Editing Gaussian beam (section)

=== Hypergeometric-Gaussian modes ===

There is another important class of paraxial wave modes in [[cylindrical coordinates]] in which the [[complex amplitude]] is proportional to a [[confluent hypergeometric function]].

These modes have a [[Mathematical singularity|singular]] phase profile and are [[eigenfunction]]s of the [[photon orbital angular momentum]]. Their intensity profiles are characterized by a single brilliant ring; like Laguerre–Gaussian modes, their intensities fall to zero at the center (on the optical axis) except for the fundamental (0,0) mode. A mode's complex amplitude can be written in terms of the normalized (dimensionless) radial coordinate {{math|1=''ρ'' = ''r''/''w''<sub>0</sub>}} and the normalized longitudinal coordinate {{math|1=''Ζ'' = ''z''/''z''<sub>R</sub>}} as follows:<ref name="Karimi et al. 2007">Karimi et al. (2007)</ref>

<math display="block">\begin{align}
  u_{\mathsf{p}m}(\rho, \phi, \Zeta) {}={}
    &\sqrt{\frac{2^{\mathsf{p} + |m| + 1}}{\pi\Gamma(\mathsf{p} + |m| + 1)}}\; \frac{\Gamma\left(\frac{\mathsf{p}}{2} + |m| + 1\right)}{\Gamma(|m| + 1)}\, i^{|m|+1} \times{} \\
    &\Zeta^{\frac{\mathsf{p}}{2}}\, (\Zeta + i)^{-\left(\frac{\mathsf{p}}{2} + |m| + 1\right)}\, \rho^{|m|} \times{} \\
    &\exp\left(-\frac{i\rho^2}{\Zeta + i}\right)\, e^{im\phi}\, {}_1F_1 \left(-\frac{\mathsf{p}}{2}, |m| + 1; \frac{\rho^2}{\Zeta(\Zeta + i)}\right)
\end{align}</math>

where the rotational index {{mvar|m}} is an integer, and <math> {\mathsf p}\ge-|m| </math> is real-valued, {{math|Γ(''x'')}} is the gamma function and {{math|<sub>1</sub>''F''<sub>1</sub>(''a'', ''b''; ''x'')}} is a confluent hypergeometric function.

Some subfamilies of hypergeometric-Gaussian (HyGG) modes can be listed as the modified Bessel-Gaussian modes, the modified exponential Gaussian modes,<ref name="Karimi et al. 2007"/> and the modified Laguerre–Gaussian modes.

The set of hypergeometric-Gaussian modes is overcomplete and is not an orthogonal set of modes. In spite of its complicated field profile, HyGG modes have a very simple profile at the beam waist ({{math|1=''z'' = 0}}):
<math display="block">u(\rho, \phi, 0) \propto \rho^{\mathsf{p} + |m|}e^{-\rho^2 + im\phi}.</math>