Thermal defocus-free Hartmann Wavefront Sensors for monitoring aberrations in Advanced Virgo

The thermal expansion or contraction of the HWS components due to environmental temperature fluctuations affects the HWS performance, resulting in the generation of a spurious curvature in the reconstructed wavefront. Spurious contributions to the wavefront curvature coupled to temperature variations are usually termed as thermal defocus [6]. In order to estimate it, we must precisely know the response of the HWS components' materials to a temperature variation—which is quantified by their thermal expansion coefficient α.

We can identify two main effects which can contribute to thermal defocus: thermal expansion/contraction of the HP and thermal expansion/contraction of the CCD. Both of them result in an increase/decrease in the distance between the holes and, therefore, between the CCD spots which, in turn, changes the wavefront gradient measured by the sensor. Because the CCD sensor is a metal-oxide-semiconductor [13], it is expected to expand/contract less than the HP—which is made of invar. Therefore, the effect of the CCD surface deformation is henceforth neglected.

To compute the thermal defocus, we consider a beam ray having divergence angle θ and radius of curvature RoC. The beam ray propagates through a HP hole and after traveling the distance $L/\mathrm{cos}(\theta)$—where the radius of curvature becomes RoC$^{^{\prime}}$—it impinges on the CCD as shown in figure 4.

Zoom In Zoom Out Reset image size

For small values of θ, we can assume RoC $\simeq$ RoC$^{^{\prime}}$ and $\theta = d/\mathrm{RoC}$. The spot displacement Δx on the CCD with respect to a reference plane wave is related to the radius of curvature of the wavefront by

$$\begin{align} \Delta x = \frac{d\cdot L}{\mathrm{RoC}^{^{\prime}}} \end{align} \tag{ 2 }$$

where d is the distance between the optical axis and the HP hole as displayed in figure 4.

The HP thermal expansion/contraction generates a transversal displacement of the hole, resulting in a variation Δd of the projection of the hole on the CCD surface and in a change Δθ of the divergence as well [6]. Moreover, also a variation ΔL of the distance between the HP and the CCD can provide a non-negligible contribution to the spot displacement on the CCD, as it is visible from figure 5.

Zoom In Zoom Out Reset image size

Taking into account all these effects, the total displacement Δx^T of the spot on the CCD can therefore be evaluated as

$$\begin{align} \Delta x^T = \left(L+\Delta L\right)\left(\theta + \Delta \theta\right) + \Delta d \end{align} \tag{ 3 }$$

with $\Delta \theta = \frac{\Delta d}{\mathrm{RoC}}$. The thermal variations for d and L due to a small temperature fluctuation ΔT are given by

$$\begin{align} \Delta d & = \alpha_d d\Delta T \end{align} \tag{ 4a }$$

$$\begin{align} \Delta L & = \alpha_L L\Delta T \end{align} \tag{ 4b }$$

where α_d and α_L are the linear thermal expansion coefficient for d and L, respectively. For small temperature changes, we can retain only the first-order terms in equation (3). Therefore, combining equations (3), (4a ) and (4b ) and assmuning θ and ΔT to be small, we can rewrite the displacement Δx^T as

$$\begin{align} \Delta x^T = \frac{d\cdot L}{\mathrm{RoC}}+d\cdot \Delta T\left(\alpha_L\cdot \frac{L}{\mathrm{RoC}}+\alpha_d+\alpha_d\cdot\frac{L}{\mathrm{RoC}} \right) \end{align} \tag{ 5 }$$

where the first term is the displacement Δx given by equation (2).

Finally, if RoC $\gg$ L, it is possible to rewrite equation (5) as

$$\begin{align} \Delta x^T\simeq\frac{d\cdot L}{\mathrm{RoC}}+\alpha_d d\Delta T = \frac{d\cdot L}{\mathrm{RoC}}+\Delta d \end{align} \tag{ 6 }$$

from which it emerges that the most relevant effect of the HWS thermal expansion is on Δd, i.e. on the distance between two adjacent holes of the HP.

