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Boundary-sensitive Gaussian overlap inversion describes the loss of stable radius information when a symmetric log-overlap ratio approaches its finite-model limit. A central reading can remain formally invertible while small measurement uncertainty reaches values for which no finite radius ratio exists.

For known $h=\log q>0$, define $D=D_h$, $u=\log \rho$ andThe inversion of a symmetric Gaussian overlap log-ratio requires $|D|<4h$. At the upper limit $D\to 4h$, the radius ratio tends to zero; at the lower limit it tends to infinity.

An original expansion near the upper limit puts $ϵ=4h-D>0$. Expanding the hyperbolic tangent givesThus the approximate relative radius uncertainty is $u(\rho )/\rho \simeq u(D)/(2ϵ)$ when linearization is still justified. The distance to the boundary, rather than the magnitude of $D$ alone, controls this sensitivity.

For stipulated $q=2$ and a central log-ratio only 0.01 below $4\log 2$, the asymptotic radius ratio is about 0.0365. A standard uncertainty 0.02 in $D$ is twice the boundary distance, so a small symmetric error bar on the inferred radius is not credible. In a hypothetical Gaussian readout model centered there, roughly 31% of readings fall beyond the finite-inversion limit. This is a statistical illustration, not an observed receiver distribution.

Interval propagation for bounded overlap data shows the asymmetry directly. A stated interval in $D$ that reaches the upper bound maps to radius ratios arbitrarily close to zero; it does not yield a symmetric finite relative interval. An interval wholly inside the domain maps monotonically, with its endpoint order reversed.

Discarding out-of-range Monte Carlo draws would condition on a selected subset and change the distribution being propagated. Instead, examine the forward measurement model, admissible parameter range and calibration uncertainty, or construct an explicitly stated inference using the measured data. JCGM 101 separates input-distribution assignment from numerical propagation and states conditions for valid Monte Carlo use.^[ref]

Finally, proximity to the mathematical boundary is not the only possible problem. A change in shared normalization, polarization or wavefront curvature can violate the one-radius model. NIST's receiver apparatus illustrates the need for optical matching and separate electrical corrections.^[ref] Numerical clipping cannot replace those physical checks.

The Gouy phase describes the additional axial phase associated with a focused wave relative to a plane wave. For a fundamental circular Gaussian beam with beam waist at $z=0$ and Rayleigh range $z_R>0$, defineWith the carrier convention $e^{i(kz-\omega t)}$, the on-axis field phase is $kz-\omega t-\psi (z)$ up to a constant. Thus the signed additional phase is $-\psi$ in this convention; defining the positive function $\psi$ does not reverse the physical prediction.^[ref]

The complex field prefactor gives a short derivation. The normalized Gaussian propagation factor isIts modulus describes the on-axis amplitude reduction; its argument is the additional phase. An intensity measurement alone loses this argument, so measuring the beam's width and brightness does not directly measure the Gouy phase.

As $z$ runs from far before to far after the waist, $\psi$ runs continuously from $-\pi /2$ to $+\pi /2$. The total additional phase change has magnitude $\pi$, but is not a discontinuous jump at the waist. For stipulated $w_0=100\mu \mathrm{m}$ and $\lambda =633\mathrm{nm}$, $z_R\approx 49.6\mathrm{mm}$. Between $z=-z_R$ and $z=+z_R$, the additional phase changes by $-\pi /2$; the far-field-to-far-field limit gives $-\pi$.

The number of confined transverse directions matters. A Gaussian beam with one transverse coordinate has prefactor $(1+iz/z_R{)}^{-1/2}$ and half this phase contribution. Simin Feng and Herbert G. Winful's primary analysis relates the effect to transverse confinement and the angular spectrum representation: a localized beam includes transverse wave-vector components, affecting axial propagation.^[ref]

An illustrative Gouy phase interferometry experiment overlaps the focused beam with a coherent reference beam and records the interference phase at different axial planes. Removing the ordinary optical-path contribution and accounting for wavefront curvature isolates the axial phase change. This is a proposed measurement outline, not an experimental result reported by the cited theoretical paper. In an optical cavity, related phase contributions help determine the resonance condition and distinguish transverse modes.

Gouy phase interferometry compares a focused field with a coherent reference to recover the additional Gouy phase. Beam brightness alone discards the phase. An interferometer converts a relative field phase into intensity modulation, but the optical paths and reference wavefront curvature must be modeled.

A published 2023 experiment by Lyubomir Stoyanov and colleagues used a Mach-Zehnder interferometer to study Bessel-Gaussian beams. Its setup included a 532-nanometre laser, beam expansion, two spatial light modulators, focusing lenses and a CCD camera on a rail to follow propagation after the focus. Its Figure 3 also describes a single-lens interferometer arrangement for stronger focusing.^[ref] Structured modes need their own phase model; that experiment should not be silently interpreted as a fundamental-Gaussian measurement.

