Changes in generalized Stokes parameters of stochastic electromagnetic beams on propagation through ABCD optical systems and in the turbulent atmosphere

Using the derived formulas for the transformation of 2 × 2 cross-spectral density matrix of the stochastic electromagnetic beams propagating through ABCD optical systems and in the turbulent atmosphere, the changes in the generalized Stokes parameters of the beams propagating under these conditions can be investigated directly. Some typical numerical calculations are illustrated relating to the electromagnetic Gaussian Schell-model beams passing through free space, focal system, dual-focus system, and the turbulent atmosphere with different structure parameters. Further extensions are also pointed out.     

Generation of thin and hollow beams by the axicon with a large open angle

We propose a method to produce diffraction-free thin and hollow beams. The method is based on Laguerre-Gaussian (LG) beams incident on a large open-angle axicon. We use the vector diffraction integrals and stationary phase method to deduce a simple and analytical formula of the propagating field of the linearly polarized LG beams through an axicon. The numerical results show that the hollow beams of whose diameter is in the order of the wavelength can be obtained by using the axicon with the refractive index n = 2 and the open angle α = 25°. These diffraction-free thin and hollow beams may be very useful to accurately trap and manipulate atoms. However, when the open angle is over large, the conversion efficiency from the LG beam to the diffraction-free hollow beam will decrease obviously.     

Extension of the van Cittert-Zernike theorem for completely polarized incoherent electromagnetic beam propagating through non-Kolmogorov turbulence

Based on the 2 × 2 cross-spectral density matrix, the van Cittert-Zernike extended theorem is developed for the completely polarized incoherent beams propagation through the paraxial non-Kolmogorov turbulence. On the consequence of the extended theorem and the definition of general spectral degree of cross-polarization of a beam, we found that the spectral degree of cross-polarization of the resultant field is independent of the refractive index structure constant of atmospheric turbulence. We investigated the influences of the propagation distance and the distance of two detection points on the degree of coherence and the spectral degree of cross-polarization.     

Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel–Gauss beams in free-space propagation

By using the generalized Huygens–Fresnel diffraction integral, the analytical expressions for the cross-spectral density matrix, spectral degree of polarization, orientation angle and degree of ellipticity of polarization ellipse of apertured stochastic electromagnetic modified Bessel–Gauss beams (MBGBs) through a paraxial optical ABCD system are derived, and used to study the changes in the state of polarization of apertured stochastic electromagnetic MBGBs propagating in free space. The invariance of the on-axis state of polarization of unapertured stochastic electromagnetic MBGBs propagating through paraxial optical ABCD systems is illustrated analytically and numerically. For apertured stochastic electromagnetic MBGBs, the on-axis spectral degree of polarization, orientation angle and degree of ellipticity of polarization ellipse increase with increasing propagation distance, and approach asymptotic values when the propagation distance is large enough. There is a uniform distribution region of the state of polarization around the center of the beams whose range decreases with increasing truncation parameter. In addition, the state of polarization of apertured stochastic electromagnetic MBGBs upon propagation can be modulated by controlling the truncation parameter.     

Nonparaxial propagation of spatially and spectrally partially coherent electromagnetic Cosh-Gaussian pulsed beams

Based on the generalized Rayleigh–Sommerfeld diffraction integral, the analytical expression for 3×3 cross-spectral density matrix of nonparaxial spatially and spectrally partially coherent electromagnetic Cosh-Gaussian (ChG) pulsed beams propagating in free space is derived, and used to formulate the spectral density and spectral degree of polarization of electromagnetic pulsed beams at the z-plane. It is found that the parameters f and f_αα are the key parameters in determining the nonparaxiality of spatially and spectrally partially coherent electromagnetic ChG pulsed beams. And the decentered parameter, pulse duration and temporal coherence length can change the nonparaxial behavior of the electromagnetic ChG pulsed beams. The effect of decentered parameter, pulse duration and temporal coherence length on the spectral density and spectral degree of polarization of electromagnetic ChG pulsed beams is illustrated through numerical calculations. Propagation of nonparaxial spatially and spectrally partially coherent electromagnetic Gaussian Schell-model pulsed beams can be treated as a special case when the decentered parameter of electromagnetic ChG pulsed beams approaches to zero.     

