Gaussian Beam Approximation Research Articles

Gaussian Beam Methods for the Helmholtz Equation

In this work we construct Gaussian beam approximations to solutions of the high frequency Helmholtz equation with a localized source. Under the assumption of nontrapping rays we show error estimates between the exact outgoing solution and Gaussian beams in terms of the wave number k, both for single beams and superposition of beams. The main result is that the relative local L 2 error in the beam approximations decay as k −N/2 independent of dimension and presence of caustics for N th order beams.

Self-mixing effect inside the fiber ring laser with different fiber-ends

In this paper, a basic issue in the research of the coupling efficiency inside fiber ring laser self-mixing interference with different fiber-ends is discussed. From the experimental section, we observe waveforms of self-mixing signal inside the fiber ring laser with different fiber-ends. In the theoretical part, we analyze the laser self-mixing process in the Gaussian beam approximation and use the ABCD matrix under paraxial approximation for the calculation of coupling efficiency inside the fiber ring laser with different fiber-ends. Theoretical and experimental results indicate that different fiber-ends would form different characteristics of the laser self-mixing signal. Furthermore, the results of the comparison between theoretical and experimental results illustrate that the plane-end usually widely used in practical application of most self-mixing interference systems and could keep a required coupling efficiency in measurements. In addition, the pigtail with sphere-end could be suitable for the occasion of high requisition of good coupling efficiency and the pigtail with cone-end is suitable for the measurement of hyperfine structure and surface roughness. • Observe self-mixing signal inside the fiber ring laser with different fiber-ends. • Use the ABCD matrix for the calculation of coupling efficiency. • Simulation show a good agreement with the experiment results.

Optimization of a Fabry-Perot Sensing Interferometer Design for an Optical Fiber Sensor of Hematocrit Level

Continuous measurement of the hematocrit level in blood can potentially be performed using optical bre sensors. The Fabry Perot interferometric sensors are a promising candidate in this application. The most important step in the design of the sensor is design of the sensing interferometer. Adequate cavity length and high interference contrast are two most important requirements in this application. The design method of the sensing interferometer presented in this paper uses a Gaussian beam approximation. In order to verify the design, an optical bre Fabry Perot interferometer with adjustable cavity length was built. Its performance was tested, con rming validity of the design approach.

PH effect on thermo-optical response of Er3+-doped silicagel prepared by the sol-gel method

We present results of the study of optical and thermo-optical properties as well as the thermo-gravimetric analysis of SiO2:Er3+ (5% mol) glasses prepared by the sol-gel method at different pH (2, 3, 4 and 5). The thermo-optical characteristics were measured at room temperature under resonant conditions for the Er3+ ions and analyzed in terms of a theoretical model for continuum wave (CW) laser induced thermal lensing under a Gaussian-beam approximation. The thermo-optical properties showed a dependence onthe pH conditions during the synthesis of the SiO2:Er3+ (5% mol.) glasses. Key words: Sol-Gel, Er3+, thermal lens, optical properties.

Analytical model for low finesse, external cavity, fiber Fabry–Perot interferometers including multiple reflections and angular misalignment

We present an analytical model for single mode, multiply reflected, external cavity, optical fiber Fabry-Perot interferometers in the low finesse regime using simple geometry and the Gaussian beam approximation. The multiple reflection model predicts attenuation of the peak-to-peak interference as the fiber to mirror distance approaches zero, as well as fringe asymmetry in the presence of nonabsorbing mirrors. A series of experiments are conducted in which a series of fiber Fabry-Perot cavities are constructed using uncoated, single mode glass fibers, and mirrors of varying reflectivity. The cavity length is swept, and the predictions of the model are found to be in good agreement with the experimental interferograms.

A Higher-Order Ray Approximation Applied to Orographic Gravity Waves: Gaussian Beam Approximation

Abstract Ray techniques are a promising tool for developing orographic gravity wave drag schemes. However, the modeling of the propagation of orographic waves using standard ray theory in realistic background wind conditions usually encounters several regions, called caustics, where the first-order ray approximation breaks down. In this work the authors develop a higher-order approximation than standard ray theory, named the Gaussian beam approximation, for orographic gravity waves in a background wind that depends on height. The analytical results show that this formulation is free of the singularities that arise in ray theory. Orographic gravity waves that propagate in a background wind that turns with height—the same conditions as in the work of Shutts—are examined under the Gaussian beam approximation. The evolution of the amplitude is well defined in this approximation even at caustics and at the forcing level. When comparing results from the Gaussian beam approximation with high-resolution numerical simulations that compute the exact solution, there is good agreement of the amplitude and phase fields. Realistic orography is represented by means of a superposition of multiple Gaussians in wavenumber space that fit the spectrum of the orography. The technique appears to give a good representation of the disturbances generated by flow over realistic orography.

