DESIGN STUDY FOR TESTING PRIMARY MIRROR SEGMENTS … · 2. The CGH is imaged onto the test surface by the projection lens 3. Two CGH diffraction orders, 0th and 1st, are selected

$Page 1: DESIGN STUDY FOR TESTING PRIMARY MIRROR SEGMENTS … · 2. The CGH is imaged onto the test surface by the projection lens 3. Two CGH diffraction orders, 0th and 1st, are selected$
DESIGN STUDY FOR TESTING PRIMARY MIRROR SEGMENTS FROM A 30-M GSMT USING A TEST PLATE

WITH COMPUTER GENERATED HOLOGRAM (CGH)

Presented to:

Larry Stepp Eric Hansen

The Association of Universities for Research in Astronomy, Inc. Tucson, AZ, 85726

Prepared By

Feenix Y. Pan Jim H. Burge

Optical Science Center University of Arizona

Tucson, AZ 85747

06/2002

Design Study for measuring GSMT primary mirror using a test plate with computer generated hologram FYP 062002 1

TABLE OF CONTENTS

Executive summary ............................................................................................................. 3 1.0 Theory of Operation................................................................................................ 5 2.0 System Optimization............................................................................................... 7

2.1 Optimization of test plate ROCs ............................................................................. 7 2.2 Optimizing System magnification M.................................................................... 10

3.0 Complete design of the optical system ................................................................. 13 3.1 Definition of GSMT primary mirror..................................................................... 14 3.2 Optical design procedure ...................................................................................... 16 3.3 Design of the projection system............................................................................ 16 3.4 Design of the imaging system............................................................................... 20 3.5 Specification of the test plate................................................................................ 22

4.0 Hologram Specification ........................................................................................ 24 4.1 Location and Layout of six representative segments............................................ 24 4.2 Balancing intensities for good fringe contrast ...................................................... 25 4.3 Test of reference sphere (segment #0) .................................................................. 26 4.4 CGH designs for the five remaining segments ..................................................... 27

5.0 Error Analysis on the most difficult segment ....................................................... 45 5.1 Error analysis for figure measurement.................................................................. 46 5.2 Error analysis for defining the optical surface relative to the parent mirror......... 55

Conclusion......................................................................................................................... 58 REFERENCE.................................................................................................................... 59


EXECUTIVE SUMMARY

We present the design and analysis for measuring the GSMT primary mirror segments using a test plate with computer generated holograms (CGHs). This test is optimal for this type of mirror because it achieves high accuracy for measuring figure errors in the different segments, while it is economical and easy to perform. The basic design of the test for the GSMT uses a 1.36-m test plate held 5 millimeters from the aspheric segment being measured. The test plate has a convex radius of curvature, which is easy to manufacture and measure, and it matches the concave 60-m curvature of the segments. The aspheric departure for each segment is accommodated using a 35.5-mm CGH placed at an image of the test plate. Using phase shift interferometry, this test will achieve 4.8-nm surface accuracy for the mirror segments. The basic parameters for the test are shown in Table 1.

Table 1. Test plate with CGH for GSMT segments Primary mirror segments

1.33-m point-point hexagonal segments from 30-m paraboloid with 60 m radius of curvature

Test plate 1.36-m diameter bi-convex lens made from Zerodur Both surfaces spherical 10 cm center thickness, 6.7 cm edge thickness

CGHs 34 mm diameter patterns written in 4 x 4 array on 150 mm fused silica substrates 15 µm nominal spacing, 2.4° diffraction angle

Test calibration A 1.4-m concave spherical reference mirror provides complete calibration of the test, as well as a verification of the CGH performance.

Projection optics 2-element system Images CGH to test plate with magnification 1/40

Imaging system 2-element optical system standard 2/3-in format CCD

An important aspect of this test is the ability to control the global parameters of each segment, in addition to the higher order surface errors. The radius of curvature and the absolute alignment of the aspheric optical surface to the mechanical segment are maintained by controlling the gap between the test plate and the segment and the mapping from the CGH to the segment. A set of alignment fiducials is provided with each CGH to provide an easy reference for aligning the test.


A complete error analysis given in this report predicts accuracy for the outer segments (the most severe) as shown in Table 2.

Table 2. Accuracy of CGH test for most severe of GSMT segments* Parameter Expected accuracy Equivalent RMS

Wavefront Error [nm]

Surface irregularity 4.8 nm 9.6nm

Relative Radius of curvature Matching

± 0.39 mm (± 24.1 nm sag)

13.9 nm

Segment radial position ± 0.106 mm 2.9 nm

Orientation of segment ± 0.006 deg (0.105 mrad)

10.86 nm

*The error for the average segment will be roughly half as large The design presented minimizes the cost and risks for implementing the test for the GSMT segments. The viewing distance of 12 meters is set as the shortest that uses a spherical surface for the back (illumination) side of the test plate. The hologram size of 35 mm is small enough that 16 holograms will easily fit on a single 150 mm substrate. So the total cost for custom holograms to provide for null tests for all 103 different segments will be less than $50,000.


1.0 THEORY OF OPERATION The proposed test is a null test. In this test, the concave segment is compared to the convex spherical reference surface of the test plate. CGHs are used to compensate for any aspherical departure of the segment from the spherical reference surface. The test plate reference surface is chosen to be spherical, since compared to aspherical surfaces, spherical surfaces are more cost effectively manufactured to high accuracy. Figure 1-1 illustrates how a CGH combined with a test plate is used to measure off-axis aspherical segments:

1. A Laser beam is first expanded to uniformly illuminate the CGH. 2. The CGH is imaged onto the test surface by the projection lens 3. Two CGH diffraction orders, 0th and 1st, are selected by placing the object stop

at the focal plane of the projection lens. 4. The reference beam has a spherical wavefront. It originates from the 0th

diffraction order of the CGH, reflects off the reference side of the test plate, and then reaches the charge-coupled device (CCD).

5. The test beam originates from the 1st order of the CGH. It has a pre-distorted wavefront that matches the shape of the aspherical mirror segment under test. After reflecting off the test surface, it too reaches the CCD.

6. When the aspherical test surface perfectly matches the wavefront prescribed by the CGH, a null fringe is observed on the CCD. When this is not the case, fringes result. For added measurement accuracy, phase shifting interferometry (PSI) should be implemented by translating the segment under test via three piezo-electric transducers (PZTs).

7. The image lens is chosen so that the test surface is imaged onto the CCD. 8. The Image stop is placed at the front focal plane of the image lens to select

appropriate orders (Figures 1-2 and 1-3).

TESTPLATE

ASPHERESURFACE

REF SURFACE

PROJECTIONLENS

COLLIMATORCGH

LASER

Object STOP

1ST ORDER (TO TEST SURF)

CCD

IMAG LENS

STOP REF AND TEST BEAMS COINSIDE

ref. beam before Test Platetest beam before Test Plateref. & test beams after Test Plate

0TH ORDER(TO REF SURF)

Figure 1-1: Layout of CGH test


Order fromCGH Reflected from Final destination

Test plate Reference beamZero order

Segment Blocked at theimage stop

Segment Test beamFirst order

Test plate Blocked at image stop

All otherorders ----------- Blocked at the stop

following the projectionlens

Figure 1-2: Order selection of the CGH test

2 spots from1st order

At the imagestop

two spotsare lined up

2 spots from0th order

Figure 1-3: Order selection at the image stop

This test is ideal for testing large quantities of off-axis aspherical segments. It produces excellent relative radii of curvatures (ROC) matching since, during the test, all concave aspheric segments are compared to the same convex spherical reference surface. It is also efficient since a single test setup can be optimized to accommodate measurement of all segments by simply replacing the CGH. In addition, this method is cost effective: 1) it ensures that both reference and test beams coincide at the CGH, so that the CGH can be written on a standard lithography substrate, and 2) it allows the test plate to be made from a non-precision transmission grade glass, like Zerodur. Finally, by employing PSI and utilizing its inherent near-common path configuration, this test achieves a high degree of measurement accuracy. Accurate axis location is achieved by implementing alignment marks on the CGH directly.


