# Daily variations of the amplitude of the fringe shifts observed when an air-glass Mach-Zehnder type interferometer is rotated

Robert Te Winkel, An M Rodriguez

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Robert Te Winkel, An M Rodriguez. Daily variations of the amplitude of the fringe shifts observed when an air-glass Mach-Zehnder type interferometer is rotated: a preliminary report. 2023. hal-04045807

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# Daily variations of the amplitude of the fringe shifts observed when an air-glass Mach-Zehnder type interferometer is rotated - a preliminary report

Robert te Winkel* and An M. Rodriguez†

Universidad Simón Bolívar
Caracas, Venezuela

(Dated: March 25, 2023)

We study the daily variations of the amplitude of the fringe shifts observed in the interference pattern of a Mach-Zehnder-type interferometer in which one arm is traversed in air while the other traverses a glass slab. The study is based on data collected over a period of years, with digital pictures of the interference pattern taken every 3 seconds, while the interferometer is being slowly rotated (120°/min) around an axis perpendicular to the ground. Fringe shifts in the interference pattern are observed and are correlated to the angle of rotation. Furthermore, we show that there is a clear, measurable daily variation in the amplitude of these fringe shifts. The frequency of the daily variation seems correlated with the sidereal day. We discuss our interpretation of these variations as well as propose a theoretical framework to understand them. Further independent studies are required to confirm our findings and to explore their implications.

## I. INTRODUCTION

Interferometers are widely used in precision measurements of the speed of light and other physical quantities. So-called one-way interferometers are interferometers in which light only travels in one direction from the source to the receiver. The interference pattern is affected by various factors that can cause a shift in the interference fringes.

In our experiment, light travels through air in one of the arms of the interferometer while on the other arm light travels through a glass slab of refractive index $n \approx 1.6$, and length 0.6m.

In this study, we focus on the daily variations of the amplitude of the fringe shifts observed when the interferometer is rotated. The amplitude of the fringe shift is correlated to the angle of rotation of the interferometer.

We aimed to investigate if whether there were any fringe shifts upon rotation, and if the amplitude of the fringe shifts would vary over time with a sidereal period. Any signal not meeting these requirements would have to be interpreted as noise.

We find that, additionally, different amplitudes of the fringe shifts are predictably expected for different configurations of the interferometer: air-glass, air-air, glass-glass, air-2xglass [1]

## II. METHODOLOGY

### A. Experimental setup

We conducted our study using a one-way interferometer at rest in a laboratory. A diagram of the experimental

<figure><img src="image_1.png" /><figcaption>FIG. 1. Diagram of experimental setup. Light is emitted by a HeNe laser seen in the middle left of the figure. The beam is divided by Beam Splitter 1 (BS1) into beams B1 and B2. B1 traverses air while being reflected by M1 and M2 to meet B2 at BS2. B2 traverses a glass slab $n=1.6$ and meets B1 in BS2, and both beams form an interference pattern that is registered by the camera. The interferometer is floating over water, and rotating slowly 120°/min while pictures are taken of the interference pattern every three seconds. This is the basic, initial configuration of the apparatus. Multiple variations were also studied (air-air, glass-glass, air-2xglass [1]).</figcaption></figure>

set up can be seen in Fig. 1.

We used a non-frequency-stabilized HeNe laser (wavelength of 632.8nm) as the light source and a digital photographic camera as the receiver.

<footnote>* rtewinkelc@gmail.com</footnote>

<footnote>† anmichel.rodriguez@gmail.com</footnote>
## B. Digital pictures

The camera takes HD (720x1280p) digital photos of the interference pattern every 3 seconds while the interferometer is being slowly rotated with respect to an axis perpendicular to the ground.

The metadata of each photo contains the cardinal orientation of the camera as well as a timestamp, among other information. Since all elements of the experimental set-up are fixed relative to each other, the orientation of the camera reveals the cardinal orientation of the beam of light that comes out of the laser.

Using this information (fringe position in the picture, timestamp and cardinal orientation of the beam of light) for each picture, we are able to study correlations between the angle of rotation of the interferometer and the observed fringe shifts.

