--- title: "Experimental Proposal -- Confirmation of a Dielectric Longitudinal Delay of a Bright Interference Fringe" subtitle: "A dielectric-first derivation and two experimental tests in a Mach-Zehnder interferometer" author: "An M. Rodriguez , Leo Marchetti " date: 2026-04-23 one-sentence-summary: "At Mach-Zehnder recombination each arm beam is the in-phase electromagnetic response to the other, so the dielectric slowing mechanism applies directly and the bright fringe propagates at c/2." summary: "Electromagnetic propagation in a dielectric slows because the medium's polarization response is an in-phase electromagnetic wave that loads the effective permittivity and permeability. At Mach-Zehnder recombination the second arm beam plays that role: both arms originate from the same coherent source and arrive in phase, making each beam the full-amplitude in-phase response to the other (k=1), which gives c_eff = c/2 by the standard dielectric formula. The ordinary output reading takes the routed output beam and predicts no delay. Two experiments discriminate the readings: refraction of the isolated bright fringe at a glass boundary — where the loaded reading predicts total internal reflection above the critical angle sin(theta_c) = n_g/2 ~ 0.75 — and time-of-flight along a propagation path." keywords: - interference - Mach-Zehnder interferometer - dielectric slowing - dielectric longitudinal delay - constructive interference - energy density - bright interference fringe - refraction - Snell's law - total internal reflection - time-of-flight - speed of light doi: https://writing.preferredframe.com/doi/10.5281/zenodo.19717816 --- ## Abstract A dielectric slows light because the medium responds to the incident electromagnetic wave with an in-phase polarization wave. Maxwell's equations then describe the combined field — incident plus response — propagating at the reduced speed $$ c_{\mathrm{eff}}=\frac{1}{\sqrt{\varepsilon_{\mathrm{eff}}\mu_{\mathrm{eff}}}} =\frac{c}{n}. $$ A dielectric is, at bottom, a coherent recombiner: one in-phase electromagnetic wave riding alongside another. At Mach-Zehnder recombination, the second arm beam is that in-phase response. Both arms originate from the same coherent source, travel equal paths, and arrive in phase at a bright point. Each beam is, for the other, the in-phase electromagnetic addition that constitutes the dielectric loading. This is not an analogy to the dielectric case — it is the dielectric mechanism, with the second arm supplying the response sector instead of the medium. At equal-beam recombination, the arm amplitudes are equal, so each arm is the full-amplitude in-phase response to the other: $k=1$. The dielectric formula then gives directly $$ c_{\mathrm{eff}}=\frac{c}{2}. $$ The ordinary reading takes the routed output beam and predicts no delay. The experiment is a direct test between these two readings, and it reduces — in the refraction version — to a binary outcome near the critical angle $\theta_c\approx 48.6^\circ$. ## Introduction In a linear dielectric, an incident electromagnetic wave induces an in-phase polarization response. Maxwell's equations, applied to the combined field of incident wave plus response, yield the reduced propagation speed $c/n$. The dielectric index encodes, at bottom, the presence of a second in-phase electromagnetic wave riding alongside the first. This observation generalizes beyond bulk media. Any configuration that places a second coherent in-phase electromagnetic wave alongside a probe wave should produce the same reduced propagation speed, by the same Maxwell derivation. A Mach-Zehnder interferometer at its recombination point is precisely such a configuration. Both arm beams originate from the same coherent source, travel equal paths, and arrive in phase at a bright point. Each beam is, from the other's perspective, a full-amplitude in-phase electromagnetic response. The mathematical structure matches the dielectric case exactly, with $k=1$, so the reduced speed is $c_{\mathrm{eff}}=c/2$. In Dirac's framing of the superposition principle — *each photon then interferes only with itself* — the two arm beams are two paths of the same coherent state. The loading one arm imposes on the other