Quantization of Gauss-Hermite and Gauss-Laguerre beams in free space

Starting from the continuous quantization of the transverse electromagnetic field in free space, we construct wavepackets from their Fourier components and form boson annihilation and creation operators for them. Then using a derived equivalence property, we express the ten conservation quantities from the Poincare invariance of space-time which are the energy, momentum, angular momentum and boost by the introduced boson operators and by the integrals over the Fourier components. After deriving the equations of the paraxial approximation with transverse and longitudinal beam shapes, we specialize this to Gauss-Hermite and Gauss-Laguerre beams where we apply the Hermite 2D and Laguerre 2D polynomials and functions. This provides a class of nonstationary beam solutions which includes the continuous transition between Gauss-Hermite and Gauss-Laguerre beams and which we discuss in detail. The quantized form in the Heisenberg representation which needs longitudinal shape propagation with group velocity is obtained by comparing the general quantum-mechanical formulae for the energy and momentum of wavepackets with the special formulae for the beam solutions considered with coefficients to be determined. The calculation of the Fourier spectrum of the quantized Gauss-Hermite and Gauss-Laguerre beam solutions leads back to the starting point of the quantization.