Editing Gaussian beam (section)

==Higher-order modes==
{{see also|Transverse mode}}

=== Hermite-Gaussian modes ===
<!--Hermite-Gaussian mode redirects here-->
[[Image:Hermite-gaussian.png|thumb|right|Twelve Hermite-Gaussian modes]]
It is possible to decompose a coherent paraxial beam using the orthogonal set of so-called ''Hermite-Gaussian modes'', any of which are given by the product of a factor in {{mvar|x}} and a factor in {{mvar|y}}. Such a solution is possible due to the separability in {{mvar|x}} and {{mvar|y}} in the [[paraxial Helmholtz equation]] as written in [[Cartesian coordinate system|Cartesian coordinates]].<ref>Siegman (1986), p645, eq. 54</ref> Thus given a mode of order {{math|(''l'', ''m'')}} referring to the {{mvar|x}} and {{mvar|y}} directions, the electric field amplitude at {{math|''x'', ''y'', ''z''}} may be given by:
<math display="block"> E(x,y,z) = u_l(x,z) \, u_m(y,z) \, \exp(-ikz), </math> 
where the factors for the {{mvar|x}} and {{mvar|y}} dependence are each given by:
<math display="block">
u_J(x,z) = \left(\frac{\sqrt{2/\pi}}{ 2^J \, J! \; w_0}\right)^{\!\!1/2} \!\! \left( \frac{{q}_0}{{q}(z)}\right)^{\!\!1/2} \!\! \left(- \frac{{q}^\ast(z)}{{q}(z)}\right)^{\!\! J/2} \!\! H_J\!\left(\frac{\sqrt{2}x}{w(z)}\right)  \, \exp \left(\! -i \frac{k x^2}{2 {q}(z)}\right) ,
</math>
where we have employed the complex beam parameter {{math|''q''(''z'')}} (as defined above) for a beam of waist {{math|''w''<sub>0</sub>}} at {{mvar|z}} from the focus. In this form, the first factor is just a normalizing constant to make the set of {{math|''u<sub>J</sub>''}} [[orthonormal]]. The second factor is an additional normalization dependent on {{mvar|z}} which compensates for the expansion of the spatial extent of the mode according to {{math|''w''(''z'')/''w''<sub>0</sub>}} (due to the last two factors). It also contains part of the Gouy phase. The third factor is a pure phase which enhances the Gouy phase shift for higher orders {{mvar|J}}.

The final two factors account for the spatial variation over {{mvar|x}} (or {{mvar|y}}). The fourth factor is the [[Hermite polynomial]] of order {{mvar|J}} ("physicists' form", i.e. {{math|1=''H''<sub>1</sub>(''x'') = 2''x''}}), while the fifth accounts for the Gaussian amplitude fall-off {{math|exp(−''x''<sup>2</sup>/''w''(''z'')<sup>2</sup>)}}, although this isn't obvious using the complex {{mvar|q}} in the exponent. Expansion of that exponential also produces a phase factor in {{mvar|x}} which accounts for the wavefront curvature ({{math|1/''R''(''z'')}}) at {{mvar|z}} along the beam.

Hermite-Gaussian modes are typically designated "TEM<sub>''lm''</sub>"; the fundamental Gaussian beam may thus be referred to as TEM<sub>00</sub> (where ''TEM'' is ''[[transverse mode|transverse electro-magnetic]]''). Multiplying {{math|''u<sub>l</sub>''(''x'', ''z'')}} and {{math|''u<sub>m</sub>''(''y'', ''z'')}} to get the 2-D mode profile, and removing the normalization so that the leading factor is just called {{math|''E''<sub>0</sub>}}, we can write the {{math|(''l'', ''m'')}} mode in the more accessible form:

<math display="block">\begin{align}
  E_{l, m}(x, y, z) ={}
    & E_0 \frac{w_0}{w(z)}\, H_l \!\Bigg(\frac{\sqrt{2} \,x}{w(z)}\Bigg)\, H_m \!\Bigg(\frac{\sqrt{2} \,y}{w(z)}\Bigg) \times {}  \exp \left( {-\frac{x^2+y^2}{w^2(z)}} \right) \exp \left( {-i\frac{k(x^2 + y^2)}{2R(z)}} \right) \times {}  \exp \big(i \psi(z)\big) \exp(-ikz).
\end{align}</math>

