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In order to benefit over the entire frequency range from the injection of squeezed vacuum light at the output of laser gravitational wave detectors, a small bandwidth high finesse cavity is required. In this paper, we investigate the light losses due to the flatness and the roughness of realistic mirrors in a 10 meters-long Fabry-Perot filter cavity. Using measurements of commercial super-polished mirrors, we were able to estimate the cavity round trip losses separating the loss contribution from low and high spatial frequencies. By careful tuning of the cavity g-factor and the incident position of the light on the mirrors, round trip losses due to imperfect mirror surfaces as low as 3 ppm can be achieved in the simulations.
© 2015 Optical Society of America
1. Introduction
The sensitivity of gravitational wave detectors such as Advanced Virgo, Advanced LIGO and KAGRA will be limited by quantum noise over a large portion of the detector frequency range [1–3]. Above a few hundred hertz quantum noise appears as photon shot noise at the interferometer output port. Below hundred hertz quantum noise appears in the form of radiation pressure noise perturbing the test masses positions. Both can be seen as due to the vacuum fluctuations entering the interferometer from the output port.
The injection of a squeezed vacuum state from the interferometer output port allows changing the amplitude of quantum noise. In particular the injection of a frequency independent squeezed vacuum is equivalent to an increase of the interferometer input power [4]. Shot noise is decreased at the cost of increasing radiation pressure. Such a solution is interesting since it will allow decreasing the input power without decreasing the detector sensitivity. Decreasing the input power allows reducing all the difficulties related to running an interferometer with a high heat load on the mirrors.
The reduction of shot noise of laser interferometer gravitational wave detectors by means of squeezed states of light was demonstrated both at GEO 600 [5,6] and at LIGO Hanford [7].
It is possible to reduce the quantum noise at all frequencies if a frequency dependent squeezed vacuum state is injected into the interferometer from the output port [8,9]. By properly changing the so called squeezing angle as a function of frequency it is possible to decrease both the radiation pressure noise at low frequency and the shot noise at high frequency. The squeezing angle frequency dependence should be such that the squeezing angle is changed at the frequency where the interferometer noise turns from being radiation pressure dominated to being shot noise dominated. According to the design sensitivities of Advanced Virgo, Advanced LIGO and KAGRA such a transition frequency should be around 50 Hz.
The envisaged solution to produce a frequency dependent squeezed vacuum state is to reflect a frequency independent squeezed vacuum state on an optical cavity [8,9]. By keeping the cavity locked off resonance by one cavity half-bandwidth, the squeezing angle of the reflected vacuum squeezed state will change as a function of the frequency. The transition frequency will coincide with the cavity bandwidth. As a consequence the cavity half bandwidth has to be around 50 Hz [10] i.e. its finesse will depend on the cavity length:
For a cavity 10 m long, the cavity finesse has to be about 150000. For a cavity ten times longer (100 m) a lower finesse (15000) would be sufficient.
The demonstration of frequency dependent squeezed light was first realized in the megahertz region a decade ago [11]. More recently frequency dependent squeezed light has been demonstrated also in the audio band [12].
The main limitation to the level of squeezing one can achieve are the optical losses in the cavity [13]. The squeezing level of a squeezed state is decreased when passing through a lossy optical element. Losses can be seen as the coherent addition of the squeezed state with an unsqueezed vacuum, thus reducing the squeezing level of the final state. In the case of an optical cavity what matters is relative weight of the cavity round trip losses compared to the cavity input mirror transmission. For a given bandwidth, longer cavities require smaller finesse and thus higher input mirror transmissions. As a consequence the impact of the cavity round trip losses will be smaller for longer cavities so that in the end the important parameter is the round trip loss per cavity unit length (ppm/m). As a consequence the total losses requirements are less stringent for longer cavities.
The need for meter long and very high finesse cavity is not limited to filtering cavities, several other modern optical experiments also requires such cavities, for example for the emission of gamma rays by Compton backscattering [14] or to measure the magnetic birefringence of the vacuum [15].
In the following of this paper we consider the case of a 10 m long cavity since a cavity of this length can be accommodated in present infrastructures such the LIGO and Virgo corner areas and the losses requirements seem still achievable [10]. The goal of this article is to identify the best optical design for such a cavity and to evaluate the round trip losses that one can achieve with present mirror technology. In section 2 we first identify the overall cavity geometry design. Then we describe the flatness and roughness measurements that we made on some commercially available substrates (Sec. 3 and Sec. 4). In section 5 we use the result of the flatness measurement to estimate the losses in the cavity using OSCAR [16], an optical simulation based on FFT field propagation. We study the losses dependence upon several parameters including the surface height RMS, the cavity finesse and the mirrors radius of curvature. Then in section 6 we used the roughness measurements to estimate the additional losses due to high spatial frequencies in the surface profile. Finally in section 7 we discuss the results and then provide some conclusions.