2.0 SYSTEM OPTIMIZATION

From the system point of view, there are three independent variables, which drive the system optimization. These variables are: two radii of curvatures (ROCs) for each of the two test plate surfaces and the system magnification factor M. The system magnification factor M is defined as the size of the test segment over the size of the CGH.

2.1 Optimization of test plate ROCs The test plate is the single most critical and most expensive optical element of this test. One side of the test plate serves as the reference surface while the other side controls the system illumination F/#. The size of the test plate has to match that of the segment under test, thus making it the largest and most expensive optical component in the test. Depending on the size, the thickness is chosen such that the test plate can be mechanically supported by actuators at its edges to maintain required surface slopes. Both the reference-side ROC and the illumination-side ROC require careful optimization to keep the test cost effective.

The Reference-side ROC optimization The optimal ROC for the test plate reference-side is the value that gives minimum amount of focus in the holograms. Minimizing the amount of focus in the hologram reduces the average hologram ruling frequency, and in turn minimizes the sensitivity to errors. To maintain a common system for testing all segments, the tilt in each segment, relative to the test plate, is held as a constant. Two main concepts underlie the selection of the optimal ROC value. The first one is to maintain minimum amount of tilt in CGH so that two diffraction orders (1st and 2nd orders) are separated. Figure 2-1 shows that to ensure no overlapping of the 1st and 2nd orders, the minimum amount of tilt is three times the maximum wavefront slope of the aberrated wave.1 This condition needs to be met whenever a CGH is used as an alternative to null optics in testing aspherics interferometrically 2.

0 +1 +2

diffracted orders-1-2

Figure 2-1: Minimum tilt needed for order separation is 3ε Design Study for measuring GSMT primary mirror using a test plate with computer generated hologram FYP 062002 7

The second key design concept is to select an optimal global ROC. Each segment has a unique local ROC value that minimizes its slope error ε (ε is measured in waves per radius). This optimal local ROC value changes as segments move toward the edge of the parent mirror (as shown in Figure 2-2, S14 is the furthest off-axis segment and s1 is the closest). The optimal local ROCs change because, as the segments become more off-axis, they become more aspherical in shape. A set of optimal local ROCs defines the bend in the curve (“elbow points”) – where minimum slope errors for each segment are located. The optimal global ROC, however, is not any of the local optimal ROCs. Instead, The optimal global ROC is where the slope error of the farthest off-axis segment is matched to that of the nearest off-axis segment. The dotted line in Figure 2-2 illustrates the location of the optimal global ROC value. Here, the slope error of the farthest segment (S14) is matched to the slope error of the closest segment (s1).

Test Plate ROC [m]

Slo

pe E

rror [

mra

d]

local optimalROC for s14

local optimalROC for s1

globaloptimal ROC

Operate system here

Figure 2-2: Location the optimal operational point for the system


Illumination-side ROC Optimization The ROC of the illumination-side controls the viewing distance, and affects the system illumination F/#. The viewing distant is the distance from the object stop to the test plate. System illumination F/# is defined as the ratio of viewing distance over the diameter of the segment. A small ROC value shortens the viewing distance at the expense of adding significant spherical aberration (SA). If not well controlled, SA significantly degrades the system performance. Spherical aberration can be eliminated by aspherizing. However this modification drives up the fabrication cost of the test plate (Figure 2-3).

Cost and Feasability Comparison for using Sphere and Asphere

100%110%120%130%140%150%160%170%180%190%200%

0.05.010.015.020.025.030.035.040.045.0

test setup length [m]

% In

crea

se in

Cos

t

sphereasphere

0.00%

0.30%

0.02%

0.00%

0.00%0.00%

0.00%0.00%

0.01%

plano-convex bi-convex

not possible

30.38%6.96%2.13%0.12% 0.23% 0.30% 0.77%0.04% 15.82%

Figure 2-3: SA blurs the spot diameter but aspherizing the test plate increases the cost

Spherical Aberration (SA) blurred the spot diameter by 30.38%

To reduce the fabrication cost of the test plate, it is desirable to determine the amount of SA the system can tolerate without significantly degrading its performance. Since SA is inversely proportional to the cubic power of F/#, a system with a faster F/# needs to have more asphericity added to its illumination surface to counter-balance SA. It is known that SA blurs the spot size of the 0th and 1st orders at the image stop, making it impossible to separate the two orders, as shown in Figures 2-4a and 2-4b. The maximum blur the system can tolerate is a 33% increasing in spot size compared to the ideal case of no SA.

stop0 +2+1-1-2

Design Study for measuring GSMT primary mirror using a test plate with computer generated hologram FYP 062002 9 Figure 2-4a: Order selection at the object stop

stopboth orders are reflected

from the ref surface

both orders are reflectedfrom the segment

Max blur can betolerated by the

zeroth order:/3 or 33%

stop

both orders are reflectedfrom the ref surface

Figure 2-4b: Order selection at the image stop (L) and SA blurs the spot diameter (R)

In another report3, we established a cost-effective way to reduce the surface slope requirement for the non-reference side, thus reducing the cost of the test. Here we simply state the result from that previous study. By redirecting both entering and returning beams off the test plate co-axial, the illumination-side surface slope can be on the order of 1.6 wv/cm, yet still yield no more than 0.002 wv wavefront error. 2.2 Optimizing System magnification M System magnification M is defined as the ratio of segment size over CGH size,

segment diameterM

CGH diameter.

Another equivalent expression for M is,

fringe spacing at the test surfaceM=s CGH line spacingΛ ≡ .

This is because CGH and test surface are conjugate pairs, i.e., every point on the CGH is imaged onto the test surface via the projection system. Fringe spacing at the test surface,


Λsyε, is calculated from results obtained earlier in finding optimal global slope error

since stem

systemsin( )λε

Λ = .

This alternate expression allows us to find the optimal system magnification M by minimizing the CGH line spacing. This approach minimized the size of CGH. Traditionally, in testing a single optical element using CGH, CGH size is not minimized to its theoretical limit. This is because, to make a hologram of 30mm or of 70mm, the associated cost is still one substrate on which the CGH is written. In testing multiple segments using multiple CGHs, keeping the CGH size at a minimum allows more CGHs to be written on the same substrate, thus reducing the total cost. For any given pre-determined wavefront (WF) test accuracy, minimum CGH line spacing (LS) can be found by,

0.1252 _ [

mLSWF accuracy

λ µλ

= × ]

(0.125-micron line pattern accuracy is assumed).

In general, current e-beam technology can maintain line pattern accuracy of 1/8 of a micron without a significant increase in cost.

In summary, optimization of system magnification factor M is equivalent to optimizing the CGH line spacing. Wider CGH line spacing increases the test accuracy but also increases the cost of the test. Figures 2-5 and 2-6 details this point for an f/1 30-m primary with 103 pieces of CGH needed to test all optically unique segments.


CGH Line spacing v.s. COST

$10

$60

$110

$160

$210

$260

$310

110100

CGH Line Spacing [um]

Cos

t of 1

03 C

GH

s ($

K)

CGH totalcost

Figure 2-5: Tighter CGH line spacing reduces the cost

CGH Line spacing v.s. Test Accuracy

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140110100

CGH Line Spacing [um]

Wav

efro

nt E

rror

[wv]

measr.accuracy

c

Figure 2-6: Wider CGH line spacing increases accuracy


3.0 COMPLETE DESIGN OF THE OPTICAL SYSTEM A detailed optical design was developed as a baseline to study the performance of this test for the GSTM primary mirror. The system performance was simulated and optimized for six of the GSTM segments. The detailed holograms were designed and are reported in Section 4.

The optical system is comprised of several subsystems or components that were designed and optimized independently before being integrated into a system model. The principal subsystems are summarized below:

Table 3-1: Optical System Layout Component Function Brief description

Computer generated

holograms

• Provide wavefront correction for aspheric departure of mirror segments

• Use sufficient tilt to allow isolation of diffraction orders 0 and 1

• Different CGH’s for different segments

• 33.5-mm diameter, chrome patterns on 3-mm fused silica substrates.

• Nominally 20 µm line spacing

• 50% duty cycle (line width = line gaps = half of period).

