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An angular spectrum propagation (ASP) algorithm with a scaling parameter to simulate optical diffraction propagation through optical systems is studied. The alterable observation size is obtained by adding the scaling parameter to the Collins formula. A directly mathematical inverse transformation of the ASP algorithm (IASP) is proposed to calculate the source optical field from the known observation optical field, and the results are proved more precise. The IASP algorithm is applied to execute the phase retrieval to derive the aberrations of optical systems from intensity profiles measured in the observation plane. The derived aberrations are fitted by Zernike polynomials under the constraint that the wavefront aberrations are smooth. Numerical simulations are performed to test the accuracy of this method.

The wide-window angular spectrum (WWAS) method is proposed to simulate field propagation through paraxial optical systems, mainly based on the Collins formula and the scaled Fourier transform (SFT). The application of the SFT algorithm makes the sampling processes in the input space, output space and spatial-frequency domains completely independent, and as a result, we can choose a larger calculation window size for simulating long-distance propagation without increasing the calculation burden. The sampling criteria are derived analytically and used in the numerical simulations to present the correctness and effectiveness of the WWAS algorithm. The advantages of the algorithm are shown by making a comparison with other angular spectrum methods for the free-space propagation case.

The angular spectrum of stray-light from holographic optical elements is measured at wavelengths of 488 nm and 1·5 μm. Holographic optical elements, for deflecting or splitting incident light beams, are recorded as volume transmission phase gratings in dichromated gelatin. The holographic optical elements are characterized with regard to their diffraction efficiency and stray-light. The diffraction efficiency is 80–85% of the incident light. The stray-light is mostly found within spurious diffraction orders or reflected beams. A signal-to-background ratio better than 1:10−4 indicates than these elements are not inherently noisy or hazy.

The phase-conjugate mirror realized by means of degenerate four-wave mixing in a Kerr-like medium is investigated using the angular spectrum representation of interacting fields. The analysed model is based on the nonlinear mixing of two infinite plane pump waves with the spatially limited probe and signal waves. Relations among harmonic components of the probe and signal fields are derived assuming non-depleted pumping and neglecting the evanescent waves. The influence of the injected signal field on the frequency spectrum of the generated phase-conjugate field is also examined.

The Bessel beam propagation characteristics are investigated in the presence of azimuthally dependent phase variations of its angular components. The phase variations influence the beam symmetry and strongly reduce an axial intensity. An experimental interpretation of presented results is also discussed.

A simple and accurate analytical model for the reflectivity of an X-ray multilayer mirror has been developed. This model can be used in the determination of interfacial roughness over large ranges of wavelength and roughness. There are no restrictions such as the roughness being less than the scale of the wavelength such as are present in theories based on the multiplication of the sharp boundary reflectivity coefficient by an exponential factor. The model is also applied to scattered radiation. Angular spectra are calculated and the use of scattered radiation in determining interfacial roughness is demonstrated.

We report here on design and computer simulation of computer-generated holograms for three dimensional (3D) imaging and display. Angular spectrum of general polygon patches was calculated in closed...

Based on the vector angular spectrum representation and the method of stationary phase, the analytical expressions for the electromagnetic fields of the cylindrical vector Laguerre–Gaussian beams with arbitrary polarization order are derived in the far field. The radially, anti-vortex and linearly polarized beams can be viewed as the special cases of our general result. The analyses indicate that the beam evolution properties and nonparaxiality are closely related to the radial mode number, the polarization order number and the ratio of the waist width to the wavelength. The high polarization order cylindrical vector beams compared with the radially polarized beams are more influenced by the nonparaxiality. This research provides a convenient approach to manipulate the cylindrical vector Laguerre–Gaussian beams by choosing the special state of polarization.

Based on the vector angular spectrum representation and stationary phase method, analytical far-field vectorial expressions of a noncanonical vortex dipole Airy beam, namely, a pair of noncanonical vortices with opposite charges embedded in an Airy beam are derived and used to investigate the phase singularities and energy flux distributions of the corresponding beam in the far-field regime, where the noncanonical characteristic of vortex is stressed. It is shown that the noncanonical strength, off-axis distance, and aperture coefficient affect the position and number of phase singularities, and the motion, creation, and annihilation of phase singularities are found by adjusting these parameters. For a low aperture coefficient, the energy flux distributions exhibit different numbers and orientations of lobes by varying the noncanonical strength. With increasing aperture coefficient, the symmetries of lobes are broken, and the energy flux distributions gradually become ellipses and the directions of their ...

