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For the X -cube prism three-charge-coupled-device (3CCD) camera, the spectra of the designed dichroic films in the X -cube prism shift with changes in the angle of incident light, producing non-uniformity of color on the image plane. We considered the influence of the incident angle on color performance in filter design and directly optimized the thin film to improve color consistency. An optical model was constructed to calculate the distribution of camera spectral sensitivity and independently correct the non-uniform color on the image plane. Results showed that the optimization and correction methods could significantly improve the color performance of the X -cube prism 3CCD camera.

The interference fringe period is an important phase-locking parameter in the scanning beam interference lithography (SBIL) system. To measure the interference fringe period accurately, a heterodyne period measurement method is proposed. Compared with traditional methods, the requirements for the stage motion characteristics are greatly reduced. In this paper, the theoretical error of the period measurement method is analyzed and relevant experiments are performed. The results show that the average period measurement value is 555.539 nm and the standard deviation of measurement repeatability is 2.5 pm. This method is significant for holographic grating fabrication using the SBIL system.

Based on the vector Fresnel diffraction integrals, analytical expressions for the electric and magnetic components of first-order Laguerre-Gaussian beams diffracted at a half-plane screen are derived and used to study the electric and magnetic polarization singularities in the diffraction field for both two- and three-dimensional (2D and 3D) cases. It is shown that there exist 2D and 3D electric and magnetic polarization singularities in the diffraction field, which do not coincide each other in general. By suitably varying the waist width ratio, off-axis displacement parameter, amplitude ratio, or propagation distance, the motion, pair-creation, and annihilation of circular polarization singularities, and the motion of linear polarization singularities take place in 2D and 3D electric and magnetic fields. The V point, at which two circular polarization singularities with the same topological charge but opposite handedness collide, appears in the 2D electric field under certain conditions in the diffraction field and free-space propagation. A comparison with the free-space propagation is also made.

This paper presents a phase-shifting algorithm based on Hough transform for sinusoidal curves. Firstly, the background intensities of phase-shifting fringe patterns are removed by calculating their differences, thus we get purely sinusoidal intensity data for each pixel; and then we implement Hough transform to the intensity difference data of each pixel. As a result, the sinusoidal parameters, including phase and amplitude, of each pixel are extracted. The simulation and experimental results demonstrate that this algorithm enables eliminating the impacts of some gross errors such as saturation of camera and impulse noise in fringe patterns, and exactly recovering the phase map from fringe patterns.

An efficient method for exploring the orbital angular momentum of an optical vortex beam is provided. The method, based on a triangular multipoint plate, can easily determine both the sign and the magnitude of the topological charge of the optical vortices. We demonstrate its feasibility by measuring the orbital angular momentum of Laguerre-Gaussian laser beams., (© 2011 Optical Society of America)

The closed-form expression for the free-space propagation of superimposed Laguerre-Gaussian beams beyond the paraxial approximation is derived, and the composite polarization singularities formed by the transverse and longitudinal electric-field components are studied in detail. It is shown that there exist composite C-points and L-lines in vector nonparaxial fields. By suitably varying a control parameter, such as the off-axis distance, relative phase, or amplitude ratio, the motion, creation, and annihilation of composite C-points may appear, and in the process the sum of topological charge remains unchanged. The shift, deformation, combination, and disappearance of composite L-lines may take place. The topological relationship holds true. The results are compared with the previous work.

The analytical far-field expressions for the TE and TM terms and energy flux distributions in two off-axis superimposed nonparaxial Laguerre-Gaussian beams with azimuthal and radial indices l(1) = -l(2) = +1, p(1) = p(2) = 0 are derived and used to study the far-field properties including phase singularities and energy flux distributions of the resulting beam, where our main attention focuses on the dependence of phase singularities on the controlling parameters such as the off-axis distance, relative phase, amplitude ratio, and waist widths of superimposed beams, and the symmetry property of edge dislocations and energy flux distributions. The results are interpreted and compared with previous work.

Phase singularities and spectral changes of spectrally partially coherent higher-order Bessel-Gauss pulsed beams in free-space propagation are studied, and the analytical expressions for the spectral degree of coherence and spectrum are derived. It is shown that there always exists an optical vortex at the center of the beam and the topological charge is conserved during the propagation. Some circular edge dislocations appear and the spectrum changes on propagation. A rapid spectral transition takes place. Numerical illustrative examples are given and the results are explained physically.

