mechanical characterization of a lisa telescope test structure observing gravitational waves and the...
TRANSCRIPT
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UNIVERSITY OF TRENTO
Faculty of
Mathematical, Physical and Natural Sciences
Undergraduate school in
Physics
Mechanical Characterization
of a
LISA Telescope Test Structure
Candidate
Ilaria Pucher
Advisors
Prof. Guido Müller, University of Florida
Dott. William Joseph Weber, University of Trento
Academic year 2009/2010
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Thanks has to be said…
first of all to Giacomo, that allowed me to do this “in campo straniero” experience, that helped and helps me
really a lot, that supports me and is near me in every moment,
to Guido, who helped me from the beginning of this “American life”, from the bureaucracy to the lab
experience,
to Bill, that, even in this busy period, found the time to care of me,
to Pep, that was very patient with me in the lab, and taught me many many things,
to Dan, Amanda, Alix and the people of the LISA lab, that helped me during my job,
to my family, that, even now that I‟m distant, was and is close to me,
to my friends and professors, that always support me, and are always willing and kind,
and to my A22 colleagues, that, even if shortly will no more be colleagues (I‟ll miss them), are part of my
good and positive experience.
THANK YOU !
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Summary 1 Introduction ............................................................................................................................................... 5
1.1 Gravitational Waves .......................................................................................................................... 5
1.1.1 What are Gravitational Waves? ................................................................................................. 5
1.1.2 Who generates gravitational waves? And how? ........................................................................ 6
1.1.3 How can we detect gravitational waves? ................................................................................... 7
1.2 GW detectors ..................................................................................................................................... 8
1.2.1 Ground- and space-based detectors ........................................................................................... 8
1.2.2 Noise Sources ............................................................................................................................ 9
1.3 LISA .................................................................................................................................................. 9
1.3.1 LISA: the winner detector? ........................................................................................................ 9
1.3.2 LISA mission ........................................................................................................................... 10
1.3.3 LISA…inside ........................................................................................................................... 10
1.3.4 Interferometry to measure GW ................................................................................................ 13
1.3.5 LISA Telescope ....................................................................................................................... 16
1.3.6 LISA Sensitivity ...................................................................................................................... 16
1.4 Experiment ...................................................................................................................................... 18
1.4.1 Power and Linear Spectral Density ......................................................................................... 20
2 Optical resonators .................................................................................................................................... 20
2.1 Resonance and spatial modes .......................................................................................................... 22
2.2 Fabry-Perot Cavity and parameters ................................................................................................. 26
2.2.1 Free spectral range and Finesse ............................................................................................... 26
3 Mode matching ........................................................................................................................................ 28
3.1 ABCD Matrices ............................................................................................................................... 28
4 Beat note .................................................................................................................................................. 30
4.1 Profiling a beam and aligning two beams........................................................................................ 30
4.2 Beat note .......................................................................................................................................... 31
5 Michelson interferometer ........................................................................................................................ 33
5.1 Interference ...................................................................................................................................... 33
5.2 Interferometer .................................................................................................................................. 33
6 Pound-Drever-Hall (PDH) Locking Techinque ....................................................................................... 35
6.1 PDH Scheme and theory.................................................................................................................. 37
6.2 Experimental Application ................................................................................................................ 41
7 Phase-Locked Loop (PLL) ...................................................................................................................... 43
7.1 Application to the experiment ......................................................................................................... 43
8 Measurements and Results ...................................................................................................................... 44
8.1 Cavity and Michelson characteristics .............................................................................................. 44
8.2 Telescope Stability .......................................................................................................................... 49
8.3 SiC CTE........................................................................................................................................... 57
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1 Introduction
LISA (Laser Interferometer Space Antenna) is a NASA–ESA mission aimed at observing sources of
gravitational waves in a low frequency range: from 0.03 mHz to 0.1 Hz. Some of the sources expected to
radiate at these low frequencies are massive black-hole binaries, galactic binaries, compact objects orbiting
massive black holes, stochastic background, etc. [1, 2, 5]
LISA consists of three identical spacecraft that form an equilateral triangle, with a 5 million kilometers side,
orbiting the sun. If a gravitational wave reaches LISA, it will stretch and compress the triangle. Therefore, by
monitoring the separation between the spacecrafts with interferometric measurements, one can detect the
wave. Also the nature and evolution of the source can be studied, by studying the shape and timing evolution
of the wave as LISA orbits around the Sun.
1.1 Gravitational Waves
1.1.1 What are Gravitational Waves?
Gravitational radiation is predicted by Einstein‟s theory of General Relativity [3]. This radiation is a
perturbation of space-time that propagates at the speed of light, carrying energy. Gravitational waves are
generated by heavy and rapidly accelerating massive bodies. Since these waves carry energy, the sources that
emits them lose energy with time.
GW are transverse waves with two possible polarizations [4]. They distort distances in a plane perpendicular
to the direction of propagation, stretching distances in one direction and shrinking in the perpendicular one.
This fact can be used to build GW detectors. However, the distortion is extremely small, that makes
detection of gravitational waves very difficult.
The polarizations of a GW are called “+” (plus) and “×” (cross) because of the orientation of the axes
associated with their force lines.
Figure 1. The two different polarizations of a GW.
GW waves are completely different in nature from electromagnetic waves. Thus, they carry different
information, and may enable us to study sources that are not visible in the EM spectrum.
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1.1.2 Who generates gravitational waves? And how?
Almost all accelerated masses create distortions of the space-time, but these are so small that the most
sensitive GW detectors can only detect relatively strong sources, like two orbiting neutron stars, or a neutron
star orbiting a black hole, or two black holes orbiting one another. In general, strong gravitational waves are
emitted by big celestial bodies changing their relative position with time (i.e. changing the quadrupolar
moment of the mass distribution).
The spectrum of GW can be seen in Figure 2 below, with indication of typical sources for each frequency
range.
Figure 2. Gravitational Waves Spectrum
Observing gravitational waves and the information they can provide us will allow us to get an insight on may
interesting scientific questions:
the birth and history of galaxies and massive black holes;
the behavior of general relativity and space-time in extreme regimes;
the expansion history of the Universe;
the dense matter physics;
and possibly new physics of the early Universe or of string theory;
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1.1.3 How can we detect gravitational waves?
Gravitational waves carry energy and information about how they were produced.
They do not strongly interact with matter, and when GW reaches Earth, they are very weak, even if they
were produced in our galaxy and in a very violent collision. This makes the detection of gravitational waves
a very difficult task.
Interferometric GW detectors contain freely falling test masses separated by a relatively large distance, often
arranged in an L shape. If a gravitational wave reaches the detector, it shrinks and stretches the two arms of
the L, and thus it changes their length. By measuring this small distortion with interferometric techniques,
GW can be detected.
Figure 3 shows the action of a GW when the size of the object that the wave acts upon is small compared to
the wavelength (as is the case for earth-based GW detectors).
Figure 4 shows an example of a typical ground-based GW interferometer.
Figure 4. Example of interferometer layout. The two additional mirrors near the beam splitter serve to create two Fabry-
Perot cavities.
The space-based detector LISA has its test masses 5x106
km distant, forming an equilateral triangle that
orbits the sun, as illustrated in Figure 5.
Figure 3. Shrinking and stretching caused by a GW propagating perpendicular to the page.
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Figure 5. LISA mission orbit around the sun. It shows always the same face to the sun.
If the interferometer in Figure 4 is arranged such that its arms lie along the x and y axis of Figure 1 and a
wave is propagating in the z direction pointing on them, with the + polarization aligned with the detector, the
force of this wave will stretch one arm and squeeze the other. Since one arm is always stretched while the
other is squeezed, one can monitor the difference in length of the two arms:
.
In this simple case, this change in length is the length of the arm times the + polarization amplitude:
.
Generally, both polarizations act on the test masses, and the total wave strain is
where F+ and F
× are the antenna response functions.
1.2 GW detectors
1.2.1 Ground- and space-based detectors
The range of frequencies spanned by ground- and space-based detectors is shown in Figure 6.
The most important distinction between LISA and the ground-based interferometers is that LISA will search
for low-frequency (milli-hertz) gravitational waves.
Ground-based detectors will probably never be sensitive well below 1 Hz, because of terrestrial gravity-
gradient noise (i.e. noise in the local gravitational field). A space-based detector is free from this kind of
noise; in addition, the interferometer can be made very large. These two characteristics allow for the
detection in the range from 10−4
Hz to .1 Hz.
