A Project Report

On

STRESS MIRROR POLISHING FOR

HEXAGONAL GEOMETRY TELESCOPE MIRRORS

Submitted to

Department of Mechanical Engineering,

Institute of Technology, Guru Ghasidas Vishwavidyalaya

In partial fulfillment of requirement for the award of degree

of

Bachelor of Technology in

Mechanical Engineering

Supervisor: Submitted by-

Dr. Rajesh Kumar Bhushan, Mr. Siddharth Maharana

Associate Professor, Mr. Utkarsh Mishra

Department of Mechanical Engineering

ABSTRACT

Primary mirrors of all large telescopes in future will be composed of a large number of smaller

hexagonal segments. The geometry of a telescope mirror is in most cases are parabolic

(sometimes hyperbolic), but the geometries of the constituent hexagonal mirror segments are

asymmetric and very difficult and time consuming to create. To overcome this challenge, Stress

Mirror Polishing (SMP) technology is used. The current procedure is to first use SMP on circular

glass blanks and then cut out the hexagon; the reason being that theory of SMP is mature only

for circular plates. One major step forward would be to develop the technology of SMP for direct

application on hexagonal plates. The purpose of this work is to explore the feasibility of this idea

through simulations in Abaqus FEA software. The simulation results in this work have shown

that the present theory of SMP can be applied on hexagonal plates successfully to within a

manageable range of deviations up to the first orthogonal warping component, α20. Further work

will be required to analyze effect on higher order orthogonal warping components, α22 and α31.

CHAPTER 1. INTRODUCTION

1.1. Early History of Telescopes

A telescope is an instrument that collects the light/electromagnetic radiation emitted by distant

astronomical sources.

The most common type is the optical telescope, a collection of lenses and/or mirrors that is used to see distant objects more clearly by magnifying them or to increase the effective brightness of a faint object.

One of the earliest telescopes was built in 1609 by Galileo Galilei with which he made the first

telescopic observations of the sky and discovered lunar mountains, four of Jupiter's moons, sunspots,

and the starry nature of the Milky Way.

Since then, telescopes have increased in size and improved in image quality. Computers are now used

to aid in the design of large, complex telescope systems.

Figure 1: A high resolution telescope view of the centre of Milky Way galaxy

1.2. Working of a Telescope :

The primary function of a telescope is that of light gathering. The larger telescopes can see further

because they can collect more light, from the very faintest of the celestial objects.

The proposed TMT (Thirty Meter Telescope) in Hawaii will have an aperture of 30 meters, and once

in operation it will enable astronomers to see distant and never seen before parts of the universe which

were too faint to be seen by smaller aperture telescopes.

Telescope sights on top of mountains are popular because the light reaching the instrument has to travel

through less air, and consequently the image has a higher resolution.

The highest telescope in the world is located at Hanle in Ladakh, which is run by Indian Institute of

Astrophysics, Bangalore.

1.3. Types of Telescopes:

• Telescopes can be broadly classified into the following two types on the basis of the nature of optical

mechanism used to collect light-

a. Refracting telescopes

b. Reflecting telescopes

• Most large telescopes built before the twentieth century were refracting telescopes because techniques

were readily available to polish lenses. But it suffers from various optical abnormalities which can be

overcome by reflecting mirrors.

• All large telescopes, both existing and planned, are of the reflecting variety. Reflecting telescopes have

several advantages over refracting designs.

Figure 2. The working of a Newtonian reflector telescope.

The features of a reflecting telescope are-

• The light gathering capacity of a telescope is

proportional to the area of the primary mirror.

• Telescope mirrors are revolution of conics-

usually parabolic sections.

Figure 3. Geometric properties of the parabola

• Parabolic sections have the property that all parallel rays meet at a point called focus.

• This is the reason that Dish TV collectors are paraboloids.

1.4. Design of Primary Mirror of Telescope

Reflecting Telescope primary mirrors are parabolic conics. Early telescopes were single piece mirrors.

Today the biggest telescopes with single mirrors are of 8m diameter.

Designing bigger telescopes than 8 meters requires that the primary mirror itself be made of many

hexagonal segments, as is shown in the picture of Keck telescope mirror below.

The geometry of the segments is asymmetric (if they are part of a larger parabolic/hyperbolic reflector).

To function, all the mirror segments have to be polished to a precise shape and actively aligned by a

computer controlled active optics system using actuators built into the mirror support cell.