Projection optics

• Project an image of the CGH onto test segment

• Pass only diffraction orders 0 and 1

• Same system and alignment for all tests

• 0.6-m long, with 2 BK7 elements

• Lens diameter -- 36 cm for 1 lens, 46 cm for 2nd.

• 300mm EFL, magnifies CGH40.0 times.

• Fully corrected for 33.5 mm CGH over 13-mm pupil.

• Telecentric at CGH

Test plate • Provide reflected wavefront from convex surface as a reference.

• Transmit test wavefront

• Minimize viewing distance

• ROC (ref side) = -6.095539000001E+004mm

• RCO (non-ref side)= 7.7918681137197E+003mm

• Made of Zerodur

• 1.340 meter diameter

• 10 cm center thickness

• weighs ~ 1000 kg

• Located 12.0 m from projection and imaging systems

Imaging optics

• Create a good image of the test part onto a CCD array.

• Pass only the reference and test beams – block others

• 58 mm long with 2 BK7 elements, 20 and 12 mm diameter.

• Creates 4.8-mm image of 1880-mm part. This fits standard 2/3” format CCD’s.

• Accommodates 6-mm stop, allowing system slope errors of ±0.2 mrad.

• Allows 1000 resolution elements across interferogram.

• Telecentric at CCD

• Distortion of image 0.1% or < 1 mm at segment.


3.1 Definition of GSMT primary mirror The primary mirror shape is based on parameters provided by E. Hansen of NOAO. The parent optical surface (Figure 3-1) is paraboloid of revolution specified by vertex radius of curvature (ROC) and conic constant K ROC= 60.0m K= -1.000 The individual segments are defined by their off-axis distance and by the size and orientation of the hexagonal part. The nominal segment dimensions are 1.153 meters across flats, 1.33 meters point to point, and 50 mm thick. A 3% deviation in size varies from the center of the aperture to the edge. The gap between segments is nominally 3 mm. The bevels on the edges are nominally 1 mm wide. [Ref: 4_51.doc from E. Hansen]. The test is designed to go outside of the hexagonal clear aperture by 5mm in every direction. In addition, to simplify the optical design, all tests are designed over a circular region 1340mm in diameter. The final CGH design will be stopped down to the correct hexagonal shape. The definitions of the segments that were used for the detailed optical designs are given below. The point design has 618 segments and the total projected area of the 618 segments is 711 square meters. Since the hexagonal pattern of the segments has six-fold symmetry, the total number of optically unique segments is equal to the total number of segments divided by six, which equals 103 pieces (Figures 3-1 and 3-2). Out of those 103 pieces, six segments (#0, #1, #4, #8, #10, and #13), shown in Table 3-2 and 3-3, were chosen for this study (#13 has the furthest off-axis distance and segment #0 is used for calibrating the system as explained in section 4.3). To properly model the testing of the six segments, off-axis location, sag, and tilt (with respect to the optical axis) need to be calculated. Table 3-3 compares the results from both IDL programming and from SolidWorks Drawing (obtained from R. Robles of NOAO).


0

analyzed segments

13

Figure 3-2: 103 pieces ofoptically unique segments

Figure 3-1: Primary mirror

(OD=30.0 meters; f/1;k=-1.000 )

SEGMENT # Y= X= ANGLE i, off_axis distance[m] sag is[mm] slope is[deg] ,0 0 0 0 0 0 01 1155.070937 11.11824058 1.10280755 1 1155.0709 11.11824 1.10287522 2310.141716 44.47295625 2.20479859 2 2310.1418 44.47296 2.20493373 3465.212182 100.0641289 3.30516022 3 3465.2127 100.06416 3.30536264 4620.28218 177.8917285 4.40308677 4 4620.2836 177.89184 4.40335595 5775.351559 277.9557136 5.49778325 5 5775.3545 277.956 5.49811856 6930.420174 400.2560316 6.58846876 6 6930.4254 400.25664 6.58886957 8085.487883 544.7926193 7.67437974 7 8085.4963 544.79375 7.6748458 9240.554553 711.5654038 8.75477299 8 9240.5672 711.56735 8.75530189 10395.62006 900.5743031 9.82892852 9 10395.638 900.57743 9.829519810 11550.68427 1111.819227 10.89615214 10 11550.709 1111.824 10.89680511 12705.7471 1345.300077 11.95577782 11 12705.78 1345.307 11.9564912 13860.80842 1601.01675 13.00716973 12 13860.851 1601.0265 13.007941

*13*added 15015.86815 1878.969136 14.04972403 13 15015.922 1878.9825 14.050552

From FYP's IDL programRICK's data

Table 3-2: Location of the segments under study

0

Table 3-3: Percentage error of locating the segment using two methods

off_axis distance sag slope (deg)1 0.00000% 0.00001% 0.00613%2 0.00000% 0.00001% 0.00613%3 0.00001% 0.00003% 0.00612%4 0.00003% 0.00006% 0.00611%5 0.00005% 0.00010% 0.00610%6 0.00008% 0.00015% 0.00608%7 0.00010% 0.00021% 0.00606%8 0.00014% 0.00027% 0.00604%9 0.00017% 0.00035% 0.00602%10 0.00021% 0.00043% 0.00599%11 0.00026% 0.00051% 0.00596%12 0.00031% 0.00061% 0.00593%

*13* 0.00036% 0.00071% 0.00589%

% of Error Between two methods:


3.2 Optical design procedure

This section summarizes the approach used to design the test set for the GTC segments. The test system was developed in the following order:

1. Preliminary 1st order system and hologram design

2. Design projection optics

3. Detailed definition of reference beam

4. Design imaging system to create image on CCD array

5. Detailed CGH design for test beam to match reference beam

It is important to understand some of the tradeoffs that were made early to avoid the need to change a basic parameter, which would have required the system to be re-designed from scratch. 3.3 Design of the projection system The projection lens transmits the two orders of diffraction from the hologram (0 and 1 order) such that no other orders propagate and the CGH is at a good image of the test mirror segments. These lenses are critical to this test because accurate compensation for the aspheric departure of the segments requires the holograms to be projected to the test surface.

A two-element set of lenses was designed to project the wavefronts from the hologram to the test plate. These lenses were actually designed backwards, i.e., they were designed to image the test mirror onto the CGH. The basic constraints on these lenses are:

1. Image scale of 33.5 mm at CGH for 1340-mm mirror segment. This came from the

preliminary design in response to concerns of sensitivity to fabrication errors and the cost of the holograms.

2. Effective entrance pupil for imaging lenses 33.5 mm diameter, located 12000mm

from test plate. The image quality was defined to be better than 0.05 waves over the entire pupil.

3. The image was constrained to be telecentric – with the ray bundle that is defined by

the 40-mm pupil hitting at normal incidence to the CGH plane. This allows the CGH to be illuminated with collimated light, which eases manufacturing difficulties in requiring a specific wavefront for illumination. This will also insure that the 1st order of diffraction is nominally at normal incidence to the CGH and that the image created by this will be centered in the projection lenses.


Several different lens configurations were designed and the best design was chosen. The projection lens uses two BK7 lens elements that must be manufactured and assembled to high accuracy. The lenses do not need to be perfect, so long as their quality is known to high accuracy. Unlike the large test plate, the two beams do not travel through these lenses together. So the surface figure and internal quality of these elements is very important.

stop

196mm

Bk7 elements

CGH

Proj. L145 mm dia

20mm center thicknessProj. L2

36 mm dia20mm center thickness

CGH

Figure 3-3. Layout of the projection lens. This shows the system as it was designed,

imaging the test plate on to the CGH. The prescription for the system is given below:

The test beam is pre-distorted by the CGH and it goes through the projection optics on-axis. The reference beam, or 0-order beam is then 2.414° off axis.