## C. Precision of the method

In each picture, fringes are usually resolved at a width of approximately 70px, so at the bare minimum we can resolve a fringe with $1.4\% = 1px/70px \times 100\%$

However, the statistical error can go as low as 0.1px.

As seen in Figure 1, in one of the arms of the interferometer (B2, aka Beam 2, in the diagram) light traverses a glass slab ($n \approx 1.6$, length=0.6m). In the other arm (B1 in the diagram) light traverses air.

We collected data, mostly continuously, since Aug-2018 to present. Pictures of the interference pattern were taken every 3s for different configurations of the interferometer (air-glass, as described above, we also measured other configuration like air-air, glass-glass and air-2xglass [1]).

The analysis starts with a typical picture like Fig. 2.

For each picture, we identify the position of each interference maxima (see Fig. 3). Notice the red vertical bars that indicate rotations of the interferometer.

We then identify the position of only one fringe, and separate into a series of rotations. A plot of individual rotation can be seen in Fig. 4. Notice that in this figure the scale on the vertical axis is still in pixels so we can still appreciate the size of the fringe in pixels.

We finally interpolate the data points to obtain 360 points (one point per degree of rotation) that we can easily average to obtain a figure like Fig. 5.

This way we are able to estimate the amplitude of the fluctuation upon rotations of the interferometer. This is obviously a very rough estimate as it ignores other seemingly important information like the angular positions of the maxima and the shape of the fringe shift profile.

We work with a stream of pictures of 30min in length (around 600 pictures) to take the average. That is why in figures such as Fig. 6 there is a point every 30 minutes.

<figure><img src="image_2.png" /><figcaption>FIG. 2. typical picture of the interference pattern.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIG. 3. Easily seen fringe shifts upon rotation. Rotations are indicated with red vertical lines. Blue line shows the altitude (right vertical axis) of HIP54589 as seen in the sky of Caracas, Venezuela at date of measurement.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 4. The plot shows the different rotations of a single fringe during the 30minute window (approximately 10 successive rotations). Notice that the vertical axis is not normalized by the fringe width, so the actual size of the fringe in pixels can be seen. The horizontal blue line shows the altitude (right vertical axis) of the star HIP54589 as seen in the sky of Caracas, Venezuela at date of measurement.</figcaption></figure>
<figure><img src="image_5.png" /></figure>

FIG. 5. 30min. average of the shift of a fringe of the interfer-
ence pattern (≈ 10 rotations, meaning it's an average of ≈ 10
points per angle), shown per angle of rotation. Fringe shift
is significantly higher than statistical error bars. Amplitude
vs Angle profile is very similar (yet inverted) to the predicted
value shown in Fig. The horizontal blue line shows the alti-
tude (right vertical axis) of the star HIP54589 as seen in the
sky of Caracas, Venezuela at date of measurement.

<figure><img src="image_6.png" /></figure>

FIG. 6. daily-variation-2020-02-20. The blue line is a vi-
sual indicator of the sidereal day frequency, by plotting the
altitude of the star HIP54589 as seen in the sky of Caracas,
Venezuela at the time of measurement. Orange dots represent
the amplitude of the fringe shift, calculated by averaging 10
rotations of the interferometer and then taking the difference
maximum-minimum of this average.

<figure><img src="image_7.png" /></figure>

FIG. 7. daily-variation-2020-05-20. The blue line is a visual indicator of the sidereal day frequency, by plotting the altitude of a fixed galactic coordinate. Orange dots represent the amplitude of the fringe shift, calculated by averaging 10 rotations of the interferometer and then taking maximum-minimum.

<figure><img src="image_8.png" /></figure>

FIG. 8. daily-variation-2020-11-19. The blue line is a visual indicator of the sidereal day frequency, by plotting the altitude of a fixed galactic coordinate. Orange dots represent the amplitude of the fringe shift, calculated by averaging 10 rotations of the interferometer and then taking maximum-minimum.

<figure><img src="image_9.png" /></figure>

FIG. 9. In this configuration of the interferometers there are no obvious visible oscillations correlated to rotation like in previous figures. Bottom blue line seems to be an error of the plot, please disregard.

