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 10 − 9 to low 10 − 8⁠, 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 < 10 %⁠. The study assigned a loss budget of 1 % from birefringence, assuming the worst case scenario that the birefringence polarization n 0   and   n e of the test mass are orientated at an angle of π / 4 to the laser polarization, the acceptable birefringence limit for ET was estimated to be 10 − 8⁠.

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.

We used the approach of Ref. 12 with a photoelastic modulator (PEM)13 and a 2D translation platform to scan a FZ Si test mass, measuring the birefringence of the entire test mass in the ⟨100⟩ axis.

The best birefringence level at the central part of the test mass is close to our measurement limit of the order Δ n ∼ 10 − 9⁠.

The degree of birefringence in a material is defined as the difference of the refractive indexes in the two polarization directions, which can be measured as the retardation phase shift when light passes through the material,
(1)
where λ is the light wavelength, d is the thickness of the material as the light passes through, and φ is the phase shift between light in the two polarization directions with refractive indices (⁠ n 1   and   n 2⁠).
The effect of the polarization loss at the interferometer output can be estimated through Jones matrix transformation. For simplicity, assuming only the test mass in one of the arms in the interferometer has birefringence. This does not affect the general conclusion as the interference is relative to the two arms. The output power at the interferometer dark port P min⁠, which cause fringe contrast defect and, thus, loss, can be expressed as14 
(2)
where P 0 is the circulating power of the interferometer, and θ is the angle between the input laser polarization and the birefringence orientation. Assuming worst case scenario that θ = π / 4 and very small birefringence, the output power became
(3)

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.

Any defect or internal residual stress in the crystal growth, as well as external force, will result in the stress-induced birefringence. The stress-induced birefringence is given in Ref. 15 as
(4)
where n 0 is the refractive index of the material in unstressed state, σ 1   and   σ 2 are the principal stresses, and (⁠ π 11 − π 12⁠) is the stress optical coefficient of the material.

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.

TABLE I.

Silicon test mass parameters used in the simulation.

Test mass dimension ∅ 100 × 30   mm 2
Young's modulus  130   GPa 
Poisson ratio  0.27 
Refractive index (⁠ n 0⁠)  3.46 
π 11 – π 12  14.4 × 10 − 7   MPa − 1 
Test mass dimension ∅ 100 × 30   mm 2
Young's modulus  130   GPa 
Poisson ratio  0.27 
Refractive index (⁠ n 0⁠)  3.46 
π 11 – π 12  14.4 × 10 − 7   MPa − 1 
View Large

For any point on the test mass, the first and second principal stresses (⁠ σ 1   and   σ 2⁠) 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 (⁠ Δ n s ≤ 10 − 10⁠) than that when sitting on a hard support (⁠ Δ n b ≤ 10 − 8⁠).

FIG. 1.
The simulated birefringence map of a Si ⟨100⟩ test mass: (a) suspended by side holes and (b) sitting on a hard surface.
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The simulated birefringence map of a Si ⟨100⟩ test mass: (a) suspended by side holes and (b) sitting on a hard surface.

FIG. 1.
The simulated birefringence map of a Si ⟨100⟩ test mass: (a) suspended by side holes and (b) sitting on a hard surface.
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The simulated birefringence map of a Si ⟨100⟩ test mass: (a) suspended by side holes and (b) sitting on a hard surface.

Close modal

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 ∅ 3 × 4   mm 2 holes on each flat side for the side-pin suspension.16 

To measure the birefringence, we follow the concept of Ref. 12. The measuring principle is shown in Fig. 2. The laser beam passes through a polarizer and then a photoelastic modulator (PEM) which modulates the light polarization state before impinging on the sample. An analyzer is placed behind the sample. The output signal is measured by a photodetector (PD) in two configurations, with the analyzer polarization axis at 45° and 0° relative to the PEM. The modulated output signal of each configuration contains a DC term, a 1F term at the modulation frequency f⁠, and other higher order harmonic frequency terms. The phase shift retardation of the sample φ can be determined by the ratio of the DC term and the 1F term,12 
(5)
(6)
FIG. 2.
Schematic diagram of the two output signal states I 45 °   and   I 0 ° used for measuring the birefringence of the silicon test mass.
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Schematic diagram of the two output signal states I 45 °   and   I 0 ° used for measuring the birefringence of the silicon test mass.

FIG. 2.
Schematic diagram of the two output signal states I 45 °   and   I 0 ° used for measuring the birefringence of the silicon test mass.
View largeDownload slide

Schematic diagram of the two output signal states I 45 °   and   I 0 ° used for measuring the birefringence of the silicon test mass.

Close modal

Here, V 45 ° ( 0 ° ) is the voltage measured from the PD, which is proportional to the intensity I 45 ° ( 0 ° )⁠, ρ is the birefringence orientation, and Δ 0 is the retardation amplitude of the PEM. The J 0   and   J 1 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.

FIG. 3.
Experimental setup for the birefringence mapping. A 2 μm laser beam passes through the polarizer, the PEM, the Si test mass, and the analyzer to the photodetector. The signal from the photodetector is collected by a fast sampler. The silicon test is on a translation stage controlled by the computer through a Raspberry Pi.
View largeDownload slide

Experimental setup for the birefringence mapping. A 2 μm laser beam passes through the polarizer, the PEM, the Si test mass, and the analyzer to the photodetector. The signal from the photodetector is collected by a fast sampler. The silicon test is on a translation stage controlled by the computer through a Raspberry Pi.

