rev. 127, 1837 (1962); Errata Phys. Rev. 134, AB1 (1964).
[CrossRef]

1961 (1)

J. Van Bladel, “Some Remarks on Green’s Dyadic for Infinite Space,” IRE Trans. Antennas Propag. AP-9, 563 (1961).
[CrossRef]

1958 (1)

C. T. Tai, “The Electromagnetic Theory of the Spherical Luneberg Lens,” Appl. Scient. Res. Sect. B 7, 113 (1958).
[CrossRef]

1951 (1)

S. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Aden, S. L.

S. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Aragon, S. R.

S. R. Aragon, M. Elwenspoek, “Mie Scattering from Thin Spherical Bubbles,” J. Chem. Phys. 77, 3406 (1982).
[CrossRef]

Asano, S.

Bellman, R.

R. Bellman, G. M. Wing, An Introduction to Invariant Imbedding (Wiley, New York, 1975).

Carnahan, B.

B. Carnahan, H. Luther, J. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Chen, K. M.

K. M. Chen, “A Simple Physical Picture of Tensor Green’s Function in Source Region,” Proc. IEEE 65, 1202 (1977).
[CrossRef]

Collin, R. E.

R. E. Collin, “On the Incompleteness of E and H Modes in Wave Guides,” Can. J. Phys. 51, 1135 (1973).
[CrossRef]

Elwenspoek, M.

S. R. Aragon, M. Elwenspoek, “Mie Scattering from Thin Spherical Bubbles,” J. Chem. Phys. 77, 3406 (1982).
[CrossRef]

Flugge, S.

S. Flugge, Practical Quantum Mechanics (Springer-Verlag, New York, 1974), problem 97, Calogero’s equation.

Howard, A. Q.

A. Q. Howard, “On the Longitudinal Component of the Green’s Function Dyadic,” Proc. IEEE 62, 1704 (1974).
[CrossRef]

Johnson, B. R.

B. R. Johnson, D. Secrest, “Quantum Mechanical Calculations of the Inelastic Cross Sections for Rotational Excitation of Para and Ortho H2 upon Collision with He,” J. Chem. Phys. 48, 4682 (1968).
[CrossRef]

Kerker, M.

S. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Luneburg, R. K.

R. K. Luneburg, The Mathemtical Theory of Optics (U. California Press, Berkeley, 1964).

Luther, H.

B. Carnahan, H. Luther, J. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Newton, R.

R. Newton, Scattering Theory of Particles and Waves (McGraw-Hill, New York, 1966).

Rahmit-Sami, Y.

Y. Rahmit-Sami, “On the Question of Computation of the Dyadic Green’s Function at the Source Region in Waveguides and Cavities,” IEEE Trans. Microwave Theory Tech. MTT-23, 762 (1975).
[CrossRef]

Secrest, D.

B. R. Johnson, D. Secrest, “Quantum Mechanical Calculations of the Inelastic Cross Sections for Rotational Excitation of Para and Ortho H2 upon Collision with He,” J. Chem. Phys. 48, 4682 (1968).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Tai, C. T.

C. T. Tai, “Comments on Electric Dyadic Green’s Functions in the Source Region,” Proc. IEEE 69, 282 (1981).
[CrossRef]

C. T. Tai, “On the Eigenfunction Expansion of Dyadic Green’s Functions,” Proc. IEEE 61, 480 (1973).
[CrossRef]

C. T. Tai, “The Electromagnetic Theory of the Spherical Luneberg Lens,” Appl. Scient. Res. Sect. B 7, 113 (1958).
[CrossRef]

C. T. Tai, Dyadic Green’s Function in Electromagnetic Theory (Intext Educational Publishers, Scranton, PA, 1971).

Van Bladel, J.

J. Van Bladel, “Some Remarks on Green’s Dyadic for Infinite Space,” IRE Trans. Antennas Propag. AP-9, 563 (1961).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wilkes, J.

