Flux Calibration

Spectrograph calibration: Flux calibration Requirements :Test the feasibility of flux calibration : Punctual Object Flat Field Specification: - measure relative flux between ≠ λ to 1 % - measure absolu flux to 10 % What means ? Correct flux losses  Calibrate flux losses to define the correction function Φ(λ) Image calibrated Source Flux Correction Optical losses: diffraction (mainly), aberrations Spectrograph Treated Image QE, noise, intra-pixel variations Detector calibration Detector Φmeasured(λ) Untreated Image Detector Image

Flux Variations Main causes of flux variations: - optical losses: diffraction, diffusion - detector: noise (thermal, readout), gain of pixel, intra-pixel variations Flux losses depend on: - position on the slicer plane: (x, y) - wavelength 2 methods to correct flux losses: create a library of reference images at different positions (x,y) and wavelength calibrate diffraction losses as a function of (x,y,λ). Which means to well-calibrate detector (dark,flat,intra/inter pixel variation).

Flux Correction from library of reference images: simulation Image at (x,y,λ), I1=kth×I0 Image at (x,y,λ) ∕ Compare the pixels distribution Method ? Image calibrated in flux Selected Reference Image Library of reference images at I0 How to calculate k ? k = ∫ image / ∫ image ref minimization chi2 How to find the best reference image ? Correlation Minimisation:chi2

PSF Library creation y x Conditions No noise Creation of library of 100 PSFs on detector at ≠ positions into one pixel (step 1/10 of pixel) : SNAP spectrograph simulation From old design of SNAP spectrograph y one pixel slice n+1 0.15 ’’ slice n x slice n-1 Conditions No noise step of position variations: 1/10 of a pixel= 0,15’’/10=0.015” Initial pixel (x0,y0)=(0”,0”)

Х2 Minimization Method For each image (p), find : (i,j) matrix index (0<i,j<N) signal and noise of the image to calibrate into pixel (i,j) Flux error signal and noise of reference image indexed p (0<p<100) associated with a single position (x,y) & λ into pixel (i,j) k : Ratio image to calibrate over reference image For each image (p), find : Deduce the index pmin of the reference image (the nearest one of the image to calibrate): Error on parameter k :

Method of Х2 Minimization : Summary Si,j =Image subtracted of the noise average Image k x Φ0 to calibrate Image (p) of library: Φ0 Determine the ONE reference image pmin the nearest Determine M reference images the nearest Interpolation :

Х2 Minimization to find coefficient {ap}0<p<M Interpolation Normalized Reference Images Normalized Image to calibrate Х2 Minimization to find coefficient {ap}0<p<M Normalization cste Solve Solve the M linear equations Minimize

X2 minimization without interpolation Si,j =Image subtracted of the noise average X2 minimization without interpolation X2 minimization with interpolation For each reference image (p), minimize: Goal - Minimize: First step: find the “nearest” reference image A first method of X2 minimization determines the M reference images the nearest (M=2 by default) The interpolation consists in computing the coefficients ap from normalized images: the method used is the Х2 Minimization. This method leads us to solve a linear system of M equations (see next slide). After solving the system of equation, we determine the ratio k (δX2/δk=0) : The one nearest reference image combination of the nearest reference images

Х2 Minimization Method in practice To compute: We need to well-know the noise B per pixel impossible (random) Hypothesis; B = 0 (first application to check the method): no detector noise, no photonic noise Generate 32 images (without noise) at λ,x given  scan along the slice width by step of 0.003 ’’ (y0={0.02 ’’:0.113’’}) Source flux= k x source flux of reference images (k=0.6)  Find k(=0.6) from 2 library of reference images (with different sampling)

Method used: minimization chi2 Library used: 10 reference images x0=0’’ fixed, λ=1,4 µm fixed y0={0.001’’:0,135”} by step 1/10 of a pixel=0.015” Library used: 40 reference images x0=0’’ fixed, λ=1,4 µm fixed y0={0’’:0,146”} by step 1/40 of a pixel=0.00375”

Method used: minimization chi2 to determine k+ interpolation Library used: 10 reference images x0=0’’ fixed, λ=1,4 µm fixed y0={0.001’’:0,135”} by step 1/10 of a pixel=0.015” Library used: 40 reference images x0=0’’ fixed, λ=1,4 µm fixed y0={0’’:0,146”} by step 1/40 of a pixel=0.00375”

λ=0.5 µm λ=1 µm (visible arm) λ=1 µm (IR arm) Erreur augmente au bord de la slice + critique ds le visible: variation de la PSF + rapide λ=1 µm (IR arm) λ=1.5 µm

Flux variation per slice as a function of source position into the slicer λ=1.5 µm slice 3 5 slice side Direction to move 3 Comprendre l’évolution de la distribution spatiale et du flux sur le détecteur en fonction de la position de la source pour ameliorer l’interpolation et la minimisation slice 2 slice 4 slice 5 slice 1 slice 0

λ=0.5 µm λ=1 µm (vis arm) λ=1 µm (IR arm) λ=1.5 µm Pente + raide ds visible (cf resultat de la minimisation ds vis) λ=1 µm (IR arm) λ=1.5 µm

slice side slice 3 slice 2 slice 4 Integration on 1 pixel per slice around the maximum λ=0.5 µm slice side slice 3 slice 2 Integration of 49 pixels per slice around the maximum slice 4 slice side slice 3 slice 2 slice 4 Integration of 9 pixels per slice around the maximum Au bord de la slice, non seulement le flux varie bcp mais aussi la distribution spatiale: interpolation lineaire suffisante ? slice 0

Variation of spatial distribution into one slice as a function of source position y (around the side of slice) y=0,065” y=0,068” y=0,071” y=0,074” y=0,086” y=0,080” y=0,083” y=0,077”

Variation of flux into one slice as a function of source position y (around the side of slice)

A faire Minimiser non pas les images sur le détecteur mais le flux total de ROI prédefini (1 roi par slice ou 2 roi par slice) Images de référence Image a calibrer flux de la slice 0 On s’affranchit d’une variation de la distribution spatiale trop rapide A minimiser

Calibration with Photonic Noise generated Φ0=0.00005 S/Nmax=106 <S/N>=1.2 Library used: 10 reference images x0=0’’ fixed, λ=1,5 µm fixed y0={0.001’’:0,135”} by step 1/10 of a pixel=0.015” Library used: 40 reference images x0=0’’ fixed, λ=1,5 µm fixed y0={0’’:0,146”} by step 1/40 of a pixel=0.00375” λ=1.5 µm λ=1.5 µm

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