Beam shape coefficient calculation for a Gaussian beam: localized approximation, quadrature and angular spectrum decomposition methods

Juncheng Qiu; Jianqi Shen

doi:10.1364/AO.57.000302

Applied Optics
Vol. 57,
Issue 2,
pp. 302-313
(2018)
•https://doi.org/10.1364/AO.57.000302

Beam shape coefficient calculation for a Gaussian beam: localized approximation, quadrature and angular spectrum decomposition methods

Juncheng Qiu and Jianqi Shen

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Juncheng Qiu and Jianqi Shen, "Beam shape coefficient calculation for a Gaussian beam: localized approximation, quadrature and angular spectrum decomposition methods," Appl. Opt. 57, 302-313 (2018)

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Abstract

Three methods for calculating the beam shape coefficients (the quadrature, the angular spectrum decomposition (ASD), and the localized approximation methods) are studied. The normalized associated Legendre function is employed in the calculation so as to prevent overflow. The two-dimensional integrations in the quadrature and the ASD methods are developed to the one-dimensional integration to speed up the calculation. The Gaussian beams reconstructed with different methods are compared, showing the difference in the remodeling effects of these methods. The reason for the presence of pseudodistribution in the localized model is given.

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$(n, m)$	Method	${A_{n m}} Re$	$Im {A_{n m}}$	$\| A_{n m} \|$
(20,0)	Quadrature	$1.75356 E - 02$	$- 5.30646 E - 02$	$5.58869 E - 02$
	ASD	$1.78143 E - 02$	$- 5.34648 E - 02$	$5.63545 E - 02$
	LA	$1.78731 E - 02$	$- 5.34912 E - 02$	$5.63982 E - 02$
(20,10)	Quadrature	$- 8.89388 E - 07$	$2.82993 E - 06$	$2.96640 E - 06$
	ASD	$- 8.71946 E - 07$	$2.81211 E - 06$	$2.94419 E - 06$
	LA	$- 8.33158 E - 07$	$2.68676 E - 06$	$2.81298 E - 06$
(20,20)	Quadrature	$- 5.75926 E - 19$	$- 2.72694 E - 17$	$2.72755 E - 17$
	ASD	$5.91641 E - 20$	$- 2.65041 E - 17$	$2.65042 E - 17$
	LA	$- 1.00122 E - 18$	$- 1.91491 E - 17$	$1.91753 E - 17$
(70,0)	Quadrature	$8.23597 E - 01$	$- 1.00690 E + 00$	$1.30083 E + 00$
	ASD	$8.21805 E - 01$	$- 1.00495 E + 00$	$1.29819 E + 00$
	LA	$8.22396 E - 01$	$- 1.00571 E + 00$	$1.29915 E + 00$
(70,15)	Quadrature	$- 5.25322 E - 04$	$9.80081 E - 05$	$5.34386 E - 04$
	ASD	$- 5.17242 E - 04$	$9.34876 E - 05$	$5.25623 E - 04$
	LA	$- 4.71133 E - 04$	$8.97063 E - 05$	$4.79597 E - 04$
(70,30)	Quadrature	$6.35585 E - 13$	$6.05894 E - 13$	$8.78109 E - 13$
	ASD	$5.68377 E - 13$	$6.12220 E - 13$	$8.35384 E - 13$
	LA	$3.38490 E - 13$	$2.94453 E - 13$	$4.48640 E - 13$
(100,0)	Quadrature	$2.18255 E - 01$	$- 3.61217 E - 01$	$4.22034 E - 01$
	ASD	$2.18990 E - 01$	$- 3.61914 E - 01$	$4.23011 E - 01$
	LA	$2.19225 E - 01$	$- 3.62601 E - 01$	$4.23721 E - 01$
(100,15)	Quadrature	$- 1.16034 E - 03$	$7.89337 E - 04$	$1.40337 E - 03$
	ASD	$- 1.15704 E - 03$	$7.77662 E - 04$	$1.39409 E - 03$
	LA	$- 1.10619 E - 03$	$7.47893 E - 04$	$1.33529 E - 03$
(100,30)	Quadrature	$2.39525 E - 10$	$4.71154 E - 11$	$2.44115 E - 10$
	ASD	$2.27244 E - 10$	$5.37485 E - 11$	$2.33514 E - 10$
	LA	$1.52335 E - 10$	$2.83589 E - 11$	$1.54952 E - 10$

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Applied Optics