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Gaussian beam

A Gaussian beam is a type of electromagnetic beam, typically produced by lasers operating in their fundamental transverse electromagnetic (TEM₀₀) mode, whose transverse intensity profile follows a Gaussian distribution, with maximum intensity at the center and exponentially decreasing toward the edges.^[1]^[2] This bell-shaped profile arises from the paraxial approximation of the wave equation, making Gaussian beams exact solutions that propagate while preserving their Gaussian form (up to scaling) in free space or through linear optical systems.^[3]^[4] Key parameters defining a Gaussian beam include the beam waist w_0, the minimum radius at the narrowest point where the intensity drops to $1/e^2 of its peak value; the Rayleigh range z_R = \pi w_0^2 / \lambda, which indicates the distance over which the beam remains roughly collimated before significant divergence; and the far-field divergence angle \theta = \lambda / (\pi w_0), determining how the beam spreads at large distances.^[5]^[6] These properties stem from the beam's complex amplitude expression, E(r,z) \propto \frac{w_0}{w(z)} \exp\left( -\frac{r^2}{w(z)^2} \right) \exp\left( i(kz + \frac{kr^2}{2[R(z)](/page/Radius_of_curvature)} - \zeta(z)) \right), where w(z) is the beam radius at distance z, R(z) the radius of curvature, and \zeta(z) the Gouy phase shift.^[1] Gaussian beams are essential in modern optics due to their diffraction-limited nature, representing the lowest possible beam divergence for a given wavelength \lambda and waist size, which enables precise focusing and efficient coupling into optical fibers or resonators.^[7] They find widespread applications in laser design, beam transformation through lenses and mirrors using ABCD matrix formalism, and technologies such as optical tweezers, laser surgery, and telecommunications.^[8]^[9]

Fundamentals

Definition and Physical Interpretation

A Gaussian beam is a type of electromagnetic wave in which the amplitude of the electric field in the transverse plane—perpendicular to the direction of propagation—exhibits a Gaussian distribution.^[1] This profile arises as the fundamental solution to the paraxial wave equation, describing the lowest-order transverse electromagnetic (TEM₀₀) mode that can be sustained in stable optical resonators.^[10] Physically, it represents an idealized light beam where the intensity, proportional to the square of the field amplitude, forms a smooth, radially symmetric bell-shaped pattern that peaks at the beam axis and decays exponentially away from it.^[11] The physical significance of Gaussian beams lies in their unique propagation characteristics, making them the preferred mode for laser applications. Unlike uniform plane waves, which diffract rapidly due to sharp edges, Gaussian beams experience minimal diffraction because their smooth intensity gradient reduces edge effects at the beam boundary.^[12] This allows for tighter focusing to small spot sizes and lower divergence over propagation distances, essential for precision optics in microscopy, telecommunications, and material processing.^[1] Additionally, Gaussian beams propagate in a self-similar manner: the transverse profile remains Gaussian at every point along the axis, with the beam width evolving predictably—contracting to a minimum at the beam waist before expanding—without distorting the overall shape.^[11] This invariance under free-space propagation, up to scaling, stems from the Gaussian function's property as its own Fourier transform, which directly relates to diffraction theory.^[12] Their stability within laser cavities further underscores their importance; the Gaussian profile naturally matches the eigenmodes of stable resonators, enabling efficient energy buildup and output coupling with low losses.^[13] For illustration, consider a typical helium-neon laser beam: its central intensity is highest, dropping to about 13.5% of the peak at the 1/e² radius, containing nearly all the power within a compact region that maintains the bell-shaped form as it travels, scaling smoothly from the waist onward.^[11] Conceptually, the Gaussian beam's transverse structure draws an analogy from quantum mechanics, where its profile mirrors the ground-state wavefunction of a one-dimensional harmonic oscillator.^[14] This parallel emerges because the paraxial approximation of the wave equation for light propagation resembles the Schrödinger equation for the oscillator, with the beam's "potential" defined by the quadratic phase terms, providing insight into the beam's confinement and evolution.^[15]

Historical Development

The invention of the first laser by Theodore Maiman in 1960, using a ruby crystal, marked a pivotal moment, as Gaussian beams were soon recognized as essential for achieving stable single-transverse-mode operation in laser cavities. In the early 1960s, significant theoretical progress occurred with G. D. Boyd and J. P. Gordon's 1961 analysis of Gaussian modes within confocal multimode resonators, demonstrating their suitability for millimeter-wave through optical masers and laying groundwork for resonator design.^[16] The first experimental verification of Gaussian beam propagation in free space came in 1966 from H. Kogelnik and T. Li, whose comprehensive review confirmed the theoretical predictions for beam evolution, resonator stability, and mode interactions.^[17] During the 1970s and 1980s, the framework expanded to astigmatic Gaussian beams, as explored by D. C. Hanna in 1969 for cavities with axial asymmetry, and higher-order modes, with A. E. Siegman's 1986 monograph "Lasers" providing a rigorous formalization of propagation characteristics across these variants.^[7]

Mathematical Description

Field Expression and Intensity Profile

The complex electric field amplitude of a fundamental Gaussian beam propagating in the positive z-direction, assuming cylindrical symmetry and linear polarization, is given by

E(r,z) = E_0 \frac{w_0}{w(z)} \exp\left( -\frac{r^2}{w(z)^2} \right) \exp\left[ i\left( kz + \frac{k r^2}{2 R(z)} - \eta(z) \right) \right],

where E_0 is the complex amplitude at the beam waist, w_0 is the waist radius, w(z) is the beam radius at axial position z, R(z) is the wavefront radius of curvature, \eta(z) is the Gouy phase, k = 2\pi/\lambda is the wavenumber, \lambda is the wavelength, and r is the radial coordinate.^[18] This formulation captures the essential features of the beam: the Gaussian envelope governs the transverse amplitude falloff, while the phase terms account for propagation, wavefront curvature, and an additional phase shift beyond the plane-wave contribution. The expression was derived as the fundamental mode solution for stable optical resonators and free-space propagation.^[18] The intensity profile I(r,z), defined as the time-averaged power flux or |E(r,z)|^2 (up to a proportionality constant), follows directly from the magnitude squared of the field, as the phase factors do not affect the local intensity:

I(r,z) = I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp\left( - \frac{2 r^2}{w(z)^2} \right),

with peak on-axis intensity I_0 \propto |E_0|^2 at the waist (z=0).^[18] This yields a radially symmetric Gaussian distribution at each z-plane, characterized by the 1/e² radius w(z), and conserves the total integrated power along the propagation axis in the absence of losses.^[18] The Gaussian transverse profile emerges as the exact lowest-order solution to the paraxial Helmholtz equation under the slowly varying envelope approximation, which in transverse coordinates takes the form of a two-dimensional diffusion-like equation: \nabla_\perp^2 u + 2 i k \partial u / \partial z = 0, where u is the envelope function.^[18] Substituting the Gaussian ansatz satisfies this equation precisely, confirming its role as the fundamental mode for paraxial beam propagation.^[18]

