Laser Beam Profiling Methods

• Laser beam profiling methods are experimental and computational techniques that characterize laser fields by measuring spot size, intensity distribution, and modal content.
• All-optical modal decomposition uses computer-generated holograms to perform real-time M² measurements, delivering rapid and accurate modal analysis.
• Advanced approaches integrate fiber-based probes, scanning methods, neural networks, and computational inversion to provide high-resolution, adaptive diagnostics in various environments.

Laser beam profiling methods are a set of experimental and computational techniques developed to characterize the spatial and modal properties of laser fields. They quantify critical parameters such as spot size, beam waist, propagation quality (M²), intensity distribution, polarization, and spatiotemporal dynamics, under a variety of conditions from open-air laboratory environments to strongly confined or ultra-high-vacuum systems. Profiling is fundamental for optimizing beam delivery in advanced photonics, quantum information systems, material processing, and high-power laser applications, and it must often conform to established international standards (e.g., ISO beam quality definitions).

1. All-Optical Modal Decomposition and Real-Time M² Measurement

A robust methodology for laser beam profiling leverages all-optical modal decomposition. The transverse electric field, $U(x,y)$, of the beam is expanded into Hermite–Gaussian (HG) orthogonal modes as $$U(x, y) = \sum_{m=0}^\infty\sum_{n=0}^\infty c_{mn} \mathrm{HG}_{mn}(x, y)$$. The modal amplitudes $c_{mn}$ are determined via

$$c_{mn} = \iint \mathrm{HG}_{mn}^*(x, y) U(x, y)\,dx\,dy\,.$$

A computer-generated hologram (CGH) acts as a spatial "correlation filter" for these modes. When illuminated and Fourier-transformed (typically in a $2f$ lens configuration), the CGH spatially separates the modal content, so that each Fourier-plane spot has intensity proportional to $\rho_{mn}^2 = |c_{mn}|^2$, providing the relative power per HG mode. Experimental implementation with a CGH designed for $(m + n) \leq 5$ retrieves the modal spectrum in a single shot for rapid, real-time analysis (measurement rates up to $4\,$Hz).

For beam propagation ratio, the standard M² parameter is reformulated in terms of modal weights: $$M_x^2 = \sum_{mn} \rho_{mn}^2 (m+n+1) + \gamma \sqrt{ \left[ \sum_{mn} \rho_{mn}^2 (m - n) \right]^2 + 4 \left[ \sum_{mn} \rho^2_{m+1, n} \rho^2_{m, n+1} \sqrt{m+1}\sqrt{n+1} \right]^2 }$$ with $\gamma$ defined via the beam's second moments. This delivers M² measures directly conforming to ISO standards, circumventing traditional, slower caustic or knife-edge scans, and crucially makes the technique accessible to fast, pulsed, or unstable beam modalities inaccessible to traditional approaches (Schmidt et al., 2011).

2. Fiber-Based and Miniaturized Profiling in Restricted Volumes

Laser beams propagating in constrained environments or ultra-high-vacuum present unique challenges for beam characterization, motivating the development of fiber-based profiling techniques. These methods use specialized probe fibers, often with micromachined slanted or flat-cleaved tips, offering micron-scale in-coupling apertures. Light enters the fiber tip by total internal reflection (TIR) or via micro-reflective coatings and is subsequently guided to a remote photodetector.

The measured profile is the convolution of the beam intensity with the probe's transmission function (TF), which for a Gaussian is: $$T(d) = \frac{4w_b^2w_f^2}{(w_b^2 + w_f^2)^2} \exp\left( -\frac{2d^2}{w_b^2 + w_f^2} \right),$$ where $w_b$ is the $1/e^2$ radius of the incident beam and $w_f$ is the probe width. For $w_f \ll w_b$, convolution effects are negligible and high spatial resolution is achievable—sub-10 micron spot-sizes with single-mode fibers have been demonstrated. The technique is particularly effective for and was experimentally validated in micro ion traps, where conventional profilers cannot be deployed (Miroshnychenko et al., 2012).

An alternative realization uses a stripped multimode fiber whose tip, acting as both scatterer and guide, is scanned through the beam. The spatial resolution is limited by a kernel

$K(x) = \mathrm{Re}\sqrt{(d/2)^2 - x^2}$

with $d \approx D \sin\theta$ (fiber diameter $D$, inclination angle $\theta$). This allows beam profiling in apertures $< 500\,\mu\mathrm{m}^2$ and at intensities $>2\,\mathrm{MW}/\mathrm{cm}^2$, compatible with UHV and high-power conditions where conventional sensors fail (Brand et al., 2019).