is therefore self-interference in the strict sense: the photon encountering its own amplitude. The present proposal tests whether that self-interference carries a measurable phase-velocity signature. The ordinary output-mode analysis disagrees. It normalizes the recombined field through the $1/\sqrt 2$ routing factor and treats the bright port as a single beam propagating at $c$. This paper derives the loaded-fringe prediction, frames the disagreement as a direct experimental fork, and proposes two tests: refraction at a glass boundary (geometric) and time-of-flight along a propagation path (temporal). The refraction test reduces the discrimination to a binary outcome near the critical angle $\theta_c\approx 48.6^\circ$, where the loaded reading predicts total internal reflection and the ordinary reading predicts standard transmission. The logic is one-sided. Constructive interference yields a denser in-phase combined field; destructive interference depletes the field and, in the dark-fringe limit, cancels it rather than producing anything faster. Only the bright-fringe direction of the fork carries a substantive prediction. ## Theory ### The Dielectric Mechanism Consider a region in which an electromagnetic probe wave $(\mathbf E_1,\mathbf H_1)$ is accompanied by an in-phase response wave with amplitude ratio $k\ge 0$, $$ \mathbf E_2=k\,\mathbf E_1, \qquad \mathbf H_2=k\,\mathbf H_1. $$ The sum fields enter Maxwell's equations through the constitutive relations $$ \mathbf D=\varepsilon_0(\mathbf E_1+\mathbf E_2)=\varepsilon_0(1+k)\,\mathbf E_1 \equiv\varepsilon_{\mathrm{eff}}\,\mathbf E_1, $$ $$ \mathbf B=\mu_0(\mathbf H_1+\mathbf H_2)=\mu_0(1+k)\,\mathbf H_1 \equiv\mu_{\mathrm{eff}}\,\mathbf H_1. $$ In a source-free region, $$ \nabla\times\mathbf E_1=-\frac{\partial\mathbf B}{\partial t} =-\mu_{\mathrm{eff}}\frac{\partial\mathbf H_1}{\partial t}, $$ $$ \nabla\times\mathbf H_1=\frac{\partial\mathbf D}{\partial t} =\varepsilon_{\mathrm{eff}}\frac{\partial\mathbf E_1}{\partial t}. $$ Taking the curl of the first equation and using $\nabla\cdot\mathbf E_1=0$ gives the wave equation $$ \nabla^2\mathbf E_1 -\varepsilon_{\mathrm{eff}}\mu_{\mathrm{eff}} \frac{\partial^2\mathbf E_1}{\partial t^2}=0, $$ so the combined field propagates at $$ c_{\mathrm{eff}} =\frac{1}{\sqrt{\varepsilon_{\mathrm{eff}}\mu_{\mathrm{eff}}}} =\frac{1}{\sqrt{\varepsilon_0\mu_0(1+k)^2}} =\frac{c}{1+k}. $$ This result depends only on the existence of an in-phase electromagnetic response with amplitude ratio $k$. It does not depend on the physical origin of that response. ### Two Physical Realizations **Linear dielectric.** In a transparent linear dielectric, $\mathbf E_2$ is the electromagnetic field of the medium's polarization response, and the amplitude ratio is the electric susceptibility, $k=\chi_e$ (analogously $\chi_m$ for the magnetic response). The standard reduced-speed formula $c/n$ follows with $n=\sqrt{(1+\chi_e)(1+\chi_m)}$. **Mach-Zehnder recombination.** At the recombination point of a Mach-Zehnder interferometer, $\mathbf E_2$ is the second arm beam. Both arms originate from the same coherent source, travel equal paths, and arrive in phase at a bright point. Each beam is, from the other's perspective, a full-amplitude in-phase electromagnetic response. At full constructive interference $\mathbf E_2=\mathbf E_1$, so $k=1$ without further substitution, and $$ c_{\mathrm{eff}}=\frac{c}{2}. $$ The physical realizations differ — medium polarization versus free-propagating beam — but the mathematical structure, and therefore the predicted propagation speed, is identical. The dielectric result is not transferred by analogy; it applies directly, because the mechanism is the same. ### Why the Split Phase Is Different {#sec:split} The split and recombination use the same physical element (a 50/50 beam splitter) but are not the same operation. The split takes one beam and produces two equal beams from it. Its purpose is to prepare coherent arm beams; no loaded interference fringe is formed. Recombination takes two coherent equal beams and concentrates them into a single signal distributed across two output channels. The bright channel receives the constructive-interference fringe; the dark channel