Evolution of the effective characteristics of high-power femtosecond laser radiation in air under optical turbulence conditions. The Gaussian beam approximation

An analytical solution to the equation for the effective radius of a high-power femtosecond laser beam in the approximation of the intensity Gaussian profile during its self-focusing in air is found. This equation was derived based on the equation for the Wigner function of the laser radiation field in developed optical turbulence conditions. Estimates are presented of the global self-focusing distance versus the peak radiation power, as well as its divergence after the nonlinear focus.

Laser-self-mixing interferometry in the Gaussian beam approximation: experiments and theory

We analyze the laser-self-mixing process in the Gaussian beam approximation and reformulate the expression of the feedback coefficient C in terms of the effective feedback power coupled back into the laser diode. Our model predicts a twenty-fold increase of the ratio between the maximum and the minimum measurable displacements judged against the current plane-wave model. By comparing the interaction of collimated or diverging Gaussian laser beams with a plane mirror target, we demonstrate that diverging beams tolerate larger wobbling during the target displacement and allow for measurement of off-axis target rotations up to the beam angular width. A novel method for reconstructing the phase front of the Gaussian beam by self-mixing scanning measurements is also presented.

Broadband terahertz detection with selected gases

We present a combined theoretical and experimental investigation of the detection of pulsed terahertz waves via field-induced second-harmonic generation in gaseous media. The effects of the probe pulse energy, bias field strength, nonlinear susceptibility of the gases, phase matching, and focusing conditions of the terahertz and optical beams are discussed. The analytical calculation, which is based on a Gaussian-beam approximation, allows for the easy identification of the parameters important for the sensitivity of the terahertz gas sensor. A figure of merit is introduced to characterize the sensitivity of gases. The experimental results are found to be in good agreement with the calculation.

Theoretical determination of the radiation force for a spherical particle illuminated by a focused laser beam

Trapping forces on dielectric spheres in single beam laser tweezers are computed. A focused beam description based on an exact solution of Maxwell's equations is compared to the 5th order Gaussian beam approximation due to Barton and Alexander. Forces on water droplets suspended in air and on polystyrene spheres suspended in water, exerted by beams focused to varying degree, are calculated. It is demonstrated that the 5th order approximation is accurate for almost paraxial beams (numerical aperture NA <∼0.25), as compared to the exact treatment. However, for strongly focused beams the 5th order approximation breaks down. Thus it is established that an accurate beam description is vital for modeling optical traps, since, in order to hold a particle effectively in a single beam trap, a strongly focused beam is required.

ABCD matrix model of quadric interface-lensed fiber and its application in coupling efficiency calculation

In order to simplify the theoretical coupling efficiency calculation of a laser diode (LD) to a single-mode fiber (SMF) with a quadric lens formed on the fiber tip, based on the Gaussian beam approximation, using the ABCD matrix method the relationship between the coupling efficiency and the structure parameters of quadric surface-lensed fiber is studied and analyzed in detail, and the coupling efficiency can reach as high as 99.87% with optimizing parameters of lensed fiber in theoretical simulation. Meanwhile, under paraxial approximation, the ABCD matrix for refraction of different quadric surface micro-lens can be formulated as a unified expression. This result can give some guidance suggestions for fabricating the quadric interface-lensed fiber.

Luminescent and thermo-optical properties of germanate glasses

The visible photoluminescence of Tb 3+-doped germanate glass of composition (in mol%) 40GeO 2–20Na 2O–40SiO 2 has been investigated for two Tb 3+ concentrations. By using steady-state and pulsed laser spectroscopy. Concentration effects on luminescence are observed for Tb 3+ concentration higher than 0.9 mol%. The thermo-optical characteristics were measured at room temperature under resonant conditions on the Tb 3 ions. The thermo-optical properties of the glasses showed a dependence on the concentration of terbium ions. The effects of TL were also analyzed in terms of a theoretical model for CW laser induced thermal lensing under a Gaussian-beam approximation. Tb 3+ concentration quenching effects on the dependence of thermo-optical properties are discussed.

The generation of mould patterns for multimode optical waveguide components by direct laser writing of SU-8 at 364 nm

A commercial direct laser writing system operating at 364 nm wavelength is used to expose 90 µm thick layers of SU-8 on silicon. The resist structures are the basis for the fabrication of moulding tools that are used for the replication of multimode optical waveguide components. The influence of the laser power on the shape as well as on the sidewall roughness of the resist structures is studied in detail. The observed profiles are compared to a mathematical model based on the Gaussian beam approximation.

Optimal design for a flat-top AWG demultiplexer by using a fast calculation method based on a Gaussian beam approximation

Passband broadening of an AWG (array waveguide grating) demultiplexer with an MMI (multimode interference) coupler connected at the end of a tapered input waveguide is considered. An explicit formula based on the field propagation of an approximate Hermit–Gaussian beam is used to calculate quickly and reliably the spectral response of the AWG demultiplexer. The widths of the input waveguide, the output waveguides and the MMI coupler are optimized. The optimal design is verified with the experimental measurement.