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Figure 6. Comparison of frequency range of sources for ground-based and space-based gravitational wave detectors.
1.2.2 Noise Sources
Some noise sources of GW interferometers that limit their sensitivity are the following. [1, 4]
• Seismic noise. Ambient or human induced seismic waves that shake the test masses of the detector.
• Thermal noise. Because the temperature is not equal to absolute zero, atoms are not at complete rest, but
rather vibrate. This introduces noise in the position of the test masses, that can be confused with the GW
signal.
• Shot noise. The number of photons in the input laser beam fluctuates (quantization)
• Radiation pressure noise. Fluctuating number of photons bouncing from the mirrors will introduce a
fluctuating force on the mirror.
• Gravity gradient noise. Fluctuating gravitational forces due to variations of the mass distribution (moving
cars, seismic waves, …) around the test masses.
• Laser intensity and frequency noise. The laser itself inevitably is somewhat noisy, with fluctuations in
both intensity and frequency.
• Residual gas. Any vacuum system contains some trace amount of gas that is extremely difficult to reduce,
and can both affect the laser beam path or directly transfer momentum to the test masses.
• Beam jitter. Jitter in the optics will cause the beam position and angle to fluctuate slightly.
• Spurious forces. Any unwanted force acting on the test masses will cause them to move, masking or
mimicking a GW signal.
However, one has to keep in mind that GW act coherently, while noise sources usually don‟t. If one can
observe a GW for long enough, the noise can be averaged out, thus improving the signal to noise ratio.
1.3 LISA
1.3.1 LISA: the winner detector?
Many certain sources of GW radiate at low frequencies, from a few μHz to about a Hz. As already said, this
is an inaccessible frequency range for ground-based interferometers because of gravitational noise and
because ground-based interferometers are limited in length to a few kilometers.
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In order to escape from these limiting factors, LISA is a space-based detector. It is a giant Michelson
interferometer that will measure fluctuations in the distance between widely separated test masses.
LISA will measure signals from several different gravitational-wave sources:
massive black holes;
binaries of compact stars in our Galaxy;
possibly, other sources of cosmological background, including the relic radiation from the very early
phase of the Big Bang.
Many known system are predicted to emit gravitational waves in the amplitude and frequency range
accessible to LISA. “…If LISA will not detect the gravitational waves from sources and with polarizations
predicted by General Relativity, it will undermine Einstein gravitational physics.”
1.3.2 LISA mission
LISA consists of three identical spacecraft that are 5×106 km distant. They form an equilateral triangle, as
shown in Figure 7. LISA is basically a giant Michelson interferometer placed in space, but with a third arm.
The latter is added to give independent information on the two gravitational wave polarizations, and for
redundancy.
Figure 7. LISA triangle orbits around the sun, 20° behind the earth and 60° with respect to the ecliptic. The center of the
triangular formation is at 1 AU (astronomical unit) from the Sun. The plane of the triangle is inclined at 60 with respect to
the ecliptic, where the center of the triangle lies. The triangle appears to rotate about the centre of the formation once per
year.
1.3.3 LISA…inside
Each LISA spacecraft contains two (almost) free-falling test masses that define the endpoints of the
measured distances, i.e. the ends of an interferometer arm.
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Figure 8. Y-shaped payload of LISA
Figure 9. Spacecraft inside: two optical benches, two telescopes.
The satellites and instrumental systems have to measure changes in the separation between test masses with
the required displacement sensitivity, and they also have to limit the disturbances acting on the test masses.
Changes in the arms length are monitored by interferometric measurements.
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Figure 10. Six test masses, six laser beams going back and forth and six telescopes.
A “drag-free” control system, consisting of an inertial sensor and a system of electrical thrusters, is
implemented in order to limit disturbances on the test masses. The position of the mass within the spacecraft
is continuously sensed. If relative position between spacecraft and test mass changes in the direction of
measurement (because external forces act on the spacecraft), the spacecraft is pushed by micro-Newton
thrusters in order to follow the test mass and keep it centered in its housing. If forces act on the spacecraft
perpendicular to the direction of measurement, electrostatic forces re-center the mass making it follow the
spacecraft. This is necessarily as the spacecraft contains two test masses that cannot be both followed in all
degrees of freedom.
Figure 11. Test mass and the electrode housing.
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Figure 12. Each spacecraft has two test masses, each one inside an electrode housing. Here are some of the possible forces
that can acts on each test mass.
The temperature stability is an important requirement for the payload [7, 8]. Optical bench fluctuations must
be kept below 10−6
K/√Hz at 1 mHz; the telescope thermal stability should be below 10−5
K/√Hz at 10−3
Hz.
To provide a thermally stable environment, each spacecraft will always have the Sun shining on the same
(“upper”) side, at an angle of incidence of 30.
1.3.4 Interferometry to measure GW
1.3.4.1 Michelson Interferometer
A Michelson interferometer in shown in Figure 14.
Figure 13. Michelson interferometer layout.
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A Michelson interferometer is used to measure length difference between two “arms”. This difference in
length will be changed if a gravitational wave pass by (with “the right” direction).
In ground-based interferometers, a laser beam is divided at the beam-splitter. Each beam travel back and
forth in one arm, then two beams are recombined at the beam-splitter. If the length of the two arms is
identical, modulo half a wavelength, the photodetector does not read anything. On the contrary, if there is a
gravitational wave, the relative phase Φ of the two beams changes because of the change in the differential
length of the two arms and a light signal will be detected by the photodiode1.
A typical estimated wave strain is h ∼ 10 −21
- 10 −22
. This strain values set the sensitivity required to
measure gravitational waves. Remembering that the length change in the interferometer arms is given by:
for a kilometer arm length L, we need to be able to measure a distance shift δL of better than 10−16
cm.
1.3.4.2 LISA Interferometer
When a gravitational wave passes through the LISA triangle, it changes the separations between the test
masses in different satellites. These changes in distance are detected by Interferometry measurements.
However, unlike what happens in ground based detector, the light coming from different arms cannot be
made to interfere directly. This is because the even relatively small differences in arm length (the triangle
equilateral triangle is not really equilateral…), and thus in light round-trip times, would cause the incoming
light of different arms to carry information about the phase emitted at different times, thus coupling the laser
phase noise in the GW signal. Instead, the phase of the light coming from different arms is measured
independently, and then recombined off-line accounting for the different delays (Time Delay
Interferometry).
The distance between two test masses (in different satellites) is determined by a total of three measurements:
one “long arm”, operating between the two optical benches in different spacecrafts, and two “short arms”,
which measure changes in the distance between the free-falling test mass and their respective optical
benches.
1 In principle, the wave strain h could be read from the photointensity of the light. In practice, a system of servo loops
control the system such that destructive interference is guaranteed; the photodiode is kept dark, and the strain signal is
inferred from the actuation needed to keep such configuration. That is why it is called the “dark port”.
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Figure 14. Interferometric measurements of distance of the test masses in the two spacecraft.
For the “long arm” measurement, a „master‟ laser beam is enlarged by a telescope and sent to the other
spacecraft, so to limit the divergence angle and the dimension of the beam at the 5 million km distant
spacecraft. A small portion of this beam is collected there by a telescope. A „secondary‟ laser on this distant
spacecraft is phase-locked to the weak incoming beam, so it can transmit back a full power beam to the
original spacecraft, which uses the same telescope to focus the incoming beam. The phase of the incoming
beam is compared to phase of the master laser by beating them. This gives information on the length of one
arm modulo the laser wavelength. The time-varying optical path caused by a gravitational wave impresses a
phase modulation on the beam that shows up in the beat signal.
The same procedure is done for the other arms. The difference between these signals, with the appropriate
delays, will thus give the difference between the arm lengths (i.e. the gravitational wave signal).
Figure 15. Optical system: optical bench and telescope.
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1.3.5 LISA Telescope
The telescope is used to transmit the laser beam to the distant spacecraft and to receive the reflected light
from that spacecraft.
Because the telescope is part of the laser beam path, if it changes its length, the path will also change, and
thus also the interferometer arm will change its length. Therefore, the dimensional stability of the telescope
is a crucial point.
In addition to this, the peripheral position of the telescope, in part near the optical bench and in part facing
the cold open space, makes it subject to temperature fluctuations and gradients between the primary (near the
optical bench) and the secondary (near the space aperture) mirrors. Its operational temperature is estimated to
be around -15C. The temperature fluctuations at the telescope must be less than 10−5
K/√Hz at 10−3
Hz.
