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Angular spectrum of quantized light beams

Andrea Aiello, Jorrit Visser, Gerard Nienhuis, J. P. Woerdman

Published Jan 1, 2006• 3 citations
OpenAlex

Abstract

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 (

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Angular spectrum of quantized light beams

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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) nonparaxial fields. For the first case the well-known paraxial results are fully recovered.

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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 (
A quantum version of the classical paraxial approximation is established. After expanding the classical free-field wave equation in powers of \ensuremath{\theta} (the characteristic opening angle of the paraxial rays), the familiar time-dependent paraxial equation appears in zeroth order. The arguments employed in deriving this approximation scheme are used in turn to identify the subspace of the photon Fock space consisting of paraxial states of the field. The quantum version of the paraxial approximation is then obtained by restricting the Maxwell field operators to this domain. Exact equal-time commutation relations are recovered as expansions in \ensuremath{\theta}. In zeroth order, the theory yields a quantized analog of the classical paraxial wave equation, and formally resembles a nonrelativistic many-body theory. This formalism is applied to show that Mandel's local-photon-number operator and Glauber's photon-counting operator reduce, in zeroth order, to the same true-number operator. In addition, it is shown that the O(${\mathrm{\ensuremath{\theta}}}^{2}$) difference between them vanishes for experiments described by stationary coherent states.
This thesis concerns a general theory of beam propa-gation in optics.Well-known basic ideas are critically reviewed, and the analogy between quantum mechanics and paraxial optics is established through a number of n~w refinements.A parabolic equation for the electrodynamics wave relation is derived using the relativistic front-description of paraxial optics.The •packet-like beam solutions to this equation are shown to be hermite functions of complex argument.1 I The role and applications of canonical transformations in paraxial optics is investigated, attention is focused on some extensions of linear transformations mapping the position and momentum operators "q, '~ to a pair sands of canonically conjugate but not necessarily Hermitian operators.It is shown that for a linear transformation a related space 'jt. of entire functions exists with an inner product over the complex plane, such that the pair s, scan be realized in the Schrbdinger representation.A unitary mapping onto the ordinary Hilbert space of square-integrable functions over :l,t through an integral transform is provided.The well known Fresnel transform is a special case when sands are the harmonic oscillator raising and lowering operators, whose eigen-' functions are the Hermite functions of complex argument.With these functions the use of mathematical distributions in Fourier optics is extended.A generalization of the Cauchy problem is presented, which expresses the field in the plane l = o in terms of the field in any plane \ =~1 > o.The solution is obtained through an extensive use of the Hermite expansion of a distribution.While this expansion is not new, the fact that is not made in the context of the Hilbert space it is.The results obtained establish the applicability of special functions in optics, as well as a precise knowledge of the meaning and accuracy of paraxial wave optics.Applications of the theory to homogeneous and inhomogeneous medium are presented in particular in a selfcontained chapter about relativistic self-focussing in a non-linear medium.This work is primarily concerned with describing 1 1 1 an alternative approach to the study of a pair of canonical operators and it provides new insight over the role of symplectic geometry in quantum mechanics and optics.Chapter I canon i ea 1 tr ans for•ms.Const rue t ion an<1 app l i ea ti ons.< 19) .
The classical concept of a paraxial wave is extended to quantum optics as a condition on the state vector, and the family of vectors obeying this condition defines a paraxial Hilbert space. Field operators acting on this space are approximated by the product of a slowly-varying envelope operator and a classical carrier wave. This permits the discussion of Gaussian pulses and paraxial wave packets. This treatment is extended to obtain an expansion of state vectors and field operators in powers of the opening angle of the cone of paraxial rays. The leading term in the expansion yields the component of the angular momentum along the propagation axis as a sum of well defined orbital- and spin-parts. With the same technique, it is shown that approximate localization of photons is possible for localization volumes much larger that the mean wavelength in the paraxial state.
This tutorial gives an overview of the use of the Wigner function as a tool for modeling optical field propagation. Particular emphasis is placed on the spatial propagation of stationary fields, as well as on the propagation of pulses through dispersive media. In the first case, the Wigner function gives a representation of the field that is similar to a radiance or weight distribution for all the rays in the system, since its arguments are both position and direction. In cases in which the field is paraxial and where the system is described by a simple linear relation in the ray regime, the Wigner function is constant under propagation along rays. An equivalent property holds for optical pulse propagation in dispersive media under analogous assumptions. Several properties and applications of the Wigner function in these contexts are discussed, as is its connection with other common phase-space distributions like the ambiguity function, the spectrogram, and the Husimi, P, Q, and Kirkwood–Rihaczek functions. Also discussed are modifications to the definition of the Wigner function that allow extending the property of conservation along paths to a wider range of problems, including nonparaxial field propagation and pulse propagation within general transparent dispersive media.