Twin photons can be entangled in transverse momentum and position, giving rise to the so-called quantum images. The term quantum image has been used for images measured with photon coincidence schemes, also called correlated images. Some kinds of correlated images are prepared by transferring the angular spectrum of the pump beam to the correlated twin beams. In this case, it is possible to show that transverse correlations can violate a classical inequality, leading to spatial photon anti-bunching. We briefly review the use of the transfer of the angular spectrum in the production of these non-classical correlations. In addition, we demonstrate experimentally that the quantum image can be prepared partially through the nonlinear interaction and partially by propagation of signal and idler beams through lenses. Using this kind of preparation it is also possible to use a telescope for transmitting the quantum image, compensating the divergence of signal and idler beams.

We discuss how it is possible to manipulate quantum spatial properties of light beams by using the transfer of the angular spectrum. Twin beams from the parametric down-conversion can be prepared in a state that carries two different quantum images. At the same time, the twin photons can be entangled in polarization. This preparation allows the selection between the detection of each one of the images at a time or a mixture by performing proper projections onto polarization sub-spaces. As another example, we show how to produce a light beam presenting spatial anti-bunching through the control of the entangled angular spectrum and reflections.

Based on the vector angular spectrum representation and the method of stationary phase, the vectorial structural characteristics of a higher-order cosh-Gaussian (HOCG) beam in the far field are investigated. Analytical expressions for the energy flux distributions of the TE term, TM term and the whole beam are derived and illustrated by numerical examples. Influences of beam orders and cosh parameters on energy flux distributions are analyzed, respectively. Furthermore, the discrepancy for the total energy flux distributions between the non-paraxial and paraxial cases is presented and analytical explanations for the phenomenon are also given.

The contribution of propagating and evanescent waves to a radially polarized beam is separated by the vector angular spectrum method, and the characteristics of a radially polarized beam close to the source in the nonparaxial regime are investigated. The analytical expressions for the propagating and evanescent waves of a radially polarized beam in the source region have been obtained without any approximation. The analyses show that the waist width and the order n of the Laguerre polynomial play an important role in the behavior of the evanescent waves. The influence of the evanescent waves on the radially polarized beam is discussed by the numerical results.

By rigorously verifying the exact expressions for separate homogeneous waves and evanescent fields in the far field of dipole radiation, it is shown that (1) the homogeneous waves alone in the angular spectrum representation of the dipole field do not generate the far field; (2) the contribution of the evanescent fields to the far field is as 1/r, or as the inverse distance from the dipole, and thus cannot be neglected; and (3) only with the combination of the two parts, i.e. the complete angular spectrum, can the far field be precisely retrieved.

A description of the cosine–Gaussian (CoG) beam is made directly starting with Maxwell's equations. By means of the full vector angular spectrum method and the method of stationary phase, the analytical expressions of the TE and TM terms of the CoG beam have been presented in the far field. The influences of the cosine parameter on the energy flux distributions of the whole beam and its TE and TM terms are investigated. The physical process of evolvement of internal structure is depicted in the far field. This research reveals the internal vectorial structure of the CoG beam in the far field and enriches the cognition of the CoG beam.

By means of the vector angular spectrum representation of the electromagnetic beam and the method of stationary phase, the analytical vectorial structure of the Ince–Gaussian beam has been presented in the far field. The amplitude distributions of the Ince–Gaussian beam and its TE and TM terms are investigated in the far field. The extreme cases of the ellipticity parameter tending to infinity or zero are also considered. Although the vectorial structures of different Ince–Gaussian beams are apparently distinct, the ratios of the amplitude distributions of the TE and TM terms to the whole beam amplitude are independent of the parity and the values of the radial and angular elliptic mode numbers. This research reveals the abundant and interesting internal details of the Ince–Gaussian beam in the far field.

By using the angular spectrum representation of the Hertz vector and the theory of nonparaxial vectorial moments of light beam propagation, expressions for the width matrix, curvature and divergence matrices and M 2 factor of nonparaxial TM flattened Gausssian beams (FGBs) are derived and their matrix elements are expressed in terms of a series sum. Generally, nonparaxial TM FGBs possess different beam widths, divergence angles and M 2 factors in the x and y directions, which are dependent on the beam order N and f parameter. For 1/f → 0 the divergence angles θ x and θ y in the x and y directions approach their maximum values θ x = 54.736° and θ y = 63.435°, respectively. The M 2 factor can be less than 1 and take arbitrarily small values if 1/f is small enough.

