Out-of-focus (OOF) holography at the Effelsberg telescope Effelsberg Science Workshop Tomas Cassanelli Max-Planck-Institut f¨ ur Radioastronomie University of Toronto 21 February 2018 Prof. Dr. K. Menten Dr. U. Bach Dr. A. Kraus Dr. B. Winkel This presentation has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 730562 [RadioNet] Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 1 / 14
Outline 1 Introduction 2 OOF holography theory α 3 OOF holography observations 4 pyoof package S9mm 5 Analysis 6 Conclusions Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 2 / 14
Introduction Motivation Motivation: Optimize the surface accuracy Normalized gain-elevation 9-mm receiver Sensitivity can be improved 1 . 00 A e π 8 k B ε aper D 2 Γ = 2 k B = 0 . 95 p G norm ( α ) 0 . 90 0 . 85 Γ ≈ 0 . 75 K/Jy @ λ = 9 mm 0 . 80 0 20 40 60 80 ε aper = ε rad ε taper ε spill ε cr ε bk ε rs ε ϕ r ε ϕ f α degress � �� � � �� � ε i ε ph Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 3 / 14
Introduction Motivation Motivation: Optimize the surface accuracy Normalized gain-elevation 9-mm receiver Sensitivity can be improved 1 . 00 A e π 8 k B ε aper D 2 Γ = 2 k B = 0 . 95 p G norm ( α ) 0 . 90 0 . 85 Γ ≈ 0 . 75 K/Jy @ λ = 9 mm 0 . 80 0 20 40 60 80 ε aper = ε rad ε taper ε spill ε cr ε bk ε rs ε ϕ r ε ϕ f α degress � �� � � �� � ε i ε ph Surface accuracy optimization Random-surface-error efficiency ε rs = e − (4 πλ − 1 δ rms ) 2 Increases Gain Decreases P obs ( u, v ) error Surface error is elevation dependent Deformation sources gravity, thermal, wind mis-collimations phase error receiver Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 3 / 14
Introduction Effelsberg telescope Effelsberg and its active surface Gregorian geometry Active surface corrects deformation in primary dish Sub-reflector equipped with active surface control system ⊥ displacement ± 5000 µ m 96 actuators → displacement ⊥ aperture Modeled by a FEM look-up table plane Sub-reflector 6.5-m D s x f F p x f D s V p Receiver z f y f Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 4 / 14
Introduction Holography Optimize surface accuracy → Holography What is holography? P obs ( u, v ) → E a ( x, y ) → ϕ ( x, y ) Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 5 / 14
Introduction Holography Optimize surface accuracy → Holography What is holography? P obs ( u, v ) → E a ( x, y ) → ϕ ( x, y ) Holography at the Effelsberg telescope Phase-coherent holography OOF holography � � S Geostationary satellite 11.7 GHz Compact source N > 200 Elevation restriction Complete elevation range Needs a reference antenna No extra equipment Time consuming ( ∼ 7 h) ∼ 45 min High accuracy (panel-to-panel) Large-scale error Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 5 / 14
Introduction OOF holography Aperture distribution E a ( x, y ) Why do we need the aperture distribution? How do we obtain the aperture from the beam pattern Aperture related to the phase error � �� 2 � � � E a ( x, y ) → ϕ ( x, y ) P ( u, v ) = E a ( x, y ) � F � Phase error has the aberration of an in-focus power pattern optical system under-determines the aperture distribution 1 in-focus and 2 out-of-focus beam maps to break degeneracy Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 6 / 14
Introduction OOF holography Aperture distribution E a ( x, y ) Why do we need the aperture distribution? How do we obtain the aperture from the beam pattern Aperture related to the phase error � �� 2 � � � E a ( x, y ) → ϕ ( x, y ) P ( u, v ) = E a ( x, y ) � F � Phase error has the aberration of an in-focus power pattern optical system under-determines the aperture distribution 1 in-focus and 2 out-of-focus beam maps to break degeneracy The aperture distribution is a collection of sub-functions e i { ϕ ( x,y )+ 2 π λ δ ( x,y ; d z ) } E a ( x, y ) = B ( x, y ) · · E a ( x, y ) Aperture Blockage Illumination Phase Optical path distribution function error difference Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 6 / 14
