Estimation of losses in a 300 m filter cavity and quantum noise reduction in the KAGRA gravitational-wave detector

Quantum noise is one of the main limiting factors for second generation gravitational-wave interferometric detectors, such as KAGRA KAGRA , Advanced LIGO aLIGO_CQG and Advanced Virgo AdV_CQG . As pointed out by Caves in 1981 Caves , the two manifestations of quantum noise, shot noise and radiation pressure noise, are created by vacuum fluctuations entering the interferometer by the antisymmetric (or dark) port. Caves also proposed the injection of a squeezed vacuum from the dark port as a strategy to decrease quantum noise without modifying the interferometer configuration.

In a squeezed vacuum state, the amplitude and phase uncertainty, equally distributed in an ordinary vacuum, are modified in order to reduce one at the expense of the other. If this is a pure state, it can be represented as an ellipse in the quadrature plane and it is characterized by the ratio of its axes (squeezing magnitude) and by its orientation (squeezing angle). Both of these parameters are functions of the Fourier frequency. If the quadrature with the reduced uncertainty is aligned in the gravitational wave signal, the signal to noise ratio is improved with respect to that achievable with an ordinary vacuum. Since the optomechanical coupling of the laser light with the interferometer test masses induces a rotation of the squeezing ellipse, the injection of a frequency-independent squeezed vacuum (i.e with constant squeezing angle) will mitigate quantum noise only in the part of the spectrum where the gravitational-wave signal is aligned with quadrature with reduced uncertainty. In the case of a phase quadrature squeezing, the improvement is the same as that achievable by increasing the laser power.

A broadband quantum noise reduction can be obtained by injecting a squeezed vacuum with an angle that varies with frequency, in such a way that the signal and reduced noise quadrature are always aligned kimble . This frequency dependent squeezing can be obtained by reflecting off a frequency-independent squeezed state by a detuned Fabry-Pérot filter cavity. The rotation of the squeezing angle has been experimentally demonstrated in the MHz regionChe and, more recently, in the kHz regionOel .

The injection of frequency dependent squeezing is particularly suitable for a future upgrade of KAGRA, since increasing laser power to reduce shot noise can be difficult due to the cryogenic temperature of the detector. Moreover, since thermal noise is low enough to make quantum noise the dominant contribution in a large part of the spectrum, the reduction of quantum noise would have a direct and significant effect on the detector sensitivity.

The achievement of a high level of frequency dependent squeezing faces two kinds of difficulties. First, the squeezing angle should undergo a rotation in the frequency region where quantum noise switches from radiation pressure noise to shot noise ($\sim 70$ Hz for KAGRA). This corresponds to a filter cavity with a very long storage time of $\sim 2$ ms. Moreover, the presence of optical losses in the filter cavity system will decrease the squeezing factor, since losses are associated with ordinary vacuum fluctuations which couple with the squeezed states reducing their squeezing level. For both these reasons, optical losses in the filter cavity system should be estimated and reduced as much as possible.

A 300 m prototype of the cavity will be developed at NAOJ, using the former TAMA infrastructure TAMA . This is, to our knowledge, the longest filter cavity prototype under construction. Previous experiments were made using a 2 m long cavity Oel . The first goal of this paper is to present the filter cavity optical design. One of the main issues in the filter cavity realization are the losses due to the mirror surface imperfections. In this work we present a study of the specifications for the mirrors, based on real mirror maps, used in the Virgo experiment. Moreover, a crucial point is to compare the squeezing degradation from the filter cavity optical losses with those originated by other mechanisms. Thus, we present a complete squeezing degradation budget for the 300 m cavity, based on the work by Kwee et al.sq2 . We finally discuss the potential improvement of KAGRA sensitivity using the 300 m filter cavity, compatible with the KAGRA infrastructure.

The interest of this work goes beyond KAGRA, since a 100-m class filter cavity is also a possible solution for a medium-term upgrade of Advanced Virgovirgo_sqz , and 300 m filter cavities are planned for the 3rd generation detector Einstein TelescopeET . The structure of the article is as follows: in section II we demonstrate the computation of the filter cavity parameters needed to obtain an optimal rotation of the squeezing angle for KAGRA. In section III the filter cavity numerical simulation, with realistic mirror maps, is presented. In section IV an estimation of the squeezing degradation, due to cavity losses and other degradation mechanisms, is obtained. Finally, the expected improvement in KAGRA sensitivity, achievable using frequency-dependent squeezing, is presented.