2. Choice of the mirrors size and mirrors radius of curvature
As the cavity mirrors have a finite size, the first optimization to be done is to find the smallest possible beam size on the mirror in order to limit the clipping loss. To this purpose we choose to have the same radius of curvature (RoC) for the two mirrors.
For a 10 m long cavity the beam diameter is always smaller than 4 mm if both RoC’s are below 20 m [17]. The minimum beam size is achieved when the RoC’s are equal to 10 m but in that configuration the cavity is degenerated. If we investigate the choice of the best radius of curvature in order to minimize the losses in the cavity and avoid degeneracy, we must also anticipate the polishing specifications and the stress induced by coating. (Here we consider the thermal deformation due to the low power of the laser beam to be negligible). The tolerances due to the manufacturing process of the mirror can add a random offset to the curvature of the mirrors. Moreover experiments done in the lab with low loss IBS coating shows that the coating induces a compression stress on the surface and increases the radius of curvature of the mirror. From those considerations, it is necessary to choose the radius of curvature above and close to 10 meters but far enough from 10 meters in order to avoid the cavity degeneracy. Since the standard specifications of the polisher are typically 1% from the target radius, we will only consider radii of curvature above 11 meters in order to keep a comfortable safety margin inside the requirement of manufacturing, and avoiding the degeneracy. The next simulations in this paper will later help us to choose a more accurate range of radius of curvature in order to minimize the losses in the cavity.
In the following loss investigation, we will focus on the physical uniformity of the substrate and the radius of curvature as specific requirement. We will not consider the other mirror surface misbehavior as the coating uniformity and the coating induced stress. We will just take into account the surface defects and aberrations due to the polishing process, that is to say the flatness and the roughness on the surface of the optics. The flatness derived from the wave-front measurement includes all the low spatial frequencies below 1400 m−1. The roughness measured with an optical profilometer represents the high spatial frequency range around 1000 m−1 up to 200 000 m−1. So, in this article, the flatness will quantify the deviation from the nominal profile with the low spatial frequencies whereas the roughness will quantify the small scale details.
Finally, we will only consider commercial micro-polished mirrors of 1 inch or 2 inches for all roughness and flatness measurements and for all cavity loss simulations. 1 inch as mirror diameter is an appropriated size since it is 6 times greater than the laser beam diameter and the light falling outside the mirror is negligible, less than 0.02 ppm per reflection.
3. Wave-front measurement
In order to estimate the round-trip losses in a 10 meters-long filtering cavity, we measure the mirror surfaces which could potentially be used in such a cavity.
First we measured four micro-polished substrates of 2 inches diameter and one of 1 inch. These measurements will help to compare the micro-polished surface uniformity. Secondly, the wave-front map measurement will be used as mirror surface in the cavity simulation.
All the measured substrates are made in standard fused silica and have the same specifications. The flat mirrors are super-polished and the manufacturers announced surface roughness better than 1 Angstrom and also a flatness quality better than λ /10 (peak-to-valley) at 633 nm of wavelength. The 2 inches mirrors are 10 mm thick versus 6 mm for the 1 inch mirror.
The low spatial frequency flatness of the substrates is measured with a Fizeau Phase Shifted Interferometer (PSI) illuminated by a laser with a 1064 nm wavelength. The PSI uses a flat reference of 6 inches diameter and has a lateral resolution of 350 micrometers. In order to check the reproducibility error of measurement, optics are measured 3 times, unmounted and mounted, between 2 consecutive measurements. Piston and tilt are subtracted from the wave-front measurement. An example of the mirror surface is given in Fig. 1.
Fig. 1 2” map flatness measured with the “Phase Shifting Interferometer” over 45mm diameter. The RMS height is 1.3 nm over 23mm of diameter.
In order to characterize the flatness of the optics, we calculate the Root Mean Square of the surface map height. The Root Mean Square (RMS) is defined as following:
where n is the number of elements in the sample (in our case the number of pixel). The values are the height of each i-point of the map.We calculate the RMS height of the wave-front measurement in the central part and over different diameters. The RMS of the different mirrors are shown in the Table 1. We see that the RMS flatness is slightly different depending on the substrate and according to the diameter of calculation. A relevant diameter of calculation is the diameter seen by the laser beam. We assume here that it is 4 mm, corresponding to twice the beam radius of the laser on the mirror surface. Concerning the 2 inches measured substrates, the RMS flatness is around 0.125 nm +/−20% according to the sample. While over the full aperture the flatness is only slightly better than the specification, over the central 4 mm diameter all the measured mirror surfaces are well below the specifications of λ/10.
