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Why only take 2 interferograms?
Why not take 2*N ?
The birth of the TAURUS concept
Note: A single interferogram only contains (1/N)th of in spatial information
Scanning the FP across a full FSR is a natural way to:
• Recover ALL of the spatial information
• Resolve the fundamental ambiguity in the individual interferograms
Scanning can be achieved in 3 ways:
a) Changing : - by tilting the FP or by moving the image across the fringes
as in photographic interferograms (to partially resolve ambiguity)
b) Changing : - by changing the pressure of a gas in the FP gap
as in 0th dimensional scanning with eg: propane
c) Scanning d: - by changing the physical gap between the plates
very difficult since need to maintain parallelism to /2N, at least
TAURUS = QI etalons + IPCS … c1980
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Queensgate Instruments (ET70)
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Queensgate Instruments
(Capacitance Micrometry)
FP etalons
Piezo transducers (3)
CS100 controller
Construction
• Super-flat /200 base
• Centre piece (optical contact)
• Top plate
Capacitors
• X-bridge
• Y-bridge
• Z-bridge (+ stonehenge)
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Scanning d over 1*FSR
m  (m-1) ; d  d + (/2)
Alec Boksenberg and the
Image Photon Counting System (IPCS)
Perfect synthesis …
Imperial College, London  QI etalons + CS100
TAURUS  16 Mbyte datacubes
University College, London  IPCS
}
Aug-Nov, 2008
IAG/USP (Keith Taylor)
How big a field?
From before, Jacquinot central spot given by:
R = 2
or: R ~ 82/2 ~ 82(d/DT)2/J2,
where J is the angle on the sky.
However, given an array detector, we can work
off-axis:
So, how far?
Answer: Until the rings get narrower than
 ~2 pixels (the seeing disk)
Now, from the Airy Function we obtain:
d/d = -1/0.sin()
Aug-Nov, 2008
The full TAURUS field, F is then:
F = 22(d/DT)2/R
F/J ~ 20, typically
IAG/USP (Keith Taylor)
or ~400 in 
TAURUS
Datacube
Recalling the phase delay equn:
m = 2l.cos
For small values of :
 goes as [1 – (2/2)]
where   tan-1[(y-y0]/[x-x0])
and (x0,y0) = centre of FP fringes
(x1,y1) is shifted in z-dirn
w.r.t. (x0,y0)
and this -shift is thus
~parabolic in 
It is also periodic in “m”. We thus
refer to this shift as a
“Phase-correction”
So the surface of constant  is a
“Nested Parabola”
Cut through a “Nested Parabola and
you get a set of rings and these
rings are the FP fringes.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Wavelength Calibration
(converting z to )
As shown, surfaces of constant , as seen in an (x,z)-slice are defined by a set
of nested parabolæ, equally spaced in z. Any (x,y)-slice within the cube cuts
through these nested paraboloids to give the familiar FP fringes (rings).
Now -calibration requires transforming z   where:
l(z) = l(0) + az
a is a constant of proportionality.
Constructive interference on axis (x0,y0) gives:
az0 = m0/2 - l(0)
but an off-axis (x,y) point transmits the same 0 at (z0 + pxy) where:
apxy = l(z0).(secxy – 1)
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Phase Correction
z
I0(z)
z
z
Phase corrected
I0()


at (x1,y1)
at (x2,y2)
The 2D phase-map, p(x,y), can be defined such that:
p(x,y) = mz0(sec – 1)
The phase-map, p(x,y), can be obtained from a -calibration data-cube by
illuminating the FP with a diffuse monochromatic source of wavelength, .
Note: Phase-map is discontinuous at each z
Aug-Nov, 2008
IAG/USP (Keith Taylor)
The Phase-Map
The phase-map is so called since it can be used to transform the raw TAURUS
cube, with its strange multi-paraboloidal iso-wavelength contours into a welltempered data cube where all (x,y)-slices are now at constant wavelengths.
The process is called phase-correction since it represents a periodic function of
period, z.
ie: If the z-value (z’) of a phase-shifted pixel exceeds the z-dimensions of the
data-cube, then the spectra is simply folded back by one FSR to (z’ - z).
It will be noted that the phase-map (as defined previously) is independent of 
and hence in principle any calibration wavelength, cal, can be used to phasecorrect an observation data-cube at an arbitary obs, remembering that:
  2
&
z = /2
But also, the phase-map can be expressed in -space as:
xy = 0(1 - cos)
and hence is also independent of gap, l, and thus applicable to all FPs at all .
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Order (m) and gap (l) determination
The periodicity of the FP interference fringes makes -calibration non-trivial. The
paraboloidal mapping from z   doesn’t exactly help, either!
Nevertheless, using 2 calibration wavelengths:
Say: 1 and 2, peak on-axis at z1 and z2
The trick is to find m1 (and hence m2), the order of interference. From m we can
infer the gap, l , and hence obtain a -calibration where:
az0 = m0/2 - l(0)
Then:
m2  m1 =
1 z2 – z1

