1. Flux Calibration

2. 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

3. 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).

4. 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

5. 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”)

6. 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 :

7. 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 :

8. 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

9. 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

10. 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 ’’ (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)

11. 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= ”

12. 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= ”

13. 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

14. 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

15. 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

16. 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

17. 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”

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

19. 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

20. Calibration with Photonic Noise generated
Φ0=
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= ”
λ=1.5 µm
λ=1.5 µm

21. Spares