Fraunhofer diffraction of a slit aperture between a knife-edge and a metal cylinder

The Fraunhofer diffraction pattern of a slit aperture formed between a reference knife-edge and a metal-cylinder surface is different from that of an ideal slit aperture. This pattern should include reflected light coming from both the front and rear sides of a cylinder surface. To investigate the influence of light reflected from the cylinder surface, we discuss the theoretical consideration based on the simple model of the reflected light on the surface. The experimental setup is designed and constructed to measure the actual diffraction pattern produced by the slit between the knife-edge and the cylinder surface. As a result it is obvious that the reflection of diffracted light on the rear side is dominant in both the simulation and the experiment.     

Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings

Expressions describing the vortex beams that are generated by the process of Fresnel diffraction of a Gaussian beam incident out of waist on fork-shaped gratings of arbitrary integer charge p, and vortex spots in the case of Fraunhofer diffraction by these gratings, are deduced. The common general transmission function of the gratings is defined and specialized for the cases of amplitude holograms, binary amplitude gratings, and their phase versions. Optical vortex beams, or carriers of phase singularity with charges mp and -mp, are the higher negative and positive diffraction-order beams. The radial part of their wave amplitudes is described by the product of the mpth-order Gauss-doughnut function and a Kummer function, or by the first-order Gauss-doughnut function and the difference of two modified Bessel functions whose orders do not match the singularity charge value. The wave amplitude and the intensity distributions are discussed for the near and far fields in the focal plane of a convergent lens, as well as the specialization of the results when the grating charge p=0; i.e., the grating turns from forked into rectilinear. The analytical expressions for the vortex radii are also discussed.     

Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium

Log-amplitude and phase-correlation and structure functions of higher-order annular laser beams in a turbulent atmosphere are derived. A higher-order annular beam source is defined as the superposition of two different higher-order Hermite-Gaussian beams. A special case of such an excitation is the annular Gaussian beam in which two beams operate at fundamental modes of different Gaussian beam sizes, yielding a doughnut-shaped (annular) beam when the second beam is subtracted from the first beam. Our formulation utilizes Rytov approximation, which makes it applicable in the weak-turbulence regime, especially for log-amplitude fluctuations. Limiting cases of our formulations correctly match with known higher-order-mode solutions that in turn reduce to the Gaussian-beam-wave (TEM00-mode) results. Our results can be applied to determine the scintillation index and the phase fluctuations in free-space optics links under higher-order annular laser beam excitation. Except for the numerical evaluation of a specific example covering an annular Gaussian beam, the results in general are left in integral form and need to be numerically evaluated in detail to obtain quantitative results.     

Electro-optic modulator capable of generating simultaneous amplitude and phase modulations

We report on the analysis and prototype characterization of a dual-electrode electro-optic modulator that can generate both amplitude and phase modulations with a selectable relative phase, termed a quadrature variable modulator (QVM). All modulation states can be reached by tuning only the electrical inputs, facilitating real-time tuning, and the device has shown good suppression and stability properties. A mathematical analysis is presented, including the development of a geometric-phase representation for modulation. The experimental characterization of the device shows that relative suppressions of 38, 39, and 30 dB for phase, single sideband, and carrier-suppressed modulations, respectively, can be obtained as well as that the device is well behaved when scanning continuously through the parameter space of modulations. The QVM is compared with existing optical configurations that can produce amplitude and phase-modulation combinations in the context of applications such as the tuning of lock points in optical-locking schemes, single-sideband applications, modulation fast-switching applications, and applications requiring combined modulations.     

Time dependence of Fresnel diffraction of ultrashort laser pulses by a circular aperture

We report on the time dependences of the Fresnel diffraction of ultrashort laser pulses by a circular aperture. The diffraction leads to a pulse delay, time shape change, pulse broadening, and peak power decrease. These effects may have to be taken into account whenever critical thresholds are encountered as, for example, in nonlinear optics or laser fusion.     

Spectral changes and spectral switches of partially coherent beams focused by an aperture lens

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.     

Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture

When a Gaussian beam is apertured, it undergoes a focal shift as well as a phase shift. The focal shift has been investigated extensively although the phase shift has seldom been discussed. We analyze the phase shift of the apertured Gaussian beam. Furthermore we point out that the phase aperture may be used to transform the intensity distribution of the Gaussian beam to obtain a more concentrated beam, to realize uniformity of the Gaussian beam, and to obtain a ring beam.     

Experiments with active phase matching of parallel-amplified Multiline HF laser beams by a phase-locked Mach-Zehnder interferometer

Active phase matching of multiline HF laser beams by means of a phase-locked Mach-Zehnder interferometer was demonstrated by locking the interferometer to the central interference fringe at zero optical path length difference. The central fringe could be found by varying the spectral content of the input beam. Laser amplification in one leg of the interferometer decreased fringe visibility without adversely affecting locking. Single-line fringe patterns produced by an array spectrometer (while the interferometer was operated in its scanning mode) were analyzed to show that no significant dispersion occurred in the amplifier. The techniques developed have potential for measuring dispersion mismatch between larger parallel amplifiers. These experiments demonstrated in principle that a number of multiline HF amplified beams can be recombined and phase-matched to produce a high beam quality output beam.     

Shape characterization of a large nonspherical particle by use of its Fraunhofer diffraction pattern

We characterize the shape of a large nonspherical particle by means of the two-dimensional Fourier transformation of its diffraction pattern, called the S function. The main properties of S functions are considered. Some ways in which to retrieve the geometric parameters of a particle by use of its S function are discussed. In particular, the parameter of nonsphericity of a particle is defined by means of the S function.     

Wavefront distortion of the reflected and diffracted beams produced by the thermoelastic deformation of a diffraction grating heated by a Gaussian laser beam

It may be advantageous in advanced gravitational-wave detectors to replace conventional beam splitters and Fabry-Perot input mirrors with diffractive elements. In each of these applications, the wavefront distortions produced by the absorption and subsequent heating of the grating can limit the maximum useful optical power. We present data on the wavefront distortions induced in a laser probe beam for both the reflected and diffracted beams from a grating that is heated by a Gaussian laser beam and compare these results to a simple theory of the wavefront distortions induced by thermoelastic deformations.     

Phase retrieval from diffraction intensities by use of a scanning slit aperture

A noniterative method of retrieving the phase of a wave field from intensities measured during scanning of a slit aperture is proposed. In the measurements, one records the diffraction intensities of wave fields transmitted through a slit that is scanned along two directions in the Fresnel-zone plane of an object's field. From these intensities, the phase in the Fresnel-zone plane can be retrieved by a method in which a novel phase calculation technique that uses Fourier transforms is included. Because the method does not require lens systems, it provides a potentially useful means for coherent imaging by use of x rays, electrons, or nuclear particles.     

Focal shift and focal switch of partially coherent flat-topped beams passing through an alignment and misalignment lens system with aperture

The focal shift and focal switch of partially coherent flat-topped (PCFT) beams passing through a lens system with an aperture are studied in detail. We have shown that a focal shift is also present for beams propagating through an aligned optical system and the amount of the focal shift depends not only on the radius aperture of the focusing system, but also on the spatial coherence and order of the flat-topped beam of the incident partially coherent light. A new phenomenon called ‘focal switch’ occurs for misaligned optical systems, i.e. the focal shift experiences a sudden transition as the aligned optical system becomes misaligned, and the influence of the spatial coherence, order of flat-topped beam and aperture size reduction, on the focal switch is investigated in detail. Finally, the necessary conditions for the focal switch to take place are investigated.     

Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations

A method for closed-loop stable control of two deformable mirrors for compensation of both amplitude and phase fluctuations is described. A generic implementation is described as well as an implementation that integrates the concept behind a point diffraction interferometer with a two-deformable-mirror system. The relationship of the closed-loop control algorithm to previously developed open-loop iterative algorithms is described. Simulation results are presented that indicate that the system is stable and provides superior performance over that of a single-deformable-mirror system. The impact of finite servo bandwidth on control of two deformable mirrors is evaluated by means of wave optical simulation, and it is found that to achieve a performance improvement attributable to compensation of amplitude fluctuations, the bandwidth of the two-deformable-mirror system must be at least twice the Greenwood frequency.     

