PSF and FoV characteristics of imaging and nulling interferometers
PSF and Field of View characteristics of imaging and nulling interferometers
François Hénault UMR CNRS 6525 H. Fizeau – UNS, CNRS, OCA Avenue Nicolas Copernic 06130 Grasse - France
Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
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PSF and FoV characteristics of imaging and nulling interferometers
Opening questions In the frame of Darwin/TPF-I exoplanet finding space missions…
- Do nulling maps depend on the type of combining optics (axial vs. multi-axial combination schemes) ? - Has it some consequence on their “nulling imaging” capacity ?
Previous publications • “Simple Fourier optics formalism for high angular resolution systems and nulling interferometry,” JOSA A 27, p. 435-449 (2010) • “Fibered nulling telescope for extra-solar coronagraphy,” Optics Letters 34, n° 7, p. 1096–1098 (2009) • “Computing extinction maps of star nulling interferometers,” Optics Express 16, 4537-4546 (2008) Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
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PSF and FoV characteristics of imaging and nulling interferometers
On-sky angular coordinates V
Entrance pupil plane (P)
Sky object
U Y
v u
s
sO
D P1
X Y’
O P2
Exit pupil plane (P’) X’
P4 P’4
O’ P3
Detection plane Y’’
P’3
B P’2 D’
P’1
B’
sO ’
X’’ s’
F’ O’’ M’
Coordinates systems
Z
MO’’ Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
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PSF and FoV characteristics of imaging and nulling interferometers
Image of a sky object O(s) projected back on sky
I(s) =
∫∫ sO ∈ ΩO
N
O(s O ) PSFT (s - s O )
∑a
n
[ (
exp[i φ n ] exp i k s O OPn − s O' P'n / m
)]
2
dΩ O
n =1
with • PSFT(s) : PSF of one individual collecting telescope, being projected back on-sky • an : amplitude transmission factor of the nth telescope • ϕn : phase-shift along the nth interferometer arm for cophasing or nulling purpose • k = 2π/λ : wavenumber of monochromatic electro-magnetic field • m : optical compression factor between telescopes and their relay optics Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
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PSF and FoV characteristics of imaging and nulling interferometers
General layout of a multi-aperture interferometer B
O
P1
Telescope 1
P2
(P)
Telescope 2 FC
Relay optics 1
Relay optics 2
APS 1 Metrology beam 1
APS 2
Metrology beam 2
B’
Converging optics Diverging optics Fold mirror Beamsplitter Acromatic Phase Shifter
O’
Combining optics
(P’)
F’
Focal plane
• All telescopes assumed to be identical • All exit pupils optically conjugated with entrance pupils
Z
Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
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PSF and FoV characteristics of imaging and nulling interferometers
PSF and Field of View characteristics • Generalized Point Spread Function (PSF) N
PSFG (s, s O ) = PSFT (s)
∑a
[
] [
(
exp[i φ n ] exp − i k s O' P' n / m exp i k s O OPn − O' P' n / m
n
)]
n =1
– Changing over the whole instrument Field of View – Object-Image relationship is not a convolution product
• Maximal achievable Field of View (FoV) – Neglecting any kind of apertures or stops, – Neglecting geometrical aberrations and diffraction effects N
FoV(s) =
∑a
n
[ (
exp[i φ n ] exp i k s OPn − O' P'n / m
)]
2
n =1
– Suitable for fast polychromatic FoV computations:
FoVδλ (s) = ∫ FoVλ (s) Bδλ (λ) dλ δλ Conference 7734 Optical and Infrared Interferometry II
∫B
δλ
(λ) dλ
δλ San Diego, June 30th 2010
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2
PSF and FoV characteristics of imaging and nulling interferometers
Golden rule of Fizeau interferometers • Only occurs when: O’P’n = m OPn • Generalized PSF becomes: N
x
PSFG (s, s O ) = PSFT (s)
∑a
n
Pupil In
[
exp[i φ n ] exp − i k s O' P'n / m
Pupil Out
]
2
n =1
– Constant over the whole FoV – Classical Object-Image relationship I(s) = O(s) * PSF(s) holds ☺
• Maximal achievable Field of View: – Becomes infinite whatever the wavelength – Constant transmission equal to:
FoV (x s) =
2
N
∑a
n
exp[i φ n ]
n =1 Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
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PSF and FoV characteristics of imaging and nulling interferometers
Numerical simulations: a 8-telescope Fizeau interferometer in imaging and nulling modes PSF (FoV center)
Input pupils
PSF (half FoV)
Maximal achievable FoV
Y D X
B
5 arcsec
Y
0
π
π 0
0 π
π
D X
0
B
5 arcsec
• Golden rule extends destructive fringe over the whole FoV, killing the central star and all its surrounding planets (this is not what we want ) Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