2
z
m1
(  2)
2 1
If m1 can be estimated (from manufacturers specs or absolute capacitance
measures) then we can search for a solution where m2 is an integer. This can
be an iterative process with several wavelength pairs. Clearly the further 1
and 2 are apart, the more accuracy is achieved.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Wavelength Calibration
If m1 can be estimated (from manufacturers specs or absolute capacitance
measures) then we can search for a solution where m2 is an integer. This can
be an iterative process with several wavelength pairs. Clearly the further 1
and 2 are apart, the more accuracy is achieved.
Once the interference order, m1, for a known wavelength, 1, has been
identified then wavelength calibration is given by:
 = 0
{ (zm zz ) + 1}
0
0
Aug-Nov, 2008
0
IAG/USP (Keith Taylor)
FPP in collimated beam
• Interference fringes formed at infinity:
• Sky and FP fringes are con-focal
• Detector sees FP fringes superimposed on sky image
IF
Aug-Nov, 2008
FP
IAG/USP (Keith Taylor)
FPI in image plane
• Interference fringes formed at infinity:
• Sky and FP fringes are not con-focal
• Detector sees FP plates superimposed on sky image
• ie: No FP fringes seen on detector
• FP is not perfectly centred on image plane (out of focus) to avoid
detector seeing dust particles on plates.
FP
IF
Aug-Nov, 2008
IAG/USP (Keith Taylor)
FP (or Interference Filter)
in image plane
The FP still acts as a periodic monochromator but the angles into the FP (or IF)
must not exceed the Jacquinot criteria, which states that:
2 < 82/R (or R = 2)
At f/8:
 = 2.tan-1(1/16) ~7.2º, so R < 500
At f/16:  = 2.tan-1(1/32) ~3.6º, so R < 2,000
Note: the FoV is determined not by the width of the fringes but by the diameter
of the FP.
Also, an IF is simply a solid FP ( ~2.1) with very narrow gap.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Data-cube science
Aug-Nov, 2008
IAG/USP (Keith Taylor)
FP observations of NGC 7793
on the 3.6m.
• Top left: DSS Blue Band
image.
• Top right: Spitzer infrared
array camera (IRAC) 3.6μm
image.
• Middle left: Hα
monochromatic image.
• Middle right: Hα velocity
field.
• Bottom: position-velocity
(PV) diagram.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
FP observations of NGC
7793 on the 36cm.
• Top left: DSS Blue
Band image.
• Top right: Spitzer IRAC
3.6μm image.
• Middle left: Hα
monochromatic image.
• Middle right: Hα velocity
field.
• Bottom: PV diagram.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
ADHOC screen shot (Henri)
Aug-Nov, 2008
IAG/USP (Keith Taylor)
ADHOC screen shot (Henri)
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Imaging Fourier Transform
Spectrographs (IFTS)
FTS = Michelson Interferometer:
IFTS = Imaging IFTS over solid angle, .
• Beam-splitter produces
2 arms;
• Light recombined to
form interference fringes
on detector;
• One arm is adjustable to
give path length
variations;
• Detected intensity is
determined by the path
difference, x, between
the 2 arms.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
IFTS theory (simple version)
Given that frequency,  = 1/ (unit units of “c”):
Phase difference between two mirrors = 2x
So recorded intensity, I, is given by:
x ) = 1 [1 + cos(2x)]
(,
I
2
2
Now, if we vary x in the range:   x/2  , continuously then:

I(x) = B().(1 + cos2x).d
-

B() = I(x).(1 + cos2x).dx
-

and

These represent
Fourier Transform pairs.
Spectrum B() is obtained from the cosine transformation of the Interferogram I(x)
Aug-Nov, 2008
IAG/USP (Keith Taylor)
IFTS reality (simple version)
• At x = 0: the IFTS operates simply as an imager;
• White light fringes – all wavelengths behave the same
• At all other x-values, a subset of wavelengths constructively/dsitructively
interfere
• For a particular , the intensity varies sinusoidally according to the simple
relationship:
1
I ( ) = [1 + cos(2x)]
2
In reality, of course, x goes from 0  xmax which limits the spectral resolving
power to:

2xmax
R0 =  =

eg: if xmax = 100mm and  = 500nm then: R0  1.105
Aug-Nov, 2008
IAG/USP (Keith Taylor)
IFTS in practice
Since we are talking here about an imaging FTS then what is it’s imaging FoV?
Circular symmetry of the IFTS is identical to the FP and hence:
2l.cos = m
And also:
R >> 2
limited only by the wavelength variation, , across a pixel:
However, in anaolgy to the FP
 Phase-correction is required in order to accommodate path difference
variations over the image surface.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Pros & Cons of an IFTS
Advantages:



Arbitary wavelength resolution to the R limit set by xmax;
A large 2D field of view;
A very clean sinc function, instrumental profile


cf: the FP’s Airy Function
A finesse N = 2/ which can have values higher than 103
Disadvantages:


Sequential scanning – like the FP. However, the effective integration time of
each interferogram image can be monitored through a separate
complementary channel, if required;
Very accurate control of scanned phase delay is required


Especially problematic in the optical
At all times, the detector sees the full spectrum and hence each
interferogram receives integrated noise from the source and the sky


This compensates for the fact that all wavelengths are observed simultaneously
which is why there is no SNR advantage over an FP;
Also sky lines produce even more noise, all the time.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Michelson Interfermeter
(N = 2 interference ; n >>1)
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Hybrid and Exotic Systems
• FP & IFTS are classical 3D imaging spectrographs
• ie: Sequential detection of images to create 3D datat cubes:
• FP = Wavelength scanning
• IFTS = Phase delay scanning
There are, however, techniques which use a 2D area detector to sample
2D spatial information with spectral information, symultaneously.
These we refer to as:
Hybrid Systems
Examples of this are: Integral Field Units (IFUs). These can use either:
 Lenslets
 Fibres
 Lenslets + Fibres
 Mirror Slicers
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Integral Field Spectroscopy
• Extended (diffuse) object with lots of spectra
• Use “contiguous” 2D array of fibres or ‘mirror slicer’ to obtain a
spectrum at each point in an image
Tiger
SIFS
MPI’s 3D
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Lenslet array (example)
LIMO (glass)
Pitch = 1mm
Some manufacturers
use plastic lenses.
Pitches down to
~50m
Used in
SPIRAL (AAT)
VIMOS (VLT)
Eucalyptus (OPD)
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Tiger (Courtes, Marseille)



Technique reimages telescope focal plane onto a micro-lens array
Feeds a classical, focal reducer, grism spectrograph
Micro-lens array:





Dissects image into a 2D array of small regions in the focal surface
Forms multiple images of the telescope pupil which are imaged through
the grism spectrograph.
This gives a spectrum for each small region of the image (or lenslet)
Without the grism, each telescope pupil image would be recorded as
a grid of points on the detector in the image plane
The grism acts to disperse the light from each section of the image
independently
So, why don’t the spectra all overlap?
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Tiger (in practice)
Enlarger
Lenslet array
Aug-Nov, 2008
Detector
Collimator
IAG/USP (Keith Taylor)
Grism
Camera
Avoiding overlap
• The grism is angled (slightly) so that the spectra can be extended
in the -direction
• Each pupil image is small enough so there’s no overlap orthogonal
to the dispersion direction
Represents a neat/clever optical trick
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Tiger constraints
• The number and length of the Tiger spectra is constrained by a combination of:
• detector format
• micro-lens format
• spectral resolution
• spectral range
• Nevertheless a very effective and practical solution can be obtained
Tiger
SAURON
OSIRIS
(on CFHT)
(on WHT)
(on Keck)
True 3D spectroscopy
– does NOT use time-domain as the 3rd axis (like FP & IFTS)
– very limited FoV, as a result
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Tiger Results (SAURON – WHT)
Aug-Nov, 2008
IAG/USP (Keith Taylor)