Rayleigh-Sommerfeld Theory Research Articles

Tight focusing of terahertz vortex beams formed by laser dielectric resonator

Wave characteristics of vortex laser beams during their tight focusing have been theoretically studied. The Rayleigh–Sommerfeld theory was used to describe propagation in free space of laser beams excited by the modes of a waveguide dielectric resonator. It is shown that at the topological charge of the spiral phase plate n = 0, the studied EH11 mode has a maximum of radiation intensity on the axis. Introduction of a topological charge leads to the appearance of a minimum of radiation intensity on the axis as well as to the increase in the size of the focal spot. However, for the TE01 mode with the topological charges n = 0 and n = 2, the intensity distribution retains a ring shape, while at n = 1 the beam profile turns into the Gaussian-like one. The wave front in the focal region of the lens for the components of the EH11 and TE01 modes transforms from spherical to spiral one with increasing the topological charge.

Theoretical Zero-Thickness Broadband Holograms Based on Acoustic Sieve Metasurfaces

Acoustic holography is an essential tool for controlling sound waves, generating highly complex and customizable sound fields, and enabling the visualization of sound fields. Based on acoustic sieve metasurfaces (ASMs), this paper proposes a theoretical design approach for zero-thickness broadband holograms. The ASM is a zero-thickness rigid screen with a large number of small holes that allow sound waves to pass through and produce the desired real image in the target plane. The hole arrangement rules are determined using a genetic algorithm and the Rayleigh–Sommerfeld theory. Because the wave from a hole has no extra phase or amplitude modulation, the intractable modulation dispersion can be physically avoided, allowing the proposed ASM-based hologram to potentially function in any frequency band as long as the condition of paraxial approximation is satisfied. Using a numerical simulation based on the combination of the finite element method (FEM) and the boundary element method (BEM), this research achieves broadband holographic imaging with a good effect. The proposed theoretical zero-thickness broadband hologram may provide new possibilities for acoustic holography applications.

CONTROL OVER HIGHER-ORDER TRANSVERSE MODES IN A WAVEGUIDE-BASED QUASI-OPTICAL RESONATOR

Subject and Purpose. The problems under consideration concern selection and focusing of higher-order modes in a waveguide-based dielectric laser. The purpose is to clarify the physics underlying the behavior of, and permitting control over, continuous terahertz-frequency laser beams of various spatial polarizations. Methods and Methodology. The mode parameters of the waveguide-based laser resonator involving an inhomogeneous phase-stepped mirror were calculated in a matrix technique. To analyze the propagation and focusing of the laser beams that can be excited in a variety of diffraction zones by the wave modes of a waveguide-based quasi-optical resonator, a vectorial Rayleigh–Sommerfeld theory was used. The pertinent experimental studies were performed with the use of known measurement methods suitable for the terahertz frequency range. Results. A method for selecting the higher-order EH12q-mode of a terahertz-range laser resonator has been suggested, substantiated theoretically and approbated in experiment. It envisages placing an additional element to perform control over the system’s modal structure, namely a (2.3…2.8) λ-wide groove on the surface of one of the resonator mirrors. This measure can significantly increase losses for all undesirable modes. At the same time, the losses for the higher EH12q-mode remain practically unchanged, which creates conditions for its predominant excitation. Theoretical and experimental studies of moderate and ‘sharp’ focusing in free space of higher-order modes with different spatial polarizations of a dielectric waveguide-based resonator have been carried out. Conclusion. As has been shown, the proposed phase-stepped mirror with a groove can effectively select the higher-order transverse modes that may be required. The linearly polarized EH12q-mode has maximum field intensity in the focal region of the lens employed. For azimuthally polarized TE02q- and TE03q-modes the central lobes, noticeably shifted from the focus of the lens, have a field maximum. An increase in the axial intensity is observed upon ‘sharp’ focusing in the field distribution of the radially polarized TM02q- and TM03q-modes. In this case their central lobes, like those of the higher TE0nq-modes, are noticeably shifted from the lens focus.