A separate original protocol illustrates four-step phase retrieval. At one detector point, apply controlled signal-arm phases $\delta =0,\pi /2,\pi ,3\pi /2$ while the beam intensities remain fixed. Write
This phase-shifting interferometry formula removes the unknown common intensity background $C$ and positive modulation amplitude $A$. It returns phase modulo $2\pi$; phase unwrapping requires continuity information between sampled positions. If the interference visibility vanishes, the phase is not observable by this formula.

For a synthetic coherent Gaussian/reference pair, use reference intensity 1 and signal intensity $1/(1+{\zeta }^2)$, with $\zeta =z/z_R$. Then $C=1+1/(1+{\zeta }^2)$, $A=2/\sqrt{1+{\zeta }^2}$ and $\varphi =-\arctan \zeta$ after common path phases are removed. At $\zeta =1$, the four intensities are $(2.5,2.5,0.5,0.5)$, giving $\hat{\varphi }=-\pi /4$. At $\zeta =-1$, the first intensity is still $2.5$, but the quadratures reverse and the retrieved phase is $+\pi /4$.

Curvature bias in Gouy-phase measurements is important away from the axis: the signal adds $k{r}^2/(2R(z))$ to its spatial phase. Fit or remove that term when using multiple pixels, retain calibrated phase steps and account for drift between frames. The recovered phase combines all coherent optical-path contributions until those controls isolate the Gouy phase.

Herbert G. Winful is an electrical-engineering researcher at the University of Michigan whose work includes nonlinear optics and photonics. His institutional record and the full byline of his 2001 paper with Simin Feng verify this identity.^[ref][ref]

Their primary analysis relates the Gouy phase of a focused wave to transverse confinement and the distribution of transverse wave vectors. It accounts for the different phase contributions in line and point focusing, rather than relying on a geometrical explanation restricted to one focusing geometry.^[ref]

A constructed interference example shows why a phase contribution is measurable even when the separate beam powers are unchanged. Let two mutually coherent beams each contribute light intensity $I_0$ at a detector. If their fields have relative phase $\varphi$, their combined intensity isA relative phase change from zero to $\pi$ changes the combined intensity from $4I_0$ to zero at that location. Each beam alone still contributes $I_0$. This is an ideal local interference calculation, not a reported experimental result from the theoretical paper; spatial overlap and polarization must also permit interference.

Gouy phase interferometry uses this phase sensitivity while separating the focusing contribution from ordinary optical-path changes and wavefront curvature. The scientific distinction is between a measured intensity pattern and the phase information inferred using a specified field model. The paper's relevance to optical cavities likewise comes from phase matching, rather than from beam width alone.

Application note explicitly credited to Francesco D'Angelo, Section 2.1 of the Edmund Optics Laser Optics Resource Guide. Equations (1)–(5) define the intensity radius, beam waist, Rayleigh range and far-field half-angle beam divergence of an ideal Gaussian beam. No publication date is asserted.

For wavefront curvature, the same primary application note gives the Gaussian beam wavefront radius in equation (6), the reciprocal complex beam parameter in equation (10), and the thin-lens transformation in equation (16): www.edmundoptics.com/knowledge-center/application-notes/lasers/gaussian-beam-propagation/ . These distinguish wavefront radius from beam radius and connect curvature to the complex beam parameter.

By Rüdiger Paschotta.

Publisher: RP Photonics AG.

The entry Wavefronts, www.rp-photonics.com/wavefronts.html , describes surfaces of constant phase, planar and spherical wavefronts, and changes of wavefront shape through optical elements. It provides the geometric context for wavefront curvature.

A symmetric beam-radius perturbation for overlap branch selection compares two controlled settings of one Gaussian beam radius: $q{w}_2$ and $w_2/q$, with known $q>1$. Under the centred, matched-wavefront curvature model of reciprocal size-ratio ambiguity in centred Gaussian overlap, writeThe hyperbolic secant form makes the symmetry explicit: the two settings evaluate the same even function at $u+h$ and $u-h$. Their ordering identifies the sign of $u$, as shown in branch selection by a controlled beam-radius change.

An original extension removes an unknown common scale and recovers the radius ratio itself. Suppose the two measured positive efficiencies satisfyThe factor $A$ is a shared multiplicative nuisance parameter, such as an unchanged response normalization. The symmetric log-overlap ratio eliminates it exactly:For the last equality, use the addition formula for hyperbolic cosine to write the fraction as $(1+t)/(1-t)$, where $t=\tanh u\tanh h$. The defining logarithmic identity for the inverse hyperbolic tangent completes the derivation. Because $h>0$, $D_h$ is positive for the initially narrower beam, negative for the wider beam, and zero at matching.

The inversion of a symmetric Gaussian overlap log-ratio now givesA finite radius ratio exists exactly when $|D_h|<4h$. This condition follows from $|\tanh u|<1$; equality is approached only as one radius ratio tends to zero or infinity. A measured log-ratio outside that interval has no finite solution in this ideal two-setting model.

