1030-CRHWP
🌀 Principle of the Continuously Rotating Half-Wave Plate (CRHWP)
A Continuously Rotating Half-Wave Plate (CRHWP) is used in CMB polarization experiments to modulate and separate the Stokes parameters (I, Q, U). It enables a single polarization-sensitive detector to measure all components and suppresses 1/f noise and systematics arising from detector mismatch.
1. Polarization modulation by HWP
A half-wave plate (HWP) rotates the polarization angle of incident radiation by twice its birefringent axis angle. If the incoming polarization angle is (\theta_{\text{in}}) and the HWP axis is at angle (\theta_{\text{HWP}}), then after transmission:
[ \theta_{\text{out}} = 2\theta_{\text{HWP}} - \theta_{\text{in}}. ]
When the HWP rotates continuously with angular velocity (\omega_{\text{HWP}}), the polarization angle is modulated as:
[ 2\theta_{\text{HWP}}(t) = 2\omega_{\text{HWP}} t. ]
In terms of Stokes parameters:
[ Q_{\text{out}}(t) - iU_{\text{out}}(t) = [Q_{\text{in}}(t) + iU_{\text{in}}(t)] , m(t), ] where the modulation function [ m(t) \equiv e^{-i\omega_{\text{mod}} t}, \quad \omega_{\text{mod}} = 4\omega_{\text{HWP}}. ]
The factor of 4 arises because linear polarization is a spin-2 quantity — a rotation of the optical element by (\theta_{\text{HWP}}) causes a (4\theta_{\text{HWP}}) phase shift in the polarization signal.
2. Detector timestream with modulation
The signal measured by a polarization-sensitive detector is:
[ d_m(t) = \delta I_{\text{in}}(t)
\varepsilon , \mathrm{Re}!\left{ [Q_{\text{in}}(t) + iU_{\text{in}}(t)] , m(t) , e^{-2i\theta_{\text{det}}} \right}
N_m(t), \tag{2.2} ]
where
(\delta I_{\text{in}}(t) = I_{\text{in}}(t) - \langle I_{\text{in}}\rangle) is the fluctuating intensity,
(\varepsilon) is the polarization modulation efficiency,
(\theta_{\text{det}}) is the detector orientation angle,
(N_m(t)) is the white noise.
Only the fluctuating component (\delta I_{\text{in}}) is measurable; the absolute baseline is typically filtered out.
3. Demodulation process
If the HWP rotation frequency is higher than the scan speed divided by the beam size, the intensity and polarization signals can be separated by frequency-domain demodulation:
[ \bar{d}m(t) \equiv \mathrm{FLPF}{d_m(t)} = \delta I{\text{in}}(t) + \bar{N}_m(t), \tag{2.3} ]
[ \bar{d}d(t) \equiv \mathrm{FLPF}{d_m(t) , 2m^*(t)e^{2i\theta{\text{det}}}} = \varepsilon[Q_{\text{in}}(t) + iU_{\text{in}}(t)]
N_{\bar{d}}^{(\mathrm{Re})}(t)
iN_{\bar{d}}^{(\mathrm{Im})}(t), \tag{2.4} ]
where
FLPF denotes a low-pass filter (cutoff ≈ rotation frequency or its double),
(m^*(t)) is the complex conjugate of (m(t)),
(\bar{d}_m(t)) and (\bar{d}_d(t)) are the recovered intensity and polarization signals, respectively.
Noise decomposition satisfies:
[ \langle [N_{\bar{d}}^{(\mathrm{Re})}]^2 \rangle = \langle [N_{\bar{d}}^{(\mathrm{Im})}]^2 \rangle = 2 \langle [\bar{N}_m]^2 \rangle \equiv 2N. ]
4. Physical picture
The CRHWP continuously modulates the sky polarization at (4f_{\text{HWP}}), shifting (Q, U) signals to a higher frequency band where 1/f noise is minimal.
The total intensity (I) remains at DC (unmodulated).
Demodulation recovers (Q, U) from the modulated timestream using the known reference function (m(t)).
In real systems, optical non-idealities and detector nonlinearity cause intensity-to-polarization (I→P) leakage and instrumental polarization, which must be characterized and corrected.
Would you like me to add a small schematic explanation (text version) — e.g. how frequency spectra look before and after modulation (I at DC, Q/U at 4fₕwₚ)? It’s useful if this note is going into a presentation or documentation.
Last updated