B. Carnahan, H. Luther, J. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Wing, G. M.

R. Bellman, G. M. Wing, An Introduction to Invariant Imbedding (Wiley, New York, 1975).

Wyatt, P. J.

P. J. Wyatt, “Scattering of Electromagnetic Plane Waves from Inhomogeneous Spherically Symmetric Objects,” Phys. Rev. 127, 1837 (1962); Errata Phys. Rev. 134, AB1 (1964).
[CrossRef]

Yaghjian, A. D.

A. D. Yaghjian, “Electric Dyadic Green’s Function in the Source Region,” Proc. IEEE 68, 248 (1980).
[CrossRef]

Yamamoto, G.

Appl. Opt. (1)

Appl. Scient. Res. Sect. B (1)

C. T. Tai, “The Electromagnetic Theory of the Spherical Luneberg Lens,” Appl. Scient. Res. Sect. B 7, 113 (1958).
[CrossRef]

Can. J. Phys. (1)

R. E. Collin, “On the Incompleteness of E and H Modes in Wave Guides,” Can. J. Phys. 51, 1135 (1973).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

Y. Rahmit-Sami, “On the Question of Computation of the Dyadic Green’s Function at the Source Region in Waveguides and Cavities,” IEEE Trans. Microwave Theory Tech. MTT-23, 762 (1975).
[CrossRef]

IRE Trans. Antennas Propag. (1)

J. Van Bladel, “Some Remarks on Green’s Dyadic for Infinite Space,” IRE Trans. Antennas Propag. AP-9, 563 (1961).
[CrossRef]

J. Appl. Phys. (1)

S. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

J. Chem. Phys. (2)

B. R. Johnson, D. Secrest, “Quantum Mechanical Calculations of the Inelastic Cross Sections for Rotational Excitation of Para and Ortho H2 upon Collision with He,” J. Chem. Phys. 48, 4682 (1968).
[CrossRef]

S. R. Aragon, M. Elwenspoek, “Mie Scattering from Thin Spherical Bubbles,” J. Chem. Phys. 77, 3406 (1982).
[CrossRef]

Phys. Rev. (1)

P. J. Wyatt, “Scattering of Electromagnetic Plane Waves from Inhomogeneous Spherically Symmetric Objects,” Phys. Rev. 127, 1837 (1962); Errata Phys. Rev. 134, AB1 (1964).
[CrossRef]

Proc. IEEE (5)

C. T. Tai, “On the Eigenfunction Expansion of Dyadic Green’s Functions,” Proc. IEEE 61, 480 (1973).
[CrossRef]

A. D. Yaghjian, “Electric Dyadic Green’s Function in the Source Region,” Proc. IEEE 68, 248 (1980).
[CrossRef]

C. T. Tai, “Comments on Electric Dyadic Green’s Functions in the Source Region,” Proc. IEEE 69, 282 (1981).
[CrossRef]

A. Q. Howard, “On the Longitudinal Component of the Green’s Function Dyadic,” Proc. IEEE 62, 1704 (1974).
[CrossRef]

K. M. Chen, “A Simple Physical Picture of Tensor Green’s Function in Source Region,” Proc. IEEE 65, 1202 (1977).
[CrossRef]

Other (10)

R. Bellman, G. M. Wing, An Introduction to Invariant Imbedding (Wiley, New York, 1975).

S. Flugge, Practical Quantum Mechanics (Springer-Verlag, New York, 1974), problem 97, Calogero’s equation.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

B. Carnahan, H. Luther, J. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

R. K. Luneburg, The Mathemtical Theory of Optics (U. California Press, Berkeley, 1964).

C. T. Tai, Dyadic Green’s Function in Electromagnetic Theory (Intext Educational Publishers, Scranton, PA, 1971).

R. Newton, Scattering Theory of Particles and Waves (McGraw-Hill, New York, 1966).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of the displaced sphere problem. The solid circle represents a spherical particle of radius 0.5 which is displaced a distance d from the origin of the coordinate system and the arrow is the propagation vector of the incident plane wave. The dotted circle R is the outer boundary of the scattering region. The circle R0 divides the scattering region into two parts: (i) the region 0 ≤ rR0 is spherically symmetric and can be solved by Mie theory, (ii) the region R0rR is not spherically symmetric and is solved using Eq. (97). The numerical values of the parameters are: (a) d = 0.3, R0 = 0.2, R = 0.8 and (b) d = 0.6, R0 = 0.1, R = 1.1.

