Based on the transfer pupil technique we develop a method to measure the lower order aberration coeffcients for optical telescopes; the method makes use of a Hartmann mask in a spectrograph for sampling the wavefront at different location

Das, H. K

Kohok, Abhay

Tandon, S. N.

English

Publications of the IAS Fellows

Bull. Astr. Soc. India (2004) 32, 121–131
An implementation of transfer-pupil in a spectrograph
on optical telescopes
H.K.Das ∗, Abhay Kohok † and S.N.Tandon ‡
Inter University Center for Astronomy and Astrophysics (IUCAA), Post Bag 4, Ganeshkhind,
Pune - 411 007, India
Received 6 February 2004; accepted 18 May 2004
Abstract. Based on the transfer pupil technique we develop a method
to measure the lower order aberration coefficients for optical telescopes; the
method makes use of a Hartmann mask in a spectrograph for sampling the
wavefront at different locations.
1. Introduction
One of the most common methods used for monitoring optics of Cassegrain telescopes
is to sample the wavefront, as obtained by a long time integration of a stellar image,
by a screen and measuring the slopes of the individual sampled portions. The classical
Hartmann test uses a full-size screen placed above the secondary mirror ( or outside
the prime focus) of the telescope; this technique is no more in common use, (Wilson II,
P150). An equivalent technique, of measurements on transfer-pupil, is capable of making
accurate measurements by using smaller components placed at the telescope focus, see
for example (Wilson II, 1999 P 150-151 and Wilson et al. 1984) for an implementation
of the technique in a spectrograph.
In the commonly used version of transfer-pupil scheme the screen (perforated mask)
is usually an array of lenslets – this is the Shack Hartmann mask (Platt et al. 1971, Shack
et al. 1971) . Each lenslet forms a sub-aperture of the transferred pupil and images it on
the detector. In this paper, we describe an equivalent scheme in which a screen placed on
a low power lens is used, instead of the array of lenslets; in this process we might be using
only a small fraction of the total light available, but it is much simpler to implement.
∗e-mail:hkdas@iucaa.ernet.in
†e-mail:abhay@iucaa.ernet.in
‡e-mail:sntandon@iucaa.ernet.in
122 H.K.Das et.al.
In section 2 details of the optical arrangement is described , and in section 3 complete
procedure used to calculate decenter and defocus of the telescope is described. In section
4 we describe the analysis part, and Appendix I describes the main algorithm of the
computer codes developed to analyze the images and also highlights a laboratory set up
for testing the procedure.
2. Experimental Set Up
To transfer the pupil of the telescope we would use the IUCAA faint object spectrograph
and camera (IFOSC), (R. Gupta et al. 2002); the pupil of the telescope is transferred on
to the grism wheel. The pupil size here is about 32 mm; see Fig.1. The set up is made
for use with IUCAA 2-m telescope which is under installation.
The observational procedure is as follows; an image of a suitably chosen star is ob-
tained at the telescope focus. The f/10 beam from the telescope then meets the collimator
(316 mm focal length) which images the exit pupil of the telescope on the grism wheel,
where a perforated mask is mounted; please refer Fig.1. The collimated beam (from the
collimator) illuminates the perforated mask, and is finally imaged by the camera (142
mm focal length) on the detector.
A similar image is then obtained by illuminating a pinhole (mounted on the aperture
wheel) with a light source in the calibration unit (explained below); the two images are
then compared to obtain the aberration coefficients, as explained in section 4.
In the laboratory we used the f/10 light from the calibration unit of IFOSC (not
shown in Fig.1). The calibration unit is an extra unit of IFOSC; it simulates the exit
pupil of the IUCAA telescope which produces an f/10 beam, including the obscuration
due to the secondary mirror.