On the basis of the vector Rayleigh-Sommerfeld diffraction integrals, the analytical expression for Gaussian vortex beams diffracted at a half-plane screen beyond the paraxial approximation is derived and used to study the polarization singularities formed by the transverse and longitudinal electric field components in the diffracted field. It is shown that there exist C-points and L-lines that depend on the off-axis parameters in the x and y directions, waist width, wavelength, and topological charge of diffracted Gaussian vortex beams, as well as on the position parameter. By a suitable variation of the off-axis and position parameters, the creation, motion, and annihilation of polarization singularities may take place, and the topological relationship holds true. Therefore, the nonparaxial beam diffraction and propagation provide a method for generating polarization singularities.

The formulas of the energy and energy flux density of partially coherent electromagnetic beams in atmospheric turbulence are derived by using Maxwell's equations. Expressions expressed by elements of electric cross spectral density matrixes of the magnetic and the mutual cross spectral density matrix are obtained for the partially coherent electromagnetic beams. Taken the partially coherent Cosh-Gaussian (ChG) electromagnetic beam as a typical example, the spatial distributions of the energy and energy flux density in atmospheric turbulence are numerically calculated. It is found that the turbulence shows a broadening effect on the spatial distributions of the energy and energy flux density. Some interesting results are obtained and explained with regard to their physical nature.

The spectral Stokes singularities are introduced to describe the polarization singularities of stochastic electromagnetic beams. The motion and annihilation, as well as the changes in the polarization of spectral Stokes singularities, take place in the free-space propagation. The results are illustrated by numerical examples.

Analytical formulas for the cross-spectral density matrix of stochastic electromagnetic Gaussian Schell-model (EGSM) beams passing through an astigmatic optical system are derived. We show both analytically and by numerical examples the effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic EGSM beams propagating through an astigmatic lens. A comparison with the aberration-free case is made, and shows that the astigmatism has significant effect on the spectra, coherence and polarization.

The closed-form propagation expressions of an edge dislocation nested in a general elliptical Gaussian beam through aligned and misaligned paraxial optical ABCD systems are derived and used to study the dynamic evolution behavior of the edge dislocation propagating in free space and through a misaligned lens. It is shown that the noncanonical vortex and dynamic inversion of the topological charge, as well as a new edge dislocation may appear under certain conditions. The results are illustrated analytically and numerically.

Taking the partially coherent Hermite-sinh-Gaussian (H-ShG) beam as a more general type of partially coherent beams, a comparative study of the beam-width spreading of partially coherent H-ShG beams in atmospheric turbulence is performed by using the relative width, normalized beam width, and turbulence length. It is shown that the relative width versus the beam parameters, such as the spatial correlation length sigma(0), beam orders m, n, Sh-part parameter Omega(0), and waist width w(0), provides a simple and intuitive insight into the beam-width spreading of partially coherent H-ShG beams in turbulence, and the results are consistent with those using the turbulence length. The validity of our results is interpreted physically.

Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, the analytical expressions of the spectrum of GSM beams propagating in dispersive media are derived, and the spectral properties are studied in detail. It is shown that, in comparison with propagation in free space and in turbulence, whether or not GSM beams satisfy the scaling law, the normalized spectrum of GSM beams in dispersive media changes on propagation in general, because the dispersive medium affects different spectral components differently. As compared with the free-space propagation, for the scaling-law GSM beams the dispersion results in spectrum change, and for the nonscaling-law GSM beams the dispersion gives rise to a further increase in spectral changes. The structure constant of the dispersive property of the media, the transverse coordinate of the observation point, the spatial correlation length of the source, and the propagation distance affect the spectral behavior of GSM beams; this effect is illustrated numerically.

The propagation of partially coherent Hermite-cosh-Gaussian (H-ChG)beams through atmospheric turbulence is studied in detail. The analytical expression for the angular spread of partially coherent H-ChG beams in turbulence is derived. It is shown that the angular spread of partially coherent H-ChG beams with smaller spatial correlation length sigma0, smaller waist width w0, smaller beam parameter Omega0, and larger beam orders m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger sigma0, w0, Omega0, and smaller m, n. Under a certain condition partially coherent H-ChG beams may generate the same angular spread as a fully coherent Gaussian beam in free space and also in atmospheric turbulence. The angular spread of partially coherent Hermite-Gaussian (H-G), cosh-Gaussian (ChG), Gaussian Schell-model (GSM) beams, and fully coherent H-ChG, H-G, ChG, Gaussian beams is studied and treated as special cases of partially coherent H-ChG beams. The results are interpreted physically.

