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.

© 2012 Optical Society of America

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References

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2011

2004

1997

1996

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[CrossRef]

1995

1994

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[CrossRef]

1993

1991

1990

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

1989

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

1986

Alda, J.

Alonso, J.

Arias, M.

Bastiaans, M. J.

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

Belanger, P. A.

Bernabeu, E.

Forbes, A.

Hu, P. H.

Lu, B.

Luo, S.

Mafusire, C.

Mahajan, V. N.

Martinez-Herrero, R.

Meijas, P. M.

Mejias, P. M.

Porras, M. A.

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[CrossRef]

Ruff, J. A.

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[CrossRef]

Siegman, A. E.

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[CrossRef]

A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32, 5893–5091 (1993).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, Lasers (University Science, 1986) pp. 663–695.

Stanley, T.

Stone, J.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[CrossRef]

Optik

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

Proc. SPIE

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Other

A. E. Siegman, Lasers (University Science, 1986) pp. 663–695.

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Figures (3)

Fig. 1.
Fig. 1.

Outline of the Hermite polynomials with m, n4.

Fig. 2.
Fig. 2.

Model of quartic aberration induced by a lens.

Fig. 3.
Fig. 3.

Beam quality factor of Gaussian beams with different Zernike aberrations (λ=632.8nm).

Tables (2)

Tables Icon

Table 1. Hermite Expansion of Common Aberrations

Tables Icon

Table 2. Hermite Expansion of 11 Former Zernike Terms

Equations (19)

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Mx2=4πx2θx2xθx2,
E˜(x,y)=u(x,y)eikφ(x,y),
x2=|E˜|x2dxdy=u2x2dxdy,
θx2=14π2θx2|E˜(θx)|2dxdy(12πθx|E˜(θx)|2dxdy)2.
θx2=14π2((ux)2+4π2λ2(uφx)2)dxdy14π2(u2φxdxdy)2,
xθx=12πu2xφxdxdy,
u(x,y)=u0ex2+y2ω2,
φ(x,y)=n=0m=0AmnHn(2xω)Hm(2yω),
Amn=2πω2φ(x,y)Hn(2xω)Hm(2yω)e(2x2+y2ω2)dxdy.
x2=ω24,
θx2=4ω2+64π2ω2λ2m,n(nAmn2)m,n=0,1,2,3,&[m,n][0,1],
xθx=42πλA02,
Mx2=1+(Mφx2)2,
My2=1+(Mφy2)2,
(Mφx2)2=16π2λ2m,n(nAmn2)m,n=0,1,2,3,&[m,n][0,1],[0,2],
(Mφy2)2=16π2λ2m,n(mAmn2)m,n=0,1,2,3,&[m,n][1,0],[2,0].
M2=Mx2=(1+8π2ω8λ2αss2)12.
φ=(ω2)4(62H40+322H20+34H00).
Mx2=(1+6π2ω8λ2αsx2)12.

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