Appendices
Fundamentals of Polarimetry
Stokes Parameters & Polarization States
Polarization Modulation Techniques (?)
SolPol Principle of Operation
SolPol measures the polarization fractions of linear polarization (expressed by the Q and U Stokes parameters) and circular polarization (V Stokes parameter) from the whole solar disk and the entirety of the atmospheric column, depending on its limiting field-of-view aperture and the choice of mounting telescope. The measurements of the degree of linear and circular polarization (i.e., and , respectively, e.g., general expression from Hansen and Travis 1974) have an absolute accuracy of 1% and precision of 1 part per million (ppm, 10-6) in polarization terms.
The first observing sequence is performed with the assembly at the rest position (zero degrees) and the rotating of the linear polarizer in steps of 90 degrees. The corresponding relative positions of the PEM and linear polarizer for the assembly rotation is shown in XXX. For a complete polarizer rotation (360 degrees) we acquire measurements for four minutes. The specific sequence provides measurements of three of the four Stokes parameters. This is followed by the second observing sequence, performed after the rotation of the entire assembly, about the PEM crystal optical axis, over 45 degrees. This observing sequence provides measurements of the fourth Stokes parameter (V), and measurements for the removal of the biases and residual polarizations due to high frequency strain of the PEM for another four minutes.
Therefore, each full measurement cycle has a duration of eight minutes in total and is comprised of two distinct sets:
i. solar irradiance measurements on four positions of the linear polarizer at 41°, 131°, 221° and 311° with the assembly being at 0°, this configuration provides measurements of the I, U and V Stokes parameters, and
ii. measurements for the same relative positions of the linear polarizer, but the whole assembly is being rotated by 45° which provides measurements of the I, Q and V Stokes parameters. Measuring I and V twice provides information for the removal of biases and residual polarization in the measurements.
The Stokes vector of light that reaches the PM can be expressed using the Mueller formulation (e.g., Van de Hulst, 1957), considering as reference coordinate system the one of the incoming sunlight, as in (1):
(1) where is the Stokes vector of the input light polarization state, is the output polarization state with the assembly at zero degrees,
(2) is the Mueller matrix of the linear polarizer at each position angle (41°, 131°, 221°, 311°) and
(3) is the Mueller matrix of the PEM that induces an input sinusoidal retardation of XX, for A the peak modulation amplitude and \(\omega\): the modulation frequency.
When the whole assembly is rotated by 45°, the reference coordinate system is rotated by the same angle, hence is the Stokes vector of the output light for this rotational scheme. The incoming sunlight at each \(\alpha\)° position, appears to be rotated by an angle of -45° and is calculated through the rotation matrix as (as in Freudenthaler, 2016, S.5.1.7; Martin et al., 2010; Supplementary material/SolPol manual, Eq. 8).
We utilize a Bessel functions expansion of the retardation \(\delta\) and derive the measurements of the Stokes parameters of the incoming sunlight, at detector level, as a function of the linear polarizer angles and the assembly rotation (see Figure XX). The XX are the n-order Bessel functions and, for the specific PEM, the modulation amplitude is fixed at A = 2.4048 so that, which makes the Q- and U-dependent direct-current (DC) terms equal to zero. Third order and above harmonic frequency terms are considered negligible, O(n >2) = 0. We, then, must sum over the four linear polarizer orientations per each assembly position, in order to eliminate the dependent terms on the I derivation, while the marginal residuals on the other Stokes parameters are also accounted for Raw Data Pre-Processing.
The following table shows the measurements at the detector, as a function of a. the Stokes components, b. the linear polarizer angles (\(\alpha\)°), and c. the rotation of the assembly.
n |
I’ |
Assembly without rotation |
Assembly rotated at 45° |
|---|---|---|---|
0 |
DC |
1/2 [I + Q cos(2\(\alpha\)) + UJ₀(A) sin(2\(\alpha\))] |
1/2 [I - QJ₀(A) sin(2\(\alpha\)) + U cos(2\(\alpha\))] |
1 |
1\(\omega\) t |
VJ₁(A) sin(\(\omega\) t) sin(2\(\alpha\)) |
VJ₁(A) sin(\(\omega\) t) sin(2\(\alpha\)) |
2 |
2\(\omega\) t |
UJ₂(A) cos(2\(\omega\) t) sin(2\(\alpha\)) |
-QJ₂(A) cos(2\(\omega\) t) sin(2\(\alpha\)) |
where Jn(A) are the n-order Bessel functions and, for the specific PEM, the modulation amplitude is fixed at A = 2.4048 so that J0(A)=0, which makes the Q- and U-dependent DC terms equal to zero. Third order and above harmonic frequency terms are considered negligible, O (n > 2) → 0.
PEM operation principle
The SolPol PEM type SII FS47 is comprised of a fused silica crystal bar with photoelastic capabilities and a piezoelectric transducer. If the optical element is compressed or stretched it induces a retardation (i.e. phase difference between the polarization components) of the incoming light. The strain stress in the PEM crystal is induced by standing mechanical acoustic waves, which are produced by the transducer, attached to the head.
The resulting retardation is time-periodic, and is provided by:
\(\delta\) (t) = Asin \(\omega\) t
where \(\delta\) is the retardation of the PEM and A is the peak amplitude. The resonant frequency, \(\omega\) of the PEM is set at 47 kHz.
The PEM controller unit performs many functions in the photoelastic modulator system. Its primary function is to control the peak retardation of the photoelastic modulator optical head. It does this by providing a DC voltage signal to the electronic head which determines the transducer vibration amplitude and thus the strain amplitude in the optical element (see pic). A current feedback loop from the electronic head enables the controller to maintain stable peak retardation levels.
For SolPol, the PEM is calibrated by the Bessel function zero methods which can be found in the manufactirers user manaul. Concerning this merhod, the DC term is kept invariable, independent of the birefringence. The DC intensity also becomes independent of changes attributed to the optical system, such as angular position of the polarizer. Normalization of the AC signals by the DC signal, renders the ratio independent of fluctuations from the intensity source.