FIG. 3.
Experimental setup for the birefringence mapping. A 2 μm laser beam passes through the polarizer, the PEM, the Si test mass, and the analyzer to the photodetector. The signal from the photodetector is collected by a fast sampler. The silicon test is on a translation stage controlled by the computer through a Raspberry Pi.
View largeDownload slide

Experimental setup for the birefringence mapping. A 2 μm laser beam passes through the polarizer, the PEM, the Si test mass, and the analyzer to the photodetector. The signal from the photodetector is collected by a fast sampler. The silicon test is on a translation stage controlled by the computer through a Raspberry Pi.

Close modal

The 1 F 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.

FIG. 4.
(a) Measured birefringence map (10k points) of the Si test mass supported by the side pins. The stress-induced birefringence is concentrated around the suspension points. There is an internal high birefringence feature. (b) The central area of the test mass with birefringence in the range of mid 10 − 8 to mid 10 − 9. The average birefringence of the central 40 mm area is 2.8 × 10−8.
View largeDownload slide

(a) Measured birefringence map (10k points) of the Si test mass supported by the side pins. The stress-induced birefringence is concentrated around the suspension points. There is an internal high birefringence feature. (b) The central area of the test mass with birefringence in the range of mid 10 − 8 to mid 10 − 9⁠. The average birefringence of the central 40 mm area is 2.8 × 10−8.

FIG. 4.
(a) Measured birefringence map (10k points) of the Si test mass supported by the side pins. The stress-induced birefringence is concentrated around the suspension points. There is an internal high birefringence feature. (b) The central area of the test mass with birefringence in the range of mid 10 − 8 to mid 10 − 9. The average birefringence of the central 40 mm area is 2.8 × 10−8.
View largeDownload slide

(a) Measured birefringence map (10k points) of the Si test mass supported by the side pins. The stress-induced birefringence is concentrated around the suspension points. There is an internal high birefringence feature. (b) The central area of the test mass with birefringence in the range of mid 10 − 8 to mid 10 − 9⁠. The average birefringence of the central 40 mm area is 2.8 × 10−8.

Close modal
FIG. 5.
(a) Measured birefringence map (5k points) of the Si test mass resting on foam. There is high stress around the drilled side holes. The high internal birefringence is still present. (b) Birefringence of the central area of the test mass with an average of 2.0 × 10−8 over central 40 mm.
View largeDownload slide

(a) Measured birefringence map (5k points) of the Si test mass resting on foam. There is high stress around the drilled side holes. The high internal birefringence is still present. (b) Birefringence of the central area of the test mass with an average of 2.0 × 10−8 over central 40 mm.

FIG. 5.
(a) Measured birefringence map (5k points) of the Si test mass resting on foam. There is high stress around the drilled side holes. The high internal birefringence is still present. (b) Birefringence of the central area of the test mass with an average of 2.0 × 10−8 over central 40 mm.
View largeDownload slide

(a) Measured birefringence map (5k points) of the Si test mass resting on foam. There is high stress around the drilled side holes. The high internal birefringence is still present. (b) Birefringence of the central area of the test mass with an average of 2.0 × 10−8 over central 40 mm.

Close modal
FIG. 6.
(a) Measured birefringence map (2k points) of the Si test mass siting on a hard cradle surface. The high stress area is at the bottom of the test mass while the central 40 mm area (b) average birefringence is now 3.5 × 10−8.
View largeDownload slide

(a) Measured birefringence map (2k points) of the Si test mass siting on a hard cradle surface. The high stress area is at the bottom of the test mass while the central 40 mm area (b) average birefringence is now 3.5 × 10−8.

FIG. 6.
(a) Measured birefringence map (2k points) of the Si test mass siting on a hard cradle surface. The high stress area is at the bottom of the test mass while the central 40 mm area (b) average birefringence is now 3.5 × 10−8.
View largeDownload slide

(a) Measured birefringence map (2k points) of the Si test mass siting on a hard cradle surface. The high stress area is at the bottom of the test mass while the central 40 mm area (b) average birefringence is now 3.5 × 10−8.

Close modal

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 Δ n = 10 − 9 − 10 − 8⁠. 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 Δ n < 5 × 10 − 9⁠. We calculated that the fractional error due to the polarizer having a finite extinction ratio α is γ = ( Δ n − Δ n α ) / Δ n < 2 α⁠. In our case, α ∼ 10 − 5⁠, so γ < 2 × 10 − 5 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 Δ n limit = ( 5.0 ± 0.3 ) × 10 − 9⁠.

The average birefringence measured in the central 40 mm is 2 ± 0.5 × 10 − 8⁠. This number is close to the upper limit tolerance of ET. Using this value, according to Eq. (3), for the worst case scenario, the loss due to such birefringence in a 50 cm thick test mass in a GW detector such as ET would be
(7)

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).

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.

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© 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Published open access through an agreement with The University of Western Australia Department of Physics