Photoinduced anisotropy in an azo-containing ionic liquid-crystalline polymer

Photoinduced anisotropy in an azobenzene ionic liquid-crystalline polymer was investigated through dichroism, birefringence and polarization holography. A dichroism degree of 1.58 and a birefringence value Δn ∼ 10⁻² were achieved in the polymer film at room temperature, and the polymer film was found to possess the characteristics of reversible and long-term optical storage. Particularly the stored birefringence could be enhanced to Δn ∼ 10⁻¹ by annealing the film, and it is attributed to the thermal self-organization of the molecules. Furthermore, linear- and circular-polarization holographic recordings were accomplished in the polymer film and pure polarization gratings were produced.     

Statistical properties of correlated radial stochastic electromagnetic array beams on propagation

A kind of array beam named the correlated radial stochastic electromagnetic array beam that is generated by an electromagnetic Gaussian Schell-model source is introduced by use of tensor method. The analytical expression for the cross-spectral density matrix of this array beam propagating through the turbulent atmosphere and in free space is obtained after performing vector integration. Some typical numerical calculations are illustrated for the changes in the spectral density, spectral degree of polarization, and spectral degree of coherence of the beam on propagation. We find that the atmospheric turbulence can destroy the correlated effect among the beamlets.     

Optimal phase steps of multi-level spiral phase plates

We analyze vortex properties of the optical beams generated by a multi-level spiral phase plate (MLSPP) and find that this kind of beams consists of vortex components with topological charges of L − kM, where k is zero or any integer, M is the level number and L is the intrinsic topological charge of the MLSPP. We proved that the orbital angular momentum of the beams generated by a MLSPP reaches its maximum only if the phase steps (or the ratio of M to L) satisfies some special conditions.     

Nonlinear optical properties and photoinduced anisotropy of an azobenzene ionic liquid-crystalline polymer

The nonlinear optical properties and photoinduced anisotropy of an azobenzene ionic liquid-crystalline polymer were investigated. The single beam Z-scan measurement showed the polymer film possessed a value of nonlinear refractive index n₂ = −1.07 × 10⁻⁹ cm²/W under a picosecond 532 nm excitation. Photoinduced anisotropy in the polymer was studied through dichroism and photoinduced birefringence. A photoinduced birefringence value Δn ∼ 10⁻² was achieved in the polymer film. The mechanism for the nonlinear optical response and the physical process of photoinduced anisotropy in the polymer were discussed.     

Topological and morphological transformations of developing singular paraxial vector light fields

The paper is devoted to establishment of the real-time topological and morphological dynamics of generic developing paraxial elliptic speckle fields generated and driven by the system ‘laser beam + photorefractive crystal LiNbO₃:Fe’. Generic space-time development of full gamut of polarization ellipse parameters (ellipticity, azimuth, morphology of C points, optical diabolos and handedness) and their combination in fixed beam cross-section was measured in details by the elaborated quick-action real-time Stokes-polarimetry. Whole field irreversible evolution is fulfilled through totality of random space/time C point pair nucleation/annihilation. The ‘life-story’ of C point and optical diabolo pairs is realized through ‘local topological/morphological transition’ with fully reversible scenario. It starts from smooth fragment of speckle field by formation of pre-nucleation local structure and finishes by after-annihilation local structure which decays to another smooth structure. Scenarios of star-monstar pair nucleation/annihilation and monstar  ↔  lemon transformation were established. Measured statistics of C point and diabolo morphological forms was in excellent agreement with theory predictions. All allowed scenarios of diabolo pair ‘life-story’ started/finished as star-hyperbolic monstar-hyperbolic pair were measured. Evolution of polarization ellipses handedness is implemented through L contours movement and reconnection with a saddle as the catalyst. Reconnection of L contour peninsula leads to birth of closed L contour delimiting island of fixed handedness ellipses with/without C points. Elaborated approach and presented results start the dynamic singular optics of time-dependent vector light fields.     

