Mathematical Optics

Correspondingly,

$\begin{array}{l}({\hat{\mathrm{p}}}_{-}+i\text{μ}{\hat{\mathcal{L}}}_{-})e^{im\text{φ}}eL{G}_{n,m}(\text{ρ},\text{ζ})\to e^{i(m-1)\text{φ}}eL{G}_{n,m-l}(\text{ρ},\text{ζ}),\\ ({\hat{\mathrm{p}}}_{+}+i\text{μ}{\hat{\mathcal{L}}}_{+})e^{im\text{φ}}eL{G}_{n,m}(\text{ρ},\text{ζ})\to e^{i(m+1)\text{φ}}eL{G}_{n-1,m+1}(\text{ρ},\text{ζ}),\end{array}$

so that one can reach ELG wavefunctions with lower values of n by

$({\widehat{\text{ρ}}}_{+}+i\text{μ}{\hat{\mathcal{L}}}_{+})({\widehat{\text{ρ}}}_{-}+i\text{μ}{\hat{\mathcal{L}}}_{-})e^{im\text{φ}}eL{G}_{n,m}(\text{ρ},\text{ζ})\to e^{im\text{φ}}eL{G}_{n-l,m}(\text{ρ},\text{ζ}).$

The preceding should be compared with (10.40).

It is accordingly easy to verify that the ELG wavefunctions can be obtained from the fundamental 2D Gaussian wavefunction (10.111) by repeated applications of ${\hat{\mathcal{L}}}_{\pm }$ through the scheme (Enderlein and Pampaloni 2004):

${\hat{\mathcal{L}}}_{-}^n{\hat{\mathcal{L}}}_{+}^{n+m}L{G}_{0,0}(\text{ρ},\text{ζ})\to e^{im\text{φ}}eL{G}_{n,m}(\text{ρ},\text{ζ}).$

Likewise, as to the sLG_n,ms, mixed relations follow by applying the operators

$\begin{array}{l}({\widehat{\text{ρ}}}_{-}-i{\text{μ}}^{\ast }{\hat{\mathcal{L}}}_{-})e^{im\text{φ}}sL{G}_{n,m}(\text{ρ},\text{ζ})\to e^{i(m-1)\text{φ}}sL{G}_{n+1,m-1}(\text{ρ},\text{ζ}),\\ ({\widehat{\text{ρ}}}_{+}+i\text{μ}{\hat{\mathcal{L}}}_{+})e^{im\text{φ}}sL{G}_{n,m}(\text{ρ},\text{ζ})\to e^{i(m+1)\text{φ}}sL{G}_{n-1,m+1}(\text{ρ},\text{ζ}),\end{array}$

while raising and lowering exclusively ...