Birefringence measurements of substrate materials and coatings for Einstein Telescope

4th GRAvitational-waves Science & technology Symposium (GRASS 2024), Trento, IT Birefringence measurements of substrate materials and coatings for Einstein Telescope Federico Della Valle1, Giovanni Di Domenico2, Aur´elie Mailliet2, Lorenzo Malagutti2, Emilio Mariotti1, Andrea Mazzolari2, Alina Mariana Soflau2and Guido Zavattini2 1Dip. di Scienze Fisiche della Terra e dell’Ambiente, via Roma 56, I-53100 Siena, Italy and INFN-Sez. di Pisa, Largo B. Pontecorvo 3, I-56127 Pisa, Italy 2Dip. di Fisica e Scienze della Terra, University of Ferrara, Italy and INFN-Sez. di Ferrara, Via Saragat 1, Edificio C, Ferrara, Italy February 14, 2025 Abstract Given the planned tenfold improvement in sensitivity of the future Einstein Telescope, the birefringence of optical elements (substrates and coatings) is becoming an important optical parameter to be measured. A sensitive optical polarimeter has been developed at the Department of Physics and Earth Sciences of the University of Ferrara and INFN - Ferrara Section, Italy to measure ellipticities and rotations induced by substrates and coatings on a linearly polarised laser beam with wavelength λ= 1064 nm both in transmission and reflection. The sensitivity in optical path difference, both in transmission and in reflection, is SD≈10−12 m/√Hz. The technique takes advantage of both heterodyne detection and of time-dependent signals (ellipticity and rotations) induced by two co-rotating half-wave plates. The method is demonstrated on some crystalline silicon samples and high-reflectivity coatings. Some examples are reported. DOI: 10.5281/zenodo.14609500 Introduction With the increasing sensitivity of future third gravitational wave detectors, the birefringence ∆nof substrates and its uniformity across the laser beam’s cross section is becoming an issue as was seen in the KAGRA interferometer [1]. One of the critical optical elements is the substrate of the input test masses of the two interferometer arms. These substrates are thick, of the order of several tens of centimeters, and the beam spot size is rather large, covering a significant portion of the substrate. Mirror coatings also present a birefringence map. The characterisation of both the substrates and the reflective coatings, in order to learn how to grow them with minimum birefringence, is mandatory. Birefringence vs. dichroism It is generally assumed that the index of refraction nof a substrate medium is uniform and that, at least in a plane perpendicular to the beam propagation, the index of refraction does not depend on 4th GRAvitational-waves Science & technology Symposium (GRASS 2024), Trento, IT the polarisation direction: given two perpendicular directions, ∥and ⊥, the difference of the index of refraction n∥−n⊥= ∆nfor light polarised along the ∥and ⊥directions is zero. The effect of a birefringence on a linearly polarised beam of light propagating through a medium of path length Lis a phase difference ∆ϕof the electric field components along the ∥and ⊥directions. This phase difference results in the electric field describing an ellipse whose ratio of the minor to major axes is the modulus of the ellipticity ψ. Given an input electric field  Eγoriented along the X axis (see Figure 1, left)  Eγ=Eγ1 0(1) and a birefringent medium whose ∥axis forms an angle ϑwith the X axis, the phase shift ∆ϕ(we will assume |∆ϕ| ≪ 1) between the ∥and ⊥components of  Eγis ∆ϕ=2π λZL ∆n dz. (2) The output electric field can then be approximated to first order as  E′γ≈Eγ1 + i∆ϕ 2cos 2ϑ i∆ϕ 2sin 2ϑ.(3) The ellipticity ψ, defined as the ratio of the minor to major axis of the ellipse generated by the electric field  E′γ, is then ψ≈iπ λsin 2ϑZL ∆n dz. (4) and can be treated as an imaginary number. Figure 1: Left: ellipticity |ψ|=a/b induced by a birefringent medium whose axis is at an angle ϑwith respect to the incident electric field. Right: rotation εinduced by a linear dichroism whose axis is at an angle ϑwith respect to the incident electric field. More in general, the index of refraction is a complex number: ˜n=n+iκ. The imaginary term κ, known as the extinction coefficient, describes the absorption of a medium. If, given two perpendicular directions ∥and ⊥, the difference ∆κ=κ∥−κ⊥= 0 the medium is said to have a linear dichroism, that is a polarisation-dependent absorption. The electric field amplitude of a plane wave propagating 4th GRAvitational-waves Science & technology Symposium (GRASS 2024), Trento, IT along the Z axis in a uniform medium is described by E(z) = Eei2π λ(n+iκ)zresulting in an exponential decay of the electric field characterised by the exponent Q=−2π λκz. In analogy to (2), by defining a differential attenuation of the electric field ∆Q ∆Q=−2π λZL ∆κ dz, (5) after a length Lthe electric field of a linearly polarised beam of light forming an angle ϑwith the ∥ direction of the dichroic medium can be approximated with  E′γ≈Eγ1−∆Q 2cos 2ϑ −∆Q 2sin 2ϑ.