Intensity-based modal analysis of partially coherent beams with Hermite-Gaussian modes

Transverse mode analysis for free-space laser beams using Bayesian analysis

Reliable and rapid assessment of the transverse mode quality of a free-space laser beam has a wide range of applications in laser development, research, and utilization. It has become even more important with recent advances in developing orbital angular momentum photon beams across a broad spectral region. In this work, a general modal analysis method for a free-space multimode laser beam has been developed based on Bayesian analysis. After transforming mode decomposition into a linear system problem, a Gaussian probabilistic model is used to find a closed-form solution. The method is found to be robust with the presence of Gaussian noise. Prior knowledge about the mode content can be incorporated into the method to improve the solution for situations when coherent disturbances or contamination are present in the laser beam. This method can be used to analyze the mode content for laser beams in different bases, such as Hermite-Gaussian (HG) modes and Laguerre-Gaussian (LG) modes. Three applications of this method are presented: a detailed modal analysis of the beam image from the incoherent intensity addition of HG modes and two examples of mode decomposition using the complex wavefront from the coherent superposition of HG and LG modes. The feasibility of this method is demonstrated using various simulation results. Based on digital images of a laser beam recorded without complex wavelength-limiting optics, in principle, this method can be used in a wide wavelength range from infrared to ultraviolet, and possibly x ray.

Genuine-field modeling of partially coherent X-ray imaging systems

A genuine representation of the cross-spectral density function as a superposition of mutually uncorrelated, spatially localized modes is applied to model the propagation of spatially partially coherent light beams in X-ray optical systems. Numerical illustrations based on mode propagation with VirtualLab software are presented for imaging systems with ideal and non-ideal grazing-incidence mirrors.

Effect of beam mode of partially Gaussian Schell-model beam on a heterodyne detection system in turbulence

A mathematical model considering the transmission of a partially coherent Gaussian Schell-model (GSM) beam in slant turbulence atmosphere of heterodyne detection was established. A closed-form expression of the weighting factor for the partially GSM beam at the receiving end was derived. The effect of the beam mode on the performance of the proposed detection system was theoretically investigated. The results show that the proportion of the fundamental mode and heterodyne efficiency can be optimized by controlling the waist radius of the signal and local oscillator beams. The inner scale of turbulence significantly affects the heterodyne efficiency and normalized M². With a larger mode order, the proportion of the fundamental mode and heterodyne efficiency are lowered.

Interferometric spatial mode analyzer with a bucket detector

A spatial mode analyzer based on a Michelson interferometer with a bucket detector is experimentally implemented. The delay line in the interferometer is an optical implementation of the fractional Fourier transform (fFT) which enables the spatial mode analysis of a given input field in the Hermite-Gaussian (HG) mode basis. Modal weights for both 1D and 2D input fields are experimentally measured. Results for input fields comprising of multiple HG modes are also presented.

Modal content of living human cone photoreceptors

Decades of experimental and theoretical investigations have established that photoreceptors capture light based on the principles of optical waveguiding. Yet considerable uncertainty remains, even for the most basic prediction as to whether photoreceptors support more than a single waveguide mode. To test for modal behavior in human cone photoreceptors in the near infrared, we took advantage of adaptive-optics optical coherence tomography (AO-OCT, λc = 785 nm) to noninvasively image in three dimensions the reflectance profile of cones. Modal content of reflections generated at the cone inner segment and outer segment junction (IS/OS) and cone outer segment tip (COST) was examined over a range of cone diameters in 1,802 cones from 0.6° to 10° retinal eccentricity. Second moment analysis in conjunction with theoretical predictions indicate cone IS and OS have optical properties consistent of waveguides, which depend on segment diameter and refractive index. Cone IS was found to support a single mode near the fovea (≤3°) and multiple modes further away (>4°). In contrast, no evidence of multiple modes was found in the cone OSs. The IS/OS and COST reflections share a common optical aperture, are most circular near the fovea, show no orientation preference, and are temporally stable. We tested mode predictions of a conventional step-index fiber model and found that in order to fit our AO-OCT results required a lower estimate of the IS refractive index and introduction of an IS focusing/tapering effect.

Sub-second mode measurement of fibers using C2 imaging

We implement cross-correlated imaging in the frequency domain (fC(2)) in order to reconstruct different modes propagating in a multi-mode optical fiber, and measure their relative powers. Our system can complete measurements in under a second (950 ms), with a maximum signal to noise ratio of 25 dB. The system is capable of group-delay temporal resolution as high as 720 fs, and this number can be tailored for a variety of modal discrimination levels by choice of apodization functions and effective bandwidths of the tunable source we use. Measurements are made on a double-clad test fiber to demonstrate simultaneous reconstruction of six guided modes. Finally, the system is used to optimize alignment into the fiber under test and achieve mono-mode purity > 95%, underscoring the utility of fC(2) imaging for near-real-time modal content analysis.

Partially coherent stable and spiral beams

Stable and spiral coherent beams, which do not change the form of their intensity distribution apart from possible scaling and rotation during propagation and therefore possess self-healing properties, are widely applied in science and technology. On the other hand, it has been found that partially coherent light often provides better output than coherent light. Here we consider two methods for the design and experimental generation of partially coherent stable and spiral beams.