Fig. 2
Fig. 2

Scattering intensity functions i1(θ) and i2(θ) for the displaced sphere test problem. The solid curves are accurate Mie theory results for the d = 0 case, the dotted curves are for d = 0.3 and the dashed curves are for d = 0.6.

Fig. 3
Fig. 3

Scattering intensity functions il(θ) and i2(θ) for the displaced Luneburg lens test problem. Each curve represents the calculated results for all three displacements d = 0, 0.3, and 0.6.

Fig. 4
Fig. 4

Scattering intensity functions i1(θ) (solid lines) and i2(θ) (dotted lines) for prolate spheroids of a/b = 2 and c = 1–7. The intensity values at 0° and 180° are shown in the margins. This figure should be compared with Fig. 3 in Ref. 24.

Fig. 5
Fig. 5

Same as Fig. 4 but for oblate spheroids. This figure should be compared with Fig. 5 in Ref. 24

Equations (160)

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× E = ( i ω / c ) H ,
× H = - ( i ω ɛ ^ / c ) E ,
ɛ ^ = ɛ + i 4 π σ ω .
× × E - k 2 E = k 2 ( ɛ ^ / ɛ 0 - 1 ) E ,
k = ɛ 0 ω / c .
u ( r ) = k 2 [ ɛ ^ ( r ) / ɛ 0 - 1 ] .
E = E inc + E sca .
E inc = ( E x e x + E y e y ) exp ( i k z ) ,
E ( v ) = e v exp ( i k z ) + E sca ( v ) ,
E sca = E x E sca ( x ) + E y E sca ( y ) .
E sca ( v ) ~ S v ( θ , ϕ ) exp ( i k r ) i k r ,
E sca ~ [ E x S x ( θ , ϕ ) + E y S y ( θ , ϕ ) ] exp ( i k r ) i k r .
E ( r ) = E inc ( r ) + G ( r , r ) u ( r ) E ( r ) d 3 r .
× × G ( r , r ) - k 2 G ( r , r ) = I δ ( r - r ) ,
× × E ( r , θ , ϕ ) - k 2 E ( r , θ , ϕ ) = 0.
( 2 + k 2 ) Ψ ( r , θ , ϕ ) = 0 ,
M ( r , θ , ϕ ) = × [ Ψ ( r , θ , ϕ ) r ] ,
N ( r , θ , ϕ ) = 1 k × M ( r , θ , ϕ )
M = 1 k × N ,
· M = · N = 0.
Ψ e o , l , m ( r , θ , ϕ ) = P l m ( cos θ ) cos sin ( m ϕ ) z l ( k r ) ,
M e o , l , m = [ - + m sin θ P l m ( cos θ ) sin cos ( m ϕ ) e θ - P l m ( cos θ ) θ cos sin ( m ϕ ) e ϕ ] z l ( k r ) ,
N e o , l , m = [ l ( l + 1 ) k r P l m ( cos θ ) cos sin ( m ϕ ) e r ] z l ( k r ) + 1 k r r [ r z l ( k r ) ] [ P l m ( cos θ ) θ cos sin ( m ϕ ) e 0 - + m sin θ P l m ( cos θ ) sin cos ( m ϕ ) e ϕ ] ,
