Download Report on waist dependence of photo

JRA3: STREGA - TASK T2
Report on waist dependence of photo-elastic noise
Leading group: Firenze, INFN (3)
Report on the high finesse cavity at low temperature
Leading group: LKB, CNRS/IN2p3 (2)
First year report (04/2004, 04/2005)
1. Introduction
The displacement sensitivity of optical interferometers is limited by several dissipative
phenomena. One of such effects is the dynamic photothermal effect: fluctuations of the laser
power absorbed by the mirrors are converted into surface displacements through thermal
expansion. A fundamental limit to the sensitivity is therefore produced by the shot noise of
the absorbed radiation.
This effect was studied theoretically by Braginsky et al. [1] and Cerdonio et al. [2]. We have
recently published the first experimental determination of the frequency dependence of the
effect [3]. This phenomenon depends on material parameters and on the geometry. As a
consequence, varying the mirror material and its temperature, the typical frequency
dependence and size can span over several orders of magnitude. Therefore it is crucial to have
an accurate model extending over such a large range.
The purpose of Task T2 is to carry out a deep experimental investigation of dynamic
photothermal effect by using the effect induced by an intensity modulation of the light
entering a Fabry-Perot cavity. This measurement can be performed over a wide frequency
range, with the temperature ranging between tenths of Kelvin up to room temperature, and
with low-losses optical coatings.
We present in this report the first-year activities, including the two deliverables on the
measurement of the photo-elastic noise (deliverable #17) and on the setup of cryogenic highfinesse cavities (deliverable #16).
We have in particular investigated the photo-elastic noise over a very wide frequency range,
including finite mirror size effects at low frequency and coating effects at high frequency. A
specific extension of the basic model (for semi-infinite homogeneous mirror) is being
developed, at the purpose of completely explaining the results. We have also built a very
high-finesse cavity (finesse  200 000) suitable for cryogenic application.
2. Waist dependence of photo-elastic noise (deliverable #17)
2.1 Experimental measurements
The measurement system is composed of a phase-modulated laser source and two Fabry-Perot
cavities: a reference cavity (C1) for the laser stabilization and a measurement cavity (C2) for
monitoring the frequency modulation due to photo-thermal effect.
The experimental setup is sketched in Figure 1. The light source is a cw Nd:YAG laser based
on a non-planar ring oscillator geometry (Lightwave, mod. 126-1064-100), emitting 100 mW
at 1064 nm. After a 40 dB optic isolator, a resonant electro-optic modulator (EOM1) is
contained in a temperature stabilized aluminum box. EOM1 provides phase modulation at
13.3 MHz with a depth of about 1 rad, used for the Pound-Drever-Hall detection scheme.
C1
QW
PD1
Frequency
servo loop
Oscilloscope
+
PC
PBS
AOM
Cavity
servo loop
13.3 MHz
C2
PD2
Laser
O.I.
PD4
EOM2
EOM1
BS
PBS
QW
PD3
Figure 1: Experimental setup for the measurement of the photo-elastic effect induced in cavity C2 by an
intensity modulation of the incident laser beam via the electro-optic modulator EOM2.
The laser radiation is then divided into two beams by a variable beam-splitter composed of an
half-way plate and a polarizing cube. Each beam is directed toward one cavity and spatially
mode-matched with suitable lenses. Optical circulators, composed of a polarizing cube beamsplitter (PBS) and a quarter-of-wave plate, allows to collect the light reflected by the cavities
on photodiodes PD1 and PD2, whose rf output are de-modulated to get the respective
dispersion signals.
The beam going to C1 is frequency displaced by an acousto-optic modulator (AOM) in
double-pass configuration. The AOM is driven by an home-made voltage-controlled oscillator
(VCO) with a central frequency of 110 MHz and a bandwidth of about 30 MHz. The VCO
linearity is accurately checked and calibrated.
The intensity of the second beam (directed to C2) is modulated by means of a wideband
electro-optic modulator (EOM2) whose axes are opportunely oriented such that, by applying a
sinusoidal signal to the electrodes of EOM2, we can change the power transmitted by the PBS
of the optical circulator. The direct reflection of this PBS is monitored by a photodiode (PD3).
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The two Fabry-Perot cavities are made by mirrors coated on fused silica substrates, which are
optically contacted to a 20 cm long Zerodur spacer. The cavities are custom products
provided by Research Electro-Optics (REO). The input mirror is flat, with a nominal
transmission of about 100 ppm, while the output mirror is concave, with a radius of 1 m and a
HR coating. The corresponding beam waist is 0.37 mm on the flat mirror and 0.41 mm on the
concave one. The measured linewidth of C2 is 20 kHz giving a finesse of 38000.
Each cavity is placed on a double stage mechanical suspension which isolates from
mechanical noise and it is enclosed in a vacuum chamber evacuated by an ion pump. The
cavity temperature is actively stabilized within 10-3 °C. By varying the cavity temperature
between 20°C and 100°C, it is possible to change the length of the cavity by roughly 0.5 m,
corresponding to one Free Spectral Range (FSR), and consequently to tune coarsely the
optical resonant frequencies of the cavities. Fine adjustments of the laser and AOM
frequencies allows to bring both laser beams into resonance with the respective cavity. The
cavities are designed as length reference for the AURIGA optical readout which is optimized
for a frequency range around 1 kHz where we had measured a relative displacement noise of
few 10-18 m/√Hz. This noise increases as 1/f for lower frequencies.