Table 1. RMS height measurements of some substrates calculated over different diameters at the center of the substrate.
While the 2 inches substrates and the 1 inch substrates have the same specification of λ/10 over the entire surface, we notice that the 2 inches substrates are better in the central part over 10 mm diameter (and also over 23 mm diameter). The 1 inch substrate, highlighted in blue in the Table 1, has the worst RMS flatness substrate over the laser beam size. For the next loss simulations, we always used this measured map, in order to have an upper limit for the RTL.
4. Roughness measurement
In principle, the local roughness of the optics is included in the RMS height calculation in previous Eq. (2). But practically the RMS calculation is always limited by the instrument lateral resolution and not all the spatial frequencies are included. In this section, we consider the “high” spatial frequency uniformity in order to estimate the additional losses in the cavity due to the roughness on the mirror surface.
We measured the high frequency roughness with an optical profilometer provided with a 5X magnification Michelson-type interference objective (The device is called “MicroMap” from the eponymus company which no longer exist). The 5X magnification of the profilometer objective provides a resolution of 2.56 µm by pixel with a field of view of 1mmx1mm. Figure 2 gives us an example of roughness measurement. We always subtract piston, tilt and power on the measurement before to display the map and before to use it for the loss calculation.
Fig. 2 Example of roughness map measured with the profilometer “Micromap” using a Micheslon-type interference objective 5X magnification, 1mm x 1mm, 2.56 μm/pixel.
For all roughness measurement, we chose a 2” substrate and we scanned an area of 12.2mmx12.2mm, 3 times greater than the laser beam diameter, in order to obtain a sufficient statistic. In total, we needed to measure 225 micro-maps of 400 pixels aside. The set of measurements represents a very extensive sample statistic of roughness on the mirrors. Using Eq. (2), we calculated the RMS roughness value of each map. The results are summarized in the histogram in Fig. 3. For this substrate, the RMS height average is 0.13 nm and then median is around 0.12 nm.
Another way to characterize the surface uniformity of the mirrors is to plot the PSD (Power Spectral Density [18]) of each mirror map. Without entering into the details here we just remind that the PSD of the mirror map is obtained from the two dimensional Fast Fourier Transform of the wave-front map by squaring the 2D FFT and then summing over a ring of constant spatial frequency. The square of the RMS value of the map is equal to the integral of the PSD curve over the spatial frequency band:
f1 is the minimal spatial frequency of the surface on which the uniformity is calculated. The PSD maximal spatial frequency f2 is exactly one half of the sampling frequency of the measurement, corresponding to the lateral resolution of the instrument. Using the profilometer, the PSD maximal frequency f2 is equal to 195000 m−1 for 2.56 µm of pixel size.In Fig. 4, we plotted all the PSD of the 225 micro-maps. The PSD are expressed in [nm2/m−1] but for more contrast concerning the amplitude of the PSD, the color scale and the unit of the color-bar are given as the logarithm of the PSD. By integration of the median of all the PSD, according to Eq. (3), we find 0.11 nm of RMS roughness very similar to the 0.12 nm given by the direct calculation from Eq. (2)
Fig. 4 2D representation of all the PSD of 225 micro-maps measurements. The color scale is the amplitude of the PSD logarithm.
At last, in Fig. 5, we superimpose the median PSD curve of the roughness maps with the PSD curve of the low spatial frequency uniformity. We observe that both PSDs combine in a satisfactory way over four spatial frequency decades.
5. Round-trip loss simulations
5.1. Procedures for the round-trip loss simulations
After description of the substrate surface measurement over a large range of spatial frequencies in previous section, we now detail the use of those maps in the simulations of the Fabry-Perot cavity.
The simulated cavity is a 10m long Fabry-Perot cavity with two mirrors of one inch diameter. For the numerical simulation, we used the package OSCAR [16] which can simulate cavities or interferometer with any arbitrary distortions. In the simulations, we set the clear aperture of the 1 inch mirror to be 21 mm in diameter to represent the real diameter of the coating. Moreover it cuts off the artifacts on the edge of the measured maps.
Surface maps used in the simulation are polished surface without any coating. However, results at LMA have shown that on small surfaces (less than 50 mm diameter), the coating is uniform enough to not degrade the quality of the polished surface. We also compared PSD before and after coating (with curvature removed) and no significant difference has been found (one more time over 2 inches diameter), the RMS being increased by 0.1 nm after the deposition of a very high reflectivity coating (transmission of 4 ppm).
For all the simulations, the input mirror and the end mirror have the same curvature and we only add the map of the wave-front measurement on the cavity end mirror, the input mirror remains perfect. The end mirror is supposed to be perfectly reflective, and the losses due to the absorption or diffusion from point defect are set to zero.
