Experimental Proposal -- Confirmation of a Dielectric Longitudinal Delay of a Bright Interference Fringe

A dielectric-first derivation and two experimental tests in a Mach-Zehnder interferometer

An M. Rodriguez <an@preferredframe.com>

Leo Marchetti <leo@preferredframe.com>

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.

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

1 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 $\theta_c\approx 48.6^\circ$ .

2 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.

3 Theory

3.1 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.

3.2 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.

3.3 Why the Split Phase Is Different

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.

4 Proposed Experiments

4.1 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 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.

4.2 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.

4.3 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:

isolate one bright interference fringe with an aperture
propagate that selected fringe over distance $L$
propagate a matched reference beam over the same distance
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 $33.4\,\mathrm{ns}$ .

5 Discussion: Energy and Flux Accounting

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.

6 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.

7 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.

8 References

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.
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.
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.”
Rodriguez, A. M. (2026). Light Speed as an Emergent Property of Electromagnetic Superposition: Polarization Without Matter. Preferred Frame. https://writing.preferredframe.com/doi/10.5281/zenodo.18396637. Precursor derivation of the effective refractive index and local light speed from coherent superposition of free electromagnetic fields.