Laser-induced changes in optical properties of H 4TPP 2+ doped silica gel prepared by sol–gel method

Optical absorption and photoluminescence of diacid tetraphenylprophyrine, H 4TPP 2+, species in silica gel glass prepared by the sol–gel (SG) method were measured in the 350–800 nm range. Under continuous (CW) laser excitation the characteristics of thermal lensing (TL) were measured at room temperature, under resonant and non-resonant conditions. Our results are analyzed based on a theoretical model for CW laser-induced thermal lensing, under a Gaussian-beam approximation. The effects of the power input and the response time for TL are explained in terms of the coupling of the H 4TPP 2+ molecules to the local vibrations, for resonant excitations. Also, the observation of laser-induced permanent changes in the refractive index of our glass sample under resonant and non-resonant excitations is reported and discussed.

Combined Raman and continuous-wave-excited two-photon fluorescence cell imaging

We demonstrate a confocal optical microscope that combines cw two-photon-excited fluorescence microscopy with confocal Raman microscopy. With this microscope fast image acquisition with fluorescence imaging can be used to select areas of interest for subsequent chemical analysis with spontaneous Raman imaging. The distribution of the UV-absorbing fluorophore Hoechst 33342 in the apoptotic HeLa cells is measured in the combined cw two-photon-excited fluorescence and Raman microscopy modes. The 647-nm line of a Kr-ion laser is used to excite both the Raman scattering and the two-photon-excited fluorescence emission. The lateral and axial resolutions in the two imaging modes are compared by use of the Gaussian beam approximation and backprojection of the focal volume through the confocal pinhole.

Theory of explosive beam spreading due to ray chaos.

In an ocean acoustic situation where rays are known to be chaotic, a theory based on a Gaussian beam approximation is used to show that the width of a narrow-angle beam increases exponentially (explosively) with range at a rate that is given by the Lyapunov exponent that is calculated in the geometric limit. This finite frequency result shows that the phenomenon of "wave chaos" is real, and might be observable experimentally.

Asymptotic analysis of silicon based Bragg fibers

We developed an asymptotic formalism that fully characterizes the propagation and loss properties of a Bragg fiber with finite cladding layers. The formalism is subsequently applied to miniature air-core Bragg fibers with Silicon-based cladding mirrors. The fiber performance is analyzed as a function of the Bragg cladding geometries, the core radius and the material absorption. The problems of fiber core deformation and other defects in Bragg fibers are also addressed using a finite-difference time-domain analysis and a Gaussian beam approximation, respectively.

Foundations for low-loss fiber gradient-index lens pair coupling with the self-imaging mechanism.

A fiber-optic collimator that emits a Gaussian beam with its beam waist at a certain distance after the exit face of the lens is labeled a self-imaging collimator. For such a collimator, the waist of the emitted Gaussian beam and its location are partly dependent on the properties of the gradient-index (GRIN) lens. Parameters for the self-imaging collimator are formulated in terms of the parameters of a GRIN lens (e.g., pitch, core refractive index, gradient index, length) and the optical wavelength. Next, by use of the Gaussian beam approximation, a general expression for the coupling power loss between two self-imaging-type single-mode fiber (SMF) collimators is, for the first time to our knowledge, derived as a function of three types of misalignment, namely, separation, lateral offset, and angular tilt misalignment. A coupling experiment between two self-imaging collimators with changing separation distance is successfully performed and matches the proposed self-imaging mechanism coupling loss theory. In addition, using a prism, lateral offset, as well as angular tilt, misalignments are experimentally simulated for a two self-imaging collimator coupling condition by a single collimator reflective test geometry. Experimental results agree well with the proposed loss formulas for self-imaging GRIN lenses. Hence, for the first time to our knowledge, the mathematical foundations are laid for employing self-imaging-type fiber collimators in SMF-based free-space systems allowing optimal design for ultra-low-loss coupling.

Ring formation of focused half-cycle terahertz pulses

Studies using a terahertz (THz) imaging system have shown that focused THz electromagnetic pulses exhibit a ring-like spatial intensity distribution. In the study, almost half-cycle THz pulses were generated from a large-aperture photoconductive antenna and focused by an off-axis parabolic mirror. Time-resolved spatial distribution of the THz electric field on the focal plane was found to form a ring before and after the time region where a single peak is formed around the time origin. On the other hand, with the delay time fixed at the origin, the field distribution was also found to exhibit a ring-like profile at positions away from the focal plane. Existence of relatively low-frequency components in the power spectrum of the THz field gives these observable phenomena. Numerical simulations using diffraction integral calculations reproduced the observed features. A simpler numerical analysis based on the Gaussian beam approximation was shown to yield the observed features and also predicted the dependence on the beam parameters.