Laguerre Functions in the Mathematical Foundations of the Electromagnetic Theory of the Paraboloidal Reflector

Pinney, Edmund

doi:10.1002/sapm194625149

Cited by 19 publications

(7 citation statements)

References 35 publications

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“…A second solution to the associated Laguerre differential equation has been investigated in the past by Bateman and Pinney in a completely different context, namely, analyzing the paraboloidal reflector [45,46]. Strictly speaking, our approach is totally different from theirs.…”

Section: Hankel Methods To Build the Laguerre Functions Of The Second...mentioning

confidence: 99%

On the physical limitations of structured paraxial beams with orbital angular momentum

Jaimes-Nájera¹,

Gómez-Correa²,

Ugalde-Ontiveros³

et al. 2022

J. Opt.

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In the description of physical systems it is common to discard singular solutions to second order differential equations due to their apparent lack of physical meaning. Nevertheless, it has been demonstrated, using a mathematical-physics approach, that singular solutions can be used in the description of optical beams. In this paper, we construct and study paraxial traveling-waves using the full set of solutions to the paraxial wave equation, and prove that they diverge at infinity. We ascribe that non-physical effect to the paraxial approximation of the Helmholtz equation. Despite this, we show that these traveling waves provide a mathematical-physics framework that unveils orbital angular momentum carrying Laguerre-Gauss beam as the superposition of these traveling waves, and permits a physical description of the self-healing process.

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Section: Hankel Methods To Build the Laguerre Functions Of The Second...mentioning

confidence: 99%

On the physical limitations of structured paraxial beams with orbital angular momentum

Jaimes-Nájera¹,

Gómez-Correa²,

Ugalde-Ontiveros³

et al. 2022

J. Opt.

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“…For integer values of m, the denominator and numerator in Eq. (19) both equal zero, in which case U (m) ν (z) is defined as a limit in m. For large |z|, an asymptotic expansion for U (m) ν (z) exists, given by [1,12]…”

Section: Radial Functions For Largecmentioning

confidence: 99%

“…( 16) and (17). The U (m) ν (z) are second solutions of the Laguerre differential equation, defined by [12]…”

Section: Radial Functions For Large Cmentioning

confidence: 99%

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Solutions to Maxwell's equations using spheroidal coordinates

Zeppenfeld

2009

New J. Phys.

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Analytical solutions to the wave equation in spheroidal coordinates in the short wavelength limit are considered. The asymptotic solutions for the radial function are significantly simplified, allowing scalar spheroidal wave functions to be defined in a form which is directly reminiscent of the Laguerre-Gaussian solutions to the paraxial wave equation in optics. Expressions for the Cartesian derivatives of the scalar spheroidal wave functions are derived, leading to a new set of vector solutions to Maxwell's equations. The results are an ideal starting point for calculations of corrections to the paraxial approximation.

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“…In this respect, it can be viewed as a generalization of the parabolic rotational surfaces and, in fact, many authors call the latter paraboloids (e.g. [8][9][10][11][12][13][14]). Nor should the paraboloidal coordinate system be confused with the parabolic cylinder one [5,15].…”

Section: Introductionmentioning

confidence: 99%

Laplace boundary-value problem in paraboloidal coordinates

Duggen¹,

Willatzen²,

Voon³

2012

Eur. J. Phys.

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This paper illustrates both a problem in mathematical physics, whereby the method of separation of variables, while applicable, leads to three ordinary differential equations that remain fully coupled via two separation constants and a five-term recurrence relation for series solutions, and an exactly solvable problem in electrostatics, as a boundary-value problem on a paraboloidal surface. In spite of the complex nature of the former, it is shown that the latter solution can be quite simple. Results are provided for the equipotential surfaces and electric field lines are given near a paraboloidal conductor.

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Laguerre Functions in the Mathematical Foundations of the Electromagnetic Theory of the Paraboloidal Reflector

Cited by 19 publications

References 35 publications

On the physical limitations of structured paraxial beams with orbital angular momentum

On the physical limitations of structured paraxial beams with orbital angular momentum

Solutions to Maxwell's equations using spheroidal coordinates

Laplace boundary-value problem in paraboloidal coordinates

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