At first, in order to reduce the calculation time, we simulate a relatively low finesse cavity (finesse of 450, input mirror transmission of 1.4% in power).
For most of the simulations, we use the mirror surface shown in Fig. 6. The 2D height map is added to the nominal curvature of the end mirror with the clear aperture of 21 mm diameter, after have to remove tilt and residual curvature. Since the wave-front measurement resolution is 350μm the map only includes the low spatial frequencies below 1430 m−1.
Fig. 6 1” substrate map of uniformity used in the simulation, as reference map on the cavity end mirror. Tilt and residual curvature were removed over the central 10 mm diameter.
For the simulation of the circulating, transmitted and reflected fields of the cavity, the simulation grid size has been set to 128 by 128 points with the same pixel size as for the surface map to avoid unnecessary resampling.
Using the simple representation of the cavity shown in Fig. 7, the round-trip losses (RTL) in the Fabry-Perot cavity are defined as: (the detailed calculation is presented in the Appendix A:
Where Pin, Pt, Pr and Pcirc are respectively the input, transmitted, reflected and circulating power of the fields traveling through the cavity.Equation (4) only takes into account the power inside and leaving the cavity independently of the mode shape. But the part of the reflected and transmitted fields made of higher order modes should be considered as losses, even if they contribute to the total power.
As a consequence, we consider the round-trip losses projected on the TEM00 mode using the following Eq. (5):
With is the reflected power contained in the TEM00 mode of the field of the cavity. The losses on all the modes are usually called the clipping losses due to the finite size of the mirrors. The losses defined on the fundamental mode, takes into account the clipping losses and the fact that the cavity is coupled to an interferometer and that only the light reflected in the fundamental mode will be useful for the rest of the system.Figure 8 shows the comparison between RTL and RTL00 as a function of the nominal radius of curvature (RoC) of the input and end mirrors (waist always in the middle of the cavity). As expected the losses are lower if we consider all the modes as useful power. From now on, for all the simulations, we will consider the round-trip loss calculation on the mode 00 according to Eq. (5).
Fig. 8 Comparison of the cavity round-trip losses when all the modes are taken into account or only when the power in the fundamental mode is considered.
In Fig. 8, one can see an excess loss for a radius of curvature of 14.5 m on the losses calculated for the fundamental mode. That is due to one higher order mode resonating at the same time as the fundamental one, the cavity is degenerated for this particular configuration. This higher order mode is rather a low order mode with only moderate excess round trip loss (by clipping of its shape outside the mirror), however its presence degrades the purity of the fundamental mode in reflection, hence increasing the RTL00.
5.2. Losses depending on the cavity finesse
If the cavity finesse is increased, the results of the round-trip losses change considerably. Figure 9 shows the RTL00 as the mirrors radius of curvature is scanned between 13 and 20 meters for two different finesses 440 and 45000.
Fig. 9 Round-trip losses in the cavity with radii of curvature of the mirrors between 13 and 20 meters. Comparison of the RTL00 according to the cavity finesse.
For historic reasons, we used a finesse of 45000, instead of 150 000 as mentioned in the introduction. However our finding about the optimal RoC for the cavity is unchanged with the change of finesse. Simulations of losses with finesse of 150 000 were also done with reduced data set and the RTL00 when the cavity is not degenerated is 5% lower compared to the value found for a cavity with a finesse of 45 000.
By increasing the cavity finesse, the Airy peaks of the degenerate modes are sharper. As a consequence, the background losses between the peaks decreases significantly. As a consequence we obtain a more comfortable range of radius of curvature for which the losses are minimal and stable.
Now with a second identical mirror as input mirror in the cavity, we checked that the minimal loss could be less than 2 ppm if the cavity finesse is 45000. That is a very good result. However the reader must keep in mind that this result is only due to the flatness of the mirrors because we have been using only a measured map with spatial frequencies above 1430 m−1.
5.3. Dependence of the loss on the RMS height of the mirror surface
From now on and up to the end of the article, the RMS height will be always given without curvature over the diameter of interest. Nevertheless for the 1 inch substrate of the Table 1, we must keep in mind that the RMS height is 0.6 nm without curvature instead of the 1.8 nm with curvature.
We compare now the RTL variation as a function of the local RMS height of the substrate surface used as cavity end mirror. We assume 3 maps where the RMS height varies from 0.3 nm to 2.4 nm. Each map is placed on the end mirror in the cavity simulation and we computed the RTL00. The scan of the loss for the 3 maps are plotted in Fig. 10 with a cavity finesse of 45000.
