Abstract

The solution of electromagnetic scattering by a homogeneous prolate (or oblate) spheroidal particle with an arbitrary size and refractive index is obtained for any angle of incidence by solving Maxwell’s equations under given boundary conditions. The method used is that of separating the vector wave equations in the spheroidal coordinates and expanding them in terms of the spheroidal wavefunctions. The unknown coefficients for the expansion are determined by a system of equations derived from the boundary conditions regarding the continuity of tangential components of the electric and magnetic vectors across the surface of the spheroid. The solutions both in the prolate and oblate spheroidal coordinate systems result in a same form, and the equations for the oblate spheroidal system can be obtained from those for the prolate one by replacing the prolate spheroidal wavefunctions with the oblate ones and vice versa. For an oblique incidence, the polarized incident wave is resolved into two components, the TM mode for which the magnetic vector vibrates perpendicularly to the incident plane and the TE mode for which the electric vector vibrates perpendicularly to this plane. For the incidence along the rotation axis the resultant equations are given in the form similar to the one for a sphere given by the Mie theory. The physical parameters involved are the following five quantities: the size parameter defined by the product of the semifocal distance of the spheroid and the propagation constant of the incident wave, the eccentricity, the refractive index of the spheroid relative to the surrounding medium, the incident angle between the direction of the incident wave and the rotation axis, and the angles that specify the direction of the scattered wave.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Mie, Ann. Phys. 25, 377 (1908).
    [CrossRef]
  2. Lord Rayleigh, Phil. Mag. 36, 365 (1918).
  3. J. R. Wait, Can. J. Phys. 33, 189 (1955).
    [CrossRef]
  4. F. Möglich, Ann. Phys. 83, 609 (1927).
    [CrossRef]
  5. F. V. Schultz, “Scattering by a Prolate Spheroid,” UMM-42, Univ. of Mich., 1950.
  6. K. M. Siegel, F. V. Schultz, B. H. Gere, F. B. Sleator, IRE Trans. Antennas Propag. AP-4, 266 (1956).
    [CrossRef]
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  8. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).
  9. C. Flammer, Spheroidal Wave Functions (Stanford U.P., Stanford, Calif., 1957).
  10. J. R. Wait, Radio Sci. 1, 475 (1966).
  11. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  12. C. W. H. Yeh, J. Math. Phys. 42, 68 (1963).
  13. C. J. Bouwkamp, J. Math. Phys. 26, 79 (1947).
  14. R. J. T. Bell, Coordinate Solid Geometry (Macmillan, London, 1948).
  15. J. M. Greenberg, A. C. Lind, R. T. Wang, L. F. Libelo, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).
  16. A. C. Lind, R. T. Wang, J. M. Greenberg, Appl. Opt. 4, 1555 (1965).
    [CrossRef]

1966

J. R. Wait, Radio Sci. 1, 475 (1966).

1965

1963

C. W. H. Yeh, J. Math. Phys. 42, 68 (1963).

1956

K. M. Siegel, F. V. Schultz, B. H. Gere, F. B. Sleator, IRE Trans. Antennas Propag. AP-4, 266 (1956).
[CrossRef]

1955

J. R. Wait, Can. J. Phys. 33, 189 (1955).
[CrossRef]

1947

C. J. Bouwkamp, J. Math. Phys. 26, 79 (1947).

1927

F. Möglich, Ann. Phys. 83, 609 (1927).
[CrossRef]

1918

Lord Rayleigh, Phil. Mag. 36, 365 (1918).

1908

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Bell, R. J. T.

R. J. T. Bell, Coordinate Solid Geometry (Macmillan, London, 1948).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).

Bouwkamp, C. J.

C. J. Bouwkamp, J. Math. Phys. 26, 79 (1947).

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U.P., Stanford, Calif., 1957).

Gere, B. H.

K. M. Siegel, F. V. Schultz, B. H. Gere, F. B. Sleator, IRE Trans. Antennas Propag. AP-4, 266 (1956).
[CrossRef]

Greenberg, J. M.

A. C. Lind, R. T. Wang, J. M. Greenberg, Appl. Opt. 4, 1555 (1965).
[CrossRef]

J. M. Greenberg, A. C. Lind, R. T. Wang, L. F. Libelo, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).

Libelo, L. F.

J. M. Greenberg, A. C. Lind, R. T. Wang, L. F. Libelo, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).

Lind, A. C.

A. C. Lind, R. T. Wang, J. M. Greenberg, Appl. Opt. 4, 1555 (1965).
[CrossRef]

J. M. Greenberg, A. C. Lind, R. T. Wang, L. F. Libelo, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Möglich, F.

F. Möglich, Ann. Phys. 83, 609 (1927).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, Phil. Mag. 36, 365 (1918).

Schultz, F. V.

K. M. Siegel, F. V. Schultz, B. H. Gere, F. B. Sleator, IRE Trans. Antennas Propag. AP-4, 266 (1956).
[CrossRef]

F. V. Schultz, “Scattering by a Prolate Spheroid,” UMM-42, Univ. of Mich., 1950.

Siegel, K. M.

K. M. Siegel, F. V. Schultz, B. H. Gere, F. B. Sleator, IRE Trans. Antennas Propag. AP-4, 266 (1956).
[CrossRef]

Sleator, F. B.

K. M. Siegel, F. V. Schultz, B. H. Gere, F. B. Sleator, IRE Trans. Antennas Propag. AP-4, 266 (1956).
[CrossRef]

Stratton, J. A.

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

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wait, J. R.

J. R. Wait, Radio Sci. 1, 475 (1966).

J. R. Wait, Can. J. Phys. 33, 189 (1955).
[CrossRef]

Wang, R. T.

A. C. Lind, R. T. Wang, J. M. Greenberg, Appl. Opt. 4, 1555 (1965).
[CrossRef]

J. M. Greenberg, A. C. Lind, R. T. Wang, L. F. Libelo, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).

Yeh, C. W. H.

C. W. H. Yeh, J. Math. Phys. 42, 68 (1963).

Ann. Phys.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

F. Möglich, Ann. Phys. 83, 609 (1927).
[CrossRef]

Appl. Opt.

Can. J. Phys.

J. R. Wait, Can. J. Phys. 33, 189 (1955).
[CrossRef]

IRE Trans. Antennas Propag.

K. M. Siegel, F. V. Schultz, B. H. Gere, F. B. Sleator, IRE Trans. Antennas Propag. AP-4, 266 (1956).
[CrossRef]

J. Math. Phys.

C. W. H. Yeh, J. Math. Phys. 42, 68 (1963).

C. J. Bouwkamp, J. Math. Phys. 26, 79 (1947).

Phil. Mag.

Lord Rayleigh, Phil. Mag. 36, 365 (1918).

Radio Sci.

J. R. Wait, Radio Sci. 1, 475 (1966).

Other

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

R. J. T. Bell, Coordinate Solid Geometry (Macmillan, London, 1948).

J. M. Greenberg, A. C. Lind, R. T. Wang, L. F. Libelo, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).

F. V. Schultz, “Scattering by a Prolate Spheroid,” UMM-42, Univ. of Mich., 1950.

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

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).