III. RESULTS

A. Amplitude of fringe shifts and their daily variation

Figures like Fig. 4 show the correlation between the
amplitude of the fringe shifts and the angle of rotation.
Figures like 6, 7 and 8 show how this amplitude varies
over time.

Furthermore, it can be seen that in the three figures
6, 7 and 8, there is a strong correlation between the blue
line (visual indicator of the sidereal day frequency) and
the measurements, indicating a strong correlation with
the sidereal day.

Figures 6, 7, and 8 are some examples of daily varia-
tions of data taken during 2020. The orange dots rep-
resent the amplitude of the fringe movement, while the
blue dots are a visual indicator of the sidereal day - more
precisely, the altitude of the star HIP54589 as seen in the
sky of Caracas, Venezuela.

The daily variations, on the other hand, were harder to
<figure><img src="image_10.png" /><figcaption>FIG. 10. In this configuration of the interferometer the average of rotations is almost under the error bars but some oscillation could be easily speculated as looking similar to Fig. 13 but on a much smaller amplitude [2].</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>FIG. 11. In this configuration of the interferometer we can see that in the individual rotations of a single fringe there are no visible oscillations correlated to rotation - at least without further, deeper analysis.</figcaption></figure>

find since we didn't expect them to be so small sometimes (for example, 0.1 fringes, February 2020), and because of natural, expected noise in the data.

These results are in stark contrast with measurements in other configurations of the interferometer. See for instance Fig. 9.

In the figure showing the average of rotations, Fig. 10, and the independent rotations of the fringe as seen in Fig. 11 movement along each independent rotation shown in Fig. ?? in this same configuration as in Fig. 9.

These alternate configuration will be reported more in depth in future, more comprehensive, reports.

Our results show not only rotational effects (fringe shifts) correlated to the angle of rotation of the interferometer - which is interesting enough. We further show that the amplitude of these fringe shifts varies with a period strongly correlated with the Sidereal Day.

We interpret our measurements as a detection of the movement of Earth through the medium that propagates light. Our interpretation is in line with Fresnel Drag effect and the Lorentzian interpretation of the Michelson-Morley experiment.

<figure><img src="image_12.png" /><figcaption>FIG. 12. Consider a glass slab (blue rectangle) that has velocity $v$ in some frame of reference and moves a time $t$ toward the black rectangle's position (for simplicity, in the same plane). For clarity in the figure angles and distances appear somewhat out of proportion. The beam of light enters the glass in $O$, with an angle $\alpha$ with respect to the normal. The light beam refracts with angle $\alpha'$ and if not by the dragging effect the beam would emerge in $B$. However, since the glass slab drags the beam along its movement with velocity $v$, the beam emerges in $D$. We need to consider the contraction of the glass in the direction of $v$, as well as the effect of Fresnel's Drag in the same direction. For us, the key insight in deriving the equations is that the dragging itself shouldn't affect light's propagation time inside the glass, so the time $OD$ is the same as the time $OB$, and thus Pythagoras' theorem can be applied for known quantities.</figcaption></figure>

## IV. THEORETICAL FRAMEWORK

As seen in Fig. 12, consider a glass slab (blue rectangle) that has velocity $v$ in some frame of reference and moves a time $t$ toward the black rectangle's position, in the same plane. The beam of light enters the glass in $O$, with an angle $\alpha$ with respect to normal. The light beam refracts with angle $\alpha'$ and if not by the dragging effect the beam would emerge in $B$. However, since the glass slab drags the beam along its movement with velocity $v$, the beam emerges in $D$. We need to consider the contraction of the glass in the direction of $v$, as well as the dragging effect in the same direction.

For us, the key insight in deriving the equations is that the dragging itself shouldn't affect light's propagation time inside the glass, so the time $OD$ is the same as the time $OB$, and thus Pythagoras' theorem can be applied for known quantities.

Also, notice that $OC \parallel E1E2$. Which is useful for calculating the time of flight along that path for the final time difference ($time_{glass} - time_{air}$, for a given angle of rotation $\theta$ of the apparatus).

Showing here the deduction of the equation for the propagation time inside and outside the glass slab (and
so to predict the number $N(\theta, v)$ of fringe shifts that are seen), is beyond the scope of this report and so will be left as an exercise to the reader.