OOF holography theory Phase error Phase error ϕ ( x, y ) P obs ( u, v ) , d − P obs ( u, v ) P obs ( u, v ) , d + z z Phase error ϕ ( x, y ) 40 20 pyoof package 0 E a ( x, y ) , d + E a ( x, y ) , d − E a ( x, y ) z z − 20 − 40 − 40 − 20 0 20 40 Phase determined by K nℓ Zernike circle polynomials coefficients ϕ ( x, y ) = 2 π � n,ℓ K nℓ U ℓ n ( x, y ) ϕ ( x, y ) → Active surface system Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 7 / 14
OOF holography observations Observed power pattern example P obs P obs P obs norm ( u, v ), d z = − 0 . 022 m norm ( u, v ), d z = 0 . 0 m norm ( u, v ), d z = 0 . 022 m 0 . 04 0 . 04 0 . 04 0 . 02 0 . 02 0 . 02 v degrees v degrees v degrees 0 . 00 0 . 00 0 . 00 − 0 . 02 − 0 . 02 − 0 . 02 − 0 . 04 − 0 . 04 − 0 . 04 − 0 . 05 0 . 00 0 . 05 − 0 . 05 0 . 00 0 . 05 − 0 . 05 0 . 00 0 . 05 u degrees u degrees u degrees Observation Radial offset Mean elevation α = 53 deg. ↑ / ↓ sub-reflector movement δ rms = 15 . 1 mJy Order of d z ∼ 2 cm S N = 1285 > 200 ϕ ( x, y ) + δ ( x, y ; d z ) → change in phase obs time ∼ 45 min Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 8 / 14
pyoof package Least squares optimization Python package: pyoof START GitHub Repository https://github.com/tcassanelli/pyoof input data: P obs , λ, d z � � resolution mesh: [ x ] , [ y ] iteration counter: j = 0 initial parameters: θ ( j ) j = j + 1 no Aperture evaluation � � [ ϕ ]+ 2 π i λ [ δ ] [ E a ] = [ B ] · [ E a ] · e yes FFT2 Estimated solution Optimality θ = θ ( j ) � Power pattern: [ P norm ] interpolation Residual Least squares END [ P obs norm ] − [ P norm ] θ ( j +1) Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 9 / 14
pyoof package Least squares optimization Least squares minimization n = 7 � c dB � ⊺ θ = K 1 − 1 K 1 1 K 2 − 2 · · · K 7 7 P obs P obs P obs norm ( u, v ), d z = − 0 . 022 m norm ( u, v ), d z = 0 . 0 m norm ( u, v ), d z = 0 . 022 m 0 . 04 0 . 04 0 . 04 0 . 02 0 . 02 0 . 02 v degrees v degrees v degrees 0 . 00 0 . 00 0 . 00 − 0 . 02 − 0 . 02 − 0 . 02 − 0 . 04 − 0 . 04 − 0 . 04 − 0 . 05 0 . 00 0 . 05 − 0 . 05 0 . 00 0 . 05 − 0 . 05 0 . 00 0 . 05 u degrees u degrees u degrees P norm ( u, v ) d z = − 0 . 022 m P norm ( u, v ) d z = 0 . 0 m P norm ( u, v ) d z = 0 . 022 m 0 . 04 0 . 04 0 . 04 0 . 02 0 . 02 0 . 02 v degrees v degrees v degrees 0 . 00 0 . 00 0 . 00 − 0 . 02 − 0 . 02 − 0 . 02 − 0 . 04 − 0 . 04 − 0 . 04 − 0 . 05 0 . 00 0 . 05 − 0 . 05 0 . 00 0 . 05 − 0 . 05 0 . 00 0 . 05 u degrees u degrees u degrees Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 10 / 14
pyoof package Phase error construction Phase error (computed) ϕ ( x, y ) n = 1 ϕ ( x, y ) n = 2 ϕ ( x, y ) n = 3 ϕ ( x, y ) n = 4 ϕ ( x, y ) n = 5 ϕ ( x, y ) n = 6 ϕ ( x, y ) n = 7 ϕ ( x, y ) n = 8 ϕ ( x, y ) n = 9 ϕ ( x, y ) n = 10 Phase error primary reflector Optimum 5 ≤ n < 10 n max = 10 ⇒ 65 coefficients Correlation + Polynomial properties lines between -2 and 2 radians Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 11 / 14
Analysis Convention in sub-reflector Adding an offset The convention problem Coordinate system Offset amplitude of 1500 µ m Numbering U ℓ n polynomials Same amplitude obtained in ϕ ⊥ calculated Labeling in actuators Angle of view sub-reflector Look-up table α = 40 deg Phase error α = 42 deg 3 3 2 2 1 1 y m y m 0 0 − 1 − 1 − 2 − 2 − 3 − 3 − 2 0 2 − 2 0 2 x m x m Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 12 / 14
Conclusions Summary Project summary Development of a OOF holography software: pyoof package Individual function selection and testing Interval Zernike circle polynomial order, 5 ≤ n < 10 Residual optimization, covariance and correlation ↓ orders → optical aberrations + orthogonality pyoof package and observations Added offset to active surface → observed in phase error Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 13 / 14
Conclusions Summary Project summary Development of a OOF holography software: pyoof package Individual function selection and testing Interval Zernike circle polynomial order, 5 ≤ n < 10 Residual optimization, covariance and correlation ↓ orders → optical aberrations + orthogonality pyoof package and observations Added offset to active surface → observed in phase error It is possible to make an OOF holography look-up table and improve the current model! Cassanelli (MPIfR) OOF holography at Effelsberg 21 February 2018 13 / 14
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