Effects of third-order aberrations on the irradiance of self-image

We examine the effects of third-order aberrations exerted on the irradiance of image that is observable in a coherent self-imaging system. Both spherical aberration and astigmatism degrade the visibility of the image of a sinusoidal-type grating as well as blur the outline of the image of a rectangular-type grating. Coma laterally shifts the image of a sinusoidal-type grating on the image plane as well as changes a rectangular-type pattern into an asymmetrically blurred pattern. According to our analysis, the self-image of a high-density grating with a period of two times the optical wavelength is not at all affected by spherical aberration. In general a self-imaging system can always be corrected for astigmatism by shifting the image plane in its normal direction. We show that the self-image with defect can be well explained by taking the third-order aberrations and the focus-shift aberration into consideration.     

Geometrical aberrations of self-imaged line gratings

We analyze the properties of a self-imaging system from the point of view of aberration theory. We examine analytically and numerically the geometrical aberrations that are observed in the self-image of a parallel-line grating. We first derive the raytracing equations for determining the optical path of a self-imaging ray with the order of diffraction l. We then obtain the third- and fifth-order contributions to ray aberrations which arise from the difference between the optical paths of a self-imaging ray and of an actual ray. We show that the overall ray aberrations are entirely undercorrected. The ray aberrations approach zero as the ratio of the grating constant to the wavelength of light becomes large enough. In a case of unit magnification, no curvatures are observed in the self-imaged lines. If the magnification is bigger than unity, the light rays passing through the point in a positive or negative domain of the aperture variable contribute to the formation of the curved images. The image evaluation technique discussed here can be useful in the various applications related to the self-image formation of a parallel-line grating and it can also provide insight into the self-imaging of other periodic objects.     

Aberrations of self-imaged patterns with two-dimensional periodicity

We develop a geometrical theory of aberration in the self-imaged patterns with two-dimensional periodicity. The patterns are considered to be a crossed-line grating or a periodic array of finite apertures. We first derive the raytracing equations for determining the optical path of a self-imaging ray. We then find the third- and fifth-order contributions to the wavefront aberration which arise from the difference between the optical paths of a self-imaging ray and of an actual ray. We also derive the expression of the ray aberration from the wavefront aberration. The ray aberration is classified into five distinct types by analogy with the cases in a refracting lens system. We show that the overall ray aberration is entirely undercorrected and the aberrated image patch is decentered from an ideal image point in the direction parallel to the direction tangent vector of a chief ray. The image evaluation technique discussed here will be useful in various applications related to self-image formation of two-dimensionally periodic patterns.     

Ray aberrations of two-dimensional oblique lattices on the self-image plane

We extend the geometrical theory of aberration for a self-imaging system to the case of two-dimensional oblique lattices. In our approach, the fundamental translation vectors of the lattice are not restricted in both length and orientation. Evaluating the disturbance of light through the oblique lattice under coherent illumination, we find the conditions of constraint which limit the self-imaging of the oblique lattice. Various types of oblique lattices are shown to obey the self-imaging conditions. We derive the equations to trace the optical paths of self-imaging rays and then analyze the ray aberrations which arise from the difference between the optical paths of a self-imaging ray and of the corresponding actual ray. The ray aberrations are shown to disappear when the periods of the lattice are large compared with the wavelength of light. We find that the ray aberrations carried by self-imaged oblique lattices are totally undercorrected and the aberrated image patches are displaced along the direction tangent vector of a chief ray.     

Measurements of axial displacements undergone by a diffusing object in speckle photography

The speckles of the image plane of an object are both radially shifted and decorrelated when the object is axially translated through an amount ?. We demonstrate that this radial shift, which is related to the position of the pupil of the optical system, disappears when the pupil lies in the back focal plane of the imaging lens. However, if the irradiance of the image plane is twice recorded on a photographic plate which is laterally shifted through ζ_H between the exposures, the minimum value of the contrast of the Fourier fringes exhibited by the plate after processing, gives the value of ? if ? is less than a particular value, ?_M which will be defined. Also we propose a new speckle measurement method in which the fringes are automatically removed when ? is greater than ?_M. We record the image of the object illuminated in convergent light through an amplitude diffuser placed in the Fourier plane of the object. The mean speckle speckle size of the diffuser is equal to the mean size of the speckles generated by the object in its Fourier plane.     

Extended AMAI-PCA technique based on multi-level Box–Behnken design

Previously, we reported a technique for in situ projection lens aberration measurement based on principal component analysis of aerial images (AMAI-PCA). The sampling approach of this technique has a great impact on the measurement range and accuracy. To meet the requirement of large aberration measurement, a multi-level Box–Behnken design approach is tested in conjunction with AMAI-PCA to build the aerial image space. Compared with regular Box–Behnken design, the new approach improves the accuracy by 30% when the amplitude of wavefront aberration is larger than 0.1λ.     

