**EPJ manuscript No.**

(will be inserted by the editor)

arXiv:0904.0229v2 [physics.gen-ph] 10 Apr 2009

BonPhysics B.V., Laan van Heemstede 38, 3297 AJ Puttershoek, The Netherlands, e-mail: victor@bonphysics.nl Version: April 2009

**Abstract. **Two optical fiber Mach–Zehnder interferometers were constructed in
an environment with a temperature stabilization of better than 1 mK per day.
One interferometer with a length of 12 m optical fiber in each arm with the main direction of the arms perpendicular to each other. Another with a length
of 2 m optical fiber in each arm where the main direction of the arms are
parallel as a control. In each arm 1 m of fiber was wound around a ring made of
piezo material enabling the control of the length of the arms by means of a
voltage. The influence of the temperature on the optical phase difference
between the interferometer arms was measured. It is attributed to the temperature
change induced variation of the interaction region of the optical fiber
couplers. Further, the influence of rotation of the interferometers at the
Earth surface on the observed phase differences was determined. For one
interferometer (with the long and perpendicular arms) it was found that the
phase difference depends on the azimuth of the interferometer.
For the other one (with the short and parallel arms) no relevant
dependence on the azimuth has been measured.

**PACS. **03.30.+p Special relativity - 06.30.-k Measurements common to
several branches of physics and astronomy - 42.25.Bs Wave propagation,
transmission and absorption

Since at the end of the 19th century methods became accurate
enough to measure the speed of light, experiments were devised to measure the
anisotropy of the speed of light at the Earth surface. This was sought to be
done by so-called first order experiments, where the effect depends in first
order on the ratio *v/c,* where v is the velocity of the observer with respect to a
preferred rest frame and c is the speed of light in this frame. When Fresnel
introduced his famous Fresnel drag coefficient it was believed that all
possible first order effects were compensated by an ether drag. Then Maxwell
[1] came along with the notion of second order experiments, where the effect
depends in second order on the same ratio. Although Maxwell thought at that
time it would be beyond any means of experimental method to measure a second
order effect, one year later in 1881 Michelson [2] devised an apparatus that
should be able to measure the change of the velocity very accurately. The
apparatus is now known as a Michelson–Morley interferometer. After some
comments on the experiment by Lorentz in 1886 [3] Michelson and Morley [4]
increased the sensitivity of the apparatus with almost a factor of ten
overcoming the accuracy objections of Lorentz. The accuracy of the apparatus
was further increased with a factor of 6 by Morley and Miller [5] and by Miller
in a series of experiments between 1905 and 1930 [6,7,8,9]. In all these
experiments the sought for magnitude of the effect was
never observed. This is satisfactorily explained by the Lorentz–Fitzgerald
contraction [10] or by Einsteins theory of relativity [11]. However, Miller in
his elaborate series of experiments, always claimed that he measured a small
second order effect and also a first order effect. The second order effects he
measured were quite small with respect to the sought for effect, but larger
than the experimental error. These second order effects were analysed by him
by combining measurements at different epochs. Combining the results from these
epochs and assuming the Sun moves relative to the preferred rest frame he was
able to find a preferred direction in space and a velocity. The first order
effect he measured depended very much on the detailed experimental settings and
were not analysed to find an anisotropy.

In
February 1927 a conference on the experiment and theoretical background was
held at the Mount Wilson Observatory [12]. This conference did not succeed in
finding a flaw in either experiment or theory, leaving the discrepancy in
tact. In 1955 Shankland, a former pupil of Miller, re-analysed Millers data
[13] and concluded that the second order effects do exist and remarks that
they *remain essential
constant in phase and amplitude through periods of several hours and are then
associated with a constant temperature pattern in the observation hut.*
Assuming that during several hours the second order effect should change
considerably, he then concludes that there is no second order effect and
contributes any other changes to temperature effects. However, it was already shown by Miller [9] that
the changes during several hours could be very small depending on the sidereal
time and the epoch. Hence, the conclusion of Shankland is unsupported and the
discrepancy between Millers results and theoretical expectations remains.

