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Aberrational Delta Magnitude

An assessment of the optical aberrations of Mag30Cam

Christopher J. Burrows1 , Robert A. Brown2 , & Erin M. Sabatke3

1 MetaJiva 2 Space Telescope Science Institute 3 Ball Aerospace3700 San Martin DriveBaltimore, MD 21218

chrisümetajiva.com rbrownüstsci.edu esabatkeüball.com

Appendix C of 182

WFC ICS Final Report Page 123 of 341

Table of Contents

Abstract 3

1 Introduction 3

2 Sharpness 5

2.1 Definition of sharpness 5

2.2 Deconstruction of sharpness 7

2.3 Sampling 9

3 Practical application 11

3.1 Computational accuracy 11

3.2 Computing steps 11

3.3 Aberrations &wavefront errors 13

3.4 Definition of Zernike polynomials 14

3.4 References 14

4 Aberrational Analysis of Mag30Cam 15

4.1 delMagAbs vs. wavelength 16

4.2 delMagAbs vs. wavefront error HWFEL 17

4.3 Detail Reports 18

4.3 .1 Window detail report 18

4.3 .2 VisSlot 33

4.3 .3 NIRSlot 48

4.4 Grid Views HdelMagAbsL 63

4.4 .1 Window grid view HdelMagAbsL 63

4.4 .2 VisSlot grid view HdelMagAbsL 88

4.4 .3 NIRSlot grid view HdelMagAbsL 113

4.5 Grid Views HWFEL 123

4.5 .1 Window grid view HWFEL 123

4.5 .2 VisSlot grid view HWFEL 148

4.5 .3 NIRSlot grid view HWFEL 173

Appendix C of 182


!"#$$%&'()%*+,#*&%+-%.)'&/0#

An assess'ent of t+e o,t-.al aberrat-ons of Mag456a'

Christopher +. -urrows, Robert 3. -rown, 5 6rin M. Sabatke

!"1&$%2&

;e assess the aberrations in our design of Mag$%&a'(to determine if they are compatible with the science program

we envision for )*+,&-s wide-field camera. ;e introduce a new metricEdelMagAbs or Faberrational delta

magnitudeFEwhich cleanly separates the effects of sampling and aberrations on the signal-to-nosie ratio. ;e also

analyGe wavefront errors in the traditional way, for comparison with delMagAbs. ;e present results for the ;in-

dow, HisSlot, and IJRSlot.

3+4)&$(0/2&'()

The cosmic volume sampled by an astronomical camera is proportional to its beneficial field of view (-FNH),

where the images have acceptable quality. ;e have developed a metric for determining -FNH on the basis of the

signal-to-noise ratio (SNR) for point sources. Called Faberrational delta magnitudeF (delMagAbs), this metric is the

degradation in SNR(due to optical aberrations (!) in the background-limited regimeQ

delMagAbs ª delMagSa'7Abs R(delMagSa'7, (S)

where delMagSa'7Abs is the SNR degradation due to the combination of the aberrations and sampling with pixel

width D, and delMagSa'7 is due to sampling alone (! ª 0).

Nur formalism for computing the terms in 6q. S is based on the quantity called FsharpnessF (Y), for which we

provide an ab(898:8; derivation, which leads to delMagSa'7Abs<

;e have developed a practical procedure for computing the terms on the right side of 6q. S at a given wavelength

(l), for a given optical design specified by !(l), and D(l), which is the Fsampling criticalityF (k(l)) of the pixel array

in the detectorQ

k(l) ª DÅÅÅÅÅÅÅÅD0

, (V)

where D0 is the width of a critically sampling pixelQ

D0 ª lGÅÅÅÅÅÅÅÅÅV =

, (3)

where > is the focal length and = is the diameter of the pupil, which is assumed circular. The aberrations !(l) are

Aberrational Delta Magnitude Section 1: Introduction

3

Appendix C of 182


coefficients of Zernike polynomials, provided by the optical design model of the camera at regular field positions.

We can study the value and variation of delMagAbs over the FOV to determine if the aberrations are acceptable. Our

criterion for the BFOV is

delMagAbs d delMagSamp, (4)

which says that the degradation in SNR due to optical aberrations alone should not be greater than the degradation

accepted in the choice of pixel size. While this documents present our results compared with this standard, we have

not yet formed a final opinion as to whether the aberrations in the current camera design are scientifically acceptable

or not.