Measurement of lens aberration based on vector imaging theory

Current aerial image based aberration measurement methods are all derived from the scalar imaging theory which is not applicable to hyper-NA lithography. Here we propose an aberration measurement method for the lithography system with an arbitrary NA based on the rigorous vector imaging theory. The retrieval error of the proposed method is clearly demonstrated by comparing with the method based on the scalar imaging theory for an immersion NA 1.35 projector using numerical results. The results show that the maximal retrieval error of our method is below 0:5mλ, while that of previous method is above 4mλ. In addition, the effect of practical aerial image metrology accuracy on the retrieval accuracy of proposed technique is analyzed. The results obtained demonstrate that the retrieval accuracy of our method is greatly improved, and the proposed technique is applicable to the retrieval of the wavefront aberration of a projector with hyper-NA.     

Optimal design of multimode interference couplers using an improved self-imaging theory

Self-imaging theory is widely accepted as a good method in designing 1 × N multimode interference (MMI) couplers, but it is also true that self-imaging theory is not suitable for low-contrast structures. An improved self-imaging theory is proposed in this paper for the optimal design of low-contrast 1 × N MMI couplers. The average effective width of MMI waveguide and the average effective propagation constant of MMI waveguide are used as the basis to modify the conventional self-imaging theory. A direct calculation of the average effective width of low-contrast MMIs is presented. We use this approach in the optimal design of a 1 × 4 silica MMI coupler, and the results show that the improved self-imaging theory is more accurate than conventional self-imaging theory for low-contrast structures, the results also show that if the material parameters and the width of an MMI waveguide are fixed, the average effective width of the MMI waveguide will increase with the decrease of the height of the core layer.     

Lower bound in energy for chaotic dynamics of ions

When a charged particle is acted upon by an electrostatic wave propagating across a uniform magnetic field, its dynamics is chaotic provided that its initial Larmor radius is larger than a threshold value ϱ_m. We analytically compute ϱ_m when the wave frequency is an integer multiple of the particle cyclotron frequency, and show that ϱ_m increases with the wave amplitude. This leads to the counter-intuitive result that the dynamics of a low energy particle can be made stochastic by decreasing the amplitude of the wave.     

Low-temperature phases obtained by linear programming: An application to a lattice system of model chiral molecules

A convenient, Peierls-type approach to obtain low-temperature phases is to use the method of an m-potential. In this paper we show that, for more complex systems where it may be rather difficult to rewrite the Hamiltonian as an m-potential and whose configurations are subject to linear constraints, the verification of the Peierls condition can be reformulated as a linear programming problem. Before introducing this novel strategy for a general lattice system, we compare it with the m-potential method for a specific model molecular system consisting of an equimolar mixture of a chiral molecule and its non-superimposable mirror image that occupy all the sites of a honeycomb lattice. In one range of interactions, we prove that a racemic low-temperature phase occurs (containing equal numbers of each enantiomer). However, in a neighboring range of interactions, we show that a homochiral low-temperature phase (containing a single enantiomer) exists, and thus chiral segregation occurs in the system. Our linear programming technique yields these results in wider ranges of interactions than the m-potential method.     

Amplitude and phase clipping: Strehl ratio versus divergence

We consider the transverse characteristics of a Gaussian laser beam subject to a phase or amplitude clipping due to a pupil which is a π-plate or an opaque disc (stop). In particular, we consider the correlation between two features, the Strehl ratio and divergence angle, usually used for characterising the focusability of a diffracted beam. It is demonstrated that the Strehl ratio does not give systematically a global view, from a divergence point of view, on the transverse properties of a Gaussian beam suffering amplitude or phase diffraction. In addition, we consider the case of self-diffraction of a Gaussian beam upon a Gaussian phase aberration of same width, and it is found that the on-axis intensity describes correctly the whole diffracted beam cross-section, from a divergence point of view, only if the central phase shift is smaller than π. Another example showing that the focusability of a pure high-order Laguerre–Gauss TEM_p0 beam, free from any clipping, cannot be correctly described by Strehl ratio is also considered.     

Non-approximate method for designing Offner spectrometers

The so-called non-approximate method was adopted to design two-mirror three-reflection concentric Offner configuration for an dispersive spectrometer, which was using geometry and ray tracing method to derive the deviation between the arbitrary ray's image point and the principal ray's image point, the system aberration was evaluated based on this deviation, then each parameter was optimized and designed to meet the using conditions by computer software. The optimal object's position with minimal aberration, which also is known as the optimal imaging annular field, could be found in a different condition. Moreover, it presented that the ratio of the radius of convex grating to the radius of concave mirror ought to be close to 0.5∼0.6, the ratio r/R changed slightly with different wavelengths, but the bigger the wavelength was, the larger the ratio r/R would be.     

Fourth order correction to a Gaussian beam focused with a parabolic index lens

We formulate the fourth order correction to a paraxial Gaussian beam propagated along the axis of symmetry of a parabolic index lens. First we examine the evolution of a complex-source-point spherical wave (equivalent paraxially to a Gaussian beam) through the lens in a two-dimensional xz plane. Taking into account the terms of up to fourth order in aperture variables, we find a ray-optical solution to the exit beam that is represented in terms of aberration function. We also analyze the effect of the lens aberration exerted on the degradation in the quality of a Gaussian beam. The fourth order-corrected wave function derived here may be used to evaluate the quality of a Gaussian beam focused with a parabolic index lens. Further it may be applied to the case of an orthogonal system in which the index variations are different in the xz and yz planes.     