The stability of the telescope should be as high as possible and should be maintained as long as possible in
order to assure correct operation of the interferometric system.
The possible materials for the telescope are the following [6]:
• CFRP (Carbon Fiber Reinforced Polymers)
• SiC (Silicon Carbide)
• Zerodur
• ULE (Ultra Low Expansion glass)
SiC have the following advantages:
• high thermal conductivity, which avoid local surface distorsions
• low price (the cheapest solution)
• high specific stiffness, which allows mass saving
• low coefficient of thermal expansion
Zerodur and ULE have similar characteristics: both have a very low CTE (on the order of 10-8
/K) and show a
stability of better than 30fm/√Hz, but, on the other hand, both are brittle, heavy and expensive.
CFRP materials outgas and shrink over time, thus this solution is discarded.
1.3.6 LISA Sensitivity
The sensitivity of the LISA antenna [1] is determined by two main factors. One is the noise in measuring
changes in the distances between the proof masses. The second is the level of spurious accelerations of the
proof masses.
Figure 16 shows the LISA sensitivity curve.
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Figure 16. The LISA sensitivity curve at frequencies above 3 mHz is determined by the distance measurement noise. At lower
frequencies, the instrumental sensitivity is limited by spurious accelerations of the proof masses.
For the changes in the distances between proof masses, the total error budget level for the round-trip arm
length difference is 40 pm/√Hz.
As it can be seen in Figure 16, the best sensitivity is between 3 and 10 mHz. In this range the sensitivity is
limited by photon shot noise and other noise sources that are assumed to be roughly white. Above about 10
mHz, the sensitivity is limited by the size of the detector: in the time the light needs to go back and forth in
one arm (about 35 s), the GW changes sign and reduces its overall effect.
At low frequencies, the sensitivity is limited by the acceleration noise on the test masses due to spurious
forces and actuation cross-talks.
The sensitivity of LISA to a given GW signal depends also on the angle of incidence of the incoming GW
with respect to the detector, which in turns depends on the position of the source in the sky and on the
orientation of the constellation as it orbits around the Sun (see Figure 7). Figure 17 gives an example of the
intensity pattern of LISA at different phases of its orbit.
Figure 17. Antenna pattern. LISA direction and polarization sensitivity for some incident angles. Directions to sources are
determined to <1° for strongest signals.
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1.4 Experiment In § 1.3.5 we talked about LISA telescope and the demands it has to fulfill.
The requirements for this telescope expressed in terms of:
Length noise Linear Spectral Density: S1/2
L ≤ 1pm/√Hz @ 1mHz
Material with CTE at -70C < T < Tamb as lower as possible, but also strong and light.
As said in § 1.3.5, one of the possible materials to build the telescope is silicon carbide. This will be the
material studied in this experiment. Thus, the aim of the experiment is:
1) to verify the stability of the LISA 60 cm length SiC telescope test structure, i.e. to measure the linear
spectral density S1/2
L of the telescope and check if it is below the requirement;
2) to measure the coefficient of thermal expansion (CTE) of the structure, for -70C < T < Tamb, and
compare it with the nominal one given by the seller (Coorstek).
1) stability
The stability of the SiC telescope will be studied by using the telescope as a spacer of a Fabry-Perot cavity,
with a high reflective mirror at each end.
For this measurement, the laser sent into the cavity has to be locked to the cavity itself; the technique used is
called Pound Drever Hall (PDH) and will be discussed in § 6.
The laser (L) will then be beat with a very stable reference laser (RL) and the frequency of the beat note will
be recorded, in order to calculate the changes in length of the cavity (i.e. the telescope) dL, through the
relation
where is the average laser frequency, L is the average cavity (telescope) length, and d is the laser
frequency change (equal to the beat note frequency change) due to a change dL in the length of the cavity it
is locked to. The beat note will be discussed in § 4.
Then, the linear spectral density of this length will give the stability trend.
2) SiC Coefficient of Thermal Expansion (CTE)
The coefficient of thermal expansion (CTE) α describes by how much a material will expand for each
degree of temperature increase and is given by the formula
where
dL = the change in length of material in the measured direction
L = overall length of material in the direction being measured
T = the change in temperature over which dL is measured
The coefficient has the unit K−1
, and its magnitude depends on the structure of the material.
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Thus the CTE of the telescope is related to its change in length by the relation
Measurement of the CTE while the temperature of the experiment is changed will be done in two different
ways:
i. Creating on the telescope a Fabry Perot cavity and measuring the change in its length, that is related
to the difference in the beam frequency that resonates into the cavity;
ii. Creating on the telescope an arm of a Michelson interferometer and measuring the change in length
by counting by how many wavelengths the interference figure has changed.
The same laser L is sent to into the Michelson, into the cavity and to beat with a stable reference laser, to
which laser L is locked through a Phase-Locked Loop (PLL) control system. PLL will be presented in § 7.
i. Cavity
As for the stability measurement, a cavity is created in the SiC telescope structure. However, in this case, the
laser L is not locked to the cavity, but to the reference laser RL (thus kept at a constant frequency).
As the cavity changes its length, the resonant frequencies of the cavity will change, alternately matchin or
not the one of the laser. Each time a new resonance in observed, the cavity has changed its length by:
Where d is the FSR of the cavity, and is the frequency of the laser (see 7.1).
Optical resonator, resonance and spatial modes will be presented in § 1.
A useful procedure to know when one works with Gaussian beams is mode matching. It was used in these
experiments to match a Gaussian laser beam into the cavity, into the interferometer and to couple two
Gaussian laser beams. Mode matching will be discussed in § 1.
ii. Michelson interferometer
As already said in § 1.3.4.1, a Michelson interferometer has two arms; the beam sent to a beam splitter is
divided in two beams, and each one goes at the end of the arm it is traveling in and comes back to the beam
splitter; when two beams encounter each other, they create an interference figure, that can change if the
length of the arms changes. Assuming that one of the two arms (reference) remains constant, we can measure
the difference in length of the telescope (that constitutes the other arm) by counting how many wavelengths
the interference pattern has changed.
An arm of a Michelson interferometer is created, in the experiment, on the telescope by mounting a beam
splitter on one extreme of the telescope and a mirror on the other extreme (first arm). The other – reference –
arm is inside the beam splitter: it is the space between the point where the input beam splits into two and the
face of the beam splitter in front of the “second” beam (this face is actually treated with an HR coating). This
second arm is much shorter than the other; thus, the difference in the interference pattern due to the change
in length of this arm is much smaller compared to the other and can be neglected. The difference in the
interference figure is therefore only due to the difference in length of the big arm (telescope).
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Looking to the interference pattern and counting how many wavelengths it has changed (due to the change in
temperature), the variation in length of the big arm (i.e. of the telescope) can be found, and from this the
CTE.
Michelson interferometer and interference will be discussed in § 5.
Figure 18. SiC Telescope setup. Cavity and Michelson mounted on the telescope.
1.4.1 Power and Linear Spectral Density
The single sided PSD is defined as the Fourier Transform of the auto-correlation function of the signal:
where
is the autocorrelation function of X(t). GX() is independent from t under the hypothesis that X(t) is a
stationary signal.
PSD is usually specified in [units of X(t)]2/Hz. It is an important quantity that gives the power distribution of
the stochastic signal X(t) as function of the frequency.
The square root of the PSD gives the Linear Spectral Density, LSD, specified in [units of X(t)]/√Hz.
2 Optical resonators
An important concept in the experiment is what an optical resonator is and how it works [9, 12].
An optical resonator is a set of optical components, which let a laser beam to circulate in a closed path.
A Fabry-Perot cavity is a two mirrors linear resonator, with a laser beam that bounces many times back and
forth inside. In the experiment there is a Fabry-Perot cavity having a monochromatic laser beam incident on
the back of a flat input mirror and reflected back/partially transmitted by a curved front mirror.
A mirror is characterized by two coefficient, that are the refection and the transmission coefficients, r and t
respectively, of the field amplitude. They are usually complex and of the form
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The relative intensity reflectivity and transmissivity are R ≡ r2, T ≡ t
2, and are related by R+T=1, for a
lossless cavity.
If the input field amplitude of the light is E0, then, inside the cavity, just above the input mirror (the light
coming from the bottom in the vertical cavity we created in the telescope, see Figure 19), the field amplitude
is
This wave then propagates up to the output mirror. A portion of it is transmitted outside the cavity and a
portion is reflected back to the input mirror. Here, a portion of it is transmitted and another portion is
reflected again. After a complete round trip, the amplitude of the wave reflected inside the cavity by the
input mirror is (assuming that it can be described as a plane wave)
where d = 60 cm is the separation between the two mirrors.
