The ABCD matrix formalism describing paraxial propagation of optical beams across linear systems is generalized to arbitrary beam trajectories. As a by-product of this study, a one-to-one correspondence between the extended ABCD matrix formalism presented here and the quantum mechanical translation operator is established.
We develop the mathematical properties of the ABCD -transformation from the classical and quantum mechanical points of view. First, we list the four types of generating function which generate the ABCD -transformation in classical mechanics. Second, we introduce the unitary operator of the ABCD -transformation. Next, we calculate the normal ordering of this unitary operator and derive the kernels in coordinate-momentum phase space. The kernels are comprised of the generating functions, which generate the ABCD -transformation in classical mechanics. This reveals a new correspondence between classical and quantum mechanics. As an application of these kernels, we show the propagation of coherent and Gaussian beams in the context of quantum optics which corresponds to the ABCD -transformation in matrix optics.
Corresponding to the Fresnel transform there exists a unitary operator in quantum optics theory, which could be known the Fresnel operator (FO). We show that the multiplication rule of the FO naturally leads to the quantum optical ABCD law. The canonical operator methods as mapping of ray-transfer ABCD matrix is explicitly shown by the normally ordered expansion of the FO through the coherent state representation and the technique of integration within an ordered product of operators. We show that time evolution of the damping oscillator embodies the quantum optical ABCD law.
We introduce a generalized angular spectrum representation for quantized light beams. By using our formalism, we are able to derive simple expressions for the electromagnetic vector potential operator in the case of: a) time-independent paraxial fields, b) time-dependent paraxial fields, and c) non-paraxial fields. For the first case, the well known paraxial results are fully recovered. c ○ 2008 Optical Society of America OCIS codes: 000.1600, 270.0270. Propagation of nonclassical states of the electromagnetic field is an issue of growing interest in quantum optics for both fundamental and technological purposes. 1 Consider, for instance, the relevance of propagation of entangled photons to quantum cryptographic systems. 2 The purpose of the present Letter is to provide a novel perfectly general formalism for the representation of quantized light beams which can be used in any regime of propagation. Such objective is achieved by using a dispersion relation that some of us have recently introduced in Ref., 3 and a generalized angular spectrum representation for the field operators. 4 The usefulness of our approach5 becomes manifest whenever one deals with quantum systems for which both paraxial and nonparaxial regimes of propagation may be relevant as, e.g., down-converted photon pairs. 6 Consider the plane-wave expansion of the positivefrequency part of the electromagnetic vector potential operator Â(r, t) = Â(+) (r, t) + H.c. in the Coulomb gauge7 Â (+) ∫ (r, t) = d 3 (
We give a brief review of the theory of quantum and optical universal invariants, i.e., certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators or the transverse phase-space coordinates of optical paraxial beams that are preserved in time (or along the axis of the beam) independently of the concrete form of the coefficients of the Hamiltonian or the parameters of the optical system, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of certain finite-dimensional algebras. Using the phase-space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation between the quantum invariants and the classical universal integral invariants of Poincaré and Cartan. The specific features of the Gaussian beams are discussed as examples.