Resolving spatial modes of lasers via matrix completion

We explain a technique that recovers the structure and the modal weights of spatial modes of lasers from a limited number of spatial coherence measurements. Our approach interpolates the unobserved spatial coherence measurements via the low-rank matrix completion algorithm based on nuclear norm minimization and then extracts the set of modes via singular value decomposition. Numerical examples are provided on a variety of lasers to demonstrate the effectiveness of the method, and it is shown that the proposed method can further reduce the number of measurements by a factor of 2 for a moderate data size.

Optical coherenscopy based on phase-space tomography

Partially coherent light provides attractive benefits in imaging, beam shaping, free-space communications, random medium monitoring, among other applications. However, the experimental characterization of the spatial coherence is a difficult problem involving second-order statistics represented by four-dimensional functions that cannot be directly measured and analyzed. In addition, real-world applications usually require quantitative characterization of the local spatial coherence of a beam in the absence of a priori information, together with fast acquisition and processing of the experimental data. Here we propose and experimentally demonstrate a technique that solves this problem. It comprises an optical setup developed for automatized video-rate measurement and a method -phase-space tomographic coherenscopy- allowing parallel data acquisition, processing, and analysis. This technique significantly simplifies the spatial coherence analysis and opens up new perspectives for the development of tools exploiting the degrees of freedom hidden into light coherence.

Spatial coherence of broad-area laser diodes

We model the spatial coherence of broad-area laser diodes (BALDs) by representing the mutual intensity as superpositions of individually fully coherent but mutually uncorrelated fields. Consideration of spectroscopic modal structure measurements and intensity-based mode recovery shows that the standard Mercer-type coherent-mode expansion can lead to unsatisfactory results for real BALDs. However, we show that a so-called shifted elementary-field method provides a sufficiently accurate tool for spatial coherence and propagation modeling even if the modal structure of the BALD is severely distorted.

Coupled simulation of chemical lasers based on intracavity partially coherent light model and 3D CFD model

Coupled simulation based on intracavity partially coherent light model and 3D CFD model is firstly achieved in this paper. The dynamic equation of partially coherent intracavity field is derived based on partially coherent light theory. A numerical scheme for the coupled simulation as well as a method for computing the intracavity partially coherent field is given. The presented model explains the formation of the sugar scooping phenomenon, and enables studies on the dependence of the spatial mode spectrum on physical parameters of laser cavity and gain medium. Computational results show that as the flow rate of iodine increases, higher order mode components dominate in the partially coherent field. Results obtained by the proposed model are in good agreement with experimental results.

High-speed modal decomposition of mode instabilities in high-power fiber lasers

A high-speed mode analysis technique is required to gain fundamental understanding of mode instabilities in high-power fiber laser systems. In this work a technique, purely based on the intensity profile of the beam, is demonstrated to be ideally suited to analyze fiber laser dynamics. This technique, together with a high-speed camera, has been applied to the study of the temporal dynamics of mode instabilities at high average powers with up to 20,000 frames per second. These measurements confirm that energy transfer between the fluctuating transversal modes takes place in millisecond-time-scale.

Numerical analysis of the effects of aberrations on coherently combined fiber laser beams

Numerical analysis of the effects of aberrations on coherently combined fiber laser beams is presented. We prove that traditional beam quality criteria, such as the M2 factor and the Strehl ratio, do not consider necessary information to evaluate the quality of a coherently combined laser beam. The beam propagation factor (BPF) is introduced and studied as a proper beam quality factor for the coherently combined beam. Two main categories of aberrations, geometry and nongeometry factors, are numerically studied to investigate their effect on beam quality by using the BPF criterion. For a ring-distributed fiber laser array with certain vacancy factor and a RMS value of tilt error, we obtain a semianalytical equation to evaluate their effect on beam quality. We present a brief discussion of those aberrations at the end of this paper. Our generalized methodology offers a good reference for investigating coherent combining of fiber laser beams in a comprehensive way.

Spatially and spectrally resolved imaging of modal content in large-mode-area fibers

A new measurement technique, capable of quantifying the number and type of modes propagating in large-mode-area fibers is both proposed and demonstrated. The measurement is based on both spatially and spectrally resolving the image of the output of the fiber under test. The measurement provides high quality images of the modes that can be used to identify the mode order, while at the same time returning the power levels of the higher-order modes relative to the fundamental mode. Alternatively the data can be used to provide statistics on the level of beam pointing instability and mode shape changes due to random uncontrolled fluctuations of the phases between the coherent modes propagating in the fiber. An added advantage of the measurement is that is requires no prior detailed knowledge of the fiber properties in order to identify the modes and quantify their relative power levels. Because of the coherent nature of the measurement, it is far more sensitive to changes in beam properties due to the mode content in the beam than is the more traditional M(2) measurement for characterizing beam quality. We refer to the measurement as Spatially and Spectrally resolved imaging of mode content in fibers, or more simply as S(2) imaging.