The round trip is repeated many times, and after n round trips the field becomes
The total field inside the cavity, just over the bottom mirror, is then a sum over an infinite number of round
trips
The corresponding intensity inside the cavity is proportional to the square of the field module
where
The transmitted field and intensity are
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The reflected field from the resonator has two contributions: one from the incident field directly reflected
from the input mirror and the other transmitted back though the input mirror from inside the resonator
−r1* coming from the imposition of energy conservation on a single mirror.
The corresponding reflected intensity is
Figure 19. Fields inside and outside the optical cavity.
2.1 Resonance and spatial modes Usually, the intensity profile of light propagating into a medium, either in free space or not, is subject to
change.
Figure 20. Beam size as function of position.
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However, there are some electric field distributions, called modes, for which it doesn‟t happen. What can
happen is that the phase changes and/or the profile is scaled, but the overall shape is maintained.
Inside a resonator, the wavefront of a mode has to be exactly equal to itself, including the phase, when the
light has done a complete round trip. This means that the phase can only change by an integer multiple of 2π.
This condition implies that the total path has to be an integer multiple of the wavelength, and thus only some
frequencies are allowed. These frequencies are called resonance frequencies.
They are defined by the relation
or
where n is an integer and d is the length of the cavity (d=60cm).
If the length of the cavity changes the resonant frequency also changes.
At a resonance frequency, the transmitted intensity become large, while the reflected intensity become small.
The simplest modes have a Gaussian profile. For modes having different intensity profiles (higher order
modes), the resonance frequencies are slightly different. Higher order modes can be Gaussian-Laguerre or
Gaussian-Hermite modes.
Gaussian Mode
Gaussian modes are waves with finite intensity profile. This profile can be rescaled, but its shape is
maintained, i.e. the shape is always Gaussian.
They are the lowest order transverse modes of an optical resonator.
The Gaussian mode appears like this:
Figure 21. 00 Gaussian Mode
Modes of higher transverse order are Hermite-Gaussian and Laguerre–Gaussian modes. They have slightly
different resonance frequencies.
The Hermite-Gaussian (HG) modes are also called Transverse Electro-Magnetic modes and are indicated as
TEMmn, where m and n are the indices of the mode. They appear like a grid of dots.
Figure 22. HG(0,2) mode.
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Laguerre-Gaussian (LG) modes appear like hollow discs.
Figure 23. HG(0,2) mode
If the propagation direction of a Gaussian beam is z and the transverse radial coordinate is ρ, the amplitude
of the beam has the following behavior
where the beam radius W(z) varies along the propagation direction according to
and the radius of curvature R of the wavefront evolves as
The phase is
and the waist size is
The Raleigh range z0 determines the distance in which the beam can propagate without diverging
significantly.
Because the intensity of the beam is the square of the modulus of the amplitude, I(r) = |U(r)|2, the transverse
profile of the intensity of the beam is also Gaussian
25
When ρ = W, the intensity drops to 1/e2 (∼ 13.5%) of its maximum value. The corresponding diameter 2W is
called the spot size.
The position z = 0 is the beam waist, where the phase profile is flat and where the beam radius is minimum
and equal to W0.
Figure 24. Evolution of the beam radius of a Gaussian beam.
Figure 25. Gaussian beam with curved wavefronts.
The divergence angle in the far field, i.e. where z >> z0, is
One can introduce the complex-valued parameter q that describes the size and the radius of curvature of the
beam at a certain z position
26
Where
It will be useful in § 3.1,where we will introduce a formalism to easily calculate the propagation of a
Gaussian beam through an optical system.
2.2 Fabry-Perot Cavity and parameters A Fabry-Perot cavity consists of two high reflective mirrors, which let the light inside it bounce back and
forth many times.
2.2.1 Free spectral range and Finesse
The free spectral range is the distance in frequency between two resonator modes, or, equivalently, the
inverse round-trip time.
As already said, resonant frequencies are given by
where n is an integer.
For a cavity in vacuum of length d, the Free Spectral Range (FSR) is
i.e., is the speed of light c divided by the round-trip path length 2d, or, more generally, is the inverse of the
round-trip travel time. It means that the shorter the cavity, the bigger the FSR.
Figure 26. Fabry-Perot parameters.
The finesse of a cavity is the FSR divided by the full width at half maximum (FWHM), or bandwidth, of the
intensity of the transmitted electric field Et
27
where E0 is the input field.
If grt is the roung-trip gain, the finesse is
If the cavity has no losses, grt = r1r2 , (r1, r2 being the reflection coefficients of the two mirrors), the finesse
can be written as2
Then, if the reflection coefficients of the mirrors are equal, the finesse is simply
where T is the transmissivity of the mirrors.
Therefore, the finesse is dependent on the cavity mirrors, and the lower the transmissivity, the bigger the
finesse. Since the finesse is independent of the resonator length, it indicates the quality of the mirrors.
Table 1 summarizes the theoretical parameters of the FP cavity built using the telescope structure.
2 Remember that ri is not qual to Ri, but they are related by Ri=ri
2
28
Table 1. Laser and Cavity theoretical parameters
parameter Value Explanation
λ 1064 nm Laser Wavelength
282 THz Frequency ( =c/λ)
k 5.9x10-6
m-1
Free-space wavenumber
(k=2π/λ)
d 60 cm Cavity length
R1 0.995 Reflectivity of bottom
mirror
R2 0.995 Reflectivity of upper mirror
T1 0.005 Transmissivity of bottom
mirror
T2 0.005 Transmissivity of upper
mirror
FSR 250 MHz Free spectral range
Ƒ 628.32 Finesse
FWHM 0.4 MHz Full width at half maximum
(=FSR/Ƒ)
3 Mode matching
Often there is the needed to transform a beam with waist size 1 at location z1, to have waist size 2 at
location z2. This can happen if one has to superimpose two beams, or to adjust a beam t to match the resonant
mode of a cavity. This procedure is called mode matching [9]. The mode matching creates an overlap of the
intensity and phase profiles of the beams that are needed to be coupled. This is for instance the case of two
beams that have to beat on a photodiode.
If a beam is mode matched into a cavity, the transmitted and reflected lights tell how good the mode
matching is. If the mode matching to a 00-mode is perfect, there will be complete transmission of the light
and no reflected one. This means that the beam, just before entering the cavity, contains no other modes than
the one resonant in the cavity (and thus completely transmitted).
ABCD matrices algebra is a useful instrument to perform mode matching calculations.
3.1 ABCD Matrices If a Gaussian beam passes through an optical element, the latter will change the parameter of the beam.
The ABCD matrix of the optical element is used together with the q-parameter to describe the effect of the
element on the laser beam.
The relation between the parameter before (q1) and after (q) the optical element is given by the ABCD law:
This 2x2 matrix can also describe the effects of more than just one optical element on the beam. In this case,
the total ABCD matrix of the optical system is the product of the matrices of the single elements, taking in
mind that each matrix sits on the left of the matrices of the previous elements.
Matrices associated with some common optical elements (including free propagation in a medium) are given
in Table 2 .
29
Table 2. ABCD matrices for common optical elements.
Optical element ABCD matrix
Propagation through a medium having index-of-
refraction n and length d
Refraction at a spherical boundary of radius R,
entering a medium of index n 2 from a medium of
index n 1. R is positive if the center of curvature lies
in the positive direction of ray propagation.
Transmission through a thin lens of focal length f
Reflection from a spherical mirror having radius
R. R is positive if the center of curvature lies in
the positive direction of incident ray propagation.
So, how can the mode matching be achieved?
In the case a beam that has to be mode matched to a cavity, if the q-parameters of both the laser beam and
the resonator beam just outside of the input mirror are known, one only need to use the ABCD law to match
them.
There is not a unique solution to the problem. Many different lens systems can give rise to the same ABCD
matrix. Simple solutions could be a lens with a particular focal length, placed in the right position, or a pair
of lenses.
As an analytical solution of the mode matching problem is not trivial, a simple interactive Mathematica
script turned out to be very useful for determining the right combination of lenses.
The experimental cavity has a flat front mirror and a curved back mirror of radius R2 = 1m.
The waist of the incoming beam should be where the flat mirror is, because the wave front of the beam at the
waist is flat, like the mirror. If the waist is not there, the beam would deform, because the points of a wave
front would arrive and be reflected at different times, so that the reflected beam would not have the same
wave fronts and it would not arrive in the right way to the curved mirror.