A Tutorial on the Classical Theories of Electromagnetic Scattering and Diffraction

AbstractStarting with Maxwell’s equations, we derive the fundamental results of the Huygens-Fresnel-Kirchhoff and Rayleigh-Sommerfeld theories of scalar diffraction and scattering. These results are then extended to cover the case of vector electromagnetic fields. The famous Sommerfeld solution to the problem of diffraction from a perfectly conducting half-plane is elaborated. Far-field scattering of plane waves from obstacles is treated in some detail, and the well-known optical cross-section theorem, which relates the scattering cross-section of an obstacle to its forward scattering amplitude, is derived. Also examined is the case of scattering from mild inhomogeneities within an otherwise homogeneous medium, where, in the first Born approximation, a fairly simple formula is found to relate the far-field scattering amplitude to the host medium’s optical properties. The related problem of neutron scattering from ferromagnetic materials is treated in the final section of the paper.

An accurate design of graphene oxide ultrathin flat lens based on Rayleigh-Sommerfeld theory

Graphene oxide (GO) ultrathin flat lenses have provided a new and viable solution to achieve high resolution, high efficiency, ultra-light weight, integratable and flexible optical systems. Current GO lenses are designed based on the Fresnel diffraction model, which uses a paraxial approximation for low numerical aperture (NA) focusing process. Herein we develop a lens design method based on the Rayleigh-Sommerfeld (RS) diffraction theory that is able to unambiguously determine the radii of each ring without the optimization process for the first time. More importantly, the RS design method is able to accurately design GO lenses with arbitrary NA and focal length. Our design is experimentally confirmed by fabricating high NA GO lenses with both short and long focal lengths. Compared with the conventional Fresnel design methods, the differences in ring positions and the resulted focal length are up to 13.9% and 9.1%, respectively. Our method can be further applied to design high performance flat lenses of arbitrary materials given the NA and focal length requirements, including metasurfaces or other two-dimensional materials.

Analytical formulae of tightly focused Laguerre–Gaussian vector fields

We develop analytical formulae for describing the tight focusing field of the Laguerre–Gaussian vector field with space-variant distribution of states of polarization in the frame of the Rayleigh–Sommerfeld theory. The analytical formulae of three radial, azimuthal and longitudinal components are very beneficial in order to understand the tightly focused field. This result is of great importance to improving the design and engineering the tightly focused field, by using phase and polarization, for applications such as optical tweezers and optical manipulation. This method is also applicable to the homogeneously polarized optical fields and optical vortex fields. In particular, these analytical formulae can also describe the dynamic process of the tightly focused fields, for example, average photon trajectories.

Diffraction of Gaussian and Hermite–Gaussian beams by finite gratings

The diffraction of Hermite–Gaussian beams by N equally spaced slits in a planar screen (a lamellar finite grating) at the scalar diffraction regime is analyzed. We start from the Rayleigh–Sommerfeld theory for two-dimensional problems with Dirichlet conditions and Kirchhoff’s approximation. The theory is presented for beams at oblique incidence; however, in the numerical simulation mainly normal incidence is considered. For Gaussian beams the ratio of the intensity diffracted at normal direction E at minimum and maximum transmitted power is studied in this paper as a function of the beam width L. Also, for Gaussian beams, the angular positions θmin of the first minimum of the diffraction patterns as a function of the L-spot diameter is analyzed. Two methods to determine L for Gaussian beams are proposed. For Hermite–Gaussian beams an interesting diffraction property previously presented for one slit [J. Opt. Soc. Am. A12, 2440 (1995)] is generalized to finite gratings, namely, τ=λ(E/N), where τ is the transmission coefficient and λ is the wavelength of the incident radiation. We study this property in detail and analyze its validity conditions in terms of the optogeometrical parameters of the system. This property might be useful in determining the total energy transmitted by a lamellar grating from knowledge of the intensity diffracted in the normal direction.

Aberrations of diffracted wave fields. II. Diffraction gratings.

The Rayleigh-Sommerfeld theory is applied to diffraction of a spherical wave by a grating. The grating equation is obtained from the aberration-free diffraction pattern, and its aberrations are shown to be the same as the conventional aberrations obtained by using Fermat's principle. These aberrations are shown to be not associated with the diffraction process. Moreover, it is shown that the irradiance distribution of a certain diffraction order is the Fraunhofer diffraction pattern of the grating aperture as a whole aberrated by the aberration of that order.

Aberrations of diffracted wave fields. I. Optical imaging.

