Abstract

Based on the T-matrix formalism, we analytically calculate derivatives of light scattering quantities by a nonspherical particle with respect to its microphysical parameters. Illustrative computations are performed for a spheroid, and the results agree with those obtained by finite differencing. The proposed formalism also predicts correctly derivatives for a sphere obtained by linearized Lorenz–Mie theory.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. O. P. Hasekamp and J. Landgraf, J. Geophys. Res. 110, D04203 (2005).
    [CrossRef]
  2. R. G. Grainger, J. Lucas, G. E. Thomas, and G. B. L. Ewen, Appl. Opt. 43, 5386 (2004).
    [CrossRef] [PubMed]
  3. P. Ginoux, J. Geophys. Res. 108, 4052 (2003).
    [CrossRef]
  4. P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
    [CrossRef]
  5. P. C. Waterman, Phys. Rev. D 3, 825 (1971).
    [CrossRef]
  6. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).
  7. F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
    [CrossRef]
  8. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
    [CrossRef]
  9. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources—Theory and Programs (Springer, 2006).
  10. G. Gouesbet, Opt. Commun. 283, 517 (2010).
    [CrossRef]
  11. D. Bernstein, Matrix Mathematics (Princeton University, 2005).
  12. M. I. Mishchenko and L. D. Travis, Opt. Commun. 109, 16 (1994).
    [CrossRef]

2010 (2)

F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
[CrossRef]

G. Gouesbet, Opt. Commun. 283, 517 (2010).
[CrossRef]

2007 (1)

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

2005 (1)

O. P. Hasekamp and J. Landgraf, J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

2004 (1)

2003 (1)

P. Ginoux, J. Geophys. Res. 108, 4052 (2003).
[CrossRef]

1994 (1)

M. I. Mishchenko and L. D. Travis, Opt. Commun. 109, 16 (1994).
[CrossRef]

1971 (1)

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

Barber, P. W.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Bernstein, D.

D. Bernstein, Matrix Mathematics (Princeton University, 2005).

Doicu, A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources—Theory and Programs (Springer, 2006).

Dubovik, O.

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

Eremin, Y. A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources—Theory and Programs (Springer, 2006).

Ewen, G. B. L.

Feng, Q.

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

Ginoux, P.

P. Ginoux, J. Geophys. Res. 108, 4052 (2003).
[CrossRef]

Gouesbet, G.

G. Gouesbet, Opt. Commun. 283, 517 (2010).
[CrossRef]

F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
[CrossRef]

Grainger, R. G.

Hasekamp, O. P.

O. P. Hasekamp and J. Landgraf, J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

Hill, S. C.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Hong, G.

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

Katawar, G. W.

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Landgraf, J.

O. P. Hasekamp and J. Landgraf, J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

Laszlo, I.

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

Lock, J. A.

F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
[CrossRef]

Lucas, J.

Mishchenko, M. I.

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

M. I. Mishchenko and L. D. Travis, Opt. Commun. 109, 16 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Sokolik, I. N.

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

Thomas, G. E.

Travis, L. D.

M. I. Mishchenko and L. D. Travis, Opt. Commun. 109, 16 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Waterman, P. C.

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

Wiscombe, W. J.

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

Wriedt, T.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources—Theory and Programs (Springer, 2006).

Xu, F.

F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
[CrossRef]

Yang, P.

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

Appl. Opt. (1)

J. Aerosol Sci. (1)

P. Yang, Q. Feng, G. Hong, G. W. Katawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, J. Aerosol Sci. 38, 995 (2007).
[CrossRef]

J. Geophys. Res. (2)

P. Ginoux, J. Geophys. Res. 108, 4052 (2003).
[CrossRef]

O. P. Hasekamp and J. Landgraf, J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

Opt. Commun. (2)

M. I. Mishchenko and L. D. Travis, Opt. Commun. 109, 16 (1994).
[CrossRef]

G. Gouesbet, Opt. Commun. 283, 517 (2010).
[CrossRef]

Phys. Rev. A (1)

F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
[CrossRef]

Phys. Rev. D (1)

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

Other (4)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources—Theory and Programs (Springer, 2006).

D. Bernstein, Matrix Mathematics (Princeton University, 2005).

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 (1)

Fig. 1
Fig. 1

Extinction cross section (dotted–dashed curve) and its derivative with respect to the axis ratio of a spheroid oriented at 45 ° with respect to the incident plane wave of wavelength λ 0 = 0.5145 μm . The derivatives are calculated both by the linearized T-matrix method (solid curve) and by the FDM (crosses). For clarity, C ext values are doubled.

