Analytical results for a Bessel function times Legendre polynomials class integrals

A A R Neves L A Padilha, A Fontes, E Rodriguez C H B Cruz L C Barbosa and C L Cesar

Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, PO Box 6165, 13083-970 Campinas, Brazil

Email: aneves@ifi.unicamp.br

Received 13 February 2006
Published 19 April 2006

PACS numbers: 41.20. - q, 42.25.Fx, 42.60.Jf

1. Introduction

In a recent work to calculate the optical force of the optical tweezers in a complete electromagnetic treatment for any beam shape focused at an arbitrary position, we encountered an integral involving Bessel and associated Legendre functions [1]. The same integral appears in fields related to vector diffraction theory where computationally intensive methods or approximations are employed [2]. This letter presents the analytical evaluation of this integral (1).

Typically, the first task in solving scattering problems is to decompose the incident beam into partial waves involving associated Legendre polynomials P^m_n(cos θ) for the angular part and spherical Bessel functions j_n(kr) for the radial part. The difficulty in doing this exactly is to determine an analytical expression for integrals of the class

which, so far, have not been reported in a closed form, as far as we know. This integral is not shown in any Integral Tables, nor in calculation packages such as Mathematica, and we do not know of any other report of this result. An analytical expression for this integral would be useful for many different fields in physics, especially for those that require partial wave decomposition. Without this result people have used all sorts of approximations to proceed forward and obtain results [3].

2. General analysis

We analysed the behaviour of such integrals for the limit of kr  =  R → 0 using Lock's result for the integral [4]

and found that the lowest order term would be compatible with the closed form and simple expression given by

valid for any n ≥ 0 and  - n ≤ m ≤ n. We then tested it numerically using Mathematica software (version 5.2) for randomly generated n, m, α and R and showed that this result is indeed true. It would also allow one to access a whole family of integrals by taking any number of derivatives with respect to R or α, such as

After discovering the expression for this integral, we proceed to prove it using the same induction procedure used in the Lock integral paper. We started by proving that I^m_n follows the same recurrence relations as F^m_n  =  2i^n - mP^m_n(cos α)j_n(R). We then proved that the results hold for n  =  m  =  0.

The recurrence relations for I^m_n can be obtained by the associated Legendre polynomials recurrence relations

where we adopted the sign convention for the associated Legendre polynomials of Abramowitz and Stegun [5], followed by the Mathematica software. The Bessel functions recurrence relation

Using relations (5) and (6), it can be readily shown that

is the desired recurrence relation for I^m_n.

Now, for F^m_n  =  2i^n - mP^m_n(cos α)j_n(R) we used the associated Legendre polynomials recurrence relation

and the spherical Bessel functions relation

to prove that

This assures that both sides of the identity follow the same recurrence relation. Therefore, the only task remaining is to prove that

holds for n  =  m  =  0. Now, the series expression for J₀ is

therefore,

The Poisson integral representation of the spherical Bessel function is

which, added to the parity null term

can be rewritten as

This leaves the initial integral as

That is still not good because the argument of the spherical Bessel function is Rcos α instead of R. To change the argument, we use the multiplication theorem for Bessel functions

To obtain

we can now make ν  =  1/2, n  =  0, z  =  Rcos α and λ  =  1/cos α. Therefore,

and finally

Comparing this result with the obtained series, it turns out that

and the proof is complete, validating the following integral for all n and m:

3. Concluding remarks

In this letter, we have shown the analytical solution to an integral involving Bessel and associated Legendre functions (23). We have obtained the solution through recurrence relations and the multiplication theorem. We believe that this result, analytical and simple, is of interest to the general community especially for problems involving electromagnetic vector diffraction of arbitrary beams.

Acknowledgments

This work was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) through the Optics and Photonics Research Center (CePOF). We thank the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for financial support of this research.

[1] 

Neves A A R, Fontes A, Padilha L A, Rodriguez E, Cruz C H B, Barbosa L C and Cesar C L 2006 Exact partial wave expansion of optical beams with respect to an arbitrary origin submitted Preprint physics/0603092  
Preprint

[2] 

Mazolli A, Neto P A M and Nussenzveig H M 2003 Theory of trapping forces in optical tweezers Proc. R. Soc. A 459 3021-41 
CrossRef

[3] 

Ren K F, Gouesbet G and Grehan G 1998 Integral localized approximation in generalized Lorenz-Mie theory Appl. Opt. 37 4218-25 
CrossRefPubMed

[4] 

Gouesbet G and Lock J A 1994 Rigorous justification of the localized approximation to beam-shape coefficients in generalized Lorenz-Mie theory: 2. Off-axis beams J. Opt. Soc. Am. A 11 2516-25 
CrossRef

[5] 

Abramowitz M and Stegun I A 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (New York: Dover)