Our techniques are quite general. Nevertheless, we are motivated by a particular issue: whether NASA's current

design for the telescope of the coronagraphic Terrestrial Planet Finder mission (TPF-C) is compatible with a

wide-field camera that could obtain a survey of ~10 deg2 of the deep universe in parallel with its primary observing

program, searching for Earth-like planets on ~100 nearby stars. That searching program use a coronagraph on the

optical axis, with a small field of view (~5 arcsec), while the wide-field camera views a surrounding and adjacent

patch of sky. Because the coronagraph must sample many roll angles around a target star for completeness, an

eccentric FOV in the wide-field camera sweeps out an area of sky proportional to the square of the off-axis extent of

the BFOV. Assuming full coverage in roll, facilitated by an azimuthal extent of the BFOV of ~30°, the survey goal

of t10 deg2 for ~100 coronagraphic target stars would be obtained if the off-axis extent of the BFOV were t 10

arcmin. Therefore, we have used Eqs. 1 and 4 to test whether the combination of our best wide-field camera design

and the current TPF-C telescope design achieves the goals BFOV ~30 arcmin2 and off-axis extent ~10 arcmin.

Aberrational Delta Magnitude Section 1: Introduction

4

Appendix C of 182


2 Sharpness

ü 2.1 Definition of sharpness

Following Burrows (2003), we define the image of a point source on the detector to be an array of photoelectron

counts recorded in pixels:

Iij ª B + A Pij + Nij , (5)

where i and j are array indices extending from -¶ to ¶; A is the total count from the point source; Pij is the point-

spread function integrated over the pixel (i,j), centered at coordinates (xi = i D, yj = j D) on the focal plane, where D

is the pixel width; Nij is noise with zero mean, no pixel-to-pixel correlation, and known variance (sij2 ) due to

Poisson noise and read noise (R):

sij2 = B + A Pij + R2 ; (6)

and B is the background count per pixel,

B = D2! + D, (7)

where ! is the background flux on the focal plane, and D is the dark count per pixel.

To represent the signal, we choose the most general, linear, unbiased estimator:

S ª ij

Wij Iij – BÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅij

Wij Pij, (8)

where Wij are weights to be determined. S is unbiased because, by construction,

S = A. (9)

When optimized, the estimator S achieves the Cramer-Rao bound, which is the best that can be done—no non-linear

estimator will give better SNR.

The noise on the signal S is given by

N = S – S 2 = ij

Wij2 sij

2

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅij Wij Pij

2 , (10)

which we obtain by direct substitution, using the assumed uncorrelated nature of the noise for reduction.

To maximize

SNR ª SÅÅÅÅÅÅÅÅÅN , (11)

we must minimize N2

ÅÅÅÅÅÅÅÅÅÅÅÅS2 with respect to Wij . Differentiating with respect to Wij and setting the result to zero, we find

for the optimal weights in the general case:

Aberrational Delta Magnitude Section 2: Sharpness

5

Appendix C of 182


Wij ! PijÅÅÅÅÅÅÅÅsij

2 = PijÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

B+A Pij +R2 . (12)

If the photon noise in the source dominates over most of the image (i.e., A Pij p B + R2 ), then Wij Ø constant and

SNR Øè!!!!

A , as expected. If the background and read-noise dominate over all the image (i.e., A Pij ` B + R2 ), which

is the case in which we are interested,

Wij ! Pij . (13)

Without loss of generality, we can normalize so that Wij = Pij . In this case,

SNR2 = A2 YÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅB + R2 (14)

where the background-limited sharpness is defined as

Y ª ij

Pij2

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅIij

Pij M2 . (15)

From the definition of Pij , Y is a function of pixel size and wavefront aberrations, !. Therefore, we can show all the

dependencies of SNR on the optical design:

SNR2 = A2 YHD, !LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D2 ! + D + R2 (16)

We now explore the dependence of Y from D and !.