Absolute spherical test towards an accuracy of one-hundredth of the wavelength

Two dominant systematic errors in the absolute interferometric test of high numerical-aperture sphericals are discussed and measured experimentally in pursuit of an accuracy of λ=100. Gravitational sag deformation of a 4-inch spherical concave surface was measured in a vertical phase-shifting Fizeau interferometer. The surface shapes of two identical transmission spherical concaves were measured via the two-surface comparison method using three positional measurements. One of the surfaces was then rotated around the optical axis and the interference phases were averaged to extract a rotationally symmetric component of the aberration. The gravitational sag was then determined by the aberration component independent of the rotation. The geometrical error in the phase shift is also estimated theoretically and corrected in the experiment. Experimental results show that the both errors amount to 6 to 7 nm peak-to-valley lengths, the magnitudes of which are comparable to that of the total aberration of the spherical surface.     

Observation of the Talbot effect for ultrasonic waves

The diffraction of ultrasonic radiation on an amplitude diffraction grating in the near-field area (Fresnel diffraction) has been studied. The effect of self-imaging of the grating (Talbot effect) has been detected for ultrasonic radiation at distances from the grating in the range from z = 0 to z = 2L _T, where L _T is the Talbot length. The fractional Talbot effect, i.e., the ultrasonic image of the grating with the period d/2, has been observed.     

Analytical solution for DC field,transparency condition,and estimation of resistivity for radially anisotropic sphere of conductivity rm

We investigate the direct current (DC) field distribution for a sphere of special conductivity. While a general case requires the introduction of either special functions or numerical approach, we show that in cases where specific conductivity of the sphere increases as r^m function, where m ≥ 0, the analytical solution can be found. This analytical solution is then used to analyze three concrete examples: a DC field distribution of a sphere submerged into a electric current field, the conditions required for the sphere to become transparent, and the resistivity of the sphere for a symmetric pair of electrodes of a variable size.     

Aberrations in fractional-Talbot images of periodic binary gratings

We discuss the ray-optical aberrations which appear in the fractional-Talbot image of a periodic binary grating with coherent illumination. First we examine the complex amplitude of an aberration-free imaging field at a fractional-Talbot plane. We then trace the path of a diffracted ray of specific order which contributes to the fractional-Talbot imaging. Next we formulate the focus-shift and third-order aberrations which arise from a focusing error and a fourth-order approximation of the path length, respectively. We then evaluate the amplitude and phase of an aberrated imaging field that are represented in terms of aberration functions. When the grating period decreases to approach the optical wavelength, the aberrations of lower-order rays are shown to be more influential on the fractional-Talbot imaging field than those of higher-order rays. The theory of aberration discussed here could be very useful in evaluating the fractional-Talbot image of a periodic binary grating.     

Axions and “light shining through a wall”: A detailed theoretical analysis

We give a detailed study of axion-photon and photon-axion conversion amplitudes, which enter the analysis of “light shining through a wall” experiments. Several different calculational methods are employed and compared, and in all cases we retain a nonzero axion mass. To leading order, we find that when the photon frequency ω is very close to the axion mass m, there is a threshold cusp which significantly enhances the photon to axion conversion amplitude, by a factor relative to the corresponding axion to photon conversion process. When m=0, the enhancement factor reduces to unity and the results of previous calculations are recovered. Our calculations include an exact wave matching analysis, which shows how unitarity is maintained near threshold at ω=m, and a discussion of the case when the magnetic field extends into the “wall” region.     

The intensity distribution of hollow Gaussian beams focused by a lens with spherical aberration

We developed an expression that describes the hollow Gaussian beams (HGBs) passing through a spherically aberrated lens by using the Collins formula. The radial intensity distribution in both spherical aberration SA free lens, lens that exhibits relatively large in both positive spherical aberration PSA, and negative spherical aberration NSA is calculated. Numerical calculations are made and the results show that the PSA and NSA have a strong influence on the intensity distribution especially at the focus. The study showed remarkable results for which there is no hollow Gaussian beam at a large NSA along the optical axis at the focus. In addition, we found that the DSS, and w_r of focused hollow Gaussian beams in the focal region depend not only on the beam radius, and beam order; but also on the spherical aberration.     

An asymmetric single-channel color image encryption based on Hartley transform and gyrator transform

A novel asymmetric single-channel color image encryption using Hartley transform and gyrator transform is proposed. A color image is segregated into R, G, and B channels and then each channel is independently Hartley transformed. The three transformed channels are multiplied and then phase- and amplitude truncated to obtain first encrypted image and first decryption key. The encoded image is modulated with a conjugate of random phase mask. The modulated image is gyrator transformed and then phase- and amplitude truncated to get second encrypted image and second decryption key. The asymmetric (decryption) keys, random phase mask, and transformation angle of gyrator transform serve as main keys. The optoelectronic encryption and decryption systems are suggested. Numerical simulation results have been demonstrated to verify the performance and security of the proposed security system.