Generation of an elliptic hollow beam using Mathieu and Bessel functions

Rijuparna Chakraborty; Ajay Ghosh

doi:10.1364/JOSAA.23.002278

Journal of the Optical Society of America A
Vol. 23,
Issue 9,
pp. 2278-2282
(2006)
•https://doi.org/10.1364/JOSAA.23.002278

Generation of an elliptic hollow beam using Mathieu and Bessel functions

Rijuparna Chakraborty and Ajay Ghosh

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Rijuparna Chakraborty and Ajay Ghosh, "Generation of an elliptic hollow beam using Mathieu and Bessel functions," J. Opt. Soc. Am. A 23, 2278-2282 (2006)

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Abstract

A new (to our knowledge) technique for the generation of a propagation-invariant elliptic hollow beam is reported. It avoids the use of the radial Mathieu function and hence is mathematically simpler. Bessel functions with their arguments having elliptic locus are used to generate the mask, which is then recorded using holographic technique. To generate such an elliptic beam, both the angular Mathieu function, i.e., elliptic vortex term, and the expression for the circular vortex are used separately. The resultant mask is illuminated with a plane beam, and the proper filtering of its Fourier transform generates the expected elliptic beam. Results with both vortex terms are satisfactory. It has been observed that even for higher ellipticity the vortices do not separate.

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Class	AMF	Fourier Series Expansion	Eigenvalue	Remarks
Class	AMF	Fourier Series Expansion	Eigenvalue	Symmetric About	Antisymmetric About
I	$c e_{2 n} (η; q)$	$\sum_{j = 0}^{\infty} A_{2 j} (q) \cos 2 j η$	$a_{2 n}$	$η = 0$ and $η = π ∕ 2$	—
II	$c e_{2 n + 1} (η; q)$	$Σ_{j = 0}^{\infty} A_{2 j + 1} (q) \cos (2 j + 1) η$	$a_{2 n + 1}$	$η = 0$	$η = π ∕ 2$
III	$s e_{2 n + 2} (η; q)$	$Σ_{j = 0}^{\infty} B_{2 j + 2} (q) \sin (2 j + 2) η$	$b_{2 n + 2}$	—	$η = 0$ and $η = π ∕ 2$
IV	$s e_{2 n + 1} (η; q)$	$Σ_{j = 0}^{\infty} B_{2 j + 1} (q) \sin (2 j + 1) η$	$b_{2 n + 1}$	$η = π ∕ 2$	$η = 0$

Class	RMF	Fourier Series Expansion	Eigenvalue
I	$J e_{2 n} (ξ; q)$	$\sum_{j = 0}^{\infty} A_{2 j} (q) \cosh 2 j ξ$	$a_{2 n}$
II	$J e_{2 n + 1} (ξ; q)$	$\sum_{j = 0}^{\infty} A_{2 j + 1} (q) \cosh (2 j + 1) ξ$	$a_{2 n + 1}$
III	$J o_{2 n + 2} (ξ; q)$	$\sum_{j = 0}^{\infty} B_{2 j + 2} (q) \sinh (2 j + 2) ξ$	$b_{2 n + 2}$
IV	$J o_{2 n + 1} (ξ; q)$	$\sum_{j = 0}^{\infty} B_{2 j + 1} (q) \sinh (2 j + 1) ξ$	$b_{2 n + 1}$

Class	AMF	Fourier Series Expansion	Eigenvalue	Remarks
Class	AMF	Fourier Series Expansion	Eigenvalue	Symmetric About	Antisymmetric About
I	$c e_{2 n} (η; q)$	$\sum_{j = 0}^{\infty} A_{2 j} (q) \cos 2 j η$	$a_{2 n}$	$η = 0$ and $η = π ∕ 2$	—
II	$c e_{2 n + 1} (η; q)$	$Σ_{j = 0}^{\infty} A_{2 j + 1} (q) \cos (2 j + 1) η$	$a_{2 n + 1}$	$η = 0$	$η = π ∕ 2$
III	$s e_{2 n + 2} (η; q)$	$Σ_{j = 0}^{\infty} B_{2 j + 2} (q) \sin (2 j + 2) η$	$b_{2 n + 2}$	—	$η = 0$ and $η = π ∕ 2$
IV	$s e_{2 n + 1} (η; q)$	$Σ_{j = 0}^{\infty} B_{2 j + 1} (q) \sin (2 j + 1) η$	$b_{2 n + 1}$	$η = π ∕ 2$	$η = 0$

Class	RMF	Fourier Series Expansion	Eigenvalue
I	$J e_{2 n} (ξ; q)$	$\sum_{j = 0}^{\infty} A_{2 j} (q) \cosh 2 j ξ$	$a_{2 n}$
II	$J e_{2 n + 1} (ξ; q)$	$\sum_{j = 0}^{\infty} A_{2 j + 1} (q) \cosh (2 j + 1) ξ$	$a_{2 n + 1}$
III	$J o_{2 n + 2} (ξ; q)$	$\sum_{j = 0}^{\infty} B_{2 j + 2} (q) \sinh (2 j + 2) ξ$	$b_{2 n + 2}$
IV	$J o_{2 n + 1} (ξ; q)$	$\sum_{j = 0}^{\infty} B_{2 j + 1} (q) \sinh (2 j + 1) ξ$	$b_{2 n + 1}$

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Equations (12)

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