Rayleigh-Sommerfeld Diffraction Research Articles

A computer algebra derivation of the Rayleigh–Sommerfeld diffraction formula by a plane screen

The Rayleigh–Sommerfeld diffraction formula is derived without resorting to generalized or Green functions starting from the plane wave or angular spectrum representation of the scalar field after a plane screen. All the steps are clarified by means of computer algebra software.

Nonparaxial propagation of flat-topped beam diffracted in the present of a rectangular aperture

On the basis of the vectorial Rayleigh–Sommerfeld diffraction integral, an analytical nonparaxial propagation equations of vectorial flat-topped beams through a rectangular aperture are derived. The far-field and paraxial propagation equations of flat-topped beams through a rectangular aperture are treated as the special cases. It is found that the beam parameters w0x, w0y and two truncation parameter a, b affect the propagation properties of flat-topped beams.

Nonparaxial propagation of a multi-Gaussian Schell-model beam

Based on the vectorial Rayleigh–Sommerfeld diffraction integral formulas, analytical expressions for the elements of 3 × 3 cross-spectral density matrix of polarized multi-Gaussian Schell-model (MGSM) beam’s nonparaxial propagation in free space are derived, and used to investigate its propagation properties. As a special case of nonparaxial propagation, the paraxial case has also been considered by the reduced paraxial approximation expressions. The distributions of spectral density, spectral degree of coherence, and spectral degree of polarization of nonparaxial MGSM beam are illustrated by numerical examples. Results show that the statistical properties of MGSM beam’s nonparaxial propagation in free space are closely related to the initial beam’s correlation width related parameter and transverse width related parameter These characteristics of MGSM beam might find applications in laser scanning, optical trapping, and free space optical communication.

Nonparaxial propagation of a rectangular multi-Gaussian Schell-model beam

Abstract According to the vectorial Rayleigh–Sommerfeld diffraction integral formulas, analytical expressions for the elements of 3×3 cross-spectral density matrix of polarized Rectangular Multi-Gaussian Schell-Model (RMGSM) beam’s nonparaxial propagation in free space are derived, and the paraxial analytical expression has also been presented as a special case of nonparaxial propagation. Its statistical properties, including the distributions of spectral density, spectral degree of coherence, and spectral degree of polarization of nonparaxial RMGSM beam are numerically demonstrated. Results reveal that the statistical properties of RMGSM beam’s nonparaxial propagation in free space are closely related to the initial beam’s transverse width σ and correlation widths δ x and δ y . These characteristics of RMGSM beam might be useful in optical material surface processing, imaging, and free space optical communication.

Propagation of a Laguerre–Gaussian correlated Schell-model beam beyond the paraxial approximation

Recently, Laguerre–Gaussian correlated Schell-model (LGCSM) beam was introduced theoretically [Mei and Korotkova, Opt. Lett. 38(2), (2013), 91–93] and generated experimentally [Wang et al., Opt. Lett. 38(11), (2013), 1814–1816]. In this paper, we treat the propagation of a LGCSM beam beyond the paraxial approximation. Based on the generalized Rayleigh–Sommerfeld diffraction integral, analytical expressions for the intensity and spectral degree of coherence of a nonparaxial LGCSM beam propagating in free space are obtained, and the corresponding results of a paraxial LGCSM beam are also derived as a special case. Our numerical results show that the nonparaxial propagation properties of a LGCSM beam are closely related to the initial beam parameters, such as the beam waist width, the coherence width and the mode order. Furthermore, in the far field, flat-topped, hollow intensity profiles can be formed through varying the initial beam parameters, which is different from that of a nonparaxial Gaussian Schell-model beam.

A novel method for the design of diffractive optical elements based on the Rayleigh–Sommerfeld integral

The original design method for diffractive optical elements (DOEs) is limited to cases of small-angle diffraction due to the Fresnel or Fraunhofer diffraction integral. In this paper, we propose a new method based on the Rayleigh–Sommerfeld diffraction integral, which does not have this limit. In this method, the target intensity distribution is first modified via coordinate transformation and intensity adjustment. Then, the modified Gerchberg–Saxton algorithm is used to achieve the phase distribution of the DOE. To verify this method, simulations and experiments are performed. The results show that the original method is effective only when the full diffraction angle of the DOE is below 25°. Conversely, this method can achieve the target with both small and large diffraction angles.