Fig. 10 Round-trip losses in the cavity as a function of the RMS value in the center of the end mirror. The clear aperture is set to a diameter of 21 mm. In the legend, the RMS height of the mirror is given over a diameter of 10 mm.
The first simulation uses a one inch substrate. The RMS height without curvature is 0.6 nm over 10 mm of diameter in center. We compare the round-trip losses due to this substrate with the losses generated by 2 other maps. One map is a 2 inches substrate with a lower RMS (0.3 nm) and another higher RMS map (2.4 nm) is the first map multiplied by a factor 4. Without surprise, we observe that the RTL are dependent on the RMS height. For this cavity design, we see that the RTL can increase by almost a factor 10 when the surface RMS increases only by a factor 2. According to the increase of the RMS, a factor of 4 should have been expected in the loss as we will see in the next section. Here we got a larger factor because the RMS calculation was not accurate since it was only taken over a diameter of 10 mm and not taking into account the exact laser beam profile, for example the RMS is 0.26 nm calculated over a 10 mm diameter but only 0.18 nm over 8mm. For Fig. 10, it was difficult to use a beam weighted RMS since the beam size changes as the radius of curvature of the cavity mirrors are scanned.
These results do not include the losses due to the roughness at spatial frequencies above 1400 m−1 which will be investigated later.
Having confirmed that the RTL changes as a function of the RMS height, we studied how the value of the local RMS uniformity as seen by the laser beam influences the losses. In other words, how the losses depend on the impact of the laser on the mirror surface.
The central 8mm × 8mm of the mirror surface is shown in Fig. 11. To estimate faithfully the local RMS seen by the laser beam, we weighted the RMS height calculation by the laser beam intensity. The map of the local RMS height as it would be seen by the laser beam is displayed in Fig. 12. The laser beam is shifted by step of 350 µm in order to match the pixel resolution of the wave-front measurement.
Fig. 11 Flatness map measured with the “Phase Shifting Interferometer” over a square of 8mmx8mm with a resolution of 350μm/ pixel.
Fig. 12 Comparison on the weighted height RMS in the 8 mm square center of the substrate (a) and the corresponding RTL00 depending on the position of the laser beam on the surface (b).
For each simulation, we slightly shifted the center of the mirror by an integer number of pixel and calculated the local RMS as well as the cavity losses. A side by side comparison of the local RMS and the local RTL00 is shown in Fig. 12. The cavity loss calculations are done on the fundamental mode, for a nominal RoC of 14.1 m and a cavity finesse of 45 000. As expected, we found a good correlation between the local RMS and the round trip losses.
According to the laser position on the mirror, we can gain a factor 2 or 3 on the RMS seen by the laser and almost factor 10 on the round-trip losses in the cavity. In the best configuration we could have only 0.1 ppm of loss by mirror. In Fig. 13, the round trip loss is plotted as a function of the RMS height over the square area of 8 mm x 8 mm at center of the substrate. We noticed that we have around 80% of probability to obtain less than 0.5 nm of RMS flatness, and around 80% of probability to obtain less than 0.5 ppm of loss (1 ppm for two cavity mirrors) with such a super-polished substrate as cavity end mirror.
Fig. 13 Round trip loss as a function of the local RMS height seen by the laser beam over a square area of 8mm by 8mm at the substrate center. (The nominal RoC of the entire area is 14.1 meters and no map is placed on the input mirror).
5.4. Losses depending on the local radius of curvature
We saw above that the local RMS changes as a function of the position of the laser beam on the mirror and hence the round trip losses. Assuming that the resonances of the higher order modes are strongly dependent on the mirror radius of curvature, we have also to check if the local curvature in the mirror center could shift significantly the peaks of the resonant modes thus changing the RTL.
The relevant radius of curvature is the radius seen by the laser beam and is depending on the laser beam size for non-perfectly spherical surface. For example, for our cavity, the fundamental mode will only sample the curvature of the mirrors within a diameter of around 4 mm from the center (twice the beam radius). The real curvature (1/RoCreal) is the combination of the local curvature due to the height non uniformity of flat mirror (1/RoClocal) with the nominal curvature (1/RoCnominal) of a perfectly spherical mirror:
For example if the nominal radius (RoCnominal) is 14 m, then the real radius (RoCreal) over 4 mm of diameter will locally stay inside 1% requirement only if │RoClocal│>1400m. It means that the additional sagitta must be less than 1.5 nm over 4 mm of diameter.
In order to take into account the effect of the random local RoC in the center of the mirrors, we simulated several positions of the laser falling on the mirror surface. Like for the RMS investigation, we scanned the mirror on an area of 8 mm x 8 mm. For each laser location, the fit of the local curvature is weighted by the power of the TEM00 in order to fit the real curvature seen by the Gaussian beam. The real radius of curvature seen by the TEM00 is depicted in Fig. 14 for a nominal RoC of 14.02 m.