By imposing $\Delta x^T = 0$ in equation (6), the wavefront RoC for which the curvature one would observe is equal and opposite to the spurious contribution due to the thermal expansion can be computed—this is the so-called critical RoC$_\mathrm{cr}$

$$\begin{align} \frac{d\cdot L}{\mathrm{RoC}_\mathrm{cr}} = \alpha_d d\Delta T \hspace{0.4 em} \rightarrow \hspace{0.4 em} \mathrm{RoC_{cr}} = \frac{L}{\alpha_d \cdot\Delta T} \end{align} \tag{ 7 }$$

which, rewritten in terms of curvature, gives the spurious curvature C due to the thermal defocus that is measured by the HWS and its direct relation with the HWS temperature fluctuations ΔT, i.e.

$$\begin{align} C = \frac{1}{\mathrm{2RoC_{cr}}} = \frac{\alpha_d\Delta T}{2L}. \end{align} \tag{ 8 }$$

The impact of local aberrations in Advanced Virgo operation has been extensively characterized in relation to the design [8]. These studies provided a set of requirements in terms of OPL residual distortion in the RC and core optics surface RoC accuracy, which is summarized in table 1 [8].

Table 1. Advanced Virgo general requirements on wavefront distortion.

Parameter	Requirement
RC residual OPL RMS	$\lt$2 nm
Core optics RoC precision	±2 m

The sensitivity of HWSs must be at least one order of magnitude better than the requirements in terms of OPL variation. This implies an additional requirement on the maximum allowable HP temperature swing, in order to keep the thermal defocus well below the required sensitivity.

2.2.1. Temperature stabilization for the RC residual OPL requirement.

The limit to the HWS thermal defocus can be computed starting from the requirement ΔW = 2 nm RMS. In the computation, we must consider—besides the requirement on the HWS sensitivity to be at least one order of magnitude better than the requirements in terms of OPL variation—also an additional factor of two due to the double-pass configuration in which the HWS operates [4]—which would give an RMS limit of 0.4 nm. While the requirement is expressed in terms of the wavefront RMS, it is simpler to put a constraint on the maximum amplitude of the thermal defocus term. It can be shown that the RMS value of a spherical wavefront over an aperture is smaller than its maximum amplitude by a factor 2$\sqrt{3}$. So, in order to translate the RMS into a limit on the amplitude of the defocus term, we must multiply it by a factor 2$\sqrt{3}$, ending with a defocus limit of 1.39 nm RMS. Therefore, the maximum allowable wavefront variation $\Delta W_\mathrm{max}$ due to the temperature fluctuation $\Delta T_\mathrm{max}$ can be written from equation (8) as

$$\begin{align} \Delta W_\mathrm{max} = \frac{\alpha_d\Delta T_\mathrm{max}}{2L}\cdot\left(\frac{D}{2}\right)^2 \unicode{x2A7D} 1.39\,\mbox{nm} \end{align} \tag{ 9 }$$

where D is the dimension of the CCD sensitive area. Rearranging equation (9) in order to constrain $\Delta T_\mathrm{max}$, assuming $\alpha_d \sim1.2\cdot10^{-6}$ K⁻¹—that is a reasonable choice for a generic invar [17]—and D = 12.28 mm for the CCD, the maximum allowed temperature fluctuation $\Delta T_\mathrm{max}$ results to be

$$\begin{align} \Delta T_\mathrm{max}\unicode{x2A7D} \frac{8L}{D^2\alpha_d}\cdot1.39\,\mbox{nm} \sim 0.61\,\mbox{K}. \end{align} \tag{ 10 }$$

2.2.2. Temperature stabilization for the TM RoC precision requirement.

An additional requirement is related to the required accuracy of ±2 m in the tuning of the core optics RoC. A wavefront reflected back by a mirror surface with a given RoC accumulates a curvature C$_\mathrm{RoC}$ given by

$$\begin{align} C_\mathrm{RoC} = 2\cdot\frac{1}{2\mathrm{RoC}} = \frac{1}{\mathrm{RoC}} \end{align} \tag{ 11 }$$

since reflection is intrinsically a double-pass process. If the RoC deviates from its nominal value by a small quantity $\epsilon \ll$ RoC, the corresponding variation in curvature is

$$\begin{align} \Delta C_{\epsilon} = \frac{1}{\mathrm{RoC}}-\frac{1}{\mathrm{RoC}^{^{\prime}}} = \frac{1}{\mathrm{RoC}}-\frac{1}{\left(\mathrm{RoC}+\epsilon\right)}\simeq\frac{\epsilon}{\mathrm{RoC}^2}. \end{align} \tag{ 12 }$$