C. Flammer, Spheroidal Wave Functions (Stanford U.P., Stanford, Calif., 1957).

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 (20)

Fig. 1
Fig. 1

Coordinate system for scattering by a prolate spheroid with semifocal distance l. The prolate spheroidal coordinates are η,ξ,ϕ. The z axis is chosen as the axis of revolution. The incident plane contains the incident direction and the z axis. The x axis is in the incident plane; for the TM mode, E is in the incident plane; for the TE mode, H is in the incident plane. The incident angle ζ is the angle in the incident plane between the incident direction and the z axis.

Fig. 2
Fig. 2

The extinction and scattering cross sections at ζ = 0° computed by truncating infinite series to finite ones including only first N terms, as a function of the termination number for the truncation N. The cross sections, normalized by those with the largest N values, for the prolate spheroids of c = 1 and 5, and of a/b = 2 (left half) and a/b = 10 (right half) are shown by solid lines for extinction and by dashed lines for scattering.

Fig. 3
Fig. 3

Angular distribution of the intensity functions i1 (solid lines) and i2 (dashed lines) at ζ = 0° for the prolate spheroids of a/b = 2 and c = 1 up to 7. The ordinate is on a logarithmic scale (1 division = a factor 10), and the abscissa is linear in the scattering angle. The values for the forescattering and backscattering are indicated in the margin.

Fig. 4
Fig. 4

Same as Fig. 3, but for the prolate spheroid of a/b = 5.

Fig. 5
Fig. 5

Same as Fig. 3, but for the oblate spheroid of a/b = 2.

Fig. 6
Fig. 6

Angular distribution of the intensity functions i11 (solid lines) and i12 (dashed lines) for the prolate spheroid of c = 1 and a/b = 2, and for ζ = 45°. The figure shows, as a function of the zenith angle θ in the circular diagram, the distribution patterns in the three scattering planes through the z axis: one parallel to the incident plane (ϕ = 0° or 180°), one inclining from it by an angle 45° (ϕ = 45° or 225°), and one normal to it (ϕ = 90° or 270°).

Fig. 7
Fig. 7

Same as Fig. 6, but for ζ = 90°

Fig. 8
Fig. 8

Angular distribution of the intensity functions i22 (solid lines) and i21 (broken lines) for the prolate spheroid of c = 1, a/b = 2, and for ζ = 45° in the three azimuth planes.

Fig. 9
Fig. 9

Same as Fig. 8, but for ζ = 90°.

Fig. 10
Fig. 10

Same as Fig. 6, but for the oblate spheroid of c = 1 and a/b = 2.

Fig. 11
Fig. 11

Same as Fig. 8, but for the oblate spheroid of c = 1 and a/b = 2.

Fig. 12
Fig. 12

Angular distribution of the intensity function 1/2(i11 + i12 + i21 + i22) for the prolate spheroid of c = 1 and a/b = 2, and for ζ = 45°. The solid, short broken, and long broken lines indicate the distribution patterns in plane (ϕ = 0° or 180°), plane 2 (ϕ = 45° or 225°), and plane 3(ϕ = 90° or 270°), respectively.

Fig. 13
Fig. 13

Same as Fig. 12, but for the oblate spheroid of c = 1 and a/b = 2.

Fig. 14
Fig. 14

Angular distribution of the intensity function 1/2(i11 + i12 + i21 + i22) for the prolate spheroid of c = 5 and a/b = 2, and ζ = 45°. The solid, short broken, and long broken lines give logarithmic values in plane 1, plane 2, and plane 3, respectively, as a function of θ.

Fig. 15
Fig. 15

Same as Fig. 14 but for ζ = 90°.

Fig. 16
Fig. 16

Same as Fig. 14, but for the oblate spheroid of c = 5 and a/b = 2.

Fig. 17
Fig. 17

Same as Fig. 15, but for the oblate spheroid of c = 5 and a/b = 2.

Fig. 18
Fig. 18

Scattering efficiency factors Qsca at ζ = 0° as a function of the parameter c for the prolate spheroids of a/b = 2, 5, and 10 (solid lines) and for the oblate spheroid of a/b = 2(long broken lines). The curve for the sphere is also shown by a short broken line.

Fig. 19
Fig. 19

Scattering efficiency factors Q1,sca (solid lines) and Q2,sca (long broken lines) as a function of c for the prolate spheroid of a/b = 2 and for ζ = 45° and 90°. Curve for ζ = 0°, or Qsca, is also shown by a short broken line.

Fig. 20
Fig. 20

Angular distribution of the intensity function 1/2(i1 + i2) for the prolate spheroid of c = 5 and a/b = 2 (dashed line) and for the spheres of the size parameter x = 4.0, 3.6, and 3.4 (solid lines).

Equations (162)

