Modes in General Gaussian Optical Systems Using Quantum-Mechanics Formalism

Feldmann, M.

doi:10.1364/josa.61.000446

Cited by 11 publications

(4 citation statements)

References 5 publications

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“…Now, the ellipse with semiaxes a and b, in x , y components, has E = (a, i b) exp(i t 0 ) for suitably chosen t 0 ; therefore it has Stokes parameters S 0 = 2H = a 2 + b 2 and S 3 = 2HZ = 2ab. Substituting appropriate expressions for H and Z in place of a and b in (14) gives…”

Section: Semiclassical Picture Of the Poincar é Sphere For Gaussian B...mentioning

confidence: 99%

“…All of the discussion of Gaussian modes will be restricted to their amplitude distribution in the focal plane (z = 0), so a fundamental Gaussian beam has amplitude (2/π ) 1/2 w −1 0 exp(−[x 2 + y 2 ]/w 2 0 ), where w 0 represents the waist width of the beam [13], and is normalized (its square, integrated over the plane, gives unity). This Gaussian has the same functional form as the ground state of a two-dimensional quantum harmonic oscillator, which is justified physically [8,[13][14][15] in terms of the curved mirrors in the laser cavity having the effect on the paraxially propagating wave, in the focal plane, of a harmonic potential.…”

Section: Introductionmentioning

confidence: 99%

“…, where w 0 represents the waist width of the beam [13], and is normalized (its square, integrated over the plane, gives unity). This Gaussian has the same functional form as the ground state of a 2D quantum harmonic oscillator, which is justified physically [8,[13][14][15] in terms of the curved mirrors in the laser cavity having the effect on the paraxially-propagating wave, in the focal plane, of a harmonic potential.…”

Section: Introductionmentioning

confidence: 99%

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Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams

Dennis

Alonso

2017

Phil. Trans. R. Soc. A.

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The connection between Poincaré spheres for polarization and Gaussian beams is explored, focusing on the interpretation of elliptic polarization in terms of the isotropic two-dimensional harmonic oscillator in Hamiltonian mechanics, its canonical quantization and semiclassical interpretation. This leads to the interpretation of structured Gaussian modes, the Hermite-Gaussian, Laguerre-Gaussian and generalized Hermite-Laguerre-Gaussian modes as eigenfunctions of operators corresponding to the classical constants of motion of the two-dimensional oscillator, which acquire an extra significance as families of classical ellipses upon semiclassical quantization.This article is part of the themed issue 'Optical orbital angular momentum'.

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Section: Semiclassical Picture Of the Poincar é Sphere For Gaussian B...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams

Dennis

Alonso

2017

Phil. Trans. R. Soc. A.

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“…As such, HG and LG modes are the most familiar and studied examples of structured light beams.This familiarity desensitizes us to a remarkable property all of these beams share-up to a scaling, their intensity profile is the same in every plane, even to the far field. This is readily accounted for mathematically by the fact that Gaussian beam families are eigenfunctions of (isotropic fractional) Fourier transformations, or in the analogy between paraxial optics and 2+1-dimensional quantum mechanics, that two-dimensional harmonic oscillator eigenfunctions spread, under evolution by the free Schrödinger equation, in a formpreserving way [4][5][6]. The two-dimensional quantum harmonic oscillator itself is analogous, via the 'Schwinger oscillator model' [7,8], to the quantum theory of angular momentum: Gaussian beams become analogous to spin directions in an abstract space (LG vertical, HG horizontal) with mode order proportional to total spin [9].…”