Table 3-4. Prescription of projection lens proj3.zmx. All units are mm. Surf Type Comment Radius Thickness Glass Diameter Conic

OBJ STANDARD 1=REF; 2=TEST Infinity 12000 1340 0

1 STANDARD Infinity 0 40 0 STO STANDARD STOP Infinity 196.0715 40 0

3 STANDARD 49.20629 20 BK7 35.48016 0 4 STANDARD 41.21586 94.98618 31.76799 0

5 STANDARD P

ROJECTION LENS 126.6618 20 BK7 45.43283 0

6 STANDARD Infinity 262.4837 44.91751 0 IMA STANDARD Infinity 33.50314 0 The quality of the projection lens is excellent, as shown below:

Figure3-4: Wavefront quality of projection lens as function of position on mirror segment.

Figure 3-5: Wavefront aberrations in projection lens.

Figure 3-6: Field curvature and distortion for projection lens.

Figure 3-7: Transverse ray aberrations in projection lens.


The system is shown to produce a diffraction-limited image at the edge of the field. This is important because the alignment marks will be projected with these lenses and aberrations in the lenses would limit the ability to determine the centroid of the image. A plot of the PSF is given below.

Figure 3-8: Point spread function for projection lens, including diffraction.

Design Study for measFYP 062002

Figure 3-9: Performance of projection lens in mode where it focuses collimated light

uring GSMT primary mirror using a test plate with computer generated hologram

19

3.4 Design of the imaging system A set of lenses was designed to image the test piece onto the CCD array. These lenses were initially designed only for the reference beam because this does not change for the different segments and because the CGH will always be designed to make the test beam identical to the reference beam. The illumination surface of the test plate introduces some spherical aberration into the beam. Figure 3-10 shows a spot diagram through focus of the reference beam at the distance of 12000 mm from the test plate. The best focus, including the spherical aberration is at 12036 mm from the test plate. The nominal fabrication errors, including index inhomogeneity, are 0.1 mrad/pass from the test plate, or 0.2 mrad in this wavefront. This corresponds to 2.4 mm image blur.

Figure 3-10: Spot diagram showing how light from the test plate comes to focus about 12.0 meters above it.


<-12000 MM TO TEST PLATE

CCD4.8 mm image

IL2 12 mm

IL1 20 mm

Figure 3-11: Layout of the imaging lens. The overall length is 68 mm.

The 2-element imager gives excellent images over the full field. It creates an image that is 4.8 mm in diameter (for the 1340 mm diameter part). Table 3-5: Prescription for imaging lens imag1.zmx R (mm) T (mm) Glass DIA (mm) Focus (= imager pupil) +10 6 IL1 63.07848 3.5 BK7 20 -56.7993 41.1354 20 IL2 28.28116 3.5 BK7 12 -637.6142 10 12 CCD Infinity 4.8 This lens provides excellent imaging over the full field, as shown in Figures below.


Figure 3-12: Image quality from imaging lens across the full segment.

igure3-13: Quality from imaging lens. F

Figure 3-14: Transverse ray aberrations in

igure 3-15: Field curvatures for imaging imaging lens. Flens.

he imaging lenses are designed and analyzed assuming the worst case of system errors

.5 Specification of the test plate

Tup to ± 0.2 mrad. These system errors define the pupil function for the imaging lenses so the pupil is sized to be 6 mm. In reality, most of the light will go through the central 1 mm of this lens so the imaging performance will be much better than described here. Although this design and analysis assumes the worst case, the performance is excellent and the optical system is still quite simple. 3


The non-reference (i.e., the illumination) side of the test plate was chosen to be spherical in shape to minimize the viewing distance and fabrication costs. The viewing distance of 12.0 m was set because the spherical aberration from the test plate starts to become significant at shorter distances.

The radius of the test plate was chosen to give optimal amount of focus in the holograms. To keep the system common for all segments, the tilt in the segment, relative to the test plate, was held as a constant. The test plate radius was chosen to minimize this slope, which minimizes the average hologram ruling frequency, thus the sensitivity to errors.

The thickness of the test plate was chosen based on experience with similar optics at the University of Arizona. There a 1.83-m test plate is used with 10 cm center and 6.4 cm edge thickness. This part is supported by 24 hydraulic actuators at its edge to maintain the requirement that surface slopes < 0.02 waves/cm. The measurement of the test plate, with mapping errors of 0.5% of the diameter can be used to achieve λ/25 P-V measurements. The GSMT TP will be about two times better, with similar mounting.


4.0 HOLOGRAM SPECIFICATION This section first gives the locations of six sample pieces that are representative of the GSMT primary segments (section 4.1). Out of the six segments studies, one of them (segment #0) is used to calibrate the system. Procedures needed for system calibration is explained in section 4.2. Detailed hologram designs were made for each of the tests described earlier. These are summarized graphically in section 4.3 4.1. Location and Layout of six representative segments Six radial pieces were chosen from the GSMT primary segments for this study (Figure 4-1). To properly model those segments, their locations were calculated. I wrote an IDL program to find the following three characteristic parameters for each of the six segments: 1) off-axis distance 2) sag and 3) slope. I then compared results from my computer program with data provided by NOAO (R. Robles). Net errors of these two methods are listed in Table 4-1 and they show that two methods are in excellent agreement.

Figure 4-1: location of the segments under study

0

13

analyzed segments


SEGMENT # Y= X= ANGLE i, off_axis distance[m] sag is[mm] slope is[deg] ,0 0 0 0 0 0 01 1155.070937 11.11824058 1.10280755 1 1155.0709 11.11824 1.10287522 2310.141716 44.47295625 2.20479859 2 2310.1418 44.47296 2.20493373 3465.212182 100.0641289 3.30516022 3 3465.2127 100.06416 3.30536264 4620.28218 177.8917285 4.40308677 4 4620.2836 177.89184 4.40335595 5775.351559 277.9557136 5.49778325 5 5775.3545 277.956 5.49811856 6930.420174 400.2560316 6.58846876 6 6930.4254 400.25664 6.58886957 8085.487883 544.7926193 7.67437974 7 8085.4963 544.79375 7.6748458 9240.554553 711.5654038 8.75477299 8 9240.5672 711.56735 8.75530189 10395.62006 900.5743031 9.82892852 9 10395.638 900.57743 9.829519810 11550.68427 1111.819227 10.89615214 10 11550.709 1111.824 10.89680511 12705.7471 1345.300077 11.95577782 11 12705.78 1345.307 11.9564912 13860.80842 1601.01675 13.00716973 12 13860.851 1601.0265 13.007941

*13*added 15015.86815 1878.969136 14.04972403 13 15015.922 1878.9825 14.050552

off_axis distance sag slope (deg)1 0.00000% 0.00001% 0.00613%2 0.00000% 0.00001% 0.00613%3 0.00001% 0.00003% 0.00612%4 0.00003% 0.00006% 0.00611%5 0.00005% 0.00010% 0.00610%6 0.00008% 0.00015% 0.00608%7 0.00010% 0.00021% 0.00606%8 0.00014% 0.00027% 0.00604%9 0.00017% 0.00035% 0.00602%10 0.00021% 0.00043% 0.00599%11 0.00026% 0.00051% 0.00596%12 0.00031% 0.00061% 0.00593%

*13* 0.00036% 0.00071% 0.00589%

From FYP's IDL programRICK's data

% of Error Between two methods:

Table 4-1: Definition of the segments under study

0

4.2. Balancing intensities for good fringe contrast

The holograms are designed with 50% duty cycle. This put 25% of the incident light into the zero order and 10% into the first order. The contrast or visibility, defined to be the ratio (Imax-Imin)/(Imax+Imin) in the fringe pattern, is easily calculated.

testref

testref

IIII

visibility+

=2

where Iref and Itest are the intensities of the reference and test wavefronts, given by the product of the diffraction efficiency and the reflectivity of the part. So for measuring bare glass segments, Iref = 0.01 (4% R and 25% diffraction efficiency) and Itest = 0.004 (4% R and 10% diffraction efficiency) and 90% contrast is achieved. The test can also be used to measure aluminized segments. Here Iref = 0.01 and Itest = 0.095 (95% R and 10% diffraction efficiency) and 58% contrast is achieved. This is still adequate to allow accurate high-resolution surface measurements.