Equations (26)

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

E ( i ) ( r ) = n = 1 m = n + n [ G m n TE M m n ( 1 ) ( k 2 r ) + G m n TM N m n ( 1 ) ( k 2 r ) ] ,
E ( in ) ( r ) = n = 1 m = n + n [ D m n M m n ( 1 ) ( k 1 r ) + C m n N m n ( 1 ) ( k 1 r ) ] ,
E ( s ) ( r ) = n = 1 m = n + n [ B m n M m n ( 3 ) ( k 2 r ) + A m n N m n ( 3 ) ( k 2 r ) ] .
[ i k 1 k 2 K m n , m n i k 2 2 J m n , m n i k 1 k 2 L m n , m n i k 2 2 I m n , m n ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... i k 1 k 2 I m n , m n i k 2 2 L m n , m n i k 1 k 2 J m n , m n i k 2 2 K m n , m n ] [ D m n ... ... C m n ] = [ G m n TE ... ... G m n TM ] ,
[ B m n ... ... A m n ] = [ i k 1 k 2 K ˜ m n , m n i k 2 2 J ˜ m n , m n i k 1 k 2 L ˜ m n , m n i k 2 2 I ˜ m n , m n ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... i k 1 k 2 I ˜ m n , m n i k 2 2 L ˜ m n , m n i k 1 k 2 J ˜ m n , m n i k 2 2 K ˜ m n , m n ] [ D m n ... ... C m n ] ,
[ B m n ... ... A m n ] = T [ G m n TE ... ... G m n TM ] .
J m n , m n = ( 1 ) m S n 2 ( r s ) d S ( r s ) · M m n ( 1 ) ( k 1 r s , θ , φ ) × N m n ( 3 ) ( k 2 r s , θ , φ ) ,
n 2 ( r s ) d S ( r s ) = d S ( r s ) = ( e r 1 r s r s θ e θ 1 r s sin θ r s φ e φ ) r s 2 ( θ , φ ) sin θ d θ d φ .
T x = V x U 1 + V ( U 1 ) x ,
( U 1 ) x = U 1 U x U 1 ,
[ B m n x ... ... A m n x ] = [ V x U 1 T U x U 1 ] [ G m n TE ... ... G m n TM ] .
( d S · M × N ) x = d S · ( M x × N + M × N x ) + ( d S ) x · M × N
M m n ( i ) ( k r , θ , φ ) ( k r ) = γ m n exp ( i m ϕ ) d j n ( i ) ( k r ) d ( k r ) × [ i m P m n ( cos θ ) sin θ e θ d P m n ( cos θ ) d θ e φ ] ,
N m n ( i ) ( k r , θ , φ ) k r = γ m n k r exp ( i m ϕ ) × { n ( n + 1 ) ( d j n ( i ) ( k r ) d ( k r ) j n ( i ) ( k r ) k r ) P m n ( cos θ ) e r + [ ( n + 1 ) ( d j n ( i ) ( k r ) d ( k r ) j n ( i ) ( k r ) k r ) k r d j n + 1 ( i ) ( k r ) d ( k r ) ] d P m n ( cos θ ) d θ e θ + [ ( n + 1 ) ( d j n ( i ) ( k r ) d ( k r ) j n ( i ) ( k r ) k r ) k r d j n + 1 ( i ) ( k r ) d ( k r ) ] i m P m n ( cos θ ) sin θ e φ } ,
γ m n = [ ( 2 n + 1 ) ( n m ) ! 4 π n ( n + 1 ) ( n + m ) ! ] 1 / 2
C sca x = 1 k 2 2 n = 1 m = n n ( A m n x A m n * + A m n A m n * x + B m n x B m n * + B m n B m n * x ) ,
C ext x = 1 k 2 2 Re n = 1 m = n n ( G m n TE B m n * x + G m n TM A m n * x ) ,
r s ( θ , φ ) = a [ cos 2 θ + a 2 b 2 sin 2 θ ] 1 / 2 ,
[ r s ( θ , φ ) ] θ = sin θ cos θ r s 3 a 2 ( a 2 b 2 1 ) .
[ r s ( θ , φ ) ] a = r s ( θ , φ ) a r s 3 a b 2 sin 2 θ ,
[ r s ( θ , φ ) ] b = r s 3 b 3 sin 2 θ ,
a { [ r s ( θ , φ ) ] θ } = 1 a [ r s ( θ , φ ) ] θ + sin θ cos θ g r s 3 a b 2 ,
( a / b ) { [ r s ( θ , φ ) ] θ } = sin θ cos θ g r s 3 b 3 ,
g = 3 ( a 2 b 2 ) sin 2 θ b 2 cos 2 θ + a 2 sin 2 θ 2.
Q v = ( a b Q a + Q b ) / 4 π a 2 ,
Q ( a / b ) = ( a b Q a 2 b 2 Q b ) / 3 a ,

Metrics