receives nothing. Together they account for the full input energy — the fringe profile $\cos^2+\sin^2=1$ sums to unity. The dielectric loading question belongs to recombination, where two in-phase equal beams combine, not to the split. ## Proposed Experiments ### The Two Readings Each arm carries amplitude $E_0$ (energy density $u$). At recombination the two coherent equal beams combine: the bright fringe has amplitude $2E_0$ and energy density $4u$; the dark fringe has $0$. The fringe profile $\cos^2+\sin^2=1$ distributes the full input energy across the two output channels. The dielectric loading applies to the combined field at the bright fringe. With $k=1$ the dielectric result gives $c_{\mathrm{eff}}=c/2$ (see @sec:energy-flux for the full energy and routing accounting). The two readings differ in the phase velocity assigned to the bright fringe: - ordinary output reading: $v_{\mathrm{bright}}=c$ - loaded-fringe reading: $v_{\mathrm{bright}}=c/2$ Two experiments can probe this phase velocity: refraction at a glass boundary (geometric) and time-of-flight along a propagation path (temporal). Both access the same underlying wavevector magnitude $|k|=n_{\mathrm{eff}}\,\omega/c$ in the overlap region; they are not independent confirmations but complementary observation channels. ### Refraction Test The simplest test to discriminate the two readings is geometric. Snell's law at a boundary between two media, $$ n_1\sin\theta_i=n_2\sin\theta_r, $$ is tangential-wavevector conservation: $|k|_{\mathrm{tangential}}$ is preserved at the boundary, and $|k|=n\,\omega/c$. The refraction angle therefore reads off the wavevector magnitude of the incident wave. The testable content of the $c_{\mathrm{eff}}=c/2$ claim is that the combined field at the bright fringe carries $|k|=2\omega/c$ in the overlap region, which at a boundary with glass of index $n_g$ bends the beam to $$ \sin\theta_r=\frac{2}{n_g}\sin\theta_i, $$ twice the ordinary prediction. **Setup.** Arrange the Mach-Zehnder so the two arm beams are collinear at the recombiner output. Isolate one bright fringe with an aperture and let it propagate toward a glass slab ($n_g\approx 1.5$) at oblique incidence $\theta_i$. A reference beam taken directly from the laser is sent to the same slab at the same $\theta_i$ for standard-refraction comparison. The two arm beams remain spatially coincident within the apertured beam, so each is still the in-phase response to the other and the dielectric loading argument persists as long as they propagate together. **Predictions.** - *Ordinary reading* ($n_{\mathrm{eff}}=1$): standard refraction, identical to the reference, $\sin\theta_r=\sin\theta_i/n_g$. - *Loaded-fringe reading* ($n_{\mathrm{eff}}=2$): $\sin\theta_r=(2/n_g)\sin\theta_i=(4/3)\sin\theta_i$. For the loaded reading a critical angle appears at $$ \sin\theta_c=\frac{n_g}{n_{\mathrm{eff}}}=0.75, \qquad \theta_c\approx 48.6^\circ. $$ Above that incidence angle the loaded reading predicts total internal reflection — no transmitted beam — while the ordinary reading still predicts standard transmission. At $\theta_i\gtrsim 49^\circ$ the experiment reduces to a binary discriminator: either the bright fringe transmits into the glass or it does not. No timing measurement is required. ### Time-of-Flight Test If the refraction test is positive, a direct temporal confirmation is to propagate the fringe and measure its group delay. Use a coherent source, modulate it, split it into two arms, and recombine the arms so they form stable fringes. Then: 1. isolate one bright interference fringe with an aperture 2. propagate that selected fringe over distance $L$ 3. propagate a matched reference beam over the same distance 4. compare delay slopes $d\tau/dL$ The ordinary reading predicts equal slopes. The loaded-fringe reading predicts a larger slope for the bright fringe. For the equal-beam limit, $$ \tau_{\mathrm{bright}}-\tau_{\mathrm{ref}}\approx \frac{L}{c}. $$ So at $1\,\mathrm{m}$ the extra delay is about $3.34\,\mathrm{ns}$, and at $10\,\mathrm{m}$ it is about $33.4\,\mathrm{ns}$. ## Discussion: Energy and Flux Accounting {#sec:energy-flux} The dielectric argument above establishes $c_{\mathrm{eff}}=c/2$ from the loading structure alone. The following energy and flux calculations are consistency checks, not the primary argument. Throughout this document $u$ denotes energy density (J/m³), not intensity (W/m²); the two are related by $I=u\,v$ where $v$ is the propagation speed, and they differ between the two readings. **Energy density at the bright center.