---
title: Experimental Proposal - Longitudinal Delay of an Isolated Interference Fringe
date: 2026-04-16
---

# Experimental Proposal

## Goal

Measure whether the center of a bright interference fringe propagates with a
different longitudinal delay than one of the incident beams that forms it.


## Rationale

The proposal tests a simple effective-advance law:

$$
J = u\,c_{\mathrm{eff}},
$$

where $J$ is the carried flux, $u$ is the recovered local density, and
$c_{\mathrm{eff}}$ is the effective longitudinal advance speed of the channel.

For one incident beam,

$$
J_0 = u\,c.
$$

For two equal incident beams, the total incoming budget is

$$
J_{\mathrm{in}} = 2u\,c.
$$

If coherent overlap recovers a bright-fringe center with local density $4u$,
then the same incoming budget concentrated into that denser channel gives

$$
c_{\mathrm{eff}}=\frac{J_{\mathrm{in}}}{4u}
=
\frac{2u\,c}{4u}
=
\frac{c}{2}.
$$

So the hypothesis is simple: coherent concentration into a denser bright
channel should produce a lower effective longitudinal speed, operationally
like a refractive slowdown.


## Complementary Outputs

Let the two equal fields arriving at the final recombination region be

$$
f_1(x,z,t)=A\,e^{i(kz-\omega t)}e^{+iqx/2},
\qquad
f_2(x,z,t)=A\,e^{i(kz-\omega t)}e^{-iqx/2},
$$

with

$$
u := |A|^2.
$$

At the final 50/50 beam splitter, the two output modes are

$$
f_+(x)=\frac{f_1(x)+f_2(x)}{\sqrt2},
\qquad
f_-(x)=\frac{f_1(x)-f_2(x)}{\sqrt2}.
$$

Therefore

$$
u_+(x)=2u\cos^2\!\left(\frac{qx}{2}\right),
\qquad
u_-(x)=2u\sin^2\!\left(\frac{qx}{2}\right).
$$

So the two outputs satisfy

$$
u_+(x)+u_-(x)=2u.
$$

A bright fringe on one output corresponds to a dark fringe on the other. This
is the basic $\cos^2+\sin^2=1$ structure of the two output branches.


## Raw Overlap Peak

Before output-mode normalization, the direct coherent overlap is

$$
f_{\mathrm{raw}} = f_1 + f_2
=
2A\cos\!\left(\frac{qx}{2}\right)e^{i(kz-\omega t)},
$$

so the raw overlap density is

$$
u_{\mathrm{raw}}(x)=4u\cos^2\!\left(\frac{qx}{2}\right).
$$

Therefore

$$
0 \le u_{\mathrm{raw}}(x) \le 4u.
$$

At a bright-fringe center

$$
x_n = \frac{2\pi n}{q},
$$

the loading reaches

$$
u_{\mathrm{raw}}(x_n)=4u.
$$

This is the local peak used in the $c_{\mathrm{eff}}$ estimate: two incident
channels of density $u$ can recover a bright-center loading of $4u$.


## Bright-Core Region

To isolate the strongest part of the fringe, require

$$
u_{\mathrm{raw}}(x) > 3u.
$$

With

$$
x=x_n+\Delta x,
$$

this becomes

$$
4u\cos^2\!\left(\frac{q\Delta x}{2}\right) > 3u,
$$

so

$$
\cos^2\!\left(\frac{q\Delta x}{2}\right) > \frac{3}{4}.
$$

Hence

$$
\left|\frac{q\Delta x}{2}\right| < \frac{\pi}{6},
$$

which gives

$$
|\Delta x| < \frac{\pi}{3q}.
$$

If the fringe period is

$$
\Lambda = \frac{2\pi}{q},
$$

then the bright-core condition is

$$
|\Delta x| < \frac{\Lambda}{6}.
$$

So the central stripe of full width

$$
\frac{\Lambda}{3}
$$

stays above $3u$.


## Crossing Angle and Fringe Width

If the two beams recombine with total crossing angle $\theta$, then the
transverse wave-number difference is

$$
q = 2k\sin\!\left(\frac{\theta}{2}\right),
\qquad
k=\frac{2\pi}{\lambda}.
$$

This is the small recombination-angle triangle: the opening angle between the
two rays sets the transverse phase gradient and therefore the fringe spacing.

