Abstract

The light-scattering problem of a sphere on or near a plane surface is solved by using an extension of the Mie theory. The approach taken is to solve the boundary conditions at the sphere and at the surface simultaneously and to develop the scattering amplitude and Mueller scattering matrices. This is performed by projecting the fields in the half-space region not including the sphere multiplied by an appropriate Fresnel reflection coefficient onto the half-space region including the sphere. An assumption is that the scattered fields from the sphere, reflecting off the surface and interacting with the sphere, are incident upon the surface at near-normal incidence. The exact solution is asymptotically approached when either the sphere is a large distance from the surface or the refractive index of the surface approaches infinity.

© 1991 Optical Society of America

Full Article  |  PDF Article

Errata

Gorden Videen, "Light scattering from a sphere on or near a surface: errata," J. Opt. Soc. Am. A 9, 844-845 (1992)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-9-5-844

References

  • View by:
  • |
  • |
  • |

  1. Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).
  2. W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem Draht,” Ann. Phys. (Leipzig) 18, 495–522 (1905).
  3. G. Mie, “Beitrage zer Optik truber Meiden speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
  4. C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).
  5. G. W. Kattawar, C. E. Dean, “Electromagnetic scattering from two dielectric spheres: comparison between theory and experiment,” Opt. Lett. 8, 48–52 (1983).
    [CrossRef] [PubMed]
  6. R. T. Wang, J. M. Greenberg, D. W. Schuerman, “Experimental results of dependent light scattering by two spheres,” Opt. Lett. 8, 543–545 (1981).
    [CrossRef]
  7. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–241 (1986).
  8. H. Yousif, “Light scattering from parallel tilted fibers,” Ph.D. dissertation (Department of Physics, University of Arizona, Tucson, Ariz., 1987).
  9. L. Lamb, Department of Physics, University of Arizona, Tucson, Ariz. 85721 (personal communication).
  10. T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane. 1. TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
    [CrossRef]
  11. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  12. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  13. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  14. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

1989

1986

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–241 (1986).

1983

1981

1967

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

1908

G. Mie, “Beitrage zer Optik truber Meiden speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

1905

W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem Draht,” Ann. Phys. (Leipzig) 18, 495–522 (1905).

1881

Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Barakat, R.

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–241 (1986).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Dean, C. E.

Greenberg, J. M.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kattawar, G. W.

Kerker, M.

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

Lamb, L.

L. Lamb, Department of Physics, University of Arizona, Tucson, Ariz. 85721 (personal communication).

Liang, C.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Lo, Y. T.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Mie, G.

G. Mie, “Beitrage zer Optik truber Meiden speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Rao, T. C.

Rayleigh,

Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Schuerman, D. W.

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).

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–241 (1986).

von Ignatowsky, W.

W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem Draht,” Ann. Phys. (Leipzig) 18, 495–522 (1905).

Wang, R. T.

Yousif, H.

H. Yousif, “Light scattering from parallel tilted fibers,” Ph.D. dissertation (Department of Physics, University of Arizona, Tucson, Ariz., 1987).

Ann. Phys. (Leipzig)

W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem Draht,” Ann. Phys. (Leipzig) 18, 495–522 (1905).

G. Mie, “Beitrage zer Optik truber Meiden speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

J. Opt. Soc. Am. A

Opt. Lett.

Philos. Mag.

Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Physica

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–241 (1986).

Radio Sci.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Other

H. Yousif, “Light scattering from parallel tilted fibers,” Ph.D. dissertation (Department of Physics, University of Arizona, Tucson, Ariz., 1987).

L. Lamb, Department of Physics, University of Arizona, Tucson, Ariz. 85721 (personal communication).

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

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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

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

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 (3)

Fig. 1
Fig. 1

Geometry of scattering system. A sphere of radius a is located a distance d from a surface. A plane wave travels in the xz plane at angle α with respect to the z axis.

Fig. 2
Fig. 2

Image coordinate surface is located a distance 2d from the sphere coordinate system along the positive z axis. Fields in the image coordinate system are inverted; e.g., the image of the incident plane wave travels in the xz plane at angle πα with respect to the z axis.

Fig. 3
Fig. 3

Maximum angle of incidence on the surface γ for an interacting ray occurs for a ray traced from the image coordinate system (located at the center of the image sphere) to the edge of the sphere and is greatest when the image sphere touches the real sphere (the sphere touches the surface). This angle can be no greater than 30°. The Fresnel equations are fairly constant near normal incidences.