The concept of aberrations of diffracted waves is revisited by using the Rayleigh-Sommerfeld theory of diffraction, and it is pointed out that these aberrations are in a class by themselves; they are only deceptively similar to the aberrations of a rotationally symmetric imaging system. Although an exact Fourier-transform expression can be written for the diffracted wave field, its numerical calculation is cumbersome because of the dependence of the aberrations of the diffracted wave on the observation point. This is true regardless of whether the diffracted wave field is observed on a plane or a hemisphere. It is shown that the Fresnel and Fraunhofer approximations, which neglect these aberrations and thereby simplify the calculations, are valid in imaging applications.

New property in the diffraction of Hermite–Gaussian beams by a finite grating in the scalar diffraction regime: constant-intensity angles in the far field when the beam center is displaced through the grating

We study the diffraction of Hermite–Gaussian beams by N equally spaced slits (finite grating) in a planar screen by means of the Rayleigh–Sommerfeld theory in the scalar diffraction regime. We find in the far field the existence of constant-intensity angles when the incident-beam position on the screen is changed. We have determined the optimal conditions under which this new diffraction property can be achieved. Also, a novel effect called the constant-intensity-angles-collapse effect is presented, in which the constant-intensity angles collapse to the minima of the diffraction pattern when the incident spot size is enlarged. For the grating case, deep dips at the maxima of the diffraction patterns of Hermite–Gaussian beams are predicted. Also, for a grating we have found that large segments of these diffraction patterns are independent of the incident-beam position (called constant-intensity curves). The results of this report may be useful for considering unstable vibrational systems in which constant intensities at some angular direction of the far field are required.

Near–field diffraction by two slits in a black screen

Measurements were made with microwaves of the near–field diffraction pattern formed by two parallel slits in a highly absorbing (‘blac’) screen. The first purpose was to examine the validity of superposing the fields of the two separate slits, as would follow from the standard Kirchhoff or Rayleigh–Sommerfeld theories. An earlier paper showed that, with a conducting screen, there is a departure from exact superposition of 7%, on average, when the polarization is parallel to the slits and 25% when it is perpendicular. The new experiments, which were done at normal incidence with slit widths of 0.5λ and separation 1.5λ, show that, with a black screen, there is a departure from exact superposition of 11% when the polarization is parallel, and 15% when it is perpendicular. Each slit generates a field behind the screen, but, unlike the case of the conducting screen, the source of the field cannot be considered to be merely the slit itself; it is as if the wave field generated by each of the two slits is affected by whether the other is open or closed. The width of the maximum immediately behind a single slit depends strongly on the polarization, an observation that has implications for resolution in, for example, the making of masks using near–field optics. All scalar theories of diffraction by a black screen with translational symmetry imply that the patterns of electric field associated with two different polarizations in the incident wave are related: one is the spatial derivative of the other. The fact that this theoretical relation is obeyed by the measurements to a close approximation shows that scalar theory is applicable, as an approximation, for the imperfect ‘black’ screen used. The relation would not be obeyed for a conducting screen.

Exact and approximate solutions for focusing of two-dimensional waves III Numerical comparisons between exact and Rayleigh–Sommerfeld theories

Recently exact and approximate solutions were given for focusing of two-dimensional scalar and electromagnetic waves through a slit aperture in a perfectly reflecting screen. We present numerical comparisons, based on these solutions, between exact results and approximate Rayleigh–Sommerfeld results for a variety of different focusing geometries. These comparisons show that, at sufficiently low Fresnel numbers and large angular apertures, approximate solutions based on the first and the second Rayleigh–Sommerfeld diffraction integrals agree well with the corresponding exact solutions for hard and soft screens, respectively. Inasmuch as these results are contrary to what one would expect from considerations of the boundary conditions, we give an analytical explanation of them through comparisons between exact and approximate near-field solutions for the corresponding half-plane problems.

Verification of partially coherent light diffraction models in x-ray lithography

The image formed in proximity x-ray lithography is normally computed using a physical optics model that takes in explicit account diffraction processes. It is important to understand the assumption implicit in the various models and the effects that they have on the image formation. We concentrate, in particular, on the effect of the type of illumination on the quality of the image that can be obtained in proximity x-ray lithography. In this paper, different methods of calculating the diffraction of partially coherent light are compared and the approximations implicit in each discussed. In a partially coherent system, the properties of the source play an intrinsic role in image formation, so they must be treated together with the imaging process. The full detailed calculations are too complex for implementation on even a supercomputer but, for some types of sources and optical systems it is possible to simplify the problem. Particularly important is the issue of the validity of the linear shift-invariant system approximation, since it allows great improvements in computational time. We prove the validity of the approximation and verify it using a full 4D numerical integration, based on the Rayleigh–Sommerfeld theory and realized on a Cray YMP8 supercomputer. Application of the results to some physical cases of 0.25 and 0.1 μm features is presented and discussed, illustrating the wide process latitude and resolution of x-ray lithography.