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

× E = i k 0 H , × H = i k 0 H 2 E , }
2 E + k 2 E = 0 , 2 H + k 2 H = 0 , }
k 0 = ω / c = 2 π / λ 0 ,
H = μ + i ( 4 π σ μ / ω ) ,
k = k 0 H .
x = l ( 1 η 2 ) 1 / 2 ( ξ 2 1 ) 1 / 2 cos ϕ , y = l ( 1 η 2 ) 1 / 2 ( ξ 2 1 ) 1 / 2 sin ϕ , z = l η ξ ,
1 η 1 , 1 ξ < , 0 ϕ 2 π
1 η 1 , 0 ξ < , 0 ϕ 2 π
e = 1 / ξ 0
e = 1 / ( ξ 0 2 + 1 ) 1 / 2
l ξ r , η cos θ , as ξ ,
2 ψ + k 2 ψ = 0
d d η [ ( 1 η 2 ) d S mn ( η ) d η ] + ( λ mn c 2 η 2 m 2 1 η 2 ) S mn ( η ) = 0 ,
c = l · k ,
S m n ( c ; η ) = r = 0,1 d r m n ( c ) P m + r m ( η ) ,
S m n ( i c ; η ) = r = 0,1 d r m n ( i c ) P m + r m ( η ) ,
1 + 1 S m n ( η ) S m n ( η ) d η = { 0 ( n n ) Λ m n ( n = n ) ,
Λ m n = r = 0,1 2 · ( r + 2 m ) ! ( 2 r + 2 m + 1 ) · r ! ( d m n r ) 2 .
d d ξ [ ( ξ 2 1 ) d R mn ( ξ ) d ξ ] [ λ mn c 2 ξ 2 ± m 2 ( ξ 2 1 ) ] R mn ( ξ ) = 0 ,
R mn ( 1 ) 1 c ξ cos ( c ξ n + 1 2 π ) ,
R mn ( 2 ) 1 c ξ sin ( c ξ n + 1 2 π ) ,
R mn ( 3 ) 1 c ξ exp [ i ( c ξ n + 1 2 π ) ] ,
R mn ( 4 ) 1 c ξ exp [ i ( c ξ n + 1 2 π ) ] ,
R mn ( j ) ( c ; ξ ) = { 1 / [ r = 0,1 ( r + 2 m ) ! r ! d r mn ( c ) ] } [ ξ 2 1 ξ 2 ] m / 2 · r = 0,1 i r + n m d r mn ( c ) ( r + 2 m ) ! r ! z m + r ( j ) ( c ξ )
R mn ( j ) ( i c ; i ξ ) = { 1 / [ r = 0,1 ( r + 2 m ) ! r ! d r mn ( i c ) ] } × [ ξ 2 + 1 ξ 2 ] m / 2 r = 0,1 i r + n m d r mn ( i c ) ( r + 2 m ) ! r ! z m + r ( j ) ( c ξ ) ,
c i c , ξ i ξ ,
ψ e o mn ( j ) ( c ; η , ξ , ϕ ) = S mn ( c ; η ) R mn ( j ) ( c ; ξ ) cos sin m ϕ ,
ψ e o mn ( j ) ( i c ; η , i ξ , ϕ ) = S mn ( i c ; η ) R mn ( j ) ( i c ; ξ ) cos sin m ϕ ,
M mn = × ( a · ψ mn ) ,
N mn = k 1 · × M mn ,
M mn = k 1 · × N mn .
M e o mn r and N e o mn r ,
M e o mn , η r ( 3 ) ( i ) n + 1 · m S mn ( cos θ ) sin θ 1 k r e i k r sin ( 1 ) cos m ϕ ,
M e o mn , ϕ r ( 3 ) ( i ) n + 1 d S mn ( cos θ ) d θ 1 k r e i k r cos sin m ϕ ,
N e o mn , η r ( 3 ) ( i ) n d S mn ( cos θ ) d θ 1 k r e i k r cos sin m ϕ ,
N e o mn , ϕ r ( 3 ) ( i ) n · m S mn ( cos θ ) sin θ 1 k r e i k r sin ( 1 ) cos m ϕ .
M e o mn r ( 1 ) ( c ; η , ξ , ϕ ) and N e o mn r ( 1 ) ( c ; η , ξ , ϕ )
e y exp [ i k ˙ ( x · sin ζ + z · cos ζ ) ] = m , n i n [ g mn ( ζ ) M e mn r ( 1 ) ( c ; η , ξ , ϕ ) + i f mn ( ζ ) N o mn r ( 1 ) ( c ; η , ξ , ϕ ) ] ,
f mn ( ζ ) = 4 m Λ mn r = 0,1 d r mn ( r + m ) ( r + m + 1 ) P m + r m ( cos ζ ) sin ζ ,
g mn ( ζ ) = 2 ( 2 δ 0 , m ) Λ mn r = 0,1 d r mn ( r + m ) ( r + m + 1 ) d P m + r m ( cos ζ ) d ζ ,
m , n
m = 0 n = m
f 1 n ( 0 ) = g 1 n ( 0 ) = 2 Λ 1 n 1 r = 0,1 d r 1 n ,
E ( i ) = e y · exp [ i k ( I ) ( x sin ζ + z · cos ζ ) ] ,
H ( i ) = ( cos ζ · e x sin ζ · e z ) H ( I ) exp [ i k ( I ) ( x sin ζ + z · cos ζ ) ] = ( i k 0 ) 1 · × E ( i ) ,
E ( i ) = m , n i n [ g m n ( ζ ) M e m n r ( 1 ) + i f m n ( ζ ) N omn r ( 1 ) ] , H ( i ) = H ( I ) m , n i n [ f m n ( ζ ) M omn r ( 1 ) i g m n ( ζ ) N e m n r ( 1 ) ] . }
E ( i ) = m , n i n [ f mn ( ζ ) M omn r ( 1 ) i g mn ( ζ ) N emn r ( 1 ) ] , H ( i ) = H ( I ) m , n i n [ g mn ( ζ ) M emn r ( 1 ) + i f m n ( ζ ) N omn r ( 1 ) ] . }
E ( s ) = m , n i n ( β 1 , mn M emn r ( 3 ) + i α 1 , mn N omn r ( 3 ) ) H ( s ) = H ( I ) m , n i n ( α 1 , mn M omn r ( 3 ) i β 1 , mn N emn r ( 3 ) } , ( ξ > ξ 0 ) ;
E ( t ) = m , n i n ( δ 1 , m n M e m n r ( 1 ) + i γ 1 , m n N o m n r ( 1 ) ) H ( t ) = H ( I I ) m , n i n ( γ 1 , m n M o m n r ( 1 ) i δ 1 , m n N e m n r ( 1 ) ) } , ( ξ < ξ 0 ) ;
E ( s ) = m , n i n ( α 2 , mn M omn r ( 3 ) i β 2 , mn N emn r ( 3 ) ) H ( s ) = H ( I ) m , n i n ( β 2 , mn M emn r ( 3 ) + i α 2 , mn N omn r ( 3 ) ) } , ( ξ > ξ 0 ) ;
E ( t ) = m , n i n ( γ 2 , mn M omn r ( 1 ) i δ 2 , mn N emn r ( 1 ) ) H ( t ) = H ( II ) m , n i n ( δ 2 , mn M emn r ( 1 ) + i γ 2 , mn N omn r ( 1 ) ) } , ( ξ < ξ 0 ) ;
E η ( i ) + E η ( s ) = E η ( t ) , E ϕ ( i ) + E ϕ ( s ) = E ϕ ( t ) H η ( i ) + H η ( s ) = H η ( i ) , H ϕ ( i ) + H ϕ ( s ) = H ϕ ( t ) } at ξ = ξ 0 .