Thus, in order to have the wavefronts with the right curvature at the mirrors, mode matching procedure was
used and a pair of lenses with the right focal length were selected.
Figure 27. Cavity beam paths.
30
4 Beat note
4.1 Profiling a beam and aligning two beams To mode match two beams to be beat together, it is necessary to know their beam parameters; these can be
measured by “profiling” the beam. One possible method to profile a beam is to use a beam profiler.
A laser beam profiler displays the spatial intensity profile of a beam at a plane perpendicular to the beam
propagation direction. The transverse intensity of a Gaussian beam is given by the equation already seen in §
2.1
As said, when =W, the intensity drops to 1/e2 (∼ 13.5%) of its maximum. At this radius, the strength of the
electric field drops to 1/e (∼ 37%) of its maximum.
For the beat note, the beams to profile in the experiment are a reference laser beam RL, coming from the
Zerodur stable cavity, and the laser beam L, which enters into the testing telescope.
The beam to profile is made enter into the beam profiler, and one takes notes of the distance z of the beam
profiler from a reference point (for this experiment it is the exit of the beam from the fiber that brings the
laser beam on an optic table, see Figure 29) and of two values that the computer shows, 2Wua and 2Wub;
these are the major axis and the minor axis, measured in μm, of the ellipse that represents the points that have
an intensity over 1/e2 ~ 37%; since the spot is supposed to be circular, the mean value of these terms is
assumed to be the radius of the beam spot at that distance. The same work is done for many points (i.e., for
many distances of the beam profiler from the fiber). Then, plotting radius versus distance and fitting with this
curve
Curved mirror:
the wave fronts
arrive with the right
curvature
Flat mirror: the wave fronts
arrive with the right curvature
Figure 28. For the wavefronts to arrive on the two mirrors with the correct curvature, the waist must be where the flat
mirror is.
31
the waist w0 and its position z0 can be found.
Applying the same procedure to both RL and L beams, the results are the following
RL: w0 = 0,2586 mm z0 = 26,5 cm
L: w0 = 0,105 mm z0 = 13,94 cm
To align these two beams, several mirrors along the path of each beam were used.
In addition, the beams were mode matched using two lenses, one in each beam path, so that, when they enter
and superimpose at the beam splitter, they have the same profile.
When they come out from the beam splitter, the beams should be one on top of each other. If not, one uses
the mirrors again to adjust the angle and the direction of the beams.
Figure 29. Beat note between L and RL.
4.2 Beat note A beat note is a signal at a frequency equal to the difference in frequency of superimposed beams. The beat
note is visible if the beams are not orthogonal.
32
Figure 30. The bottom curves show two different electric field strength. The upper one shows the sum of the two.
If the frequencies of the two beams are reasonably close, the beat note will not have a frequency on the order
of THz (that could be the frequency of a beam alone), but rather MHz or GHz, and thus can be measured by
a photodetector.
The two laser beams, RL and L, are superimposed into a photodiode, using a polarizing beam splitter. The
two beams have to be very well aligned. The polarizing beam splitter is used so that the electric fields of the
two beams propagate in the same direction, and to polarize one of the two beams in the same way of the
other.
Connecting the photodiode (DC output) to an oscilloscope and blocking one of the two beams, the beam
signal that enters in the photodiode can be maximized by careful alignment, while looking to how the signal
changes on the oscillator. Blocking the first one, the other one can also be maximized. After this, the AC
output of the photodiode (now hit by both beams) is connected to a spectrum analyzer. It shows the spectrum
of the signal, that is roughly analogous to the Fourier transform of the signal, and thus the beat note can be
seen.
Figure 31. Beat note between L and RL.
When the laser L (“testing” laser) is sent into the cavity, its temperature is changed till the spike appears
(changes in temperature generate changes in frequency) on the display of the spectrum analyzer: this means
33
that the laser is resonating in the cavity with a frequency close enough to that of RL to generate a beat note at
a frequency low enough to be measured. After the beat note (the peak, at certain frequency) is found, the
photodetector (AC output) is connected to the oscilloscope to maximize the beat by further alignment of one
of the two beams.
5 Michelson interferometer
5.1 Interference When two coherent light beams at the same frequency are superimposed in a point in space, the intensity of
the combination at that point can be bigger or smaller than the intensity of a single beam. This effect is called
interference.
For this effect to take place, the two beams should not have orthogonal polarizations.
The total intensity can be from zero to four times the intensity of a single beam.
If the total intensity is bigger than the original single intensity, there is constructive interference. If, on the
other hand, the total intensity is smaller than the single one, there is destructive intensity.
The total energy is conserved in any case.
Figure 32. Interference: if two waves are in phase, there is constructive interference; if the two waves are 180° out of phase,
there is destructive interference.
5.2 Interferometer A simple interferometer consists of two high reflective mirrors, a beam splitter and a photodetector.
A beam is split by a beam splitter; the resulting two coherent beams travels into two different (most of the
times perpendicular) arms and are then reflected back, at the end of the arms, by two high reflective mirrors.
These reflected beams recombine together on the same (or another) beam splitter, where they generate an
interference pattern, and are gathered by a photodetector.
One of the two mirrors can be movable. This allows very precise measurements of length variation by
counting how many fringes pass through the visual field when the moving mirror moves.
Waves in phase Waves 180° out of phase
34
Figure 33. Michelson interferometer.
A Michelson interferometer is an interferometer that uses a single beam splitter to separate and recombine
the laser beams. An interference pattern is generated by these two coherent beams when they recombine.
In the experiment, the Michelson Interferometer has two very different arms. One arm is very short and its
length is the distance from the center of the beam splitter, that splits the input beam, and one of its side face,
which is actually a mirror. Since the beam splitter is located at one end of the telescope and the mirror of the
second arm is at the other end, the length of this arm is equal to the length of the telescope under test (60cm).
Length variations due to the cooling or heating of the telescope are supposed to be negligible for the short
arm that is thus considered constant; therefore, only the long arm, i.e. the telescope itself, is responsible for
changes in the interference pattern. Its length variation can be calculated by counting the fringes that pass
through the visual field while the telescope is under cooling/heating.
Figure 34. interference fringes of Michelson interferometer. The center of the screen is bright when the optical path
difference is an integral number of wavelengths. The condition for a maximum of intensity at the center of the screen is:
2dn
35
6 Pound-Drever-Hall (PDH) Locking
Techinque
Pound-Drever-Hall Locking technique was invented in order to stabilize the frequency of a laser by locking
it to a Fabry-Perot cavity, but it can actually also be used to lock a cavity to a laser [10, 11]. This technique
also allows measuring small changes in length with very good precision, thus can be used to measure small
changes in length of the cavity to study its stability.
The PDH technique is part of a control system. To lock a laser to a cavity, the control system changes the
frequency of the laser in order to keep the laser in resonance with the cavity.
Briefly: locking a laser to a cavity using PDH technique.
A laser light is phase modulated by an EOM (Electro-Optic Modulator) at 27.9MHz. This means the two
sidebands are added to the carrier frequency (at + 27.9 and – 27.9 MHz with respect to the carrier). The two
sidebands are outside the FWHM of the cavity, so that if the laser is on resonance with cavity, the sidebands
are not, and thus they do not enter into the cavity, but are rather reflected back by the input mirror. The
reflected beam, consisting of the two sidebands at the modulation frequency and a shifted carrier component,
is gathered by a photodetector (PD). The output of this PD is compared in a mixer with the modulation
signal. The output of the mixer (product of its inputs) is an error signal that is sent to the laser to adjust its
phase, in order to have the laser always on resonance with the cavity.
If the feedback is correctly set up, the system should adjust the frequency of the laser until it is on resonance
with the cavity and then hold the laser locked there, compensating for any little noise.
The PDH is usually implemented by modulating in phase, but thinking about changes in frequency instead of
phase is probably easier to understand how it works. The power of the beam reflected by a cavity is at a
minimum when the beam is on resonance with the cavity. Outside resonance, i.e. above and below it, the
power is equal. Thus one has to find a way to discover which way to change the frequency in order to push
the system back on resonance.
Looking to Figure 38, it can be seen that when the frequency of the beam is above resonance, an increase of
the frequency means an increase of the power in the reflected beam; on the contrary, below resonance an
increase of the laser frequency couples in a decrease of the reflected beam power. So, if the frequency is
modulated, the system is above resonance (that means that the mean value of the frequency of the laser is
above the resonance frequency) if the reflected power varies in phase with the modulation; similarly, it is
power
frequency
Figure 35. Reflected light intensity from a Fabry-Perot cavity as a function of the laser frequency.