8

Appendix C of 182


ü 2.2 Deconstruction of Sharpness

Because the point-spread function is the Fourier transform (fourier) of the optical transfer function (!) (Goodman

2005), the pixel-integrated point-spread function is the transform of the convolution of ! and the transform of the

pixel function (pix):

P[x, y] = fourier[! [ fx , fy ] ! fourier[pix]], (17)

where ! indicates convolution, and fx and fy are spatial frequencies on the focal plane in the x- and y-directions, and

pix(x, y) = rect! xÅÅÅÅÅD

- px " rect! yÅÅÅÅÅD

- py ", (18)

where the (x, y) coordinate system is centered on the pixel, 0 ! px ! 1 and 0 ! py ! 1 are fractional pixel phases, and

the rectangle function is defined as

rect(s) ª 1 for #s# ! 0.5 and zero otherwise. (19)

Therefore,

fourier[pix] = D2 sinc!D fx " sinc!D fy " ‰-Â 2 p px fx ‰-Â 2 p py fy , (20)

where

sinc!s" ª sin$p s%ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

p s(21)

has unit integral, is zero for non-zero integer arguments, and is defined to be 1 at s = 0.

! is the normalized autorcorrelation of the amplitude transfer function (H):

! ª 16 D2

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅp

corr[H, H], (22)

where

H[ fx , fy ] ª circ[4D0&#####################

fx2 + fy

2 ] ‰Â f! fx , fy " , (23)

where

circ[s] ª 1 for s ! 1, and zero otherwise, (24)

and we parameterize the aberrations on the pupil using fx and fy :

f[ fx , fy ] ª 2 p !$2 D D0 fx , 2 D D0 fy %ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅl , (25)

where !(X, Y) is distribution of path-length errors, and X and Y are Cartesian coordinates on the pupil. The normal-

ization of Eq. 22 is chosen so that

![0, 0] =1. (26)


6

Appendix C of 182


From Eqs. 22, 23, and 24,

! ª 0 for » fx » > 1ÅÅÅÅÅÅÅÅÅÅÅ2 D0

or » fy » > 1ÅÅÅÅÅÅÅÅÅÅÅ2 D0

, (27)

which means that its Fourier transform, the point-spread function, as well as P, which is a convolution of the

point-spread function, are band-limited, with the same band limit. Because the Nyquist frequency of the pixels is1ÅÅÅÅÅÅÅÅ

2 D, they will critically sample the highest frequency in P if D ! D0 ,

The formalities for reconstructing Y(D, !, px , py ) from these elements are as follows. We obtain ! given ! from

Eqs. 22, 23, 24, and 25; we obtain P given D, px , and py from Eqs. 17 and 20; and then we obtain Y from Eq. 15,

using:

Pij [D, D, !, px , py ] ª P[D, D, !, px , py ; xi , yj ], (28)

where the two occurrences of the parameter D are needed to keep separate track mathematically of sampling effects

(first occurrence) and pixel-integration effects (second occurrence). When referring to the detector, the two occur-

rences must be equivalent.


7

Appendix C of 182


ü 2.3 Sampling

According to the sampling theorem, PKL ª PKL [D, D, !, px , py ] is completely determined by Pij ª Pij [D0 , D, !, 0, 0].

(The notation is now carrying an additional burden: capitalized indices refer to the grid of pixels with spacing D and

phase (px , py ) with respect to the point-spread function, and lower-case indices refer to the critically sampling grid

on the point-spread function, with spacing D0 and zero pixel phase.) We normalize and center Pij such that

ijPij =1 , (29)

and P00 is the maximum value. Then we can interpolate using sinc to obtain

PKL = ijPij sinc[(px + K) k – i] sinc[(py + L) k – j]. (30)

When k < 2 (at least; proof pending for k ! 2; results verified by numerical integration to 4% for k = 4),

k sinc[k q – s] = 1ÅÅÅÅq

. (31)

Then from Eqs. 29, 30, and 31,

KLPKL = 1ÅÅÅÅk

. (32)

From this expression we can compute the sharpness for any pixel phase:

Y( px , py ) = k4ijmnKL PijPmn

ä sinc[(px + K )k – i] sinc[(px + K) k – m]

ä sinc[(py + L) k – j] sinc[(py + L) k – n] (33)

To compute average sharpness over pixel phase, observe that

-¶

¶sincs - t sinct „ t = sinc[s], (34)

which means the average sharpness over pixel phase is simply

< Y > = 1ÅÅÅÅÅÅÅk2

0

1

0

1 Ypx , py „ px „ py = k2 ijPij

2 , (35)

where we have used k 0

1 Fp + k D „ p = -¶

¶Fx „ x and sinc[i – j] = dij for integer (i, j), where dij is the

Kronecker delta function.