We observed that the real curvature locally increases or decreases by about 2% from the nominal curvature. For example, instead of having a real curvature equal to the nominal curvature at 14.02 m, we have 13.76 m of radius of curvature as seen by the TEM00.
However, when we scan the losses as a function of the end mirror nominal radius of curvature, we always observe a loss peak at 14.02 m of nominal radius whatever the laser location on the mirror (as seen in Figs. 8, 9 or 10). In other words, the peaks location of the resonant high order modes is not dependent on the local radius of curvature seen by the TEM00, but it depends only on the nominal radius.
In Fig. 15, we can see the field amplitude of the degenerated high order modes when the nominal radius of curvature is 14.02 m. The order of this mode is 31. As depicted in the Table 2, the real radius seen by the TEM00 can vary from 13.760 m to 14.017 m, depending on the location of the laser on the mirror, but the higher order mode will always resonate for the same nominal curvature i.e. the radius of curvature seen by the high order mode does not change significantly.
Table 2. Comparison between the radii of curvature seen by the TEM00 and by one HOM when they are resonant at the same time at 14.02 m of nominal mirror radius of curvature (as seen on one RTL peak), corresponding to 2 positions of the beam on the mirror.
In fact, we checked in Fig. 16 that the curvature seen by the resonant high order mode is very close to the nominal curvature. That is because the higher order mode has a very large spatial extend and hence the random fluctuations of the mirror surfaces average out.
We conclude in this section that the local variation of the radius of curvature does not affect the losses significantly. However the losses can be significantly modulated by the local defects (the local RMS height) depending on the position of the laser beam on the mirror.
5.4. Conclusion about the low spatial frequency losses
Concerning the low spatial frequency uniformity, we first have to consider that the relevant RMS flatness is the local flatness obtained by weighting the surface height with the laser beam power distribution over the mirror. For the substrates considered in this study we checked that the local flatness seen by the laser beam is below 1 nm with some large area having local flatness smaller than 0.5 nm on average. In some cases, we found that the flatness seen by the laser can be very dependent on the impact of the beam on the mirror surface. We saw that moving the laser beam over a distance less than the waist size in the center of the mirror can easily change the RMS value by a factor 2 or 3 between the best and the worst position of the laser. At the same time, the simulated round-trip losses due to the scattered light in the cavity can increase by a factor 10. But if we carefully tune the laser in an ideal position, we can achieve losses as small as 0.1 ppm.
So far, we never take into account the losses due to the input mirror in the simulations. But we checked that, as expected, adding the defect on the input mirror increases the minimal loss by a factor 2 (input and end mirror maps being different and uncorrelated). As a consequence, we can reach around 0.2 ppm of round-trip loss in the cavity due to the low spatial frequencies flatness.
In the next section we will consider the effect of the surface roughness (the high spatial frequency surface defects).
6. Additional loss due to the high spatial frequencies
The wave-front measurement is only sampled with a pixel size of 350 µm. That means that the maps used previously in order to simulate the round-trip losses, do not include the spatial frequencies above 1400 m−1.
Since the previous simulations are not suitable to take into account the high spatial frequencies, a simple analytical model will be derived to estimate the scattering loss for those frequencies.
First, let’s understand how the losses are dependent on the light diffracted by the mirror surface. The diffraction angle θ of a light ray which meets a surface at normal incidence is proportional to the wavelength λ and to the spatial frequency f of the defects. The angle θ is expressed as follows:
Also, we define θmax the maximal angle of diffraction above which a diffracted ray in the mirror center will exit the cavity. The angle θmax is shown in Fig. 17 and is expressed as following:
Using both the Eqs. (7) and (8), we deduce flimit the spatial frequency of the roughness above which the scattered light exits the cavity. We find then:
Equation (9) is an approximate formula based on simple geometry and the grating formula which does not take into account the complex Gaussian profile.
For a ten-meter long cavity made of one inch diameter mirrors, we obtain flimit around 1200 m−1.