The closed-form expression for the mean-squared beam width of partially coherent Hermite-Gaussian (H-G) beams propagating through atmospheric turbulence is derived. The influence of turbulence on the spreading of partially coherent H-G beams is studied quantitatively by examining the mean-squared beam width. It is found that the smaller the coherence length sigma(0) of the source is, and the larger the beam order m and the wavelength lambda are, the less partially coherent H-G beams are affected by the turbulence, although the beams with smaller sigma(0), larger m, and larger lambda have greater spreading in free space. In addition, it is shown that two partially coherent H-G beams may generate the same angular spread and that there exist equivalent partially coherent H-G beams that may have the same directionality as a fully coherent Gaussian beam in free space and also in turbulence. The results are illustrated by examples, and a comparison with previous work is also made.

The analytical expression for the spectral degree of coherence of partially coherent flat-topped beams propagating through the turbulent atmosphere is derived, and the spatial correlation properties are studied in detail. In particular, we find that the oscillatory behavior and phase singularities of the spectral degree of coherence may appear when partially coherent flat-topped beams propagate through the turbulent atmosphere; this behavior is very different from the behavior of Gaussian Schell-model beams. But the oscillatory behavior becomes weaker with increasing turbulence and even disappears when the turbulence is strong enough. The width of the spectral degree of coherence is always smaller than that of the spectral density in the far field when the turbulence is strong enough, whereas the width of the spectral degree of coherence in free space can be either larger or smaller than that of the spectral density in the far field.

By introducing a new kind of Green function, we formulate an improved diffraction integral, which can be used to numerically evaluate the diffracted field of a microlens of plane-convex shape. Analytical expressions for the diffracted field of microlens are derived for the case where the curvature radius of the convex surface is larger than the dimension of the microlens aperture. The validity of the results and the diffracted field of the microlens are illustrated with numerical examples. The focal shifts of the diffracted field are found to depend mainly on the Fresnel number N of the microlens.

Based on the three-dimensional transient heat conduction equation and the elastic stress-strain equation, the temperature rise, distortion, and equivalent stress distributions of a high-reflectivity silicon reflector and a white bijou window irradiated by a high-power sloped annularly distributed laser beam are simulated using a three-dimensional finite element model (FEM). The effects of laser intensity, output duration, beam obscure ratio, and laser intensity spatial gradient on the results are especially investigated. The effects of mirror and window thermal distortion on laser beam phase aberrations are also evaluated. This noncylindrosymmetric three-dimensional FEM can be used to evaluate high-power, high-energy, laser beam-induced thermal effects on optical components.

On the basis of the vectorial Rayleigh-Sommerfeld formulas and by means of the relation between Hermite and Laguerre polynomials, the analytical expressions for the propagation of the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams beyond the paraxial approximation are derived, with the corresponding far-field propagation expressions and that for the Gaussian beams being given as special cases of the results. Some detailed comparisons of our results with the expansion series and paraxial expressions are made, which show the advantages of our results over the expansion series. With the results obtained, some typical intensity patterns of nonparaxial HG and LG beams are shown.

The polarization properties of vectorial coherent nonparaxial Gaussian beams are studied. It is shown that, when the source of a nonparaxial Gaussian beam is completely polarized, the degree of polarization of the propagation field maintains a constant value of 1. However, when the source is completely unpolarized, the degree of polarization does not maintain a constant value of 0. In the far field, vectorial nonparaxial Gaussian beams can become partially polarized, except on the propagation axis, and are spatially nonuniform across a typical cross section of the beam even though the source field is completely unpolarized.

On the basis of the Rayleigh-Sommerfeld diffraction integral, a closed-form propagation expression for the Wigner distribution function of partially coherent nonparaxial beams in free space is derived for what is to our knowledge the first time. The propagation of spatially fully coherent nonparaxial beams is treated as a special case of our general result. Application of the result is illustrated with the nonparaxial propagation of partially coherent anisotropic Gaussian-Schell-model beams and TEM11-mode Hermite-Gaussian beams.