** With $k=1$ and arm amplitude $E_0$ (energy density $u$), $$ \mathbf E_{\mathrm{tot}}=2E_0, \qquad \mathbf H_{\mathrm{tot}}=2H_0, $$ and the instantaneous energy density is $$ u_{\mathrm{tot}}=4u. $$ This is twice the input laser energy density $2u$ and four times each arm's energy density $u$. Across the fringe profile, $$ u(x)=4u\cos^2\!\left(\frac{\Delta\phi(x)}{2}\right), $$ averaging to $2u$ over a full fringe period. The dark fringe carries $0$, so the spatial redistribution accounts for the full input energy. **Instantaneous derivation.** No time averaging is needed. At a full constructive-interference point, $\mathbf E_1(t)=\mathbf E_2(t)=E_0(t)$ and $\mathbf B_1(t)=\mathbf B_2(t)=B_0(t)$, so $\mathbf E_{\mathrm{tot}}(t)=2E_0(t)$, $\mathbf B_{\mathrm{tot}}(t)=2B_0(t)$, and $$ u_{\mathrm{tot}}(t) = \frac{\varepsilon_0}{2}\lvert \mathbf E_{\mathrm{tot}}(t)\rvert^2 + \frac{1}{2\mu_0}\lvert \mathbf B_{\mathrm{tot}}(t)\rvert^2 =4u. $$ The factor of four is instantaneous and exact: amplitude doubles, energy density quadruples. **Output routing.** The recombiner maps the overlap into two output spatial modes. For a lossless 50/50 recombiner, $$ \mathbf E_+=\frac{\mathbf E_1+\mathbf E_2}{\sqrt 2}=\sqrt{2}\,E_0, \qquad \mathbf E_-=\frac{\mathbf E_1-\mathbf E_2}{\sqrt 2}=0, $$ and likewise for the magnetic fields. The $1/\sqrt 2$ routing factor, combined with the $1/\sqrt 2$ that already reduced the arm amplitudes at the initial split, returns the full input energy to the bright port: $$ u_+=2u, \qquad u_-=0. $$ The ordinary reading starts from this $2u$ output and finds no anomalous refraction or delay. The loaded-fringe reading starts from the $4u$ raw overlap and predicts $c/2$. These energy-accounting relations are consistent with both readings; they do not by themselves decide which propagation speed is physical. That discrimination is what the refraction and time-of-flight experiments provide. ## Conclusion The experiments test which object should be treated as the propagating fringe after recombination: - the ordinary normalized output mode, with $n_{\mathrm{eff}}=1$ - or the isolated raw constructive-interference fringe, with $n_{\mathrm{eff}}=2$ For the refraction test, if the bright fringe transmits into the glass at $\theta_i\gtrsim 49^\circ$ together with the reference, the ordinary reading wins. If the bright fringe undergoes total internal reflection while the reference still transmits, the loaded-fringe reading is supported. For the time-of-flight test, if the measured delay matches the reference, the ordinary reading wins. If the delay approaches the $c/2$ prediction in the equal-beam limit, the loaded-fringe reading is supported. ## Acknowledgments We thank Celso L. Ladera of Universidad Simón Bolívar, Caracas, for introducing one of us (A.M.R.) to Dirac's dictum on self-interference during his undergraduate optics course. ## References 1. Jackson, J. D. (1999). *Classical Electrodynamics*, 3rd ed. Wiley. Chapter 7 covers dielectric polarization, wave propagation in linear media, Snell's law, reflection and refraction at plane interfaces, and total internal reflection with evanescent decay — the full toolbox used in the derivation and in both proposed tests. 2. Hecht, E. (2017). *Optics*, 5th ed. Pearson. Standard undergraduate treatment of interference (including the Mach-Zehnder geometry), Snell's law, reflection and refraction, and total internal reflection with evanescent waves. 3. Dirac, P. A. M. (1958). *The Principles of Quantum Mechanics*, 4th ed. Oxford University Press. §I.3 on the superposition principle: "each photon then interferes only with itself." 4. Rodriguez, A. M. (2026). *Light Speed as an Emergent Property of Electromagnetic Superposition: Polarization Without Matter*. Preferred Frame. . Precursor derivation of the effective refractive index and local light speed from coherent superposition of free electromagnetic fields.