At the end of the last century some new interests in the theory of the interferometric method to determine the anisotropy (or its absence) of the speed of light at the Earth surface emerged. Múnera [14] showed that the interpretation of the amplitude and phase of the second order effect should be done for each rotation of the interferometer separately, not by averaging on forehand. Further, following Hicks [15] and Righi [16] De Miranda Filho describes possible first order effects in the Michelson–Morley interferometer [17]. This is quite a brave assumption as it is always believed that first order effects could not be detected with light interference measurements. This was also derived by Lorentz [3]. However, in his derivation he did not explicitly take into account the Doppler effect, as has been ignored by many of his contemporary researchers. Hicks [15] and Righi [16] took the Doppler effect into account, finding a first order effect. In the above mentioned conference Lorentz acknowledges that the discrepancy between his derivation and that of Hicks or Righi should be resolved and he promises to do so. Unfortunately one year later he dies.

In view of this discrepancy several researchers try to find experimental evidence of first or second order effects with Michelson–Morley interferometer type instruments. This has been done by, for instance, Piccard [18,19], Illingworth [20] and Joos [21]. All these authors report the absence of the sought for effect. However, according to Múnera these experiments all have results comparable with those of Miller. Hence, experimental evidence is not conclusive whether or not some first or second order effect exists.

Recently Múnera [22] reported an experiment claiming to see second order effects. He used a Michelson–Morley interferometer being stationary in the laboratory frame. The rotation of the Earth was used to change the direction of the velocity of the apparatus with respect to the preferred frame. This idea was followed by Cahill [23] using a fiber optical Mach–Zehnder interferometer. In these experiments the influence of the temperature on the signal was acknowledged. Múnera corrects his data for it and Cahill claims that the temperature can not influence the signal significantly. The present paper reports on an experiment where both the influence of temperature on a fiber Mach–Zehnder interferometer and the phase differences changes due to rotation around a vertical axis were measured, to determine the possible origin of the effect measured by Cahill.

With
a fiber Mach–Zehnder interferometer (see figure 1) the light injected into a
fiber by means of a laser is split into two equal and in phase parts by a 2x2 directional coupler
[24]. These two light beams travel trough two (perpendicular) fiber arms of
the interferometer and are rejoined by a second 2x2 directional coupler.
Depending on their mutual phase, the light beams interfere constructively for
one output fiber and destructively for the other one or vice versa. The two
outputs of the second 2x2 directional coupler are fed into two detectors where
their intensities *I*_{1} and *I*_{2} are measured. The sum of the intensities is proportional to the
laser output power. It would be equal to the laser output power if all losses
would be negligible. However, the coupling of the laser light into the fiber is
not perfect, the absorption of the fiber is not nil and the coupling of the
fibers are not perfect either. The difference of the two intensities relative
to their sum is called the visibility, V. For an ideal interferometer the
visibility would change between –1
and +1, depending on the phase difference, A$ of the light beams in the second
directional coupler according to

The phase difference is determined by the optical path of the
light, while traveling along the fibers from the first directional coupler to
the second one. This optical path depends on the length of the arms and the
wavelength of the light moving through the fibers of the arms. The wavelength,
A of the light depends on the speed of light in the fibers and hence on the
refractive index, *n *of the fibers and (possibly) on the velocity and motion direction,
**v **of the Earth relative to a preferred rest frame. Hence,

wheredenotes the line integral along arm *i* from the first directional coupler to the second one. The
determination of the change in phase difference due to the motion of the Earth
is the objective of this experiment. Therefore the whole interferometer is put
on a rotation table to be able to change the azimuth of the interferometer. In
this way the motion direction of the Earth relative to the direction of the
arms changes during rotation. When tracking the phase difference changes during
the rotation of the interferometer, one can get the magnitude of and the
direction in which the projection of the Earth velocity with respect to the
preferred rest frame is maximum. These values, relative to the local North,
change with time of year and the sidereal day. If there is any effect due to
the rotation of the Earth this should give a change in this signal depending on
sidereal time (for a detailed discussion on the dependence on this effect see
for instance [14]). Of course for a perfect interferometer it is not needed to
rotate it as the Earth rotates around its axis every 24 hours. However, for a
real-life fiber interferometer it is very difficult to get a stable signal for
more than several hours, worse for days and virtual impossible for a year.