36
below resonance if the reflected power is 180° out of phase with the modulation (that means that the non-
modulated frequency is below resonance) (see Figure 36).
Figure 36. Comparison between the error signals of a modulated and non-modulated system. If the system is non-modulated
(left column), the reflected power P is above the minimum value regardless of the beam frequency being above or below
resonance, and the control loop has no way of telling in which direction to change to frequency. If the system is modulated, on
the other hand, the power is still (obviously) above it minimum value, but it is either in phase on in anti-phase with the
modulation signal depending on the mean frequency being above or below resonance. The sign of the demodulated power
tells the feedback in which direction to push the frequency.
When the reflected power is demodulated using the modulation frequency as a reference, the sign of the error
signal generated by the mixer is different on either side of resonance, and it is zero when the system is
exactly on resonance, as shown in Figure 39.
In actual implementations, phase modulation is preferred to frequency modulation because it is much easier
to realize.
37
6.1 PDH Scheme and theory
Figure 37. Sample of a PDH setup.
The laser is sent from the laser to a Faraday isolator, that is a device which transmits light in a certain
direction while blocking it in the opposite one (thus preventing reflected beams to go back in the laser). From
there, part of the beam is intercepted for use to beat with a very stable beam (this beat is used to measure
changes in frequency of the laser, and thus also changes in length of the cavity under test, related by / =
L/L), and part is phase modulated using an electro-optic modulator (EOM) at 27,9 MHz and then sent into
the cavity. The reflected light is gathered by a photodetector, which output is mixed with the modulation
signal. The error signal produced is then sent to a controller and from there to the pzt (piezo) and the
temperature controllers of the laser, which control the frequency of the laser. At this point, the feedback loop
is complete and the laser is locked on resonance.
A little bit of theory…
Theoretically, a simple electric field is given by
After it has passed through an EOM, it can be written as
where m is the modulation index, or modulation depth, of the EOM, and Ω = 27,9 MHZ is the driven
frequency of the modulator.
For m << 1, we can write
Written in this way, it can be seen that the EOM adds two sidebands, located at ω ± Ω, to the carrier beam,
as shown in Figure 38.
38
Figure 38. Carrier and sidebands of a beam after passing through an EOM with modulation frequency Ω.
The beam is then sent to a polarizing beam splitter, which transmits only the light that is polarized in a
particular direction, while it reflects the light polarized in the perpendicular one. The transmitted light is sent
to a λ/4-plate, which transforms a linear polarization at 45° with respect to its axis in a circular polarization.
Finally, the beam passes through a vacuum window into the cavity.
The reflected beam hit again the λ/4-plate that now converts the circular polarization in a linear one, rotated
by π/2 with respect to the beam hitting the plate on the other side. When the beam reaches the polarizing
beam splitter, it has the right polarization to be reflected and be sent to a photodiode.
If both the sidebands and the carrier are on resonance, the field incident on the photodiode is
where Tr is the transfer function of the reflected field, given by
where L is the length of the cavity (60cm).
If the modulation frequency, Ω, is larger than the cavity linewidth (i.e., Ω is not in the Gaussian peak of the
resonance frequency), when the carrier is on resonance, the sidebands will not, and they will be totally
reflected by the first mirror. In this case, the transfer functions of the sidebands are the main terms of the
reflected field incident on the photodiode.
The intensity on the photodiode is the magnitude of the EPD field
The term of interest is the time-varying (AC) component, thus, neglecting the DC term and second-order
terms, the intensity on the photodiode is
39
When the output signal of the photodiode is mixed with the modulation signal, the term proportional to
sin(Ωt) is pulled out from the mixer and the error signal is
Figure 39. PDH error signal: /mE02 versus ω/FSR, when the modulation frequency is smaller than the cavity line-width.
Figure 40. Example of PDH error signal: /mE02 versus ω/FSR , when the modulation frequency is bigger than the cavity line-
width.
Since the goal of PDH locking technique is to keep the laser on resonance with the cavity, let‟s consider the
error signal near resonance.
As already said, if the carrier is resonant but the modulation frequency is outside the cavity linewidth, the
sidebands will be totally reflected back and the transfer function for the sidebands will be nearly one. In this
case, the error signal can be written as
For small ω yields, Tr(ω) can be linearized and it gives, to first order,
ε /
m E
02
ε /
m E
02
40
The error signal is thus
For a cavity where r1 = r2 = r, rewriting the equation for the finesse
such that
the error signal can be written as
where Δ cavity is the FWHM of the cavity. The error signal is, in this case, proportional to the change in
length of the cavity ( / =L/L).
Figure 41. Example of error signal: a cavity with FSR = 600 MHz, E0 = 1, m = 0.1, r1 = r2 = r = 0.99.
Figure 41 shows clearly that when the laser is on resonance with the cavity, the error signal is zero. The error
signal becomes very big as soon as the frequency of the laser moves even only a little bit out of resonance.
41
The error signal is sent to some control electronics that will control the slow (temperature) and the fast (PZT)
laser actuators.
The laser will be locked to the cavity as long as the gain and the bandwidth of the control electronics are
adequate.
6.2 Experimental Application PDH technique is used to lock Laser L to the SiC cavity under test and reference laser RL to a Zerodur
cavity. This allows the stability measurement of the SiC cavity.
Zerodur very low CTE (on the order of 10-8
K-1
) is the reason why Zerodur cavities are used as reference
cavities. The measured stability of the reference cavity are about one-two order of magnitude better than the
requirements for the telescope. Figure 45 shows the length stability of the Zerodur cavity [11]; it can be seen
that length variations are on the order of 10-100 fm/√Hz in the interesting frequency range.
Both the SiC and the Zerodur cavities are separately shielded with gold-coated stainless-steel thermal layers,
as stability studies require an environment with temperature stability such that expansions and contractions
related to temperature are smaller than the requirements. Each cavity is also enclosed in a separate vacuum
chamber.
The frequency stability of the SiC cavity is measured by recording the beat frequency between L and RL.
Frequency [Hz]
No
ise
spec
tru
m [
m/√
Hz]
Figure 42. Zerodur length stability.
42
Figure 43. L and RL setup and beat note
If each of the two cavities has a very stable resonant frequency, then the frequency difference between the
lasers will remain constant. By looking at the changes of the beat note, one can measure the relative stability.
As already said, a glass plate is used to pick up part of the beam of each laser. The two beams are combined
at a power beamsplitter and are mode matched to maximize the beat signal on a photodiode. The beat note
between the beams is measured with a frequency counter.
Table 3. Laser model
Laser Serial # PZT tuning coefficient Experimental PZT coefficient
L 2751 5.2369 MHz/V 5.36 MHz/V
RL 2258 4.653366 MHz/V
Table 4. Modulator model
EOM model Driven frequency Vpi (voltage required to produce
a phase shift of 180°)
m (modulation
depth)
New Focus 4003 27.9 MHz 16 V @ 1060 nm >0.2 rad/V
Table 5. Beat note and operating parameters of the L and RL lasers
Beat note
frequency
Voltage
[V]
L temperature
[°C]
L power
[mW]
L voltage
[V]
RL temperature
[°C]
RL power
[mW]
RL voltage
[mV]
90 MHz -1.55 47.1490 216 -1.06 49.208 186 - 400
43
7 Phase-Locked Loop (PLL)
For the SiC CTE measurement, the PDH locking technique is not suitable. Thus, a different technique is used
to measure the CTE while the telescope is cooled down till -70°C. Instead of keeping the laser L locked to
the cavity and measuring the change in its frequency by comparison with the reference RL, the laser L is
locked to RL using a Phase-Locked Loop. As the cavity changes in length, its resonances will change too
and will correspond or not to the frequency of L, depending on the length of the cavity.
A PLL is a circuit that creates an error signal. The phase of this error signal has a fixed relation with the
input reference signal. If there is a phase shift, a control system acts such that this shift is reduced. The phase
of the output error signal is thus locked to the input signal.
7.1 Application to the experiment A PLL control system is used to keep laser L locked to the reference laser RL, with a constant difference in
frequency between them, in order to measure the coefficient of thermal expansion (CTE) of SiC in a
temperature range between room temperature and -70° C.
Since L is locked to RL, changes in the length of the telescope can be measured by monitoring the resonant
modes of the cavity and the fringes of the Michelson.