Beam Width Evolution and Waist

The beam waist of a Gaussian beam is defined as the minimum radius w_0 achieved by the transverse intensity profile at the focal point, typically located at z = 0 along the propagation axis. This waist radius represents the smallest possible spot size for a diffraction-limited beam in paraxial optics, arising as the fundamental solution to the Helmholtz equation for free-space propagation. The evolution of the beam radius w(z) with propagation distance z from the waist is described by the formula

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

where z_R = \frac{\pi w_0^2}{\lambda} is the Rayleigh range and \lambda is the wavelength. This expression, derived from the paraxial approximation of the wave equation, shows that the beam cross-section remains Gaussian in shape but expands hyperbolically along the axis. In the near-field regime, where |z| \ll z_R, the term \left( \frac{z}{z_R} \right)^2 is negligible, resulting in w(z) \approx w_0 and a nearly collimated beam with constant width over the depth of focus. Conversely, in the far-field regime, where |z| \gg z_R, the beam radius approximates w(z) \approx w_0 \frac{|z|}{z_R}, leading to linear divergence with propagation distance. This behavior underscores the trade-off between tight focusing (small w_0, short z_R) and beam collimation (large w_0, long z_R). As the beam width w(z) varies, the on-axis intensity scales inversely with w(z)^2, maintaining the total power while redistributing it over the expanding profile. For example, a He-Ne laser beam (\lambda = 633 nm) focused to a waist radius of w_0 = 10 μm has a Rayleigh range of approximately 0.58 mm, such that the beam width doubles to $2w_0 after propagating about 1 mm from the waist.

Wavefront Curvature and Phase

The wavefronts of a Gaussian beam exhibit spherical curvature that varies along the propagation direction z, described by the radius of curvature

R(z) = z \left[ 1 + \left( \frac{z_R}{z} \right)^2 \right],

where z_R is the Rayleigh range.^[19] At the beam waist (z = 0), R(z) approaches infinity, corresponding to planar wavefronts; as the beam propagates, the wavefronts curve positively for z > 0, reaching a minimum radius R(z_R) = 2z_R at the Rayleigh range, beyond which the curvature gradually flattens again.^[19] This quadratic phase variation arises from the parabolic term in the complex field expression, which imparts a position-dependent phase delay that shapes the diverging or converging spherical surfaces.^[19] In addition to the curvature-induced phase, Gaussian beams experience the Gouy phase shift, an extra longitudinal phase accumulation given by

\eta(z) = \arctan\left( \frac{z}{z_R} \right).

This results in a total phase advance of \pi over a distance of $2z_R centered at the waist, distinguishing the beam from a uniform plane wave.^[19] The Gouy phase originates as a topological effect from the transverse spatial confinement of the beam, which, via the uncertainty principle, introduces a spread in longitudinal momentum components that manifests as this anomalous shift. These phase structures significantly influence interference phenomena. The varying wavefront curvature and Gouy phase can introduce spatial and axial phase mismatches when combining multiple Gaussian beams, leading to reduced fringe visibility or shifted interference patterns in setups like Michelson interferometers or nonlinear interactions. For instance, unequal propagation distances or waist locations between beams exacerbate dephasing, impacting applications such as coherent beam coupling.

Key Parameters

Rayleigh Range and Confocal Parameter

The Rayleigh range, denoted z_R, quantifies the longitudinal extent over which a Gaussian beam maintains its narrowest profile near the waist, serving as a fundamental measure of beam collimation. It is defined as

z_R = \frac{\pi w_0^2}{\lambda},

where w_0 is the beam waist radius at its minimum and \lambda is the wavelength of the light. This parameter originates from the paraxial wave equation solutions for Gaussian beams and represents the propagation distance from the waist where the beam radius expands to \sqrt{2} times w_0, effectively doubling the beam area. Beyond this distance, the beam begins significant diffraction, marking the transition from the near-field (collimated) to the far-field (diverging) regime; thus, z_R characterizes the depth of focus, indicating how far the beam remains suitable for applications requiring tight focusing.^[20] Closely related is the confocal parameter, b = 2 z_R, which spans the full symmetric region around the waist where the beam stays relatively collimated, with the radius varying by no more than \sqrt{2} from w_0.^[3] This parameter, also termed the depth of focus, provides a practical metric for the total axial distance over which the beam's cross-section remains approximately constant, aiding in the design of optical systems like resonators or focusing setups.^[1] The confocal parameter arises naturally in Gaussian beam theory as twice the Rayleigh range, emphasizing the balanced propagation on either side of the waist. The physical significance of these parameters hinges on the ratio w_0 / \lambda: for tightly focused beams with small w_0 / \lambda (e.g., subwavelength waists), z_R is correspondingly small, resulting in rapid divergence and a shallow depth of focus; in contrast, larger ratios produce extended z_R and b, enabling highly collimated beams over greater distances, as seen in long-range laser applications.^[20] For instance, in microscopy techniques such as light-sheet imaging, the Rayleigh range directly governs axial resolution, limiting the effective imaging depth to approximately z_R before beam spreading degrades focus and contrast.^[21] The Rayleigh range also informs the beam width evolution w(z), where it sets the scale for quadratic broadening near the waist.

Beam Divergence and Far-Field Behavior

In the far field, where the propagation distance z greatly exceeds the Rayleigh range, a Gaussian beam exhibits a linear increase in its beam radius w(z) with distance, approximated as w(z) \approx \theta |z|, where \theta is the half-angle divergence.^[22] This divergence angle quantifies the angular spread of the beam and is given by \theta = \frac{\lambda}{\pi w_0}, with \lambda as the wavelength and w_0 the beam waist radius at z=0.^[22] The formula arises from the paraxial approximation of Gaussian beam propagation, reflecting the fundamental diffraction limit for a beam with minimum uncertainty in transverse position and momentum.^[12] The intensity profile in the far field maintains a Gaussian form, but expressed in angular coordinates rather than spatial ones, with the angular width determined by \theta.^[23] Specifically, the far-field intensity distribution I(\phi) as a function of the angle \phi from the beam axis follows I(\phi) \propto \exp\left(-2 \phi^2 / \theta^2\right), where the beam spreads conically with Gaussian weighting.^[23] This angular Gaussian profile ensures that the beam quality remains preserved over long distances, unlike higher-order modes which diverge more rapidly. The half-angle divergence \theta establishes the diffraction limit for focusing Gaussian beams, as the smallest achievable spot size upon focusing is fundamentally tied to this angle and the wavelength.^[24] For an ideal Gaussian beam, the beam quality factor M^2 = 1, indicating diffraction-limited performance where no tighter focus is possible without violating optical principles.^[24] This property is crucial in applications requiring precise beam control; for instance, semiconductor lasers, which typically emit from small apertures resulting in large \theta (often tens of degrees), employ aspheric or cylindrical lenses to collimate and reduce divergence for efficient coupling into optical fibers or free-space systems.^[25] Note that \theta relates inversely to the Rayleigh range z_R, such that the divergence scales with the beam waist relative to this depth-of-focus parameter.