M 1 2 , l , m = K l , m M o e , l , m ,
N 1 2 , l , m = K l , m N e o , l , m ,
K l , m 2 = ( 2 - δ m , o ) ( 2 l + 1 ) ( l - m ) ! 4 π l ( l + 1 ) ( l + m ) ! .
( a r a θ a ϕ ) a r e r + a θ e θ + a ϕ e ϕ .
A 1 2 , l , m ( θ , ϕ ) = K l , m ( 0 + - m sin θ P l m ( cos θ ) cos sin ( m ϕ ) - P l m ( cos θ ) θ sin cos ( m ϕ ) ) ,
B 1 2 , l , m ( θ , ϕ ) = K l , m ( 0 P l m ( cos θ ) θ cos sin ( m ϕ ) - + m sin θ P l m ( cos θ ) sin cos ( m ϕ ) ) ,
C 1 2 , l , m ( θ , ϕ ) = K l , m ( [ l ( l + 1 ) ] 1 / 2 P l m ( cos θ ) cos sin ( m ϕ ) 0 0 ) .
X ˜ p , l , m ( Ω ) X p , l , m ( Ω ) d Ω = δ p , p δ l , l δ m , m Δ ,
X ˜ p , l , m ( Ω ) Y p , l , m ( Ω ) d Ω = 0 ,
M p , l , m ( r , Ω ) = A p , l , m ( Ω ) z l ( k r ) ,
N p , l , m ( r , Ω ) = B p , l , m ( Ω ) 1 k r r r z l ( k r ) + C p , l , m ( Ω ) [ l ( l + 1 ) ] 1 / 2 k r z l ( k r ) .
e x exp ( i k z ) = l = 1 i l [ 2 π ( 2 l + 1 ) ] 1 / 2 [ M 1 , l , 1 ( 1 ) - i N 1 , l , 1 ( 1 ) ] ,
e y exp ( i k z ) = - l = 1 i l [ 2 π ( 2 l + 1 ) ] 1 / 2 [ M 2 , l , 1 ( 1 ) + i N 2 , l , 1 ( 1 ) ] .
E sca ( v ) = p , l , m i l [ 2 π ( 2 l + 1 ) ] 1 / 2 [ a p , l , m ( v ) M p , l , m ( 3 ) - i b p , l , m ( v ) N p , l , m ( 3 ) ] .
G ( r , r ) = G 0 ( r , r ) + G 1 δ ( r - r ) ,
G 0 ( r , Ω ; r Ω ) = i k p = 1 2 l = 1 m = 0 l M p , l , m ( 3 ) ( r , Ω ) · M ˜ p , l , m ( 1 ) ( r , Ω ) + N p , l , m ( 3 ) ( r , Ω ) · N ˜ p , l , m ( 1 ) ( r , Ω ) M p , l , m ( 1 ) ( r , Ω ) · M ˜ p , l , m ( 3 ) ( r , Ω ) + N p , l , m ( 1 ) ( r , Ω ) · N ˜ p , l , m ( 3 ) ( r , Ω ) ,
j 1 ( l ; r ) = j l ( k r ) ,
j 2 ( l ; r ) = 1 k r r [ r j l ( k r ) ] ,
j 3 ( l ; r ) = [ l ( l + 1 ) ] 1 / 2 k r j l ( k r ) ,
h 1 ( l ; r ) = h l ( 1 ) ( k r ) ,
h 2 ( l ; r ) = 1 k r r [ r h l ( 1 ) ( k r ) ] ,
h 3 ( l ; r ) = [ l ( l + 1 ) ] 1 / 2 k r h l ( 1 ) ( k r ) .