4.3. Test of reference sphere (RS) The test system will be calibrated using a concave spherical mirror that has the same diameter as the segments. This part is designated as the reference sphere RS. The purpose of this auxiliary optic is two-fold. It allows a direct measurement of the convex surface of the test plate (designated TP). The RS must have its figure measured off line. Also, this will provide the reference for the radius of curvature of the mirror, so the radius of this RS must be measured. The slope errors in the system will be calibrated using two additional tests on the reference sphere RS and this study specifies one of the two tests. These test use holograms with different amounts of tilt in them. By comparing the direct measurement above with these two CGH measurements, the shape of the reference sphere, and system slope errors in the y- and x- directions can be determined. This study addresses the slope error in y-direction only (hologram Rsy). The hologram and system designs for one of the tests are given below. The hologram is plotted above in Table 4-3. The RSy test uses a simple hologram to put tilt in the wavefront so that the test beam comes to focus on the optical axis of the projection lens. The other orders are easily separated here.

0 order

1st order

2nd order

Figure 4-2: Appearance at the intermediate focus of the projection lens for the RSy

hologram. The 2nd order of diffraction is easily blocked and all other orders are blocked by the 13-mm aperture of the projection lens.


Table 4-2: Definition for holograms RSy

RSy Number of terms: 24 Maximum rad ap : 35.2 Zernike Term 1: 0 Zernike Term 2: 0.023432009 Zernike Term 3: -576.2 Zernike Term 4: 0.043884061 Zernike Term 5: -0.13444454 Zernike Term 6: 0 Zernike Term 7: 0 Zernike Term 8: -0.60129546 Zernike Term 9: 0.027782846 Zernike Term 10: 0 Zernike Term 11: -0.0061931876 Zernike Term 12: -0.027340775 Zernike Term 13: 0 Zernike Term 14: 0 Zernike Term 15: 0.05568552 Zernike Term 16: 0.00062574732 Zernike Term 17: 6.7814289e-005 Zernike Term 18: 0 Zernike Term 19: 0 Zernike Term 20: -0.00021232869 Zernike Term 21: -0.00063126542 Zernike Term 22: 0 Zernike Term 23: 0 Zernike Term 24: 0.00091276619

4.4. CGH designs for the five remaining segments The holograms are specified as phase function using Zernike polynomials. These coefficients can be directly imported to the fracturing software at Diffraction International. In the optical design code Zemax, the Zernike phase surface is interpreted to compute a phase function from the coefficients, then to a phase function is simply add to the wavefront. This is exactly how the CGH performs. The CGH, acting at mth order adds m waves to the phase for each line in the pattern.


The design method for optimizing these Zernike coefficients was quite cumbersome. I used the multi-configuration mode of Zemax and set the program up so configuration 1 was the reference beam. It ignores the phase terms in the hologram and light is reflected off the test plate reference surface. The test beam is defined in configuration 2. Here the hologram adds the specified phase function and light is transmitted through the test plate, reflects off the mirror segment, and then goes back through the test plate.

The hologram is designed to make the test beam at the CCD look exactly like the reference beam. Both do not have to be perfect but they must match. This was done in Zemax in two steps: First, added a dummy Zernike surface after the CCD and then optimized coefficients to create a perfect image for the reference light from the test plate. Then optimized the phase coefficients of the hologram to create a near-perfect image, using the same dummy surface as was used for the reference beam.

Table 4-3: Summary of hologram designs

Test # Pattern at focus of Projection lens Wavefront Contour (without dominating tilt terms)

(Shown at scale where 1 contour line equals 500 waves in CGH.)

T1

T5


T8

T10

T13


T1 CGH design The layout for the T1 test is given below in Figure 4-3.

Full parent

Top-end optics

Segmen

Figure 4-3: Layout of T1 test, showing parent primary.

CGHobj stop

projection optics

1st order

0th order

Figure 4-4: Layout of the top-end optics


Figure 4-5: Appearance at the intermediate focus of the projection lens for the T1 hologram. The 2nd order is isolated from the 1st, but must be blocked. All orders

other then 0 and 1 are blocked by the 50-mm aperture of the projection lens.


The prescription for the T1 hologram is given below: STORED IN \S1_zerns.txt stored in\S1_zerns.txt MAX TERM OF ZERN USED IS 37 NORMAL RADIUS IS 17.20 Zernike # 1 = 0.0000000000 Zernike # 2 = -0.9177405923 Zernike # 3 = -573.5882320367 Zernike # 4 = 55.4969861175 Zernike # 5 = -0.0000387115 Zernike # 6 = -0.9346763574 Zernike # 7 = -0.9026334317 Zernike # 8 = -0.3269647982 Zernike # 9 = 0.0029013343 Zernike #10 = 0.0000315173 Zernike #11 = 0.1716170178 Zernike #12 = -0.0089349554 Zernike #13 = -0.0000283137 Zernike #14 = -0.0018793480 Zernike #15 = -0.0000023131 Zernike #16 = -0.0015687856 Zernike #17 = -0.0011964692 Zernike #18 = -0.0000210333 Zernike #19 = 0.0003314996 Zernike #20 = -0.0000321551 Zernike #21 = 0.0006761080 Zernike #22 = -0.0018532925 Zernike #23 = -0.0000091385 Zernike #24 = 0.0003076835 Zernike #25 = 0.0000014417 Zernike #26 = -0.0005445395 Zernike #27 = -0.0000362967 Zernike #28 = -0.0000154181 Zernike #29 = 0.0051634621 Zernike #30 = -0.0000477750 Zernike #31 = -0.0003776462 Zernike #32 = -0.0000175046 Zernike #33 = 0.0006661852 Zernike #34 = -0.0000165006 Zernike #35 = -0.0002280410 Zernike #36 = -0.0000208988 Zernike #37 = 0.0000348657


T5 CGH design The layout for the T5 test is given below:

Figure 4-6: Layout of T5 test, showing parent primary.


The T2 hologram design is described below. The image from the projection lens is shown below. The CGH pattern was shown in Table 4-3.

Figure 4-7: Appearance at the intermediate focus of the projection lens for the T5 hologram. The 2nd order is isolated from the 1st, but must be blocked. All orders other then 0 and 1 are blocked by the

50-mm aperture of the projection lens.


The Zernike coefficients specifying the CGH are given below. STORED IN \S5_zerns.txt stored in\S5_zerns.txt MAX TERM OF ZERN USED IS 37 NORMAL RADIUS IS 17.20 Zernike # 1 = 0.0000000000 Zernike # 2 = -5.3465889440 Zernike # 3 = -574.1664401950 Zernike # 4 = 9.2682298068 Zernike # 5 = -0.0022212161 Zernike # 6 = -33.3974920525 Zernike # 7 = -0.9052961428 Zernike # 8 = -1.9075895317 Zernike # 9 = 0.0010739350 Zernike #10 = 0.0131931135 Zernike #11 = 0.0162956187 Zernike #12 = -0.0918786794 Zernike #13 = -0.0001688691 Zernike #14 = -0.0028580720 Zernike #15 = 0.0000154444 Zernike #16 = -0.0093717972 Zernike #17 = -0.0011065403 Zernike #18 = -0.0001231728 Zernike #19 = 0.0003241494 Zernike #20 = 0.0000142404 Zernike #21 = 0.0006574374 Zernike #22 = -0.0010833622 Zernike #23 = 0.0000021019 Zernike #24 = 0.0004616137 Zernike #25 = 0.0000587124 Zernike #26 = -0.0005503913 Zernike #27 = 0.0000087255 Zernike #28 = 0.0000490709 Zernike #29 = 0.0052201008 Zernike #30 = -0.0000330061 Zernike #31 = -0.0003345798 Zernike #32 = -0.0000593920 Zernike #33 = 0.0006437138 Zernike #34 = 0.0000000000 Zernike #35 = 0.0000000000 Zernike #36 = 0.0000000000 Zernike #37 = 0.0000000000


T8 CGH design

Figure 4-8: Layout of test T8


Figure 4-9: Appearance at the intermediate focus of the projection lens for the T8 hologram. The 2nd order is isolated from the 1st, but must be blocked.

All orders other then 0 and 1 are blocked by the 50-mm aperture of the projection lens.


