# Reply to "Experiment to observe an optically induced change of the vacuum index" (Robertson et al., *Phys. Rev. A* 103, 023524 (2021)) – Re-examination using structured electromagnetic fields **An M. Rodriguez*** Preferred Frame Laboratory, Barcelona, Spain **James E. Callister** Optical Topology Group, University of York, UK (*Corresponding author: [an@preferredframe.com](mailto:an@preferredframe.com)*) ## One-word-summary Topology ## Abstract We respectfully reply to the work of Robertson et al. (*Phys. Rev. A* 103, 023524 (2021)), which reported no detectable refractive-index modification of the vacuum under intense Gaussian laser pulses. We show that (i) a Gaussian pump beam has *zero* optical topological charge and lacks the phase structure required to generate a persistent electromagnetic energy confinement, and (ii) experiments on structured (vortex and toroidal) light demonstrate that non-trivial field topology produces localized, self-sustained electromagnetic energy distributions. We therefore propose that future measurements use structured fields—Laguerre-Gaussian or toroidal modes—to re-examine the vacuum-index effect predicted by Maxwellian field polarization. ## 1. Introduction Robertson et al. designed an elegant interferometric test of vacuum nonlinearity, modeling the pump as a Gaussian pulse. Within quantum electrodynamics, such a null result constrains photon-photon scattering. However, in a **classical-field (Maxwellian)** ontology where space is intrinsically polarizable by electromagnetic energy itself, the *geometry* and *topology* of the field distribution determine whether a measurable refractive perturbation can persist. A purely Gaussian field, with no phase circulation and rapidly vanishing spatial recurrence, may fail to generate the stationary polarization required to refract a probe. ## 2. Critique of the Gaussian Pump Configuration Robertson et al. explicitly assume a spatiotemporal Gaussian pump intensity $$ I_{\mathrm{pump}}(\mathbf r,t)=I_0 \exp\left[-\frac{x^2+y^2}{W_0^2}\right] \exp\left[-\frac{(t-t_0)^2}{T_0^2}\right], $$ yielding an induced index perturbation $$ \delta n(\mathbf r,t)=n_2 I_{\mathrm{pump}}(\mathbf r,t), $$ where (W_0) and (T_0) are the waist and pulse duration. They evaluate probe deflection from (\partial_y \delta n) via Fermat's principle. This model is entirely consistent with a *phase-flat* Gaussian envelope carrying **zero optical vortex charge**. ## 3. Topological charge and field structure ### 3.1 Definition In structured electromagnetism, the **topological charge** (l) (winding number) measures the net (2π) phase circulation around a closed loop encircling a phase singularity: $$ \ell=\frac{1}{2\pi}\oint_{\mathcal C}\nabla\Phi\cdot d\mathbf l =\frac{1}{2\pi}\Delta\Phi_{\mathcal C}. $$ For a complex field (\mathbf{E}(\mathbf r)=\tilde{\mathbf E}(\mathbf r)e^{-i\omega t}), non-zero (ℓ) corresponds to **helical wavefronts** and a central intensity null (an optical vortex). A Gaussian beam satisfies (\partial_\varphi\Phi=0 \Rightarrow \ell=0); a Laguerre-Gaussian or toroidal beam satisfies (\Phi=\ell\varphi+\cdots \Rightarrow \ell\neq0). ### 3.2 Physical meaning Topological charge quantifies phase circulation and hence the **orbital angular momentum** (OAM) density of the electromagnetic field. For beam power (P), $$ L_z=\frac{\ell P}{\omega}. $$ Non-zero (ℓ) implies an azimuthal Poynting component (S_\varphi\propto \ell/r), creating a *recirculating toroidal energy flow*. This self-contained energy loop behaves as a localized electromagnetic inertia, capable of polarizing space analogously to matter's bound fields. Conversely, a Gaussian beam ((ℓ=0)) transports energy linearly without confinement. ### 3.3 Experimental confirmations Structured beams with quantized (ℓ) are routinely produced and measured: | Year | Reference | Result | | ---- | ---------- | ------- | | 2003 | **M. R. Dennis et al.**, *Nature* 426 (2003) | Optical vortex knots confirming quantized topological charge. | | 2010 | **K. O'Holleran et al.**, *New J. Phys.* 12 (2010) | Stable vortex lattices with integer winding numbers. | | 2024 | **J. Zhong & Q. Zhan**, *Nat. Commun.