3. Camera-, Knife-edge-, and Scanning-Based Spot Size Characterization

Conventional imaging methods, including bare CMOS/CCD sensors without lenses, remain widespread for laboratory-scale profiling. Images are acquired along the $z$-axis and intensity profiles are fitted with Gaussians or error functions, extracting $1/e^2$ radii and beam waists. Automation using translation stages and embedded computing (e.g., Raspberry Pi, Pi NoIR camera, stepper motor) has enabled high-resolution, cost-effective, in-situ spot size and focal position measurement with sub-micron pixel resolution and M² estimation via fits to

$$w(z) = w_0\sqrt{1 + \left(\frac{z - z_0}{z_R}\right)^2}\,,$$

with $z_R = \pi w_0^2/\lambda$ and $M^2 = \pi w_0^2/(\lambda z_R)$. Measurement uncertainties of 1–3% on waist/radii are attainable with careful calibration, and in-situ measurement supports applications such as overlap quantification for counter-propagating beams (Keaveney, 2018, Alamo et al., 2020).

Scanning knife-edge or pinhole techniques serve for cases where camera saturation or non-Gaussian beam structure limits imaging. Here, translation of a photodiode or LDR sensor across the beam yields integrated intensity curves fitted with cumulative Gaussian functions. Step resolutions of $10\,\mu$m are typical; spot size precisions better than 2% are feasible. However, LDRs exhibit coarse error ($\sim$11%), and all such methods require mechanical motion and, in general, are less suited to small or complex beams (Alamo et al., 2020).

A related analytic-extrapolation method derives the beam waist $\omega_0$ from a spot size $\omega(z)$ measured away from focus: $$\omega_0^2 = \frac{\omega(z)^2 \pm \sqrt{\omega(z)^4 - 4(z\lambda/\pi)^2}}{2},$$ selecting the root according to position relative to the Rayleigh length. This is particularly suitable for high-power or ultrafast lasers where direct waist measurements risk damaging sensors (Rao et al., 2018).

4. Advanced and Computational Profiling: Phase Diversity, Neural Networks, and Inverse Methods

For highly non-Gaussian or phase-distorted beams, classical amplitude-only methods are inadequate. The CAMELOT (Complex Amplitude MEasurement by a Likelihood Optimization Tool) approach extends phase diversity to retrieve both amplitude and phase of the beam from a minimal set of intensity images at various focal planes. The method optimizes a Maximum A Posteriori criterion over the propagated field, directly reconstructing the complex amplitude with sub-percent residuals between model and experiment. Errors in modulus are on the order of $10^{-2}$ under typical experimental signal-to-noise conditions, establishing its utility for laser system monitoring, adaptive optics, and imaging in the presence of strong amplitude/phase perturbations (Védrenne et al., 2014).

Spatial beam parameters may also be inferred via deep neural networks, which can be trained on synthetic and experimental datasets (e.g., ResNet50-based CNNs with Rotated Region Proposal Networks, RRPNs). These networks regress per-beam spatial parameters ($x_0, y_0, w_x, w_y, \theta$) directly from images. After fine-tuning to experimental data, root-mean-square-errors drop below 1.1%. NN profiling is advantageous for automated, multi-beam, overlapping, or high-noise situations and can operate in real time as part of beam diagnostics pipelines (Hofer et al., 2022).

Inverse heat transfer methods represent a further class: temperature distributions on the rear of a laser-illuminated plate are measured and used to reconstruct laser intensity profiles via either Levenberg–Marquardt optimization or trained artificial neural networks. Both approaches rely on detailed forward models (e.g., 3D transient heat transfer in ANSYS Fluent) and can reconstruct parameters such as laser power, beam shape coefficient (super-Gaussian exponent), and pulse timing. LM can reach mean errors under 2%; ANN-based solvers provide immediate parameter estimation but exhibit higher sensitivity to noise. These indirect methods facilitate profiling for high-power, pulsed beams in destructive environments, with performance validated via extensive synthetic data (Pietrak et al., 20 Oct 2025, Pietrak et al., 21 Oct 2025).