(6) The Xand Ycomponents of  E′γ(the Ycomponent is now a real quantity) are still in phase resulting in a linear polarisation whose direction, though, is rotated by an angle ε≈ −(∆Q/2) sin 2ϑ ε≈ −π λsin 2ϑZL ∆κ dz. (7) and is treated as a real number. This is shown in Figure 1, right. Note that the measurement of the output power after an analyser oriented along the Ydoes not allow to discriminate the contributions of a birefringence and a dichroism. In the following we will describe how to distinguish between an ellipticity and a rotation. Measurement method Two crossed polarisers set to maximum extinction (with extinction ratio σ2) with a birefringent medium between them generating an ellipticity ψgiven by equation (4) will transmit a power Pout Pout =P0σ2+|ψ|2.(8) A typical value for the extinction ratio of a pair of good polarisers is σ2≲10−7. A static measurement proves to be difficult in that σ2∼ |ψ|2. Furthermore any optical element between the two crossed polarisers will contribute to ψmaking it very difficult to separate the effect of the substrate from the other optical elements. From equation (2) it is clear that small ellipticities add up algebraically. Therefore by adding a time dependent known ellipticity η(t) to ψand modulating in time the substrate ellipticity ψ=ψ(t) itself, Pout now results in Pout(t) = P0hσ2+|ψ(t) + η(t)|2i=P0σ2+|η(t)|2+Re[2η(t)ψ∗(t)] + |ψ(t)|2(9) linearising the output power as a function of the desired ellipticity. Furthermore, the time dependence (typically sinusoidal) of both η(t) and ψ(t) generates a well-defined Fourier spectrum with the components deriving from the double product η(t)ψ(t) far from the DC component where noise is generally higher. Indeed, if η(t) = iη0cos ωmtand ψ(t) = iψ0cos ωψtthe dominant Fourier components of Pout(t) will be (here we neglect terms of order |ψ(t)|2and higher) ˜ Pout(DC) = P0σ2+η2 0 2(10) ˜ Pout(ωm+ωψ) = ˜ Pout(ωm−ωψ) = P0η0ψ0(11) ˜ Pout(2ωm) = P0 η2 0 2(12) 4th GRAvitational-waves Science & technology Symposium (GRASS 2024), Trento, IT from which one can determine ψ0=π λR∆n dL as ψ0=π λZ∆n dL =˜ Pout(ωm+ωψ) + ˜ Pout(ωm−ωψ) ˜ Pout(2ωm) η0 4.(13) The ellipticity modulation η(t) is obtained with a photo-elastic resonant modulator working at νm= 50 kHz and the modulation of ψ(t) is obtained by rotating the polarisation direction using two halfwave plates [2]. In practice, two lock-in amplifiers demodulating Pout(t) at ωmand 2ωmare used to extract the components at ωm±ωψand at 2ωm. It is interesting to note that the presence of a rotation ε(t) does not affect the ellipticity measurement when using an ellipticity modulator. In fact, ellipticity is an imaginary quantity whereas a rotation is real. Therefore the power output in the presence of both a rotation and an ellipticity with an ellipticity modulator will be Pout(t) = P0nσ2+ε(t) + ψ(t) + η(t) 2o=P0σ2+ε2(t) + |η(t)|2+Re[2η(t)ψ∗(t)] + |ψ(t)|2: (14) the modulation η(t) does not beat with the rotation ε(t). Introducing a rotation modulation φ(t) at a frequency ωFwith a Faraday rotator, for example, one can simultaneously and independently measure rotations and ellipticities: Pout(t) = P0nσ2+|ε(t) + φ(t) + ψ(t) + η(t)|2o= (15) =P0σ2+φ2(t) + |η(t)|2+ε2(t) + |ψ(t)|2+ 2φ(t)ε(t) + Re[2η(t)ψ∗(t)].