The magnetic field of radiation emitted by an electric dipole contains travelling and evanescent waves when represented as an angular spectrum. The evanescent waves decay exponentially away from the xy-plane, and will therefore not contribute to the detectable radiation in the far field, in general. It is well known, however, that in a small region around the z-axis the evanescent waves of the electric field do end up in the far zone. We have studied the corresponding magnetic evanescent waves, and we have found that the evanescent waves of the magnetic field do not contribute to the far zone in the neighbourhood of the z-axis. When considering the neighbourhood of the xy-plane, it appears that both the electric and magnetic evanescent waves end up in the far field, and the travelling and evanescent waves contribute equally to the radiation in the far zone. Close to the dipole the radiation field diverges, and we have shown that this is entirely due to the evanescent waves.

The optical near field of a localized source has been studied by means of the angular spectrum representation of the electromagnetic Green's tensor. This Green's tensor can be expressed in terms of four auxiliary functions, which depend on the field point through the dimensionless radial distance q to the source, or origin of coordinates, and the polar angle ρ with the z axis. Each function separates into a part containing travelling (radiative) waves and a part which is a superposition of evanescent (decaying) waves. We have derived series expansions in q of the four functions, both for the travelling and for the evanescent parts. It is shown that the travelling waves are finite at the origin of coordinates, and that all singular behaviour of the radiation field is governed by the evanescent waves. It is illustrated numerically that the series expansions are applicable up to about five wavelengths from the origin. In order to extend the range to also cover larger values of q, we have derived ser...

The grating diffraction of beams is theoretically investigated by applying an electromagnetic method (the Integral Equation System Method with Parametrization of the grating profile IESMP) to their plane wave components. For the first time, explicit values for the displacement of grating diffracted Gaussian beams are calculated with this method. For total reflection this displacement of beams is known as the Goos-Hanchen shift. A maximum shift of 36 μm has been found for the investigated sinusoidal grating near an anomaly which is much greater than the known Goos-Hanchen shift of about 1 μm for the total reflection case. The replacement of the angular spectrum of plane waves with constant wavelength by a wavelength spectrun of plane waves of constant direction allows an analogous treatment of short-time pulses. Surprisingly, the above anomaly causes a maximum temporal shift of 80 fs for the pulse diffraction. These temporal shifts and additional effects like pulse deformations can influence ultra short-time pulse experiments. Furthermore, the behaviour of temporally and spatially Gaussian shaped light pulses (TSG pulses) by grating diffraction are studied considering the diffraction of an angular and wavelength dependent spectrum of plane waves. The diffraction of a short TSG pulse at the above grating deforms the pulse and creates an additional smaller satellite pulse. All described effects occur only at positions of the space-time complex filtering function in the angular-wavelength frequency space with high gradient of the phase.

It is shown that the angular spectrum of the cross-spectral density is a Wigner distribution function, which is equivalent to the marginal cross-spectral density of the field. The value of the Wigner distribution function at each position of the observation plane results from the contributions of aperture radiator pairs, around each position into the aperture, properly projected along the corresponding direction of propagation. The contributions are weighted by the complex degree of spatial coherence of the illuminating optical field, and their relative weights will be estimated by introducing a ‘visibility’ parameter. There are also forbidden directions, along which certain radiator pairs into the coherence patch cannot contribute depending on the phases of both the complex degree of spatial coherence and the aperture transmittance, but all radiator pairs provide their maximal contributions along orthogonal directions to the aperture plane.

One of the most basic optical ‘components’ is free-space propagation. A common approximation used when calculating the resultant field distribution after propagation is the Fresnel integral. This integral can be evaluated in two ways: directly or by using the angular spectrum. In this paper, we estimate the regions in which each mode of evaluation is preferable according to computing efficiency and accuracy considerations. A fast numerical algorithm is introduced for each region. The result is relevant also for the evaluation of the Rayleigh-Sommerfeld diffraction formula.