TELESCOPE FOCAL PLANE
ILLUMINATED PINHOLE / STAR IMAGE IN THE 
PERFORATED 
MASK
TELESCOPE
 PUPIL OF THE 
DETECTOR
GRISM WHEEL
 CAMERA
OF SPECTROGRAPH
COLLIMATOR
APERTURE   WHEEL
f/10 f/4.5
Figure 1. The position of the transferred pupil of the telescope in the grism wheel of the
spectrograph is shown.
Transfer pupil implementation in spectrograph 123
(Das, et al.)
IUCAA
MASK
PLANO CONVEX
LENS
(a) (b)
Figure 2. (a) The perforated mask on the lens of diameter 40 mm, the holes are in a circle of
30 mm diameter; (b) Images of the pinhole on the detector.
As shown in Fig. 2a, the perforated mask is placed on a plano convex lens, of focal
length 1000 mm, and the entire assembly is mounted on the grism wheel of IFOSC; the
mask is made of a 50 micron copper foil and it contains 1 mm diameter holes on a grid
of 3X3 mm placed on a circle of 30 mm diameter. The lens slightly decollimates the
collimated beam, and images of the pin-hole are therefore separated from each other as
shown in Fig.2b.
3. Procedure of obtaining decentering and defocus
3.1 Finding the aberration coefficients
The main idea of wavefront analysis is that local tilts in the wavefront would cause the
light ray to reach a point away from the point where the light ray would have reached had
it been an ideal wavefront. So an optical arrangement (Hartmann mask) is introduced to
monitor tilts in different parts of the wavefront. Comparison of the ideal and aberrated
wavefront would give us an estimate of the wavefront tilts.
The wavefront aberration W, used in this analysis can be represented as (Wilson II
p-148,284):
W = K11ρcos(φ+ θ11) +K02ρ
2 +K13ρ
3cos(φ+ θ13) +K04ρ
4 (1)
where we have considered only lower order terms. In Eqn.1, ρ is the radial distance of the
holes of the perforated mask from the center normalized to unity at the edge, Knm is the
optical aberration coefficient, φ the azimuth angle in the pupil plane , θnm the phase angle
of the aberration and n,m are positive integers. The different terms, K11ρcos(φ+ θ11)
corresponds to wavefront tilt, K02ρ
2 corresponds to defocus, K13ρ
3cos(φ+ θ13) corre-
sponds to decentering coma and K04ρ
4 to spherical aberration.
124 H.K.Das et.al.
IDEAL WAVEFRONT
RAY POSITION
IDEAL RAY 
   POSITION
R X
ABERRATED WAVEFRONT
ABERRATED 
Figure 3. Figure showing the actual and the ideal ray position.
From Eqn.1, the wavefront slopes can be obtained as,
W ′x = K11cosθ11 + 2K02x+ 3K13x
2cosθ13 − 2K13xysinθ13 +K13y2cosθ13
+ 4K04x
3 + 4K04xy
2 (2)
W ′y = −K11sinθ11 + 2K02y −K13x2sinθ13 + 2K13xycosθ13 − 3K13y2sinθ13
+ 4K04x
2y + 4K04y
3 (3)
where W ′x = 4x/R, W ′y=4y/R, (Malacara and Fig.3); x and y are the cartesian com-
ponents of ρ, ρcosφ and ρsinφ; 4x and 4y are the shifts in the centroids of the pinhole
images between the ideal and the aberrated image and R is the focal length of the lens
which re-images the pupil.
In order to find solution to these equations using a linear least square fit to the images,
a parametrization is done as follows:
4x/R = a1 + b1x+ c1(3x2 + y2) + 2d1xy + e1(x3 + xy2) (4)
4y/R = a2 + b1y + 2c1xy + d1(x2 + 3y2) + e1(x2y + y3) (5)
where, a1=K11cosθ11, a2=−K11sinθ11 , b1=2K02, c1=K13cosθ13,
d1=−K13sinθ13, e1=4K04
The aberration coefficients (Ks) and phase angles (θs) can be obtained in terms of
a,b,c,d,e as follows:
K11=
√
a12 + a22, K02=b1/2, K13=
√
c12 + d1
2, K04=e1/4, sinθ11=−a2/K11,
sinθ13=−d1/K13, cosθ11= a1/K11, cosθ13=c1/K13
Transfer pupil implementation in spectrograph 125
3.2 Finding decentering
The decentering coma can be equated to the lateral tangential Coma by equating K13 to
−3/d1(y1/r1)2[1 + (1− 2d1/r1)(µ22 − 1)]δ in radians (6)
(Wilson I, p 253), where y1 is the half diameter of the primary, r1 is the radius of
curvature of the primary, d1 is the distance between primary and secondary, µ2 is the
inverse of the magnification by the secondary, and δ is the decentering (in mm) of the
secondary mirror with respect to the primary. From known values of y1, r1, d1, and m2
and measured K13 decentering can be calculated.