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PSF and FoV characteristics of imaging and nulling interferometers
Sheared-Pupil Telescopes (SPT) Secondary Mirror
Monolithic telescope Primary Mirror
(P)
APS 1
Relay optics
Metrology May be replaced beam 2 with a modified Beam Mach-Zehnder densifier combiner Metrology beam 1
Beam combiner (exit pupil plane)
APS 2 and OPD compensation
P’1
O’
O’
(P’)
F’
(P’)
P’2
[or Michelson B’ equipped with cube-corners]
Metrology beams
Focal plane F’
Z
Focal plane Z
Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
9
PSF and FoV characteristics of imaging and nulling interferometers
Two different types of Sheared-Pupil Telescope Input pupil plane
Output pupil plane
Y
Y ’
Unmasked output sub-pupils
O
D’
X
O’
X’
D
Y
Masked output sub-pupils
D
O
B Conference 7734 Optical and Infrared Interferometry II
Monolithic pupil telescope
X
D’
B’ Y ’
O’
Lyot stop on Exit pupil X’
B’ San Diego, June 30th 2010
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PSF and FoV characteristics of imaging and nulling interferometers
Interest and limitations of nulling SPTs • Usable for exploratory science missions: exo-zodi characterization, Jupiter-like planets… • If rotating, allow to validate most of the Darwin/TPF-I algorithms envisaged for planets finding and characterization • When unmasked, they concentrate energy in very small core, overcoming Rayleigh’s diffraction limit ☺ • But no real super-resolving power , since PSF are sharpened after sub-aperture filtering: Specific ObjectΙ(s) = Image relationship:
N
∑a
n
[
2
exp[i φ n ] exp − i k s O' P'n / m
] × [PSF (s) * O(s)] T
n =1
Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
11
PSF and FoV characteristics of imaging and nulling interferometers
5 arcsec
Masked SPT B=1m D=3m
Masked SPT B = 0.5 m D=4m
Sheared-Pupil Telescope D=5m
PSF and Field of View simulations of SPTs
• Unmasked SPT: high throughput, residual star leakage • Masked SPT: no leakage, enlarged nulled area of low throughput Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
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PSF and FoV characteristics of imaging and nulling interferometers
Axially Combined Interferometer (ACI) B
O
P1
Telescope 1
P2
(P)
Telescope 2
Relay optics 1
Relay optics 2
APS 1
APS 2
Metrology beam 1 Metrology beam 2
Axial beam combiner
Fringe tracker
• Co-axial recombination by means of a balanced set of beamsplitters • Equivalent to the previous monolithic, masked SPT • Nulls all diffracted light originating from central star • Specific Object-Image relationship:
(P’) O’
N Ι(s) = PSFT (s) * ∑ a n exp[i φ n ] exp i k s OPn n =1
[
F’
Focal plane
]
2
O(s)
Z
Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
13
PSF and FoV characteristics of imaging and nulling interferometers
Masked SPT B=1m D=3m
Masked SPT B = 0.5 m D=4m
2 arcsec
Fictitious sky object
Nulling imaging capacities of SPT
• •
No gain in angular resolution, but diffracted starlight cleaned before final image blurring Progressive leakage from central objects (to be traded against throughput) Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
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PSF and FoV characteristics of imaging and nulling interferometers
ACI B = 20 m D=5m
ACI B = 10 m D=5m
2 arcsec
Fictitious sky object
Nulling imaging capacities of ACI
•
For longer baselines, nulling ACI behaves as a single-dish telescope Nulling capacity seems to be lost
Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
15
PSF and FoV characteristics of imaging and nulling interferometers
Conclusions • Nulling maps and nulled images produced by different types of multi-aperture optical systems can be rapidly evaluated by means of a simple Fourier optics formalism • From the results of theory and first numerical computations:
[
– “Golden rule for interferometric imaging” extends the destructive fringe pattern of nulling interferometers over their whole Field of View – A nulling monolithic, sheared-pupil telescope is an attractive solution Requires further tradeoff on throughput/leakage – Theoretical Object-Image relationship of the Bracewell interferometer allows full extinction of diffracted starlight, but no super-resolution is possible Please this is nulling all very interferometers, preliminary, eventually – In the caseretain: of fibered best anecdotal throughput are andusing somewhat Furtherschemes work is required and achieved axial heuristic. recombination (nulling ACIs) cooperations are welcome
Conference 7734 Optical and Infrared Interferometry II
San Diego, June 30th 2010
]
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