In order to derive the curvature variation measured by the HWS, we should include in equation (12) the square of the telescope magnification factor M and an additional factor of two because of the double-pass measurement scheme in the off-axis layout [6], which gives

$$\begin{align} \Delta C_{\epsilon}^\mathrm{HWS} = 2M^2\Delta C_{\epsilon} = \frac{2M^2\epsilon}{\mathrm{RoC}^2}. \end{align} \tag{ 13 }$$

As for the previous section, in order to compute the maximum permissible temperature variation $\Delta T_\mathrm{max}$ we impose a measurement accuracy one order of magnitude better than the requirement, which results in

$$\begin{align} C\unicode{x2A7D} \Delta C_{\epsilon/10}^\mathrm{HWS} = \frac{2M^2}{\mathrm{RoC}^2}\frac{\epsilon}{10}. \end{align} \tag{ 14 }$$

Using equations (8) and (14), M = 7, RoC = 1500 m, ε = 2 m and assuming again $\alpha_d \sim1.2\cdot10^{-6}$ K⁻¹, the maximum temperature variation can be computed as

$$\begin{align} \Delta T_\mathrm{max}\unicode{x2A7D} \frac{2 M^2 L \epsilon}{5\alpha_d \mathrm{RoC}^2}\sim \mbox{0.14 K}, \end{align} \tag{ 15 }$$

which results to be the most stringent sensing requirement.

The HWS CCD case is installed on an aluminum support—mounted on the optical bench of TeTis—in order to dissipate the heat produced when it is switched on. As it will be described later, the base can be either a simple post or a larger custom-made support.

In order to monitor the CCD temperature, PT100 [18] resistance temperature detectors (RTDs) have been used. Two PT100 sensors have been placed on the left and on the bottom of the sensor front invar cover, respectively. They have been labeled as the Witness and the Controller, respectively, with the former (out-of-loop sensor) measuring the actual temperature of the invar cover, while the latter is in-loop with the control system actuators. An additional PT100 is anchored to the optical bench to measure environmental temperature fluctuations. To stabilize the CCD temperature within the requirements, an actuator capable of subtracting and/or supplying the right amount of heat is needed. The selected actuator is a thermoelectric Peltier cell, which operates exploiting the Peltier effect [19, 20].

In order to properly control the temperature fluctuations of the HWS, a proportional-integrative-derivative (PID) control loop has been designed and implemented. The output u(t) of this control can be written as [21]

$$\begin{align} u\left(t\right) = K_\mathrm{P}\cdot e\left(t\right)+K_\mathrm{I}\int_0^t e\left(\tau\right)\mathrm{d}\tau+K_\mathrm{D}\frac{\mathrm{d}}{\mathrm{d}t}e\left(t\right) \end{align} \tag{ 16 }$$

where $K_\mathrm{P}$, $K_\mathrm{I}$ and $K_\mathrm{D}$ are the proportional, integral and derivative gain, respectively, and e(t) is the error signal defined as the difference between the chosen setpoint and the process variable. The output u(t) can be rewritten as a function of the integral and derivative time $T_\mathrm{I} = \frac{K_\mathrm{P}}{K_\mathrm{I}}$ and $T_\mathrm{D} = \frac{K_\mathrm{D}}{K_\mathrm{P}}$, respectively, as

$$\begin{align} u\left(t\right) = K_\mathrm{P}\left[e\left(t\right)+\frac{1}{T_\mathrm{I}}\int_0^t e\left(\tau\right)\mathrm{d}\tau +T_\mathrm{D}\frac{\mathrm{d}}{\mathrm{d}t}e\left(t\right) \right]. \end{align} \tag{ 17 }$$

Using the Ziegler–Nichols method [22], a preliminary estimation of the PID parameters has been computed. The control loop has been then tested with the Ziegler–Nichols settings and subsequently fine-tuned by changing the PID parameters until the best performance in terms of temperature stability has been reached. The final optimized PID parameters and the comparison with the Ziegler–Nichols settings are summarised in table 2.