Equations (38)

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

2 E - μ 2 E t 2 - μ σ E t = 0 , 2 H - μ 2 H t 2 - μ σ H t = 0 ,
1 r 2 r ( r 2 u r ) + 1 r 2 sin ϑ ϑ ( sin ϑ u ϑ ) + 1 r 2 sin 2 ϑ 2 u φ 2 + k 2 u = 0.
u ( r , ϑ , φ ) = R ( r ) Θ ( ϑ ) Φ ( φ ) .
Φ ( ϑ ) = exp ( i m φ ) , Θ ( ϑ ) = P ˜ n m ( cos ϑ ) = [ ( 2 n + 1 ) ( n - m ) ! 2 ( n + m ) ! ] 1 / 2 P n m ( cos ϑ ) , R ( r ) = z n ( k r ) = ( π 2 r ) 1 / 2 Z n + 1 / 2 ( k r ) ,
E = n , m , p m n M n m ( i ) + q n m N n m ( i ) , H = k i ω μ n , m p n m N n m ( i ) + q n m M n m ( i ) .
M n m ( i ) = φ ^ [ i m sin θ z n ( k r ) P ˜ n m ( cos ϑ ) exp ( i m φ ) ] - φ ^ { z n ( k r ) d d ϑ [ P ˜ n m ( cos ϑ ) ] exp ( i m ϑ ) } , N n m ( i ) = r ^ [ 1 k r z n ( k r ) n ( n + 1 ) P ˜ n m ( cos ϑ ) exp ( i m φ ) ] + φ ^ { 1 k r d d r [ r z n ( k r ) ] d d ϑ [ P ˜ n m ( cos ϑ ) ] exp ( i m φ ) } + φ ^ { 1 k r d d r [ r z n ( k r ) ] i m sin ϑ P ˜ n m ( cos ϑ ) exp ( i m φ ) } .
E = n , m × ( r p n m u n m ( i ) ) + 1 k × × ( r q n m u n m ( i ) ) , H = k i ω μ n , m × ( r q n m u n m ( i ) ) + 1 k × × ( r p n m u n m ( i ) ) .
E inc = n = 0 m = - n n a n m M n m ( 1 ) + b n m N n m ( 1 ) H inc = k i ω μ n = 0 m = - n n b n m M n m ( 1 ) + a n m N n m ( 1 ) .
E sca = n = 0 m = - n n e n m M n m ( 3 ) + f n m N n m ( 3 ) , H sca = k i ω μ n = 0 m = - n n f n m M n m ( 3 ) + e n m N n m ( 3 ) .
E sph = n = 0 m = - n n c n m M n m ( 1 ) + d n m N n m ( 1 ) , H sph = k sph i ω μ sph n = 0 m = - n n d n m M n m ( 1 ) + c n m N n m ( 1 ) .
E int = n = 0 m = - n n g n m M n m ( 1 ) + h n m N n m ( 1 ) , H int = k i ω μ n = 0 m = - n n h n m M n m ( 1 ) + g n m N n m ( 1 ) .
e n m = - ( a n m + g n m ) × k sph μ ψ n ( k sph a ) ψ n ( k a ) - k μ sph ψ n ( k a ) ψ n ( k sph a ) k sph μ ψ n ( k sph a ) ξ n ( k a ) - k μ sph ξ n ( k a ) ψ n ( k sph a ) = ( a n m + g n m ) Q e n , f n m = - ( b n m + h n m ) × k sph μ ψ n ( k a ) ψ n ( k sph a ) - k μ sph ψ n ( k sph a ) ψ n ( k a ) k sph μ ξ n ( k a ) ψ n ( k sph a ) - k μ sph ψ n ( k sph a ) ξ n ( k a ) = ( b n m + h n m ) Q f n ,
ϕ n ( r ) = r j n ( r ) ,             ξ n ( r ) = r h n ( 1 ) ( r ) ,
P ˜ n m [ cos ( π - ϑ ) ] = ( - 1 ) n + m P ˜ n m [ cos ( ϑ ) ] .