The polarisation properties of the light beams and the non-ideal
directional couplers changes equation (1) into [25]:

where,and Vo are determined by the exact properties of the
directional couplers, the fibers used, beam polarisation and wavelength of the
light. One of the problems of this equation is that the visibility is
determined by the cosine of the phase difference between the light beams.
There is an ambiguity considering the sign and exact value of when only the visibility is determined. To overcome this
problem an optical phase shifter is inserted in the arms of the interferometer,
introducing an extra phase shift. This phase shifter is just an optical fiber
wrapped around a thin walled cylinder made of piezo material. When applying a
voltage over the wall of the cylinder the piezo material is either contracted
or expanded, reducing or extending the length of the fiber wrapped around it.
When the fiber stretches the optical path length changes accordingly and the
phase of one light beam with respect to the other is changed. This added phase
shift is controlled in such away that the visibility remains zero and hence the
argument of the cosine remains. The control signal can be calibrated
and translated into an effective phase difference of the interferometer. In
this way, also the sensitivity of the visibility to the polarisation direction
of the light is greatly reduced, because when the visibility is reduced to zero
*V* only determines the resulting accuracy of
the control signal and not the control signal itself [25].

To determine the influences on the phase difference we simplify equation (2) by assuming the refractive index of the fiber, and the wavelength of the light in the fiber does not depend on the position in the fiber, then:

where *n*
is the refractive index of the fibers used, Li is the length of arm *i* andthe wavelength of the He–Ne laser in vacuum (633 nm).

Except for the sought for effect, several other factors can influence the phase difference of the light beams. For instance a change in the Earth magnetic field, in the relative humidity of the air or in the atmospheric pressure theoretically all can influence the phase difference. These influences have relatively small gradients and hence affect both arms in the same way. Hence, only the length difference between the arms must be taken into account. The experiment has been constructed with equal length fibers in the arms and hence the length difference, although not accurately measured, was much less than 1 cm. As the Earth magnetic field is very small and the fibers used are insensitive to magnetic fields the influence of the Earth magnetic field is neglected. Further, as the fibers are made of glass the influence of the relative humidity of air can also be neglected.

According to [26] the influence due to a change in the atmospheric
pressure can be calculated from the compressibility of the fiber core and
cladding material. The pressure sensitivity of the optical phase in a fiber is
defined aswhereis the change in the phase a due to a pressure changeFor fiber optic materials this pressure sensitivity
depends on the core diameter and cladding diameter and has a value between —10^{-6}
and —10^{-5} per bar. The phase change for a pressure change of 10 mbar
and a difference in length of the two arms of 1 cm results for a wavelength of 633 nm in a change of phase between 1.5 mrad and 15 mrad. Although
not quite negliable, it can be taken into account by measuring the atmospheric
pressure and assuring that during the measurements the pressure change is well
below 10 mbar.

Another important factor is the frequency, *v* of the He– Ne laser. According to equation (4) the
phase difference is determined by the wavelength of the light, generated by the
laser. Although the frequency of the He–Ne
laser is quite stable it can have mode changes due to changes in the length of
the cavity introducing a frequency change. A standard laboratory He–Ne laser has a longitudinal mode sweep between 0.3 and 1.1
GHz. A stabilized laser reduces this mode sweep to 10 MHz or below. The
influence due a change in frequency can be determined from equation (4)

where it was used thatand c is velocity of light

in vacuum. The first term is due to the change of the wavelength
of the light and the second term is due to the dispersion of the refractive
index of the fiber material. For fuse silica at 300 K and a wavelength of 633 nm this dispersion is= –0.030 *μ*m^{–}^{1} [27] so that it

can be neglected with respect to the first term. Hence, for

a length difference of 1 cm the phase change is approximately 0.305 rad/GHz.