As the effect of angular decenter of the secondary is usually small compared to that
of lateral decentering (Wilson I, sec. 3.7.2.1 and 3.7.2.2), we only consider decentering.
3.3 Finding defocus
The defocus in mm = −b1R2/(pupil radius) (7)
where “pupil radius” is the radius of the reimaged pupil in mm.
4. Analysis
We need a set of two images (as in Fig. 2b) for our analysis; (a) an ideal image obtained
by illuminating a pinhole at the focal plane and (b) an image obtained with a long time
( > 20 secs.) exposure to a suitable star.
These two images need to be compared to obtain the wavefront deviations, i.e. 4x
and 4y in Eqns. 4 and 5. To solve these equations computer programs were developed
in IDL programming language. The IDL program does the following.
1. Calculates the centroids of all the pin holes from the ideal and aberrated images.
2. Calculates 4x and 4y of the two set of centroids.
3. Obtains the coefficients, a1, a2, b1, c1, d1 and e1 by the method of least square fit.
4. Interprets these different coefficients in terms of the aberration coefficients.
5. Calculates decenter and defocus.
For a discussion on the results obtained in a simulation experiment and their inter-
pretation please refer to Appendix I.
126 H.K.Das et.al.
5. Conclusion
In this paper we have described the theory, experimental method and analysis techniques
to measure the decenter and defocus of large optical telescopes through the use of a
Hartmann mask placed in the collimated beam of a focal plane instrument.
6. Acknowledgment
We are thankful to Paramita Barai of IIT Kharagpur, who had initially worked on this
problem as our project student.
References
Gupta R., et al., 2002, BASI, 30, 785.
Malacara Daniel, 1978, Optical Shop testing, Wiley, New York, p. 326.
Platt B., Shack R.V., 1971, Optical sciences center newsletter, Univ. of Arizona, Tucson, 5(1),
15.
Shack R.V., Platt B.C., 1971, JOSA, 61, 656.
Wilson R.N., 1996, Reflection telescope optics I, A&A Library, Berlin, Springer–Verlag,
New York.
Wilson R.N., 1999, Reflection telescope optics II, A&A Library, Berlin, Springer–Verlag,
New York.
Wilson R.N., Franza F., Noethe L., Tarenghi M., 1984, in Proc. IAU colloq. No.79: Very Large
telescopes, their Instrumentation and Programs, ESO, Garching, 119.
Transfer pupil implementation in spectrograph 127
Appendix I
A.1 Brief description of the main program
The main program needs the following parameter files and a subroutine:
• A parameter file for calculating the centroids. This file contains the following pa-
rameters; background, minimum good data, maximum good data, small box size,
big box size.
• A parameter file for reading the ideal and aberrated image names and the rough
coordinates of both the images; (these rough coordinates are used to calculate the
centroids).
• A parameter file for the telescope and CCD parameters; it contains the pixel size in
mm, magnification by CCD in pixel/mm, focal length of the camera lens of IFOSC,
pupil radius in pixels, radius of the primary mirror, in mm, radius of curvature of
the primary mirror in mm, distance between primary and secondary mirror, in mm,
and magnification by secondary mirror.
• A subroutine for calculating the centroid, (flowchart given in Fig.4).
A.2 Laboratory set up for testing the procedure
To test the validity of the analysis technique, and also to test the codes developed, we
illuminated the pinhole with white light from the calibration unit, as explained in section
2. Two images of the pinhole were taken as described below; the images appear as shown
in Fig. 2b.
• First image - camera lens of IFOSC focused: Ideal wavefront.
• Second image - camera lens of IFOSC shifted by 100 µm: Aberrated wavefront;
a check on raytracing with Zeemax software shows that this shift of 100µm corre-
sponds to a defocus of 140µm.
These two images were analyzed on the basis of the program in Fig.5.
The test results obtained are as follows:
a1 = 1.66e− 05
a2 = −3.46e− 05
b1 = −9.2e− 05
c1 = 1.32e− 06
128 H.K.Das et.al.
    
