Modulator

The modulator is made of three mirrors which rotate as a block. Each reflexion impacts the beam, represented by the Stokes vector and can be modelled by the reflexion Mueller matrix:

with the ratio of Fresnel amplitude reflexion coefficients and with τ the difference of phase shift between both polarizations. This matrix is obviously dependent on the angle of incidence i and the wavelength λ but also on the optical index of the mirror, the one of the coating and its thickness. The Mueller matrix of the modulator is then M_modulator . When modulating, the rotation is given by the rotation matrix: R(θ) = .

Analyzer

To polarize the light, we do not have any other choice than to use a mirror at the Brewster angle. This creates a nonnegligible issue: a chromatic effect. Indeed, the Brewster angle depends on the wavelength and we cannot have a 100% efficient polarizer at all wavelengths. The Mueller matrix of the analyzer is then a reflexion matrix.

Global Matrix and (de)modulation matrix

The global Mueller matrix of the polarimeter is then:

Unfortunately, we can measure only intensities, which are given by the first line of the matrix. We introduce the Modulation Matrix which is the first line of the global matrix for each modulation angle. The modulation matrix answers the equation: I_out = M_modulation * S_in, with I_out the matrix of the measurement made for each angle of modulation and S_in the Stokes vector we want to measure. To retrieve the Stokes vector, we have to used the demodulation matrix, which is the pseudo-inverse of the modulation matrix: . We then have S_in = M_demodulation * I_out, to retrieve directly the initial Stokes vector from the intensity measurements. The demodulation matrix will be used to determine the efficiency of the polarimeter, see section 3.2.1.

2.1.1

Without coating

Here we present the modulation of a K-mirror made of a metal without coating, we use the following equations based on [2]:

with θ the angle of incidence and with

and

2.1.2

With coating

Here we consider the addition of a coating on the three mirrors.

To determine χ and τ as introduced in section 2.1, we need to calculate the reflectivity and the phase shift for each polarization (‖ and ⊥). In [3], the reflexion with a transparent film on an absorbing substrate is given by:

As shown in figure 2, we call 1 the environment of the mirror, vacuum in our case. We call 2 the coating and 3 the mirror. To model the polarimeter, we need complex indices of the materials we want to use. Little can be found in the literature, therefore we set up our own experiment to measure them.

Figure 2:

Reflection on a surface with a coating.

3.2.1

Modulation of the gold polarimeter

The modulation is given by the same expressions as in section 2.1.1 as we are in the case of pure gold without coating. For example for a 70° incidence and 4 modulation angles (30°, 74°, 105° and 149°), we obtain the modulation shown in figure 4.

Figure 4:

Modulation by a gold K-mirror rotating at 4 angles with an incidence of 70°.

Efficiency

To determine the efficiency of the modulation, we introduce polarimetric efficiency based on the demodulation matrix [5]: , n the number of measurements (number of modulation angles).

To achieve the determination of all the Stokes parameters simultaneously, the optimal efficiency is is (because . For this experiment, the coefficients can be measured independently which improves the efficiencies. For example for a 70° incidence and 4 modulation angles (30°, 74°, 105° and 149°) and simultaneous measurements, the polarimetric efficiencies are shown in figure 5. The efficiencies are quite low for this example because of the gold. The optimal material for the FUV domain, which we will use for Pollux, will improve the efficiencies and the reflectivity.

Figure 5:

Polarimetric efficiencies for a gold polarimeter with 4 angles of modulation (30°, 74°, 105° and 149°).

3.2.2

Analyzer

To polarize the light, as explained in section 2.0.1, the Brewster angle is being considered. We are looking for the metal equivalent that has the best contrast between s and p polarization for gold. The contrast is defined as . We consider wavelengths from 90 to 135 nm. In figure 6, by displaying the contrast as a function of the angle of incidence and as a function of wavelength, we can find the incidence for which the contrast is maximal for most wavelengths. We can also see that the contrast is more or less constant with wavelength, which means the chromatic effect is negligible. By averaging the contrast results over wavelength, a maximum is easily spotted at i_max = 61°. The mean contrast is then 70%.

Figure 6:

Contrast between s and p light as a function of incidence angle (in abscissa, from 0° to 90°) and as a function of wavelength (in ordinate, from 90 to 200 nm) for a gold mirror.

To evaluate the global efficiency of the polarizer, we introduce the figure of merit ε is maximum for a perfect polarizer and equals 0.7071. In figure 7, we show the figure of merit as a function of wavelength. The mean of the figure of merit is 0.3610. This means that our polarizer is not perfect and does not have a 100% efficiency. We have to take this into account for the measurements.

Figure 7:

Figure of merit of the gold Brewster-like analyzer (gold mirror at 61°) as a function of wavelength.