G 0 = i k p , l , m ( A h 1 ) ( A ˜ j 1 ) + ( B h 2 + C h 3 ) ( B ˜ j 2 + C ˜ j 3 ) ; r > r ( A j 1 ) ( A ˜ h 1 ) + ( B j 2 + C j 3 ) ( B ˜ h 2 + C ˜ h 3 ) ; r < r .
G 0 = i k p , l , m ( A , B , C ) ( h 1 j 1 0 0 0 h 2 j 2 h 2 j 3 0 h 3 j 2 h 3 j 3 ) ( A ˜ B ˜ C ˜ ) ;             r > r ,
G 0 = i k p , l , m ( A , B , C ) ( j 1 h 1 0 0 0 j 2 h 2 j 2 h 3 0 j 3 h 2 j 3 h 3 ) ( A ˜ B ˜ C ˜ ) ;             r < r .
Y p , l , m ( Ω ) = [ A p , l , m ( Ω ) , B p , l , m ( Ω ) , C p , l , m ( Ω ) ] .
g l ( r , r ) = i k ( h 1 j i 0 0 0 h 2 j 2 h 2 j 3 0 h 3 j 2 h 3 j 3 ) ;             r > r ,
g l ( r , r ) = i k ( j 1 h i 0 0 0 j 2 h 2 j 2 h 3 0 j 3 h 2 j 3 h 3 ) ;             r < r .
G 0 ( r , Ω ; r , Ω ) = p , l , m Y p , l , m ( Ω ) g l ( r , r ) Y ˜ p , l , m ( Ω ) ,
J l ( r ) = ( j 1 ( l ; r ) 0 0 j 2 ( l ; r ) 0 j 3 ( l ; r ) ) ,
H l ( r ) = ( h 1 ( l ; r ) 0 0 h 2 ( l ; r ) 0 h 3 ( l ; r ) ) .
g l ( r , r ) = i k H l ( r ) J ˜ l ( r ) ; r > r , i k J l ( r ) H ˜ l ( r ) ; r < r .
[ M p , l , m ( 1 ) ( r , Ω ) , N p , l , m ( 1 ) ( r , Ω ) ] = Y p , l , m ( Ω ) J l ( r ) ,
[ M p , l , m ( 3 ) ( r , Ω ) , N p , l , m ( 3 ) ( r , Ω ) ] = Y p , l , m ( Ω ) H l ( r ) ,
D p , l , m ( x ) = i l [ 2 π ( 2 l + 1 ) ] 1 / 2 δ 1 , m δ 1 , p ( 1 - i ) ,
D p , l , m ( y ) = - i l [ 2 π ( 2 l + 1 ) ] 1 / 2 δ 1 , m δ 2 , p ( 1 i ) .
e v exp ( i k z ) = p , l , m Y p , l , m ( Ω ) J l ( r ) D p , l , m ( v ) .
E ¯ ( r ) = E inc ( r ) + d 3 r G ( r , r ) · u ( r ) E ¯ ( r ) .
E ( r ) = E inc ( r ) + d 3 r G 0 ( r , r ) · u ( r ) E ¯ ( r ) .
E ¯ ( r ) = Z ( r ) E ( r ) ,
Z ( r ) = ( ɛ 0 / ɛ ^ ( r ) 0 0 0 1 0 0 0 1 ) .
E ( r ) = E inc ( r ) + d 3 r G 0 ( r , r ) u ( r ) Z ( r ) E ( r ) .
E n ( r , Ω ) = Y n ( Ω ) J n ( r ) + r 2 d r d Ω n = 1 L Y n ( Ω ) g n ( r , r ) Y ˜ n ( Ω ) u ( r , Ω ) Z ( r , Ω ) E n ( r , Ω ) .
F n , n ( r ) = r 2 d Ω Y ˜ n ( Ω ) u ( r , Ω ) Z ( r , Ω ) E n ( r , Ω ) .
E n ( r , Ω ) = Y n ( Ω ) J n ( r ) + 0 R d r n = 1 L Y n ( Ω ) g n ( r , r ) F n , n ( r ) ,