The
most important influence however is that of ambient temperature, *T*. Both the
refractive index and the length of the fiber arms are dependent on the
temperature. Then, from equation (2) for a homogeneous temperature change it
follows that

whereandThe first

term
represents the change in refractive index due to a temperature change and the
second term represents the change in length of the fiber arms. For fused silica
fibers as used here: *n*
= 1.457, α* _{n}* = 8.6 × 10

Another
important temperature effect is that the properties of the directional coupler
itself depend on the temperature via the coupling length, *l* and the
refractive index. These properties influenceandIf the
polarisation influence is neglected and only the coupling length of the
directional couplers and a different absorption in both arms are taken into
account, it is possible to determine equation (3) by using the transfer matrix
approach [24]. Let *i*_{1}
and *i*_{2}
be the amplitudes of the light waves at the two inputs of the directional
coupler and *o*_{1}
and *o*_{2}
the same for the outputs, then for couplers with identical waveguides:

whereand *l _{i}* is the
coupling length of either

the
first coupler or the second one. The 2×2 matrix is
referred to as *transfer matrix*
and describes the propagation of the beam through the coupler [24]. The
propagation of the beam through the arms of the interferometer can be described
by a simple multiplication of the amplitude at the beginning of the arm by the
transmission and a phase factor describing the optical path length. This can be
written in another transfer matrix, that is diagonal as there is no interaction
between the beams when travelling through the arms. The elements of the diagonal
matrix can be written aswhereis the
root of the

transmission
of the fiber (as we are dealing with amplitudes, not with intensities) andis
optical path length of arm *i*.
The complete transfer from the inputs of the first coupler (where *i*_{1} = 1 and
i_{2} = 0) is given by

Takingand inserting them in equation (1), yields

equation (3) with

and

whereandInterestingly,

if
one of the couplers is an ideal 3dB directional coupler, with, then
V_{o} =0. Here *m _{i}*
is an integer

determined by the design of the directional coupler. However, forto be ±1, both couplers should be ideal and the transmission of the arms should be equal. If the transmissions are not the same the amplitude of the visibility change is reduced to. For non-ideal couplers

a small offset for the visibility can occur and a diminished amplitude. Let us assume that

whererepresents the influence of a change in temperature, frequency or another parameter. Then upto first order in

and

From this equation is is clear that first order small changes can only influence the visibility via a change in transmission of one of the arms or when the transmission of the arms are not equal.

In the measurements a phase shifter is used, always adjusting the phase so thatSmall changes in

anddue to temperature changes can occur, resulting in a shift ofHence,

using equation (10), this becomes

The first term to the right describes the temperature influence via transmission change of the fibers. The transmission can change, but not more than a few percent so

that this term can be neglected. The second term describes the temperature influence via the change of either the length of the coupler or the refractive index. This term can be reduced by usingand

to

where
the + holds when *m*_{2}
is odd and the — for *m*_{2}
is even. The directional coupler is protected by a rigidly connected metal
shield. The expansion coefficient of metal is larger than that of fiber glass
increasing the temperature sensitivity. For aluminium for instance *α** _{L}* = 23 ×10

change in due to a temperature change is 42 mrad. Further, it should be noted that for the derivation of this formula it was assumed that the couplers were almost ideal and that polarisation effects do not play a role. Also it was assumed that the transfer matrix has the ideal form as given above. In reality these assumptions are not realized and the temperature influence might be much larger. It shows however, that although at first sight the temperature has no or limited influence on the working of a directional coupler, in combination with a variation in another parameter, in this case the transmission difference of the arms, it can have an important influence.

From the above considerations it is clear that the effects of several influences on the phase difference can be kept under control as long as the gradient of these influence is so small as that they affect both arms of the interferometer in the same way. Then, only the length difference of the arms has to be taken into account reducing all influences to acceptable and controllable limits. See also table 1 for a summary of the magnitude of the influences.

