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Liu P, Yan J, Li W, Wu YK. Transverse mode analysis for free-space laser beams using Bayesian analysis. APPLIED OPTICS 2021; 60:3344-3352. [PMID: 33983238 DOI: 10.1364/ao.420217] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2021] [Accepted: 03/19/2021] [Indexed: 06/12/2023]
Abstract
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.
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Li Y, Guo J, Liu L, Wang T, Shao J. Linear equations method for modal decomposition using intensity information. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2013; 30:2502-2509. [PMID: 24323010 DOI: 10.1364/josaa.30.002502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
The linear equations method is proposed to calculate the complete modal content of the partially coherent laser beam using only the intensity information. This method could give not only the incoherent expansion coefficients of the modal decomposition but also the cross-correlation expansion coefficients using the intensity profiles in several planes of finite distance along the propagation direction. A simulation is also presented to verify the validity of this theory. In our algorithm, the minimum and maximum mode orders should be known a priori, so we provide an estimation method for the two parameters.
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Partanen H, Tervo J, Turunen J. Spatial coherence of broad-area laser diodes. APPLIED OPTICS 2013; 52:3221-3228. [PMID: 23669833 DOI: 10.1364/ao.52.003221] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2013] [Accepted: 04/05/2013] [Indexed: 06/02/2023]
Abstract
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.
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Affiliation(s)
- Henri Partanen
- University of Eastern Finland, Department of Physics and Mathematics, P.O. Box 111, Joensuu FI-80101, Finland.
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Schulze C, Ngcobo S, Duparré M, Forbes A. Modal decomposition without a priori scale information. OPTICS EXPRESS 2012; 20:27866-27873. [PMID: 23262731 DOI: 10.1364/oe.20.027866] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
The modal decomposition of an arbitrary optical field may be done without regard to the spatial scale of the chosen basis functions, but this generally leads to a large number of modes in the expansion. While this may be considered as mathematically correct, it is not efficient and not physically representative of the underlying field. Here we demonstrate a modal decomposition approach that requires no a priori knowledge of the spatial scale of the modes, but nevertheless leads to an optimised modal expansion. We illustrate the power of the method by successfully decomposing beams from a diode-pumped solid state laser resonator into an optimised Laguerre-Gaussian mode set. Our experimental results, which are in agreement with theory, illustrate the versatility of the approach.
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Affiliation(s)
- Christian Schulze
- Institute of Applied Optics, Friedrich Schiller University Jena, Jena, Germany
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Litvin IA, Dudley A, Roux FS, Forbes A. Azimuthal decomposition with digital holograms. OPTICS EXPRESS 2012; 20:10996-1004. [PMID: 22565722 DOI: 10.1364/oe.20.010996] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
We demonstrate a simple approach, using digital holograms, to perform a complete azimuthal decomposition of an optical field. Importantly, we use a set of basis functions that are not scale dependent so that unlike other methods, no knowledge of the initial field is required for the decomposition. We illustrate the power of the method by decomposing two examples: superpositions of Bessel beams and Hermite-Gaussian beams (off-axis vortex). From the measured decomposition we show reconstruction of the amplitude, phase and orbital angular momentum density of the field with a high degree of accuracy.
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Schmidt OA, Schulze C, Flamm D, Brüning R, Kaiser T, Schröter S, Duparré M. Real-time determination of laser beam quality by modal decomposition. OPTICS EXPRESS 2011; 19:6741-6748. [PMID: 21451700 DOI: 10.1364/oe.19.006741] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
We present a real-time method to determine the beam propagation ratio M2 of laser beams. The all-optical measurement of modal amplitudes yields M2 parameters conform to the ISO standard method. The experimental technique is simple and fast, which allows to investigate laser beams under conditions inaccessible to other methods.
