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Aleksander Stefanov works across mathematical physics, mechanics and optical measurement. A 2022 Sofia University academic statement identifies his education in theoretical and mathematical physics, doctoral work on nonlinear dynamical systems associated with Lie algebras, and work in the Faculty of Mathematics and Informatics. It describes research on integrable models, walking-robot dynamics and educational physical-constant measurements.^[ref]

The same document identifies Korteweg-De Vries equation and nonlinear Schrödinger-type models among his theoretical interests. He also coauthored the 2023 theoretical and experimental comparison of Gouy phase for Bessel-Gaussian beams with Lyubomir Stoyanov, Alexander Dreischuh and Gerhard G. Paulus.^[ref]

An original example illustrates the quantitative idea behind a noise-based constant measurement. For ideal independent charge-transfer events, the one-sided current shot noise spectral density is $S_I=2eI$. With mean current $I=100\mu \mathrm{A}$, elementary charge $e=1.602176634\times 10^{-19}$ C and a rectangular effective noise bandwidth $B=10$ kHz, the predicted root-mean-square fluctuation isThese stipulated numbers are not data from Stefanov's apparatus. They explain why estimating charge from noise requires a calibrated gain, known bandwidth and a model for other fluctuations. Thermal noise and detector noise must be distinguished from the charge-transfer contribution; a measured noise amplitude alone is insufficient.

The attraction of this work is the bridge between mathematical models and observable signals. The academic statement describes physical walking-robot tests as well as electronic measurements; those are different ways to test a theoretical prediction. The available sources identify this Sofia researcher, and do not justify merging him with unrelated researchers who share a surname.

Alexander Dreischuh is a physicist associated with the Department of Quantum Electronics at Sofia University. His author-hosted curriculum vitae records a professorship there beginning in 2004.^[ref] His publication list documents experimental and theoretical work in nonlinear optics and structured light, including joint work with Lyubomir Stoyanov and Gerhard G. Paulus.

A useful theme is that an optical field has a spatial wave phase as well as a light intensity. An optical vortex has a phase singularity around which the wave phase winds. In an original local model $u(r,\phi )=A(r)e^{im\phi }$, with integer $m$ and vanishing central amplitude when $m\ne 0$, a complete circuit changes the continuous phase by $2\pi m$. The optical vortex charge is therefore $m$; $m=2$ gives a $4\pi$ phase winding. The visible dark center alone does not determine the sign or magnitude of that winding.

Dreischuh coauthored the 2023 comparison of Gouy phase theory with measurements on Bessel-Gaussian beams. One apparatus used a 532 nm laser, two spatial light modulators, a Mach-Zehnder interferometer and a rail-mounted camera behind the focusing lens.^[ref]

The apparatus separates preparation from diagnosis: a spatial light modulator controls an optical field's spatial structure, while interference with a reference converts phase differences into recorded intensity fringes. Changing the longitudinal camera position tests propagation through and beyond the focus. That makes the observed phase evolution a quantitative experimental question rather than something inferred solely from a single ring-shaped intensity image.

A Bessel-Gaussian beam is a structured optical beam whose Bessel-like transverse oscillations are limited by a Gaussian envelope. For a circularly symmetric zero-order waist-plane model,where $J_0$ is a Bessel function of the first kind, $k_t$ sets the radial oscillation scale and $w_0$ sets the Gaussian envelope. This specifies the initial field; propagating it requires a wave model rather than keeping the same product unchanged at every plane.

An ideal Bessel beam can be constructed from a ring of plane waves with the same longitudinal wavenumber and different transverse directions. Their coherent angular sum gives $J_0(k_{t}r)$. The ideal infinitely extended beam has unbounded transverse power. Multiplication by the Gaussian envelope makes the power integral finite, since $|J_0|\le 1$ andFinite transverse extent also means that indefinitely unchanged propagation is no longer guaranteed.

For an original paraxial scale example, let $\lambda =532$ nm, $w_0=0.50$ mm and $k_t=k\sin \theta$ with $\theta =1{}^{\circ }$. The first radial zero of $J_0$ is at $k_{t}r\simeq 2.4048$, giving $r\simeq 11.67\mu \mathrm{m}$. The geometric overlap estimate $w_0/\tan \theta \simeq 28.6$ mm supplies a finite propagation-distance scale, not an exact invariant-range theorem or a measured beam length. The envelope and central-lobe scales answer different questions.

A primary 2023 experiment compared Bessel-Gaussian Gouy phase theory with interferometric measurements. Its apparatus used a 532 nm laser, beam expansion, spatial light modulators, focusing lenses and a camera translated along a rail; Figure 3 documents the interferometer geometry.^[ref] Its beam parameters should not be identified with the invented example above.

A ringed light intensity image alone does not determine the whole propagated complex field. Measuring wave phase, aperture effects and longitudinal evolution distinguishes a finite structured beam from the mathematical ideal. In particular, the Gouy phase of such a mode should not automatically be replaced by the fundamental Gaussian beam formula.

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.

Lyubomir Stoyanov is an optical researcher associated with the Department of Quantum Electronics at Sofia University. His author-hosted profile and publication list identify work on structured laser fields and their propagation.^[ref] His coauthored work includes measurements of the Gouy phase of Bessel-Gaussian beams and higher-order Hermite-Gaussian optical modes.

A 2024 experiment used spatial light modulator preparation of higher-order Hermite-Gaussian optical modes and interferometry to compare their Gouy phase with the familiar mode-order prediction.^[ref] Measuring the phase supplies information that a beam's spot pattern alone cannot provide.

For an original ideal-mode example, define the positive Gouy-phase function bywhere $z_R>0$ is the Rayleigh range. In a field convention containing $e^{i(kz-\omega t)}$, this factor contributes $e^{-i{\psi }_{mn}}$. Thus the $(m,n)=(2,0)$ mode at $z=z_R$ has ${\psi }_{20}=3\pi /4$, three times the fundamental mode's $\pi /4$. The formula states a convention explicitly so that the sign of the additional field phase is not confused with the sign of the named positive function.

Comparing modes at matched $z/z_R$ is important. A change of focusing geometry can change the Rayleigh range, while the mode-order factor changes the phase at a fixed normalized distance. An interferometric comparison therefore needs both mode identification and calibrated propagation position. These ideal numbers illustrate the model; they are not fitted results from the paper.