Influences that operate on one arm of the fiber alone could increase the influence to much higher values and must be avoided as much as possible. One, not so easy controlled, influence is stress on the fibers. Changes in stress can occur due to rotation of the whole set up, which might not be completely horizontal. Or by a change in temperature the support of the fibers might change its size and introduces stress. All these factors should be reduced as much as possible. The exact magnitude of the effect of stress on the fibers is hard to predict. The best way to determine them is by direct measurement.

Parameter | Range | Influence mrad |

Atmospheric Pressure | 10 mbar | 1.5 - 15 |

Ambient Temperature | 0.01 K | 13 |

Temperature directional coupler and interferometer arms with different transmissions, η = sqrt(2) | 0.01 K | 42 |

Frequency of Laser | 1 GHz | 305 |

Frequency of Stabilized Laser | 10 MHz | 3.1 |

**Table 1. **Influences on the phase difference between the two arms of a fiber
interferometer with 1 cm length difference.

**Fig. 2. **Basic set-up of the Mach–Zehnder interferometer. At the bottom and
left the *effect* interferometer with the perpendicular arms is wrapped around
glass discs with diameters of 40 mm. At the top the *control* interferometer with the parallel arms is
positioned. The four metal bars in the middle of the arms are the directional
couplers. The fiber stretchers have been omitted in this photo for clarity
reasons.

The
experiment has been designed with the above raised issues kept in mind. Two
interferometers were coupled by means of an additional directional coupler to
the same laser (a stabilized He-Ne laser type Coherent 200, linear polarized,
0.5 mW, maximum mode sweep of 10 MHz). One with its arms perpendicular to each
other as shown in figure 1. Another with its arms parallel as a control. Any
anisotropy in the speed of light should turn up in the first one
and should cancel in the second one. The fibers used were single mode fibers
SM600. The five 2x2 directional couplers used were all FC632-50B-FC Split Ratio
Coupler 632 nm, 50/50 from Thorlabs Inc. The detectors used are amplified
Silicon detectors (PDA36A from Thorlabs Inc). To prevent optical feedback from
the interferometer into the laser a polarisation dependent optical isolator
(IO- 2D-633-VLP from Thorlabs Inc, isolation at least 35 dB) was included (see
figure 1). The length of the fibers in the arms of the first interferometer was
12 m. The effective arm length however was just 4 m as the fibers were mounted on a glass support of 20 by 20 cm and had to be wrapped around glass discs with 40 mm diameter (see also figure 2). The length of the fibers in the
arms of the second interferometer was 2 m, where the effective arm length was 0.7 m. The glass support is made of ultra-low expansion glass (CCZ) with an
expansion coefficient of less than 10^{–7} 1/K.

**Fig. 3. **Visibility of both interferometers (red *control* and blue *effect)* as function of the voltage applied to the
fiber stretchers.

In
all arms an extra 1 meter long fiber was inserted wrapped around a thin-walled
cylinder made of piezo material
acting as a fiber stretcher. For one interferometer the fiber stretchers in the
different arms were used with opposite voltage to simultaneously increase the
length of the fiber in one arm and reduce it in the other, so that the effect
on the phase difference is twice as large. The typical response of the fiber
stretchers as function of applied voltage is shown in figure 3. The lines are
cosine fits to the measured points with a frequency of 0.887(1) rad/V and an
amplitude of 0.891(1) for the stretcher in the *control *interferometer and 0.940(1)
rad/V respectively 0.492(1) for the other one. The frequency corresponds to a
change in the circumference of the cylinders of about 100 nm/V. The standard
deviation in the measured visibility is almost as large as the symbols. Note that
the visibility amplitude of both interferometers is quite different. This can
be due to a bad transmission of one of the arms or a misaligned polarisation
[25].