( a ) ( 1 η 2 ) 1 / 2 S mn ( η ) = t = 0 A t mn · P m 1 + t m 1 ( η ) ,
( b ) ( 1 η 2 ) 1 / 2 S mn ( η ) = t = 0 B t mn · P m 1 + t m 1 ( η ) ,
( c ) η ( 1 η 2 ) 1 / 2 S mn ( η ) = t = 0 C t mn · P m 1 + t m 1 ( η ) ,
( d ) η ( 1 η 2 ) 1 / 2 S mn ( η ) = t = 0 D t mn · P m 1 + t m 1 ( η ) ,
( e ) ( 1 η 2 ) 3 / 2 S m n ( η ) = t = 0 E t m n · P m 1 + t m 1 ( η ) ,
( f ) η ( 1 η 2 ) 3 / 2 S m n ( η ) = t = 0 F t m n · P m 1 + t m 1 ( η ) ,
( g ) ( 1 η 2 ) 1 / 2 d S m n ( η ) d η = t = 0 G t m n · P m 1 + t m 1 ( η ) ,
( h ) η ( 1 η 2 ) 1 / 2 d S m n ( η ) d η = t = 0 H t m n · P m 1 + t m 1 ( η ) ,
( i ) ( 1 η 2 ) 3 / 2 d S m n ( η ) d η = t = 0 I t m n · P m 1 + t m 1 ( η ) .
( c ) η ( 1 η 2 ) 1 / 2 S 0 n ( η ) = t = 0 C t 0 n · P 1 + t 1 ( η ) ,
( f ) η ( 1 η 2 ) 3 / 2 S 0 n ( η ) = t = 0 F t 0 n · P 1 + t 1 ( η ) ,
( g ) ( 1 η 2 ) 1 / 2 d S 0 n ( η ) d η = t = 0 G t 0 n · P 1 + t 1 ( η ) ,
( i ) ( 1 η 2 ) 3 / 2 d S 0 n ( η ) d η = t = 0 I t 0 n · P 1 + t 1 ( η ) .
A t m n = N m 1 , m 1 + t 1 · r = 0,1 d r m n × 1 + 1 ( 1 η 2 ) 1 / 2 P m + r m ( η ) P m 1 + t m 1 ( η ) d η ,
= { 0 , ( n m ) + t = odd , ( t + 2 m 1 ) ( t + 2 m ) ( 2 t + 2 m + 1 ) d t m n t ( t 1 ) ( 2 t + 2 m + 3 ) d t 2 m n , ( n m ) + t = even , B t m n = N m 1 , m 1 + t 1 · r = 0,1 d r m n × 1 + 1 ( 1 η 2 ) 1 / 2 P m + r m ( η ) P m 1 + t m 1 ( η ) d η ,
= { 0 , ( n m ) + t = odd , ( 2 t + 2 m 1 ) r = t d r m n , ( n m ) + t = even , C t m n = N m 1 , m 1 + t 1 · r = 0,1 d r m n 1 + 1 η ( 1 η 2 ) 1 / 2 P m + r m ( η ) P m 1 + t m 1 ( η ) d η ,
= { 0 , ( n m ) + t = even , ( t + 2 m 1 ) ( t + 2 m ) ( 2 t + 2 m + 1 ) [ ( t + 2 m + 1 ) ( 2 t + 2 m + 3 ) d t + 1 m n + t ( 2 t + 2 m 1 ) d t 1 m n ] t ( t 1 ) ( 2 t + 2 m 3 ) [ ( t + 2 m 1 ) ( 2 t + 2 m 1 ) d t 1 m n + ( t 2 ) ( 2 t + 2 m 5 ) d t 3 m n ] , ( n m ) + t = odd , D t m n = N m 1 , m 1 + t 1 · r = 0,1 d r m n × 1 + 1 η ( 1 η 2 ) 1 / 2 P m + r m ( η ) P m 1 + t m 1 ( η ) d η ,
= { 0 , ( n m ) + t = even , t d t 1 m n + ( 2 t + 2 m 1 ) r = t + 1 d r m n , ( n m ) + t = odd , E t m n = N m 1 , m 1 + t 1 r = 0,1 d r m n × 1 + 1 η ( 1 η 2 ) 3 / 2 P m + r m ( η ) P m 1 + t m 1 ( η ) d η ,
= { 0 , ( n m ) + t = odd , ( t + 2 m 1 ) ( t + 2 m ) ( t + 2 m + 2 ) ( t + 2 m + 1 ) ( 2 t + 2 m + 1 ) ( 2 t + 2 m + 3 ) × [ d t m n ( 2 t + 2 m + 1 ) d t + 2 m n ( 2 t + 2 m + 5 ) ] 2 t ( t 1 ) ( t + 2 m ) ( t + 2 m 1 ) ( 2 t + 2 m 3 ) ( 2 t + 2 m + 1 ) × [ d t 2 m n ( 2 t + 2 m 3 ) d t m n ( 2 t + 2 m + 1 ) ] + t ( t 1 ) ( t 2 ) ( t 3 ) ( 2 t + 2 m 3 ) ( 2 t + 2 m 5 ) × [ d t 4 m n ( 2 t + 2 m 7 ) d t 2 m n ( 2 t + 2 m 3 ) ] , ( n m ) + t = even , F t m n = N m 1 , m 1 + t 1 · r = 0,1 d r m n × 1 + 1 η ( 1 η 2 ) 3 / 2 P m + r m ( η ) P m 1 + t m 1 ( η ) d η ,
= { 0 , ( n m ) + t = even , ( t + 2 m 1 ) ( t + 2 m ) ( t + 2 m + 1 ) × ( t + 2 m + 2 ) ( t + 2 m + 3 ) ( 2 t + 2 m + 1 ) ( 2 t + 2 m + 3 ) ( 2 t + 2 m + 5 ) × [ d t + 1 m n ( 2 t + 2 m + 3 ) d t + 3 m n ( 2 t + 2 m + 7 ) ] t ( t 2 m ) ( t + 2 m + 1 ) ( t + 2 m ) ( t + 2 m 1 ) ( 2 t + 2 m 3 ) ( 2 t + 2 m + 1 ) ( 2 t + 2 m + 3 ) × [ d t 1 m n ( 2 t + 2 m 1 ) d t + 1 m n ( 2 t + 2 m + 3 ) ] t ( t 1 ) ( t 2 ) ( t + 2 m 1 ) ( t + 4 m 1 ) ( 2 t + 2 m 5 ) ( 2 t + 2 m 3 ) ( 2 t + 2 m + 1 ) × [ d t 3 m n ( 2 t + 2 m 5 ) d t 1 m n ( 2 t + 2 m 1 ) ] + t ( t 1 ) ( t 2 ) ( t 3 ) ( t 4 ) ( 2 t + 2 m 3 ) ( 2 t + 2 m 5 ) ( 2 t + 2 m 7 ) × [ d t 5 m n ( 2 t + 2 m 9 ) d t 3 m n ( 2 t + 2 m 5 ) ] , ( n m ) + t = odd , G t m n = N m 1 , m 1 + t 1 r = 0,1 d r m n × 1 + 1 ( 1 η 2 ) 1 / 2 d P m + r m ( η ) d η P m 1 + t m 1 ( η ) d η ,
= { 0 , ( n m ) + t = even , t ( t + m 1 ) d t 1 m n + m ( 2 t + 2 m 1 ) r = t + 1 d r m n , ( n m ) + t = odd , H t m n = N m 1 , m 1 + t 1 r = 0,1 d r m n 1 + 1 η ( 1 η 2 ) 1 / 2 d P m + r m ( η ) d η P m 1 + t m 1 ( η ) d η ,