The FM sideband signal from PD1 is used as discriminator for frequency locking the laser
with a servo loop that has a unit gain frequency of 30 kHz and a steep slope below it. We
have checked that the gain of this loop is high enough in the whole range of measurement,
since the residual modulation in the error signal is always negligible. The de-modulated signal
from PD2 is used in a servo loop acting on the VCO with a bandwidth of about 10 kHz, so
that the resonant condition is kept in both cavities.
To observe the dynamic photo-thermal effect we apply a sinusoidal voltage to EOM2. The
power incident on C2 is thus modulated around an average value, as well as the light power
absorbed by the mirrors, and a cavity length change is originated due to their non-zero
thermal expansion coefficient.
We simultaneously acquire the signal from PD3, to monitor the power modulation (the
intensity variations detected by PD3 are exactly complementary to the ones of the radiation
impinging on C2), and the correction signal at the output of the VCO. These two signals are
successively elaborated with a numerical lock-in to extract amplitude and phase of the
component synchronous with the modulation. The frequency ranges between 0.3 mHz and
8 kHz, with a modulation depth of about 30%.
Special care is taken for the measurements at high frequency, for two different effects. First,
we must consider that a sinusoidal voltage applied to EOM2 gives directly a phase
modulation of the beam, i.e., a frequency modulation with a depth proportional to the
frequency. We have verified that this direct modulation becomes more important than the
photo-thermal effect above 200 Hz. To get rid of that we have set the EOM2 axes and its
voltage offset in order to modulate around a working point where the transmission is
maximum. In this way, the first-order amplitude modulation is minimized, but we have a
strong modulation at the double frequency that is used to monitor the photo-thermal effect.
On the contrary, the direct phase modulation is still linear and it is negligible at the double
detection frequency. The second important effect is due to the low gain of the servo loop on
the VCO at high frequencies (above few kHz), implying that the modulation amplitude
deduced from the correction signal is under-estimated. For the measurements in this region
we have also acquired the error signal (i.e., the de-modulated output from PD2) and we have
used it together with the correction signal to obtain the correct modulation amplitude and
phase.
The final results are reported in Figure 2, together with the basic theoretical curve (a)
calculated as in [2]. The agreement is very good in the middle range, and a deviation of the
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experimental data from this curve is very clear both at low frequencies (below 10 mHz) and in
the high frequency region (above 30 Hz). Such effects are due to physical phenomena not
considered previously and we have developed a new theoretical description to account for
these discrepancies.
2.2 Theoretical description of the photo-elastic effect
Thermoelastic effects are computed in [2] by solving the heat propagation equation in the
substrate, assuming that the heating is produced at the surface of the mirror by the laser
absorption, and for a semi-infinite mirror. We have adapted this approach in order to account
for the two deviations experimentally observed. One actually has two assumptions in the
theoretical approach which may not be valid at very low and very high frequencies: at very
low frequencies one has to take into account the finite size of the mirror, whereas the
photothermal effect occurring in the coatings may become dominant at high frequencies.
Figure 2: Observed photo-elastic displacement (top) and phase (bottom) as a function of the modulation
frequency of the incident intensity. Solid curves are theoretical results from the basic model
(curve a), taking into account the finite size effects (curves b), and the coating effects (curve c)
2.2.a Finite size effect
At very low frequency, namely below 10 mHz for our silica substrates, the thermal length
becomes of the same order as the mirror size, and the description of the mirror as a semi4
infinite space is no longer valid. To take into account the finite size of the mirrors, we have
considered the following simple model. The mirrors are actually optically glued on their
coated face to the Zerodur spacer which has a very low expansion coefficient (less than 10 9)
compared to the one of the silica substrate (around 106). As a consequence, when a thermal
wave reaches the mirror edge, the induced expansion is modified by the Zerodur spacer. The
main effect is to push back the whole mirror, inducing an increase of the cavity length and a
reduction of the photothermal effect measured at the center of the mirror.
For an infinite mirror, the displacement X measured by the optical beam corresponds to a
spatial average of the mirror deformation u z r  at point r of the surface by the profile of the
optical beam:
X
1
d 2 r u z r e  r
2 
r0
2
r02
,
where r0 is the waist of the optical beam. Finite-size effects are taken into account by a
redefinition of the measured quantity, where we subtract the displacement of the mirror edge
calculated in the semi-infinite approximation:
X   X  u z r  R ,
where R is the mirror radius. This quantity can be computed as in ref. [2] and related to the
various experimental parameters. The result is plotted in Figure 2 (curves b), both for the
amplitude and phase of the photo-elastic response. The main effect is to suppress the non
physical logarithmic divergence at low frequency. The new response amplitude now presents
a flat behavior, in very good agreement with the experimental results in the frequency range
from 1 to 10 mHz. One can also observe a faster variation of the phase at low frequency,
which is in better agreement with the experimental data.