Gouy Phase Shift

The Gouy phase shift represents a characteristic phase anomaly in the propagation of focused Gaussian beams, arising specifically from their transverse confinement. For the fundamental Gaussian mode, the Gouy phase is expressed as

\eta(z) = \arctan\left(\frac{z}{z_R}\right),

where z is the propagation distance from the beam waist and z_R is the Rayleigh range. As the beam propagates from z = -\infty to z = +\infty, the Gouy phase evolves from -\pi/2 to +\pi/2, yielding a total Gouy shift of \pi over the full confocal distance. This total shift stems from a phase velocity difference between the interior of the beam and an ideal plane wave outside it: the transverse momentum spread induced by spatial confinement reduces the effective axial wave number, resulting in a superluminal phase velocity v_p > c within the beam.^[26] The derivative of the Gouy phase, \frac{d\eta}{dz} = \frac{1}{1 + (z/z_R)^2}, quantifies this anomalous velocity effect locally, equaling the ratio of the waist spot size squared to the local beam width squared and reaching a maximum of unity at the waist.^[26] This phase anomaly originates in the mathematical derivation of Gaussian beam solutions to the paraxial wave equation, either through saddle-point evaluation of the angular spectrum integral or via normalization conditions for the mode fields. In practical applications, the Gouy phase shift alters resonance frequencies in optical cavities by contributing an additional round-trip phase that depends on the beam's confocal parameter, necessitating precise accounting in cavity stability and mode selection analyses. It also plays a critical role in interferometry, where differential Gouy shifts due to beam focusing or misalignment can introduce systematic phase errors in fringe patterns and signal interpretation.^[27]

Intensity and Power

Peak Intensity and Total Power

The peak intensity of a Gaussian beam occurs on the optical axis and is inversely proportional to the squared beam radius at position z along the propagation direction. For the fundamental TEM_{00} mode, the on-axis intensity is expressed as

I_0(z) = \frac{2P}{\pi w(z)^2},

where P is the total power of the beam and w(z) is the $1/e^2 intensity radius at z. This relation follows directly from the radial intensity profile I(r,z) = I_0(z) \exp\left(-2r^2 / w(z)^2\right), normalized such that the peak value scales with the inverse of the effective beam area. The total power P remains constant during free-space propagation of an ideal, untruncated Gaussian beam and can be obtained by integrating the intensity distribution over the transverse plane:

P = \int_0^\infty I(r,z) \, 2\pi r \, dr = \frac{\pi w(z)^2 I_0(z)}{2}.

This integral yields the inverse of the peak intensity formula, confirming power conservation independent of z. In practice, approximately 86% of the total power is contained within the $1/e^2 radius w(z).^[1] In laser sources, the total power P corresponds to the device's output power, determined by factors such as the pump source efficiency, gain medium properties, and cavity design.^[7] For instance, a continuous-wave laser emitting 1 W with a beam waist radius w_0 = 100 \, \mu\mathrm{m} (at z=0) achieves a peak intensity of approximately $6.4 \times 10^3 \, \mathrm{W/cm^2} at the waist.^[11]

Power Through Apertures and Truncation Effects

The fraction of total power transmitted through a circular aperture of radius a for a Gaussian beam with radius w (defined at the $1/e^2 intensity contour) is given by

T = 1 - \exp\left(-2 \frac{a^2}{w^2}\right).

This expression results from integrating the radial Gaussian intensity profile I(r) = I_0 \exp(-2 r^2 / w^2) over the aperture area, normalized to the total infinite-plane power.^[28] As a \gg w, T approaches unity, indicating negligible loss for sufficiently large apertures.^[1] Representative values illustrate the rapid containment of Gaussian beam power: an aperture of radius a = w transmits approximately 86% of the total power, while a = 2w transmits about 99%.^[29] These thresholds guide practical aperture sizing to minimize losses while avoiding oversized optics. Significant truncation, where a is comparable to or smaller than w, clips the beam tails and introduces components of higher-order modes, distorting the pure fundamental Gaussian profile. This leads to increased far-field divergence beyond that of an ideal Gaussian beam and elevates the beam quality factor M^2 > 1, quantifying the degradation in propagation characteristics.^[23]^[30] In laser resonators, if mirror dimensions are smaller than a few times the local beam radius w(z), clipping induces round-trip power losses, reducing resonator efficiency and potentially exciting unwanted higher-order modes that further destabilize operation.^[31]

Complex Beam Parameter

Definition and Propagation

The complex beam parameter q(z) provides a compact mathematical description of a Gaussian beam's properties along its propagation axis at position z. It is defined by the relation

\frac{1}{q(z)} = \frac{1}{R(z)} - i \frac{\lambda}{\pi w(z)^2},

where R(z) is the radius of curvature of the beam's wavefront, w(z) is the beam radius at position z, and \lambda is the wavelength of the light. This formulation encodes both the real part, associated with wavefront curvature, and the imaginary part, related to the beam's transverse intensity profile and phase structure.^[1] At the beam waist, where the wavefront is flat (R(0) = \infty) and the beam radius is minimum (w(0) = w_0), the parameter simplifies to q(0) = i z_R, with the Rayleigh range z_R = \pi w_0^2 / \lambda representing the axial distance over which the beam remains roughly collimated. This imaginary value at the waist highlights the beam's minimal divergence point, serving as a reference for propagation analysis.^[1] In free space, the complex beam parameter evolves linearly with propagation distance according to q(z) = q(0) + z. This simple additive rule arises from the paraxial wave equation solutions for Gaussian beams, allowing straightforward tracking of beam evolution without separate computations for width and curvature.^[1] The use of q(z) unifies multiple beam parameters into a single complex quantity, facilitating efficient calculations of Gaussian beam propagation and transformations.^[1] This q-parameter approach, often termed the q-transform, streamlines analysis by avoiding decoupled equations for individual parameters like w(z) and R(z).

Transformation Through Optical Elements

The complex beam parameter q, which encapsulates the Gaussian beam's waist size and wavefront curvature, transforms through paraxial optical systems according to the ABCD matrix formalism.^[32] For an optical element or system characterized by the ray-transfer matrix \begin{pmatrix} A & B \\ C & D \end{pmatrix}, the output parameter q_2 relates to the input q_1 via

q_2 = \frac{A q_1 + B}{C q_1 + D},

where the matrix elements satisfy AD - BC = 1 for lossless systems.^[33] This transformation preserves the Gaussian form of the beam while updating its spot size and curvature.^[34] Consider a thin lens of focal length f as a representative example. The ABCD matrix for such a lens is \begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix}, leading to the output parameter

q_\text{out} = \frac{q_\text{in}}{1 - q_\text{in}/f}.