Changes in the spectrum, polarization, and coherence of stochastic spatially and spectrally partially coherent electromagnetic J0-correlated Schell-model pulsed beams propagating in free space

Taking the stochastic electromagnetic J ₀-correlated Schell-model pulsed (JSMP) beam as a typical example of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams, the analytical expressions for the cross-spectral density matrix, spectral density, spectral degree of polarization and spectral degree of coherence of stochastic electromagnetic JSMP beams propagating in free space are derived, and used to study the changes in the spectrum, polarization, and coherence of stochastic spatially and spectrally partially coherent electromagnetic JSMP beams. It is shown that the on-axis spectrum is blue-shifted in free-space propagation and is dependent on the pulse temporal coherence length and spatial correlation parameter. The distribution of the on-axis spectral degree of polarization depends on the frequency and spatial correlation parameter. The spectral degree of coherence increases with increasing pulse temporal coherence length. The results derived are interpreted physically.     

Identification of optical nonlinearities of dye-doped nematic and polymer-dispersed liquid crystals using Z-scan technique

This study investigates the nonlinear optical properties of azo-dye-doped nematic and polymer-dispersed liquid crystal (ADDPDLC) films with nano-sized LC droplets using the Z-scan technique, which is a simple but powerful technique for measuring the optical Kerr constants of materials. The results indicate that the optical Kerr constant (n₂) of the azo-dye-doped nematic LC (ADDLC) film is large because of the photoisomerization effect and the thermal effect. Therefore, the optical Kerr constant of this material can be modulated by varying the temperature of the sample and the direction of polarization of incident laser. The range of n₂ modulated is from −5.26 × 10⁻³ to 1.62 × 10⁻³ cm²/W. The optical Kerr constants of ADDPDLC films at various temperatures are also measured. The experimental results reveal that liquid crystals in the ADDPDLC film strengthen the nonlinearity. The n₂ of the ADDPDLC film is maximal at ∼35 °C, because of the decrease in the clearing temperature of the ADDPDLC films. The clearing temperatures of the liquid crystals (E7), and the ADDPDLC film used in this work were found to be 61 °C and 43 °C, respectively.     

Bandwidth tuning of hybrid liquid-crystal Šolc filters based on an optical cancelling technique

We demonstrate that in addition to their role in tuning the wavelength of an N-stage hybrid liquid-crystal Šolc filter, liquid-crystal cells can also be used to vary the transmission bandwidth of such filter around any of the tuned wavelength. This bandwidth tuning is based on the variation of the number of stages by what we call here an “optical cancelling technique”. This is achieved by varying the birefringence of the liquid-crystal cells whose optical path difference switches between two particular values. We show that for a 10-stage filter and at λ_i = 1.532 μm, the calculated 3-dB bandwidth varies from 2.6 to 11.8 nm when the number of “optically-cancelled” hybrid plates increases from 0 to 8. During the tuning process, the contrast ratio remains equal to that of the equivalent classical Šolc filter.     

Propagation of astigmatic stochastic electromagnetic beams in oceanic turbulence

In this paper, we investigate the characteristics of astigmatic stochastic electromagnetic beams through oceanic turbulence. Taking the electromagnetic Gaussian Schell-model (GSM) beam as an example, the analytic expressions for the spectral density and the spectral degree of polarization of the beam propagating the oceanic turbulence are derived. It is indicated that the spectral density along the z-axis of the GSM beam in the oceanic turbulence is severely influenced by the source correlation properties, as well as by the sea-related parameters. We show that the characteristics of the spectral density along the x-axis, y-axis and z-axis of astigmatic electromagnetic GSM beams passing through the oceanic turbulence are qualitatively different. Furthermore, we find that as the astigmatic coefficient becomes larger, the maximum value of the spectral density along the z-axis increases rapidly and the width of the spectral density becomes shorter rapidly. Finally, the results have shown that different strengths of astigmatism have different effects on the spectral degree of polarization.     