Pixel size adjustment in coherent diffractive imaging within the Rayleigh-Sommerfeld regime.

The reconstruction of the smallest resolvable object detail in digital holography and coherent diffractive imaging when the detector is mounted close to the object of interest is restricted by the sensor's pixel size. Very high resolution information is intrinsically encoded in the data because the effective numerical aperture (NA) of the detector (its solid angular size as subtended at the object plane) is very high. The correct physical propagation model to use in the reconstruction process for this setup should be based on the Rayleigh-Sommerfeld diffraction integral, which is commonly implemented via a convolution operation. However, the convolution operation has the drawback that the pixel size of the propagation calculation is preserved between the object and the detector, and so the maximum resolution of the reconstruction is limited by the detector pixel size, not its effective NA. Here we show that this problem can be overcome via the introduction of a numerical spherical lens with adjustable magnification. This approach enables the reconstruction of object details smaller than the detector pixel size or of objects that extend beyond the size of the detector. It will have applications in all forms of near-field lensless microscopy.

The Far Properties of Nonparaxial Cosine-Squared Gaussian Beams

Based on the Rayleigh-Sommerfeld diffraction integrals, the closed-form expression of nonparaxial cosine-squared Gaussian beams in free space is derived. The power in the bucket (PIB) of nonparaxial regime is introduced to characterize the beam quality of nonparaxial beams in the far field. It is found that the intensity and PIB of nonparaxial cosine-squared Gaussian beams depends onfparameter, decentered parameter, and the PIB is additionally depends on the bucket’s size taken. For a given bucket’s size, the smaller decentered parameter andfparameter result in the better power focusability in the far field and the better beam quality.

Vector Fourier optics of anisotropic materials

The Fourier optics technique is founded on the transfer function derived from the scalar wave equation and thus has traditionally been applied to monochromatic scalar fields propagating paraxially in isotropic lossless materials. We review scalar and vector diffraction theory to show that the scalar Fourier optics technique can be seamlessly extended to the case of time-dependent nonparaxial and evanescent vector fields propagating in anisotropic and/or absorbing materials. The missing piece is shown to be the Fourier-domain vector boundary conditions that have recently been derived from the real-domain vector Rayleigh–Sommerfeld diffraction integral. These boundary conditions, complemented by the anisotropic transfer functions provided by the roots of the Booker quartic, combine the rigor of Green’s function integrals with the intuitive simplicity and general applicability of Fourier optics. We illustrate the power of this technique via analytically simple solutions to traditionally complex problems such as Fresnel transmission between arbitrary media, uniaxial conoscopy, and biaxial conical refraction. The accuracy of vector near-field diffraction is validated via comparison with the finite-difference time-domain method.

Manipulation of acoustic focusing with an active and configurable planar metasurface transducer.

It has a pivotal role in medical science and in industry to concentrate the acoustic energy created with piezoelectric transducers (PTs) into a specific area. However, previous researches seldom consider the focal resolution, whose focal size is much larger than one wavelength. Furthermore, there is to date no such design method of PTs that allows a large degree of freedom to achieve designed focal patterns. Here, an active and configurable planar metasurface PT prototype is proposed to manipulate the acoustic focal pattern and the focal resolution freely. By suitably optimized ring configurations of the active metasurface PT, we demonstrate the manipulation of focal patterns in acoustic far fields, such as the designed focal needle and multi foci. Our method is also able to manipulate and improve the cross-sectional focal resolution from subwavelength to the extreme case: the deep sub-diffraction-limit resolution. Via the acoustic Rayleigh-Sommerfeld diffraction integral (RSI) cum the binary particle swarm optimization (BPSO), the free manipulation of focusing properties is achieved in acoustics for the first time. Our approach may offer more initiatives where the strict control of acoustic high-energy areas is demanding.