This relation demonstrates how the lens alters the beam's divergence and focusing properties without changing the total power.^[32] Reflection from a mirror introduces an additional phase flip, effectively transforming the parameter as q' = -q for the backward-propagating beam after the matrix application, which accounts for the reversal in propagation direction.^[33] For a curved mirror with radius of curvature R, the ABCD matrix \begin{pmatrix} 1 & 0 \\ -2/R & 1 \end{pmatrix} further adjusts the wavefront curvature accordingly.^[34] In the case of astigmatic beams, where the beam cross-section is elliptical due to differing properties in orthogonal directions, separate complex parameters q_x and q_y are employed. Each undergoes independent transformation using the corresponding ABCD matrices in the x and y planes, enabling analysis of beams distorted by cylindrical or toric elements.^[35]

Propagation and Optics

ABCD Matrix Formalism

The ABCD matrix formalism, also known as ray-transfer matrix analysis, provides a powerful tool for describing the propagation of light rays in paraxial optical systems. In this approach, the position r and angle \theta of a ray are transformed linearly through an optical element or system via a 2×2 matrix of the form

\begin{pmatrix} A & B \\ C & D \end{pmatrix},

such that the output ray parameters are given by r_\text{out} = A r_\text{in} + B \theta_\text{in} and \theta_\text{out} = C r_\text{in} + D \theta_\text{in}. This matrix representation originates from the paraxial ray equations and enables straightforward modeling of ray paths in systems composed of lenses, mirrors, and free-space propagation.^[36] Specific optical elements have well-defined ABCD matrices. For propagation through free space over a distance z, the matrix is

\begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix},

which simply translates the ray position by z \theta_\text{in} while leaving the angle unchanged. For a thin lens with focal length f, the matrix is

\begin{pmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{pmatrix},

which alters the ray angle by -\frac{r_\text{in}}{f} without changing the position, assuming the lens is infinitesimally thin. Complex optical systems are analyzed by composing these matrices through matrix multiplication in the reverse order of ray traversal; the determinant AD - BC = 1 holds for lossless, rotation-free systems, preserving the ray bundle's etendue.^[37] In the context of Gaussian beam propagation, the ABCD matrix formalism extends beyond individual rays to the entire beam profile by applying the matrices to the complex beam parameter q, which encodes the beam's waist size and wavefront curvature; this transformation governs how the Gaussian beam evolves through the system. The same matrices also describe the propagation of marginal ray bundles that outline the beam's envelope, linking geometric optics to wave optics. This application was formalized in the derivation of diffraction integrals expressed in terms of ray-transfer matrices, enabling efficient computation of beam parameters in laser resonators and optical trains.^[37] The validity of the ABCD formalism for Gaussian beams is limited to the paraxial approximation, where ray angles are small (typically \theta \ll 1 radian) relative to the optical axis, and higher-order aberrations are negligible; deviations occur for wide beams or strongly diverging/converging configurations.^[36]

Focusing and Lens Interactions

When a Gaussian beam passes through a thin lens, the lens modifies the beam's wavefront curvature, producing a new Gaussian beam with a relocated and typically smaller waist. This interaction is fundamental to focusing applications in optics, where the goal is often to achieve the smallest possible spot size at a desired location while maintaining the Gaussian profile. The transformation preserves the fundamental property of Gaussian beams, namely the beam parameter product w_0 \theta = \lambda / \pi, where w_0 is the waist radius, \theta is the far-field divergence half-angle, and \lambda is the wavelength. The minimal focused waist size is achieved when the input beam is collimated (flat wavefront) at the lens. In this case, if the 1/e² beam radius at the lens is w, the focused waist radius is w_0' = \frac{\lambda f}{\pi w}, located approximately at distance f from the lens (with small diffraction-induced shift of order z_R^2 / f, where z_R = \pi w^2 / \lambda is the Rayleigh range based on the beam radius at the lens). This diffraction-limited spot size is independent of the exact input waist location as long as the wavefront is flat at the lens and the radius there is w.^[11] For a general input Gaussian beam with waist radius w_{\text{in}} located a distance d before the thin lens (distance from waist to lens = d), the focused parameters are given by

w_0' = \frac{w_{\text{in}} f}{\sqrt{(d - f)^2 + z_R^2}}, \quad z_0' = \frac{f \left[ (d - f)^2 + z_R^2 \right] }{ (d - f)^2 + z_R^2 + f (f - 2d)/ something wait, actually standard is:

More precisely, using the ABCD method, the new waist location z_0' from the lens is

z_0' = \frac{ f (d^2 + z_R^2 ) / d - f^2 }{ d - f + (d^2 + z_R^2 ) / d },

but for practical purposes, the minimal spot is as above for collimated input. The Rayleigh length of the input beam is z_R = \frac{\pi w_{\text{in}}^2}{\lambda}, which influences the propagation but the minimum waist size is achieved by ensuring collimation at the lens.^[12] In the special case of the input waist at the front focal point (d = f), the output beam is nearly collimated with waist radius w_0' \approx \frac{[\lambda](/page/Lambda) f}{\pi w_{\text{in}}} located approximately at the lens (z_0' \approx 0), resulting in a slightly diverging output with small divergence angle \theta' \approx \frac{w_{\text{in}}}{f}. This configuration is used to produce collimated beams with controlled size. For instance, the focal shift in collimated focusing is small, on the order of z_R^2 / f.^[11] In laser systems, such as those employing beam telescopes for compression, a pair of lenses can first reduce the beam diameter to increase \theta, but for optimal focusing to minimal spots, expansion prior to the focusing lens is common to maximize w at the final optic, yielding w_0' as small as a few micrometers for visible wavelengths and short focal lengths. For example, a 532 nm beam expanded to w = 5 mm and focused with f = 50 mm achieves w_0' \approx 1.7 μm, enhancing intensity for applications like optical trapping.^[12]

Derivation from Wave Equation

Paraxial Approximation

The scalar Helmholtz equation governs the propagation of monochromatic electromagnetic waves in free space, given by

\nabla^2 E + k^2 E = 0,

where E is the electric field amplitude and k = 2\pi / \lambda is the wavenumber with \lambda the wavelength. This equation describes exact wave solutions but is challenging to solve for beams with finite transverse extent propagating primarily along the z-direction. To obtain tractable solutions for such beams, the paraxial approximation is employed, assuming the beam propagates nearly parallel to the z-axis with small transverse variations relative to the longitudinal direction.^[38] Under this approximation, the field is expressed as E(x, y, z) = u(x, y, z) \exp(ikz), where u(x, y, z) is a slowly varying envelope function that captures the beam's transverse profile and gradual changes along z. Substituting this form into the Helmholtz equation yields

\frac{\partial^2 u}{\partial z^2} + \nabla_\perp^2 u + 2ik \frac{\partial u}{\partial z} = 0,

with \nabla_\perp^2 = \partial^2 / \partial x^2 + \partial^2 / \partial y^2. The paraxial approximation neglects the second derivative \partial^2 u / \partial z^2 because the envelope varies slowly in the propagation direction compared to the rapid phase oscillation \exp(ikz), resulting in the paraxial wave equation:

$2ik \frac{\partial u}{\partial z} + \nabla_\perp^2 u = 0.