Eq. 35 says that <Y > is the product of the square of the sampling criticality k2 and a sharpness-like quantity, which

we call the "sharpness factor" (Y f (k, !)). The sharpness factor involves the sums in Eq. 15, but the terms are from

the critically sampled, pixel-integrated point-spread function at zero pixel phase. (The sharpness and the sharpness

factor are identical for critical sampling.) Furthermore, from Eq. 16, the SNR associated with the average sharpness

is directly proportional to Y f k, ! , with no other dependences on optical design if the background noise domi-

nates the dark noise and read noise.

We can now compute the terms in Eq. 1:


9

Appendix C of 182


delMagSampAbs = – 2.5 log $%%%%%%%%%%%%%%%Y f Hk, !LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅY f H1, 0L (36)

and

delMagSamp = – 2.5 log $%%%%%%%%%%%%%%%Y f Hk, 0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅY f H1, 0L . (37)


10

Appendix C of 182


3 Practical Application

ü 3.1 Computational accuracy

We use discrete Fourier transforms of arrays with dimension M ä M , where M is a power of two. We choose M/2

samples across the aperture for the array representing H, which ensures critical sampling. The accuracy of the results

varies with M, which we can explore in the absence of aberrations, because we can compute ! analytically in that

case (Goodman p. 145) and compare results. We find:

Y f (1, 0)»analytic = 0.08001 (38)

and

DY fÅÅÅÅÅÅÅÅÅÅÅY f

ª X » Y f H1,0L »sampled - Y f H1,0L »analyticÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅY0 H1,0L »analytic

» \ > 1ÅÅÅÅÅÅÅÅÅ400

, 1ÅÅÅÅÅÅÅÅÅÅÅÅ1000

, and 5ÅÅÅÅÅÅÅÅÅÅÅÅ8000

, (39)

for M = 128, 256, 512, respectively.

ü 3.2 Computing steps

Here we describe the computational steps for computing Y f (k, !), from which we can compute delMagAbs, using

Eqs. 1, 36, and 37.

Step 1. Compute the array representing the amplitude transfer function:

atfArray(iX, jY) = ‰Â 2 p !HriX,jY ,jiX,jY L for riX,jY ! 1 and 0 otherwise. (40)

Step 2. Compute the autocorrelation of atfArray to obtain the array representing the optical transfer function:

otfArray(iFX, jFY) = ‚

iX, jYatfArrayHiX,jYL atfArrayHiX+iFX–1,jY+iFY-1L

êêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêêê

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅiX,jY H»atfArrayHiX,jYL»L2 , (41)

where the indices are cyclic over the range 1 to M.

Step 3. Multiply otfArray by the (unnormalized) Fourier transform of a pixel to obtain the array representing the

(unnormalized) Fourier transform of the pixel-integrated, critically sampled point-spread function:

fpIcSpSFArray(iFX, jFY) = otfArray(iFX, jFY) sinc[ HiFX-1L kÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅM

] sinc[HjFY-1L kÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

M]. (42)

Step 4. Compute the discrete Fourier transform of fpIcSpSFArray, then take the absolute value, to obtain the array

representing the (unnormalized) pixel-integrated, critically sampled point-spread function:

pIcSpSFArray(iX, jY) = 1ÅÅÅÅÅÅM

»iFX,jFY pIcSpSFArray(iFX, jFY) ‰Â 2 p HiFX-1L HiX-1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

M ‰Â 2 p HiFY-1L HiY-1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

M ». (43)

Step 5. Compute the sharpness-type summations on pIcSpSFArray to obtain an estimate of the sharpness factor:

Aberrational Delta Magnitude Section 3: Practical Application

11

Appendix C of 182


Y f Hk, !L = iX,jY

pIcSpSFArray HiX, jYL2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅIiX,jY

pIcSpSFArray HiX, jYLM2, (44)

(which renderes the normalization of Eq. 43 moot).

Numerically, we find:

Y f (1, 0) = 0.08001, (45)

which is a universal constant for an unobscured, circular aperture. Figure 1 shows delMagSamp for a range of k.