   
WITHIN THE SMALL BOX REGION, DIVIDES EACH 
           
DRAWS SMALL BOX AROUND IT, TO CALCULATE 
THE FINE CENTROIDS.
  THEN
K=K+1
           IF K > 5 
ELSE
ELSE
READS IMAGE NAME 
READS ROUGH COORDINATES FROM
A FILE
     READS CENTROD CALCULATING 
PARAMETERS FROM A FILE
CALCULATES COARSE CENTROID BY
DRAWING BIG BOX ACROSS EACH GIVEN
ROUGH COORDINATE, AFTER SUBTRACTING 
BACKGROUND, CENTX, CENTY.
XCENTRD = CENTX
YCENTRD = CENTY
K=0
TAKING  XCENTRD, YCENTRD AS REFERENCE 
PIXEL  10X10 TIMES TO RETAIN THE DECIMAL PART
OF THE REFERENCE CENTROID.
CALCULATES FINE CENTROID ACROSS SMALL BOX
IF  | NWXCNTRD − XCENTRD | > 0.01 
IF  | NWYCNTRD − YCENTRD | > 0.01
XCENTRD = NWXCNTRD
YCENTRD = NWYCNTRD
THEN
TO A FILE
PRINT NWXCNTRD AND NWYCNTRD
AFTER SUBTRACTING BACKGROUND, NWXCNTRD,
NWYCNTRD
Figure 4. Flow chart of the centroid calculating program.
Transfer pupil implementation in spectrograph 129
PERFORATED  MASK IMAGES
X AND Y
R R
READS THE IDEAL AND ABBERATED 
A FILE
FROM A FILE
CALCULATES THE CENTERS OF THE TWO 
CALCULATES THE RADIAL DISTANCE OF
ALL THE HOLES OF THE MASK FROM 
THE CENTER AS A FUNCTION OF PUPIL 
RADIUS.
READS THE TELESCOPE AND CCD PARAMETERS
CALCULATES  THE CENTROIDS OF ALL THE
IMAGE NAME AND COORDINATE FILE FROM 
PIN  HOLE  IMAGES
OBTAINS  COEFFICIENTS  BY
PRINTS RESULT 
 LEAST  SQ.  FIT.
CALCULATES  THE  WAVEFRONT  SLOPE
Figure 5. Flow chart of the main wavefront calculating program.
130 H.K.Das et.al.
d1 = 4.46e− 07
e1 = −2.94e− 06 (8)
K11 (TILT, in rad.) is 3.84e-05
Tilt az(deg.)(θ11) is 64.32
K02 (DEFOCUS, in rad.) is -4.6e-05
Defocus(mm) is 0.12
K13(DECENTRING COMA,in rad.) is 1.40e-06
Secondary decenter(in mm) is 0.39
Coma az(deg.)(θ13) is -18.61
K04 (3rd ORDER SPHERICAL ABERRATION, in rad.) is -7.35e-07
Let us try to understand the value of each coefficient listed above with reference to
the situation that the experiment only involved a defocus.
• Defocus: As explained earlier the expected defocus is 0.14 mm, whereas a value
of 0.12 mm is obtained; this error of 20µm would deteriorate the image quality by
≈ 4µm for our f/4.5 camera lens, which corresponds to 0.09′′ in the telescope focal
plane.
• Tilt: The returned value of tilt corresponds to a shift of ≈ 5µm on the CCD, which
corresponds to 0.1′′ on the telescope focal plane.
We know from our previous testing of IFOSC that the focus adjustment of the
camera has some lateral movements associated with it, and this could be the source
of it.
• Decentering Coma: In the linear scale this error is 0.2 µm (0.013 pixel) corre-
sponding to 0.005 ′′ in the telescope focal plane, whereas the expected value is nil.
The value returned represents error in the method.
• Spherical aberration: In the linear scale this error is 0.1 µm (0.007 pixel) corre-
sponding to 0.002 ′′ in the telescope focal plane, whereas the expected value is nil.
The value returned represents error in the method.
A.3 Error estimate of the coefficients
To estimate the error level in the coefficients given in Eqn.8, and also the Knm
coefficients, we did the following. We added normally distributed random numbers to the
values of 4x and 4y (please refer Eqns.4 and 5); as our error in the measurement of the
centroids was 0.01 pixel, the rms value of the random numbers was chosen at 0.02 and
the values of the coefficients calculated. This procedure was repeated 30 times and the
Transfer pupil implementation in spectrograph 131
rms error on the coefficients was found from these 30 trials. The rms error on the Knm
coefficients are in the range 3− 5×10−7 radians; i.e. the errors in the CCD image plane
are ∼ 0.1 ′′ which corresponds to < 0.003 ′′ in the focal plane of the telescope.


An implementation of transfer-pupil in a spectrograph on optical telescopes

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An implementation of transfer-pupil in a spectrograph on optical telescopes

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