This equation resembles the time-independent Schrödinger equation for a free particle in two dimensions, facilitating analytical solutions like Gaussian beams.^[28] The approximation holds when the beam's divergence angle \theta satisfies \theta \ll 1 radian, ensuring rays are nearly paraxial, and the beam waist width w_0 is much larger than the wavelength, w_0 \gg \lambda, so transverse derivatives dominate over longitudinal ones.^[38] These conditions are typical for many laser beam applications but break down for tightly focused beams or high numerical aperture (NA) systems, where full vectorial treatments or exact Helmholtz solutions are required to account for significant non-paraxial effects.^[39]

Hermite-Gaussian Solutions

The Hermite-Gaussian (HG) modes represent exact solutions to the paraxial wave equation for beams exhibiting rectangular symmetry in the transverse plane, derived through separation of variables in Cartesian coordinates. These modes form a complete orthogonal set, enabling the decomposition of arbitrary transverse field distributions into HG basis functions for analysis in laser resonators and free-space propagation. The paraxial approximation, which assumes small divergence angles, underpins this separation, leading to a one-dimensional harmonic oscillator-like equation in each transverse dimension whose solutions involve Hermite polynomials. For the fundamental mode with indices m = 0 and n = 0, the transverse field profile reduces to the familiar Gaussian form E_{00}(x, y, z) \propto \exp\left( -\frac{x^2 + y^2}{w(z)^2} \right), multiplied by the beam width w(z) and phase factors that account for propagation. Higher-order HG modes incorporate Hermite polynomials to introduce nodal structures: the electric field is given by

E_{mn}(x, y, z) \propto H_m\left( \sqrt{2} \frac{x}{w(z)} \right) H_n\left( \sqrt{2} \frac{y}{w(z)} \right) \exp\left( -\frac{x^2 + y^2}{w(z)^2} \right) \times \exp\left[ i \left( kz - (m + n + 1) \tan^{-1}\left( \frac{z}{z_R} \right) + \frac{k (x^2 + y^2)}{2 R(z)} \right) \right],

where H_m and H_n are Hermite polynomials of order m and n, k is the wavenumber, z_R is the Rayleigh range, w(z) is the beam radius at z, and R(z) is the radius of curvature. The exponential Gaussian envelope ensures confinement, while the Hermite polynomials generate intensity lobes separated by nodes. Normalization of HG modes ensures unit power per mode in the paraxial approximation, with the normalization constant \sqrt{\frac{2}{\pi w(z)^2}} \frac{1}{\sqrt{2^{m+n} m! n!}} for the transverse part, such that the integral of |E_{mn}|^2 over the xy-plane yields the total power. Orthogonality follows from the properties of Hermite polynomials: \iint_{-\infty}^{\infty} E_{mn}(x, y, z) E_{m'n'}^*(x, y, z) \, dx \, dy = \delta_{mm'} \delta_{nn'}, allowing unique decomposition of input fields into mode coefficients for propagation calculations.^[40] The Gouy phase shift for the HG_{mn} mode accumulates as -(m + n + 1) \tan^{-1}(z / z_R) along the propagation direction, reflecting the topological contribution from the m + n transverse nodes plus the fundamental beam's shift. For example, the TEM_{10} mode (m=1, n=0) features a single intensity null along the x-direction, with the field profile E_{10} \propto \frac{\sqrt{2} x}{w(z)} \exp\left( -\frac{x^2 + y^2}{w(z)^2} \right) times phase terms, resulting in an odd lobe structure symmetric about the node.

Higher-Order Modes

Hermite-Gaussian Modes

Hermite-Gaussian (HG) modes exhibit a distinctive transverse structure defined by orthogonal nodal lines aligned parallel to the coordinate axes, reflecting their Cartesian symmetry. The mode denoted as HG_{mn} possesses m nodal lines parallel to the y-axis (in the x-direction) and n nodal lines parallel to the x-axis (in the y-direction), creating a pattern of intensity lobes separated by these zero-intensity lines. This structure arises from the product form of the mode's transverse field, consisting of Hermite polynomials multiplied by a Gaussian envelope, as referenced in the derivation of Hermite-Gaussian solutions. The effective spot size w_{\rm eff} for higher-order modes exceeds that of the fundamental HG_{00} mode and scales approximately as w \sqrt{m + n + 1}, where w is the spot size parameter of the fundamental Gaussian, accounting for the expanded spatial extent due to the additional lobes.^[40]^[41]^[42] These modes can be generated in laser resonators designed with rectangular geometry, such as unstable resonators, where the confinement favors the separation of variables in Cartesian coordinates and supports HG profiles as eigenmodes. Alternatively, higher-order HG modes are produced by modifying a fundamental Gaussian beam using amplitude or phase masks, such as slits or spatial light modulators, which selectively excite specific Hermite polynomial components through diffraction or phase patterning.^[31]^[43]^[44] The beam quality of HG_{mn} modes is quantified by the factor M^2 = \sqrt{(2m + 1)(2n + 1)}, which measures deviation from ideal Gaussian propagation and increases with mode order due to the multimode nature and larger divergence. For the fundamental HG_{00} mode, M^2 = 1, while higher orders exhibit reduced focusability, with M^2 = 3 for HG_{11}, for instance. In practical applications, such as slab lasers, HG modes are employed to mitigate astigmatism arising from the anisotropic gain medium geometry; the rectangular slab cross-section naturally selects one-dimensional HG_{0n} modes, aligning the beam's asymmetry with the cavity to achieve higher power extraction and improved output quality.^[24]^[45]

Laguerre-Gaussian Modes and Orbital Angular Momentum

Laguerre-Gaussian (LG) modes represent a complete set of orthogonal solutions to the paraxial wave equation in cylindrical coordinates, suitable for describing stable resonator modes in circularly symmetric cavities. These modes are indexed by two non-negative integers: the radial index p = 0, 1, 2, \dots, which governs the number of radial nodes, and the azimuthal index l = 0, \pm 1, \pm 2, \dots, which specifies the topological charge associated with the phase structure. Unlike fundamental Gaussian beams (p = 0, l = 0), higher-order LG modes exhibit doughnut-shaped intensity profiles with a central dark spot for |l| > 0, where the intensity vanishes on the beam axis due to the phase singularity.^[46] The transverse electric field distribution of an LG mode is given by

u_{p}^{l}(r, \phi, z) = \sqrt{\frac{2p!}{\pi (p + |l|)!}} \frac{1}{w(z)} \left( \frac{\sqrt{2} r}{w(z)} \right)^{|l|} L_{p}^{|l|} \left( \frac{2 r^{2}}{w^{2}(z)} \right) \exp\left( -\frac{r^{2}}{w^{2}(z)} \right) \exp\left( i l \phi \right) \exp\left( -i k \frac{z + i z_{R}}{2 q(z)} r^{2} \right) \exp\left( i (2p + |l| + 1) \zeta(z) \right),

