Simple interferometric technique for generation of a radially polarized light beam

We present a theoretical and experimental investigation of an interferometric technique for converting a linearly polarized Gaussian beam into a radially polarized doughnut beam. The experimental setup accomplishes the coherent summation of two orthogonally polarized TEM01 and TEM10 beams that are obtained from the transformation of a TEM00 beam by use of a simple binary diffractive optical element. We have shown that the degree of radial polarization is maximum at a given distance from the interferometer output port that depends on the diameter of the incident beam at the interferometer input port.     

Formation of a sub-wavelength longitudinally polarized beam using a radially polarized controllable dark-hollow beam

On the basis of vector diffraction theory, the tightly focusing properties of radially polarized controllable dark-hollow (CDH) beams are examined theoretically. Calculation results demonstrate that by choosing the initial parameters of the proposed light beams suitably, a sub-wavelength (0.422λ) longitudinally polarized light beam with high beam quality (82.2%) can be formed without any filters. Meanwhile, we find that a relatively long depth of focus benefits from larger beam order. The dependence of the focal spot size on the parameters such as truncation parameter, variation constant, and beam order is also explored in detail. Moreover, an alternative method to generate the CDH beams is proposed.     

Generation of radially polarized beams with an image-rotating resonator

We show how a passive image-rotating optical resonator can be used to convert a linearly polarised, lowest-order Gaussian beam into a radially polarized beam. The image and polarization rotation of the cavity removes the frequency degeneracy of the modes, making it possible to select the radially polarized mode by cavity tuning. With the addition of gain, the same cavity should operate as a radially polarized laser when injection seeded at the proper wavelength.     

Axial separation of orthogonally polarized focal field components due to a radially polarized beam

In this paper, we investigate the field distribution in the focal volume of an aberrated radially polarized beam. Using two different forms of the vectorial diffraction theory, we show that the presence of defocus in the beam displaces both the axially and the radially polarized fields parallel to the optical axis of the focusing lens, while the presence of spherical aberration primarily shifts the longitudinally polarized field only. This facilitates axial separation of the two orthogonally polarized field components, resulting in a significant boost to the ratio of the peak longitudinally polarized field to the peak laterally polarized field in the focal plane. We further show that with an appropriate combination of oppositely signed defocus and spherical aberration, the energy density in the focal volume due to the longitudinally polarized field can be caused to peak at the focal plane. The results obtained are expected to be beneficial to the applications requiring a stronger longitudinally polarized focal field relative to the laterally polarized focal field component.     

Description of the propagation of a radially polarized beam with the scalar Kirchhoff diffraction

Since radially polarized beams have only one magnetic field component, the azimuthal component, a scalar Kirchhoff diffraction integral can be used to describe the propagation of the magnetic field. In the far-field zone, this diffraction formula gives an analytic expression for the magnetic field from which the electric field component expressions are derived by the Faraday relation. Numerical results from these expressions correctly reflect the properties of a radially polarized beam.     

Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere

Based on the Huygens–Fresnel principle and the unified theory of coherence and polarization of partially coherent beams, we investigate the propagation characteristics of a partially coherent radially polarized doughnut (PCRPD) beam in a turbulent atmosphere. It is found that, after propagating through a turbulent atmosphere, the doughnut beam spot is changed into a circular Gaussian beam. Moreover, the degree of coherence, the degree of polarization and the degree of cross-polarization of the beam will change on propagation, and this change is dependent upon the degree of coherence of the source and atmospheric turbulence.     

Propagation properties of a partially coherent radially polarized beam in atmospheric turbulence

Based on the extended Huygens–Fresnel integral, second-order moments of the Wigner distribution function of a partially coherent radially polarized beam propagating through atmospheric turbulence are derived. Besides, propagation properties such as the mean-squared beam width, angular width, effective radius of curvature, beam propagation factor and Rayleigh range can also be obtained and calculated numerically. It is shown that the propagation properties are dependent on the spatial correlation length, refraction index structure constant and propagation distance.     