Laser beam quality and pointing measurement with an optical resonator

We present a compact diagnostic breadboard that is based on an optical ring resonator for measuring beam quality and pointing of single-frequency continuous wave lasers at a wavelength of 1064 nm. To determine the beam quality of the coherent test beam, this optical resonator is used to perform a mode decomposition into Hermite-Gaussian modes. For our laser system, a power fraction in the fundamental Gaussian mode of 97.2%+/-0.2% was measured. Residual misalignment and mis-mode-matching to the resonator as well as the astigmatism and/or ellipticity of the test beam have been determined. Numerical simulations showed that measurements of the M(2) factor and transversal intensity distribution are not suitable for determining this power fraction. To measure the beam pointing, the fundamental mode of the optical resonator was used as a stable reference. The pointing of the test beam was measured with the differential wave front sensing technique up to Fourier frequencies of 1 kHz with a sensitivity to relative pointing of /epsilon/=1x10(-6)/sqrt[Hz]. Pointing measurements with an alternative method were performed and showed good agreement.

Modal power decomposition of beam intensity profiles into linearly polarized modes of multimode optical fibers

We calculate the modal power distribution of a randomly and linearly polarized (LP) multimode beam inside a cylindrical fiber core from knowledge of spatial-intensity profiles of a beam emitted from the fiber. We provide an exact analysis with rigorous proofs that forms the basis for our calculations. The beam from the fiber end is collimated by a spherical lens with a specific focal length. The original LP-mode basis is transformed by the spherical lens and forms another orthogonal basis that describes the free-space beam. By using this basis, we calculate the modal power distribution from the mutual-intensity profile. This is acquired by adopting a well-known mutual-intensity-profile-retrieving technique based on measurements of the intensity patterns several times after two orthogonal cylindrical lenses with varying separation. The feasibility of our decomposition algorithm is demonstrated with simulations.

Evaluation of the spatial coherence of a light beam through transverse intensity measurements

The problem of recovering the coherence features of a partially coherent quasi-monochromatic scalar optical source, starting solely from intensity measurements on the emitted beam, is addressed in the most general way, under the paraxial approximation. In particular, it is shown that on expanding the beam emitted by the source as a bundle of partially correlated Hermite-Gaussian beams, the correlation coefficients can be recovered, in principle, simply by performing scalar products between transverse intensity distributions and suitably defined functions.

Decentered elliptical Hermite-Gaussian beam

A new kind of laser beam called the decentered elliptical Hermite-Gaussian beam (DEHGB) is defined by use of a tensor method. The propagation formula of the DEHGB passing through a nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be easily reduced to the propagation formula of an aligned elliptical Hermite-Gaussian beam and that of a decentered elliptical Gaussian beam under certain conditions. By use of this formula, the propagation characteristics of the DEHGB through free space are presented graphically. As application examples, we construct a generalized laser array using the DEHGB as the fundamental mode. We also obtain the decentered elliptical flattened Gaussian beam by expressing it as superposition of a series of DEHGBs by using polynomial expansion.

Coherent-mode decomposition of partially polarized, partially coherent sources

It is shown that any partially polarized, partially coherent source can be expressed in terms of a suitable superposition of transverse coherent modes with orthogonal polarization states. Such modes are determined through the solution of a system of two coupled integral equations. An example, for which the modal decomposition is obtained in closed form in terms of fully linearly polarized Hermite Gaussian modes, is given.

Modal decomposition of partially coherent beams using the ambiguity function

Phase-space representations of optical beams such as the ambiguity function or the Wigner distribution function have recently gained considerable importance for the characterization of coherent and partially coherent beams. There is growing interest in beam properties such as the beam propagation factor and the coherence and phase information that can be extracted from these phase-space representations. A method is proposed to decompose a partially coherent beam into Hermite-Gaussian modes by using the ambiguity function. The modal weights and the possible phase relations of the Hermite-Gaussian modes are retrieved. The method can also be applied for the decomposition of the Wigner distribution function. Some examples are discussed in the scope of beam characterization.

Beam analysis by fractional Fourier transform

A method of spatial modal decomposition for optical beams by fractional Fourier transform, and its practical implementation with reduced complexity by use of modal interleavers, are discussed.

Intensity-based modal decomposition of optical beams in terms of Hermite-Gaussian functions

We show that when an arbitrary optical beam is decomposed into a superposition of Hermite-Gaussian functions, it is sufficient to record a number of intensity profiles sampled at various transverse planes to uniquely determine the relative modal weights. This result follows from the parity relation and the nature of the Gouy phase, in addition to the orthogonality of the Fourier-transformed intensity profiles associated with the Hermite-Gaussian modes.

Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite-Gaussian modes

A new, to our knowledge, technique for determining the modal content of partially coherent beams that are made up of an incoherent superposition of Hermite-Gaussian modes is studied. The algorithm makes use of the intensity profile of the beam at an arbitrarily chosen transverse plane. Analytical derivations are presented for a Gaussian Schell-model source and flat-topped beams, as well as an analysis of their performances in the presence of experimental errors and noise. Numerical simulations are performed to test the accuracy and the stability of the recovery algorithm.