Excellent mechanical and thermal properties of silicon make it a promising material for the test masses in future gravitational wave detectors. However, the birefringence of silicon test masses, due to impurity and residual stress during crystal growth or external stress, can reduce the interference contrast in an interferometer. Using the polarization–modulation approach and a scanning system, we mapped the birefringence of a float zone silicon test mass in the ⟨100⟩ crystal orientation to assess the suitability of such material for future gravitational wave detectors. Apart from the stress-induced birefringence at the supporting area due to the weight of the test mass, the high resolution birefringence map of the silicon test mass revealed a high birefringence feature in the test mass. At the central 40 mm area, birefringence is in the range of mid to low , which satisfy the requirement for future gravitational wave detectors.
The next generation of gravitational wave (GW) detectors will likely be operating at cryogenic temperatures1 to reduce the thermal noise and other thermally induced effects in the detectors. In addition, future detectors will have high laser power to reduce quantum phase noise.2 The high optical power in the arm cavities of the interferometer detectors causes temperature gradients in the test masses due to absorption, resulting in thermal deformation and thermo-optic distortion. Silicon (Si) is one promising material for the test masses.1,3,4 Si has excellent mechanical, optical, and thermal properties at cryogenic temperatures. Si thermal expansion approaches zero around 18 and 123 K,5 resulting in negligible substrate thermo-elastic noise and negligible thermal surface distortion around these temperatures.
Its high thermal conductivity at cryogenic temperatures makes it easier to cool and results in lower thermal gradients.
Silicon is classified as a diamond cubic crystal, which in theory does not have birefringence. However, due to impurities and dislocations in the crystal or residual mechanical/thermal stress during the crystal growth, the test mass will have a certain level of birefringence. In addition, there will be a stress-induced birefringence due to the stress around the suspension points.
In an interferometer GW detector, the input light is linearly polarized. If the test masses have spatially dependent birefringence, some light from the two arms that pass through the input test masses will not be in the same polarization when combined at the beam splitter and will not interfere. This light will be equally split between the dark detection port and the bright input port of the interferometer where normally interference would result in almost all going to the bright input port, this is a contrast defect. Contrast defect allows light to go from the input to the output bypassing the interferometers common mode rejection, resulting in input laser frequency and intensity noise coupling to the output. The light at the input port of GW detectors is recycled with a power recycling mirror; therefore, contrast defect also results in loss from the power recycling cavity and hence increased shot noise. The KAGRA detector uses sapphire test masses, and it was found that the birefringence of sapphire increased shot noise by 16% and also increased laser frequency noise by an order of magnitude compared to non-birefringent test masses.6 In addition, the squeezing technique7 in GW detectors requires lower optical losses. A study for the Einstein Telescope (ET)4 showed that the total optical loss required to achieve a squeezing level of 10 dB is . The study assigned a loss budget of from birefringence, assuming the worst case scenario that the birefringence polarization of the test mass are orientated at an angle of to the laser polarization, the acceptable birefringence limit for ET was estimated to be .
There are many previous studies on Si birefringence for samples with different crystal growth methods and different crystal orientations.4,8–11 Most of the measurements were done by measuring a few points or small areas on the sample. It will be challenging to produce large, uniform silicon test masses for future GW detectors. There are two main single crystal silicon manufacturing techniques: the float zone (FZ) method and the Czochralski (CZ) method. We chose FZ silicon test masses due to the lower impurity concentration than CZ silicon. Impurity is generally non-uniformly distributed, resulting in non-uniform birefringence. It is essential to measure the clear aperture of the test mass to ensure future GW detector requirements are met.
The best birefringence level at the central part of the test mass is close to our measurement limit of the order .
In a GW detector, the test masses are suspended. There will be mechanical stress on the test mass due to the suspension, which will result in extra stress-induced birefringence. Although the test mass suspension points are generally far away from the central area where the laser beam passes, there will still be certain level of the external stress-induced birefringence present that needs to be carefully analyzed.
Estimation of the stress-induced birefringence due to the suspension with gravity was performed by first simulating the stress distribution with finite element modeling (FEM) COMSOL MultiPhysics and then calculating the birefringence with the stress information. The test mass investigated is a 100 mm diameter and 30 mm thick cylinder which has two side flat surfaces, and two small holes on each flat surface for the purpose of suspension. The holes were considered fixed constraints in the simulation. We used the Si ⟨100⟩ photoelastic properties. Table I lists the parameters used in the simulation.
|Test mass dimension .||.|
|Refractive index ( )|
|Test mass dimension .||.|
|Refractive index ( )|
For any point on the test mass, the first and second principal stresses ( ) are obtained from FEM. Then, the difference in the refractive index due to stress is estimated using Eq. (4). Figure 1 shows the calculated birefringence map of the Si test mass from the simulated stress, supported by side pins and a hard cradle, respectively. The stress-induced birefringence around the supporting points is dominant. The calculated minimum birefringence occurs at the center of the test mass. The value is smaller when supported by the side pin ( ) than that when sitting on a hard support ( ).