Statistical properties of a radially polarized twisted Gaussian Schell-model beam in a uniaxial crystal

We derive the analytical formulas for the elements of the cross-spectral density matrix of a radially polarized partially coherent beam with a twist phase named radially polarized twisted Gaussian Schell-model (RPTGSM) beam propagating in a uniaxial crystal, and explore the statistical properties, such as the intensity distribution, the degree of polarization (DOP) and the state of polarization (SOP) of such beam in a uniaxial crystal with the help of the derived formulas. It is found that the statistical properties of a RPTGSM beam are closely related with the twist phase and the anisotropy of the uniaxial crystal, e.g. the twist phase leads to the rotation of the intensity, DOP and SOP distributions, and the anisotropy of the uniaxial crystal leads to the asymmetry distributions of these statistical properties. Our results will be useful for designing light field with prescribed intensity, DOP and SOP distributions, and may be useful in optical manipulations and free-space optical communications.     

Polarization changes in a partially coherent radially polarized doughnut beam through turbulent ocean

On the basis of the extended Huygens–Fresnel integral principle and unified theory of coherence and polarization of light, we studied the effects of oceanic turbulence on polarization properties of a partially coherent radially polarized doughnut (PCRPD) beam. The ocean-induced fluctuations in the refractive index are assumed be driven by temperature and salinity fluctuations. Numerical examples of changes in polarization properties, such as the degree of polarization, the degree of ellipticity, and the orientation angle in the oceanic turbulence for the PCRPD beam, are given. Our analysis demonstrates how polarization of the PCRPD beam is affected by statistical properties of the source and by several parameters of oceanic turbulence. We find that the propagation of the PCRPD beam is different from that of stochastic beams in oceanic turbulence. The degree of polarization for the PCRPD beam approaches a certain steady value, and the elliptical polarized state of the fully polarized portion of the beam will become fully linear in the far field.     

Generation of dark hollow femtosecond pulsed beam by phase-only liquid crystal spatial light modulator

Based on the refractive laser beam shaping system, the dark hollow femtosecond pulse beam shaping technique with a phase-only liquid crystal spatial light modulator (LC-SLM) is demonstrated. The phase distribution of the LC-SLM is derived by the energy conservation and constant optical path principle. The effects of the shaping system on the temporal properties, including spectral phase distribution and bandwidth of the femtosecond pulse, are analyzed in detail. Experimental results show that the hollow intensity distribution of the output pulsed beam can be maintained much at more than 1200 mm. The spectral phase of the pulse is changed, and the pulse width is expanded from 199 to 230 fs, which is caused by the spatial-temporal coupling effect. The coupling effect mainly depends on the phase-only LC-SLM itself, not on its loaded phase distribution. The experimental results indicate that the proposed shaping setup can generate a dark hollow femtosecond pulsed beam effectively, because the temporal Gaussian waveform is unchanged.     

Multipole theory for tight focusing of polarized light, including radially polarized and other special cases

A multipole expansion, based on spherical harmonics, provides an efficient method for calculating the field in the focal region of a lens for radially polarized illumination, or other illumination polarization and phase distributions, including vortex beams. The multipole approach also has the benefit of providing a simple measure of the purity of the longitudinal field mode. The method is also convenient for calculation of fields scattered by particles and calculation of optical trapping forces.     

Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators

We report on the high-quality holographic generation of higher-order Laguerre-Gaussian (LG) beams using a liquid-crystal-on-silicon spatial light modulator. The effects of the input beam pattern on the output LG beam quality are investigated in detail through theoretical discussions and experiments. Correlation analyses between observed beam patterns and theoretical mode profiles reveal that higher beam quality is achieved for output LG beams generated from a top-hat input beam than from a Gaussian input beam.     

Generation of linearly polarized transition radiation X-ray beam

We have successfully produced almost linearly polarized X-ray beam by extracting Transition Radiation (TR) X-rays through a rectangular slit 3 mrad long and 0.2 mrad wide placed downstream of a 7.5-μm thick Kapton foil stack radiator bombarded with a 1-GeV electron beam. Our calculation predicted that the linearity of TR photon was 94% for both a vertical and a horizontal slit. The measured results for the vertical slit agreed with the calculation, and the linearity of obtained TR X-ray beam was more than 90%. For the horizontal slit, the measured results differed from our expectations due to a prolonged electron beam profile. In the polarization measurement of TR photon we used the Bragg reflection on a lithium fluoride crystal [LiF(200)] which acted as both a polarimeter and a spectrometer.     