STORED IN \S8_zerns.txt stored in\S8_zerns.txt MAX TERM OF ZERN USED IS 37 NORMAL RADIUS IS 17.20 Zernike # 1 = 0.0000000000 Zernike # 2 = 0.0158158369 Zernike # 3 = -572.3489054128 Zernike # 4 = -26.9071510131 Zernike # 5 = -0.0045921578 Zernike # 6 = -58.6408320612 Zernike # 7 = -0.9050036473 Zernike # 8 = -2.4757460359 Zernike # 9 = -0.0001927577 Zernike #10 = 0.0317411032 Zernike #11 = -0.1075150359 Zernike #12 = -0.1577585797 Zernike #13 = -0.0002136560 Zernike #14 = -0.0038288406 Zernike #15 = 0.0000194760 Zernike #16 = -0.0119024688 Zernike #17 = -0.0017857955 Zernike #18 = 0.0001943988 Zernike #19 = 0.0003481587 Zernike #20 = 0.0002230864 Zernike #21 = 0.0006069800 Zernike #22 = -0.0015740254 Zernike #23 = 0.0000258095 Zernike #24 = -0.0001550209 Zernike #25 = 0.0000174877 Zernike #26 = -0.0008449205 Zernike #27 = 0.0000073887 Zernike #28 = 0.0001090043 Zernike #29 = 0.0060862131 Zernike #30 = 0.0000000000 Zernike #31 = 0.0000000000 Zernike #32 = 0.0000000000 Zernike #33 = 0.0000000000 Zernike #34 = 0.0000000000 Zernike #35 = 0.0000000000 Zernike #36 = 0.0000000000 Zernike #37 = 0.0000000000 All Done!


T10 CGH design The T10 design follows the same scheme as the others.

Figure 4-10: Layout of segment #10 (T10)


1st order

2nd order

0 order

Figure 4-11: Appearance at the intermediate focus of the projection lens for the T10

hologram. The 2nd order is isolated from the 1st, but must be blocked. All orders other then 0 and 1 are blocked by the 50-mm aperture of the projection lens.


Executing C:\Documents and settings\feenix\Z_macro\GETCGHZERN_TEXT.ZPL. STORED IN \S10_zerns.txt MAX TERM OF ZERN USED IS 37 NORMAL RADIUS IS 17.20 Zernike # 1 = 0.0000000000 Zernike # 2 = 0.0000000000 Zernike # 3 = -572.3393204300 Zernike # 4 = -72.3699726529 Zernike # 5 = -0.0087756252 Zernike # 6 = -90.1426875166 Zernike # 7 = -0.8957849325 Zernike # 8 = -3.0029231187 Zernike # 9 = -0.0001660526 Zernike #10 = 0.0642469432 Zernike #11 = -0.2561638185 Zernike #12 = -0.2305721258 Zernike #13 = -0.0001873379 Zernike #14 = 0.0047491707 Zernike #15 = 0.0001298846 Zernike #16 = -0.0132831798 Zernike #17 = -0.0015854617 Zernike #18 = 0.0014838778 Zernike #19 = 0.0012774214 Zernike #20 = 0.0009526985 Zernike #21 = 0.0000761497 Zernike #22 = -0.0038170966 Zernike #23 = 0.0000257814 Zernike #24 = -0.0006626338 Zernike #25 = 0.0000168936 Zernike #26 = 0.0000000000 Zernike #27 = 0.0000000000 Zernike #28 = 0.0000000000 Zernike #29 = 0.0000000000 Zernike #30 = 0.0000000000 Zernike #31 = 0.0000000000 Zernike #32 = 0.0000000000 Zernike #33 = 0.0000000000 Zernike #34 = 0.0000000000 Zernike #35 = 0.0000000000 Zernike #36 = 0.0000000000 Zernike #37 = 0.0000000000 All Done!


T13 CGH design T13 is the furthest off-axis segment.

Figure 4-12: Layout of segment #13 (T13)


Figure 4-13: Appearance at the intermediate focus of the projection lens for the T13 hologram. The 2nd order is isolated from the 1st, but must be blocked. All

orders other then 0 and 1 are blocked by the 50-mm aperture of the projection lens.

1st order

2nd order

0 order


STORED IN \S13_zerns.txt stored in\S13_zerns.txt MAX TERM OF ZERN USED IS 37 NORMAL RADIUS IS 17.20 Zernike # 1 = 0.0000000000 Zernike # 2 = -10.3496306150 Zernike # 3 = -576.0090048576 Zernike # 4 = -156.0843108150 Zernike # 5 = -0.0184402239 Zernike # 6 = -147.3530826644 Zernike # 7 = -0.9030393963 Zernike # 8 = -3.6580486680 Zernike # 9 = 0.0026052681 Zernike #10 = 0.1724650633 Zernike #11 = -0.4563860382 Zernike #12 = -0.2569628693 Zernike #13 = 0.0001897872 Zernike #14 = 0.1024076351 Zernike #15 = 0.0006272052 Zernike #16 = -0.0045390582 Zernike #17 = -0.0010773908 Zernike #18 = 0.0108717689 Zernike #19 = 0.0024347132 Zernike #20 = 0.0053815814 Zernike #21 = 0.0012966812 Zernike #22 = -0.0019389065 Zernike #23 = 0.0000902547 Zernike #24 = 0.0017169562 Zernike #25 = 0.0001022227 Zernike #26 = 0.0000000000 Zernike #27 = 0.0000000000 Zernike #28 = 0.0000000000 Zernike #29 = 0.0000000000 Zernike #30 = 0.0000000000 Zernike #31 = 0.0000000000 Zernike #32 = 0.0000000000 Zernike #33 = 0.0000000000 Zernike #34 = 0.0000000000 Zernike #35 = 0.0000000000 Zernike #36 = 0.0000000000 Zernike #37 = 0.0000000000 All Done!


5.0 ERROR ANALYSIS ON THE MOST DIFFICULT

SEGMENT The error analysis is divided in two sections, figure errors and errors in the definition of the segment location with respect to the parent primary. The figure errors are dealt with in section 5.1 and are defined as shape errors that have more than 1 cycle across the aperture. The low-frequency errors equivalent to misalignment of the segment are described in section 5.2. Summary of the tolerance analysis is shown in the ‘road map of errors’ (Figure 5-1).

Figure 5-1: Summary of Error Analysis

System Tolerance Analysis

∆x: 0.106 mm

∆θ: 0 . 006deg(0.105mrad)

∆sag: 24.1nm

Proj. system error0.0089 wv

CGH fab error0.0083 wv

uncertainty in segmentlocation wrt the parent

primary

WF Figure Error0.015 wv

(4.8 nm surface)

∆w=13.8 nm

∆w=10.86nm

∆w=2.9 nm

Equivalent RMSWavefront Error

reference surface0.0083 wv

test sphere0.0058 wv

test plate (ref.side) 0.0058wv

ability to back out measurement: 0.005 wv

interferometric measurement: 0.003 wv

ability to back out measurement: 0.005 wv

interferometric measurement: 0.003 wv

test plate0.0037 wv

surface slope0.003 wv

ref. index slope0.0022 wv

fiducial marks0.00035 wvAlignment Error

0.0030 wvdue to segment

0.003 wv interferometric measurement: 0.003 wv

2nd order magnification error: 0.00035 wv

(1-m segmentdiameter)


5.1 Error analysis for figure measurement There are several different components to the figure error in the measurement of a segment. These are listed below:

• Wavefront errors from CGH or the test plate that directly couple into surface errors (section A).

• Errors in the projections optics, which cause a distortion when projecting the CGH image to the aspheric surface (section B).

• Slope errors in the system, typically from refractive index variations in the test plate, coupled with the slope differences between the two beams (section C).

• Mapping errors that limit the ability to completely back errors out, even if they are known (sections D and E).