F n , n ( r ) = U n , n ( r ) J n ( r ) + 0 R d r m = 1 L U n , m ( r ) g m ( r , r ) F m , n ( r ) ,
U n , n ( r ) = r 2 d Ω Y ˜ n ( Ω ) u ( r , Ω ) Z ( r , Ω ) Y n ( Ω ) .
g n ( r , r ) = i k H n ( r ) J ˜ n ( r ) .
E n ( r , Ω ) = Y n ( Ω ) J n ( r ) + n = 1 L Y n ( Ω ) H n ( r ) T n , n ,
T n , n = i k 0 R d r J ˜ n ( r ) F n , n ( r ) .
U ( r ) = [ U n , n ( r ) ] ,
F ( r ) = [ F n , n ( r ) ] .
J ( r ) = [ J n ( r ) δ n , n ] ,
H ( r ) = [ H n ( r ) δ n , n ] ,
g ( r , r ) = [ g n ( r , r ) δ n , n ] .
g ( r , r ) = i k H ( r ) J ˜ ( r ) ; r > r , i k J ( r ) H ˜ ( r ) ; r < r .
T = [ T n , n ] .
F ( r ) = U ( r ) J ( r ) + 0 R d r U ( r ) g ( r , r ) F ( r ) ,
T = i k 0 R d r J ˜ ( r ) F ( r ) .
F ( r i ) = U ( r i ) J ( r i ) + j = 1 N w j U ( r i ) g ( r i , r j ) F ( r j ) ,
T = i k j = 1 N w j J ˜ ( r j ) F ( r j ) ,
g l ( r , r ) = i k ( h 1 j 1 0 0 0 h 2 j 2 ½ ( h 2 j 3 + j 2 h 3 ) 0 ½ ( h 2 j 3 + j 2 h 3 ) h 3 j 3 ) ,
g ( r , r ) = i k 2 [ H ( r ) J ˜ ( r ) + J ( r ) H ˜ ( r ) ] .
g ( r , r ) = i k J ( r ) H ˜ ( r )
g ( r , r ) = i k H ( r ) J ˜ ( r )
F ( n r i ) = U ( r i ) J ( r i ) + j = 1 n w j U ( r i ) g ( r i , r j ) F ( n r i ) ,
T ( r n ) = i k j = 1 n w j J ˜ ( r j ) F ( n r j ) .
[ I - w n U ( r n ) g ( r n , r n ) ] Q = w n U ( r n ) ,
Q 1 , 1 = i k J ˜ ( r n ) QJ ( r n ) ,
Q 1 , 2 = i k J ˜ ( r n ) QH ( r n ) ,
Q 2 , 2 = i k H ˜ ( r n ) QH ( r n ) .
F ( n r i ) = F ( n - 1 r i ) ( I + p ) ,
F ( n r n ) = w n - 1 Q [ J ( r n ) + H ( r n ) q ] ,
F ( n - 1 r i ) ( I + p ) = U ( r i ) J ( r i ) + j = 1 n - 1 w j U ( r i ) g ( r i , r j ) × F ( n - 1 r j ) ( I + p ) + U ( r i ) g ( r i , r n ) × Q [ J ( r n ) + H ( r n ) q ] ,
w n - 1 Q [ J ( r n ) + H ( r n ) q ] = U ( r n ) J ( r n ) + j = 1 n - 1 w j U ( r n ) g ( r n , r j ) × F ( n - 1 r j ) ( I + p ) + U ( r n ) g ( r n , r n ) × Q [ J ( r n ) + H ( r n ) q ] .
g ( r j , r n ) = i k J ( r j ) H ˜ ( r n ) ,
g ( r n , r j ) = i k H ( r n ) J ˜ ( r j )