(16) Transmission measurements In Figure 2 a scheme of the polarimeter for measuring the birefringence of a substrate in transmission is shown. Figure 2: Scheme of the polarimeter for transmission measurements. The linearly polarised 1064 nm beam passes through the rotating half-wave plate L1making the polarisation rotate at a frequency 2νw. The induced ellipticity due to the substrate will be time dependent resulting in a Fourier component at a frequency 4νw. The second half-wave plate L2stops the rotating polarisation allowing measurements in extinction. The ellipticity modulator adds a known ellipticity η(t) to ψ(t) linearising ψ(t). By demodulating the signal from PDE at the modulator frequency one can extract ψ(t). The 532 nm beam is used for alignment purposes and the magnetic field for calibration. Both 1064-nm and 532-nm wavelengths are used. The actual ellipticity measurement of the sample is performed with the 1064 nm laser whereas the 532 nm wavelength is used to align the rotating half-wave plates L1and L2as will be described shortly. Both wavelengths are polarised by the input polariser. To modulate the ellipticity induced by the sample at 1064 nm, a half-wave plate for 1064 nm 4th GRAvitational-waves Science & technology Symposium (GRASS 2024), Trento, IT rotating at a frequency νw= 2 Hz generates a rotating polarisation at a frequency 2νw. Given the dependence of the induced ellipticity of equation (4) on the angle ϑbetween the ∥axis and the polarisation direction, the ellipticity signal will be modulated at a frequency 4νw= 8 Hz. A second rotating half-wave plate L2at 1064 nm, phase locked to L1, stops the rotating polarisation allowing extinction. The photo-elastic ellipticity modulator adds an ellipticity η(t) to the ellipticity induced by the sample ψ(t). Finally, the light passes through the analyser set to extinction. A Faraday rotator for rotation measurements is also present in the polarimeter between the ellipticity modulator and the analyser but is not shown and will not be discussed further in this paper. A rotating 2.5 T, 82 cm long, dipolar magnetic field typically rotating at νB= 0.5 Hz is also present between the two rotating half-wave plates to generate a known ellipticity for calibration purposes at a frequency 4νw±2νB(the sign depends on the relative rotation direction of the half-wave plates and the magnetic field). Systematics As discussed in reference [3], the rotating half-wave plates generate a spurious ellipticity ψspurious(t) at various harmonics of the rotating wave plates, due to several causes. A general expression of the total ellipticity signal observed with the scheme in Figure 2, excluding the magnetic field effect, is ψtotal(t) = ψ(t) + ψspurious(t) = iψ0sin 4ϑ(t) + iα1(t) 2sin [2ϑ(t)+2ϑ1] + iα2(t) 2sin [2ϑ(t)+2ϑ2] (17) where ϑ(t) = ωwtin the frame of Figure 1, α1,2(t) represent the instantaneous phase differences from πgiven by the two half-wave plates and ϑ1,2represent the angles of the ∥axes of the two wave plates with respect to the input polarisation direction at t= 0. The α’s are time dependent due to the structural and alignment defects that are summarised below and thoroughly discussed in reference [3]; their time dependence contains harmonics of the rotation frequency νw. The most dangerous harmonic in ψspurious(t) is clearly the 4-th where the sample signal ψ(t) appears. From equation (17) a 4-th harmonic of νwcan be generated in ψspurious(t) if α1,2(t) has a second harmonic. By expanding α1,2(t) in terms of cos ϑ(t) one can write α1,2(ϑ(t), T) = α(0) 1,2(T) + α(1) 1,2cos ϑ(t) + α(2) 1,2cos 2ϑ(t) + ... (18) where α(0) 1,2(T) are the static average deviations from πof the retardation of the two wave plates, which in turn depends on temperature T, and α(1) 1,2and α(2) 1,2are the deviations from πof the retardation with, respectively, a one-revolution periodicity and a half-revolution periodicity. Contributions can be due to a wedge β≲10−6rad between the front and back surfaces of each wave plate coupled to a non-centered beam, a misalignment of the laser beam, the rotation axis and the normal to the wave plate surface and finally to a periodic de-centering of the wave plate during rotation. These effects are depicted in Figure 3. Each rotating wave plate must be aligned independently to minimise the above-mentioned spurious effects but this cannot be done at the nominal wavelength of the halfwave plates: a single rotating half-wave plate does not allow a measurement in extinction. The frequency doubled 532-nm beam is used for this reason. At 532 nm, 1064-nm half-wave plates behave as full-wave plates – with an extra contribution to α(0) 1,2(T) due to dispersion; in this case, then, the presence of a single rotating 1064 nm half-wave plate leaves the polarisation direction fixed, allowing to work in extinction. The spurious effects in Figure 3, though, remain