Abstract

Computations of light scattering require finding Bessel functions of a series of orders. These are found most easily by recurrence, but excessive rounding errors may accumulate. Satisfactory procedures for cylinder and sphere functions are described. If argument z is real, find Yn(z) by recurrence to high orders. From two high orders of Yn(z) estimate Jn(z). Use backward recurrence to maximum Jn(z). Correct by forward recurrence to maximum. If z is complex, estimate high orders of Jn(z) without Yn(z) and use backward recurrence.

© 1972 Optical Society of America

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References

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  1. F. W. J. Olver, H. A. Antosiewicz, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds., Applied Mathematics Series 55 (U.S. Govt. Printing Office, Washington, D.C., 1964), Chaps. 9, 10.
  2. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Antosiewicz, H. A.

F. W. J. Olver, H. A. Antosiewicz, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds., Applied Mathematics Series 55 (U.S. Govt. Printing Office, Washington, D.C., 1964), Chaps. 9, 10.

Kerker, M.

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

Olver, F. W. J.

F. W. J. Olver, H. A. Antosiewicz, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds., Applied Mathematics Series 55 (U.S. Govt. Printing Office, Washington, D.C., 1964), Chaps. 9, 10.

Van de Hulst, H. C.

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

Other (3)

F. W. J. Olver, H. A. Antosiewicz, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds., Applied Mathematics Series 55 (U.S. Govt. Printing Office, Washington, D.C., 1964), Chaps. 9, 10.

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).

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Equations (32)

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j n ( z ) = ( π / 2 z ) 1 2 J n + 1 2 ( z )
y n ( z ) = ( π / 2 z ) 1 2 Y n + 1 2 ( z ) .
ψ n ( z ) = z j n ( z ) and χ n ( z ) = - z y n ( z ) .
α = 2 π a m 2 / λ 0 ,             β = 2 π a m 1 / λ 0 ,
a n = m 2 ψ n ( β ) ψ n ( α ) - m 1 ψ n ( β ) ψ n ( α ) m 2 ψ n ( β ) ζ n ( α ) - m 1 ψ n ( β ) ζ n ( α ) , b n = m 1 ψ n ( β ) ψ n ( α ) - m 2 ψ n ( β ) ψ n ( α ) m 1 ψ n ( β ) ζ n ( α ) - m 2 ψ n ( β ) ζ n ( α ) ,
ζ n ( α ) = ψ n ( α ) + i χ n ( α )
( 2 n + 1 ) ψ n ( z ) = ( n + 1 ) ψ n - 1 ( z ) - n ψ n + 1 ( z )
χ 0 ( α ) = cos α ,
χ 1 ( α ) = sin α + ( cos α ) / α ,
χ n + 1 ( α ) = - χ n - 1 ( α ) + ( 2 n + 1 ) χ n ( α ) / α .
ψ 0 ( α ) = sin α ,
ψ 1 ( α ) = - cos α + ( sin α ) / α ,
ψ n ( α ) = α / [ 2 l χ n ( α ) ] ,
t = [ ( n + 0.5 ) 2 - α 2 ] 1 2 .
f = ψ m ( α ) ( forward ) / ψ m ( α ) ( backward ) .
j n ( α ) = - 1 / [ 2 t α y n ( α ) ] .
Y n - 1 ( α ) + Y n + 1 ( α ) = 2 n Y n ( α ) / α ,
J n ( α ) = - 1 / [ π t Y n ( α ) ] ,
t = ( n 2 - α 2 ) 1 2 .
β = u - i v .
χ 0 ( β ) = cos u cosh v + i sin u sinh v ,
ψ 0 ( β ) = sin u cosh v - i cos u sinh v .
n = abs ( β ) + 1 + 7 ( u ) 1 3 ,
ψ n ( β ) = e t 2 ( β t ) 1 2 ( β n + 0.5 + t ) ( n + 0.5 ) ,
t = [ ( n + 0.5 ) 2 - β 2 ] 1 2 .
ln ψ n ( β ) = t + ( n + 1 ) ln β - ( n + 0.5 ) ln ( n + 0.5 + t ) - 0.5 ln t - ln 2.
limit = - 19 + v ,
J n ( β ) = e t ( 2 π t ) 1 2 ( β n + t ) n ,
t = ( n 2 - β 2 ) 1 2 ,
ln J n ( β ) = t + n ln β - n ln ( n + t ) - 0.5 ln t - 0.5 ln ( 2 π ) .
n = u + 0.5 v + 12 ( u ) 1 3 ,
Y n ( β ) = - e - t ( π t / 2 ) 1 2 ( n + t β ) n .

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