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Journal ArticleDOI

Focusing of electromagnetic waves into a dielectric slab: I. Exact and asymptotic results

TL;DR: In this article, exact and asymptotic results for the electromagnetic field inside a dielectric slab in terms of the given field or source current outside the slab were given.
Abstract: We present exact and asymptotic results for the electromagnetic field inside a dielectric slab in terms of the given field or source current outside the slab. Exact results for the reflected and transmitted fields are also given. In particular, we provide explicit exact and asymptotic results for the case in which the incident wave is a converging electromagnetic wave generated by a time-harmonic aperture current. In a separate paper our results are used to obtain exact and asymptotic numerical results for the focused field inside the slab.
Citations
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Journal ArticleDOI
TL;DR: In this paper, Torok et al. derived the Debye approximation for high aperture confocal point spread functions when focusing through a dielectric interface using the Huygens-Fresnel principle.
Abstract: Summary As discussed in recent work (Sheppard, C J R & Torok, P, J Microsc, 185, 366–384; Torok et al, J Microsc, 188, 158–172), two approaches have been used extensively for vectorial computations of high aperture confocal point-spread functions when focusing through a dielectric interface Whereas the equation by Hell, Reiner, Cremer & Stelzer (J Microsc, 169, 391–405) is based on the Huygens–Fresnel principle, the more recent approach by Torok, Varga & Booker (J Opt Soc Am A, 12, 325–332; J Opt Soc Am A, 12, 2136–2144) is based on the Debye approximation While the earlier theory considers a large but finite focal length the second theory is derived for an infinitely high Fresnel number In a high aperture microscope, a high Fresnel number is equivalent to assuming that the focal length be infinitely large with respect to the wavelength So far, the two theories are regarded as different, with the one by Torok et al being rigorous In this paper, we demonstrate that, if the same conditions are applied, the equation by Torok et al can be analytically derived from that by Hell et al Producing the same results, the benefit brought about by the equation by Torok et al is improved flexibility and computational speed for cases with azimuthal symmetry

62 citations


Cites background from "Focusing of electromagnetic waves i..."

  • ...The ubiquity of this problem has spurred many researchers to contribute to this field, both experimentally and theoretically (Ling & Lee, 1984; Wilson & Carlini, 1989; Carlsson, 1991; Gibson & Lanni, 1991; Sheppard & Cogswell, 1991; Dhayalan & Stamnes, 1998; Egner et al., 1998)....

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Journal ArticleDOI
TL;DR: In this article, an integral representation for a possibly high-aperture, finite-Fresnel-number lens and a homogeneous medium of propagation is obtained for vectorial diffraction of electromagnetic waves.
Abstract: The problem of vectorial diffraction of electromagnetic waves is addressed. An integral representation is obtained for a possibly high-aperture, finite-Fresnel-number lens and a homogeneous medium of propagation. The solution is given in terms of coherent superposition of plane electromagnetic waves with position coordinates scaled with the well-known Li–Wolf scaling factor [J. Opt. Soc. Am. A1, 801 (1984)]. This integral representation is then used to obtain formulas for the case in which light is focused through a plane dielectric interface. The solution is given by the linear combination of three functions, each of which consists of only a single integral. The aberration function, representing spherical aberration, is shown to be analytical. Numerical examples are given to demonstrate the effectiveness of the solution.

53 citations

Journal ArticleDOI
TL;DR: In this paper, the authors derived integral representations suitable for studying the focusing of electromagnetic waves through a plane interface into a uniaxial crystal and derived explicit expressions for the dyadic Green's functions associated with the transmitted fields.

48 citations

Journal ArticleDOI
Sjoerd Stallinga1
TL;DR: The effects of birefringence on the light distribution in the focal region of a high-NA optical system are investigated with use of the Debye approach to vector diffraction theory.
Abstract: The effects of birefringence on the light distribution in the focal region of a high-NA optical system are investigated with use of the Debye approach to vector diffraction theory. The attention is limited to uniaxially birefringent media with symmetry axis along the optical axis of the imaging system. The radially (p) and tangentially (s) polarized fields in the exit pupil produce spots in the focal region that are defocused with respect to each other. For small birefringence values the relative defocus causes a distortion and broadening of the spot; for larger values the two spots separate completely. As a corollary to the theory it is shown that there is a tangential tornadolike flow of energy in the focal region when the polarization in the entrance pupil is elliptical.