Combined total wavefront error is summarized below:

E

Cprte

te

rese

R

DeFY06

Table 5-1: Figure error budget for test of the T13 segment ( the most difficult). Derivation for all terms are given in this section.

ffect Magnitude Wavefront Surface Figure

λ rms nm rmsGH fabrication errors 0.125 µm 0.0083 2.626ojection optics Table 5-2 0.0089 2.816st plate inhomogeneity ±0.15 mrad 0.0030 0.937

st plate illumination surface 2 fringes/cm 0.0022 0.696

ference surface figure Table 5-3 0.0083 2.626gment alignment & test error Table 5-4 0.0030 0.954

oot Sum Squared 0.015 4.80

sign Study for measuring GSMT primary mirror using a test plate with computer generated hologram P

2002 46

Section A: wavefront errors from CGH fabrication error The CGH patterns have period of about 15 µm center-to-center and will be written with 50% duty cycle. These holograms can be made standard with accuracy of ±0.125 µm (3σ), which gives wavefront accuracy of 0.125/15 = 0.0083 λ. This corresponds to surface accuracy of 0.0042 λ or 2.6 nm. Section B: wavefront errors from projection optics For the purposed of this tolerance analysis, the projection optics includes, not only the projection system of two BK7 lens, but also the laser beam and the test plate. To first order, errors in the projection optics do not affect the test at all because both wavefronts go through the optics together. However, there exists significant second-order coupling between the projection optics and the separation of the two beams from the CGH, since in reality, the two beams are not coincident through the optical system. The dominant type of error in the projection optics comes from the distortion of the CGH image at the test plate. The typical line spacing is 15 µm at the CGH and the system magnification for the putting the image of the CGH onto the test plate is 40.0, so the image of the CGH at the test plate has a period of 600 µm (=15 µm x 40.0). If this image is distorted by an amount equivalent to one line, the resulting wavefront error is distorted by 1 wave. So, to keep the error to 0.01 λ, the distortion must be less then 6.0 µm over the 1340mm image (image of CGH fills the test plate diameter). The error analysis for the projection lens was performed by simulating the complete system. A multi-step procedure was developed for calculating the sensitivities that define the error budget. The steps for simulating the effect of each parameter tolerance are given below. The tolerance table was built up one item at a time using this procedure. To simulate the effect of a single error (like radius of curvature on a lens), the following steps were followed: 1. In Zemax, a complete simulation of the system is made and the parameter of choice is

perturbed by the amount of the tolerance.

2. Since perturbation affects both the test and reference wavefronts, the reference wave is re-optimized by changing the coefficients of a dummy aspheric function that is defined at the CCD array. Since the measurement uses only the difference between the reference and test beams, and since the reference beam has been re-optimized, the equivalent residual measurement error shows up as the aberrations left in the test wavefront. This exactly corresponds to the interference that would be measured by the real system.

3. The position of the test surface is optimized to minimize the rms wavefront in the test beam. The adjustment required for this, as well as the rms wavefront itself are then recorded.

Unfortunately, the effect of projection system errors is varies for the different segments. Rather than to develop a complex error budget that is different for each segment, we


developed the error budget for the most difficult segment, segment #13, and the accuracy for the others will be better.

Table 5-2 shows all parameters, tolerances, and sensitivities. The radii of curvature, thickness, spacing, tolerances correspond to precision fabrication. The element tilt and wedge are defined in degrees. It was assumed that the lenses would be made from high grade BK7. The refractive index is measured to ±0.00001 by the glass manufacturer and this data is usually provided with the glass. Refractive index variations of ±1E-6 are available as H4 quality material. Results from Table 5-2 can be improved slightly if alignment of the test plate to the fiducials is also simulated. This step should precede the re-optimization of the reference beam stated in step 2 of the procedure listed earlier. The test plate position should be adjusted to optimally place the projected fiducial images in the correct place.


Table 5-2: Error analysis of the projection system

parameter value tolerance units ∆r ∆ x ∆θ rms WF (λ)

mm mm deg LASER beam 1 wv wv P-V 0.0000 0.0000 0.0000 0.0023 0.002 deg -0.0153 -0.0278 0.0000 0.0015 CGH decenter 0.01 mm 0.1847 0.0372 -0.0004 0.0009 tilt 0.005 mm 0.2041 0.0379 -0.0003 0.0009 rotation 0.002 deg 0.0040 0.0005 0.0020 0.0008 0.0000 CGH - L1 spacing 259.9157 0.01 mm -0.0882 0.0014 0.0000 0.0010 L1 R1 inf 0.05 mm 0.0000 0.0000 0.0000 0.0001 ct 20 0.01 mm -0.0541 0.0014 0.0000 0.0007 R2 -126.6618 0.002 mm 0.0781 0.0021 0.0000 0.0008 index 0.00001 0.01910 -0.00011 -0.00001 0.0013 surface 1 surf PV 0.125 wave -0.0063 -0.0005 0.0000 0.0011 surface2 surf PV 0.0125 wave -0.0166 0.0010 -0.0001 0.0009 inhomogeneity PV 2.00E-06 0.0000 decenter 0.005 mm -0.1028 -0.0042 0.0001 0.0012 tilt /100mm 0.005 mm 0.0054 0.0002 -0.0001 0.0008 wedge /100mm 0.005 mm -0.0624 -0.0017 0.0018 0.0018 0.0000 L1 - L2 spacing 94.9862 0.005 mm -0.0476 -0.0010 0.0002 0.0008 L2 0.0000 R1 -41.21586 0.005 mm 0.0000 0.0031 -0.0001 0.0017 ct 20 0.005 mm -0.0323 0.0068 -0.0002 0.0002 R2 -49.20629 0.005 mm 0.0197 0.0036 0.0000 0.0015 index 0.00001 0.0000 surface 1 surf PV 0.125 wave -0.0140 0.0018 -0.0002 0.0009 surface2 surf PV 0.125 wave -0.0001 0.0039 -0.0001 0.0013 inhomogeneity PV 2.00E-06 0.0000 decenter 0.01 mm 0.0175 -0.0024 0.0000 0.0010 tilt /200mm 0.01 mm 0.0087 -0.0269 0.0000 0.0024 wedge /200mm 0.005 mm -0.0708 -0.0023 0.0021 0.0018 0.0000 Test plate 0.0000 R1 7791.8688 11.7 mm 0.0175 -0.0006 0.0000 0.0020 thick 5 mm 0.0024 0.0000 0.0000 0.0008 wedge /1340mm 2 mm 0.0010 0.0300 0.0020 0.0060 RSS 0.3429 0.073 0.0040 0.0089


Section C: wavefront errors from the test plate A potential limitation for this CGH test would be from slope of error introduced to the test via severe test plate refractive index variations and from the surface slope of the illumination surface. This is problematic for two reasons: (a) test beam travel through 5mm air gap whereas the reference beam does not, so any slope error from the test plate leads a net shear difference between the two beams (section C.1), and thus leads to wavefront error; and (b) test beam and reference beam are not completely common path at the illumination surface (section C.2), so any slope error from the test plate leads to wavefront error. In section C.3, all the errors are summed by root sum square (RSS) method. Section C.1: net shear difference due to the air gap In order to calculate the net shear difference induced by any slope error from the test plate, we make the following two assumptions on the quality of the test plate:

1. Max index change is on the order of , with spatial variation of 4 cycles across the test plate diameter. This leads to slope error of 0.019mrad as calculated below:

max| n| 1E-5∆ = ±

[ ]

[ ]

maxtest plate

test plate

ref index erro

test plate maxtest plate test plate

ref index error

2 4n=| n| cosDia

W=t nd( W)

dx

2 4 2 4t | n| sinDia Dia

Therefore,

| | 100

x

x

mm

π

θ

π π

θ

∆ ∆ ×

∆ ×∆

∆∆

= × ∆ × × −

∆ = ×[ ]

ref index error

2 41E-51340mm

or,| | 0.019mrad

π

θ

± ×

∆ =

2. The surface slope of the illumination side is on the order of 2 fringes per cm or 1 wave per cm.


[ ]

ref index erro

test plate

d( W)dx

W ds= (ndx dx

2 1.45 1cm

0.029mrad

θ

λ

∆∆

∆ × −

= × − =

1)

Any test plate slope error shear the reference and test beam the same amount before the air gap. Since only the test beam goes through the air gap, the net shear difference between the two beams comes only from the additional shear experienced by the test beam accumulated when it goes through the air gap. This shear can be easily calculated:

net air gap order separationx 2 t

2 5 1.05310

mm mradum

ε∆ = × ×

= × ×=

This shear multiplied by the wavefront slope errors (added in root sum square) gives wavefront error of

