Generator of spatial evolution of the electromagnetic field

Abstract

Starting with Maxwell's equations and defining normal variables in the Fourier space, we write the equations of temporal evolution of the electromagnetic field with sources in the Hamiltonian and Lagrangian forms, making explicit all intermediate steps often omitted in standard textbooks. Then, we follow the same steps to write the equations of evolution of this field along a spatial dimension in the Hamiltonian and Lagrangian forms. In this way, we arrive at the explicit form of the generator of spatial evolution of the electromagnetic field with sources and show that it has a physical meaning of the modulus of momentum transferred through a given plane orthogonal to the direction of propagation. In a particular case of free field this generator coincides with the projection of the full momentum of the field on the propagation direction, taken with a negative sign. The Hamiltonian and Lagrangian formulations of the spatial evolution are indispensable for a correct quantization of the field when considering its spatial rather than temporal evolution, in particular, for a correct definition of the equal-space commutation relations.

arXiv comment17 pages, 2 figures, final accepted version

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Cited by in corpus (9)

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• Erecting time telescope for photonic quantum networks
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• Time-to-space ghost imaging
• Few-mode squeezing in type-I parametric downconversion by complete group velocity matching
• Spectral and temporal properties of type-II parametric down-conversion: The impact of losses during state generation
• Isoenergetic model for optical downconversion and error-specific limits of the parametric approximation
• Continuous-variable spatio-spectral quantum networks in nonlinear photonic lattices
• Time-resolved second-order autocorrelation function of parametric downconversion