As was shown in the previous section a stable temperature is of the utmost importance. First to ensure that there is no phase change due to a change in temperature and second a stable temperature reduces the possible temperature gradients. Therefor the whole set-up has been positioned in a temperature controlled environment consisting of three compartments. The first compartment is the laboratory, where the temperature is controlled to be constant within ±2 K. The second compartment consists of a box containing the detectors and electronics for the temperature sensors. The walls of this box are made of 1 mm thick aluminum shields, 5 cm of a poly urethane foam isolation and again an aluminum shield of at least 1 mm thickness. The heat conductivity of aluminum is 150 W/(m K) and of poly urethane 0.038 W/(m K). This large difference ensures a homogeneous distribution of the temperature. In this box the air temperature is controlled to be constant within ±0.2 K. The third compartment consists of a smaller box containing the interferometers and optical isolator. The walls are constructed in a similar way as the walls of the second compartment for the same reason. The temperature of this box is stable for days within 3 mK. For shorter times the temperature is stable within 1 mK. The heaters are glued to the outside of the aluminum shields. The temperature sensors are PT100s. The heat applied to control the temperature of the middle

**Fig. 4. **Top: laser power as function of warm up time showing mode changes
as changing intensities. Middle: Visibility of both interferometers (red *control* and blue *effect,* shifted with +0.25) during the same
period. Bottom: Phase difference of both interferometers during the same
period, showing the influence of the changes of 0.565 GHz in the frequency of
the laser.

compartment varies between 1 and 4 W. The heat applied to control the temperature of the inner compartment is below 150 mW. This ensures that the possible temperature gradients within the inner compartment are less than 1 mK/m.

The second compartment together with the laser and control electronics are put on a rotation stage (LT360 Precision Turntable from LinearX Systems Inc with a rotation accuracy of 0.1 degree) to enable the change of azimuth.

**Fig. 5. **Top: set-point (red line) and control temperature as function of
time. Bottom: phase difference of interferometers (red *control* and blue *effect*) during the same period as the upper
graph. The phase oscillations due to the temperature oscillations are
reproduced imposed on a linear drift of unknown origin.

First,
to find the effect of the laser frequency on the phase difference of the
interferometers, the laser was switched off and switched on again after a cool
down period of one
hour. It takes the laser sometime to become stabilized again. During this warm
up period the laser experiences regular mode changes, depending on the size of
laser cavity. These mode changes of 0.565 GHz change the phase difference of
the interferometers and are shown in figure 4. The laser was turned on and
after 2 minutes the phase control was turned on. The visibilities then rapidly
decrease to around 0 and the phase difference is controlled by the stretchers.
The visibility then becomes an error signal, indicating the accuracy of the
controlled phase. Due to a time constant of the control of 1 s, rapid changes
in phase difference create an error signal, which is used to change the voltage
applied to the shifters. From this graph it is clear that both interferometers
are sensitive to a change in laser frequency. For the *control* interferometer it is 0.86
radians or 1.5 rad/GHz, for the other its about half of that value. Hence,
their sensitivity to a frequency change is much larger than expected due to a
difference in length of the arms of the interferometers. The reason for this
discrepancy is sought for in the interaction length of the directional
couplers. However, as long a the laser is stabilized the mode changes are less
than 10 MHz, reducing the effect to negliable proportions.

Second,
the influence of the temperature on the phase difference has been determined by
varying the temperature of the most inner container as a cosine with an amplitude
of 0.02 K and a period of 6 hours. The results are shown in figure 5. The red
line in the top graph shows the set-point temperature. The black circles and
error bars represent the control temperature. The deviations are due to the
limitations of the temperature control. The curves in the bottom graph show the
corresponding oscillations of the phase difference of the *control* interferometer (in red) and
for the other one (in blue). The oscillations correspond to a sensitivity to
temperature variations of -130 rad/K of the *control* interferometer and for the
other one to a sensitivity of +90 rad/K. These values are much large than estimated
in section 3. This could indicate that the coupling length of the directional
couplers is much larger than assumed, or that there remains a considerable
temperature gradient over the arms that changes for a changing temperature.
The fact that such a small and slow change of temperature of the interferometer
results in such a large change in the phase difference, much larger than
expected from the simple considerations above, could explain the results of
Cahill [23] as he took great care to minimize the temperature gradients, but
did not control the temperature of his instrument. An extra linear decrease or
increase in the phases as function of elapsed time is also observed. For the *effect*
interferometer, this change is larger than for the *control* interferometer. This is
typical for all experiments done with these interferometers, indicating that
building a fiber interferometer with a stability over days is very hard to
accomplish. The mechanisms causing these changes have not been explicitly
determined. A possibility is the long time behavior of fibers properties and
fiber stretcher properties. For instance, the fibers used are glass fibers and
glass essentially remains a very thick liquid. Further, the stretchers are
made of piezo material that has some long term relaxation.