= { 0 , ( n m ) + t = odd , t ( t 1 ) ( t + 2 m 2 ) ( 2 t + 2 m 3 ) d t 2 m n t ( t 1 ) ( 2 t + 2 m + 1 ) + ( t + 2 m ) ( t + 2 m 1 ) 2 ( 2 t + 2 m + 1 ) d t m n + m ( 2 t + 2 m 1 ) r = t + 2 d r m n , ( n m ) + t = even , I t m n = N m 1 , m 1 + t 1 r = 0,1 d r m n 1 + 1 ( 1 η 2 ) 3 / 2 d P m + r m ( η ) d η P m 1 + t m 1 ( η ) d η ,
= { 0 , ( n m ) + t = even , ( t + 2 m ) ( t + 2 m 1 ) ( 2 t + 2 m + 1 ) [ ( t + m + 2 ) ( t + 2 m + 1 ) ( 2 t + 2 m + 3 ) d t + 1 m n t ( m + t 1 ) ( 2 t + 2 m 1 ) d t 1 m n ] t ( t 1 ) ( 2 t + 2 m 3 [ ( t + m ) ( t + 2 m 1 ) ( 2 t + 2 m 1 ) d t 1 m n ( t 2 ) ( t + m 3 ) ( 2 t + 2 m 5 ) d t 3 m n ] , ( n m ) + t = odd ,
N m 1 , m 1 + t = 1 + 1 [ P m 1 + t m 1 ( η ) ] 2 d η , = 2 · ( t + 2 m 2 ) ! ( 2 t + 2 m 1 ) · t ! .
C t 0 n = N 1,1 + t 1 r = 0,1 d r 0 n 1 + 1 η ( 1 η 2 ) 1 / 2 P r 0 ( η ) P 1 + t 1 ( η ) d η ,
= { 0 , n + t = even , 1 ( 2 t + 1 ) [ ( t + 1 ) ( 2 t + 3 ) d t + 1 0 n + t ( 2 t 1 ) d t 1 0 n ] 1 ( 2 t + 5 ) [ ( t + 3 ) ( 2 t + 7 ) d t + 3 0 n + ( t + 2 ) ( 2 t + 3 ) d t + 1 0 n ] , n + t = odd , F t 0 n = N 1,1 + t 1 r = 0,1 d r 0 n 1 + 1 η ( 1 η 2 ) 3 / 2 P r 0 ( η ) P 1 + t 1 ( η ) d η ,
= { 0 , n + t = even , ( t + 3 ) ( t + 4 ) ( t + 5 ) ( 2 t + 5 ) ( 2 t + 7 ) [ d t + 1 0 n ( 2 t + 3 ) ( 2 t + 5 ) 2 d t + 3 0 n ( 2 t + 5 ) ( 2 t + 9 ) + d t + 5 0 n ( 2 t + 9 ) ( 2 t + 11 ) ] + 3 t ( t + 3 ) ( 2 t + 1 ) ( 2 t + 5 ) [ d t 1 0 n ( 2 t 1 ) ( 2 t + 1 ) 2 d t + 1 0 n ( 2 t + 1 ) ( 2 t + 5 ) + d t + 3 0 n ( 2 t + 5 ) ( 2 t + 7 ) ] t ( t 1 ) ( t 2 ) ( 2 t 1 ) ( 2 t + 1 ) [ d t 3 0 n ( 2 t 5 ) ( 2 t 3 ) 2 d t 1 0 n ( 2 t 3 ) ( 2 t + 1 ) + d t + 1 0 n ( 2 t + 1 ) ( 2 t + 3 ) ] , n + t = odd , G t 0 n = N 1,1 + t 1 r = 0,1 d r 0 n 1 + 1 ( 1 η 2 ) 1 / 2 d P r 0 ( η ) d η P 1 + t 1 ( η ) d η ,
= { 0 , n + t = even , d t + 1 0 n , n + t = odd , I t 0 n = N 1,1 + t 1 r = 0,1 d r 0 n 1 + 1 ( 1 η 2 ) 3 / 2 d P r 0 ( η ) d η P 1 + t 1 ( η ) d η ,
= { 0 , n + t = even , 1 2 t + 1 [ ( t + 1 ) ( t + 2 ) ( 2 t + 3 ) d t + 1 0 n t ( t 1 ) ( 2 t 1 ) d t 1 0 n ] 1 2 t + 5 [ ( t + 3 ) ( t + 4 ) ( 2 t + 7 ) d t + 3 0 n ( t + 1 ) ( t + 2 ) ( 2 t + 3 ) d t + 1 0 n ] , ( n + t ) = odd ,
N 1,1 + t = 1 + 1 [ P 1 + t 1 ( η ) 2 ] d η , = 2 ( t + 1 ) ( t + 2 ) ( 2 t + 3 ) ,
U m n ( j ) t ( c ( h ) ) = m ξ 0 R m n ( j ) ( c ( h ) ; ξ 0 ) [ ( ξ 0 2 1 ) 2 B t m n ( c ( h ) ) + 2 ( ξ 0 2 1 ) A t m n ( c ( h ) ) + E t m n ( c ( h ) ) ] ,
V m n ( j ) , t ( c ( h ) ) = i c ( h ) { m 2 ( ξ 0 2 1 ) R m n ( j ) ( c ( h ) ; ξ 0 ) [ ( ξ 0 2 1 ) 2 D t m n ( c ( h ) ) + 2 ( ξ 0 2 1 ) C t m n ( c ( h ) ) + F t m n ( c ( h ) ) ] R m n ( j ) ( c ( h ) ; ξ 0 ) [ λ m n ( c ( h ) ) ( c ( h ) ξ 0 ) 2 + m 2 ξ 0 2 1 ] × [ ( ξ 0 2 1 ) C t m n ( c ( h ) ) + F t m n ( c ( h ) ) ] + ξ 0 ( ξ 0 2 1 ) × [ d R m n ( j ) ( c ( h ) ; ξ ) d ξ ] ξ 0 [ 2 C t m n ( c ( h ) ) + ( ξ 0 2 1 ) G t m n ( c ( h ) ) + I t m n ( c ( h ) ) ] + R m n ( j ) ( c ( h ) ; ξ 0 ) [ ( ξ 0 2 1 ) 2 G t m n ( c ( h ) ) + ( 3 ξ 0 2 1 ) I t m n ( c ( h ) ) ] } ,
X m n ( j ) , t ( c ( h ) ) = ξ 0 R m n ( j ) ( c ( h ) ; ξ 0 ) G t m n ( c ( h ) ) [ d R m n ( j ) ( c ( h ) ; ξ ) d ξ ] ξ 0 · C t m n ( c ( h ) ) ,
Y m n ( j ) , t ( c ( h ) ) = 1 c ( h ) m ( ( ξ 0 2 1 ) 1 R m n ( j ) ( c ( h ) ; ξ 0 ) [ A t m n ( c ( h ) ) + H t m n ( c ( h ) ) ] + { R m n ( j ) ( c ( h ) ; ξ 0 ) + ξ 0 [ d R m n ( j ) ( c ( h ) ; ξ 0 ) / d ξ ] ξ 0 } B t m n ( c ( h ) ) ) ,
U m n ( j ) , t ( i c ( h ) ) = m ξ 0 R m n ( j ) ( i c ( h ) ; i ξ 0 ) [ ( ξ 0 2 + 1 ) 2 × B t m n ( i c ( h ) ) 2 ( ξ 0 2 + 1 ) A t m n ( i c ( h ) ) + E t m n ( i c ( h ) ) ] ,
V m n ( j ) , t ( i c ( h ) ) = i c ( h ) { m 2 ( ξ 0 2 + 1 ) R m n ( j ) ( i c ( h ) ; i ξ 0 ) × [ ( ξ 0 2 + 1 ) 2 D t m n ( i c ( h ) ) 2 ( ξ 0 2 + 1 ) C t m n ( i c ( h ) ) + F t m n ( i c ( h ) ) ] + R m n ( j ) ( i c ( h ) ; i ξ 0 ) [ λ m n ( i c ( h ) ) ( c ( h ) ξ 0 ) 2 m 2 ξ 0 2 + 1 ] [ ( ξ 0 2 + 1 ) C t m n ( i c ( h ) ) F t m n ( i c ( h ) ) ] + ξ 0 ( ξ 0 2 + 1 ) [ d R m n ( j ) ( i c ( h ) , i ξ ) d ξ ] ξ 0 × [ 2 C t m n ( i c ( h ) ) + ( ξ 0 2 + 1 ) G t m n ( i c ( h ) ) I t m n ( i c ( h ) ) ] + R m n ( j ) ( i c ( h ) ; i ξ 0 ) [ ( ξ 0 2 + 1 ) 2 × G t m n ( i c ( h ) ) ( 3 ξ 0 2 + 1 ) I t m n ( i c ( h ) ) ] } ,