The concept and necessary technologies were initially developed under the leadership of Dr. Jerry

Nelson in the construction of Keck telescopes.

FIGURE 3: Picture of the primary mirror of Keck Telescope. It was the first telescope to use

segmented mirrors. It has primary mirror of 10 m aperture, which is made of 30 hexagonal

segments.

The next generation large telescopes are the following-

1. Thirty Meter Telescope (TMT) – (30 m diameter) to be built at Mauna Kea in Hawaii in which India

is a partner with 10% of funding and technology.

2. European Extra Large Telescope (EELT) - 39 m aperture diameter by European Space Agency.

Figure 4: Sketch of the

proposed primary mirror of

TMT.

1.5. Thirty Meter Telescope

• TMT (Thirty Meter Telescope) is the next generation big telescope being constructed at Mauna Kea

in Hawaii which when complete will be one the largest telescopes in the world.

• India is a partner country in this multinational project with 10% funding and technology development,

and recently the Indian Government has sanctioned rupees 1300 crores for this project in December

2014. The overall budget of the project is over 1 billion dollars and includes USA, China, Japan,

Canada and India as members. It is scheduled to be completed by 2022.

• The TMT primary mirror design is a nearly parabolic conic of revolution with a 60 meter paraxial

Radius of Curvature (ROC) and a 30 meter diameter (Stephen F. Sporer, 2007).

There are 123 unique mirror segments (each a slightly

irregular hexagon) with 6-fold rotational symmetry (plus one set of

spares) requiring a total of 861 segments.

All of the 123 different mirror surfaces match a single best-fit

sphere within about 180 um Peak-to-Valley (P-V)

• Figure 5. TMT design layout

Challenges of designing TMT segmented mirrors:

• The primary mirror being a parabolic (or hyperbolic), the different hexagonal segments are of

asymmetric and complicated geometry.

• In general, we are provided spherical glass blanks of circular aperture, which we then fabricate to obtain

the required geometry of the mirror.

• The fastest and the most accurate polishing techniques available today are those that polish out

spherical surfaces (i.e. they generate spherical surface to very good accuracy).

• So directly creating these asymmetric and seemingly irregular mirror geometries from spherical glass

blanks using conventional methods is very difficult and too slow to meet the speed requirements of

TMT.

• In light of this enormous challenge, it is imperative to find faster and economical methods of fabricating

mirror blanks. SMP (Stress Mirror Polishing) is a technology that promises to meet these challenges,

as it this same technology that was used to fabricate the Keck telescope mirrors in the 1990s.

CHAPTER 2. SMP (Stress Mirror Polishing) LITERATURE REVIEW

2.1. Introduction

Directly creating these asymmetric and seemingly irregular mirror geometries from spherical glass

blanks using conventional methods is very difficult and too slow to meet the speed requirements of

TMT.

The fastest and the most accurate polishing techniques available today are those that polish out

spherical surfaces (i.e. they generate spherical surface to very good accuracy).

To overcome this challenge Nelson and Lubliner proposed the SMP process (Lubliner and Nelson,

1980).

The basic procedure of SMP is as follows-

1. If we want to create any asymmetric surface on a mirror blank of spherical curvature, we apply

mechanical loads (bending moment and shear forces) on the mirror to give its surface an inverse

deformation, i.e. same magnitude of deformation as required to create the desired surface but in the

opposite direction.

2. Then by conventional polishing methods, this warped surface can be polished to a spherical surface to

a very high order of accuracy.

3. The warping loads are then removed, the glass elastically reverts back and we obtain the desired

asymmetric geometry of the mirror.

Figure 6: Depiction of the core idea of SMP in case of flat mirrors

(Hull, A et al: January 2010). If our objective is to obtain the shape

in figure d, one way of doing it is-

1. To warp the flat plate in the inverse direction as in b by using

loads.

2. Then this plate is polished flat in c, and we can do flat polishing to a very high degree of surface finish

accuracy.

3. Then the warping loads are removed, and we get the desired geometrical shape in d.

• The current technology of SMP is mature for application to circular plates only.Upon applying SMP

on circular plates, a hexagon is cut out of it to fit into the large primary mirror. This process is shown

in figure 7.

• After the hexagon has been cut, it is subjected to finishing processes that give high surface accuracy.

Ion Beam Figuring (IBF) is the commonly used process which can give surface accuracies to within

few micrometres.

