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On elliptic trigonometric form of the Zernike system and polar limits
摘要： The Zernike system provides orthogonal polynomial solution bases on the unit disk that separate in coordinates that are generically elliptic. This is a superintegrable system whose optical realization is a scalar wavefield on the plane of a circular pupil. Here we describe the solution set in the trigonometric form of elliptic coordinates expressed in terms of special functions, and examine closely the two limits where the explicit form of the wavefunctions in elliptic coordinates reduce to wavefunctions in polar coordinates.
关键词： separation of variables,elliptic trigonometric coordinates,Zernike system,limits to polar spherical coordinates
更新于2025-09-23 15:22:29
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Scattering of a Plane Electromagnetic Wave by a Multilayer Spherical Lens
摘要： We propose an analytical solution of the problem of diffraction of a plane electromagnetic wave by a multilayer dielectric (including plasmon) sphere. The solution is obtained using the method of separation of variables. New efficient recurrence relationships are obtained for calculations of the fields in layers, as well as formulas for the fields in the near and far diffraction zones. The novelty of the proposed solution is connected with the way of representing its radial part in the form of normalized functions. It is shown that as the number of the lens layers, which approximate the smooth profile of dielectric permittivity, grows, the electric field at the focusing point increases and reaches the maximum value. This allows one to determine the minimum required number of layers in practical problems. Resonance properties of metal-dielectric nanoparticles are studied in the optical band.
关键词： diffraction,multilayer dielectric sphere,plane electromagnetic wave,separation of variables,optical band,metal-dielectric nanoparticles,resonance properties,recurrence relationships,focusing point
更新于2025-09-10 09:29:36