Near-field acoustic resonance scattering of a finite bessel beam by an elastic sphere

The near-field acoustic scattering from a sphere centered on the axis of a finite Bessel acoustic beam is derived stemming from the Rayleigh-Sommerfeld diffraction surface integral and the addition theorems for the spherical wave and Legendre functions. The beam emerges from a finite circular disk vibrating according to one of its radial modes corresponding to the fundamental solution of a Bessel beam J0. The incident pressure field's expression is derived analytically as a partial-wave series expansion taking into account the finite size and the distance from the center of the disk transducer. Initially, the scattered pressure by a rigid sphere is evaluated, and backscattering pressure moduli plots as well as 3-D directivity patterns for an elastic PMMA sphere centered on a finite Bessel beam with appropriate tuning of its half-cone angle, reveal possible resonance suppression of the sphere only in the zone near the Bessel transducer. Moreover, the analysis is extended to derive the mean spatial incident and scattered pressures at the surface of a rigid circular receiver of infinitesimal thickness. The transducer, sphere and receiver are assumed to be coaxial. Some applications can result from the present analysis since all physically realizable Bessel beam sources radiate finite sound beams as opposed to waves of infinite extent.

Nonparaxial circular grating diffraction properties of radially polarized beams

The Rayleigh–Sommerfeld diffraction integral is employed to study the circular grating nonparaxial diffraction properties of the radially polarized beams, and the analytical expression for the diffraction electric field is derived. The analyses indicate that the properties of the nonparaxial diffraction field strongly depend on the grating parameters and beam waist. When the radially polarized beam is diffracted by the sub-wavelength grating, only the 0th diffraction ring is obtained, or the distinct higher order diffraction rings will emerge. Compared with the paraxial diffraction, the nonparaxial diffraction intensity field is weaker. But when the beam waist is larger than 3λ the paraxial approximation is valid; thus their distinction will be negligible. At last, it shows that in the radially polarized beams nonparaxial diffraction field, the longitudinal component not only leads to stronger diffraction intensity than the azimuthally polarized beams, but also makes the polarization degenerate.

Full field analysis of critical dimension uniformity due to focal variation for contact features in extreme ultraviolet lithography.

Extreme ultraviolet (EUV) lithography is the leading candidate for 22 nm node technology and beyond. This research studies the influence of focal contributor on critical dimension (CD) variation in the EUV lithography and calculates the CD sensitivity to focal contributor based on the resist CD. EUV lithography parameters used in the simulation include NA = 0.25, 6 degrees oblique incident on the mask and source wavelength at 13.6 nm. The reflection design of EUV mask consists of 40 alternating Si/Mo layers, a ruthenium capping layer, and an absorber layer. The Rayleigh-Sommerfeld diffraction theory is adapted to solve the aerial image of contact hole (CH) feature layouts on the EUV mask. Then the resist CDs for the CH feature are measured at the position of the 10% from the bottom of resist profile. The target CD is 35 nm on wafer. The simulation results reveal the 2 nm discrepancy of spatial position for the aerial image curves measured between on the X metrology and Y metrology planes. The CDs of CH features with the iso-pitch are more sensitive to focal variation than the CDs of the features with the dense pitch. The maximum CD uniformity is 3.1 nm for the focal range at 90 nm.

Effects of turbulent medium refractive index structure constant Cn2 on the propagation characteristics of double-half hollow Gaussian beams

Based on the Rayleigh-Sommerfeld diffraction integral theory, the propagation analytical expression of double-half Gaussian hollow beams in turbulent atmosphere is derived by using the cross spectral density function, and the effects of turbulent medium refractive index structure constant Cn2 on the propagation characteristics of double half a hollow Gaussian beams are studied and the intensity distributions under the different conditions are obtained as well. The research results show that the increased Cn2 can exacerbate the near-field diffraction effects of double-half Gaussian hollow beams, which not only shortens the propagation distance of hollow beams fully evolved into the Gaussian beams, but also increases the extent of the Gaussian beam extend outward.

Nonparaxial accelerating Bessel-like beams

A new class of nonparaxial accelerating optical waves is introduced. These are beams with a Bessel-like profile that are capable of shifting laterally along fairly arbitrary trajectories as the wave propagates in free space. The concept expands on our previous proposal of paraxial accelerating Bessel-like beams to include beams with subwavelength lobes and/or large trajectory angles. Such waves are produced when the phase at the input plane is engineered so that the interfering ray cones are made to focus along the prespecified path. When the angle of these cones is fixed, the beams possess a diffraction-free Bessel profile on planes that stay normal to their trajectory, which can be considered as a generalized definition of diffractionless propagation in the nonparaxial regime. The analytical procedure leading to these results is based on a ray optics interpretation of Rayleigh-Sommerfeld diffraction and is presented in detail. The evolution of the proposed waves is demonstrated through a series of numerical examples and a variety of trajectories.

