Computing the Casimir energy using the point-matching method

Lombardo, Fernando C.; Mazzitelli, Francisco D.; Vázquez, Mariano; Villar, Paula I.

doi:10.1103/physrevd.80.065018

Cited by 6 publications

(12 citation statements)

References 67 publications

(47 reference statements)

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“…The agreement between dots and dotted lines is extremely good. Similar results can be obtained for the Neumann (TE) modes (see 6 for details). It is worth to remark that, when the amplitude of the corrugation is not very small, the exact results cannot be reproduced with a simple fit of the form y(x) = A * cos(x).…”

Section: Cylindrical Rack and Pinionsupporting

confidence: 82%

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Using Boundary Methods to Compute the Casimir Energy

Lombardo

Mazzitelli

Villar

2010

Quantum Field Theory Under the Influence of External Conditions (QFEXT09)

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We discuss new approaches to compute numerically the Casimir interaction energy for waveguides of arbitrary section, based on the boundary methods traditionally used to compute eigenvalues of the 2D Helmholtz equation. These methods are combined with the Cauchy's theorem in order to perform the sum over modes. As an illustration, we describe a point-matching technique to compute the vacuum energy for waveguides containing media with different permittivities. We present explicit numerical evaluations for perfect conducting surfaces in the case of concentric corrugated cylinders and a circular cylinder inside an elliptic one.

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Section: Cylindrical Rack and Pinionsupporting

confidence: 82%

“…Thus, the matrices P 1 → M 1 and P 2 → M 2 , reobtaining in this way, the usual perfect conductor wave-guide case studied in. 6 For the system of Eq. (11) to have non trivial solutions, the determinant must be zero, i.e.…”

Section: Point-matching Numerical Approachmentioning

confidence: 99%

Using Boundary Methods to Compute the Casimir Energy

Lombardo

Mazzitelli

Villar

2010

Quantum Field Theory Under the Influence of External Conditions (QFEXT09)

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“…Micromechanical torsion oscillators, which have already been used for Casimir experiments [18], seem readily adaptable for testing Eq. (17). Because the overall strength of the Casimir effect is weaker for a disk than for a sphere, observing Casimir forces in this geometry will require greater sensitivities or shorter separation distances than the sphere-plane case.…”

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confidence: 99%

Casimir force at a knife’s edge

Graham¹,

Shpunt²,

Emig³

et al. 2010

Phys. Rev. D

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The Casimir force has been computed exactly for only a few simple geometries, such as infinite plates, cylinders, and spheres. We show that a parabolic cylinder, for which analytic solutions to the Helmholtz equation are available, is another case where such a calculation is possible. We compute the interaction energy of a parabolic cylinder and an infinite plate (both perfect mirrors), as a function of their separation and inclination, H and θ, and the cylinder's parabolic radius R. As H/R → 0, the proximity force approximation becomes exact. The opposite limit of R/H → 0 corresponds to a semi-infinite plate, where the effects of edge and inclination can be probed.PACS numbers: 42.25. Fx, 03.70.+k, Casimir's computation of the force between two parallel metallic plates [1] originally inspired much theoretical interest as a macroscopic manifestation of quantum fluctuations of the electromagnetic field in vacuum. Following its experimental confirmation in the past decade [2], however, it is now an important force to reckon with in the design of microelectromechanical systems [3]. Potential practical applications have motivated the development of numerical methods to compute Casimir forces for objects of any shape [4]. The simplest and most commonly used methods for dealing with complex shapes rely on pairwise summations, as in the proximity force approximation (PFA), which limits their applicability.Recently we have developed a formalism [5,6] that relates the Casimir interaction among several objects to the scattering of the electromagnetic field from the objects individually. (For additional perspectives on the scattering formalism, see references in [6].) This approach simplifies the problem, since scattering is a well-developed subject. In particular, the availability of scattering formulae for simple objects, such as spheres and cylinders, has enabled us to compute the Casimir force between two spheres [5], a sphere and a plate [7], multiple cylinders [8], etc. In this work we show that parabolic cylinders provide another example where the scattering amplitudes can be computed exactly. We then use the exact results for scattering from perfect mirrors to compute the Casimir force between a parabolic cylinder and a plate. In the limiting case when the radius of curvature at its tip vanishes, the parabolic cylinder becomes a semi-infinite plate (a knife's edge), and we can consider how the energy depends on the boundary condition it imposes and the angle it makes to the plane.The surface of a parabolic cylinder in Cartesian coordinates is described by y = (x 2 − R 2 )/2R for all z, as shown in Fig. 1, where R is the radius of curvature at the tip. In parabolic cylinder coordinates [9], defined through x = µλ, y = (λ 2 − µ 2 )/2, z = z, the surface is simply µ = µ 0 = √ R for −∞ < λ, z < ∞. One advantage of the latter coordinate system is that the Helmholtz equationwhich we consider for imaginary wavenumber k = iκ, admits separable solutions. Since sending λ → −λ and µ → −µ returns us to the same point, we restr...

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“…[32]. (Interior geometries were also considered in [33], using large-scale computation, and for the Casimir-Polder interaction in Ref. [34], using the exact tensor Green's function.)…”

Section: Interior Geometriesmentioning

confidence: 99%

Electromagnetic Casimir forces of parabolic cylinder and knife-edge geometries

et al. 2011

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An exact calculation of electromagnetic scattering from a perfectly conducting parabolic cylinder is employed to compute Casimir forces in several configurations. These include interactions between a parabolic cylinder and a plane, two parabolic cylinders, and a parabolic cylinder and an ordinary cylinder. To elucidate the effect of boundaries, special attention is focused on the "knife-edge" limit in which the parabolic cylinder becomes a half-plane. Geometrical effects are illustrated by considering arbitrary rotations of a parabolic cylinder around its focal axis, and arbitrary translations perpendicular to this axis. A quite different geometrical arrangement is explored for the case of an ordinary cylinder placed in the interior of a parabolic cylinder. All of these results extend simply to nonzero temperatures.

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Computing the Casimir energy using the point-matching method

Cited by 6 publications

References 67 publications

Using Boundary Methods to Compute the Casimir Energy

Using Boundary Methods to Compute the Casimir Energy

Casimir force at a knife’s edge

Electromagnetic Casimir forces of parabolic cylinder and knife-edge geometries

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