at 532 nm. Therefore, each rotating wave plate can be independently aligned with the 532-nm beam keeping the other 4th GRAvitational-waves Science & technology Symposium (GRASS 2024), Trento, IT Figure 3: Different alignment issues which will generate spurious ellipticity harmonics of νw. In particular the left most and right most effects will generate a 4-th harmonic in ψ(t) still. For alignment, the rotating wave plates are mounted on a 4-degree optical mount. A typical acceptable value for the spurious 4-th harmonic ellipticity induced by the rotating half-wave plates is |ψ(4−th) spurious(t)|≲10−5. This spurious ellipticity (amplitude and phase) can be determined by removing the substrate sample from the optical path; its value is to be vector-subtracted from the measurement with the sample. This subtraction determines the ultimate sensitivity of the polarimeter which is estimated to be about |ψ(4−th) spurious(t)|/5≈2×10−6corresponding to an optical path difference sensitivity S∆D=Rsample ∆n dz ≲10−12 m. However, as will be shown in the section Results, the sample contribution ψ(t) is typically much larger than ψ(4−th) spurious(t). Calibration The rotating magnetic field is used to calibrate the polarimeter taking advantage of the Cotton-Mouton effect in air. Gases exhibit magnetic birefringence ∆nCM which is parametrised as ∆nCM = ∆nuPB2(19) where Pis the gas pressure (in atmospheres), B2is the square of the external dipolar magnetic field (in tesla) and ∆nuis the Cotton-Mouton specific for each gas. Assuming that air is mainly composed of 20% oxygen and 80% nitrogen one finds [4] ∆n(air) u= 6.4×10−13 T−2atm−1.(20) With the magnetic field installed on the setup parameterised by Rmagnet B2dL = 5.13 T2m the induced ellipticity at 1064 nm is |ψ(air) CM |= 1.0×10−5. The magnetic field is typically rotated at νB= 0.5 Hz so that the frequency at which the ellipticity ψ(air) CM appears is 4νw±2νB= 7 Hz or 9 Hz depending on its relative rotation direction with respect to the half-wave plates. 2D mapping For mapping the optical path difference induced by the birefringence of a sample, the sample is mounted on a horizontal-vertical mount with micrometer screws with a 2.5 cm range. 2-D maps with a typical step size of 2 mm of the induced ellipticity can therefore be taken. The spatial resolution is 4th GRAvitational-waves Science & technology Symposium (GRASS 2024), Trento, IT limited by the beam spot size of about 0.5 mm diameter. For each sample, one can extract both the amplitude of the average birefringence along the sample thickness and its direction. It must be noted that given the high refractive index of silicon of nSi = 3.55 at 1064 nm there is a significant etalon effect (no anti-reflection coatings must be used). This effect multiplies the induced ellipticity by a factor in the range 0.8÷1.2 but will not be considered in this paper. Reflection measurements Figure 4 shows a simplified scheme for reflection measurements. The principle is very similar to the transmission scheme. Figure 4: Simplified scheme for reflection measurements. The beam enters the polarimeter through the output port of the polariser and is linearly polarised. The beam then passes through the ellipticity modulator and rotating half-wave plate reaching the mirror under investigation. After reflection the polarisation rotation is stopped by a second passage through the half-wave plate. The beam then passes a second time through the modulator and is then anaylised by the polariser. The rejected beam is collected on a position sensitive photodiode to ensure that the reflected beam passes through the half-wave plate always in the same position. The laser beam enters the polarimeter through the output polariser (analyser) shown in Figure 2 and is therefore linearly polarised. The beam then passes through the ellipticity modulator and the rotating half-wave plate HWP thereby reaching the sample to be measured in reflection. After reflection the beam returns through the half-wave plate and the polarisation stops rotating. A second pass through the ellipticity modulator doubles the effective modulation to 2η(t). Finally, the beam is analysed in extinction. In this scheme the output power is P(refl.) out (t) = P0nσ2+|ψ(t)+2η(t)|2o=P0nσ2+ 4 |η(t)|2+Re[4η(t)ψ∗(t)] + |ψ(t)|2o(21) Here too the rotating half-wave plate will introduce harmonics in the total