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Affiliation(s)
- Oliver A Schmidt
- Institute of Applied Optics, Friedrich-Schiller-University Jena, Jena, Germany
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Dyer PE, Walton CD, Akeel KA. Coherence effects in surface roughness induced by vacuum ultraviolet F2 laser ablation. OPTICS LETTERS 2005; 30:1336-8. [PMID: 15981525 DOI: 10.1364/ol.30.001336] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
We analyze statistical fluctuations in the pulse-to-pulse spatial fluence distribution of a highly multimode F2 laser beam and the influence that coherence effects have on the attainable smoothness of an ablated surface. The magnitude of fluctuations is predicted to be a few percent, with associated roughness increasing as m1/2, where m is the number of ablation pulses. By use of a white-light optical interferometer, measurements have been made of roughness induced on N-BK7 glass surfaces ablated with a 157-nm laser, and reasonable agreement with predictions based on this roughening mechanism has been found.
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Affiliation(s)
- P E Dyer
- Department of Physics, University of Hull, Hull HU6 7RX, UK
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Tervo J, Setälä T, Friberg AT. Theory of partially coherent electromagnetic fields in the space-frequency domain. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2004; 21:2205-2215. [PMID: 15535379 DOI: 10.1364/josaa.21.002205] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We construct the coherent-mode representation for fluctuating, statistically stationary electromagnetic fields. The modes are shown to be spatially fully coherent in the sense of a recently introduced spectral degree of electromagnetic coherence. We also prove that the electric cross-spectral density tensor can be rigorously expressed as a correlation tensor averaged over an appropriate ensemble of strictly monochromatic vectorial wave functions. The formalism is demonstrated for partially polarized, partially coherent Gaussian Schell-model beams, but the theory applies to arbitrary random electromagnetic fields and can find applications in radiation and propagation and in inverse problems.
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Affiliation(s)
- Jani Tervo
- University of Joensuu, Department of Physics, P.O. Box 111, FI-80101 Joensuu, Finland.
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Borghi R, Guattari G, de la Torre L, Gori F, Santarsiero M. Evaluation of the spatial coherence of a light beam through transverse intensity measurements. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2003; 20:1763-1770. [PMID: 12968649 DOI: 10.1364/josaa.20.001763] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
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.
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Affiliation(s)
- Riccardo Borghi
- Dipartimento di Elettronica Applicata and Istituto Nazionale per la Fisica della Materia, Universita Roma Tre, Via della Vasca Navale 84, 1-00146 Rome, Italy
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Gori F, Santarsiero M, Simon R, Piquero G, Borghi R, Guattari G. Coherent-mode decomposition of partially polarized, partially coherent sources. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2003; 20:78-84. [PMID: 12542320 DOI: 10.1364/josaa.20.000078] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
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.
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Affiliation(s)
- Franco Gori
- Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica, Università Roma Tre, Via della Vasca Navale 84, 1-00146 Rome, Italy
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Laabs H, Eppich B, Weber H. Modal decomposition of partially coherent beams using the ambiguity function. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2002; 19:497-504. [PMID: 11876313 DOI: 10.1364/josaa.19.000497] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
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.
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Affiliation(s)
- H Laabs
- Technische Universität Berlin, Optisches Institut, Germany.
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Xue X, Wei H, Kirk AG. Beam analysis by fractional Fourier transform. OPTICS LETTERS 2001; 26:1746-1748. [PMID: 18059685 DOI: 10.1364/ol.26.001746] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
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.
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Warnky CM, Anderson BL, Klein CA. Determining spatial modes of lasers with spatial coherence measurements. APPLIED OPTICS 2000; 39:6109-6117. [PMID: 18354616 DOI: 10.1364/ao.39.006109] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
We explain a technique that extracts both the structure and the modal weights of spatial modes of lasers by analyzing the spatial coherence of the beam. This is the first time, to our knowledge, that an experimental method is being used to measure arbitrary forms of the spatial modes. We applied this method to an edge-emitting Fabry-Perot semiconductor laser with a stripe width of 5 mum and extracted fundamental and first-order lateral modes with relative power weights of 96.2% and 3.8%. There was a single transverse mode.
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Xue X, Wei H, Kirk AG. Intensity-based modal decomposition of optical beams in terms of Hermite-Gaussian functions. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2000; 17:1086-1091. [PMID: 10850479 DOI: 10.1364/josaa.17.001086] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
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.
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Affiliation(s)
- X Xue
- Department of Electrical and Computer Engineering, McGill University, Montreal, Quebec, Canada.
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