A.

Design description

The instrumental concept is studied to form four images in phase quadrature. The Mach-Zehnder (MZ) interferometer is placed in a parallel beam. The four outputs of the MZ are folded on a single detector by the four corner mirrors (M1, M2, M3, M4) and form four images thanks to four identical objectives. From these four images, we can use the “ABCD” method (Wyant 1975) which permits to get rid of the continuum component of the incident light. These images are obtained thanks to a specific optical design, based on a modified MZ interferometer (Fig. 1).

Figure 1:

Optical design of the Mach-Zehnder

The “empty MZ” is composed of two blocks (K1 and K2) made in K5G20. These blocks are composed of two optical elements glued together and separated by a semi reflective treatment. Because the size of these blocks is different, an “empty” Optical Path Difference (OPD) is introduced.

The sensitivity of the MZ to velocity field is the result of an optimization increasing the sensitivity of the phase and decreasing the contrast with the OPD change. By exploration of the solar spectrum, we looked for features giving contrast maxima. The best sensitivity is obtained for an OPD of 5048 μm at 519.5 nm. This OPD is created by introducing glass plates in the arms of the interferometer. A tilted plate in BK7G18 is placed in each arm in order to eliminate stray light and to make the fringes regular and rectilinear (B1-B2 and B3). One of the plates is sliced into two parts, the prismatic one (B2) being mounted on a piezoelectric plate in order to achieve the OPD modulation. The modulation will be helpful to blur the fringes during the pixel sensitivity calibration processes, but its main function is to create phase shifts of kπ/2 (k = 0, 1, 2, 3) to calibrate the response of each channel. Finally, a quarter wave plate, based on two quartz plates with perpendicular optical axes (Q1 and Q2), achieves the phase quadrature. Then, polarizer cubes are placed at each output and produce a kπ/2 phase shift. So we obtain four images in phase quadrature:

Where I₀ is the continuum component of the incident light, γ is the fringe contrast and ϕ is the fringe phase that we want to measure. From the phase measurement, we can obtain the radial velocity v_r which contains the information:

where λ₀ is the central wavelength, Δ the OPD and c the speed of light.

The choice of the optical glasses respects two criteria. First, materials must be resistant to radiations. Then, thermal dilatation coefficients must compensate refractive index variations. That is why we have selected: K5G20 and BK7G18 from Schott, and crystalline quartz wave plates.

B.

Optimization of the thermal stability

In order to ensure the best thermal stability, the MZ must operate at a precise temperature, called stability temperature. Simulations of the phase versus temperature result in a minimization of the thermal effect for a value of the temperature of 18.9°C. If the experimental temperature does not respect this condition, the optical parameters of the MZ have to be redefined. Indeed, there is a relation between glass plate thicknesses and stability temperature.

The protocol consists in measuring the phase of the fringes at different temperatures (5 measurement points are sufficient). The Figure 2 represents the theoretical phase in function of the temperature, and the fit of this curve.

Figure 2:

Fringe phase versus temperature

From the equation of the fit curve, we can deduce the stability temperature which is located at the minimum of the curve. The image processing allows the stability temperature determination with a precision of 0.01°C. A modification of the plate thickness will lead in a change in this stability temperature. An interesting point is that the BK7G18 plates have an effect on the stability temperature and on the OPD whereas the quartz plates only impact the OPD. The protocol will consist in adjusting the stability temperature T_s to the required value by modifying the thicknesses of the BK7G18 plates e_B. Then, the OPD will be retrieved by the adapted modification on the quartz plate thickness e_Q.

As the thicknesses will be modified, the first version of the prototype will not be glued. The optical elements will be maintained mechanically or by molecular adherence.

C.

Theoretical performances

The aim of Echoes is to analyze the solar spectral lines reflected at the surface of Jupiter. That is why it must operate in the visible domain. On top of that, the quantum efficiency of the detectors is better in the visible. Considering this requirement, the maximum of sensitivity is reached with the optimized parameters given in Table 1:

Table 1:

Theoretical performances

Considering these performances, the theoretical fringe contrast is 5%. The FoV is determined to observe the whole planet at a minimal distance of 0.02 AU. It corresponds to an input pupil size of 2 cm.

A.

Control of the transmissions

A first test bench for the control of the transmission of the blocks has been implemented (Fig. 4). This bench consists in measuring the photometry at the two outputs of each prism.

Figure 4:

Test bench for the transmission control

We occult one path and we acquire series of images. We repeat the process for the other arm. Measurements have been achieved separately on each prism block. The values of transmissions and reflections are measured in each polarization, P and S, and are given in Table 3.

Table 3:

Measured transmissions and reflections of the coatings

From these values, we can determine the transmission and the contrast at each output of the Mach-Zehnder, for each polarization (Table 4).

Table 4:

Transmission and contrast at each output

We can see that the differential transmission is less than 2% which respects the specification of 5%. The absorption of the coatings is around 12% which does not respect the specification of 3%. Section (4.3.) gives the consequence of these measured values on the expected performance.

B.

Control of the polarization

The dedicated test bench (see section 3.) is now used to control the polarization of the coatings. A light source is injected through an adjustable slit and directed to the MZ. A combination of mirrors is studied to image easily each output separately. At the output of the MZ, we have the “periscope – polarizer” module, the diffraction grating and finally, the interferential filter which is placed upstream of the detector (Fig. 5).

Figure 5:

Test bench for the phase shift control

The Fig. 6 represents parts of the images obtained at each output of the MZ, resized for the image processing.

Figure 6:

Images obtained at each output of the MZ

The “empty MZ” produces interference fringes because the size of the two blocks K1 and K2 is different. Fringes are obtained in the spectral and spatial dimensions (the OPD varies with the wavelength and with the FoV).

Protocol of the phase shift calculation:

The theoretical phase shift between the two outputs of the MZ is 180°. The error to the phase quadrature is ΔΦ₁ in the polarization 1 and ΔΦ₂ in the polarization 2. The results (averages over several images) are the following:

These results give an idea of the precision of the measurement, of the order of a few degrees, sufficient for our task. The specification of the error to the phase quadrature is reached. We notice that the difference between the two polarizations is low, so it will not increase the noise level in the phase measurement. These preliminary results will be supplemented by other measurements.

C.

Acceptance

The standard deviation on the velocity variations due to the photon noise is given by:

where σ₀ is the central wavenumber and N the total number of photons. By considering the measured transmissions (4.1.) and the errors to the phase quadrature (4.2.), the degradation of the performances in terms of velocity measurement is 23% (Eq. 8).

The study being limited to a R&D, we decided to accept the coatings with present values, although slightly out the specification range, in order to achieve the complete test set. Indeed, the most crucial part of the test is the thermal sensitivity and the final adjustment of the MZ, and the determination of the calibration procedure.