The frequency flimit represents a threshold beyond which the light is lost after the first round-trip in the cavity. That is to say it concerns the defects of roughness whose spatial period is below 830 µm. For all the frequencies higher than flimit, the following analytical formula gives us a value of the losses per reflection which only depends on the wavelength and of the amplitude of the defects [19]:
σrms is here the RMS roughness whose spatial frequencies are greater than 1200 m−1. This micro-roughness is measured with the “MicroMap” (see section 4).The field of view 1mmx1mm and the pixel size of 2.56 µm allow the measurement of the spatial frequencies between 1000 m−1 and 200 000 m−1. As a consequence the MicroMap is the ideal tool for the computation of the roughness σrms because it can measure spatial frequencies starting from 1000 m−1, which is very close to flimit, the spatial frequency above which the scattered light is definitively lost. Thus, knowing the median of the RMS height σrms of the micro-maps measured in the section 4 (equal to 0.12 nm), and using Eq. (10), we find an additional loss of 2.0 ppm per mirror due to the light diffracted outside the cavity.Using Eq. (3), we can also compute the RMS value σrms by integration of the PSD given in Fig. 5. If the curve is integrated in the range frequency above 1000 m−1, we find now 0.10 nm RMS of roughness leading to 1.4 ppm of additional loss per mirror.
Even if the roughness is quite homogeneous on the entire scanned surface, we nevertheless see in Fig. 3 that some roughness maps are 2 or 3 times higher than the average. Assuming that one map of 1 mm aside measures 25% of the beam size on the mirror, maybe the diffraction of the light due to the local roughness on the mirror could provide more losses than expected previously. So we quantified the exact loss value due to the diffracted light when the laser reaches a large area on the mirror. The ideal area on the mirror must be 3 times greater than the beam diameter of the laser. Using specific software we stitched all the 225 micro-maps seen in the section 4. Thus, we obtained a large roughness map 12.2 mm aside. The large stitched map includes all the roughness over a real surface of 12.2 mm aside, but low spatial frequency artifacts appeared as shown in Fig. 18. Indeed the total height RMS was 5 times greater than the RMS of one micro-map. However, the low frequency artifact and the mirror flatness have a similar height standard deviation (0.6 nm RMS once the residual focus is removed) and we checked that the PSD of the stitched map is coherent with the other PSD in Fig. 5.
Fig. 18 Roughness map stitched with 225 maps of 1mm aside measured with the optical profilometer “micromap” and a Michelson-type interference objective of 5x magnification. The map size is 12.2 mm x 12.2 mm.
Using now the map in Fig. 18 as surface of the end mirror, we estimated the losses due to the roughness by a numerical method. Still with OSCAR as simulation tool, we computed the power of the electric field propagated over 10 meters after the light (a perfect TEM00) has been reflected off the mirror. The picture of the amplitude of the electric field is shown in Fig. 19. We find then 1.5 ppm of the reflected light falls outside the clear aperture of one inch and is thus lost.
Fig. 19 : Amplitude of the electric field falling outside the cavity after reflection off the mirror and propagation over 10 meters. The total amount of light falling off the mirror is 1.5 ppm.
In order to simulate more finely the cavity round-trip loss, we combine the roughness map shown in Fig. 18 and the flatness map shown in Fig. 6 into a single map. Before directly adding the maps, some adjustments are necessary. First they must have the same pixel size. Then, in order to avoid a gap between both maps, the edges of the roughness map were smoothly reduced until the pixels of the periphery fall to zero. At last, the final map used in the FFT simulation had to be adapted to the simulation computing capability. Indeed, using OSCAR, about one million points is considered as reasonable grid size to reduce the computation time. The 12.2 mm aside roughness map was under-sampled to 638 X 638 pixels, while the 21.0 mm diameter low frequency map was over-sampled to 1098 pixels X 1098 pixels.
Finally, the lateral spatial resolution of the “Micromap” which was initially 2.56μm/pixel was reduced to 19.2μm/pixel. However it is still 18 times greater than the spatial resolution accessible with the wave-front measurement. After combining the 2 maps, the maximal spatial frequency of the full map reaches 26000 m−1 instead of 1430 m−1 when taking into account only the low spatial frequency map. The RMS height of the new map is 0.77 nm in the region where both the maps overlapped, to be compared to the 0.63 nm RMS of the map with only the low frequency.
Using OSCAR, first we placed the full map as end mirror of the cavity and we computed the round-trip losses (RTL) on a grid size of 1024 x 1024 pixels. Then the same simulation is done but using now the down-sampled map with a pixel size is 350µm as the wave-front map lateral resolution. In both cases, the nominal radius of curvature of the mirrors is set to 14.1 meters. As all the simulations, the clear aperture of the mirrors is set to 21 mm (slightly less to one inch in order to clip the edge defects of measurement and also to consider the realistic coating diameter on the substrate). The losses of both maps are computed on the mode 00 (RTL00). The results are given in the Table 3 for different values of the cavity finesse. We noticed that by adding the high spatial frequency in the simulations, the round trip losses increased by a factor 2. That is an important result, meaning that for realistic estimation of the loss, the wavefront map is not enough and that additional measurements with an optical profilometer are necessary.