Figure 1. delMagAbs versus sampling criticality. delMagAbs is the loss in SNR due to sampling alone, computed for

zero aberrations.


12

Appendix C of 182


ü 3.3 Aberrations & wavefront errors

For the circular pupil we describe the aberrations (path-length errors) by coefficients (ak ) of Zernike standard

polynomials (Zk ):

!(r, j) > k ak Zk r, j, (46)

Where " is the fractional radius on the aperture, and j is the azimuthal angle. Table 1 shows our normalization of

the first 37 Zernike polynomials normalized to produce the total RMS wavefront error (s):

s = k ak2 . (47)

However, before applying Eq. 41 to the Zernike coefficients received from the design process, we make three

corrections, as follows. We set the piston term (k = 1) to zero; it is pure phase, with no effect on image quality. The

tip and tilt terms (k = 2 & 3) move the image but have no effect on its photometric quality for monochromatic light.

Nevertheless, tip and tilt can be removed at only one wavelength, and a lateral chromatic effect will remain. (The

focus term (k = 4) could be zeroed at one wavelength over all the bandpasses, if we can arbitrarily warp the focal

plane. However, we have chosen not to include this option in our current analysis.)

The corrections for tip and tilt in the computation of RMS wavefront error are determined by introducing two free

parameters (c2 , c3 ) such that the new wavefront is described by

!(l; r, j) > a2 l - c2ÅÅÅÅÅÅÅl Z2 r, j + a3 l - c3ÅÅÅÅÅÅÅ

l Z3 r, j + k>3 ak Zk r, j, (48)

where the wavelength dependence is explicit, and c2 and c3 are divided by the wavelength because the optical path

is expressed in waves. We must optimize c2 and c3 over a bandpass represented by weights (Wi ) for each docu-

mented wavelength within the bandpass. The parameters are determined by minimizing the quantities

i

Wi a2 li - c2ÅÅÅÅÅÅÅli2 and

iWi a3 li - c3ÅÅÅÅÅÅÅ

li2 . (49)

which means

c2 = i

Wi a2 li ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

i

WiÅÅÅÅÅÅÅÅÅli

and c3 = i

Wi a3 li ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

i

WiÅÅÅÅÅÅÅÅÅli

. (50)

Properly, the weights Wi should be proportional to the signals—the product of throughput and spectral energy

distribution—at each wavelength li . For the current exercise on Mag30Cam, where we have Zernike coefficients

for only five wavelengths per filter, and where we have no detail on typical signals versus wavelength, assuming

Wiª 1 is a good-enough approximation.

Figures 3a and 3b show scatter plots of the corrected RMS wavefront errors versus delMagAbs for the 625 points in

the Window and VisSlot Zernike coefficients. For Window aberrations of 12/06/05, the median wavefront error is

0.2 waves RMS, and the median delMagAbs is 0.7.

The value of delMagAbs that corresponds to 'diffraction limited' or 1/14 waves RMS is

delMagAbsdiffraction limited > 0.18.