For an original numerical example, take $q=1.5$, true $\rho =0.8$ and unknown scale $A=0.75$. The ideal readings are $V_{+}=0.7256114$ and $V_{-}=0.5172352$. They give $D_h=0.3385169$, recovering $\rho =0.8$ without supplying $A$ to the inversion. The physical bound is $4h=1.6218604$.

Finite measurement noise still limits branch selection. For small relative errors, the delta method givesA perfectly shared multiplicative fluctuation cancels sample by sample, paralleling common-gain cancellation in heterodyne overlap uncertainty. Independent relative standard uncertainties of $1\%$ in the two readings instead give $u(D_h)\simeq 0.014142$. At the worked point, the derivative of the inverse gives $du/d{D}_h=-0.6780385$, hence about $0.959\%$ relative standard uncertainty in $\rho$. At exact matching the two branches coincide; a noisy sign cannot identify a distinct physical branch there.

These conclusions require $A$ and the other optical mode matching conditions to remain shared across the settings. Changing a lens can alter wavefront curvature and polarization as well as radius. NIST's physical heterodyne apparatus explicitly controlled polarization and spatial matching.^[ref] Larger radius changes may strengthen the log-ratio response while lowering either measured efficiency and changing its noise. Step choice therefore depends on the apparatus and noise model.

A symmetric log-overlap ratio compares two measured Gaussian-beam overlap efficiencies after equal and opposite logarithmic changes of one radius. It removes a common multiplicative response scale and retains the sign of the initial radius mismatch.

In the centered, matched-curvature model, let $\rho =w_2/w_1$, $u=\log \rho$ and $h=\log q>0$. The two controlled settings have positive efficiency readingsHere $q$ and $1/q$ multiply the second radius. The readings are efficiencies proportional to squared field overlap, not field amplitudes. The shared nuisance scale cancels inUsing the addition formula for hyperbolic cosine, put $b=\tanh h$ and $z=\tanh u$ to obtainThus $D_h$ is odd in $u$, positive when $\rho <1$ and negative when $\rho >1$. Its finite-radius range is $|D_h|<4h$. As $u$ tends to either infinity, the ratio approaches a limiting value even though individual efficiencies can become very small.

The derivative makes the sensitivity explicit:At matching it is $-4\tanh h$; far from matching it approaches zero. A saturated log-ratio consequently supplies little information about additional radius mismatch.

For an original example, take $q=2$, $\rho =1/2$ and $A=0.7$. The positive setting gives $V_{+}=0.7$, while the negative setting gives $V_{-}=0.7\cdot 64/289\approx 0.15502$. Their log-ratio is $\log (289/64)\approx 1.50754$, independent of the chosen scale. The allowed bound is $4\log 2\approx 2.77259$. Taking logarithms of field-amplitude readings instead would give half this log-ratio.

A symmetric-overlap normalization drift breaks the simple cancellation: different scales $A_{+},A_{-}$ add $\log (A_{+}/A_{-})$ to the ideal $D_h$. Interleave the settings and monitor conditions to test that sharing assumption. NIST's receiver apparatus explicitly controlled spatial mode and polarization, but did not establish this particular symmetric-radius diagnostic.^[ref] The displayed relations are model derivations whose use requires fixed displacement, wavefront curvature and polarization across both settings.

Wavefront curvature describes the change in the normal direction across a surface of equal wave phase. A plane wavefront has zero curvature; a spherical wavefront has curvature magnitude $1/|R|$ for radius $R$. A general wavefront can have different principal curvatures in different transverse directions. The sign convention must be stated when distinguishing converging and diverging waves.^[ref]

In a paraxial transverse plane, choose the convention in which a diverging spherical wave has $R>0$. Its quadratic phase variation is approximatelyThe corresponding local ray angle is approximately $r/R$. For illustrative $R=0.50\mathrm{m}$ and $r=2\mathrm{mm}$, the angle is $0.004$ radians and the optical path difference is $r^2/(2R)=4\mu \mathrm{m}$. At wavelength $500\mathrm{nm}$, that is eight phase cycles. A modest geometrical departure can therefore produce a substantial optical phase difference.

For an ideal Gaussian beam with its waist at $z=0$,The reciprocal form remains finite at the waist: its value is zero, so the wavefront there is locally planar even though the beam radius is minimal. At $z=z_R$, $R=2z_R$. Beam size and curvature are distinct beam parameters.^[ref]

A thin lens changes the curvature according to $1/R^{\prime }=1/R-1/f$ in this convention, while the beam radius at the lens is unchanged in the thin-element approximation.^[ref] Thus a lens can alter convergence without an immediate jump in the beam width.

Experimentally, curvature can be inferred from a measured phase profile or local wavefront slopes. Phase unwrapping and the chosen reference matter because an intensity image in one plane does not itself provide the quadratic phase. A curved wavefront is not necessarily aberrated: the appropriate ideal reference can itself be spherical or Gaussian-beam curvature.