Generalized vectorial Rayleigh-Sommerfeld diffraction integrals are developed for the cross-spectral-density matrices of spatially partially coherent beams. Using the Gaussian Schell-model (GSM) beam as an example, we derive the expressions for the propagation of cross-spectral-density matrices and intensity of partially coherent vectorial nonparaxial beams, and the corresponding far-field asymptotic forms, beyond the paraxial approximation. The propagation of the vectorial nonparaxial GSM beams are evaluated and analyzed. It is shown that a 3 x 3 cross-spectral-density matrix or a vector theory is required for the exact description of nonparaxial GSM beams.

Based on the vectorial Rayleigh-Sommerfeld diffraction integrals, an analytical propagation equation of vectorial, nonparaxial, elliptical Gaussian beams through a rectangular aperture is derived. Unlike in previous work, the aperture effect and nonrotational symmetry of the beam and aperture are considered in our theoretical model. The results of the far-field and paraxial approximation for the apertured case are treated as special cases of our general expression. It is found that two f parameters fx, fy and two truncation parameters deltax, deltay in the x and y directions, respectively, have to be introduced that affect the beam nonparaxial evolution behavior in both the near field and the far field.

The concept of a partially coherent nonparaxial beam is proposed. A closed-form expression for the propagation of nonparaxial Gaussian Schell model (GSM) beams in free space is derived and applied to study the propagation properties of nonparaxial GSM beams. It is shown that for partially coherent nonparaxial beams a new parameter f(sigma) has to be introduced, which together with the parameter f, determines the beam nonparaxiality.

The truncated second-order moments and generalized M2 factor (M(G)2 factor) of two-dimensional beams in the Cartesian coordinate system are extended to the case of three-dimensional rotationally symmetric hard-edged diffracted beams in the cylindrical coordinate system. It is shown that the propagation equations of truncated second-order moments and the M(G)2 factor take forms similar to those for the nontruncated case. The closed-form expression for the M(G)2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams is derived that depends on the truncation parameter beta and beam order N. For N --> infinity, the M(G)2 factor equals 4/square root of 3 corresponding to the value of truncated plane waves, which guarantees consistency of the formalism.

Starting from the propagation law of partially coherent polychromatic light in the space-frequency domain, detailed numerical results and physical analysis are given to elucidate spectral changes and spectral switches at the geometrical focal plane of Gaussian Schell-model beams focused by an aperture lens. It is found that, in contrast to the aperture-induced spectral anomalies of spatially fully coherent polychromatic light, for partially coherent polychromatic light aperture diffraction plays an important role in spectral switching, but the truncation parameter, spectral correlation, and bandwidth all affect its spectral behavior.

Based on the vectorial Rayleigh diffraction integral, the nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture is studied. The far-field and paraxial cases are treated as special cases of our general result. It is shown that for the apertured case the f parameter still plays an important role in determining the nonparaxiality of vectorial diffracted Gaussian beams, but both the f parameter and truncation affect the beam evolution behavior.

The subject is the spectral characteristics of partially coherent light whose spectral degree of coherence satisfies or violates the scaling law in diffraction by a circular aperture. Three kinds of spectral correlations of the incident light are considered. It is shown that no matter whether the partially coherent light satisfies or violates the scaling law, a spectral switch defined as a rapid transition of spectral shifts is always found in the diffraction field. Different spectral correlations of the incident field in the aperture result in different points at which the spectral switch occurs. With an increment in the correlations, the position at which the spectral switch takes place moves toward the point at which the phase of the center frequency component omega0 becomes singular for illumination by spatially fully coherent light. For light that satisfies the scaling law, the spectral switch is attributed to the diffraction-induced spectral changes; for partially coherent light that violates the scaling law, the spectral switch is attributed to both the diffraction-induced spectral changes and the correlation-induced spectral changes.

On the basis of angular spectrum representation and the stationary-phase method, a far-field expression for nonparaxial Gaussian beams diffracted at a circular aperture is derived, which permits us to study the far-field nonparaxial properties of apertured Gaussian beams both analytically and numerically. It is shown that for the apertured case, the f -parameter and the truncation parameter affect the beam's far-field properties. The f -parameter plays the more important role in determining the beam nonparaxiality than does the truncation parameter, whereas the truncation parameter additionally influences the beam diffraction. A comparison with the paraxial case is made. For the unapertured case our results reduce to the previous ones.

