Method to evaluate beam quality of Gaussian beams with aberrations

Yuntao Qiu; Lei Huang; Mali Gong; Liu Qiang; Ping Yan; Haitao Zhang

doi:10.1364/AO.51.006539

Applied Optics
Vol. 51,
Issue 27,
pp. 6539-6543
(2012)
•https://doi.org/10.1364/AO.51.006539

Method to evaluate beam quality of Gaussian beams with aberrations

Yuntao Qiu, Lei Huang, Mali Gong, Liu Qiang, Ping Yan, and Haitao Zhang

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Yuntao Qiu, Lei Huang, Mali Gong, Liu Qiang, Ping Yan, and Haitao Zhang, "Method to evaluate beam quality of Gaussian beams with aberrations," Appl. Opt. 51, 6539-6543 (2012)

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Abstract

Zernike polynomials are commonly used to describe the aberration of light beams, and the beam quality of Gaussian beams with aberrations can be deduced when the order of Zernike polynomials is limited. In this paper, Hermite polynomials are utilized to reconstruct the aberrations of Gaussian beams. The beam quality factor is directly related to the coefficient and terms of Hermite polynomials and has no limit on its index. We analyzed the beam quality of a Gaussian beam with a quartic aberration and other former 11th Zernike aberrations by Hermite polynomial expansion. The result corresponds with the published research work.

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Name	Algebraic Expressions	Hermite Terms
Piston	$α_{p}$	$α_{p} H_{00}$
$x$ -tilt	$α_{t}^{x} x$	${(\frac{ω^{2}}{2})}^{\frac{1}{2}} α_{t}^{x} H_{01}$
$y$ -tilt	$α_{t}^{y} y$	${(\frac{ω^{2}}{2})}^{\frac{1}{2}} α_{t}^{y} H_{10}$
Defocus	$α_{d} (x^{2} + y^{2})$	$\frac{ω^{2}}{2} α_{d} (H_{20} + H_{02} + \sqrt{2} H_{00})$
0° astigmatism	$α_{at}^{0} (x^{2} - y^{2})$	$\frac{ω^{2}}{2} α_{at}^{0} (H_{02} - H_{20})$
45° astigmatism	$α_{at}^{45} x y$	$\frac{ω^{2}}{2} α_{at}^{45} H_{11}$
$x$ -primary coma	$α_{co}^{x} x (x^{2} + y^{2})$	${(\frac{ω^{2}}{2})}^{\frac{3}{2}} α_{co}^{x} (\frac{\sqrt{3}}{2} H_{03} + \frac{1}{2} H_{21} + \sqrt{2} H_{01})$
$y$ -primary coma	$α_{co}^{y} y (x^{2} + y^{2})$	${(\frac{ω^{2}}{2})}^{\frac{3}{2}} α_{co}^{x} (\frac{\sqrt{3}}{2} H_{30} + \frac{1}{2} H_{12} + \sqrt{2} H_{10})$
Spherical aberration	$α_{s s} {(x^{2} + y^{2})}^{2}$	${(\frac{ω^{2}}{2})}^{2} α_{s s} (\frac{\sqrt{6}}{2} H_{40} + \frac{\sqrt{6}}{2} H_{04} + H_{22} + 2 \sqrt{2} H_{20} + 2 \sqrt{2} H_{02} + 2 H_{00})$

Zernike Terms	Hermite Terms
$Z_{0}^{0}$	$H_{00}$
$Z_{1}^{- 1}$	$\frac{1}{γ} \frac{\sqrt{2}}{2} H_{10}$
$Z_{1}^{1}$	$\frac{1}{γ} \frac{\sqrt{2}}{2} H_{01}$
$Z_{2}^{- 2}$	$\frac{1}{γ^{2}} \frac{\sqrt{3}}{2} H_{11}$
$Z_{2}^{0}$	$\frac{1}{γ^{2}} \sqrt{3} (H_{20} + H_{02}) + (\frac{1}{γ^{2}} \sqrt{2} - \sqrt{3}) H_{00}$
$Z_{2}^{2}$	$\frac{1}{γ^{2}} \frac{\sqrt{6}}{2} (H_{02} - H_{20})$
$Z_{3}^{- 3}$	$\frac{1}{γ^{3}} (\frac{\sqrt{3} ω^{3}}{2} H_{30} - \frac{3 ω^{3}}{2} H_{21})$
$Z_{3}^{- 1}$	$\frac{1}{γ^{3}} (\frac{3 \sqrt{3}}{2} H_{30} + \frac{3}{2} H_{12}) + (\frac{3 \sqrt{8}}{2} \frac{1}{γ^{3}} + \sqrt{8} \frac{1}{γ}) H_{10}$
$Z_{3}^{1}$	$\frac{1}{γ^{3}} (\frac{3 \sqrt{3}}{2} H_{03} + \frac{3}{2} H_{21}) + (\frac{3 \sqrt{8}}{2} \frac{1}{γ^{3}} + \sqrt{8} \frac{1}{γ}) H_{01}$
$Z_{3}^{3}$	$\frac{1}{γ^{3}} (- \frac{\sqrt{3} ω^{3}}{2} H_{03} + \frac{3 ω^{3}}{2} H_{12})$
$Z_{4}^{0}$	$\frac{1}{γ^{4}} (\frac{3 \sqrt{30}}{4} H_{40} + \frac{3 \sqrt{30}}{4} H_{04} + \frac{3 \sqrt{5}}{2} H_{22}) + (\frac{3 \sqrt{10}}{γ^{4}} - \frac{3 \sqrt{5}}{γ^{2}}) (H_{20} + H_{02}) + (\frac{3 \sqrt{5}}{γ^{4}} - \frac{3 \sqrt{10}}{γ^{2}} + \sqrt{5}) H_{00}$