At lower frequencies, other phenomena can play an important role and this simple model may
no longer be valid: one has to take into account thermal losses induced by the Zerodur spacer
or due to black body radiation, drifts of the cavity length, etc. A better characterization at very
low frequency would certainly require a theoretical approach via a finite element model to get
a better understanding of these somewhat complex loss mechanisms.
2.2.b Photothermal effect in the coatings
At high frequencies, namely above 100 Hz, the thermal wavelength becomes comparable to
the size of the coatings (of the order of 7 m). In this case, photothermal effects in the coating
cannot be neglected. The temperature profile inside the coating also becomes non uniform and
filters the heat flow sent into the substrate. One can clearly observe this effect in the
experimental data, both on the amplitude and phase of the response.
We have developed a first model, in the framework of a mono-dimensional description. This
approach is justified since transverse dimensions are huge compared to the thermal length at
high frequency (centimeters compared to hundreds of microns). We have then taken into
account the reflection of the thermal wave at the interface between the layer and the substrate.
Although the calculation is in progress, Figure 2 shows a preliminary result both for the
amplitude and phase of the response (curves c). One gets a very nice agreement with
experimental data, both for the behavior of the amplitude and for the dramatic change in the
phase of the response.
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3. High-finesse cavity at low temperature (deliverable #16)
Cavities with very high finesses (600 000 or more) have already been built with mirrors
having a transmission and loss of the order of a few ppm. In our experiments, however, the
transmission of the coupling mirror of the cavity has to be large as compared to losses to
preserve the sensitivity of the measurement to mirror displacements. We have thus decided to
build a cavity with a front mirror having a transmission of 20 ppm, leading to a cavity finesse
in the range between 200 000 and 300 000.
The mirrors have been made by Jean-Marie Mackowski at the LMA in Lyon. We have first
measured the centring of the coatings with test mirrors since we are using a Titane-Sapphire
laser working at 810 nm, instead of a Yag laser (see Figure 3). This laser can easily be swept
to find the optical resonance of our rigid high-finesse cavity.
Figure 3: Measured variation of transmission for test mirrors as a function of the laser wavelength
The mirrors made at LMA are based on super-polished substrates from REO. We have
measured the transmissions of the front and end mirrors of the cavity: the front mirror has a
transmission varying from 18 ppm to 25 ppm for a 50 nm change of the laser wavelength
around 800 nm, whereas the transmission of the end mirror is always smaller than 3 ppm.
Losses are expected to be on the order of a few ppm. The possibility to vary the coupling
transmission will allow us to tune the finesse and the bandwidth of the cavity, without
appreciable change of the losses.
We have built a new single-ended high-finesse cavity based on these mirrors. Both mirrors
are fixed to annular pieces of copper suitable for operation in a cryostat at low temperature.
The cavity is very compact, with a length equal to 250 m. The parallelism between the two
mirrors is better than 10 m so that the optical axis of the cavity is centered on the mirror with
an accuracy better than 1 mm.
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Figure 4: Compact and rigid high-finesse cavity, with mirrors assembled in copper rings suitable for
cryogenic application
In order to avoid any spurious displacement noise, the cavity is rigid and cannot be scanned.
Its resonance is then obtained by a sweep of the laser frequency. We have determined all the
optical characteristics of the cavity by measuring its bandwidth, its free spectral range, its
reflection coefficient at resonance. As an example, Figure 5 shows the cavity bandwidth
obtained by measuring with a photodiode the intensity transmitted through the back mirror
when the laser frequency is scanned around the optical resonance of the cavity.
Figure 5: Observed bandwidth of the high-finesse cavity, leading to a finesse of 230 000
Losses and transmissions of the cavity are compatible with the values measured on individual
mirrors (coupling transmission of 18 ppm at 810 nm, total losses of 9 ppm), and lead to a
cavity finesse of 230 000. The incident light power can be larger than a few milliwatts
without any damage on the mirrors.
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Figure 6: Cryogenic facility adapted to the high-finesse cavity.
The next step is to test the cavity at cryogenic temperature. We have thus built a specific
cryostat for the high-finesse cavity, where the cavity is thermally connected by copper wires
to the liquid helium tank, but mechanically connected to the input window so that vibrations
are minimized (see Figure 6). The temperature can be swept down to 2 K.
4. References
[1] "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave
antennae", V. B. Braginsky, M. L. Gorodetsky and S. P. Vyatchanin, Phys. Lett. A 264, 1
(1999)
[2] "Thermoelastic effects at low temperatures and quantum limits in displacement
measurements", M. Cerdonio, L. Conti, A. Heidmann and M. Pinard, Phys. Rev. D 63,
082003 (2001)
[3] "Experimental Measurement of the Dynamic Photothermal Effect in Fabry-Perot Cavities
for Gravitational Wave Detectors", M. De Rosa, L. Conti, M. Cerdonio, M. Pinard, and F.
Marin, Phys. Rev. Lett. 89, 237402 (2002)
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