In the first part of this section we discuss the choice of finesse and detuning of the filter cavity. In the second part, we address the problem of the choice of cavity length, mirror dimension and radius of curvature.

The presence of losses in the filter cavity, caused by mirror defect and mirror finite size, induces squeezing degradation. In this section we will detail the procedure needed to find the specifications on mirror dimensions and surface flatness.

Mirror defects, namely deviations of the mirror surface from a perfect spherical one, are described by the mirror map, which is a square matrix of $n$ elements with $n=l_{mir}/res$, where $l_{mir}$ is the length of the surface described, and $res$ is the resolution of the map. Each matrix element, corresponding to a pixel with area $res^{2}$, contains a measure of the mirror surface height $h$ (with respect to a perfect spherical one).

We can associate to each mirror map (or to a part of it) the Root Mean Square (RMS) of the height, defined as:

Another useful number to quantify surface flatness is the peak-to-valley value (PV): a measure of the difference between the highest and the lowest point.

Mirror defects can be studied in the spatial frequency domain by applying a 2D Fourier transform to the mirror map. The lowest spatial frequency $f_{min}$ coincides with the inverse of $l_{mir}$, while the maximum spatial frequency $f_{max}$ is given by$1/(2\cdot res)$. A 1D Power spectral density (PSD) can be associated with the 2D Fourier transform map hiro . For such a 1D PSD we have the relation:

The low frequency defects, those which have a spatial frequency up to $10^{3}m^{-1}$, contribute to the so-called mirror flatness while higher frequency defects are associated with the mirror roughness. This distinction has no physical motivation, but it is simply due to different techniques used to measure spatial defects in the two cases nicolas .

The scattering angle for light at normal incidence of wavelength $\lambda$ can be written as a function of the frequency of spatial defect as sto :

This equation means that a defect at spatial frequency $f$ will scatter a fraction of the light at angles $\theta$ or larger. As a consequence, the amount of light reflected back at normal incidence will be reduced by the same amount. This fraction is given by $(4\pi\times\sigma(f)/\lambda)^{2}$ where $\sigma(f)$ is the amplitude of the defect at spatial frequency $f$.

For a given cavity length $L$ and mirror diameter $d$, there is a maximum scattering angle $\theta_{\mathrm{limit}}$ above which light is scattered out of the cavity:

Using Eq. 16 and 17 we can then find a spatial frequency $f_{\mathrm{limit}}$ for the mirror defects above which the light is scattered out of the cavity:

From the equation above, the losses due to defects with spatial frequency above $f_{\mathrm{limit}}$ can be estimated as sto :

where $\sigma$ is the RMS for frequencies above $f_{\mathrm{limit}}$. For the filter cavity we are considering $f_{\mathrm{limit}}=157m^{-1}$.

It is important to note that the RMS for frequencies lower than $f_{\mathrm{limit}}$ also contributes to losses. In fact, even if light is not immediately scattered out of the cavity, it is likely to be transferred on higher order modes and eventually exit the cavity. An accurate estimation of the round trip losses has been done using a numerical simulation of the fields in the cavity after adding realistic maps to the mirror surfaces. This allowed us to set specifications for the surface defects flatness needed to keep losses below a desired threshold.

The round trip losses in a Fabry-Pérot cavity are defined as rtl_ref :

where $P_{in}$ is the input power (which is assumed to be a fundamental mode), $P_{circ}$, $P_{t}$, $P_{r}$ are the powers circulating in the cavity, transmitted and reflected, respectively. Since we can only take advantage of the light reflected on the fundamental mode, the definition of Eq. 20 has been modified to:

where $P^{00}_{r}$ is fraction of the reflected power which is on the fundamental mode.

We used the MATLAB package OSCAR oscar to simulate the fields and compute the values of $P_{circ}$ and $P^{00}_{r}$ to be used in Eq.21.

We ran the simulation using five different mirror maps, for mirrors used in the Virgo experiment. Four of these mirrors were produced for the initial Virgo experiment (2007-2011) with a standard polishing technology. The fifth belongs to the Advanced Virgo experiment (which will become operational in 2016) and is obtained with an ion beam polishing technique. The Virgo maps have a resolution of about $350\,\mathrm{\mu}$m, while the Advanced Virgo map has a resolution of $378.4\,\mathrm{\mu}$m. One example of the Virgo maps and the Advanced Virgo map are shown in Fig. 1 along with their relative PSD.