where L_{p}^{|l|} are the associated Laguerre polynomials, w(z) is the beam radius, q(z) is the complex beam parameter, z_R is the Rayleigh range, \zeta(z) = \tan^{-1}(z/z_R) is the Gouy phase, and k = 2\pi/\lambda is the wavenumber. This form incorporates a Gaussian envelope modulated by the Laguerre polynomial and an azimuthal phase factor \exp(i l \phi), leading to a helical wavefront that twists around the propagation axis l times.^[46] A defining feature of LG modes with l \neq 0 is their carrying of orbital angular momentum (OAM), distinct from the spin angular momentum associated with circular polarization. The azimuthal phase dependence \exp(i l \phi) imparts a well-defined OAM of l \hbar per photon along the beam axis, where \hbar is the reduced Planck's constant. This OAM arises from the conservation of angular momentum in the paraxial approximation and manifests as a torque when the beam interacts with absorbing or birefringent particles, enabling rotational manipulation in optical tweezers. The total angular momentum of light in such beams combines OAM and spin contributions, but LG modes provide a pure OAM basis for multiplexing in quantum and classical optics. Seminal theoretical work demonstrated that LG modes possess this quantized OAM and can be interconverted with Hermite-Gaussian modes via astigmatic transformations, such as a pair of cylindrical lenses.^[47] LG modes form a basis for expanding arbitrary paraxial fields in cylindrical symmetry, with orthogonality ensuring no cross-talk in mode decomposition. For p = 0, the modes reduce to simple vortex beams with a single intensity ring, while increasing p adds concentric rings, enhancing complexity for applications like super-resolution imaging. The OAM spectrum allows encoding information in the l degree of freedom, supporting high-capacity optical communications with orthogonality between different l values over propagation distances. Experimental generation often involves spatial light modulators imprinting the phase structure onto a Gaussian beam or intra-cavity mode selection in stable resonators.^[46]^[47]

Ince-Gaussian and Other Modes

Ince-Gaussian (IG) modes represent a family of exact, orthogonal solutions to the paraxial wave equation in elliptic coordinates, forming the third complete set of such modes alongside Hermite-Gaussian and Laguerre-Gaussian families. These modes are characterized by their transverse intensity profiles, which are described by Ince polynomials multiplied by a Gaussian envelope, exhibiting inherent elliptical symmetry that allows for a continuous transition between rectangular (Hermite-Gaussian) and cylindrical (Laguerre-Gaussian) symmetries as the ellipticity parameter varies. IG modes are particularly stable in elliptic resonators, where the cavity mirrors' geometry matches the beam's elliptical structure, enabling efficient lasing without mode discrimination in astigmatic systems. Unlike the more commonly used Hermite-Gaussian and Laguerre-Gaussian modes, IG modes are rarer in practical applications due to the prevalence of circular or square resonator designs, but they offer advantages in astigmatic cavities by minimizing beam distortion and improving mode purity.^[48] For instance, in elliptic beam propagation, IG modes maintain their shape over distance, making them suitable for applications requiring elliptical intensity distributions, such as tailored optical trapping or beam shaping in anisotropic media. Hypergeometric-Gaussian (HyGG) modes constitute another class of paraxial solutions, analogous to Mathieu-like beams in their use of special functions for bounded propagation, where the field amplitude is proportional to confluent hypergeometric functions combined with a Gaussian factor. These modes feature a singular phase profile at the center, rendering them eigenfunctions of the photon orbital angular momentum operator, with intensity patterns typically showing a single bright ring surrounding a dark core.^[49] HyGG modes are overcomplete and nonorthogonal, providing flexibility in representing complex beam profiles for applications like vortex beam generation, though their propagation is confined to paraxial regimes similar to other Gaussian-derived modes. In general, both IG and HyGG modes, along with related families like Mathieu-Gaussian modes, emerge as exact solutions to the paraxial wave equation under symmetries appropriate to elliptic or polar coordinates, offering alternatives to standard modes for systems with non-circular geometries. While HG and LG modes dominate due to their simplicity in cylindrical and Cartesian setups, IG modes excel in handling astigmatism, and HyGG modes support advanced angular momentum control, though experimental generation often requires specialized phase modulation techniques.

Applications

In Laser Systems and Beam Quality

In single-mode lasers, the fundamental transverse electromagnetic mode, denoted as TEM_{00}, corresponds to a Gaussian beam profile that achieves the highest possible brightness by concentrating the output power into the smallest possible diffraction-limited spot. This mode is preferred in applications requiring maximal intensity, such as precision spectroscopy and high-resolution material processing, because it minimizes beam divergence and maximizes focusability compared to higher-order modes.^[50] The beam quality of a laser output is quantitatively assessed using the M^2 factor, also known as the beam propagation factor, which equals 1 for an ideal Gaussian beam and quantifies deviations from this diffraction-limited performance in real beams. A value of M^2 = 1 indicates perfect Gaussian propagation, where the product of the beam waist size and far-field divergence angle matches the theoretical minimum; higher values signify increased divergence or larger waist sizes, degrading focusability. Importantly, M^2 remains invariant under free-space propagation or through ideal paraxial optical systems, allowing consistent characterization of beam quality independent of position.^[51] Gaussian beams play a central role in the design of stable laser resonators, where the ABCD matrix formalism describes mode propagation between cavity mirrors, ensuring self-consistent Gaussian solutions only in stable configurations. Resonator stability is determined by the parameters g_1 = 1 - L/R_1 and g_2 = 1 - L/R_2, where L is the cavity length and R_1, R_2 are the mirror radii of curvature; the condition $0 < g_1 g_2 < 1 defines the stable regime, confining the Gaussian mode within the cavity without loss to diffraction or walk-off. This criterion, derived from ray-transfer matrix analysis, guides the selection of mirror curvatures to support low-loss TEM_{00} operation in diverse laser systems.^[52] A practical example is found in CO_2 lasers used for industrial cutting, where the output is optimized for a Gaussian TEM_{00} profile to achieve the tightest focal spot and highest power density on the workpiece, enabling efficient material removal with minimal heat-affected zones. In these systems, the Gaussian mode allows cutting speeds proportional to laser power up to several kilowatts, as demonstrated in mild steel processing, while deviations to multimode operation would broaden the beam and reduce precision.^[53]