13

Appendix C of 182


ü 3.4 Definition of Zernike Polynomials

1 1

2 2 r Cos@jD3 2 r Sin@jD

4è!!!!3 H-1 + 2 r2L

5è!!!!6 r2 Cos@2 jD

6è!!!!6 r2 Sin@2 jD

7 2è!!!!2 H-2 r + 3 r3 L Cos@jD

8 2è!!!!2 H-2 r + 3 r3 L Sin@jD

9 2è!!!!2 r3 Cos@3 jD

10 2è!!!!2 r3 Sin@3 jD

11è!!!!5 H1 - 6 r2 + 6 r4 L

12è!!!!!!10 H-3 r2 + 4 r4 L Cos@2 jD

13è!!!!!!10 H-3 r2 + 4 r4 L Sin@2 jD

14è!!!!!!10 r4 Cos@4 jD

15è!!!!!!10 r4 Sin@4 jD

16 2è!!!!3 H3 r - 12 r3 + 10 r5 L Cos@jD

17 2è!!!!3 H3 r - 12 r3 + 10 r5 L Sin@jD

18 2è!!!!3 H-4 r3 + 5 r5L Cos@3 jD

19 2è!!!!3 H-4 r3 + 5 r5L Sin@3 jD

20 2è!!!!3 r5 Cos@5 jD

21 2è!!!!3 r5 Sin@5 jD

22è!!!!7 H-1 + 12 r2 - 30 r4 + 20 r6 L

23è!!!!!!14 H6 r2 - 20 r4 + 15 r6 L Cos@2 jD

24è!!!!!!14 H6 r2 - 20 r4 + 15 r6 L Sin@2 jD

25è!!!!!!14 H-5 r4 + 6 r6 L Cos@4 jD

26è!!!!!!14 H-5 r4 + 6 r6 L Sin@4 jD

27è!!!!!!14 r6 Cos@6 jD

28è!!!!!!14 r6 Sin@6 jD

29 4 H-4 r + 30 r3 - 60 r5 + 35 r7 L Cos@jD

30 4 H-4 r + 30 r3 - 60 r5 + 35 r7 L Sin@jD

31 4 H10 r3 - 30 r5 + 21 r7 L Cos@3 jD

32 4 H10 r3 - 30 r5 + 21 r7 L Sin@3 jD

33 4 H-6 r5 + 7 r7 L Cos@5 jD

34 4 H-6 r5 + 7 r7 L Sin@5 jD

35 4 r7 Cos@7 jD

36 4 r7 Sin@7 jD

37 3 H1 - 20 r2 + 90 r4 - 140 r6 + 70 r8L

ü 3.5 References

Burrows, C. J. 2003, in The Design and Construction of Large Optical Telescopes, ed. P. Y. Bely (New York:

Springer), 433

Goodman, J. W., Introduction to Fourier Optics. Roberts & Company, Englewood, Colorado, 2005. 138


14

Appendix C of 182


4 Aberrational analysis of Mag30Cam

This document reports our analysis of the SNR performances of Window, VisSlot, and NIRSlot using the delMag-

Abs and delMagSamp metrics, using coeffients of Zernike polynomials supplied by the Mag30Cam optical designer,

Erin Sabatke of Ball Aerospace.

Table 1 summarizes the FOVs of Mag30Cam in the current optical design. The Slot FOVs are slightly different than

our focal-plane detector layout.

Table 1. FOVs analyzed for delMagAbs

Window

Sampling criticality k 2

Wavelength range l 400 900 nm

Filters 5

Wavelengths per filter 5

Field positions 25

Zernike sets 625

VisSlot



Filters 5


Field positions 25

Zernike sets 625

NIRSlot



Filters 5


Field positions 25

Zernike sets 625

Table 1. FOVs analyzed for delMagAbs.

Aberrational Delta Magnitude Section 4: Aberrational Analysis of Mag30Cam

15

Appendix C of 182


ü 4.1 delMagAbs vs. wavelength

Figures 2. delMagAbs vs. wavelength. Each black dot (625 in all) represents a unique combination of field position,

filter, and wavelength where delMagAbs was computed. The 25 dots at one value of wavelength are the 5x5 field

positions per filter, shown in the "grid views." The groups of five connected dots are the wavelength variation at one

field position. The blue curve is delMagSamp. The red points (and red type) are described in the "detail reports."


16

Appendix C of 182


ü 4.2 delMagAbs vs. wavefront error

!"#$%&' 3. +,-..&% /01. 12 ,1%%&,.&3 45+ 6-7&2%18. &%%1% -83 delMagAbs 21% -00 '&.' 12 9&%"8:& ,1&22","&8.'. ;101%

"83",-.&' 6-7&0&8#.<, 2%1> 400 8> (B0$&) .1 900 8> (%&3). E<& .<&1%&.",-0 ,$%7& '<16' .<& &F/&,.&3 3&#%-3-."18 "8

SNR 3$& .1 %&3$,."18 "8 .<& +.%&<0 %-."1 21% '>-00, <"#<G2%&H$&8,I -B&%%-."18', 6<",< %&>17& 0"#<. 2%1> .<& /1"8.G

'/%&-3 2$8,."18 6".<1$. -22&,."8# ".' '<-/&. E<& ',-..&% /1"8.' 2-00 B&016 .<"' 0"8& B&,-$'& 12 .<& /%&7-0&8,& 12

016G2%&H$&8,I -B&%%-."18', 6<",< %&>17& 0&'' 0"#<. 2%1> .<& /"F&0' .<-. ,1$8. 21% ,1>/$."8# .<& '"#8-0.