• The surface accuracy needed from SMP is such that it should be within the capture range of IBF.

Figure 7: Actual process flow in practice. (Hull, A et al: January 2010).

2.2. Description of Mirror geometry in Local Coordinates

Figure 8. Local segment surface in cylindrical coordinates.

• The geometry of the hexagonal mirror segments is most conveniently expressed in local cylindrical

coordinate system with origin at the centre of the mirror. The description of the surface in local segment

coordinates is given by Nelson and Temple-Raston, 1982.

• The segment coordinate system, shown in Figure 8, is centred on the part with the z-axis normal to the

mirror surface. In this orientation each term of the segment shape can be represented by a single value,

or polar monomial coefficient α (m, n) SEG, because the orthogonal terms vanish.

• When these terms are expanded as in equation 1, we obtain a generic equation that can represent a wide

range of surfaces on circular apertures. The terms m and n are called order and azimuthal periodicity

respectively, and together they describe the geometrical surface of the mirrors.

The segment surface is described in the segment coordinate system by:

Z (ρ, θ) = ∑ α𝑚,𝑛 (𝑚, 𝑛)cos (nθ) , for even m-n, and m ≥ n≥ 0 (1)

Where α (m, n) SEG is represented by α (m, n) in the equation 1.

The explicit equations for calculating α (m, n) up to fourth order are given in the paper by Nelson and

Temple-Raston (Nelson, and Temple-Raston, 1982).

Power 𝛼20𝑠𝑒𝑔 =

𝑎2

𝑘× {

2−𝐾𝑒2

4(1−𝐾𝑒2)3/2}

Astigmatism 𝛼22𝑠𝑒𝑔 =

𝑎2

𝑘× {

𝐾𝑒2

4(1−𝐾𝑒2)3/2}

Coma 𝛼31𝑠𝑒𝑔 =

𝑎3

𝑘2 × {Ke[1− (K+1)e2 ]1/2 (4−𝐾𝑒2)

4(1−𝐾𝑒2)3/2 }

Where, where k = the parent mirror paraxial ROC (60000 mm),

K = the conic constant (-.999114),

R = the off-axis distance of the segment centre,

e=R/k,

a = the segment radius.

• The amount of warping required is the difference between the spherical curvature of the mirror blank

and the desired aspheric surface shape. If kB is the radius of curvature (ROC) of the blank in its stress-

free state, then the stressing coefficients are:

Power α(2,0) STR= a2/(2kB) - α(2,0)SEG (2)

Astigmatism α(2,2)STR= -α(2,2)SEG (3)

Coma α(3,1)STR= -α(3,1)SEG (4)

• The successive higher order terms make very insignificant contribution to the geometry of the plate to

be considered for SMP theory.

• In this work we have assumed the layout, centre position of the hexagonal array of mirrors as given in

the paper by Tinsley Lab (Stephen F. Sporer, 2006).

• Table 1 shows the resultant stressing coefficients and P-V (peak to valley) stressing departures for the

extreme (innermost and outermost) TMT segments, using the minimum stress blank ROC established

(Stephen F. Sporer, 2006).

• Table 1. Stressing coefficients and P-V stressing departures for the extreme (innermost and outermost)

TMT segments

Innermost

Segment

Outermost

Segment

Description m n α(m,n)STR

(in mm)

P-Vmn

(µm)

α(m,n)STR

(in mm)

P-Vmn

(µm)

power 2 0 -0.1152 115.2 0.0514 15.4

astigmatism 2 2 0.00135 2.7 0.0828 165.6

coma 3 1 0.00090 1.8 0.00627 12.54

2.3. Warping Theory

• The theory of warping for SMP is based on Kirchhoff’s thin plate theory.

• The typical dimensions of mirror plate have thickness to diameter ratio in the order of 1/20, which is

where the above theory is applicable.

• The detailed application of thin plate theory to SMP and calculation of the loads for warping are given

in the pioneering papers of Nelson and Lubliner (Lubliner and Nelson, 1982(1)).

• The following is the key feature of the theory of warping of circular plates-

I. It turns out that, for an extremely wide range of optical surfaces including TMT, the desired

surface shapes can be achieved by the application of appropriate, continuously varying,

tangential moments and shear (normal) forces only around the perimeter of the part.

II. No warping loads are required in the interior of the part. In the same paper, given is the method

for calculating the continuous moment and shear force densities required around a circular

segment perimeter to achieve those desired warpings.