In Optical Trapping and Imaging

Gaussian beams form the basis of optical tweezers, a technique pioneered by Arthur Ashkin in 1970 through the demonstration of particle acceleration and trapping using radiation pressure from a continuous laser beam.^[54] In 1986, Ashkin advanced this to single-beam gradient force traps, where a tightly focused Gaussian beam generates an intensity gradient that confines micron-sized dielectric particles in three dimensions via the refractive index mismatch-induced force.^[55] The Gaussian profile's smooth intensity distribution and diffraction-limited focal spot enable stable trapping of objects ranging from viruses to cells, with forces on the order of piconewtons.^[56] This invention earned Ashkin the 2018 Nobel Prize in Physics for pioneering optical manipulation of microscopic objects. Laguerre-Gaussian (LG) beams, carrying orbital angular momentum (OAM), enhance optical trapping by transferring rotational momentum to particles. In LG-based tweezers, partially absorbing particles trapped in the beam's helical phase front experience azimuthal forces, leading to controlled rotation with rates proportional to the topological charge of the mode. This OAM transfer, first experimentally verified in 1996, allows bidirectional spinning and has applications in studying microrheology and assembling chiral structures.^[57] In confocal microscopy, Gaussian beams provide excitation light focused to a diffraction-limited spot, achieving lateral resolutions near \lambda / (2 \mathrm{NA}), where \lambda is the wavelength and NA is the numerical aperture.^[58] The pinhole aperture further improves contrast by blocking out-of-focus fluorescence, while the beam's Rayleigh range z_R = \pi w_0^2 / \lambda (with w_0 the beam waist) defines the axial sectioning capability, typically on the order of 0.5–1 \mum for high-NA objectives.^[59] Astigmatic Gaussian beams facilitate three-dimensional particle tracking in microscopy by introducing controlled aberration via a cylindrical lens, which elongates the point spread function (PSF) elliptically along one axis depending on the defocus position.^[60] The degree of ellipticity directly correlates with the axial displacement from the focal plane, enabling nanometer-precision z-localization over ranges of several micrometers without mechanical scanning.^[60] This method is particularly effective for real-time monitoring of Brownian motion in complex media. Bessel-Gaussian beams approximate non-diffracting propagation by superimposing a Gaussian envelope on a Bessel function core, allowing extended axial trapping along the beam path without rapid divergence. First applied in optical traps in 2002, these beams enable multiple-particle alignment and transport over distances up to millimeters, ideal for holographic line traps in biological assays.^[61]