In this form, the parameter {{math|''w''<sub>0</sub>}}, as before, determines the family of modes, in particular scaling the spatial extent of the fundamental mode's waist and all other mode patterns at {{math|1=''z'' = 0}}. Given that {{math|''w''<sub>0</sub>}}, {{math|''w''(''z'')}} and {{math|''R''(''z'')}} have the same definitions as for the fundamental Gaussian beam described [[#Evolving beam width|above]]. It can be seen that with {{math|1=''l'' = ''m'' = 0}} we obtain the fundamental Gaussian beam described earlier (since {{math|1=''H''<sub>0</sub> = 1}}). The only specific difference in the {{mvar|x}} and {{mvar|y}} profiles at any {{mvar|z}} are due to the Hermite polynomial factors for the order numbers {{mvar|l}} and {{mvar|m}}. However, there is a change in the evolution of the modes' Gouy phase over {{mvar|z}}:
<math display="block"> \psi(z)  = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right), </math>

where the combined order of the mode {{mvar|N}} is defined as {{math|1=''N'' = ''l'' + ''m''}}. While the Gouy phase shift for the fundamental (0,0) Gaussian mode only changes by {{math|±''π''/2}} radians over all of {{mvar|z}} (and only by {{math|±''π''/4}} radians between {{math|±''z''<sub>R</sub>}}), this is increased by the factor {{math|''N'' + 1}} for the higher order modes.<ref name="gouy_phase_shift">{{cite encyclopedia  |title=Gouy Phase Shift |url=https://www.rp-photonics.com/gouy_phase_shift.html |encyclopedia=Encyclopedia of Laser Physics and Technology |publisher=RP Photonics |first=Rüdiger |last=Paschotta |date=12 December 2006 |access-date=May 2, 2014}}</ref>

Hermite Gaussian modes, with their rectangular symmetry, are especially suited for the modal analysis of radiation from lasers whose cavity design is asymmetric in a rectangular fashion. On the other hand, lasers and systems with circular symmetry can better be handled using the set of Laguerre-Gaussian modes introduced in the next section.

=== Laguerre-Gaussian modes ===
<!--Several terms redirect here.-->
[[Image:Intensity profiles of Laguerre-Gaussian modes.pdf|thumb|right|Intensity profiles of the first 12 Laguerre-Gaussian modes.]]
Beam profiles which are circularly symmetric (or lasers with cavities that are  cylindrically symmetric) are often best solved using the  Laguerre-Gaussian modal decomposition.<ref name="goubau"/> These functions are written in [[cylindrical coordinates]] using [[Laguerre polynomials#Generalized Laguerre polynomials|generalized Laguerre polynomials]]. Each transverse mode is again labelled using two integers, in this case the radial index {{math|''p'' ≥ 0}} and the azimuthal index {{mvar|l}} which can be positive or negative (or zero):<ref name="orbital momentum of light">{{cite journal |title=On the properties of circular beams: normalization, Laguerre–Gauss expansion, and free-space divergence |journal=Optics Letters |volume=40 |issue=8 |date=April 8, 2015 |pages=1717–1720 |url=https://doi.org/10.1364/OL.40.001717 |first1=G. |last1=Vallone |doi=10.1364/OL.40.001717|pmid=25872056 |arxiv=1501.07062 |bibcode=2015OptL...40.1717V |s2cid=36312938 }}</ref><ref>{{Cite journal |last1=Miatto |first1=Filippo M. |last2=Yao |first2=Alison M. |last3=Barnett |first3=Stephen M. |date=2011-03-15 |title=Full characterization of the quantum spiral bandwidth of entangled biphotons |url=https://link.aps.org/doi/10.1103/PhysRevA.83.033816 |journal=Physical Review A |language=en |volume=83 |issue=3 |page=033816 |doi=10.1103/PhysRevA.83.033816 |issn=1050-2947|arxiv=1011.5970 |bibcode=2011PhRvA..83c3816M }}</ref>
[[File:Focused Laguerre-Gaussian beam.webm|thumb|right|A Laguerre-Gaussian beam with l=1 and p=0]] 
<math display="block">\begin{align}
  u(r, \phi, z) ={}
    &C^{LG}_{lp}\frac{1}{w(z)}\left(\frac{r \sqrt{2}}{w(z)}\right)^{\! |l|} \exp\! \left(\! -\frac{r^2}{w^2(z)}\right)L_p^{|l|}  \! \left(\frac{2r^2}{w^2(z)}\right) \times {} \\
    &\exp \! \left(\! - i k \frac{r^2}{2 R(z)}\right) \exp(-i l \phi) \, \exp(i \psi(z)) ,
\end{align}</math>

