Angular momentum and the geometrical gauge of localized photon states

Hawton, Margaret; Baylis, W. E.

doi:10.1103/physreva.71.033816

Cited by 23 publications

(32 citation statements)

References 15 publications

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“…In spherical coordinates, the transversality condition (2.2) becomes particularly simple: 0 Spherical coordinates provide a description of a plane wave in the field spectrum in the local basis attached to its k -vector. Akin to the laboratory circular-polarization basis 0  u , the circular polarizations defined with respect to the spherical vectors form the helicity basis [82,83]…”

Section: Field Representations and Rotationsmentioning

confidence: 99%

Goos–Hänchen and Imbert–Fedorov beam shifts: an overview

Bliokh

Aiello

2013

J. Opt.

540

514

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We consider reflection and transmission of polarized paraxial light beams at a plane dielectric interface. The field transformations taking into account a finite beam width are described based on the plane-wave representation and geometric rotations. Using geometrical-optics coordinate frames accompanying the beams, we construct an effective Jones matrix characterizing spatial-dispersion properties of the interface. This results in a unified self-consistent description of the Goos-Hänchen and Imbert-Fedorov shifts (the latter being also known as spin-Hall effect of light). Our description reveals intimate relation of the transverse Imbert-Fedorov shift to the geometric phases between constituent waves in the beam spectrum and to the angular momentum conservation for the whole beam. Both spatial and angular shifts are considered as well as their analogues for the higher-order vortex beams carrying intrinsic orbital angular momentum. We also give a brief overview of various extensions and generalizations of the basic beam-shift phenomena and related effects.

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Section: Field Representations and Rotationsmentioning

confidence: 99%

Goos–Hänchen and Imbert–Fedorov beam shifts: an overview

Bliokh

Aiello

2013

J. Opt.

540

514

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“…For unit vectors of the form (9) the z-component of total angular momentum and photon position operators satisfy [13] …”

Section: Position Operatormentioning

confidence: 99%

Photon wave mechanics and position eigenvectors

Hawton

2007

Phys. Rev. A

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One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of qunatum mechanics, this wave function gives the probability to count a photon at any position in space.

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“…The derivation of Hawton's position operator does not rely on the construction of a genuine Hilbert space for a photon. There is also very limited information about the behavior of the corresponding localized states 14,[18][19][20] and position wave functions. In particular, the nature of the electric and magnetic field configurations for a localized photon is not known.…”

Section: Introductionmentioning

confidence: 99%

Quantum mechanics of a photon

Babaei

Mostafazadeh

2017

Journal of Mathematical Physics

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A first-quantized free photon is a complex massless vector field A = (A µ) whose field strength satisfies Maxwell's equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.

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Angular momentum and the geometrical gauge of localized photon states

Cited by 23 publications

References 15 publications

Goos–Hänchen and Imbert–Fedorov beam shifts: an overview

Goos–Hänchen and Imbert–Fedorov beam shifts: an overview

Photon wave mechanics and position eigenvectors

Quantum mechanics of a photon

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