X m n ( j ) , t ( i c ( h ) ) = ξ 0 R m n ( j ) ( i c ( h ) ; i ξ 0 ) G t m n ( i c ( h ) ) + [ d R m n ( j ) ( i c ( h ) ; i ξ ) d ξ ] ξ 0 C t m n ( i c ( h ) ) ,
Y m n ( j ) , t ( i c ( h ) ) = i c ( h ) m ( ( ξ 0 2 + 1 ) 1 R m n ( j ) ( i c ( h ) ; i ξ 0 ) [ A t m n ( i c ( h ) ) + H t m n ( i c ( h ) ) ] + { R m n ( j ) × ( i c ( h ) ; i ξ 0 ) + ξ 0 [ d R m n ( j ) ( i c ( h ) , i ξ ) d ξ ] ξ 0 } × B t m n ( i c ( h ) ) ) ,
U 0 n ( j ) , t = Y 0 n ( j ) , t = 0 ,
V 0 n ( j ) , t = i c ( h ) { R 0 n ( j ) [ λ 0 n ( c ( h ) ξ 0 ) 2 ] [ ( ξ 0 2 1 ) C t 0 n ± F t 0 n ] + ξ 0 ( ξ 0 2 1 ) ( d R 0 n ( j ) d ξ ) ξ 0 [ ± 2 C t 0 n + ( ξ 0 2 1 ) G t 0 n ± I t 0 n ] + R 0 n ( j ) [ ( ξ 0 2 1 ) 2 G t 0 n ± ( 3 ξ 0 2 1 ) I t 0 n ] } ,
X 0 n ( j ) , t = ξ 0 R 0 n ( j ) · G t 0 n [ d R 0 n ( j ) d ξ ] ξ 0 C t 0 n .
E η : n = m i n [ V m n ( 3 ) , t ( c ( I ) ) · α 1 , m n + U m n ( 3 ) , t ( c ( I ) ) · β 1 , m n V m n ( 1 ) , t ( c ( I I ) ) · γ 1 , m n U m n ( 1 ) , t ( c ( II ) ) · δ 1 , m n ] = n = m i n [ f m n ( ζ ) · V m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) · U m n ( 1 ) , t ( c ( I ) ) ] ,
E ϕ : n = m i n [ Y m n ( 3 ) , t ( c ( I ) ) · α 1 , m n + X m n ( 3 ) , t ( c ( I ) ) · β 1 , m n Y m n ( 1 ) , t ( c ( II ) ) · γ 1 , m n X m n ( 1 ) , t ( c ( II ) ) · δ 1 , m n ] = n = m i n [ f m n ( ζ ) · Y m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) · X m n ( 1 ) , t ( c ( I ) ) ] ,
H η : n = i n [ U m n ( 3 ) , t ( c ( I ) ) · α 1 , m n + V m n ( 3 ) , t ( c ( I ) ) · β 1 , m n H U m n ( 1 ) , t ( c ( II ) ) · γ 1 , m n H V m n ( 1 ) t ( c ( II ) ) · δ 1 , m n ] = n = m i n [ f m n ( ζ ) · U m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) · V m n ( 1 ) , t ( c ( I ) ) ] ,
H ϕ : n = m i n [ X m n ( 3 ) , t ( c ( I ) ) · α 1 , m n + Y m n ( 3 ) , t ( c ( I ) ) · β 1 , m n H X m n ( 1 ) , t ( c ( II ) ) · γ m n H Y m n ( 1 ) , t ( c ( II ) ) · δ 1 , m n ] = n = m i n [ f m n ( ζ ) · X m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) Y m n ( 1 ) , t ( c ( I ) ) ] , ( m = 0,1,2 , ; t = 0,1,2 ,… )
E η : n = m i n [ U m n ( 3 ) , t ( c ( I ) ) · α 2 , m n + V m n ( 3 ) , t ( c ( I ) ) · β 2 , m n U m n ( 1 ) , t ( c ( II ) ) γ 2 , m n V m n ( 1 ) , t ( c ( II ) ) δ 2 , m n ] = n = m i n [ f m n ( ζ ) U m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) V m n ( 1 ) , t ( c ( I ) ) ] ,
E ϕ : n = m i n [ X m n ( 3 ) , t ( c ( I ) ) · α 2 , m n + Y m n ( 3 ) , t ( c ( I ) ) · β 2 , m n X m n ( 1 ) , t ( c ( II ) ) · γ 2 , m n Y m n ( 1 ) , t ( c ( II ) ) · δ 2 , m n ] = n = m i n [ f m n ( ζ ) · X m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) Y m n ( 1 ) , t ( c ( I ) ) ] ,
H η : n = m i n [ V m n ( 3 ) , t ( c ( I ) ) · α 2 , m n + U m n ( 3 ) , t ( c ( I ) ) · β 2 , m n H V m n ( 1 ) , t ( c ( I I ) ) · γ 2 , m n H U m n ( 1 ) , t ( c ( I I ) ) · δ 2 , m n ] = n = m i n [ f m n ( ζ ) V m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) · U m n ( 1 ) , t ( c ( I ) ) ] ,
H ϕ : n = m i n [ Y m n ( 3 ) , t ( c ( I ) ) · α 2 , m n + X m n ( 3 ) , t ( c ( I ) ) β 2 , m n H Y m n ( 1 ) , t ( c ( II ) ) γ 2 , m n H X m n ( 1 ) , t ( c ( II ) ) · δ 2 , m n ] = n = m i n [ f m n ( ζ ) Y m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) · X m n ( 1 ) , t ( c ( I ) ) ] , ( m = 0,1,2 , ; t = 0,1,2 , )
H = H ( II ) / H ( I ) .
U m n ( j ) , t = Y m n ( j ) , t = 0 , for ( n m ) + t = odd ,
V m n ( j ) , t = X m n ( j ) , t = 0 , for ( n m ) + t = even ,
E 1 , η ( s ) = H 1 , ϕ ( s ) / H ( I ) = i λ ( I ) 2 π r exp i 2 π r / λ ( I ) × m , n [ α 1 , m n d S m n ( cos θ ) d θ + β 1 , m n · m S m n ( cos θ ) sin θ ] sin m ϕ ,
E 1 , ϕ ( s ) = H 1 , η ( s ) / H ( I ) = i λ ( I ) 2 π r exp i 2 π r / λ ( I ) × m , n [ α 1 , m n · m S m n ( cos θ ) sin θ + β 1 , m n d S m n ( cos θ ) d θ ] cos m ϕ ,