We present a paraxial wave formulation of three-wave mixing in a negative uniaxial crystal, including effects of diffraction and transverse walkoff. The theory, though general, is applied specifically to second harmonic generation in KDP with Nd:glass laser radiation, followed by the mixing of the second harmonic with the fundamental in a second KDP crystal to produce third harmonic radiation near 0·35 μm. For applications of interest, which can involve third harmonic conversion efficiencies approaching 90%, walkoff is potentially much more deleterious to conversion than diffraction. However, walkoff is negligible when the angular spectrum of the pump field does not have substantial contributions from spatial frequencies ≳ (aL)−1, where L is the crystal length and a depends upon the derivative of each extraordinary refractive index with respect to the angle between the optic axis and the direction of wave propagation. A similar result, scaling as L− 1/2 rather than L −1, holds for diffraction. We...

New conservation laws are derived for free electromagnetic fields propagating into a half-space. Such fields can be represented by an angular spectrum of electromagnetic plane waves that contains only homogeneous plane waves. The similarity between some of these new conservation laws and the Poynting theorem is examined in detail.

Weyl's formula describes the decomposition of a monochromatic spherical wave into an angular spectrum of plane waves. In this paper, explicit formulae for the separate contributions of homogeneous and inhomogeneous plane waves are given, and the results used to obtain an exact solution to the half-space boundary-value problem for the focusing of a converging spherical incident wave in an infinite-aperture system.

The angular spectrum representation of a scalar wave field separates naturally into two parts–the contribution of homogeneous plane waves and the contribution of inhomogeneous (or evanescent) plane waves. Exact diffraction formulae are derived for the separate contributions for an arbitrary incident field, when the field is monochromatic, and also when the field has arbitrary time-dependence.

An important problem in optics involves the computation of the field amplitude in the Fresnel region of a circular lens. When the observation point lines in the image plane, it is well known that the field can be written in closed form in terms of Bessel functions, i.e. the Airy pattern. In this paper, it is shown that away from the image plane the field can be expressed in terms of incomplete Weber integrals. This closed-form representation is much more accurate than the commonly used technique involving fast Fourier transforms (FFTs). This is demonstrated by comparing the closed-form results with those produced by an FFT. The closed-form solution is used to study the resolution of two point sources under the assumption that the observation plane is misaligned from the image plane.

In a series of papers [1], it has been claimed that evanescent waves contribute to the far field of a radiating dipole, i.e. that the amplitude of the sum of the evanescent waves decreases as 1/r with increasing distance, r, from the dipole. In spite of the fact that three recent papers [2] have demonstrated that these claims are incorrect and that they contradict well-established results relating to the farzone behavior of the angular spectrum representation of wavefields [3], the author continues to repeat this claim, most recently in a note with the title ‘Evanescent waves do contribute to the field’. It is the purpose of this note to point out an error in the previous publications [1] which has led to this erroneous conclusion.

The description of a Lorentz beam is directly obtained from Maxwell's equations. Based on the vector angular spectrum representation of Maxwell's equations and the method of stationary phase, the analytical TE and TM terms of the Lorentz beam have been presented in the far field. The TE term is located at the y axis, and the TM term the x axis. The light intensity distributions of the Lorentz beam and its TE and the TM terms are depicted in the far field reference plane. The influences of the inequality of the transverse beam waists on the light intensity distributions of the Lorentz beam, its TE and TM terms are also analyzed. Moreover, the vectorial structure of the Lorentz beam is compared with that of the Gaussian beam. Due to the relatively small diffraction effect, the beam spot of the Lorentz beam is smaller than that of the Gaussian beam. This research reveals the vectorial composition of the Lorentz beam.

Lateral displacement and its mechanism in asymmetric layered configuration are investigated. It is shown by the stationary-phase approach that displacements of both the transmitted and reflected beams can be positive as well as negative, and their magnitude can be greatly enhanced at transmission resonances. Numerical simulations show that for a well-collimated incident beam, if the norms of transmission and reflection coefficients are approximately constant within the peak of the angular-spectrum distribution of the beam, the transmitted and reflected beams can undergo lateral displacement without distortion, as long as the thickness of the layer satisfies a restriction. It is concluded that lateral displacement is not caused by beam distortion, but the result of reshaping the multiply transmitted (or reflected) beams.

The non-diffractive vector Bessel beams of an arbitrary order are examined as both the solution to the vector Helmholtz wave equation and the superposition of vector components of the angular spectrum. The transverse and longitudinal intensity components of the vector Bessel beams are analysed for the radial, azimuthal, circular and linear polarizations. The radially and azimuthally polarized beams are assumed to be formed by the axicon polarizers used with the initially unpolarized or linearly polarized light. Conditions in which the linearly polarized Bessel beams can be approximated by the scalar solutions to the wave equation are also discussed.