Hypergeometric-Gaussian Modes

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1D
2D
3D
m
3
p
5
Hypergeometric-Gaussian modes are paraxial wave modes with cylindrical symmetry. The complex amplitude involves a confluent hypergeometric function and a gamma function. Such modes are of importance in laser beam optics.

Details

In cylindrical coordinates, the paraxial equation is
1
r
∂
∂r
r
∂
∂r
+
1
2
r
2
∂
∂
2
ϕ
+2ik
∂
∂z
u(r,ϕ,z)=0
,
and its solutions are of the form
f(m,p,r,ϕ)=
m+p+1
2
πΓ(p+m+1)
m+1
i
m
r
exp(imϕ)
Γ
p
2
+m+1
Γ(m+1)
1
F
1
-
p
2
;m+1;
2
r
.

External Links

Cylindrical Coordinates (Wolfram MathWorld)
Gamma Function (Wolfram MathWorld)
Hypergeometric Function (Wolfram MathWorld)

Permanent Citation

Enrique Zeleny
​
​"Hypergeometric-Gaussian Modes"​
​http://demonstrations.wolfram.com/HypergeometricGaussianModes/​
​Wolfram Demonstrations Project​
​Published: April 29, 2014