E 2 , η ( s ) = H 2 , ϕ ( s ) / H ( I ) = i λ ( I ) 2 π r exp i 2 π r / λ ( I ) × m , n [ α 2 , m n · m S m n ( cos θ ) sin θ + β 2 , m n d S m n ( cos θ ) d θ ] cos m ϕ ,
E 2 , ϕ ( s ) = H 2 , η ( s ) / H ( I ) = i λ ( I ) 2 π r exp i 2 π r / λ ( I ) × m , n [ α 2 , m n d S m n ( cos θ ) d θ + β 2 , m n m S m n ( cos θ ) sin θ ] sin m ϕ ,
T 11 ( θ , ϕ ) = m , n [ α 1 , m n · σ m n ( θ ) + β 1 , m n · χ m n ( θ ) ] cos m ϕ ,
T 12 ( θ , ϕ ) = m , n [ α 1 , m n · χ m n ( θ ) + β 1 , m n · σ m n ( θ ) ] sin m ϕ ,
T 21 ( θ , ϕ ) = m , n [ α 2 , m n · χ m n ( θ ) + β 2 , m n · σ m n ( θ ) ] sin m ϕ ,
T 22 ( θ , ϕ ) = m , n [ α 2 , m n · σ ( θ ) + β 2 , m n · χ m n ( θ ) ] cos m ϕ ,
σ m n ( θ ) = m [ S m n ( cos θ ) / sin θ ] ,
χ m n ( θ ) = ( d / d θ ) [ S m n ( cos θ ) ] ,
σ m n ( 0 ) = χ m n ( 0 ) = { 0 , ( m 1 ) , 1 2 r = 0,1 ( r + 1 ) ( r + 2 ) d r 1 n , ( m = 1 ) .
I 1 , ϕ = E 1 , ϕ ( s ) · E 1 , ϕ ( s ) * = ( λ 2 ( I ) / 4 π 2 r 2 ) i 11 ( θ , ϕ ) ,
I 1 , η = E 1 , η ( s ) · E 1 , η ( s ) * = ( λ 2 ( I ) / 4 π 2 r 2 ) i 12 ( θ , ϕ ) .
I 2 , ϕ = E 2 , ϕ ( s ) · E 2 , ϕ ( s ) * = ( λ 2 ( I ) / 4 π 2 r 2 ) i 21 ( θ , ϕ ) ,
I 2 , η = E 2 , η ( s ) · E 2 , η ( s ) * = ( λ 2 ( I ) / 4 π 2 r 2 ) i 22 ( θ , ϕ ) ,
i 11 ( θ , ϕ ) = | T 11 ( θ , ϕ ) | 2 ,
i 12 ( θ , ϕ ) = | T 12 ( θ , ϕ ) | 2 ,
i 21 ( θ , ϕ ) = | T 21 ( θ , ϕ ) | 2 ,
i 22 ( θ , ϕ ) = | T 22 ( θ , ϕ ) | 2 .
I ϕ = ( λ 2 ( I ) / 4 π 2 r 2 ) · i 1 ( θ ) · sin 2 ϕ ,
I η = ( λ 2 ( I ) / 4 π 2 r 2 ) · i 1 ( θ ) · cos 2 ϕ ,
i 1 ( θ ) = | n = 1 [ α 1 n · χ 1 n ( θ ) + β 1 n · σ 1 n ( θ ) ] | 2 ,
i 2 ( θ ) = | n = 1 [ α 1 n · σ 1 n ( θ ) + β 1 n · χ 1 n ( θ ) ] | 2 ,
( s ) E 1 , ϕ | θ = ζ , ϕ = 0 = i λ ( I ) 2 π r exp i 2 π r / λ ( I ) · T 11 ( ζ , 0 ) , = i · λ ( I ) 2 π r exp i 2 π r / λ ( I ) · m , n [ α 1 , m n σ m n ( ζ ) + β 1 , m n χ m n ( ζ ) ] ;
C 1 , ext = λ 2 ( I ) π Re m , n [ α 1 , m n · σ m n ( ζ ) + β 1 , m n · χ m n ( ζ ) ] ,
C 2 , ext = λ 2 ( I ) π Re m , n [ α 2 , m n · σ m n ( ζ ) + β 2 , m n · χ m n ( ζ ) ] .
C ext = λ 2 ( I ) π · n = 1 [ r = 0,1 ( r + 1 ) ( r + 2 ) 2 d r 1 n ] × Re ( α 1 n + β 1 n ) .
C 1 , sca = 1 k ( I ) 2 [ i 11 ( θ , ϕ ) + i 12 ( θ , ϕ ) ] d Ω , = 1 k ( I ) 2 0 2 π 0 π [ | T 11 ( θ , ϕ ) | 2 + | T 12 ( θ , ϕ ) | 2 ] sin θ d θ d ϕ .
0 2 π sin m ϕ · sin m ϕ d ϕ = 0 2 π cos m ϕ · cos m ϕ d ϕ = { 0 , ( m m ) π , ( m = m )
0 π [ σ m n ( θ ) · χ m n ( θ ) + σ m n ( θ ) · χ m n ( θ ) ] sin θ d θ = 0 ,
C 1 , sca = λ 2 ( I ) 4 π m = 0 n = m n = m × n n m Re ( α 1 , m n · α 1 , m n * + β 1 , m n · β 1 , m n * ) ,
n n m = 0 π [ σ m n ( θ ) · σ m n ( θ ) + χ m n ( θ ) · χ m n ( θ ) ] sin θ d θ , = { 0 , | n n | = odd , r = 0,1 2 ( r + m ) ( r + m + 1 ) ( r + 2 m ) ! ( 2 r + 2 m + 1 ) r ! d r m n d r m n , | n n | = even .
C 2 , sca = λ 2 ( I ) 4 π m = 0 n = m n = m n n m Re ( α 2 , m n · α 2 , m n * · + β 2 , m n · β 2 , m n * ) .
C sca = λ 2 ( I ) 4 π n = 1 n = 1 Π n n 1 · Re ( α 1 n · α 1 n * + β 1 n · β 1 n * ) ,
Π n n 1 = { 0 , | n n | = odd , r = 0,1 2 ( r + 1 ) 2 ( r + 2 ) 2 ( 2 r + 3 ) d r 1 n d r 1 n , | n n | = even .
( 1 η 2 ) 1 / 2 S m n ( c ( h ) ; η ) = t = 0 A ¯ t m n ( c ( h ) ) · S m 1 , m 1 + t ( c ( I ) ; η ) ,
A ¯ t m n ( c ( h ) ) = Λ m 1 , m 1 + t 1 ( c ( I ) ) . 1 + 1 ( 1 η 2 ) 1 / 2 · S m n ( c ( I ) ; η ) · S m 1 , m 1 + t ( c ( I ) ; η ) d η ,
= { 0 , ( n m ) + t = odd , Λ m 1 , m 1 + t 1 ( c ( I ) ) λ = 0,1 d λ m 1 , m 1 + t ( c ( I ) ) ( 2 λ + 2 m 1 ) [ 2 ( λ + 2 m ) ! ( 2 λ + 2 m + 1 ) λ ! d λ m n ( c ( h ) ) 2 ( λ + 2 m 2 ) ! ( 2 λ + 2 m 3 ) ( λ 2 ) ! d λ 2 m n ( c ( h ) ) ] , ( n m ) + t = even ,
A ¯ t m n ( c ( h ) ) = Λ m 1 , m 1 + t 1 ( c ( I ) ) · λ = 0,1 d λ m 1 , m 1 + t ( c ( I ) ) · N m 1 , m 1 + λ · A t m n ( c ( h ) ) ,
E η + ( s ) E η = 0 , E ϕ + ( s ) E ϕ = 0 , } at ξ = ξ 0 ,
n = m i n [ V m n ( 3 ) , t ( c ( I ) ) · α 1 , m n + U m n ( 3 ) , t ( c ( I ) ) · β 1 , m n ] = n = m i n [ f m n ( ζ ) · V m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) · U m n ( 1 ) , t ( c ( I ) ) ] ,