The cavity parameters used in the simulation are those reported in Tab. 2. The maps were only applied to the end mirror, while the input mirror has been considered perfect. We checked that the round trip losses for a cavity where both the mirrors have defects can be obtained by simply multiplying by two the previous result. Results presented here have already been multiplied by two.

Each surface has been characterized by its RMS and its PV over different diameters. The measured values are reported in Tab. 3. Round trip losses for the various mirrors have been calculated in terms of the radius of curvature and are reported in Fig. 2. The losses floor for each mirror has also been reported in the last column of Tab. 3 in order to be directly compared with the mirror flatness. The presence of peaks in the plots of Fig.2 is due to power transferred to higher order modes which are partially resonant along with the fundamental mode for certain values of the curvature radius.

Maps used in the simulation account for mirror defects with spatial frequency going from 10 $m^{-1}$ to $2\cdot 10^{3}m^{-1}$. This means that losses caused by mirror roughness are not included in this estimation. A map of the the roughness has been measured for the Advanced Virgo mirror. this map, obtained by scanning an area of 0.3 mm x 0.3 mm with a resolution of $1.28\,\mathrm{\mu}$m, scans frequencies from $3.3\cdot 10^{3}$ to $3.9\cdot 10^{5}m^{-1}$. From its RMS, under the assumption that it is uniform on the surface, we can estimate additional losses due to roughness, which should be added to those already measured. The RMS is 0.08 nm, which corresponds to 0.89 ppm of additional losses for a single reflection. For the initial Virgo mirrors, the specification on the roughness RMS was 0.1 nm which corresponds to 1.4 ppm of additional losses for a single reflection. In both cases the losses are dominated by flatness defects.The roughness map for Advanced Virgo and the relative PSD are shown in Fig.3. The measurement of the light scattered at angles larger than a few degrees gives losses of the order of 5 ppm. These include both the losses due to the roughness discussed above and those due to point defects. The corresponding additional round-trip losses will be about 10 ppm. We remark that, even including all these effects, still difference exists between the measured losses in Advanced LIGO and simulation results. Losses due to scattering at angles between mrad and a few degrees are being investigated as a possible cause of these differenceshiroprivate . These hypothetical losses are not included in this paper.

The conclusions of this study are that, for his filter cavity (length $300$ m, mirror diameter $10$ cm, RoC $\sim 400$ m):

• •

  An Advanced Virgo-class mirror will produce floor losses $<10$ ppm and a Virgo-class mirror will produce losses $\sim 40-80$ ppm. In order to determine the final specifications on the mirror flatness, the squeezing degradation given by the cavity losses has to be compared to the other degradation mechanisms. This analysis is performed in the next section.

• •

  The accidental degeneracy can amplify the losses by more than an order of magnitude. The simulation gives the safe regions, where the losses are at the floor level.

• •

  The precise RoC value should be chosen in some of the floor losses regions. A precision on the RoC of $\sim 1\%$ is necessary to guarantee the RoC to be in these regions.

The quantum noise in an interferometer, normalized with respect to shot noise, can be simply expressed as

where $K$ is the optomechanical coupling defined in Eq.2.

In an ideal system, the injection of a frequency dependent squeezed vacuum from the interferometer dark port will reduce the quantum noise to

where $\sigma$ is connected with the squeezing magnitude usually expressed in decibel by $\sigma_{\mathrm{dB}}=-20\,\sigma\times\log_{10}e$.

In a real setup, two factors prevent this optimal noise reduction: first, optical losses will decrease the squeezing level, introducing a non-squeezed vacuum; second, fluctuations of the squeezing angle will preclude an optimal rotation of the squeezed state.

In sq2 a detailed analysis of several of these mechanisms has been applied to a 16 m filter cavity, with round trip losses of 1 ppm/m, which is considered a possible short-term solution for Advanced LIGO sq1 . The same analysis is performed here for the 300 m filter cavity, with round trip losses of 80 ppm, corresponding to a conservative estimate for Virgo-class quality mirrors. To ease the comparisons between the two cases, the numerical values for other sources of squeezing degradation are exactly the same of sq2 , as reported in Tab. 5.