**Fig.
9. **Top: first
(red) and second (blue) order amplitude of the signal of the *effect* interferometer as function of sidereal time. Bottom: first (red) and
second (blue) order azimuth of the maximum of the same signal. For clarity
reason the error bars have been omitted. They have approximately the same
length as the spread in points.

**Fig.
8. **Top: first
(red) and second (blue) order amplitude of the signal of the *control* interferometer as function of sidereal time. Bottom: first (red) and
second (blue) order azimuth of the maximum of the same signal. For clarity
reason the error bars have been omitted. They have approximately the same
length as the spread in points.

**Fig.
6. **Top: angle of
set-up as function of time. Middle: phase difference of *control* interferometer during the same period as the upper
graph. Bottom: phase difference of the *effect* interferometer again during the same period.

**Fig.
7. **Top: second
order amplitude of the signal of the **effect **interferometer
as function of sidereal time as calculated for the same times and epoch as with
the experiment. Bottom: second order azimuth of the maximum of the same signal.

Finally,
the set-up is rotated along a vertical axis to find out if there is an
influence due to the velocity of the Earth in a preferred rest frame. Every 3
hours the set-up rotates from 0 to -180 degrees, from -180 to +180 and from
+180 to 0 with steps of 15 degrees. For 0 degrees the interferometer with the
parallel arms points to the local North. For 90 degrees it points to the local
East. The longitude and latitude of the location of the interferometers on
Earth are 4.7 degrees and 51.4 degrees. At every step the phase of both
interferometers is measured during half a minute to average out statistical
variations. A complete rotation takes about 40 minutes. A typical response of
the interferometers as function of angle is shown in figure 6. The top graph of
this figure shows the angle of the setup as function of time. The middle one
shows the phase difference of the *control*
interferometer during the same period as the upper graph. The bottom graph
shows the phase difference of the *effect*
interferometer again during the same period. In the lower graph the phase
oscillations due to the rotation of the set-up are clearly visible, again
imposed on a linear drift of unknown origin. At first, it seems that this
interferometer gives an indication that there might be some effect on the phase
difference due to the motion of the Earth. However, if such an effect exists
it should depend on the time of day and year. Upon rotation of the set up, the
amplitude and azimuth of the maximum should vary between certain minima and
maxima depending on the orientation of the Earth velocity with respect to the
preferred frame as discussed by Múnera [14].
The Fourier transform of the phase difference (corrected for the linear
assumed drift) as function of rotation angle gives this amplitude and azimuth
of the maximum phase difference for all orders. The zeroth order is just the
average phase difference during a rotation. The first order represents the
amplitude and azimuth of that part of the signal that varies with the cosine of
the angle of the set-up, corresponding to first order effects in v/c. The
second order represents the amplitude and azimuth of that part of the signal
that varies with the cosine of twice the angle of the set-up, corresponding to
second order effects, and so on. The error in the values can be estimated from
the difference between the Fourier transform of the data points measured for
increasing set-up angles and the one measured for decreasing set-up angles (to
find the systematic error due to the unknown drift) combined with the Fourier
transform of the variances (to find an estimate of the statistical error). The
amplitude and azimuth should vary with the sidereal time and epoch. For the
month March 2009 the effect is calculated using the same rotations of the
set-up as used in the experiment. The theoretical values were treated in
exactly the same way as the measurement data were treated, except that instead
of using the measured results for the phase difference, the theoretical values
for the phase differences are used. The theoretical values were calculated
following Múnera
[14] using the Miller data for the velocity of the Sun (Right ascension 4.9
hours and declination –70.6 degrees with a velocity
of 205 km/s). Miller assumes that due to some unknown cause the effect is
reduced by a factor of 20 [9] in the velocity. Here the effect is reduced by a
factor of 100 and hence the second order term coefficient becomes 0.0001 x 4π*nL*/*λ*_{0} = 1.2
× 10^{4}. The results are shown in
figure 7. A clear dependence of the amplitude and maximum azimuth appears. The
irregular shape of the curves is due to the change of the projection of Earth
velocity with respect to the preferred frame on surface of the Earth during one
rotation (as a complete rotation of the set-up takes about 40 min). During the
experiment the set-up was rotated every 3 hours. This would cover 2/3 of the
sidereal time scale in one month as the difference between a sidereal day and
a normal day is 1/365 of a day, which adds up to 2 hours per month. However, at
the middle of the series, the rotation interval was started again, shifting the
times at which the rotations occurred so that an overlap occurred with
previously measured sidereal time. The same happened again at the end of the
series when the computer clock was advanced with one hour due to the end of
daylight saving time. The results is a smaller coverage of the sidereal time
scale and not overlapping results. The slightly different values for sidereal
times that are the same, are due to the difference of the position of the Earth
in its orbit around the Sun at the time the data point were taken. Hence, for
instance, the effect for 10 hours sidereal time at the beginning of the month
is a bit different from the effect for 10 hours sidereal time at the end of the
month. During a year this leads to much larger differences [14], and should be
taken into account for the analysis of these kinds of measurements. However,
during the observation period, these irregularities are quite small with
respect to the full oscillation of the effect during a sidereal day and are
further ignored.