For the cavity, the distance between two 00-mode is equal to one FSR (250MHz), that is associated with a
length change of
For the Michelson, the separation between a maximum and the nearest minimum is λ/2, that means, since the
beam has to travel back and forth along the arm, a change in the long arm length of
44
Figure 44. PLL setup to keep L locked to RL while cooling down the tank.
The beat signal between L and RL is gathered by a photodetector (PD). The difference in frequency
between L and RL is made to be roughly 10-15 MHz by changing the temperature of the laser L. A sine
wave of frequency F = 10 MHz is then mixed with the AC output of the PD. The output of the mixer can be
considered an error signal and is connected to the PLL, which controls the frequency of the laser L acting on
its slow (temperature) and fast (PZT) frequency actuators. Thus, the PLL locks the laser L to the reference
laser RL, forcing the difference in frequency, , between them to be equal to F, and keeps it locked. This
are the operating temperatures obtained in our experiment:
Temperature of L: TL = 41.1568 °C
Temperature of RL: TRL = 49.2081 °C
The PLL is a feedback control system that doesn‟t work (like the PDH) if the difference in frequency is
too far from F (the PDH doesn‟t work if the laser frequency is too far from the resonance frequency of the
cavity); that‟s why the frequency of L is tuned to be near the one of RL before setting the PLL.
8 Measurements and Results
8.1 Cavity and Michelson characteristics Some important parameters, both of the cavity and of the Michelson, have been measured, in order to
characterize the testing setup.
Visibility of the Cavity
The visibility of the cavity tells how well the system can distinguish between a resonance and non-resonance
condition and affects the quality of the error signal and eventually the stability of the laser locking.
DC Voltage of reflected beam (laser locked to the cavity): VDC
L= 380 mV
45
DC Voltage of reflected beam (laser NOT locked to the cavity): VDC
NL= 405 mV
The ratio of the two voltages is: VDC
L/VDC
NL = 380/405 = 0,94
The visibility of the cavity is given by: 1- ratio = 1-0,94 = 0,06 6%
This visibility is definitely not good. However, it was sufficient to perform a stability measurement near the
pm/√Hz level.
After the first stability measurement and cool down, the alignment of the cavity was revisited and improved.
The visibility was increased to 65%. Preliminary measurements show that this did not improve the measured
stability noise, suggesting that the low visibility was not the limiting factor for the measurement presented
here.
PZT coefficient
The PZT controller is a fast actuator for the laser frequency. The peak of the beat note between L and RL
measured by the spectrum analyzer changes as a different voltage is applied to the piezo of the laser.
Figure 45. Beat note at different voltages applied to the piezo.
Applying 0 pk-pk voltage, the beat note peak is at 838,1 MHz; applying 10 V pk-pk, the beat note is at 891,7
MHz.
The PZT coefficient tells how much the frequency changes if the voltage is changed, thus it is given by
This is not so far from the nominal value written in the datasheet (that is 5,2369 MHz/V), but this difference
is probably due to the fact that the Laser is “old”, and this characteristics changes with time.
Gain of the PZT controller
The error signal used to lock the laser to the cavity is sent to the controller, which has to outputs: one feeds
the fast (PZT) actuator and one feeds the slow (temperature) actuator of the laser.
The error signal that feed the PZT is amplified in the controller with a certain gain. To find this gain G, a
square wave is sent to the controller (and to an oscilloscope), and the error signal that feed the PZT controller
is monitored with an oscilloscope.
46
Figure 46. Electronics used to find the PZT controller gain.
Table 6. Gain of PZT controller
Input Output Gain Total gain G
Square wave 2 V -11,4 V 5
~ 5 0 V -1,4 V
Square wave 200 mV -220 mV 4,9
0 V -1,2 V
FSR
A peak can be seen on the spectrum analyzer when the laser L is on resonance with the cavity. The frequency
of L can be changed using the temperature actuator of the laser to find adjacent resonances.
Once a beat note (between L and RL) is seen on the spectrum analyzer, another resonance can be found
changing the frequency of the laser L until another peak appears. The difference in frequency between the
two peaks is the free spectral range FSR of the cavity.
First peak: 1 = 92 MHz
Second peak: 2 = -158 MHz
Figure 47. FSR
47
Finesse
The cavity finesse tells how the cavity is a good filter. In fact, since the finesse is defined as the FSR divided
by the FWHM, the bigger the finesse, the smaller the frequency range allowed to resonate with the cavity.
A high finesse means a big slope of the error signal.
A resonant condition is found, by hand, changing the temperature of the laser that in turn changes the
frequency.
A triangular wave is applied to the piezo of the laser in order to change the laser frequency around the
resonance. The transmitted light from the cavity shows a Gaussian peak on the oscilloscope, where the
maximum is reached when the laser is exactly on resonance.
By measuring the FWHM (Full Width at Half Maximum) of this peak and the slope of the triangular wave
that feeds the piezo, the finesse can be found.
Figure 48. Finesse measurement. Above: the triangular wave applied to the piezo. Below: the Gaussian peak of the
transmitted light.
The slope is the ratio between the pk-pk Voltage and half the period:
The Gaussian peak has:
Height: V = 564 mV
Half height: V/2 = 282 mV
FWHM = 352 μs
Converting this width in frequency
the finesse is finally
V
t
48
This finesse is very low, compared to the usual finesse of a good cavity. Again, it was improved to about 600
after the first stability measurement and the first cool down. However, this did not apparently improve the
quality of the measurement.
Discriminator of the Error signal
The discriminator D is a parameter that tells how much the voltage of the error signal changes when the
frequency changes by a given quantity.
Figure 49. PD DC signal vs time and error signal.
The same triangular wave used before is used to measure this parameter. The voltage from 10% to 90% of
the error signal and the correspondent time interval of the Gaussian are measured.
V = 70 mV
t = 45 μs
The time interval is then converted in frequency
The discriminator D is then
Michelson Visibility
The Michelson visibility is a measure of how well dark and bright fringes can be distinguished.
Looking at the reflected beam of the Michelson, the maximum and the minimum voltages of the sine wave
shown on the oscilloscope can be measured.
The visibility is then given by
t
V
49
Vmax [mV] Vmin [mV] Visibility %
47,2 15,2 51,28
A quite good visibility is 50-60%, thus this visibility is not bad.
8.2 Telescope Stability As said in § 1.4, the SiC telescope stability can be measured by mounting a cavity on the telescope, with one
mirror at each end.
To do this measurement, a laser L has to be locked to the cavity with the PDH technique described in § 6, in
order to be able to measure the dimensional stability at the picometer level by monitoring the frequency
stability of L.
The telescope is shielded, as said in § 6.2, with gold-coated stainless-steel thermal layers, and is closed in a
vacuum chamber, that is pumped down with a turbo pump. See Figure 50.
50
Figure 50. Telescope shielded with thermal layers. In front of it, the optic table.
The stability is measured both at room temperature and at -70°C.
The data taken are:
- The frequency of the beat note between the laser L that is sent inside the SiC cavity and the reference
laser RL. This can be easily converted in a length stability of the cavity by the relation:
- The time.
- The temperature reading of the seven sensors mounted in different positions on the telescope and of
the one outside the tank;
The sensors are sampled by a computer and the frequency of the beat note is read by a frequency counter.
Other measured parameters are the reflected signal and the error signal.
Thermal
layers
51
Figure 51. Experimental telescope setup.
Photodetector for the reflected beam: EOT amplified InGaAs Detector, ET-3000A-FC-DC
Photodetector for the beat note between L and RL: V 1611 IR, 1GHz Low Noise Photoreceiver
BEAT NOTE
L (Laser) Temperature 47,1490 C
Power 216 mW
RL (Reference Laser) Temperature 49,2081 C
Power 186 mW
Voltage of the beat note (on the oscilloscope) -1,55 V
L Voltage -1,06 V
RL Voltage -0,49 V
Mean frequency of the beat note: 93,5 MHz
Amplitude: 1,5 dBm
Data have been analyzed using Matlab.
Cavity Upper
mirror
Cavity Bottom
mirror
Sensor‟s
cables
sensors
SiC structure
Michelson
mount
52
The stability of the SiC cavity obtained by converting the beat note frequency in length change is shown in
Figure 52, along with the LISA requirement.
It can be seen that measured stability of the SiC cavity does not match the LISA requirement, except in the
highest frequencies, around 1 Hz. However, the measure could be limited by instrumental noise, and not by
intrinsic SiC cavity length stability.