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confidence: 99%

Gaussian mode families from systems of rays

Dennis

Alonso

2019

J. Phys. Photonics

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Gaussian mode families, including Laguerre-Gaussian, Hermite-Gaussian and generalized Hermite-Laguerre-Gaussian beams, can be described via a geometric optics construction. Ray families crossing the focal plane are represented as one-parameter families of ellipses, parametrized by curves on an analog Poincaré sphere for rays. We derive the optical path length weighting the rays, and find it to be related to the Pancharatnam-Berry connection on the Poincaré sphere. Dressing the rays with Gaussian beams, the approximation returns the Laguerre-, Hermite-and generalized Hermite-Laguerre-Gaussian beams exactly. The approach strengthens the connection between structured light and Hamiltonian optics, opening the possibility to new structured Gaussian beams. IntroductionGaussian beams are ubiquitous in contemporary optics as the simplest model of paraxially propagating, monochromatic light beams, as are the mode sets based on them, the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes [1-3]. The fundamental Gaussian and HG modes are familiar from the laser laboratory as modes of laser cavities, and LG modes epitomize structured light beams, carrying quantized orbital angular momentum and optical vortices on their axis. As such, HG and LG modes are the most familiar and studied examples of structured light beams.This familiarity desensitizes us to a remarkable property all of these beams share-up to a scaling, their intensity profile is the same in every plane, even to the far field. This is readily accounted for mathematically by the fact that Gaussian beam families are eigenfunctions of (isotropic fractional) Fourier transformations, or in the analogy between paraxial optics and 2+1-dimensional quantum mechanics, that two-dimensional harmonic oscillator eigenfunctions spread, under evolution by the free Schrödinger equation, in a formpreserving way [4][5][6]. The two-dimensional quantum harmonic oscillator itself is analogous, via the 'Schwinger oscillator model' [7,8], to the quantum theory of angular momentum: Gaussian beams become analogous to spin directions in an abstract space (LG vertical, HG horizontal) with mode order proportional to total spin [9]. The simplest example of mode order N=1 therefore resembles the Poincaré sphere for polarization [10], giving rise to the celebrated Poincaré sphere for modes [11][12][13][14][15][16]. Modes corresponding to an arbitrary abstract spin direction are the generalised Hermite-Laguerre-Gaussian (GG) beams [17,18].In [19] we proposed a new and broad definition of structured Gaussian beams, using the mathematics of semiclassical ray optics, based on the self-similarity of propagating Gaussian beams. This involved identifying each Gaussian beam with a two-parameter family of light rays, potentially overlapping, each of which is weighted by a complex amplitude: the sum of ray amplitudes at each point approximates the complex amplitude of the propagating wave field. This approximation can be improved by replacing each linear ray with a Gaussian beam whose axis correspon...

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Quantum‐optical analogies using photonic structures

Longhi¹

2009

Laser & Photonics Reviews

672

579

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Engineered photonic waveguides have provided in the past decade an extremely rich laboratory tool to visualize with optical waves the classic analogues of a wide variety of coherent quantum phenomena encountered in atomic, molecular or condensed-matter physics. As compared to quantum systems, optics offers the rather unique advantage of a direct mapping of the wave function evolution in coordinate space by simple fluorescence imaging or scanning tunneling optical microscopy techniques. In this contribution recent theoretical and experimental advances in the field of quantum-optical analogies are reviewed. Special attention is devoted to some relevant optical analogies based on the use of curved photonic structures, including: coherent destruction of tunneling in driven bistable potentials; coherent population transfer and adiabatic passage in laser-driven multilevel atomic systems; quantum decay control and Zeno dynamics; electronic Bloch oscillations and Zener tunneling, Anderson localization and dynamic localization in crystalline potentials. Curvilinear Propagation distance (cm) u u ( m) m 8 6 4 2 0 0 20 40 60 Curvilinear Propagation distance (cm) u Beam Center of Mass n > > 0 2 4 6 8 0 2 4 6 8 10 u u 0 Input Beam R a Optical analogue of quantum particle bouncing on a lattice under the gravitational field. A light wave packet periodically bounces the edge of a circularly-curved waveguide array, showing collapses and revivals.

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Modes in General Gaussian Optical Systems Using Quantum-Mechanics Formalism

Cited by 11 publications

References 5 publications

Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams

Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams

Gaussian mode families from systems of rays

Quantum‐optical analogies using photonic structures

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