Periodic sub-micrometric structures using a 3D laser interference pattern

A method to obtain three-dimensional sub-micrometric periodic structures is presented. The experimental set-up consists in a pulsed UV laser beam source (λ = 355 nm) coming into an interferometer in order to generate four beams converging inside a chamber. According to the directions, to the relative intensities and to the polarizations of these four beams, a 3D interference pattern can be obtained inside the overlapping volume of these four beams; the characteristics of the four laser beams have been optimized in order to obtain a maximal contrast of intensity. In order to visualize the interference pattern, its contrast and its stability at each laser pulse, a video camera coupled to an oil immersion microscope objective has been installed above the interferometer. By suppressing the central beam, it is also possible to generate a bidimensional interference pattern which defines an hexagonal structure in the (1 1 1) plane with a period of 377 nm.This optical set-up has been used to obtain 3D sub-micrometric periodic structures in negative photoresists. Experiments consist in a one- or multi-pulse irradiation of the photoresist followed by a development procedure which leads to a sub-micrometric face-centred cubic structure cut in a (1 1 1) plane with a cell parameter of 650 nm. The optimization of the experimental conditions is presented for two kinds of photoresists; the role of the substrate according to its reflectivity at the laser wavelength and its influence on the interference pattern is also discussed.     

Method for measurements of second-order nonlinear optical coefficient based on Z-scan

The method for measuring second-order nonlinear optical coefficients based on well-known Z-scan is presented. The influence of linear absorption coefficients on normalized transmittance is discussed. Using this method, we obtained the second-order nonlinear coefficient d₃₁(5%MgO:LiNbO₃) = 4.5 × 10⁻¹² m/v at 1064 nm, which agrees well with theoretical calculations and previous well-known values.     

Propagation of a stochastic electromagnetic Gaussian Schell-model beam through an optical system in turbulent atmosphere

A generalized diffraction integral formula for stochastic electromagnetic beams propagating through an optical system in turbulent atmosphere is derived with the help of tensor method. Some analyses are illustrated by a numerical example relating to changes in the average intensity and the degree of polarization of an electromagnetic Gaussian Schell-model beam propagating through a double-lenses system. It is shown that the optical system has strong influence on the propagation properties of the beam. The method used in this paper can be widely applied to the propagation of astigmatic beams through an optical system in turbulent atmosphere.     

Effect of polarization on the evolution of electromagnetic hollow Gaussian Schell-model beam

Based on the theory of coherence, an analytical propagation formula for partially polarized and partially coherent hollow Gaussian Schell-model beams (HGSMBs) passing through a paraxial optical system is derived. Furthermore, we show that the degree of polarization of source may affect the evolution of HGSMBs and a tunable dark region may exist. For two special cases of fully coherent and partially coherent δ_xx = δ_yy, normalized intensity distributions are independent of the polarization of source.     

Flat-topped spectrum obtained from partially coherent hollow Gaussian Schell-model beams passing through a slit

The anomalous spectral behaviors of partially coherent polychromatic hollow Gaussian Schell-model beams (HGSMBs) diffracted by a slit are investigated. Besides spectral switches, asymmetrical dual-peak spectral split and multi-peak spectral split are observed. In the vicinity of the positions where spectral switches occur, flat-topped spectrum can be obtained, the bandwidth of which is adjustable by changing the transverse coherence width and the observation position. The flat-topped spectrum has potential application for creating probe beams in optical metrology systems and generating broadband optical sources in fiber optic sensor interface systems.     

Parametric study of uniformly polarized stochastic electromagnetic beam and its imaging

A parametric study is performed in investigating the stochastic electromagnetic beam generated by a uniformly polarized electromagnetic Gaussian Schell-model source and passing through ABCD optical systems. Through theoretical analysis, the requirement is derived that the uniformly polarized electromagnetic field can be obtained at the output plane of the imaging optical system. Furthermore, the general imaging formula of the stochastic electromagnetic beam is derived. Numerical examples are also presented to illustrate the application.