Fresnel diffraction by a circular aperture with off-axis illumination and its use in deconvolution of microscope images

The Fresnel approximation for off-axis illumination of a circular aperture is reexamined. The point-spread function for an aberration-free system can be expressed in terms of redefined optical coordinates. An improved expression is given for contours of constant intensity in the focal plane. The variation in axial width of the focal spot with angle of offset is discussed. The predictions are compared with exact calculations of the Rayleigh-Sommerfeld diffraction integral. Limitations for application in deconvolution of microscope images formed with objectives of finite tube length are discussed.     

Transverse intensity distributions of a broadband laser modulated by a hard-edged aperture

By means of the Huygens-Fresnel diffraction integral, the field representation of a laser beam modulated by a hard-edged aperture is derived. The near-field and far-field transverse intensity distributions of the beams with different bandwidths are analyzed by using the representation. The numerical calculation results indicate that the amplitudes and numbers of the intensity spikes decrease with increasing bandwidth, and beam smoothing is achieved when the bandwidth takes a certain value in the near field. In the far field, the radius of the transverse intensity distribution decreases as the bandwidth increases, and the physical explanation of this fact is also given.     

Single-scattering theory of light diffraction by a circular subwavelength aperture in a finitely conducting screen

A perturbation theory based on a single-scattering approximation is developed from the rigorous differential theory of diffraction in cylindrical coordinates. It results in analytical formulas in the inverse space for the field amplitudes providing results that are in quantitative agreement with the results of the rigorous method, in both the near- and far-field regions, when a proper correction to the incident field inside the aperture is made by using the renormalized Born approximation. When working in reflection by a screen having permittivity high in modulus, the method proposes an equivalence with the simple model consisting of the emission by a single magnetic dipole excited inside the pierced layer, emission that is then transferred back into the cladding following the Fresnel's coefficients of transmission from the layer into the cladding. The theory predicts a directivity of the radiation pattern that increases for smaller values of modulus of permittivity, both for dielectrics and metals, thus independently of the possibility of plasmon surface wave excitation along the interface. The theory can take into account such surface wave resonances, as well as the waveguide supported by a dielectric slab, but cannot implicitly recognize the modes carried out by the cylindrical waveguide corresponding to the aperture. This fact limits its domain of validity when used in transmission, although the far- and near-field maps can be reconstructed sufficiently well within a multiplicative factor corresponding to the enhanced transmission due to the excitation of these modes.     

Particle-size analysis by laser diffraction with a complementary metal-oxide semiconductor pixel array

Existing laser-diffraction instruments that use photodiode detectors have a limited resolution for particle sizing. We attempt the implementation of a complementary metal-oxide semiconductor pixel sensor for particle-size measurement by laser diffraction. The sensor has unique features: high resolution, no blooming, and a wide dynamic range (i.e., direct measurement of the scattering pattern). The calibration of the sensor is based on each pixel. The signal-processing and the inversion schemes for obtaining the particle-size distribution are described. The results indicate an improved size resolution and an increased flexibility of application.     

Measurements of laser phase fluctuations induced by atmospheric turbulence over 2 km and 17.5 km distances

A laser heterodyne system was used to measure the phase fluctuations imposed on a 1.5?μm wavelength laser beam when double-passed over long atmospheric paths. Two distances were used: 2 and 17.5?km. Results are given for intensity scintillation, phase fluctuation time series and spectra, and phase structure function. The results are found to agree well with theory: the spectrum of phase fluctuations follows the 8/3 power law predicted for Kolmogorov turbulence over 3 orders of magnitude in frequency. The methods reported here could be used to investigate large-scale temperature variations in the atmosphere.     

Compensation for time fluctuations of phase modulation in a liquid-crystal-on-silicon display by process synchronization in laser materials processing

We demonstrate the adverse influence of temporal fluctuations of the phase modulation of a spatial light modulator (SLM) display device on nanosecond laser micromachining. We show that active cooling of the display reduces the amplitude of these fluctuations, and we demonstrate a process synchronization technique developed to compensate for these fluctuations when applying the SLM to laser materials processing. For alternative SLM devices developed specifically for laser wavefront control (which do not exhibit such flickering problems), we show that our process synchronization approach is also beneficial to avoid machining glitches when switching quickly between different phase profiles (and hence beam patterns).