Rotation of nanowires with radially higher-order Laguerre-Gaussian beams produced by computer-generated holograms

In this paper, the influence of radially higher index p of Laguerre-Gaussian (LG) beams on the rotation of nanowires is studied. Radially higher-order LG beams are produced by computer-generated holograms, which are displayed on a spatial light modulator. A series of experiments on manipulating ZnO nanowires was performed on our holographic optical tweezers platform. The experiments demonstrated that radially higher-order LG beams could effectively rotate nanowires along the innermost bright ring of themselves. Compared with radially low-order LG beams, they have larger torques exerted on nanowires and can make nanowires rotate more quickly.     

Determination of complex coefficients of radially polarized piezoelectric ceramic cylindrical shells using thin shell theory

A method is presented to determine the complex coefficients E/sub 33//sup T/, s/sub 11//sup E/, s/sub 12//sup E/, and d/sub 31/ of piezoelectric materials. The real parts of these coefficients are determined using axially polarized thin discs in the ANSI/IEEE Standard but are determined here using radially polarized cylindrical shells. The coefficients are determined by iteratively refining them until the values of the low-frequency complex admittance, three resonance frequencies, and three band-widths computed using a thin-shell analytical model and the coefficients are very nearly equal to measured values. The accuracy of the method is determined by using quantities computed using a finite-element model in place of measured values. Measurement errors are accounted for by using a resolution of 10 Hz to compute the critical frequencies. The differences between the coefficients input to the finite-element model and those obtained using the iteration method are the errors. It is shown that the method is sufficiently accurate to use thin radially polarized cylindrical shells to determine the properties of new materials as well as characterize those used in hydrophones or other devices.     

Propagation analysis of the Laguerre-Gaussian beam with astigmatism

The entire range of transformations that a Laguerre-Gaussian (LG) beam with astigmatism can go through in free space is clarified. The transformations are governed by the relative phase between the astigmatic Hermite-Gaussian components. Formulas describing the behavior of this relative phase are obtained and used to classify and map the transformation patterns to initial beam parameters. The difference between an LG beam and a phase singular beam generated by a hologram under astigmatic conditions is also investigated.     

Tight focusing induces pulse delay and pulse compression of double-ring-shaped radially polarized ultrashort light pulses

The focusing of double-ring-shaped radially polarized ultrashort light pulses by a high-numerical aperture objective is investigated using vectorial Debye theory. After focusing, the double-ring-shaped radially polarized ultrashort light pulses slow down near the focus, and this pulse delay induces pulse compression in the propagation direction. That is, without changing the pulse duration, the spatial pulse length of the ultrashort light pulse is decreased near the focus. The velocity of the longitudinal component of the light pulse near the focus is lower than that of the radial component. The simulation results demonstrate that control of the ratio of the pupil radius to the beam radius affects not only the longitudinal component of the light pulse, but also the velocity and spatial pulse length of the light pulse because of the destructive interference between the longitudinal components of the inner and outer rings of the light pulse.     

Focusing of radially and azimuthally polarized beams through a uniaxial crystal

We calculated and measured the difference between focal positions of radially and azimuthally polarized beams after passing through a uniaxial crystal. Calculations were carried out on the basis of the ray optics and the vector diffraction theory. The results of the calculations were in good agreement with those of the experiment. In addition, we discussed the polarization selection in a hemispherical laser cavity that was used for the generation of a radially polarized beam by use of the birefringence of a c-cut Nd:YVO4 laser crystal [Opt. Lett. 31, 2151 (2006)]. The stability range of the laser cavity length for the generation of a radially polarized beam was also in good agreement with the differences mentioned above.     

Rotating light fields of an azimuthally polarized light beam generated by two-belt spiral phase modulation

The tightly focused light fields of an azimuthally polarized light beam through a two-belt spiral phase plate were investigated. The focused light fields are presented in accordance with vectorial diffraction theory. The results show that a rotating light field with different intensity patterns can be produced by altering the azimuthal polarization state and modulating the two-belt spiral phase. A concurrent change in spiral handedness in the two-belt phase plate causes the rotation to occur along the direction of propagation, and the relative angular offset in the two-belt spiral phase plate can be exploited to rotate the light fields. The proposed method is useful for engineering the intensity distribution near the focal plane and related applications.     

On finite oscillations of a gas-filled radially isotropic elastic spherical shell

The periods of finite radial oscillations of a transversely isotropic gas-filled spherical shell have been obtained. Numerical calculations have been presented using a particular form of strain energy function and have been compared with the isotropic case.