p = Q ˜ 1 , 2 + Q 2 , 2 q ,
q = T ( r n - 1 ) [ I + p ] .
T ( r n ) = Q 1 , 1 + ( I + Q 1 , 2 ) q .
T ( r n ) = Q 1 , 1 + ( I + Q 1 , 2 ) [ I - T ( r n - 1 ) Q 2 , 2 ] - 1 T ( r n - 1 ) ( I + Q ˜ 1 , 2 ) .
w n = Δ r .
Q = U ( r ) Δ r .
d T ( r ) d r = i k [ J ( r ) + H ( r ) T ( r ) ] T U ( r ) [ J ( r ) + H ( r ) T ( r ) ] ,
E sca ( v ) = n Y n ( Ω ) H n ( r ) X n ( v ) ,
X n ( v ) = i l [ 2 π ( 2 l + 1 ) ] 1 / 2 ( a n ( v ) - i b n ( v ) ) .
E ( v ) = e v exp ( i k z ) + n n Y n ( Ω ) H n ( r ) T n , n D n ( v ) ,
T n , n = ( T 1 , 1 T 1 , 2 T 2 , 1 T 2 , 2 ) n , n .
E sca ( v ) = n n Y n ( Ω ) H n ( r ) T n , n D n ( v ) ,
X n ( v ) = n T n , n D n ( v ) .
( a n ( x ) - i b n ( x ) ) = n i ( l - 1 ) ( 2 l + 1 2 l + 1 ) 1 / 2 × δ 1 , m , δ 1 , p ( T 1 , 1 T 1 , 2 T 2 , 1 T 2 , 2 ) n , n ( 1 - i ) .
a n ( x ) = n i ( l - l ) ( 2 l + 1 2 l + 1 ) 1 / 2 δ 1 , m δ 1 , p [ ( T 1 , 1 ) n , n - i ( T 1 , 2 ) n , n ] ,
b n ( x ) = n i ( l - l ) ( 2 l + 1 2 l + 1 ) 1 / 2 δ 1 , m δ 1 , p [ ( T 2 , 2 ) n , n + i ( T 2 , 1 ) n , n ] .
a n ( y ) = - n i ( l - l ) ( 2 l + 1 2 l + 1 ) 1 / 2 δ 1 , m δ 2 , p [ ( T 1 , 1 ) n , n + i ( T 1 , 2 ) n , n ] ,
b n ( y ) = n i ( l - l ) ( 2 l + 1 2 l + 1 ) 1 / 2 δ 1 , m δ 2 , p [ ( T 2 , 2 ) n , n - i ( T 2 , 1 ) n , n ] .
H l ( r ) ~ ( - i ) l exp ( i k r ) i k r ( 1 0 0 i 0 0 ) .
S v ( Ω ) = n [ 2 π ( 2 l + 1 ) ] 1 / 2 [ a n ( v ) A n ( Ω ) + b n ( v ) B n ( Ω ) ] .
E = e x exp ( i k z ) + E sca .
E sca ~ S ( θ , ϕ ) exp ( i k r ) i k r .
A l ( Ω ) = K l ( 0 π l ( cos θ ) cos ϕ - τ l ( cos θ ) sin ϕ ) ,
B l ( Ω ) = K l ( 0 τ l ( cos θ ) cos ϕ - π l ( cos θ ) sin ϕ ) ,
C l ( Ω ) = K l ( [ l ( l + 1 ) ] 1 / 2 P l 1 ( cos θ ) cos ϕ 0 0 ) ,
K l 2 = 1 2 π 2 l + 1 [ l ( l + 1 ) ] 2 ,
π l ( cos θ ) = 1 sin θ P l 1 ( cos θ ) ,
τ l ( cos θ ) = d d θ P l 1 ( cos θ ) .
S ( Ω ) = l [ 2 π ( 2 l + 1 ) ] 1 / 2 [ a l A l ( Ω ) + b l B l ( Ω ) ] ,
a l = l i ( l - l ) ( 2 l + 1 2 l + 1 ) 1 / 2 [ ( T 1 , 1 ) l , l - i ( T 1 , 2 ) l , l ] .