Expressions for these, along with the relationships between the polar monomial coefficients, and P-V

amplitudes for the various terms are summarized in Table 2.

Table 2. Polar monomial coefficient relationships.

Description order Azimuthal

periodicity

Polar monomial

coefficient

Continuous moment

density

Shear force density

m n α(m,n) Mmm (in

Nm/m)

Vmm

(in N/m)

power 2 0 α(2,0) (2D/a2)(1+v)

α(2,0)STR

0

astigmatism 2 2 α(2,2) (2D/a2)(1-v)

α(2,2)STR

2M22/a

coma 3 1 α(3,1) (2D/a2)(3+v)

α(3,1)STR

-M31/a

Here a is the radius of the circular glass plate.

D = Et3 / 12(1-v2), where

E= Young’s modulus of elasticity,

v= Poisson’s ratio,

t= thickness of the mirror plate.

• In practise applying instead of continuously varying moment and continuously varying shear force,

discrete bending moments and shear forces are applied (Lubliner and Nelson, 1982(2)).

• The number of levers used will depend on the dimensions of the diameter of glass mirror. In the Keck

Mirror fabrication, 24 levers were used, although further work have hinted that as less as 12 to 16

levers may suffice.

• For N equally spaced levers at θL (L=1 to N), the discrete moment and shear force required at each

lever is:

ML (θL) = (2πa/N) ∑ M𝑚,𝑛 (𝑚, 𝑛)cos (nθ) for even m-n, and, (5)

VL (θL) = (2πa/N) ∑ V𝑚,𝑛 (𝑚, 𝑛)cos (nθ) for even m-n. (6)

Where Mmn and Vmn are the continuous moment and shear force densities respectively from Table 2.

CHAPTER 3: SIMULATIONS OF PLATE WARPING

3.1. Objective of work:

• The objective of our work is to study the viability of SMP application directly on hexagonal plates.

• We have used Abaqus FEA mechanical analysis software to simulate warping loads on the circular and

hexagonal plates and compare the results.

• Abaqus is a very common and widely used analysis software, and we have used Free Student’s version

of the product.

• .The loading on the circular plate will be based on the same theory as developed for circular plates.

• This work is an exploratory study done with the purpose of gauging the feasibility of the concept of

direct SMP application on hexagonal plates

3.2. Procedure:

• We have assumed Zerodur (Ralf Jedamzik et al: 2011) as the material of the glass with the following

values of E (Young’s Modulus) and v (Poisson’s ratio):

E= 88290N/mm2 v= 0.25

• Other mechanical properties of the blank have not been considered, especially density because we are

not considering the deformations of the glass due to its own weight.

In this work, we have the used the following methodology-

I. We have referred to the Tinsley paper (Stephen F. Sporer, 2006), and taken their calculations directly

for the geometry and location of different plate segments,

II. Using the plate theory (Lubliner and Nelson, 1982(1)), calculated the required bending moments and

shear forces to obtain those geometries by warping.

III. The warping loads for the deformations can be and have been calculated separately for each of the

orthogonal components α20, α22 and α31.

IV. We have taken the innermost plate, and considered α20 component first.

V. The loads are highest for the innermost plate, and decreases as we go away from primary mirror’s

geometrical centre. So if the current theory stands good for α20 component of the innermost plate, it

would mean it can be applied to all other segments of the primary mirror.

VI. The warping load required for this component consist of only continuous constant bending moment

around the plate and the required shear load is 0, as can be seen from Table 2.

VII. We have applied the required continuous bending moment M20 on a circular plate first. The

deformations are analysed to ensure that the plate theory predictions meets the simulations results.

VIII. Then we calculate the required discrete bending moments for 16 levers from equations 5 and 6.

Applying these point loads and doing the simulations, we observed and analysed the difference in

deformations obtained in this case and them compared to the case of continuous bending moment.

IX. Next we have taken a hexagonal plate and then applied the warping loads (continuous bending moment)

on the hexagon

X. The above step was followed by application of discrete bending moments instead of continuous

moment on hexagonal plate.

XI. This exercise can be repeated for each of the other components α22 and α31, and then for all the

components together.

• The warping loads in Table 2 have been obtained from theoretical analysis, and their simple and elegant

forms is a consequence of circular symmetry of the plates. There are two points to notice-

A. The circumference of a hexagon is not equidistant from the centre of the plates, so the warping loads

calculated for circular plates is not directly applicable to the hexagon.