Table 2. Optimized parameters for the HWS temperature control system compared with the ones computed with the Ziegler-Nichols method.

Parameter	Ziegler–Nichols	Optimized
$K_\mathrm{P}$	3	8
$T_\mathrm{I}$	10 min	2.5 min
$T_\mathrm{D}$	0.27 min	2 min
Sampling time	—	10 s
Number of averages	—	6

Using these optimal settings, the control loop has been tested with the HWS in three different configurations:

• Configuration A: the HWS is mounted on an aluminum cylindrical post, with the Peltier cells placed between the copper layer and the fins.
• Configuration B: the HWS is mounted on a bulky support made of aluminum, while the Peltier cells are left in the same position as in the previous configuration.
• Configuration C: the HWS is mounted on the same support as in the previous configuration, but in order to try to reduce the dependence of the HWS temperature from the fluctuations of the environment, the Peltier cells have been placed below the HWS—i.e. between it and the aluminum support.

The whole assembly of the HWS in all the three configurations with the PT100 RTDs mechanically anchored on the front cover and the two Peltier cells is shown in figure 6.

Zoom In Zoom Out Reset image size

The efficiency and the stability of the control loop were tested on timescales from few hours to week-long measurements. In all cases, the control is able to provide a temperature stabilization $\unicode{x2A7D}$0.01 K and to maintain it on long timescale, compensating for the day/night temperature variations. An example of the provided stability is shown in figures 7 and 8, while the typical results obtained for each configuration are summarized in table 3.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 3. Typical results obtained in terms of temperature stability in all the tested configurations, expressed as standard deviation values for both the PT100 sensors.

Configuration	$\sigma_\mathrm{contr}$	$\sigma_\mathrm{witn}$	Duration
A	0.0088 ∘C	0.0095 ∘C	22 h and 23 min
B	0.0044 ∘C	0.0051 ∘C	114 h and 55 min
C	0.0072 ∘C	0.0078 ∘C	22 h and 45 min

The measurements are aimed at highlighting the expected correlation between the curvature of the observed wavefronts and the HWS temperature variation when changing the setpoint of the control loop. The measured curvature in two different measurements is compared in figure 10 with the nominal value predicted by equation (8) assuming a free standing invar HP, that is, a value of $\alpha_d \sim1.2\cdot10^{-6}$ K⁻¹. The correlation is clearly evident, confirming that the observed curvature is indeed a thermal defocus. However, with this choice of α_d , the measured curvature does not match the expected one.

Zoom In Zoom Out Reset image size

To recover an optimal matching, it is necessary to multiply the nominal curvature by an effective factor k estimated, in a least-squares sense, by finding the value that minimizes the χ²

$$\begin{align} \chi_\mathrm{Curv}^2 = \sum_{i = 1}^{N_\mathrm{WF}}\left[\mathrm{Curv}\left(i\right)-k\cdot \mathrm{Curv}_\mathrm{th}\left(i\right)\right]^2 \end{align} \tag{ 18 }$$

where $N_\mathrm{WF}$ is the number of wavefront acquired during a measurement, $\mathrm{Curv}(i)$ are the measured curvature values and $\mathrm{Curv_{th}}(i)$ are the theoretical ones. In order to quantify the best estimate and the uncertainty associated with this definition of the factor k, a statistical analysis has been applied. Curvature measurements taken in the same conditions can be regarded as a set of measurements of a random variable with Gaussian distribution. Therefore, for each temperature step the curvature mean value was computed with the relative standard deviation

$$\begin{align} \sigma_m\left(i\right) = \frac{\sigma\left(i\right)}{\sqrt{N_i}} \end{align} \tag{ 19 }$$

where $\sigma(i)$ is the standard deviation for the ith step and N_i is the total number of curvature samples in that step. These values as a function of the relevant $\Delta T_\mathrm{wit}$ have been fitted with a function $y(x) = Bx$ using the least-squares method, as shown in figure 11.