On the exact Kirchhoff and Rayleigh-Sommerfeld theories for the focusing of an infinite scalar spherical wave-field

This communication discusses the exact analytic solutions to three well-known scalar diffraction theories - the Kirchhoff and first and second Rayleigh-Sommerfeld theories - for the focusing of a monochromatic spherical incident wave in an infiniteaperture system.

Axial irradiance and optimum focusing of laser beams

Using the Rayleigh-Sommerfeld theory of diffraction we obtain an exact expression for the axial irradiance of a focused annular laser beam valid for all axial points. Conditions for the validity of the Fresnel theory are obtained. We discuss why and how the depth of focus and asymmetry of focused fields about the focal plane depend on the Fresnel number of the beam aperture as observed from the geometric focus. When a beam is focused on a distant target so that the Fresnel number is small (≲5), the principal maximum of axial irradiance occurs at a point which is significantly away from the geometric focus in the direction of the aperture. We discuss how to optimally focus a beam to illuminate a moving distant target in terms of the encircled energy on it. We show that, to obtain the maximum possible concentration of energy on a target, the beam must be focused on it, thus requiring active focusing for a moving target. However, if energy concentration is adequate for a beam focused on a target at a certain distance, it is more than adequate for a considerable range of the distance of a moving target without active focusing. In a shared-aperture optical system the aperture used for focusing a beam on the target is also used for imaging the target. Hence, in such a system the optical transfer function is also more than adequate over a wide range of the target distance without active focusing.

Particle-Size Measurement Using Forward-Scatter Holography*

This study is an experimental and theoretical investigation of the application of forward-scatter Fresnel holography to water droplets. The theoretical holograms were constructed on a digital computer by determining the irradiance of the interference pattern formed by the light scattered from a droplet and a reference source, at discrete points on a hologram. The forward-scattered light is evaluated by assuming that the droplet is an opaque disk and utilizing the Rayleigh–Sommerfeld theory of diffraction. The reconstructed wave front is evaluated by performing the Fresnel transform on the hologram matrix. The corresponding experimental holograms were made with a helium–neon continuous-wave laser and using glass beads from 80 to 250 μm in diameter as models of the water droplets. The half-radiance width of the reconstructed wave-front radiance distribution is shown theoretically to increase linearly as the distance of the object from the hologram is decreased. Also, the half-radiance width increases logarithmically with hologram area. The theoretical and experimental radiance distributions in the plane of reconstruction were compared. Proper alignment of the hologram during reconstruction is necessary for the reconstruction of the original radiance distribution. Size can be measured by noting edge enhancement (radiance peak at the edge of the droplet) in the reconstructed image. A relationship between the diameter of the peak and the actual droplet diameter is determined for sphere diameters from 20 to 250 μm and object distances from 10 to 20 cm. Experimental accuracy to within 4% is achieved with this technique.

Comparison of the Kirchhoff and the Rayleigh–Sommerfeld Theories of Diffraction at an Aperture

With a view to elucidating the effect of a well-known mathematical inconsistency in Kirchhoff’s diffraction theory, a comparison is made of the predictions relating to the field diffracted at an aperture, based on Kirchhoff’s theory (UK) and on formulas due to Rayleigh and Sommerfeld (UR). It is shown that, when the incident wave is plane or spherical, the difference δ = UK−UR represents a boundary wave, i.e., a wave which may be thought of as originating at each point of the edge of the aperture. It is shown further that, when the linear dimensions of the aperture are large compared with the wavelength, the boundary values of δ in the plane of the aperture change very rapidly and almost periodically from point to point, with the mean period close to the wavelength of the incident radiation. This result is shown to imply that if the linear dimensions of the aperture are large compared with the wavelength, the two theories predict essentially the same behavior for the diffracted field in the far zone, at moderate angles of diffraction.