B. We expect different warping loads for different points on hexagon circumference.

The warping loads for a circular plate depends directly on the radius of the plate, so we have calculated

all the loads for hexagon using the radius corresponding to the point of application of the load on the

hexagon.

The validity of such as approximation have not been confirmed by us to be correct on theoretical

grounds, but we have assumed that to be correct and based my work on those calculations.

All calculations are done taking the following dimensions of the primary mirror and the blanks-

Radius of Curvature of the primary mirror, k= 60,000mm

Initial curvature of the blank in free state, kB= 62,000mm

Thickness of the plate, t= 40mm.

Radius of blank, a= 600mm

3.3. Simulations:

Thin plates can be easily simulated in Abaqus, as it has a special feature under the name “Shells” that

specially deals with thin plates using Kirchhoff’s thin plate theory.

The procedure for the simulations in Abaqus are as follows-

1. We choose Shell section, and create a circular plate of diameter 1200mm and 40mm thickness.

2. We have used the local coordinate system of the mirror segments in the simulations also, so we have

fixed the origin of the plate in the simulations as a boundary condition, which is why in all simulations

the warping at the origin of the plate is zero.

3. We have then applied the warping loads round the circumference, and then allowed Abaqus to simulate

the warping displacement that will occur due to the combination of fixed origin and the warping loads.

4. The warping occurring is shown in the figures that follow for each case. In the top-left hand box, the

contours are given which show the quantitative value of warping with their corresponding colours. The

warping is expressed in z direction in millimetres.

5. In table 1, the values of α(m,n) give the maximum value of warping required for each of the segments.

These are values that are expected at the circumference of the circular plate, and inside the plate smaller

magnitude of warping occurs.

6. In this work, we are going to check whether the theoretical loads calculated as per SMP theory in table 1

and table 2 give us the required deformations. The basis for such conclusions will be the warping

occurring on the circumference of the plate due to the calculated loads in the simulations as compared to

that required in table 1.

As can be seen from table 1, for the innermost plate, α20= -0.1152mm.

The corresponding loads for circular plates which can be obtained from table 2 are:

M20= -401.8176 N.mm/mm

V20= 0 N/mm

From equations (5) and (6),

m20= -94676 N.mm. (16 levers)

= -126235 N.mm (12 lever)

M20 and correspondingly m20 has the very simple geometrical property that its magnitude remains same

throughout the circumference of the plate.

In this work, only simulations for α20 term of the innermost plate has been done as this is only an

exploratory work. All the simulations done in this work have used the following sequence:

I. Simulation 1: Continuous bending moment M20 is applied on a circular flat plate of diameter

1200mm and thickness 40mm.

II. Simulation 2: We now load the circular plate with discrete bending moment m20 to check how this

effects the warping induced.

III. Simulation 3: continuous moment M20 is applied on a regular hexagonal plate with side 600mm

each.

IV. Simulation 4: Discrete bending moments m20 are applied on the hexagonal plate with 24 levers

spaced at equal intervals of 𝜃 = 15 degrees.

Stimulation 1

Loading a circular plate with M20 along the circumference, we get α20 in the simulation exactly equal

to the theoretical value.

The simulation obtained in form of image is given below.

As can be seen in the box at the top left corner, the warping along the circumference of the plate is -

0.1152 mm, which is exactly the α20 component of the warping required at the innermost segment in

table 1.

Figure 8: Simulation of circular plate loaded with continuous bending moment M20.

Simulation 2

Now we load the circular plate with m20 moments- bending moment in form of discrete loads which

what is actually done in practise as applying a continuous bending moment is not practically

possible.

In the present work, the circular plates are assumed to have been warped by 16 levers. The required

load is m20 = -94676 N.mm. The simulation thus done is shown below.

The maximum warping obtained at the circumference has the value -0.1180mm as opposed to the

theoretical value of -0.1152mm.

This amounts to an error of 2.43% which can be corrected in practise.

Another important point to notice is that the locus of equally warped regions is circular meaning that

our approximation of applying discrete bending moments is a sound engineering choice.

Figure 9: Simulation of a circular plate loaded with discrete bending moment M20 along the circumference.

Simulation 3

Next continuous moment M20 is applied on a regular hexagonal plate with each edge/ side of length

of 600mm each. M20= -401.8176 N.mm/mm as in the case of the circular plate.