R TE ( ϑ i ) = Z 2 cos ϑ i - Z 1 [ 1 - ( n 1 / n 2 ) 2 sin 2 ϑ i ] 1 / 2 Z 2 cos ϑ i + Z 1 [ 1 - ( n 1 / n 2 ) 2 sin 2 ϑ i ] 1 / 2 , T TE ( ϑ i ) = 2 Z 2 cos ϑ i Z 2 cos ϑ i + Z 1 [ 1 - ( n 1 / n 2 ) 2 sin 2 ϑ i ] 1 / 2 , R TM ( ϑ i ) = - Z 1 cos ϑ i - Z 2 [ 1 - ( n 1 / n 2 ) 2 sin 2 ϑ i ] 1 / 2 Z 1 cos ϑ i + Z 2 [ 1 - ( n 1 / n 2 ) 2 sin 2 ϑ i ] 1 / 2 , T TM ( ϑ i ) = 2 Z 2 cos ϑ i Z 1 cos ϑ i + Z 2 [ 1 - ( n 1 / n 2 ) 2 sin 2 ϑ i ] 1 / 2 ,
Z 1 Z 2 = μ 1 k 2 μ 2 k 1 ,
g n m u n m ( 3 ) = R ( 0 ° ) ( - 1 ) n + m e n m u n m ( 3 ) , h n m u n m ( 3 ) = R ( 0 ° ) ( - 1 ) n + m f n m u n m ( 3 ) ,
g n m u n m ( 3 ) = R ( 0 ° ) ( - 1 ) n + m e n m n = m c n ( n , m ) u n m ( 1 ) , h n m u n m ( 3 ) = R ( 0 ° ) ( - 1 ) n + m f n m n = m c n ( n , m ) u n m ( 1 ) .
g n m = R ( 0 ° ) n = m ( - 1 ) n + m e n m c n ( n , m ) , h n m = R ( 0 ° ) n = m ( - 1 ) n + m f n m c n ( n , m ) .
e n m = [ a n m + R ( 0 ° ) n = m ( - 1 ) n + m e n m c n ( n , m ) ] Q e n , f n m = [ b n m + R ( 0 ° ) n = m ( - 1 ) n + m f n m c n ( n , m ) ] Q e n .
c n ( 0 , 0 ) = - ( 2 n + 1 ) 1 / 2 h n ( 1 ) ( 2 k d ) ,
c n ( - 1 , 0 ) = - ( 2 n + 1 ) 1 / 2 h n ( 1 ) ( 2 k d ) ,
[ ( n - m + 1 ) ( n + m ) ( 2 n + 1 ) ] 1 / 2 c n ( n , m ) = [ ( n - m + 1 ) ( n + m ) ( 2 n + 1 ) ] 1 / 2 c n ( n , m - 1 ) - 2 k d [ ( n - m + 2 ) ( n - m + 1 ) ( 2 n + 3 ) ] 1 / 2 c n + 1 ( n , m - 1 ) - 2 k d [ ( n + m ) ( n + m - 1 ) ( 2 n - 1 ) ] 1 / 2 c n - 1 ( n , m - 1 ) ,
c n ( n , m ) = c n ( n , - m ) ,
n c n ( n - 1 , 0 ) ( 2 n + 1 2 n - 1 ) 1 / 2 - ( n + 1 ) c n ( n + 1 , 0 ) ( 2 n + 1 2 n + 3 ) 1 / 2 = ( n + 1 ) c n + 1 ( n , 0 ) ( 2 n + 1 2 n + 3 ) 1 / 2 - n c n - 1 ( n , 0 ) ( 2 n + 1 2 n - 1 ) 1 / 2 .
E TE inc = [ 1 + R TE ( α ) exp ( 2 i k d cos α ) ( - 1 ) n + m ] × n = 0 m = - n n a n m TE M n m ( 1 ) + b n m TE N n m ( 1 ) , H TE inc = k i ω μ [ 1 + R TE ( α ) exp ( 2 i k d cos α ) ( - 1 ) n + m ] × n = 0 m = - n n b n m TE M n m ( 1 ) + a n m TE N n m ( 1 ) , E TM inc = [ 1 + R TM ( α ) exp ( 2 i k d cos α ) ( - 1 ) n + m ] × n = 0 m = - n n a n m TM M n m ( 1 ) + b n m TM N n m ( 1 ) , H TM inc = k i ω μ [ 1 + R TM ( α ) exp ( 2 i k d cos α ) ( - 1 ) n + m ] × n = 0 m = - n n b n m TM M n m ( 1 ) + a n m TM N n m ( 1 ) .
a n m TE = i n n ( n + 1 ) { [ ( n - m ) ( n + m + 1 ) ] 1 / 2 P ˜ n m + 1 ( cos α ) - [ ( n - m + 1 ) ( n + m ) ] 1 / 2 P ˜ n m - 1 ( cos α ) } = 2 i n + 2 n ( n + 1 ) P ˜ n m ( cos α ) α ,
b n m TE = i n + 2 ( 2 n + 1 ) n ( n + 1 ) { P ˜ n + 1 m - 1 ( cos α ) × [ ( n - m + 1 ) ( n - m + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) ] 1 / 2 + P ˜ n + 1 m + 1 ( cos α ) × [ ( n + m + 1 ) ( n + m + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) ] 1 / 2 } = 2 i n + 2 n ( n + 1 ) m P ˜ n m ( cos α ) sin α ,