Figure 52. Linear spectral density of the SiC cavity length. 10 windows means that the data are obtained mediating on 10
windows with an overlap of 50%. The red line is the LISA requirement.
In Figure 53 is shown the temperature noise spectrum. Temperature changes in the lab could propagate
through the thermal shields into the cavity, causing contraction and expansion of the cavity. The cavities of L
and RL are both in the same lab, but there is probably a different temperature effect on them due the different
vacuum chamber they are installed in.
Below, in Figure 54, can be seen that there could be a relation, at low frequencies (below 1mHz), between
the beat note frequency and the temperature.
10-5
10-4
10-3
10-2
10-1
100
10-4
10-2
100
102
104
106
108
S1/2
L [
pm
/Hz
1/2
]
Frequency [Hz]
Cavity Length Spectrum @ Room Temp
single window
10 windows
LISA requirement
53
Figure 53. Temperature noise spectrum. The blue line shows the spectrum analyzed with a single window; the green line
shows the spectrum analyzed with 10 windows.
Figure 54. Relation between beat note frequency and temperature. The plots suggest that there could be a relation between
them..
10-5
10-4
10-3
10-2
10-1
100
10-6
10-5
10-4
10-3
10-2
10-1
100
101
S1/2
T [
K/H
z1/2
]
Frequency [Hz]
Temperature Spectrum
single window
10 windows
0 1 2 3 4 5 6
x 104
6
7
8
9
10x 10
7 beat note vs time
time [s]
beat
note
[H
z]
0 1 2 3 4 5 6
x 104
24.84
24.86
24.88
24.9
24.92temperature vs time
time [s]
Tem
pera
ture
[C
]
54
Figure 55 shows that there is actually a relation between the beat note frequency and the temperature. Thus,
it can be said that the thermal shields do not suppress well enough temperature fluctuations at very low
frequencies.
Excluding the data at the very beginning of the measurement (the blue ones in the plot), where probably the
temperature has to stabilize, the relation appears to be linear, as demonstrated by the fit (black line).
Figure 55. Linear relation between the beat note frequency and the temperature.
Another plot of the linear spectral density of the cavity, with the temperature noise subtracted from the beat
note, is shown in Figure 56.
It can be seen that, because of the noise in the temperature measurement, at high frequencies the plot is worst
than before. However, for low frequencies the plot is closer to the requirement, although it does not match it
yet.
24.85 24.86 24.87 24.88 24.89 24.9 24.916.5
7
7.5
8
8.5
9
9.5
10x 10
7 Beat Note vs Temperature
Temperature [C]
Beat
Note
[H
z]
excluded data
good data
fit y=-6.1e+008x+1.5e+010
55
Figure 56. Upper: Linear spectral density of the cavity, subtracting the temperature noise. Lower: LSP with original data
and with temperature corrected data.
10-5
10-4
10-3
10-2
10-1
100
100
101
102
103
104
105
S1/2
L [
pm
/Hz
1/2
]
Frequency [Hz]
Cavity Length Spectrum @ Room Temp
single window
10 windows
LISA requirement
10-5
10-4
10-3
10-2
10-1
100
10-4
10-2
100
102
104
106
108
S1/2
L [
pm
/Hz
1/2
]
Frequency [Hz]
Cavity Length Spectrum @ Room Temp
temperature corrected data
original data
LISA requirement
56
Other sources of noise could affect the measurements. Some of them are the electronics, the radio frequency
amplitude modulation (RFAM) – that can affect the PDH error signal through a misalignment of the EOM –,
the feedback controllers, the heating of the mirrors due to the power of the laser that is stored in the cavity.
These and other sources are under investigation and other measurements are planned, with different setup
and electronics.
During the cool down, a misalignment of the beam into the cavity caused the cavity signal to be lost and did
not allow a stability measurement at -70°C.
Other measurements are planned, in order to be able to keep a better alignment and to take a stability
measurement at temperatures below 0°C. The cool down will be done in steps of 10°C, with an adjustment of
the alignment from outside the tank (optics on the optic table) at every step.
Figure 57. SiC Telescope with Michelson interferometer and Fabry-Perot Cavity setup and optic board.
cavity
Michelson
Optic board
57
8.3 SiC CTE As already said in § 7, for the CTE measurement, the locking setup used is the PLL, which lock laser L to
laser RL.
Once L is locked to the reference laser RL, the output of the detector that gathers the transmitted light from
the cavity is connected to a multimeter, which in turn is connected to a computer that shows the trend of the
voltage with respect to time. Peaks are shown every time the cavity passes through a mode resonant at the
frequency of L (and RL), with the 00-mode distinguishable because bigger (i.e., with higher intensity) than
the other modes.
While the tank that contains the telescope is cooled down, using liquid nitrogen, till -70C, data are taken of:
- Intensity transmitted by the cavity;
- Temperature of the telescope;
- Fringes of the Michelson;
- Time.
The length of the telescope, that is both the length of the cavity and the length of the long arm of the
Michelson, changes during the cool down.
CTE with the cavity.
Counting how many peaks of the 00-mode are registered (there is one every 0,2C temperature decrease),
and taking in mind that the FSR, i.e. the distance in frequency between two 00-mode peaks, is 250 MHz and
that
we can measure the change in length of the cavity L.
CTE with the Michelson.
The distance between a maximum and a minimum in an interference figure is equal to λ/2. If the long arm of
the Michelson moves by λ/4, the path of the light moves by λ/2.
No modes 00-mode and higher
mode
Figure 58. In the right figure of the first picture it can be seen that no modes are detected. In the right figure of the second
picture can be seen two modes: the deepest one is a 00-mode, the other one is a higher mode.
58
Thus, counting by how many fringes the interference figure of the Michelson changes, the change in length
of the arm is:
Where both maximum and minimum are taken into account (i.e. in equal to 2 in going from a
bright fringe to the next)
Then, the coefficient of thermal expansion α of the SiC can be calculated from this change in length L, of
both the cavity and the Michelson arm, using the relation
The coefficient of thermal expansion is measured both with the data of the cool down and the data of the
heating (back to room temperature).
COOLING
Because during the cool down the 00-mode of the cavity was lost, the CTE was calculated only with the data
of the Michelson.
Figure 59 shows the trend of the CTE measured with the Michelson, along with the seller (Coorstek)
nominal trend.
Figure 59. CTE measured with the Michelson while cooling down to -70°C
The experimental CTE seems to be lower than the nominal one. The spikes in the plot could be due to the
athermal mounts that hold the beam splitter and the mirror of the Michelson (see Figure 60).
-70 -60 -50 -40 -30 -20 -10 0 10 20 300
0.5
1
1.5
2
2.5
3x 10
-6
temperature [oC]
S
iC [K
-1]
CTE vs temperature - cooling
Michelson (w/mounts)
Coorstek
59
Figure 60. Michelson and Cavity.
HEATING
After the first measurement, the cavity was better aligned, reaching a visibility of 65%.
The measure of the CTE while heating, with the cavity better aligned, are shown in Figure 61. It can be seen
that none of the experimental CTE values match the nominal (Coorstek) ones.
The noise of the Michelson plot is probably due, as before, to the mounts. Also the CTE measured with the
Michelson is lower than that measured with the cavity: again, we suspect that thermal expansion in the
mounts could affect the overall measurement.
Because the peaks of the 00-modes were not very well distinguishable from the other modes, the 00-modes
were counted “by hand”, looking at the pictures that a camera took every 1 second of the transmitted beam.
The spikes in the cavity results are probably due to this fact: there are for sure human mistakes in the
“counting by hand”.
Figure 61. CTE measurement with cavity and Michelson while heating.
The mounts of the Michelson are now removed and other measurements are planned.
-80 -60 -40 -20 0 200
0.5
1
1.5
2
2.5
3x 10
-6
temperature [oC]
S
iC [K
-1]
CTE vs temperature - heating
Michelson (w/mounts)
Cavity
Coorstek
L(T)
Athermal
mounts
60
61
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1. P. Bender et al. LISA: System and Technology Study Report. Technical Report LISA ESA-
SCI(2000)11, ESA (2000)
2. NASA LISA website, http://lisa.nasa.gov/index.html
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May 4, 2009
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9. Cogswell et al. Optical Resonators and Mode Matching. Advanced Optics Lab-ECEN 5606 notes,
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April 16, 1998.
11. Rachel J. Cruz. Development of the UF LISA Benchtop Simulator for Time Delay Interferometry.
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