Name	Algebraic Expressions	Hermite Terms
Piston	$α_{p}$	$α_{p} H_{00}$
$x$ -tilt	$α_{t}^{x} x$	${(\frac{ω^{2}}{2})}^{\frac{1}{2}} α_{t}^{x} H_{01}$
$y$ -tilt	$α_{t}^{y} y$	${(\frac{ω^{2}}{2})}^{\frac{1}{2}} α_{t}^{y} H_{10}$
Defocus	$α_{d} (x^{2} + y^{2})$	$\frac{ω^{2}}{2} α_{d} (H_{20} + H_{02} + \sqrt{2} H_{00})$
0° astigmatism	$α_{at}^{0} (x^{2} - y^{2})$	$\frac{ω^{2}}{2} α_{at}^{0} (H_{02} - H_{20})$
45° astigmatism	$α_{at}^{45} x y$	$\frac{ω^{2}}{2} α_{at}^{45} H_{11}$
$x$ -primary coma	$α_{co}^{x} x (x^{2} + y^{2})$	${(\frac{ω^{2}}{2})}^{\frac{3}{2}} α_{co}^{x} (\frac{\sqrt{3}}{2} H_{03} + \frac{1}{2} H_{21} + \sqrt{2} H_{01})$
$y$ -primary coma	$α_{co}^{y} y (x^{2} + y^{2})$	${(\frac{ω^{2}}{2})}^{\frac{3}{2}} α_{co}^{x} (\frac{\sqrt{3}}{2} H_{30} + \frac{1}{2} H_{12} + \sqrt{2} H_{10})$
Spherical aberration	$α_{s s} {(x^{2} + y^{2})}^{2}$	${(\frac{ω^{2}}{2})}^{2} α_{s s} (\frac{\sqrt{6}}{2} H_{40} + \frac{\sqrt{6}}{2} H_{04} + H_{22} + 2 \sqrt{2} H_{20} + 2 \sqrt{2} H_{02} + 2 H_{00})$

Zernike Terms	Hermite Terms
$Z_{0}^{0}$	$H_{00}$
$Z_{1}^{- 1}$	$\frac{1}{γ} \frac{\sqrt{2}}{2} H_{10}$
$Z_{1}^{1}$	$\frac{1}{γ} \frac{\sqrt{2}}{2} H_{01}$
$Z_{2}^{- 2}$	$\frac{1}{γ^{2}} \frac{\sqrt{3}}{2} H_{11}$
$Z_{2}^{0}$	$\frac{1}{γ^{2}} \sqrt{3} (H_{20} + H_{02}) + (\frac{1}{γ^{2}} \sqrt{2} - \sqrt{3}) H_{00}$
$Z_{2}^{2}$	$\frac{1}{γ^{2}} \frac{\sqrt{6}}{2} (H_{02} - H_{20})$
$Z_{3}^{- 3}$	$\frac{1}{γ^{3}} (\frac{\sqrt{3} ω^{3}}{2} H_{30} - \frac{3 ω^{3}}{2} H_{21})$
$Z_{3}^{- 1}$	$\frac{1}{γ^{3}} (\frac{3 \sqrt{3}}{2} H_{30} + \frac{3}{2} H_{12}) + (\frac{3 \sqrt{8}}{2} \frac{1}{γ^{3}} + \sqrt{8} \frac{1}{γ}) H_{10}$
$Z_{3}^{1}$	$\frac{1}{γ^{3}} (\frac{3 \sqrt{3}}{2} H_{03} + \frac{3}{2} H_{21}) + (\frac{3 \sqrt{8}}{2} \frac{1}{γ^{3}} + \sqrt{8} \frac{1}{γ}) H_{01}$
$Z_{3}^{3}$	$\frac{1}{γ^{3}} (- \frac{\sqrt{3} ω^{3}}{2} H_{03} + \frac{3 ω^{3}}{2} H_{12})$
$Z_{4}^{0}$	$\frac{1}{γ^{4}} (\frac{3 \sqrt{30}}{4} H_{40} + \frac{3 \sqrt{30}}{4} H_{04} + \frac{3 \sqrt{5}}{2} H_{22}) + (\frac{3 \sqrt{10}}{γ^{4}} - \frac{3 \sqrt{5}}{γ^{2}}) (H_{20} + H_{02}) + (\frac{3 \sqrt{5}}{γ^{4}} - \frac{3 \sqrt{10}}{γ^{2}} + \sqrt{5}) H_{00}$

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Applied Optics