Table 3. Simulated Round-trip loss in the cavity using as end mirror the low frequency map associated with the additional high frequencies roughness on the full map. (Radius of curvature = 14.1 m)
The additional losses we found due to the high spatial frequency are coherent with the simple model from Eq. (10) which predicted losses from high frequency to be between 1.4 ppm and 2 ppm.
7. Loss budget and conclusion
In this article, we highlighted the influence of surface maps to overall round trip losses in the cavity. Different numerical tools have been presented in order to derive in a realistic way the cavity losses.
Using some real data of the surface of standard super-polished optics, we saw that according to the geometry of the cavity, a large number of higher order modes could resonate at the same time as the fundamental one, generating some excess loss. As a result, we showed that for a high finesse cavity, the losses can change by two orders of magnitude if the radii of curvature of the mirrors shifts by only 1%.
Even if the polisher can guarantee the nominal radius of curvature, stress and thickness non-uniformity due to the coating process or thermal effect may shift the mirror radius. In such a cavity, it will be wise to provide a thermal compensation system in order to change the curvature in situ if necessary. Assuming an optimized curvature of the mirrors, we can expect less than 0.2 ppm of loss only due to the low frequency flatness of both mirrors for our 10 m long filter cavity. In the square area of 8 mm x 8 mm at the center of the substrate, we have 80% of chance to reach less than 0.5 ppm of loss and around the same probability to have less than 0.5 nm of local height RMS on the substrate surface.
Finally the additional losses due to the light diffracted outside the cavity by the roughness defects of the mirrors have been estimated. We saw that the high frequency defects could add 3 ppm of loss in the cavity. The finite mirror sizes associated with the roughness of the surface bring around 50% of the overall losses. In our simulations, we did not include possible point defects on the coating or substrate, measurement of large angle scattering indicates that additional loss as low as 2 ppm per mirror could be obtained [20].
If we select mirrors twice larger for the cavity, we checked that the amount of light falling outside the cavity is lower as predicted with Eq. (10). Indeed using the map containing both high and low frequencies, the RTL simulation with 2-inches mirrors gives us 0.7 ppm instead of 3.2 ppm for a finesse of 45000. Using a full flatness and roughness map on each 2-inches mirror of the cavity, we found that the RTL is now 1.3 ppm.
Regarding the mirror specifications, the polishing companies that provided the micro-polished substrates used for this study announce a flatness quality better than λ /10 (λ = 633 nm), in the standard range. Our measurement showed that the optics well exceed those specifications, so we recommend to require the same surface quality we have measured in order to guarantee the losses predicted in this study. For example, the two-inch substrate flatness must be less than λ /20 peak to valley. We also saw that the RMS flatness must be less than 0.5 nm over the central diameter of 10 mm. At last, the most important requirement is the RMS roughness with the spatial frequencies above 1000 m−1, which must be less than 0.15 nm over the central part.
As a conclusion, we recommend that the polished substrates in the 10 meters-long cavity must have a diameter of 2 inches and a radius of curvature of 14.25+/−0.1 meters (cavity g-factor of 0.09). This curvature allows obtaining stable and minimal loss lower than 1 ppm by simulation and without adding the losses due to the roughness. Alternatively a thermal compensation system can be used in order to adjust in situ the radius of curvature.
Appendix A Defining the cavity round-trip loss
In this section, we detailed the calculation used to derive the formula for the loss (Eq. 4).
For the following sections, we define Pin, Pcirc, Pr and Pt as respectively the input power, the circulating power, the reflected power and the transmitted power of the cavity. We also define P00circ, P00r, P00t as the circulating power, the reflected power and the transmitted power in the fundamental mode TEM00. The projection of the electric field to the fundamental mode is numerically done with the overlap integral. The input beam of the cavity is always assumed to be in the fundamental mode (i.e. P00in = Pin).
Assuming the conservation of the optical energy by the cavity, what enters in the cavity is equal to what is going out plus what is lost:
Dividing Eq. (11) of the optical energy conservation by the circulating power, we obtain:
Introducing RTL, the round trip power loss and r1, r2 and t1, t2 the amplitude reflectivities and transmissions of respectively the input and end mirror of the cavities, Eq. (12) can be written as:
Typically since r2 is close to 1, it can be neglected. This method to calculate the round trip loss is one of the most widely used because it is relatively easy to implement numerically.
Acknowledgments
The authors gratefully acknowledge the support of the European Gravitational Observatory (EGO). N. Straniero is supported by the EGO collaboration convention for the funding of a fellowship at LMA. G. Cagnoli is grateful to the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the “Université de Lyon” for its financial support within the program “Investissements d'Avenir” (ANR-11-IDEX-0007) of the French government operated by the National Research Agency (ANR).
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