45 citations

Journal ArticleDOI
TL;DR: Experimental results for focusing of a three-dimensional electromagnetic wave through a plane interface into two different uniaxial crystals, a positive MgF2 crystal and a negative LiNbO3 crystal are presented.

35 citations

References
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Journal ArticleDOI
TL;DR: In this article, an investigation of the structure of the electromagnetic field near the focus of an aplanatic system which images a point source is made, and the results are illustrated by diagrams and in a tabulated form based on data obtained by extensive calculations on an electronic computor.
Abstract: An investigation is made of the structure of the electromagnetic field near the focus of an aplanatic system which images a point source. First the case of a linearly polarized incident field is examined and expressions are derived for the electric and magnetic vectors in the image space. Some general consequences of the formulae are then discussed. In particular the symmetry properties of the field with respect to the focal plane are noted and the state of polarization of the image region is investigated. The distribution of the time-averaged electric and magnetic energy densities and of the energy flow (Poynting vector) in the focal plane is studied in detail, and the results are illustrated by diagrams and in a tabulated form based on data obtained by extensive calculations on an electronic computor. The case of an unpolarized field is also investigated. The solution is riot restricted to systems of low aperture, and the computational results cover, in fact, selected values of the angular semi-aperture a on the image side, in the whole range 0 ≤ α ≤ 90°. The limiting case α → 0 is examined in detail and it is shown that the field is then completely characterized by a single, generally complex, scalar function, which turns out to be identical with that of the classical scalar theory of Airy, Lommel and Struve. The results have an immediate bearing on the resolving power of image forming systems; they also help our understanding of the significance of the scalar diffraction theory, which is customarily employed, without a proper justification, in the analysis of images in lowaperture systems.

2,636 citations

Journal ArticleDOI
TL;DR: In this article, an integral representation for the electromagnetic field in the image space of an optical system is obtained in the form of an angular spectrum of plane waves, which is closely related to that introduced by Luneberg (1944) as a vector generalization of well-known formulae of Debye (1909) and Picht (1925).
Abstract: An integral representation is obtained for the electromagnetic field in the image space of an optical system . This representation, which is not restricted to systems of low angular aperture, is in the form of an angular spectrum of plane waves, and is closely related to that introduced by Luneberg (1944) as a vector generalization of well-known formulae of Debye (1909) and Picht (1925). It is shown that the representation has a simple physical interpretation in terms of a modified Huygens—Fresnel principle which operates with secondary plane waves rather than with secondary spherical waves.

761 citations

Journal ArticleDOI
TL;DR: In this paper, the diffraction problem for a planar interface between two isotropic and homogeneous materials with this interface perpendicular to the optical axis is solved in a rigorous mathematical manner, and it satisfies the homogeneous wave equation.
Abstract: The diffraction of electromagnetic waves for light focused by a high numerical aperture lens from a first material into a second material is treated. The second material has a different refractive index from that of the first material and introduces spherical aberration. We solve the diffraction problem for the case of a planar interface between two isotropic and homogeneous materials with this interface perpendicular to the optical axis. The solution is obtained in a rigorous mathematical manner, and it satisfies the homogeneous wave equation. The electric and magnetic strength vectors are determined in the second material. The solution is in a simple form that can be readily used for numerical computation. A physical interpretation of the results is given, and the paraxial approximation of the solution is derived.

434 citations

Journal ArticleDOI
TL;DR: In this paper, an integral representation for the electromagnetic field in the region of focus of a coherent light beam that emerges from an aplanatic optical system has been derived, which is used to analyze the structure of the focal region.
Abstract: An integral representation for the electromagnetic field in the region of focus of a coherent light beam that emerges from an aplanatic optical system has been derived by Ignatowsky (1919) and by Richards and Wolf (1959). In the present paper this representation is used to analyze the structure of the focal region in a typical case. Contours of the time-averaged electric energy density in the focal plane, in one defocused plane and in two meridional sections of the focal region of a system with angular semi-aperture 45\ifmmode^\circ\else\textdegree\fi{} are presented. The meridional diagrams refer to axial sections through a cylindrical region around the axis near focus, of length $16\ensuremath{\lambda}$ and cross-sectional diameter $10\ensuremath{\lambda}$, where $\ensuremath{\lambda}$ is the wavelength of the light. It is found that the field has a strong longitudinal component at certain points of the focal plane and that longitudinal electric field strengths of the order of ${10}^{5}$ V/cm could now be attained with focused laser beams. A diagram illustrating the complete behavior of the longitudinal component in the focal plane is also given.

166 citations