Section C.2: beam shear The reference and the test beam are laterally sheared at the illumination side of the test plate. The amount of shear is proportional to the tilt need for order separation:

order sperationtest plate

test plate

x= t

1.053mrad=100mm1.45

72.6

n

m

ε

µ

∆ ∆ ×

×

=

Section C.3: combined wavefront error Combined wavefront can now be calculated:


[ ] [ ]ref index error ref index error slope error slope error

2 22 2

beam shear beam shear air gap shear beam shear air gap shear

2 2

W = x x x x

0.019 72 0.019 10 0.0285 72mrad m mrad m mrad

θ θ θ θ

µ µ

∆ ∆ × ∆ + ∆ × ∆ + ∆ × ∆ × ∆ × ∆

= × + × + ×

[ ] [

]

[ ] [ ]

2 2

2 2

2 2

0.0285 10

1.9E-6 m +3.5E-6 m

0.0022 0.00296

0.0037

m mrad mµ µ

µ µ

λ λ

λ

+ ×

=

= +

=

Section D: wavefront errors from testing the reference surface Figure errors in the reference surface are measured and removed by using the concave reference sphere. This in turn has its errors measured using an interferometer that has been calibrated. The ability to remove these errors is limited by the coupling between surface errors and distortion. All measurements are made with distortion less than 1 pixel (corresponds to about 5 mm on the mirror). The reference surface figure errors are specified to have slope errors less than 0.01 λ/cm rms for the test plate and the concave sphere used to calibrate it. Table below shows the error budget for measuring figure from the test plate’s reference surface and the reference sphere.

Surface Figure Figureλ rms nm rms

Measurement of concave reference using interferometer

0.003 0.95

Effect of distortion backing out interferometer errors

0.005 1.58

Measurement of convex test plate 0.003 0.95Effect of distortion backing out reference sphere errors

0.005 1.58

Root sum squared 0.00825 2.61

Effect

Table 5-3: Error budget for measuring figure from the test plate and reference sphere.


Section E: wavefront errors from alignment errors Two factors contributing to this error are alignment error of the test plate by using six fiducial marks and interferometric measurement error of the segment. Table5-4 summarized the combined wavefront error. Figure 5-2 below depicts the errors accumulated through the processes of calibrating the system and measuring the aspherical segment (steps A and B are described in the previous section).

Table 5-4: Error budget for measuring surface figure from the test plate and reference sphere.

Magnitude Surface Figure

Figure

λ rms nm rmsinterferometric measurement 0.003

0.0030.95

alignment to fiducials -- coupled through magnification effect 6 @ 0.1mm 0.0003 0.10

Root Sum Squared 0.0030 0.95

Effect


ERROR:interferometric accuracy

of 0.003wv

ERROR:inability to copmletelyback out test sphere

figure error 0.005wv


of 0.003wv

A:measurement of

concave referencesphere

C:measurement of

concave asphericalsegment

B:measurement of convex

test plate referencesurface

ERROR:inability to copmletelyback out test plate's

reference surface error 0.005wv


of 0.003wv

Surface under test:concave reference sphere

Surface under test: testplate (TP) reference

surface

Surface under test:aspherical segment

+

+

Figure 5-2: Error accumulated through calibration of reference sphere (RS) and test plate (TP)

reference surface

Alignment mark contributes to the total wavefront error since their limited position accuracy of 31ppm (Section 5.1) is coupled though the system as a magnification error. Through ZEMAX simulation, it is determined that this equivalent magnification error leads to and wavefront error of 0.0003 wv.


5.2 Error analysis for defining the optical surface relative to the parent mirror

The CGH test allows accurate definition of the optical surface including radius of curvature and absolute position of off-axis segment on parent hyperboloid. This is accomplished by imaging, through the projection optics, 6 fiducial marks written on the CGH substrate to the test plate. Several factors limit the ability to accurately position the segment with respect to the parent mirror and they are:

Imperfect projection system Fabrication error on etching the fiducials Misalignment to the fiducials Inherent accuracy limitation in alignment marks Mechanical error in transferring alignment marks

System of fiducials is explained in section A. Section B summarizes the error budget for segment position ( ), segment rotation ( ) and relative radius of curvature . x∆ θ∆ sag∆ Section A: System of fiducials The alignment procedure requires the fiducials to be located to 0.1 mm. If we use 6 reference patterns, each of which will give 0.1 mm accuracy, the overall accuracy of the alignment of the ring to the CGH will be 0.041 mm over diameter (31 ppm scale over 1.34 m) since0.1 / 6 0.041mm mm= 0.058 mm lateral positioning since there are x- and y-movement, so each is allocated

roughly 0.041 2 0.058mm mm× = 0.046 mrad rotation about axis since 0.041 / 670 0.061 (0.0035deg)mm mm mrad= The images themselves project to be 300 µm wide (FWHM) at the test plate, so determining the center to 100 µm will not be difficult. This could be done with a simple optical loupe or with a CCD camera. The effect of the lateral position and the rotation are direct. The effect of the 31 ppm scale error was simulated for segment 3, the most severe. This simulation was done the same way as the projection lens analysis. The net effect of the 31 ppm scale error from the alignment procedure is

Segment lateral shift ∆x 0.002mm Segment radius error ∆sag 10.54nm

Residual wavefront 0.0003 λ rms This is in addition to the 0.041 mm uncertainty in the lateral position from the direct effect of the fiducial alignment.


There will also be a limitation to the ability to transfer the reference measured above to the segment reference surfaces. I have assumed that this can be done to 0.05 mm. This can be readily achieved with standard micrometers. Section B: Segment position, orientation and relative radius of curvature relative to

the parent mirror The analysis to determine the accuracy of defining the absolute segment position and orientation is all given above. It is summarized and added for the case of segment T3, the most difficult one. The overall test accuracy is shown to be 0.18 mm for segment position and 0.2 mrad for orientation.

Table 5-5. Error budget for position and angle for the T13 segment test (the most

extreme). Derivation for all terms are given elsewhere in this report.

here are two specifications for radius of curvature – absolute radius and the degree of

Effect Magnitude Segment position ∆x

Segment rotation ∆θ

mm Deg

CGH fabrication errors 0.125 um 0.003 0.0002projection optics Table 5-2 0.073 0.0040alignment to fiducials - coupled by scale effect

6@ 0.1 mm 0.002 1.4E-05

alignment to fiducials - direct effect

6@ 0.1 mm 0.058 0.0035

mechanical measurements 0.05 mm 0.05 0.0032


Tmatching between segments. The overall radius of curvature will be limited by the measurement of the concave reference sphere. This analysis assumes this is known to ±2 mm. The analysis to determine the radius of curvature matching is given above. It is summarized here for the case of segment T13.


Table 5-6 Error budget for radius of curvature matching for the T13 segment test

(the most extreme). Derivation for all terms are given elsewhere in this report.

Effect Magnitude ∆sag [nm], Radius of curvature

matching

∆R [mm], Radius of curvature matching

nm mmprojection optics Table 6 21.38 0.34alignment to fiducials - coupled through scale effect

6@ 0.1 mm10.54 0.17

mechanical measurements 0.05 mm 3.12 0.05



CONCLUSION A detailed design study has been performed to assess the feasibility of the method of using CGHs and a test plate with spherical reference surface to measure the GSMT primary mirror segments. The system can be designed and all the key parameters can be optimized in the design with values that fit with fabrication methods. To first order, errors in the test caused by limitations in the optical fabrication of the components and the internal quality of the optics can be measured and removed using a reference spherical part and several holograms. The alignment of the parts to the test will rely on fiducial marks being projected and measured. All optical designs were done using Zemax optical design code.


REFERENCE

1. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1968

2. Wyant, J. C. and V. P. Bennett, “Using Computer Generated Holograms to Test

Aspherical Wavefronts,” Applied Optics, 11(12), pp2833, 1972

3. F. Pan, J. Burge and D. Anderson, “Laboratory demonstration of interferometric measurement using a test plate and CGH”, NOAO, June 2002,