For
the *control*
interferometer the amplitude and azimuth of the maximum measured as function
of sidereal time in the month March 2009 are shown in figure 8. Similar data
for the *effect*
interferometer are shown in figure 9. There is no apparent sidereal signal in
any of the graphs. Hence, the conclusion is justified that the built Mach–Zehnder fiber
interferometer does not measure any anisotropy of the velocity of light on the
Earth surface.

When the second order signal of the interferometer with the long fibers is interpreted in a similar way as was done for the Michelson–Morley experiments, the result would be that the second order amplitude of the signal is less than 0.06 radians corresponding to 0.01 parts of a fringe. Using the simple formula the result is a maximum possible velocity of 7 km/s. This is comparable to the results obtained from all previous Michelson–Morley type interferometer experiments.

The two optical fiber interferometers built in a temperature controlled environment enable the determination of temperature effects on the phase difference between the light beams traveling through the two arms of the interferometer. The temperature influence on this phase difference is much larger than theoretically expected. It has been shown, that although at first sight the temperature has no or limited influence on the working of a directional coupler, in combination with a variation in another parameter, in this case the transmission difference of the arms, it can have an important influence. So the measured phase difference could be due to the interaction region of the directional coupler or to a temperature gradient of unknown origin.

Upon rotation of the interferometers around a vertical axis in one of the interferometers an oscillation in the phase difference as function of the azimuth of the set-up is observed. With regard to the measurement accuracy, this oscillation is constant and is attributed to changing stresses in the arms of the interferometer. Analysis of the signal of the interferometer with the perpendicular arms show that the fiber optical Mach–Zehnder interferometer built, is unable to detect an anisotropy of the velocity of light at the Earth surface. When the signal is interpreted in a similar way as Michelson and Morley did in their famous experiment, the resulting maximum velocity of Earth with respect to the preferred frame would be 7 km/s.

In view of the experimental difficulties getting a stable signal it is questionable that the accuracy of this type of measurements of the anisotropy of the speed of light on the Earth surface could be increased.

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