Periodic Orbits for Billiards on an Equilateral Triangle

Baxter, Andrew M.; Umble, Ronald

doi:10.1080/00029890.2008.11920555

Cited by 15 publications

(9 citation statements)

References 8 publications

(5 reference statements)

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“…Here, we examine triangular optical billiards with sharp corners as open optical systems. It has been shown that the properties of trajectories in generic triangular billiards display a rich behavior depending crucially on the realized geometry [13,14,15,16,17].…”

Section: Introductionmentioning

confidence: 99%

Ray picture and ray-wave correspondence in triangular microlasers

Stockschläder¹,

Kreismann²,

Hentschel³

2016

J. Opt.

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We apply ray-optical methods to dielectric optical microcavities in the shape of triangles made of low refractive index material. We find ray trajectories that maximize the intensity inside the cavity to determine the far-field emission characteristics and to complement the concept of the unstable manifold applicable to chaotic microlasers. As these maximum intensity trajectories need not to be periodic, we suggest that they provide a more general explanation for emission patterns of microlasers than short periodic orbits. Further, the geometrical optics description is extended by the inclusion of intensity amplification along the optical path to achieve a better description of active, lasing cavities. Far-field emission patterns of equilateral triangle cavities obtained in this way agree well with our full electromagnetic wave simulations and with previously reported experimental results.

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Section: Introductionmentioning

confidence: 99%

Ray picture and ray-wave correspondence in triangular microlasers

Stockschläder¹,

Kreismann²,

Hentschel³

2016

J. Opt.

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“…If you happen to be conversing with a mathematician, however, the term may instead bring to mind the wellknown dynamical system involving the motion of a particle within a closed domain. The billiard dynamical system has a great deal of intrinsic beauty (e.g., see Figure 1), is a source of mathematical connections (e.g., to number theory [1] and Cantor sets [6]), and has applications to phenomena like the classical motion of gas particles in a closed container [12], optics, and quantum chaos [5]. Even non-mathematicians may be interested in the mathematical version of billiards when they learn that Lewis Carroll investigated billiard trajectories within certain polyhedra [14].…”

Section: Introductionmentioning

confidence: 99%

Sizing Up Outer Billiard Tables

Doğru¹,

Otten²

2011

AJUR

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The outer billiard dynamical system models the motion of a particle around a compact domain, such as a planet orbiting a star. When considering outer billiards in hyperbolic space, an interesting problem is to determine precisely the conditions in which an orbiting particle breaks orbit and escapes to infinity. Past work has classified triangular and Penrose kite billiard tables according to whether or not their orbiting particles escape. This article presents a classification of regular polygonal tables.I.

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“…An equilateral triangle billiard is also well-studied in both its classical and quantum incarnations [49][50][51][52][53][54]. The eigenspectrum for a quantum particle of mass µ trapped in an equilateral triangular box of side length L was obtained in [49,53,54].…”

Section: The Equilateral Triangle Billiardmentioning

confidence: 99%

From Classical Periodic Orbits in Integrable $π$-Rational Billiards to Quantum Energy Spectrum

Panda¹,

Maulik²,

Chakraborty³

et al. 2018

Preprint

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In the present note, we uncover a remarkable connection between the length of periodic orbit of a classical particle enclosed in a class of 2-dimensional planar billiards and the energy of a quantum particle confined to move in an identical region with infinitely high potential wall on the boundary. We observe that the quantum energy spectrum of the particle is in exact oneto-one correspondence with the spectrum of the amplitude squares of the periodic orbits of a classical particle for the class of integrable billiards considered. We have established the results by geometric constructions and exploiting the method of reflective tiling and folding of classical trajectories. We have further extended the method to 3-dimensional billiards for which exact analytical results are scarcely available -exploiting the geometric construction, we determine the exact energy spectra of two new tetrahedral domains which we believe are integrable. We test the veracity of our results by comparing them with numerical results.

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Periodic Orbits for Billiards on an Equilateral Triangle

Abstract: Using elementary methods, we find, classify and count the classes of periodic orbits of a given period on an equilateral triangle.

Cited by 15 publications

References 8 publications

Ray picture and ray-wave correspondence in triangular microlasers

Ray picture and ray-wave correspondence in triangular microlasers

Sizing Up Outer Billiard Tables

From Classical Periodic Orbits in Integrable $π$-Rational Billiards to Quantum Energy Spectrum

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