A detailed study of ultrashort pulsed Bessel beams in linear dispersive media is performed. The spatial and temporal parts of pulsed Jn beams are separable in dispersive media, provided that the parameter a is independent of frequency omega. The spatial part keeps the Jn shape unchanged during propagation. The temporal evolution behavior of pulsed Jn beams depends on the material's dispersion and diffraction. The pulses can be broadening and become negatively chirped while propagating in anomalous dispersive media. In normal dispersive media, the pulses can be broadening and positively or negatively chirped; even dispersion-free propagation can be achieved if the beam and material parameters are suitably chosen. The condition under which higher-order dispersive effects can be neglected is also discussed.

The generalized Huygens-Fresnel diffraction integral for misaligned asymmetric first-order optical systems is derived by using the canonical operator method, which enables us to study propagation properties of anisotropic Gaussian Schell-model (AGSM) beams through misaligned asymmetric first-order optical systems. It is shown that under the action of misaligned asymmetric first-order optical systems AGSM beams do not preserve the closed property. Therefore generalized partially coherent anisotropic Gaussian Schell-model beams called decentered anisotropic Gaussian Schell-model (DAGSM) beams are introduced, and AGSM beams can be regarded as a special case of DAGSM beams.

We present a simple formula to evaluate the relative focal shift in a circular-aperture lens system illuminated by a nonuniformly polarized (NUP) light wave. Specifically, it is shown that the relative focal shift is determined by the effective Fresnel number. The effective Fresnel number is equal to the product of the Fresnel number of the lens aperture and the parameter sigma, which describes the uniformity of the polarization distribution of the NUP beam across the lens aperture. Some examples are given to illustrate the use of this approach. The influence of the polarization distribution of the incident NUP light wave on the polarization distribution in the axial points of the focused field is also presented.

On the basis of generalized truncated second-order moments, a closed-form expression for the generalized M2 factor of hard-edge diffracted flattened Gaussian beams is derived that is determined by the beam order and the truncation parameter. Special cases are discussed. Moreover, it is shown that the M2 factor of truncated plane waves is equal to 4sqaure root3/3, independent of the aperture width.

The closed-form propagation equation of flattened Gaussian beams passing through a paraxial optical ABCD system, in which the linear gain and absorption media are included, is derived, and its general applicable advantage is illustrated with numerical examples.

A detailed study of phase-locked and non-phase-locked radial laser arrays is presented. The closed-form propagation expressions for the beamlets and the resulting beam are given, which enable us to study beam propagation properties of radial laser arrays for both phase-locked and non-phase-locked cases. Numerical calculation examples are given to illustrate the application of our analytical results and the differences between phase-locked and non-phase-locked radial arrays.

A detailed study of the coherent and the incoherent combinations of two-dimensional off-axis Hermite-Gaussian beams with rectangular symmetry is made. The closed-form propagation formulas of the resulting beam are derived, and the resulting beam quality in terms of the M(2) factor and power in the bucket is discussed and compared for the coherent and the incoherent combinations. In addition, it is shown that the resulting astigmatic beam can be symmetrized in the sense of the second-moment definition of beam width. However, the symmetrizing transformation of the resulting astigmatic beams is incomplete, because there exist different irradiance profiles.

Explicit expressions for the M(2) factor and field distribution of flattened Gaussian beams that propagate through an unapertured paraxial ABCD system are derived and illustrated with numerical examples. We show that the far-field expression of flattened Gaussian beams can be readily obtained by use of our results.

Analytical expressions for the M(2) factor and the mode-coherence coefficients of hyperbolic-cosine- (cosh-) Gaussian beams have been derived, and a simple method of producing cosh-Gaussian beams in an experiment, superposition of two decentered Gaussian beams, has been proposed. Our results will be useful for studying more-general Hermite-sinusoidal-Gaussian beams.

Nondegenerate four-wave mixing (NDFWM) in a double-L system of Rb atomic vapor was achieved for collinear pump fields. By comparison of different pump-probe configurations, direct experimental evidence of influence of coherent population trapping on the output power of the NDFWM signal was established. A maximal efficiency for generating the NDFWM signal exists for a moderate pump power in this double-L system.