The general idea is that after considering the dragging effects on light by the moving medium, as well as the length contraction effects due to our motion relative to the medium that propagates light.

Denoting by $n$ the refractive index of the glass, $L$ the contracted length of the glass traversed by the beam of light, $\theta$ the angle of rotation, $\theta'$ is the angle of rotation corrected by light's refraction inside the glass; and importantly, $v$ the speed of movement of the laboratory relative to the medium that propagates light, we arrive at:

$$t(v, \theta) = \frac{2Lv \cos(\theta') + R(v, \theta)}{2(V_n^2 - v^2 \cos^2(\theta') - (vsin(\theta)/n^2)^2)} \quad (1)$$

Where we have abbreviated:

$$R(v, \theta) = \sqrt{(2Lv \cos(\theta'))^2 + 4L^2(V_n^2 - v^2 \cos^2(\theta')) - (vsin(\theta)/n^2)^2} \quad (2)$$

Also, denoting $\theta' = \theta - \alpha'$, we have that,

$$\alpha' = \arctan \left( \frac{(vsin(\theta)/n^2)}{[c/n + v \cos(\theta')(1 - 1/n^2 - dn/d\lambda)]} \right) \quad (3)$$

and the speed of light in the medium is given by

$$V_n = \frac{c}{n} + vcos(\theta')(1 - \frac{1}{n^2} - \frac{\lambda}{n} \frac{dn}{d\lambda}) \quad (4)$$

Notice that if $n=1$ then $V_n = c$.

$\frac{dn}{d\lambda}$ is derived from tables with experimental characteristics of the material [3].

The term $-\frac{\lambda}{n} \frac{dn}{d\lambda}$ was introduced by Lorentz[4] to account for the change in wavelength introduced by the dragging of the light by the moving medium.

To calculate $N(n,v)$ - the amplitude of the expected fringe movement given in number of fringes, $N$, for a value of $v$, for we can use the formula:

$$N(\theta) = \frac{c}{\lambda} \left[ \begin{array}{l} (t_g^{\theta_2} - t_v^{\theta_2} - t_{v\perp}^{\theta_2+90^\circ} - t_{v\perp}^{\theta_2-90^\circ}) \\ - (t_g^{\theta_1} - t_v^{\theta_1} - t_{v\perp}^{\theta_1+90^\circ} - t_{v\perp}^{\theta_1-90^\circ}) \end{array} \right] \quad (5)$$

Where we have used superscripts to indicate the angle of rotation and lowerscripts to indicate if the time $t$ corresponds to an optical path through glass (g) or vacuum (v). (For simplicity, here we are simplifying the equations by using $n=1$ for air, instead of its real value.)

We can see a plot of the expected fringe shift profile in Fig. 13.

<figure><img src="image_13.png" /><figcaption>FIG. 13. Simulation of the expected fringe shift profile vs angle of rotation. In this simulation we used $v = 368000 \text{km/s}$. It is not adjusted for any other factors like latitude might have. The profile is very similar, yet inverted, to Fig. 5</figcaption></figure>

## V. DISCUSSION

### A. Meaning of the results

The amplitude of the measured effect is in line with our predictions.

We expected to measure daily variations in the amplitude of the fringe shifts observed upon rotation (because of Earth's rotation and hence our perspective of the sky), but we expected a greater amplitude in the intraday variation of the amplitude of these fringe shifts.

Since mainstream, accepted physics predict no fringe shifts (other than caused by noise) for an interferometer at rest in the laboratory, we are forced to find alternate explanations for our measurements of fringe shifts in correlation with the angle of rotation and, importantly, in daily variations of this amplitude seemingly in correlation with the sidereal day.

One such alternate explanation is accepting the existence of a medium that propagates light, or perhaps equivalently that vacuum is indeed 'something' we can move relative to (much like absolute space).

If we can move relative to such a medium, then this would explain our results by means of Eq. 1. This equation is in line with the Lorentzian interpretation of the Michelson-Morley experiment, as well as the theory of Fresnel Drag (please refer to the Suggested Reading section).

It has been shown recently that the existence of a 'modern' 'ether' doesn't necessarily contradict the General Theory of Relativity [5].

