Phase-shifting fringe tracking method for sparse aperture interferometer arrays F. Hénault Institut de Planétologie et d’Astrophysique de Grenoble, Université Joseph Fourier, CNRS, B.P. 53, 38041 Grenoble – France
Conventional fringe tracking
Phase-shifting mode • Inject three successive phase-shifts φ = 0, 2π/3 and –2π/3 into a reference waveguide
VLTI beams injection
Hardware
Image plane
Multi-axial IO combiner
Principle Polychromatic interferogram
λ
λ3
X’ Monochromatic interferogram
λ1
X’
• For a given spectral channel [λ-δλ/2, λ+δλ/2] • Compute complex OTF via inverse Fourier transform
X
0
• Extract phase differences over the first N(N–1)/2 OTF peaks
X
• Compute piston errors from phase differences
a
12a
7a
2a
Fringe pattern
- Conventional fringe tracking is achievable by means of spectral decomposition of the interferograms generated by a multi-axial, non redundant arrangement of the fringe tracker exit sub-pupils - The fringe tracker can also be operated in phase-shifting mode: Three interfeerograms are measured, for different values of the phase-shifts φ1 = 0, 2π/3 and 4π/3 introduced into a reference sub-pupil - The OPDs are directly sensed on all other sub-pupils by means of simple algorithms (linear combination, FFT…) -Those sub-pupils can be arranged into a fully redundant geometry, allowing one to decrease dramatically the total number of required pixels (see Table below) and thus the measurement noises - Numerical simulations were carried out, based on the VLTI parameters and the optical concept of POPS, a 2nd-generation fringe tracker proposed to ESO in 2010, using an Integrated Optics Combiner (IOC)
Low dispersion spectrograph
λ2
Multi-axial IO combiner
Input spectrograph
X’ • Obtain three phase-shifted interferograms
X’
• Combine interferograms with complex coefficients
1, exp[2iπ/3] and exp[-2iπ/3] • Compute synthetic OTF via inverse Fourier transform
1
X • Extract phase differences over the first N–1 OTF peaks
X
• Compute piston errors from direct phase measurements
Software
3a
6a
4a
2a
Non-redundant vs. fully redundant pupil arrangement: Allows for wider interfringe distance and less sampling pixels
X
2a
2a
2a
2a
2a
Software
2a
X Interferograms
-188
-141
-94
-47
0
47
94
141
188
235
282
0.75
Phase extraction on 28 channel pairs
0.5 0.25 0 -282
-235
-188
-141
-94
-47
0
47
94
141
188
235
282
1 0.5 0 -0.5
255
565
4.64
4.81
0 -57
-188
-141
-94
-47
0
47
94
141
188
235
282
IOC exit aperture
F / 4.5
Total number of pixels Number of pixels per fringe
F / 3.3
F / 2.5
41
75
115
4.67
4.55
4.49
50
40
100
50
80
40
RMS error (nm)
Success ratio (%)
With phase-shifting
IOC exit aperture
F / 6.5
F / 5.5
20
20
0
0
0
0
4
5
6
7
8
Magnitude of guide star
4 Telescopes, Group delay mode
9
10
0
1
2
3
4
5
6
7
8
Magnitude of guide star
6 Telescopes, Group delay mode
57
9
10
0 -57
-38
-19
0
38
57
Reference channel location
19
38
57
0.5 0 -0.5 -1 -1.5 -57
-38
-19
0
19
38
57
Pixel #
0.3
0.15
0 -57
-38
-19
0
19
38
57
Pixel #
Results
80
40
40
20
0 4
19
0.45
60
3
0
Pixel #
No phase-shifting
2
-19
1
60
1
-38
1.5
100
0
0.5
0.25
0.2
0
F / 4.7
0.75
0.4
With phase-shifting
40
0 3
80
10
20
2
80
20
40
1
100
60
10
0
100
With phase-shifting
20
38
Pixel #
30
60
19
0.6
-57
No phase-shifting
No phase-shifting
30
0
0.6
Fringe tracker parameters after preliminary optimization
Pixel #
-19
Pixel #
RMS error (nm)
-235
-38
0.8
-1 -1.5 -282
RMS error (nm)
Normalized phase
Pixel # 1.5
95 4.8
Success ratio (%)
MTF (arbitrary units)
Pixel #
1
0.2
MTF (arbitrary units)
-235
1
Phase extraction on 7 waveguide channels
0.4
Normalized phase
-282
0.6
5
6
7
8
9
Success ratio (%)
0
Total number of pixels Number of pixels per fringe
Intensity (arbitrary units)
0.2
Intensity (arbitrary units)
0.4
Intensity (arbitrary units)
CASE
0.6
fringe tracking Conventional in phasefringe tracking shifting mode
Intensity (arbitrary units)
0.8
4 6 8 telescopes telescopes telescopes
0.8
- No clear advantage for 4 and 6 telescopes, but one magnitude gain with eight telescopes - Gain expected to increase as more and more telescopes are added (10, 12 etc…)
10
Magnitude of guide star
8 Telescopes, Group delay mode
Contact
[email protected]