Zoom In Zoom Out Reset image size

The resulting slope of the fit is B = 4.26 · 10⁻⁴ K⁻¹ m⁻¹ and the uncertainty on this value is σ_B = 2.69 · 10⁻⁶ K⁻¹ m⁻¹. The coefficient B is related to the thermal expansion coefficient α by equation (8)

$$\begin{align} B = \frac{\alpha}{2L}. \end{align} \tag{ 20 }$$

Considering the obtained value for B with its uncertainty and a lever arm of L = ($9\pm1$) mm, the measured curvature is compatible with an effective thermal expansion coefficient equal to $\alpha = (7.7\pm0.9)\cdot 10^{-6}$ K⁻¹. Thus, the ratio between the experimental value of α and the expected one, expressed by the previously-defined factor k, is given by

$$\begin{align} k = \frac{\alpha}{\alpha_{\mathrm{invar}}} = 6.4\pm0.8. \end{align} \tag{ 21 }$$

A possible explanation of the observed value k > 1 is the presence of a coupled thermal expansion between the CCD aluminum case and the invar components (HP, spacer, front plate) that are mechanically constrained to the case by four screws. Since the aluminum has an expansion coefficient one order of magnitude higher than that of the invar, the HP and the plates are expected to be dragged by the CCD case deformation. In order to investigate the hypothesis, the HWS was installed in the horizontal configuration shown in figure 9. In this way, it was possible to loosen the screws and then to mechanically disentangle the CCD case from the HP. In order to maintain a good thermal contact between the RTDs and the invar front plate, the PT100s were glued on the plate through a conductive paste—this ensured a good control loop stability.

As for the vertical setup measurements, the setpoint of the control loop was changed a few times during the wavefront acquisition. Noticeably, in this configuration the discrepancy between the expected and experimentally measured curvature values was lower with respect to the previous set of measurements, as shown in figure 12 in two different measurements.

Zoom In Zoom Out Reset image size

Since all the data collections were carried out in almost the same experimental conditions, the same statistical analysis used for the vertical setup measurements was performed in order to assess the best estimate of the k factor. The resulting least-squares fitting is shown in figure 13.

Zoom In Zoom Out Reset image size

For this experimental setup, the slope of the fit is B = $1.42 \cdot 10^{-4}$ K⁻¹ m⁻¹ and the related uncertainty on this value σ_B = 0.42 · 10⁻⁶ K⁻¹ m⁻¹. The obtained effective thermal expansion coefficient is $\alpha = (2.6\pm1.0)\cdot 10^{-6}$ K⁻¹. Therefore, the ratio between the effective value of α and the expected one is

$$\begin{align} k = \frac{\alpha}{\alpha_{\mathrm{invar}}} = 2.2\pm0.2. \end{align} \tag{ 22 }$$

Moving from the vertical configuration—the one in which the HWSs are currently mounted in Advanced Virgo—to the horizontal one, the discrepancy between the measured curvature and the thermal defocus obtained by assuming a nominal value of $\alpha_d \simeq 1.2 \cdot 10^{-6}$ K⁻¹ was actually reduced, as demonstrated by the drop of the matching factor from k ∼ 6 down to k ∼ 2. This reduction confirms the hypothesis of a coupling between the invar components and the CCD case.

We therefore assume that, in the measurements with the horizontal setup, the HP was efficiently decoupled from the CCD case. In this case, the estimated thermal expansion coefficient is $\alpha_d \simeq (2.6\pm1.0) \cdot 10^{-6}$ K⁻¹. Assuming that the invar type of the HP is the most common one—the so-called invar 36 containing 64$\%$ of Iron and 36$\%$ of Nickel [23–26]—the thermal expansion coefficient is expected to be in the range [0.5–2.0] · 10⁻⁶ K⁻¹. Given the associated uncertainty in the final computed α_d , the obtained values are compatible with this range.

The maximum allowed temperature swing can be recalculated from the coefficients obtained in the two different configurations by using equations (10) and (15) as constrained by the Advanced Virgo requirements. Since the measured expansion coefficients are larger than the assumed value for α_d , the final requirements on the maximum allowed temperature fluctuation are tighter—see table 4.

Table 4. The maximum allowed temperature fluctuation $\Delta T_\mathrm{{max}}$ as constrained by the Advanced Virgo requirements, computed from the experimentally measured linear thermal expansion coefficients.

Configuration	αd (10−6 K−1)	$\Delta T_{\mathrm{max}}$ (K)
Vertical	7.7±0.9	0.02
Horizontal	2.6±1.0	0.05