b l = l i ( l - l ) ( 2 l + 1 ) 2 l + 1 ) 1 / 2 [ ( T 2 , 2 ) l , l + i ( T 2 , 1 ) l , l ] .
S ( θ , ϕ ) = ( 0 S θ ( θ ) cos ϕ S ϕ ( θ ) sin ϕ ) ,
S θ ( θ ) = l = 1 2 l + 1 l ( l + 1 ) [ a l π l ( cos θ ) + b l τ l ( cos θ ) ] ,
S ϕ ( θ ) = - l = 1 2 l + 1 l ( l + 1 ) [ a l τ l ( cos θ ) + b l π l ( cos θ ) ] .
E θ ~ S θ ( θ ) cos ϕ exp ( i k r ) i k r ,
E ϕ ~ S ϕ ( θ ) sin ϕ exp ( i k r ) i k r .
a l = - b ¯ l * and b l = - a ¯ l * ,
i 1 ( θ ) = S ϕ ( θ ) 2 , i 2 ( θ ) = S θ ( θ ) 2 .
C ext = - 2 π k 2 l = 1 ( 2 l + 1 ) Re ( a l + b l ) ,
C sca = 2 π k 2 l = 1 ( 2 l + 1 ) ( a l 2 + b l 2 ) .
T l , l = ( T 1 , 1 0 0 T 2 , 2 ) l , l δ l , l ,
a l = ( T 1 , 1 ) l , l ,
b l = ( T 2 , 2 ) l , l .
T l , l = ( a l 0 0 b l ) δ l , l .
U l , l ( r ) = k 2 r 2 [ ɛ ^ ( r ) / ɛ 0 - 1 ] d Ω ( A ˜ l B ˜ l C ˜ l ) ( ɛ 0 / ɛ ^ ( r ) 0 0 0 1 0 0 0 1 ) ( A l , B l , C l ) ,
U l , l ( r ) = k 2 r 2 [ ɛ ^ ( r ) / ɛ 0 - 1 ] ( 1 0 0 0 1 0 0 0 ɛ 0 / ɛ ^ ( r ) ) δ l , l .
d a l ( r ) d r = i k 3 r 2 [ ɛ ^ ( r ) / ɛ 0 - 1 ] [ j 1 ( l ; r ) + h 1 ( l ; r ) a l ( r ) ] 2 ,
d b l ( r ) d r = i k 3 r 2 [ ɛ ^ ( r ) / ɛ 0 - 1 ] { [ j 2 ( l ; r ) + h 2 ( l ; r ) b l ( r ) ] 2 + [ ɛ 0 / ɛ ^ ( r ) ] [ j 3 ( l ; r ) + h 3 ( l ; r ) b l ( r ) ] 2 } .
a l = i k 3 r 0 2 ( ɛ - 1 ) j 1 ( l ; r 0 ) 2 Δ r ,
b l = - i k 3 r 0 2 ( ɛ - 1 ) [ j 2 ( l ; r 0 ) 2 + ɛ - 1 j 3 ( l ; r 0 ) 2 ] Δ r .
a l = i δ ( m 2 - 1 ) Ψ l ( α ) 2 ,
b l = i δ ( m 2 - 1 ) [ Ψ l ( α ) 2 + l ( l + 1 ) Ψ l ( α ) 2 m 2 α 2 ] ,
ɛ ( r ) = 2 - ( r / r 0 ) 2 ,
c = 2 π λ ( a 2 - b 2 ) 1 / 2 .
E = Z - 1 E ¯ .
Z - 1 ( r ) E ¯ ( r ) = E inc ( r ) + d 3 r G 0 ( r , r ) u ( r ) E ¯ ( r ) .
Z - 1 ( r ) = I - r ^ r ^ k 2 u ( r ) ,
r ^ r ^ = ( 1 0 0 0 0 0 0 0 0 ) ,
E ¯ ( r ) = E inc ( r ) + d 3 r [ G 0 ( r , r ) + r ^ r ^ k 2 δ ( r - r ) ] u ( r ) E ¯ ( r ) .
G ( r , r ) = G 0 ( r , r ) + r ^ r ^ k 2 δ ( r - r ) ,

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