17

Appendix C of 182


Window '()*+', -.id /oint1 = ,2

W34*l*n6t7 in 8i9.on1 = ':2;

S38/lin6 9.iti93lit= 3t w34*l*n6t7 = >:'2'?>

)ilt*. n@8+*. = >:

A Bi*ld /o1ition = ':'>CD('>

E Bi*ld /o1ition = ':'CCCCCC

S38/lin6 9.iti93lit= 3t D'' n8 = >

F@31iS73./n*11 HGS) i1 not /iH*l-int*6.3t*dL I+*..3tion1 = ':,2,?'2

F@31iS73./n*11 HGS) i1 not /iH*l-int*6.3t*dL JK I+*..3tion1 = ':,D2,'?

S73./n*11 HLnt*6.3t*d GS) o4*. /iH*lL I+*..3tion1 = ':CM'C?M

S73./n*11 HLnt*6.3t*d GS) o4*. /iH*lL JK I+*..3tion1 = ':CM2,,(

S73./n*11)INOKP HLnt*6.3t*d GS) o4*. /iH*lL I+*..3tion1 = ':'D(C2?

S73./n*11)INOKP HLnt*6.3t*d GS) o4*. /iH*lL JK I+*..3tion1 = ':'D(???>

G731*-34*.36*d 173./n*11 I+*..3tion1 = ':>C??DM

G731*-34*.36*d 173./n*11 JK 3+*..3tion1 = ':>2MM

Q*lt3 836nit@d* d@* to 138/lin6 3lon* H@1in6 ,2H,2 6.idL = ':M(D,C>

SJP d*6.3d3tion in 836nit@d*1 B.o8 138/lin6R JK 3+*..3tion1 = ':M(D,;C

SJP d*6.3d3tion in 836nit@d*1 d@* to 3+*..3tion1 3lon* = ':''D'???2

SJP d*6.3d3tion in 836nit@d*1 B.o8 138/lin6 S 3+*..3tion1 = ':M?'(?>

+M: +':''DD'M( +M2: -':'''>M; +>(: -':'''>(D;

+>: +':''C>D'C +MD: +':''';C>2 +>?: -':'''MC(>

+C: +':''>D('( +M,: +':'''CC'D +>;: +':''''2?M

+2: +':''M?2;C +M(: -':''''M(( +C': +':'''M,D

+D: +':''D?;,; +M?: +':'''>?>( +CM: +':'''M'?(

+,: +':''M?,,? +M;: +':''''?2; +C>: +':''''?2D

+(: +':'''('D? +>': +':''''2, +CC: +':'''M>(

+?: +':''M>D>2 +>M: +':'''>,(C +C2: +':'''''(M

+;: +':''2MM2, +>>: -':'''M,,D +CD: +':'''M>'2

+M': +':''M(2(; +>C: -':''''MD2 +C,: -':''''>2(

+MM: -':''M>'?D +>2: -':'''>D'? +C(: -':''''?DC

+M>: -':''MCM>; +>D: -':'''MD'M

+MC: +':''MM(DC +>,: -':'''>M2

Aberrational Delta Magnitude Detail Reports: Window Section 4: Aberrational Analysis of Mag30Cam

18

Appendix C of 182


0 10 20 30 40 50 60

0

10

20

30

40

50

60

Aberrated PSF

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Unaberrated PSF

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Aberrated - Unaberrated OTF

0 0.2 0.4 0.6 0.8 1Normalized Spatial Frequency

0

0.2

0.4

0.6

0.8

1

FT

M

Modulation Transfer Function

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Aberrated OTF

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Unaberrated OTF


19

Appendix C of 182


0 10 20 30 40 50 60

0

10

20

30

40

50

60

Pixelated Aberrated PSF

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Pixelated Unaberrated PSF

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Pixelated Aberrated OTF

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Pixelated Unaberrated OTF

0 2 4 6 8 10

0

0.02

0.04

0.06

0.08

0.1Aberrated PSF

0 2 4 6 8 10

0

0.02

0.04

0.06

0.08

0.1Unaberrated PSF


20

Appendix C of 182


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