Figure 6 represents the squeezing degradation budget for the 300 m filter cavity. It shows the ratio between the quantum noise of a dual recycled Fabry-Pérot Michelson interferometer without squeezing (for instance, initial KAGRA) and the quantum noise in the presence of frequency dependent-squeezing. An initial realistic squeezing level of 9 dB has been considered and the various degradation mechanisms have been taken into account separately and combined (black curve). In this analysis, as for sq2 , the contribution of the interferometer losses has been neglected. The squeezing degradation mechanisms considered are (for a detailed description see sq2 ):

This estimation of the achievable level of frequency-dependent squeezing using a 300 m filter cavity can be used to quantify the related improvement in KAGRA sensitivity. In Fig. 7 the quantum noise for KAGRA, without squeezing, is compared with the quantum noise obtained using 9 dB of frequency-dependent squeezed light, when all degradation mechanisms previously described are taken into account. We considered both the case of a filter cavity with round trip losses of 80 ppm and that of a perfect filter cavity. The comparison shows that no major improvements can be obtained by reducing round trip losses under the level of $\sim 80$ ppm since their effect becomes comparable with that of other degradation mechanisms.

Fig. 8 shows the improvement in KAGRA sensitivity using 9 dB frequency-dependent squeezing, considering a filter cavity with round trip losses of 80 ppm and other degradation mechanisms. We remark that the use of squeezing allows us to reach a sensitivity beyond the standard quantum limit around $70$ Hz. In Fig. 9 the quantum noise of a realistic lossy system is shown along with other KAGRA noise sources. We observe how a reduction in the only quantum noise will result in an improvement in almost the whole KAGRA observation bandwidth.

We also remark that:

• •

  In the frequency region below 100 Hz, thermal noise is close to quantum noise and there is little to be gained from a further significant reduction in quantum noise (obtainable by improving mismatching, injection/readout losses and decreasing losses in the filter cavity).

• •

  In the frequency region above 100 Hz, which is dominated by quantum noise, there is much more room for improvements. However, a reduction of the filter cavity optical losses would not provide a higher squeezing level, since in this region the squeezing degradation is mainly caused by injection and readout losses.

This suggests that a further reduction of the filter cavity losses would not be necessary, while a major benefit can be obtained by protecting squeezing from losses due to injection, detection and mismatching.

We have presented the optical design of a 300 m filter cavity used for the KAGRA experiment. The cavity will be tested at NAOJ, inside the TAMA infrastructure. First, given KAGRA parameters, we have found the detuning and finesse values needed to produce an optimal rotation of the squeezing angle. In order to find the surface quality needed for the filter cavity, we have computed the round trip losses using a numerical simulation with realistic mirror maps. The test bench maps used are those of Virgo and Advanced Virgo.

An analysis of the squeezing deterioration mechanism shows that for a round trip loss value of 80 ppm, corresponding to a conservative estimate for Virgo-class mirrors, the squeezing degradation due to the filter cavity are comparable or lower than contributions from the cavity mismatching and injection/detection losses. Moreover frequency-dependent phase noise becomes completely negligible.

We then computed the improvements in KAGRA sensitivity with a 9 dB frequency dependent squeezing and realistic loss mechanisms. We found a sensitivity improvement in almost the full KAGRA signal bandwidth, with a factor $\sim 2$ at high frequency. As already mentioned, the interferometer internal losses have been neglected in this analysis. In order to have a more precise estimation of the sensitivity improvement achievable using frequency dependent squeezing, the contribution of these losses should be estimated.

We have also shown that, without an improvement in other noise sources, a further reduction of the filter cavity losses will not lead to a considerable improvement in the detector sensitivity.

The results of this work can be easily extended to any gravitational-wave interferometric detector planning the use of a 100-m class filter cavity, and in particular for a medium term upgrade of Advanced Virgo and for Einstein Telescope.

We would like to thank Lisa Barsotti, John Miller and Matteo Tacca for useful discussions. This work was supported by the JSPS Grant-in-Aid for Scientific Research the JSPS Core-to-Core Program, A. Advanced Research Networks and the European Commission under the Framework Program 7 (FP7) ’People’, project ELiTES (grant agreement 295153). E.C. is supported by the European Gravitational Observatory.