Therefore the fringe period is

$$
\Lambda = \frac{2\pi}{q}
=
\frac{\lambda}{2\sin(\theta/2)}
\approx
\frac{\lambda}{\theta}
\quad
(\theta \ll 1).
$$

A $1\,\mathrm{m}$ Mach-Zehnder arm is practical, but the fringe width is set by
the recombination angle $\theta$, not by the arm length.

For a HeNe laser,

$$
\lambda = 632.8\,\mathrm{nm}.
$$

Choosing a fairly wide fringe with

$$
\theta = 0.2\,\mathrm{mrad},
$$

gives

$$
\Lambda \approx \frac{632.8\times 10^{-9}}{2\times 10^{-4}}
\approx
3.16\,\mathrm{mm}.
$$

Then the bright-core region above $3u$ has full width

$$
\frac{\Lambda}{3} \approx 1.05\,\mathrm{mm}.
$$

That is large enough for straightforward spatial isolation.

If the detector or entrance slit is centered on the fringe maximum and has
active width $a$, then the worst-case sampled loading is

$$
u_{\mathrm{edge}} = 4u\cos^2\!\left(\frac{\pi a}{2\Lambda}\right).
$$

For the same HeNe example:

- if $a=0.25\,\mathrm{mm}$, then $u_{\mathrm{edge}}\approx 3.94u$;
- if $a=0.50\,\mathrm{mm}$, then $u_{\mathrm{edge}}\approx 3.76u$.

So a practical target is a sensor or slit width in the range

$$
0.25\,\mathrm{mm} \;\text{to}\; 0.50\,\mathrm{mm},
$$

centered on the bright-fringe maximum.


## Measurement Model

1. Split a coherent laser beam into two equal beams.
2. Recombine them at a small angle in a Mach-Zehnder interferometer.
3. Use one output branch, where stable straight fringes are visible.
4. Spatially isolate the center of one bright fringe ridge.
5. Propagate that bright ridge over a known distance $L$.
6. In parallel, propagate one incident beam as a reference over the same
   distance.
7. Amplitude-modulate both channels from the same source.
8. For each channel, measure the delay $\tau$ relative to the common
   modulation signal on an oscilloscope or phase meter.

For each channel separately, collect data

$$
(L_1,\tau_1),\ (L_2,\tau_2),\ (L_3,\tau_3),\ \ldots
$$

and fit

$$
\tau(L)=mL+b.
$$

With the zero-length reference chosen appropriately, $b$ should be close to
zero. The local slope estimates are

$$
m_i = \frac{\delta \tau_i}{\delta L_i},
$$

and the regression returns the mean slope

$$
\langle m \rangle \approx \frac{d\tau}{dL}.
$$

Since

$$
\tau = \frac{L}{v},
$$

the speed is

$$
v = \frac{1}{\langle m \rangle}.
$$

This is done independently for the incident-beam reference and for the
isolated bright-fringe channel:

$$
v_{\mathrm{ref}} = \frac{1}{\langle m_{\mathrm{ref}} \rangle},
\qquad
v_{\mathrm{fringe}} = \frac{1}{\langle m_{\mathrm{fringe}} \rangle}.
$$


## Experimental Question

Does the isolated bright-fringe channel yield

$$
v_{\mathrm{fringe}} < v_{\mathrm{ref}} \; ?
$$

If yes, the bright fringe carries an additional longitudinal delay.  
If no, the fringe behaves like the reference beam within experimental error.


## Minimum Requirements

- stable coherent source
- Mach-Zehnder interferometer
- controlled small crossing angle at recombination
- stable fringe pattern
- spatial filter for one bright ridge
- common amplitude modulation source
- oscilloscope or phase-delay readout
- variable path length


## Summary

The proposal is simple:

1. create stable fringes in a Mach-Zehnder output,
2. isolate the center of one bright fringe ridge,
3. measure its delay for several lengths,
4. fit $\tau(L)$,
5. recover $v = 1/\langle m \rangle$,
6. compare that speed against one incident beam.

This gives a direct experimental test of whether the bright fringe channel
acquires an additional longitudinal delay.