5. Specialized Profiling: Spectral, 3D, and Real-Time Modal Content

Profiling methods have been extended to address high-resolution mapping of not only intensity but also spectral and spatio-temporal beam features. Double-channel imaging, employing a micrometer-scale fiber-coupled photodetector and parallel optical spectrum analyzer, synchronously maps both spatial beam intensity and emission wavelength at each point across the beam's transverse cross-section. Such synchronous methods have revealed that below threshold, semiconductor lasers display spectral inhomogeneity correlated with deviations from ideal Gaussian intensity, whereas above threshold the central region displays spectral uniformity; wings, often affected by spontaneous emission and fabrication defects, retain fluctuations (Wang et al., 2015).

Fluorescence imaging with tomographic reconstruction enables noninvasive mapping of complete 3D intensity distributions. By imaging fluorescence from an atomic vapor (e.g. rubidium) as a shaped beam propagates through and using inverse Radon transforms on side-view projections, full volumetric profiles with transverse resolution $200 \times 200$ and longitudinal sampling over 659 points can be reconstructed. This is invaluable for applications in optical tweezers, atom traps, and complex pattern formation, especially for counterpropagating configurations incompatible with direct imaging (Radwell et al., 2013).

Modal decomposition by CGH not only allows M² determination but yields the entire modal spectrum in real time, offering a powerful diagnostic for time-varying or multi-modal beams. Direct observation of higher-order mode content is crucial for real-time feedback in cavity alignment and laser system health monitoring (Schmidt et al., 2011).

6. Profiling in Adaptive and Industrial Contexts

Active control of beam profiles is achievable via adaptive optics. Segmented electrical heaters applied to transmissive elements (e.g., SF57 glass), exploiting the thermo-refractive effect, enable creation of tunable thermal lenses. By mapping the actuator voltages via a $4 \times 4$ transfer matrix to beam size and position shifts, feedback can stabilize beam parameters against thermal lensing disturbances at rates limited by actuator response. Spherical and cylindrical lensing, as well as beam steering by controlled differential heating, have been experimentally demonstrated, with application to precision interferometry (e.g., Advanced LIGO) and industrial beam stabilization (Liu et al., 2013).

In laser material processing and additive manufacturing, beam shaping—modifying top-hat, Gaussian, or arbitrary transverse profiles—directly controls melt pool thermodynamics, bead formation, and microstructure morphology. High-fidelity simulations with experimentally validated input demonstrate that, for identical process parameters, different beam shapes yield significant variations in peak temperature, Marangoni flow, bead geometry, and solidification (columnar/equiaxed transition) (Ebrahimi et al., 2023). Feedback-enabled, real-time beam shaping (using spatial light modulators with direct phase mapping) has been developed for rapid patterning, lithography, and ultracold atom manipulation, supporting speckle-free, arbitrary profiles with real-time correction mechanisms (Silva et al., 2022).

7. Data Processing, Noise Reduction, and Computational Enhancement

Advanced post-processing methodologies, such as principal component analysis (PCA) coupled with computer vision algorithms (e.g., MSER for centering), have been employed to enhance beam profiling in noisy or aberration-rich environments. By acquiring image series during controlled mechanical movement (translation or rotation), background noise—fixed pattern, stationary fringes—can be separated from the beam signal. PCA reconstruction yields denoised profiles with reduced root-mean-square error by 40–60%, critical for reducing systematic uncertainties in precision atom interferometry or quantum sensors (MAGIS-100 context) (Jachinowski et al., 2022).

In tilt and angular metrology applications, accurate centroid determination via nonlinear regression on the Gaussian beam profile allows for sub-microradian tilt measurement accuracy, supporting high-dynamic-range position/angle tracking for alignment stabilization in high-power or large-scale optics (Šarbort et al., 2022).

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These diverse methods collectively provide a comprehensive toolkit for laser beam profiling, ranging from direct optical modal decomposition to computational inversion, suited for a wide array of physical environments and measurement regimes. Selection among them depends on criteria including spatial resolution, sensitivity to power and wavelength, compatibility with beam environment (open air, vacuum, confined volumes), modality (intensity, phase, spectral content), and required acquisition rate. Each technique, validated against international standards or simulation-experiment benchmarks, underpins the long-term optimization and reliability of contemporary laser systems across research and industrial domains.