* 15 (2024) | Toroidal beams with controlled ((m,n)) winding topology. | | 2024 | **Physics World**, "Vortex cannon generates toroidal electromagnetic pulses" | Microwave-scale toroidal EM pulses carrying vortex charge. | These confirm that (ℓ) is an experimentally measurable invariant and that toroidal electromagnetic modes exist across optical and microwave regimes. ### 3.4 Relevance to the Robertson experiment Because the 2021 pump field had (ℓ=0), it lacked the phase topology required for a stationary electromagnetic polarization of space. Repeating the measurement with **structured fields**—counter-propagating Laguerre-Gaussian or toroidal modes of charge (ℓ≠0)—would create a **stationary energy density** and an effective index perturbation $$ \Delta n_{\mathrm{eff}} \propto n_2,\langle |E_{\mathrm{pump}}|^2\rangle_\text{toroidal}, $$ potentially orders of magnitude larger than in the transient Gaussian case. ## 4. Proposed experimental modification We suggest adapting the original setup as follows: 1. **Pump:** replace the Gaussian pulse with a structured beam (LG(_0^{\ell}) or toroidal cavity mode) of moderate power, forming a standing or recirculating toroidal field. 2. **Probe:** a weak Gaussian traversing the high-energy region; phase shift measured by a Fabry-Perot or Mach-Zehnder interferometer. 3. **Parameters:** scan (ℓ), polarization, and relative phase; record fringe displacement (\Delta N = (L/\lambda)\Delta n_{\mathrm{eff}}). 4. **Control:** repeat with (ℓ=0) to isolate topological contribution. This configuration directly tests whether structured electromagnetic energy can act as a polarizing medium in free space. ## 5. Discussion Structured electromagnetic modes—vortices, toroidal beams, and knotted fields—have been experimentally verified through **phase and interference diagnostics**, which are effectively measurements of spatially varying optical path length. These diagnostics are conceptually identical to the refractive-index detection method used by Robertson *et al.* Thus, rather than constituting a distinct phenomenon, the observation of topological field structure may be viewed as a *special case* of the same optical test applied to an intrinsically structured field. In this sense, the experiment of Robertson *et al.* and the optical-vortex literature occupy adjacent regions of the same empirical landscape: both probe how electromagnetic energy modifies propagation through space. The null result of Robertson *et al.* for a topologically trivial (Gaussian) field contrasts with the reproducible detection of toroidal and vortex electromagnetic modes in other optical experiments, where phase gradients and deflection are the primary observables. This complementary evidence suggests that **field topology, not merely intensity**, is the determining factor in whether an electromagnetic configuration can modify light propagation. In that sense, the Robertson result *confirms by exclusion* that the absence of vorticity yields no measurable refractive effect, while structured fields with nonzero topological charge do manifest such effects through observable phase evolution. It would be of great interest to see a systematic **family of measurements parameterized by the vorticity or topological charge of the electromagnetic flow**, extending the Robertson geometry from Gaussian to vortex and toroidal regimes. ## 6. Conclusion We commend Robertson *et al.* for their precise work and encourage extending the measurement using **structured electromagnetic fields**. Existing technologies for optical vortices and toroidal waves make this feasible. A comparative series of measurements across increasing topological charge would clarify whether the vacuum's optical properties depend only on field strength, or also on **field topology**—connecting electromagnetic structure with curvature and polarization in a purely Maxwellian framework. ## References 1. M. R. Dennis et al., *Nature* 426 (2003). 2. K. O'Holleran et al., *New J. Phys.* 12 (2010). 3. J. Zhong and Q. Zhan, *Nat. Commun.* 15 (2024). 4. "Vortex cannon generates toroidal electromagnetic pulses," *Physics World* (2024). 5. M. Robertson et al., "Experiment to observe an optically induced change of the vacuum index," *Phys. Rev. A* 103, 023524 (2021).