### B. Rotational effects

As evidenced by the figure in the Results section, there is a correlation between the amplitude of the fringe shifts and the angle of rotation. This rotational effect can be
explained by the aforementioned assumption.

Further, the presence of daily variations with seem-
ingly sidereal period, puts a high bar in trying to discard
our results as noise.

C. Small daily variations

Initially, our hypothesis was that any detected move-
ment would likely be aligned with the CMBR dipole.
However, given the observed daily variations of 0.1-0.3
fringes, it is unlikely that our motion is directed towards
the CMBR (or perhaps some geometric effect of Earth's
axis orientation). If our velocity were directed towards
the CMBR dipole, we would expect a greater decrease in
amplitude in accordance with the trajectory of the galac-
tic coordinates as seen in the sky ([6]).

While we cannot rule out this possibility, it seems un-
likely. The detection of daily variations as small as 0.1
fringes (on fringe shifts of 0.6 fringes per rotation) sug-
gests a relatively small angle of the sky where our velocity
could be pointing to ([7]).

D. Sidereal day frequency

The period of the daily variations is seemingly corre-
lated with the sidereal day. This has to be studied in
much more detail. More data and other analysis are yet
to be done by the authors.

There are some yet unexplained phase differences, but
it can be argued that the correlation holds.

VI. CONCLUSIONS

In our investigation, we measured the amplitude of the
fringe shifts observed upon rotation of the interferome-
ter. We interpret these as being caused by the Earth's
rotation and our perspective of the sky.

Once we detected the fringe shifts on rotation, we nat-
urally expected to measure daily variations in the ampli-
tude of these fringe shifts, because of Earth's rotation.

Given the trayectory of HIP54589 as seen in the sky of
Caracas (which crosses the horizon and mostly passes
through the cenit/nadir), we expected to measure a
greater amplitude of the daily variation than what we
ended up measuring.

However relatively small (as compared to our expec-
tations) the amplitude of the daily variations was mea-
sured, as expected. In the opinion of the authors, this
indicates that the measurements obtained can be hardly
discarded as noise. We obviously remain open to this
possibility.

We initially hypothesized that any detected movement
would likely be towards the CMBR dipole, but the ob-
served daily variations of 0.1-0.3 fringes seem to suggest
otherwise. We also observed unexplained phase changes
between the period of the daily variations and the side-
real day.

We have discussed potential causes for these daily vari-
ations. However, further research is necessary to confirm
these findings and investigate their broader implications
in physics.

These results are sure to provide new insights into our
understanding of the universe.

VII. SUGGESTED REFERENCES

... To be updated ...

ACKNOWLEDGMENTS

This draft manuscript and research wouldn't have been
possible without the hard-work and collaboration of Dr.
E.D. Greaves - both as a researcher and as a dear pro-
fessor of one of the co-authors. We also acknowledge the
unwavering support, for so many years, from the USB
(thanks again to Dr. Greaves and all the leads and col-
laborators of the students' optics lab), for supporting
my education and our fundamental research into the fron-
tiers of what we know. ...

<footnote>[1] Authors, (), the light beam traverses twice the optical path in glass, in the same direction, giving double the effect. Usually this means 1.2m. (glass slab is 60cm).</footnote>

<footnote>[2] Authors, (), please refer to Dr. Greaves for information about specifics about the configuration on this date. This will be detailed in a subsequent report.</footnote>

<footnote>[3] E. D. Greaves, Private study of books and manuals (2018).</footnote>

<footnote>[4] R. te Winkel, *Please contact Robert te Winkel for references on the Lorentz dn/dL term* (2018).</footnote>

<footnote>[5] F. J. T. Maurice J. Dupre, General relativity as an aether theory, arXiv:1007.4572 (2010), 'Most early twentieth cen-</footnote>

<footnote>tury relativists — Lorentz, Einstein, Eddington, for exam-
ples — claimed that general relativity was merely a theory
of the aether.'.</footnote>

<footnote>[6] The galactic coordinates is seen traversing the horizon and almost 80° altitude.</footnote>

<footnote>[7] For example, if our movment were towards the Polar star,
the wouldn't be any visible daily variation.</footnote>