ellipticity ψtot(t) just as in the transmission case. In this case, the contribution of ψ(4−th) spurious to the 4-th harmonic of the total ellipticity ψtot(t) cannot be determined by removing the sample. Therefore for each position of the beam on the sample two measurements are taken: one with the sample at ϑ0= 0◦and the second at ϑ0= 90◦. For the two measurements the total ellipticities after the demodulation at νmwill be ψ(0◦) total(t) = ψ(t) + ψspurious(t) = iψ0sin 4ϑ(t) + iα(t) sin [2ϑ(t)+2ϑ2] (22) ψ(90◦) total (t) = −ψ(t) + ψspurious(t) = −iψ0sin 4ϑ(t) + iα(t) sin [2ϑ(t)+2ϑ2] (23) where the sample ellipticity changes sign between the two sample orientations. 4th GRAvitational-waves Science & technology Symposium (GRASS 2024), Trento, IT By further demodulating ψtotal(t) at 4ωw, the semi-sum and semi-difference of the in −phase and quadrature components are taken thereby extracting separately ψ(4−th) 0with its phase and ψ(4−th) 0,spurious with its phase too. During reflection measurements, attention must be taken to ensure that the reflected beam always passes through the same position on HWP otherwise ψspurious(t) will change when changing the position of the sample or its azimuthal angle. For this reason a position-sensitive photo-diode (PSD) is mounted on the non-extinguished beam as can be seen in Figure 4. Results Fourier spectrum Figure 5 shows a typical Fourier amplitude spectrum of the demodulated signal at νm. This particular spectrum refers to a transmission measurement but is very similar for reflection measurements. The amplitude is reported in optical path difference ∆D=R∆n dz. In red is a spectrum without a sample and in blue is the same spectrum with a 1 mm thick silicon sample. The half-wave plates were rotating at νw= 2 Hz and the magnetic field was rotating at νB= 0.5 Hz in the same direction as the half-wave plates, resulting in a Cotton-Mouton peak of air at ν(air) CM = 7 Hz. Figure 5: Typical demodulated optical path difference spectra, with the wave plates rotating at νw= 2 Hz. Blue spectrum: 1-mm thick Si(100) sample; red spectrum: no sample. As can be seen the first three harmonics of νware completely superimposed in the red and the blue spectra, and do not depend on the presence of the silicon sample and nor does the Cotton-Mouton signal peak. The Cotton-Mouton peak indicates an optical path difference ∆DCM = 3.2×10−12 m resulting in a measured ∆n(air) u= ∆DCM/Rmagnet B2dL = (6.3±0.2) ×10−13 T−2m−1compatible with the value in (20). The fourth harmonic of νwis where the silicon sample generates an ellipticity signal. As can be seen, without the sample, the spurious peak corresponds to an optical path difference ∆Dspurious = 3.2×10−12 m whereas with the sample the optical path difference ∆Dsample = 1.1× 10−10 m corresponds to an average birefringence over the thickness of ∆n= 1.1×10−7. 4th GRAvitational-waves Science & technology Symposium (GRASS 2024), Trento, IT In reflection measurements the ellipticity spectrum contains the same harmonics and is very similar except that a spectrum without the sample cannot be taken. 0 2 4 6 8 10 12 14 16 X (mm) 0 2 4 6 8 10 Y (mm) < n > = 1.42e-07 0.5 1.0 1.5 2.0 2.5 3.0 n 1e 7 0 2 4 6 8 10 12 14 16 X (mm) 0 2 4 6 8 10 Y (mm) < n > = 1.31e-07 0.5 1.0 1.5 2.0 2.5 3.0 n 1e 7 Figure 6: Birefringence maps of two 1-mm thick Si(100) crystalline samples. For all the points the same no-sample value has been vector subtracted. 2-D transmission maps Figure 6, left and right, shows examples of birefringence maps integrated along the beam path of two Si(100) crystalline samples 2.5 cm ×2.5 cm, 1 mm thick. The samples were supported vertically without stress by a holder with a 1 mm groove. The birefringence maps were taken with a 2 mm step. The maps show a clear nonuniformity in birefringence both in value and in direction specific for each sample. All angles are covered, indicating that there is no connection to the crystalline planes. Note that here the birefringence axis directions for the two samples are at the moment undetermined to an additive angle. Figure 7: Optical path difference maps for two reflecting samples samples. Left: a commercial silver mirror (protected); right: a dielectric mirror with 10−3transmission. To get rid of spurious signal, each point is half the vector difference of two measurements taken with the sample oriented at 0◦and 90◦, respectively.