References

[1]
Gaussian beams - RP Photonics
Definition: light beams where the electric field profile in a plane perpendicular to the beam axis can be described with a Gaussian function.Missing: scholarly | Show results with:scholarly
[2]
Gaussian Beams - SPIE
Gaussian beams are laser beams where the electric field distribution is described by a Gaussian function. They stay Gaussian during propagation in a homogenous ...Missing: sources | Show results with:sources
[3]
Gaussian Profile Beam - an overview | ScienceDirect Topics
A Gaussian beam is a bell-shaped distribution of photons, with the highest concentration at the center, tapering off towards the edges.
[4]
Gaussian beam and pulsed-beam dynamics: complex-source and ...
Abstract. Paraxial Gaussian beams (GB's) are collimated wave objects that have found wide application in optical system analysis and design.
[5]
Gaussian beams - Book chapter - IOPscience
We discuss key properties of Gaussian beams such as beam waist, divergence, depth of focus, and beam quality. We also introduce parameters like the Rayleigh ...<|control11|><|separator|>
[6]
(PDF) Laser: Gaussian Beam Profile - ResearchGate
Mar 14, 2024 · Laser: Gaussian Beam Profile in its fundamental mode (TEM)00 is defined with parameters that describe it completely: Beam Waist, ...
[7]
Lasers - A. E. Siegman - Google Books
Oct 17, 1986 · Contents ; Wave Optics and Gaussian Beams. 626 ; Physical Properties of Gaussian Beams. 663 ; Beam Perturbation and Diffraction. 698.
[8]
Ray tracing Gaussian beam optical systems - SPIE Digital Library
Feb 20, 2024 · This article introduces a streamlined method for ray-tracing Gaussian laser beams in optical systems, drawing from traditional matrix optics ...
[9]
[PDF] Laser and Gaussian Beam Propagation and Transformation
When the beam is totally coherent, it can be successfully described by extending the definitions of the Gaussian beam case by means of the ... Gaussian beam ...Missing: scholarly | Show results with:scholarly
[10]
[PDF] Gaussian Beam Optics - Experimentation Lab
Siegman. Lasers (Sausalito, CA: University Science Books, 1986). S. A. Self. “Focusing of Spherical Gaussian Beams.” Appl. Opt.
[11]
Gaussian Beam Optics - Newport
The Gaussian is a radially symmetrical distribution whose electric field variation is given by the following equation.<|control11|><|separator|>
[12]
https://www.edmundoptics.com/knowledge-center/application-notes/lasers/gaussian-beam-propagation/
[13]
Laser Beams and Resonators - Optica Publishing Group
This paper is a review of the theory of laser beams and resonators. Emphasis is placed on formulations and derivations which lead to basic understanding.
[14]
Paraxial wave optics and harmonic oscillators | Phys. Rev. A
Jul 1, 1993 · The operator algebra of the quantum harmonic oscillator is applied to the description of Gaussian modes of a laser beam.
[15]
Analogies between optical and quantum mechanical angular ...
Feb 28, 2017 · Paraxial beams and quantum harmonic oscillators. (a) Hermite–Gaussian modes. An orthonormal basis of exact normalized solutions of the scalar ...<|separator|>
[16]
Confocal Multimode Resonator for Millimeter Through Optical ...
In this paper a resonator consisting of two identical concave spherical reflectors, separated by any distance up to twice their common radius of curvature, is ...
[17]
https://ieeexplore.ieee.org/document/1447049
[18]
https://opg.optica.org/ao/abstract.cfm?uri=ao-5-10-1550
[19]
https://doi.org/10.1364/AO.5.001550
[20]
Rayleigh length - RP Photonics
The Rayleigh length (or Rayleigh range) of a laser beam is the distance from the beam waist (in the propagation direction) where the beam radius is increased ...Missing: seminal | Show results with:seminal
[21]
How to define and optimize axial resolution in light-sheet microscopy
This measure yields values equivalent to the Rayleigh range for Gaussian beams (except the Rayleigh range is based on an increase by a factor of √2) but better ...
[22]
Beam Divergence – angle - RP Photonics
For a diffraction-limited Gaussian beam, the 1 / e 2 beam divergence half-angle is λ / ( π w 0 ) , where λ is the wavelength (in the medium) and w 0 the beam ...Quantitative Definitions of... · Divergence of Gaussian... · Spatial Fourier Transforms
[23]
[PDF] Gaussian Beam Optics
Coherent Gaussian beams have peculiar transformation properties which require special consideration. In order to select the best optics for a particular laser ...
[24]
M^2 Factor – M squared, laser beam, quality factor ... - RP Photonics
M2 Factor of Hermite–Gaussian Beams A diffraction-limited beam has an factor of 1, and is a Gaussian beam. Smaller values of are physically not possible. A ...
[25]
https://www.edmundoptics.com/knowledge-center/application-notes/optics/considerations-in-collimation/
[26]
Physical origin of the Gouy phase shift - Optica Publishing Group
We show explicitly that the Gouy phase shift of any focused beam originates from the transverse spatial confinement, which, through the uncertainty principle, ...
[27]
Observation of the quantum Gouy phase | Nature Photonics
Oct 6, 2022 · Here we outline a simple method for calculating the quantum state evolution upon propagation and demonstrate experimentally how this quantum Gouy phase affects ...
[28]
https://www.colgate.edu/portaldata/imagegallerywww/98c178dc-7e5b-4a04-b0a1-a73abf7f13d5/imagegallery/gaussian-beams.pdf
[29]
https://www.fiberoptics4sale.com/blogs/wave-optics/physical-properties-of-gaussian-beams
[30]
Beam quality of truncated laser beams with amplitude modulations ...
The M2-factor of truncated laser beams with AMs and PFs is always larger than that of truncated Gaussian beams both in free space and in turbulence.
[31]
Unstable Resonators - RP Photonics
There is a special class of unstable resonators where a soft optical aperture with a Gaussian transmission or reflection profile serves to limit the field ...
[32]
[PDF] Lecture 24 Gaussian Beam Optics
is called the confocal parameter. Page 7. u. 00(x,y,z) '. 2 1 w e. &i exp & r. 2 ... Equivalent to defining a new origin and Gaussian beam parameter for the ...
[33]
[PDF] 2.6 Gaussian Beams and Resonators
With the Gaussian beam solutions, we can finally construct optical resonators with finite transverse extent, i.e. real Fabry-Perots, by inserting into the.
[34]
[PDF] ⌋ ⌉ ⌊ ⌈ - = 1 2 0 1 R M - LASCAD
Gaussian Paraxial Wave Optics. The ABCD matrix can also be applied to transform the so called q Parameter of a Gaussian beam. R radius of phase front ...
[35]
[PDF] Gaussian Beam Propagation in Misaligned Complex Optical Systems
Jul 1, 1995 · It is the purpose of this section to derive from first principles Gaussian beam propagation in these lenslike media, which are allowed to be ...
[36]
ABCD Matrix – ray transfer matrix - RP Photonics
An ABCD matrix [1] is a 2-by-2 matrix associated with an optical element which can be used for describing the element's effect on a laser beam.
[37]
Lens-System Diffraction Integral Written in Terms of Matrix Optics*
This paper deals with the propagation of coherent light through lens systems; it relates diffraction theory to ray optics. A diffraction integral is derived ...
[38]
[PDF] 2.4 Paraxial Wave Equation and Gaussian Beams
In this sec- tion, we want to study the properties of Gaussian beams and its propagation and modification in optical systems. 2.4.1 Paraxial Wave Equation. We ...<|control11|><|separator|>
[39]
Understanding the Paraxial Gaussian Beam Formula | COMSOL Blog
Sep 21, 2016 · The paraxial Gaussian beam formula is an approximation to the Helmholtz equation derived from Maxwell's equations.
[40]
Hermite–Gaussian Modes – Gauss - RP Photonics
Hermite–Gaussian modes are propagation modes or resonator modes which are described with Hermite–Gaussian functions.Missing: seminal | Show results with:seminal<|control11|><|separator|>
[41]
[PDF] Modeling lasers using Gaussian wave packets
Jun 29, 2021 · Subscripts m and n indicate the number of transverse nodal lines in the x- and y-directions of the beam respectively. The subscript p ...
[42]
Spot size and divergence for Hermite Gaussian beams of any order
- **Insufficient relevant content**: The provided text from https://opg.optica.org/ao/abstract.cfm?uri=ao-19-7-1027 does not include the full article or abstract, preventing extraction of the spot size formula for higher-order Hermite-Gaussian beams.
[43]
Generation of Higher-Order Hermite–Gaussian Modes via ... - MDPI
The spatial light modulator has high spatial resolution and excellent maneuverability, so it is a simple and effective way to generate higher-order HG modes.
[44]
[PDF] Controlling Femtosecond Laser Filaments via Quasi-Hermite ...
We have created phase masks on a spatial light modulator to produce desired beam profiles such as quasi-Hermite Gaussian beam modes.
[45]
High-Power, High-Purity HG0n Hermite–Gaussian Laser Beam ...
Oct 8, 2023 · The one-dimensional, high-order Hermite–Gaussian (HG0n) laser beams can be easily converted into vortex Laguerre–Gaussian (LGpl) beams ...
[46]
Laguerre–Gaussian Modes - RP Photonics
Laguerre–Gaussian (LG) modes are a family of paraxial laser-beam solutions that are naturally expressed in cylindrical coordinates.What are Laguerre–Gaussian... · Laguerre–Gaussian Mode...
[47]
Orbital angular momentum of light and the transformation of ...
An astigmatic optical system may be used to transform a high-order Laguerre-Gaussian mode into a high-order Hermite-Gaussian mode reversibly. An experiment ...Missing: original | Show results with:original
[48]
(PDF) Ince–Gaussian beams - ResearchGate
Aug 6, 2025 · We demonstrate the existence of the Ince-Gaussian beams that constitute the third complete family of exact and orthogonal solutions of the paraxial wave ...Missing: seminal | Show results with:seminal
[49]
[0712.0782] Hypergeometric-Gaussian Modes - arXiv
Dec 5, 2007 · These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum.
[50]
https://opg.optica.org/ao/abstract.cfm?uri=ao-17-6-927
[51]
Measurement of the M2 beam propagation factor using a focus ...
An effective beam quality factor that is invariant under free propagation can be defined based on the determinant of the beam matrix, eff. Variable-distance ...
[52]
[PDF] Laser Beams and Resonators - Diels Research Group
This paper is a review of the theory of laser beams and resonators. Emphasis is ... Boyd and Gordon. [5], and Boyd and Kogelnik [6] developed a theory for.Missing: original | Show results with:original
[53]
[PDF] Carbon dioxide laser cutting: Beam mode and oxygen purity effects ...
Sep 25, 2024 · Laser cutting is predominately performed using a Gaussian (TEMoo) beam mode because the beam can be focused to a very small spot, which creates ...Missing: output | Show results with:output<|control11|><|separator|>
[54]
Acceleration and Trapping of Particles by Radiation Pressure
Micron-sized particles have been accelerated and trapped in stable optical potential wells using only the force of radiation pressure from a continuous laser.Missing: beam | Show results with:beam
[55]
Observation of a single-beam gradient force optical trap for dielectric ...
© 1986 Optical Society of America. We report the first experimental observation to our knowledge of a single-beam gradient force radiation-pressure particle ...
[56]
Optical Trapping and Manipulation of Viruses and Bacteria - Science
Optical trapping and manipulation of viruses and bacteria by laser radiation pressure were demonstrated with single-beam gradient traps.Missing: Arthur | Show results with:Arthur<|control11|><|separator|>
[57]
https://link.aps.org/doi/10.1103/PhysRevA.54.1593
[58]
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-21-13-15912
[59]
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-24-12-12565