where {{math|''L<sub>p</sub><sup>l</sup>''}} are the [[Laguerre polynomials#Generalized Laguerre polynomials|generalized Laguerre polynomials]]. {{math|''C{{su|p=LG|b=lp}}''}} is a required normalization constant:<ref name="LG_normalization">Note that the normalization used here (total intensity for a fixed {{math|''z''}} equal to unity) differs from that used in section [[#Mathematical form]] for the Gaussian mode. For {{math|1=''l = p = 0''}} the Laguerre-Gaussian mode reduces to the standard Gaussian mode, but due to different normalization conditions the two formulas do not coincide.</ref>
<math display="block">C^{LG}_{lp} = \sqrt{\frac{2 p!}{\pi(p+|l|)!}} \Rightarrow \int_0^{2\pi}d\phi\int_0^\infty dr\; r \,|u(r,\phi,z)|^2=1,</math>.

{{math|''w''(''z'')}} and {{math|''R''(''z'')}} have the same definitions as [[#Beam parameters|above]]. As with the higher-order Hermite-Gaussian modes the magnitude of the Laguerre-Gaussian modes' Gouy phase shift is exaggerated by the factor {{math|''N'' + 1}}:
<math display="block">\psi(z) = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right) ,</math>
where in this case the combined mode number {{math|1=''N'' = {{mabs|''l''}} + 2''p''}}. As before, the transverse amplitude variations are contained in the last two factors on the upper line of the equation, which again includes the basic Gaussian drop off in {{mvar|r}} but now multiplied by a Laguerre polynomial. The effect of the [[rotational modes|rotational mode]] number {{mvar|l}}, in addition to affecting the Laguerre polynomial, is mainly contained in the ''phase'' factor {{math|exp(−''ilφ'')}}, in which the beam profile is advanced (or retarded) by {{mvar|l}} complete {{math|2''π''}} phases in one rotation around the beam (in {{mvar|φ}}). This is an example of an [[optical vortex]] of topological charge {{mvar|l}}, and can be associated with the [[orbital angular momentum of light]] in that mode.

=== Ince-Gaussian modes ===
[[File:Ince Gaussian Modes.jpg|thumb|Transverse amplitude profile of the lowest order even Ince-Gaussian modes.]]
In [[elliptic coordinates]], one can write the higher-order modes using [[Ince polynomial]]s. The even and odd Ince-Gaussian modes are given by<ref name="ince-beams"/>

<math display="block">
u_\varepsilon \left( \xi ,\eta ,z\right) = \frac{w_{0}}{w\left(
z\right) }\mathrm{C}_{p}^{m}\left( i\xi ,\varepsilon \right) \mathrm{C}
_{p}^{m}\left( \eta ,\varepsilon \right) \exp \left[ -ik\frac{r^{2}}{
2q\left( z\right) }-\left( p+1\right) \zeta\left( z\right) \right] ,
</math>
where {{mvar|ξ}} and {{mvar|η}} are the radial and angular elliptic coordinates defined by
<math display="block">\begin{align}
x &= \sqrt{\varepsilon /2}\;w(z) \cosh \xi \cos \eta ,\\
y &= \sqrt{\varepsilon /2}\;w(z) \sinh \xi \sin \eta .
\end{align}</math>
{{math|''C{{su|b=p|p=m}}''(''η'', ''ε'')}} are the even Ince polynomials of order {{mvar|p}} and degree {{mvar|m}} where {{mvar|ε}} is the ellipticity parameter. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for {{math|1=''ε'' = ∞}} and {{math|1=''ε'' = 0}} respectively.<ref name=ince-beams/>

=== 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>