Laguerre-Gaussian Modes of Paraxial Wave Equation

l

3

p

5

view

2D

3D

Laguerre–Gaussian modes are solutions of the paraxial wave equation. They have circular symmetry and can be written in terms of the Laguerre polynomials

l

L

p

(r)

, where

p

is the radial index and

l

is the azimuthal index.

Details

The paraxial equation in cylindrical coordinates is given by

1

r

∂

∂r

r

∂

∂r

+

1

2

r

2

∂

2

ϕ

+2ik

∂

∂z

u(r,ϕ,z)=0

.

A simple separable solution can be written in the form

u(r,ϕ)=

l

(

2

r)

exp(-

2

r

)

l

L

p

(2

2

r

)exp(ilϕ)exp



2

r

2

exp(i(2p+l+1))

.

External Links

Laguerre Polynomial (Wolfram MathWorld)

Permanent Citation

Enrique Zeleny

"Laguerre-Gaussian Modes of Paraxial Wave Equation"
http://demonstrations.wolfram.com/LaguerreGaussianModesOfParaxialWaveEquation/
Wolfram Demonstrations Project
Published: May 6, 2014