n = m i n [ Y m n ( 3 ) , t ( c ( I ) ) · α 1 , m n + X m n ( 3 ) , t ( c ( I ) ) · β 1 , m n ] = n = m i n [ f m n ( ζ ) · Y m m ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) · X m n ( 1 ) , t ( c ( I ) ) ] , ( m = 0,1,2 , ; t = 0,1,2 , )
n = m i n [ U m n ( 3 ) , t ( c ( I ) ) · α 2 , m n + V m n ( 3 ) , t ( c ( I ) ) · β 2 , m n ] = n = m i n [ f m n ( ζ ) · U m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) V m n ( 1 ) , t ( c ( I ) ) ] ,
n = m i n [ X m n ( 3 ) , t ( c ( I ) ) · α 2 , m n + Y m n ( 3 ) , t ( c ( I ) ) · β 2 , m n ] = n = m i n [ f m n ( ζ ) · X m n ( 1 ) , t ( c ( I ) ) + g m n ( ζ ) · Y m n ( 1 ) , t ( c ( I ) ) ] , ( m = 0,1,2 , ; t = 0,1,2 , )
n = m { i n [ V m n ( 3 ) , 2 s + 1 ( c ( I ) ) · α 1 , m n V m n ( 1 ) , 2 s + 1 ( c ( II ) ) · γ 1 , m n ] + i n + 1 [ U m n + 1 ( 1 ) , 2 s + 1 ( c ( I ) ) · β 1 , m n + 1 U m n + 1 ( 1 ) , 2 s + 1 ( c ( II ) ) · δ 1 , m n + 1 ] } = n = m [ i n f m n ( ζ ) V m n ( 1 ) , 2 s + 1 ( c ( I ) ) + i n + 1 g m n + 1 ( ζ ) U m n + 1 ( 1 ) , 2 s + 1 ( c ( I ) ) ] ,
n = m { i n [ X m n ( 3 ) , 2 s + 1 ( c ( I ) ) · α 1 , m n H X m n ( 1 ) , 2 s + 1 ( c ( II ) ) · γ 1 , m n ] + i n + 1 [ Y m n + 1 ( 3 ) , 2 s + 1 ( c ( I ) ) · β 1 , m n + 1 H Y m n + 1 ( 1 ) , 2 s + 1 ( c ( II ) ) · δ 1 , m n + 1 ] } = n = m [ i n f m n ( ζ ) · X m n ( 1 ) , 2 s + 1 ( c ( I ) ) + i n + 1 g m n + 1 ( ζ ) · Y m n + 1 ( 1 ) , 2 s + 1 ( c ( I ) ) ] ,
n = m { i n + 1 [ U m n + 1 ( 3 ) , 2 s + 1 ( c ( I ) ) · α 1 , m n + 1 H U m n + 1 ( 1 ) , 2 s + 1 ( c ( II ) ) · γ 1 , m n + 1 ] + i n [ V m n ( 3 ) , 2 s + 1 ( c ( I ) ) · β 1 , m n H V m n ( 1 ) , 2 s + 1 ( c ( II ) ) · δ 1 , m n ] } = n = m [ i n + 1 f m n + 1 ( ζ ) U m n + 1 ( 1 ) , 2 s + 1 ( c ( I ) ) + i n g m n ( ζ ) V m n ( 1 ) , 2 s + 1 ( c ( I ) ) ] ,
n = m { i n + 1 [ Y m n + 1 ( 3 ) , 2 s + 1 ( c ( I ) ) · α 1 , m n + 1 Y m n + 1 ( 1 ) , 2 s + 1 ( c ( II ) ) · γ 1 , m n + 1 ] + i n [ X m n ( 3 ) , 2 s + 1 ( c ( I ) ) · β 1 , m n X m n ( 1 ) , 2 s + 1 ( c ( II ) ) · δ 1 , m n ] } = n = m [ i n + 1 f m n + 1 ( ζ ) · Y m n + 1 ( 1 ) , 2 s + 1 ( c ( I ) ) + i n g m n ( ζ ) X m n ( 1 ) , 2 s + 1 ( c ( I ) ) ] ,
n = m { i n + 1 [ V m n + 1 ( 3 ) , 2 s ( c ( I ) ) · α 1 , m n + 1 V m n + 1 ( 1 ) , 2 s ( c ( II ) ) · γ 1 , m n + 1 ] + i n [ U m n ( 3 ) , 2 s ( c ( I ) ) · β 1 , m n U m n ( 1 ) , 2 s ( c ( II ) ) · δ 1 , m n ] } = n = m [ i n + 1 f m n + 1 ( ζ ) V m n + 1 ( 1 ) , 2 s ( c ( I ) ) + i n g m n ( ζ ) U m n ( 1 ) , 2 s ( c ( I ) ) ] ,
n = m { i n + 1 [ X m n + 1 ( 3 ) , 2 s ( c ( I ) ) · α 1 , m n + 1 H · X m n + 1 ( 1 ) , 2 s ( c ( II ) ) · γ 1 , m n + 1 ] + i n [ Y m n ( 3 ) , 2 s ( c ( I ) ) · β 1 , m n H Y m n ( 1 ) , 2 s ( c ( II ) ) · δ 1 , m n ] } = n = m [ i n + 1 f m n + 1 ( ζ ) · X m n + 1 ( 1 ) , 2 s ( c ( I ) ) + i n g m n ( ζ ) · Y m n ( 1 ) , 2 s ( c ( I ) ) ] ,
n = m { i n [ U m n ( 3 ) , 2 s ( c ( I ) ) · α 1 , m n H U m n ( 1 ) , 2 s ( c ( II ) ) · γ 1 , m n ] + i n + 1 [ V m n + 1 ( 3 ) , 2 s ( c ( I ) ) · β 1 , m n + 1 H V m n + 1 ( 1 ) , 2 s ( c ( II ) ) · δ 1 , m n + 1 ] } = n = m [ i n f m n ( ζ ) U m n ( 1 ) , 2 s ( c ( I ) ) + i n + 1 g m n + 1 ( ζ ) · V m n + 1 ( 1 ) , 2 s ( c ( I ) ) ] ,
n = m { i n [ Y m n ( 3 ) , 2 s ( c ( I ) ) · α 1 , m n Y m n ( 1 ) , 2 s ( c ( II ) ) · γ 1 , m n ] + i n + 1 [ X m n + 1 ( 3 ) , 2 s ( c ( I ) ) · β 1 , m n + 1 X m n + 1 ( 1 ) , 2 s ( c ( II ) ) · δ 1 , m n + 1 ] } = n = m [ i n f m n ( ζ ) Y m n ( 1 ) , 2 s ( c ( I ) ) + i n + 1 g m n + 1 ( ζ ) · X m n + 1 ( 1 ) , 2 s ( c ( I ) ) ] ,
cos Ө = cos ζ · cos θ + sin ζ · sin θ · cos ϕ .
F 11 1 2 ( i 11 + i 12 + i 21 + i 22 ) ,
P = F 21 F 11 = ( i 11 + i 21 ) ( i 12 + i 22 ) ( i 11 + i 21 ) + ( i 12 + i 22 ) ,
Q sca = C sca / G ( ζ ) ,
G ( ζ ) = π a b 2 ( a 2 · cos 2 ζ + b 2 · sin 2 ζ ) 1 / 2
G ( ζ ) = π a 2 b ( b 2 · cos 2 ζ + a 2 · sin 2 ζ ) 1 / 2 ,

Metrics