1. Wavelet-based Faraday Rotation Measure Synthesis (funded by RFBR & DFG) Peter Frick, Rodion Stepanov, Institute of Continuous Media Mechanics, Perm, Russia Dmitry Sokoloff, Department of Physics, Moscow University, Russia Rainer Beck Max-Planck-Institut für Radioastronomie, Bonn, Germany

2. RM Synthesis RM synthesis works on observed Q,U cubes in wavelength to produce a cube of the complex Faraday dispersion function (pol. amplitude and intrinsic angle) in Faraday depth Φ: QU FrFrFrFr FiFiFiFi RM Synthesis λ2λ2λ2λ2 λ2λ2λ2λ2 ΦΦ

3. Background The complex-valued intensity of polarized radio emission for a given wavelength is defined by the emissivity ε and the intrinsic position angle χ along the line of sight: If the Faraday depth φ is a monotonic function of z (which means that z is a single-valued function of φ), we can define the Faraday dispersion function („Faraday spectrum“) as a function of the Faraday depth: z

4. Background Burn (1966) noted that the complex polarized intensity P obtained from a radio source is related to the Faraday dispersion function F(φ) as: The Fourier transform is defined as ?

5. Broad box structure in Φ appears as double horns Phase (polarisation angle) not well reconstructed Bad reconstruction for small spectral windows Brentjens & de Bruyn (2005), A&A,441

6. Box structures in Φ appear as double horns Gaussian structure more difficult to reconstruct Wrong phases (polarisation angle) for narrow and wide spectral windows

7. Improving the algorithm by symmetry arguments (Frick et al. 2010, MNRAS 401, L24) 1. Object is symmetric with respect to the point then Excellent reconstruction of phase of one structure for wide spectral windows Good reconstruction of phase of one structure for narrow spectral windows

8. Wavelets Continuous wavelet transform (A.Grossmann, J.Morlet, 1984) Test example Translation and dilation - Morlet wavelet b a

9. RM Synthesis and wavelets Wavelet transform (direct and inverse): Symmetry argument in wavelet space: (Frick et al. 2010, MNRAS 401, L24)

10. Excellent reconstruction of amplitude and phase of both structures for wide spectral windows Good reconstruction of amplitude and phase of both structures for narrow spectral windows Frick et al. MNRAS 401, L24

11. RM Spread Function: Standard: B&B: Wavelet:

12. Cleaning and Wavelets

13. Distributions of regular (galactic) magnetic fields and corresponding PI and Faraday spectra

14. The Role of the Spectral Range

15. Regular + turbulent magnetic fields