Calculation of the volumetric diffracted field with a three-dimensional convolution: the three-dimensional angular spectrum method

The first Rayleigh-Sommerfeld diffraction formula is treated in an exact form as a three-dimensional (3D) convolution in the spatial domain. Therefore, a 3D Fourier transform can be employed to convert the 3D diffracted electromagnetic field to the reciprocal space without approximations, which we call the 3D angular spectrum (3D-AS) method. It is also demonstrated that if evanescent waves are neglected, the 3D-AS method can be readily implemented numerically, with the results in good agreement with theoretical predictions.

Numerical characterization of an ultra-high NA coherent fiber bundle part II: point spread function analysis

Straightforward numerical integration of the Rayleigh-Sommerfeld diffraction integral (R-SDI) remains computationally challenging, even with today's computational resources. As such, approximating the R-SDI to decrease the computation time while maintaining a good accuracy is still a topic of interest. In this paper, we apply an approximation for the R-SDI that is to be used to propagate the field exiting a Coherent Fiber Bundle (CFB) with ultra-high numerical aperture (0.928) of which we presented the design and modal properties in previous work. Since our CFB has single-mode cores with a diameter (550 nm) smaller than the wavelength (850 nm) for which the CFB was designed, we approximate the highly divergent fundamental modes of the cores with real Dirac delta functions. We find that with this approximation we can strongly reduce the computation time of the R-SDI while maintaining a good agreement with the results of the full R-SDI. Using this approximation, we first determine the Point Spread Function (PSF) for an 'ideal' output field exiting the CFB (identical amplitudes for cores on a perfect hexagonal lattice with the phase of each core determined by the appropriate spherical and tilted plane wave front). Next, we analyze the PSF when amplitude or phase noise is superposed onto this 'ideal' field. We find that even in the presence of these types of noise, the effect on the central peak of PSF is limited. From these types of noise, phase noise is found to have the biggest impact on the PSF.

Average intensity and spectral shifts of a partially coherent standard or elegant Laguerre-Gaussian beam beyond paraxial approximation

With the help of the Rayleigh-Sommerfeld diffraction integral, an analytical formula for the spectral density of a nonparaxial polychromatic partially coherent (NPPC) standard or elegant Laguerre-Gaussian (LG) beam propagating through free space is derived. The evolution properties of the intensity distribution of a NPPC standard or elegant LG beam are studied numerically. It is found that the behavior of the intensity distribution of the NPPC standard or elegant LG beam is determined by the initial beam parameters (i.e., beam waist size, spatial coherence width and beam orders). Furthermore, we also study the influence of the initial beam parameters on the spectral change of a NPPC standard or elegant LG beam upon propagation in free space. It is shown that the on-axis spectral shift of a NPPC standard LG beam is almost the same with that of a NPPC elegant LG beam when the initial beam parameters of two type beams are identical. However, the difference between the off-axis spectral shift between a NPPC standard LG beam and a NPPC elegant LG beam is enhanced with the decrease of beam waist size or with the increase of spatial coherence width and beam mode orders.

Vectorial design of super-oscillatory lens

A design and optimization method based on vectorial angular spectrum theory is proposed in this paper for the vectorial design of a super-oscillatory lens (SOL), so that the radially polarized vector beam can be tightly focused. The structure of a SOL is optimized using genetic algorithm and the computational process is accelerated using fast Hankel transform algorithm. The optimized results agree well with what is obtained using the vectorial Rayleigh-Sommerfeld diffraction integral. For an oil immersed SOL, a subwavelength focal spot of about 0.25 illumination wavelength without any significant side lobe can be created at a distance of 184.86 μm away from a large SOL with a diameter of 1mm. The proposed vectorial design method can be used to efficiently design a SOL of arbitrary size illuminated by various vector beams, with the subwavelength hotspot located in a post-evanescent observation plane.

Rayleigh–Sommerfeld diffraction formula in k space

An angular spectrum representation in three dimensions is used to develop three-dimensional Fourier forms of the first and second Rayleigh-Sommerfeld diffraction formulae and the Kirchhoff diffraction formula. For forward-propagating waves, these reduce to three-dimensional Fourier representations for diffraction in the forward half-space.