The float zone Si test mass we measured has the symmetry ⟨100⟩ axis aligned with the cylindrical axis. The test mass is polished with no optical coatings. It has two flats on the circumference, with two holes on each flat side for the side-pin suspension.16
Here, is the voltage measured from the PD, which is proportional to the intensity , is the birefringence orientation, and is the retardation amplitude of the PEM. The are the 0th order and 1st order Bessel functions, respectively. The birefringence magnitude and angle can be obtained using Eq. (1).
Figure 3 shows the experimental setup. A polarized 2 μm laser beam17 passes through the polarizer, the PEM, the test mass, and the analyzer before it is collected by a photodetector. The relative polarization directions of the polarizer, the analyzer, and the retardation axis of the PEM are as indicated. The test mass is on a translation platform from a 3D-printer and the analyzer was mounted on a motorized rotation stage to allow automatic birefringence mapping.
The signal is extracted from the power spectrum at the modulation frequency, and the DC term is extracted from the mean DC level of the time trace. The 3D-printer translation stage is controlled via a Raspberry Pi with Python 3 code.18 The automatic mapping was realized by synchronizing the translation stage step and signal acquisition from the fast sampler. The data were processed in Matlab19 and the map was generated in Python.
A square grid of up to 10 000 points was scanned to cover the entire test mass.
Figures 4–6 show the birefringence map of the Si test mass supported with side pins, on a foam, and on a hard cradle, respectively.
It can be seen that there are areas around the supporting pins with high birefringence due to the support stress. The manufacturing of the side holes also introduce residual stress which is apparent in Fig. 5 when the test mass sits on a foam. The points where the beam overlaps with the hole positions and the edge of the test mass are not reliable as the beam direction is changed by the hole or edge geometry. There is a clear high birefringence “strip” feature in the upper part of the test mass, possibly due to impurity. This feature is always present when we rotate the test mass 180°. At the center area of the test mass, the birefringence is in the range of . It is apparent that the support stress has some effect in the central area.
It should be noted that the test mass has a small wedge for anti-reflection purpose in an interferometer. This means that once the experimental apparatus is aligned with the test mass in place, the beam is not aligned anywhere else. Therefore, the data outside the test mass (empty space) do not give meaningful value and could not be used as a comparison in situ.
The measurement limit of the system is likely due to misalignment, laser beam jitter, and the residual stress in the fused silica crystal of the PEM or imperfection of the polarizer, as well as the electronic noise. The intrinsic birefringence of the PEM was estimated to be <0.2 nm,12 corresponding to . We calculated that the fractional error due to the polarizer having a finite extinction ratio α is . In our case, , so which is negligibly small.
To determine the measurement limit, 20 measurements were made with the same experimental setup without the test mass over 30 days, which resulted in a limit of .
This is in the same order of advanced LIGO requirement for the contrast defect.2,20 In summary, we have mapped the birefringence of a float zone silicon test mass. The results indicated that the birefringence of float zone silicon will meet the requirement for GW detectors. The stress-induced birefringence was clear near the suspension points but was at an acceptable level in the central area. The map also showed some high level birefringence feature that is most likely due to defects created in the manufacturing. It is, thus, very important that detailed birefringence mappings will be carried out for the quality control of the test masses.
This project was supported by the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (No. CE170100004).
Conflict of Interest
The authors have no conflicts to disclose.
Vahid Jaberian Hamedan: Formal analysis (equal); Investigation (lead); Software (equal); Writing – original draft (lead); Writing – review & editing (equal). Alexander James Adam: Methodology (equal); Software (equal); Writing – original draft (supporting). Carl Blair: Investigation (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Li Ju: Conceptualization (supporting); Investigation (supporting); Supervision (equal); Validation (lead); Writing – original draft (equal); Writing – review & editing (equal). Chunnong Zhao: Conceptualization (equal); Formal analysis (supporting); Investigation (supporting); Supervision (equal); Validation (equal); Writing – review & editing (supporting).
The data that support the findings of this study are available from the corresponding author upon reasonable request.