a n m TM = i b n m TE ,
b n m TM = i a n m TE .
e n m TE = { [ 1 + R TE ( α ) ( - 1 ) n + m exp ( 2 i k d cos α ) ] a n m TE + R TE ( 0 ° ) n = m ( - 1 ) n + m e n m TE c n ( n , m ) } Q e n , e n m TM = { [ 1 + R TM ( α ) ( - 1 ) n + m exp ( 2 i k d cos α ) ] a n m TM + R TM ( 0 ° ) n = m ( - 1 ) n + m e n m TM c n ( n , m ) } Q e n , f n m TM = { [ 1 + R TM ( α ) ( - 1 ) n + m exp ( 2 i k d cos α ) ] b n m TE + R TE ( 0 ° ) n = m ( - 1 ) n + m f n m TE c n ( n m ) } Q f n , f n m TM = { [ 1 + R TM ( α ) ( - 1 ) n + m exp ( 2 i k d cos α ) ] b n m TM + R TM ( 0 ° ) n = m ( - 1 ) n + m f n m TM c n ( n m ) } Q f n .
h n ( 1 ) ( k r ) ~ ( - i ) n i k r e i k r .
[ E ϑ sca E ϕ sca ] = e i k r - i k r [ S 2 S 3 S 4 S 1 ] [ E TM inc E TE inc ] .
S 1 = n = 0 m = - n n ( - i ) n e i m φ × [ 1 + R TE ( π - ϑ ) ( - 1 ) n + m exp ( - 2 i k d cos ϑ ) ] × [ f n m TE m sin ϑ P ˜ n m ( cos ϑ ) + e n m TE ϑ P ˜ n m ( cos ϑ ) ] , S 2 = - i n = 0 m = - n n ( - i ) n e i m φ × [ 1 + R TM ( π - ϑ ) ( - 1 ) n + m exp ( - 2 i k d cos ϑ ) ] × [ e n m TM m sin ϑ P ˜ n m ( cos ϑ ) + f n m TM ϑ P ˜ n m ( cos ϑ ) ] , S 3 = - i n = 0 m = - n n ( - i ) n e i m φ × [ 1 + R TM ( π - ϑ ) ( - 1 ) n + m exp ( - 2 i k d cos ϑ ) ] × [ e n m TE m sin ϑ P ˜ n m ( cos ϑ ) + f n m TE ϑ P ˜ n m ( cos ϑ ) ] , S 4 = n = 0 m = - n n ( - i ) n e i m φ × [ 1 + R TE ( π - ϑ ) ( - 1 ) n + m exp ( - 2 i k d cos ϑ ) ] × [ f n m TM m sin ϑ P ˜ n m ( cos ϑ ) + e n m TM ϑ P ˜ n m ( cos ϑ ) ] .
S 1 = n = 0 m = - n n ( - i ) n e i m φ T TE ( ϑ ) × [ f n m TE m sin ϑ P ˜ n m ( cos ϑ ) + e n m TE ϑ P ˜ n m ( cos ϑ ) ] , S 2 = - i n = 0 m = - n n ( - i ) n e i m φ T TM ( ϑ ) × [ e n m TM m sin ϑ P ˜ n m ( cos ϑ ) + f n m TM ϑ P ˜ n m ( cos ϑ ) ] , S 3 = - i n = 0 m = - n n ( - i ) n e i m φ T TM ( ϑ ) × [ e n m TE m sin ϑ P ˜ n m ( cos ϑ ) + f n m TE ϑ P ˜ n m ( cos ϑ ) ] , S 4 = n = 0 m = - n n ( - i ) n e i m φ T TE ( ϑ ) × [ f n m TM m sin ϑ P ˜ n m ( cos ϑ ) + e n m TM ϑ P ˜ n m ( cos ϑ ) ] .
e n m ¯ TE = ( - 1 ) m e n m TE , e n m ¯ TM = ( - 1 ) m + 1 e n m TM , f n m ¯ TE = ( - 1 ) m + 1 f n m TE , f n m ¯ TM = ( - 1 ) m f n m TM ,
P ˜ n - m ( x ) = ( - 1 ) m P ˜ n m ( x ) .
S 11 = ½ ( S 1 2 + S 2 2 ) , S 12 = ½ ( S 2 2 - S 1 2 ) , S 33 = Re ( S 1 S 2 * ) , S 34 = Im ( S 2 S 1 * ) .

Metrics