The reason why the same value of M20 has been used even for hexagonal plates is that M20 is

independent of the radius of the circular plate for the same plate.

From the simulation, the maximum displacement at the circumference of the hexagon is -0.1153mm,

as compared to the required warping of –0.1152mm. This result is very encouraging. The geometry of

the warped plate is what we wanted it to be in table 1, and the error is less than 1%.

The locus of equally warped regions in the hexagon form circles, implying that there has been not

enough stress concentrations due to the sharp edges of the hexagon so as to cause a change in the

warping induced.

Figure 10: Simulation of continuous moment M20 on a hexagonal plate with each edge/ side of 600mm

Simulation 4

Next we apply on the hexagon discrete bending moments taking 24 levers (spread equally along the

plate plane).

The loading points are of three types as per their distances from the centre of the plate and correspond

to different bending moments m20, m’20 and m”20; m20 being the farthest point and m”20 being the nearest

points of application of the bending moment.

The distances of the loading points from the centre of the plate in decreasing order are 600mm,

569.31mm and 520mm respectively.

For a plate the discrete bending moment corresponding to α20 term depend linearly on the distance from

the centre of the plate.

The values of m20, m’20 and m”20 are:

m20 = -63117.3 N.mm

m’20 = -56931.86 N.mm

m”20 = -54701.66 N.mm

Upon doing the simulations with these loads, we find:

The outermost points of the hexagons are warped by -0.1243mm, which corresponds to a deviation

from the required geometry by 7.89%. This deviation can further be reduced by taking more number

of levers.

The locus of equally warped regions is still circular, which is an indication that there has been

insignificant stress concentrations.

Figure 11: Simulation of applying discrete bending loads on the hexagon in as discrete bending moments

m20, m’20 and m”20 rather than continuous bending moments.

CHAPTER 4: RESULT

• So as far as α20 component is concerned, as per our analysis the current SMP theory can be effectively

used to a satisfactory degree of accuracy directly on the hexagons.

• The deviation with continuous bending moment along the edges is much less 0.1%, and that with 24

levers is about 7.80%.

• SMP for the α20 can be applied directly on hexagonal plates, but with a higher number of levers than

that required for SMP on circular plates for the first monomial term α20.

• There has been no effect on warping due to stress concentrations as can be expected due to sharp

vertices of the hexagon. The locus of equally warped regions is still circular in the case of hexagonal

plates.

CHAPTER 5: CONCLUSIONS

The simulations show that the current SMP theory can be applied for hexagonal plates but with

higher number of levers.

• All the loads calculated and applied in the simulations are based on thin plate theory for circular plates,

and have been extrapolated to be applied to the hexagons. Whether such an extrapolation is

theoretically correct is something that has to be reviewed, and hence further theoretical work in this

direction is suggested.

• One very encouraging result is that there has been no effect on warping due to stress concentrations as

can be expected due to sharp vertices of the hexagon. This observation is highly critical, and says that

there is no significant stress concentration at the hexagonal vertices.

• Further work is needed to be carried out to check if the other two terms α22 and α31 can also behave the

same way for hexagonal plates under warping loads applied as per the current SMP theory.

References:

1. Lubliner and Nelson, "Stressed Mirror Polishing. 1: A technique for producing non-axisymmetric

mirrors", TMT Report No. 61. [Applied Optics, Vol. 19, No. 14, 15 July 1980]. (Keck Observatory

Report No. 21)

2. Nelson, Lubliner, Gabor, Hunt, Mast, "Stressed Mirror Polishing. 2: Fabrication of an off-axis

section of a paraboloid", TMT Report No. 62. [Applied Optics, Vol. 19, No. 14, 15 July 1980].

(Keck Observatory Report No.22)

3. Hull, A et al: January 2010: Tinsley Progress on Stress Mirror Polishing (SMP) for the Thirty

Meter Telescope (TMT) Primary Mirror Segments; BAAS 42, No. 1, 441.27 (2010)

4. http://www.tmt.org/observatory

5. http://keckobservatory.org/gallery

6. Nelson and Temple-Raston, "The Off-Axis Expansion of Conic Surfaces", UC TMT (Keck)

Report No. 91, November 1982

7. Ralf Jedamzik, Clemens Kunisch and Thomas Westerhoff, "ZERODUR® for stress mirror

polishing", Proc. SPIE 8126, (2011)

8. Mast and Nelson, "TMT Primary Mirror Segment Shape", TMT Report 58, November 2004.

……