introduction and manufacturing of oblate spheroid...

Introduction and manufacturing of oblate spheroid mirror

Li Depei, Yuan Lvjun, He Li and Wang Bin

National Astronomical Observatories / Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing 210042, China

Key Laboratory of Astronomical Optics & Technology, Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing 210042, China

ABSTRACT

The oblate spheroid mirror is one of the two dimension rotation surface. It is an axially symmetrical aspheric surface, but not free from spheric aberration surface. It can’t be used individually in optical systems. The oblate spheroid mirror can make up with another mirror as a free from spheric aberration optic systems. So understand the oblate spheroid mirror is essential.

This paper introduced the oblate spheroid mirror and some testing method, furthermore, it also introduced how to design and choose from those method with author’s experience when we face different oblate spheroid surface.

Key words: oblate spheroid mirror, testing method

1. WHAT IS OBLATE SPHEROID we can know oblate spheroid from several viewpoints.

1.1 Viewed from 2D equation

We took a vertex in the origin, the x axis is the rotation axis of symmetry, and then the public equation of the conical is 0

2 2 22 (1 )y Rx e x= − − ………(1) or

20 02 2

2

(1 )1

R R e yx

e− − −

=−

………(2)

Among them 0R is the osculating circle of the curve vertex. We can draw a group of curve by taking different range of

value e2

When e2＝0 (circle),

0<e2<1 (elliptic),

e2＝1 (parabolic),

e2>1 (hyperbolic),

e2<0 (oblate spheroid), ( Figure 1)

From figure 1, we can see that for elliptic, parabolic and hyperbolic, the osculating circles of the vertex are on the right side of the curve, which are included inside the curve.

For flat sphere, its osculating circles of the vertex, is located in the left curve, in the outside of the curve.

6th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Advanced OpticalManufacturing Technologies, edited by Li Yang, Eric Ruch, Shengyi Li, Proc. of SPIE Vol. 8416, 84161Z

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Figure 1 Figure 2

1.2 With these curves rotated round the x axis, we can get the corresponding quadric surface

To be worth to emphasize, when the long axis of the elliptic rotated around the x axis, we can get ellipsoid. It has two conjugate points S and S ', or called no aberration points, which are all in the x axis. Such as showed figure 2. Then if we took out a surface form the shorter axis direction. Taking note that this plane is not a axial symmetry plane, but it is still a no aberration surface, its no aberration points are S and S '. We call it off-axis ellipsoid or partial shaft ellipsoid. If we cut it into round, in its vertex M the surface is astigmatic surface, not axial symmetry surface, namely on each direction the radius of curvature is different. The direction of the biggest radius of curvature is perpendicular to the direction of the shortest radius of curvature.

The no aberration point is if we put point light in S, the ray will converge at S 'point with no aberration by reflecting from the concave surface, and the reverse is also true.

1.3 If we take the short axis of the ellipse and the x abscissa axle coincidence, and make the short axis rotate around the x abscissa axle, we can get the oblate spheroid. From figure 1 we can see that it is an axial symmetry surface, and the symmetry axis is x axis. At this time the track of the two no aberration points formed cirque. Surely the oblate spheroid is not a no aberration surface.

1.4 Taking the view from the normal aberration of flat spherical. Normal aberration ΔR y ＝M0y-M00, M is vertex of mirror. For concave quadric surface, the distribution of its normal aberration sees Figure 3(equivalent to negative spherical aberration).

Figure 3 Figure 4

Among them x is the vector height of aspheric surface, 00 is the curvature center of the vertex, 0y is the cut point of the normal (the normal of aspheric surface half diameter H) and x axis (note: it is not the center of curvature of besides binning). We took a triangle HNOy, there is

2

2 00 0 022 2 2 2 2 2 2 2 2

0 02 2 2 2 2 2 2

(1 ) (1 )

(1 ) (3)

y R e x x y R e x y R R R e y

y R e y R e y

ρ⎧ ⎫⎡ ⎤⎪ ⎪⎡ ⎤ ⎢ ⎥⎡ ⎤= + + − = + − − = + − − − −⎨ ⎬⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭

= + − − = + KK



So normal length ρ is not the curvature radius, but when e2=0, ρ＝0R＝R. On the other line the absolute value of the

curvature radius is: 22 3(1 )yR

y′+

=′′

…………(3)

On any point of the curve, the coordinates of centre curvature is: 2(1 )y yX x

y′ ′+

= −′′ ,

21 yY yy′+

= +′′

The distance from 00 to 0y is called "normal aberration ", equal to e2x. For an ellipsoid ΔRy <x. For a paraboloid ΔRy＝x. For a hyperboloid ΔRy >x, and for a concave oblate spheroid, the distribution of its normal aberration sees figure 4.The distance from the cut point of skew normal to the vertex is longer than the distance from the cut point of half aperture normal to the vertex (equivalent to the positive spherical aberration). So ΔRy＝M0y-M00＝e2x, equal to a negative value. As long as e2＜0, it is oblate spheroid, no longer to point equal, greater than, or less than 1.

2．PROCESSING METHOD OF OBLATE SPHEROID

For processing concave or convex oblate spheroid, it has three methods, see figure 5, 6. Now we take a look at the concave oblate spheroid showed in figure5.

2.1 If we take the sphere which is tangent to the vertex to transit, see figure 5 upper left. The sphere is tangent to the oblate spheroid in vertex and then separated most at H. Now we move the sphere to the right side until it connect with H, then we will polish largest amount of glass at the center of the mirror and gradually reduced till zero in the edge. We can see the diagram in figure 5 left lower.

2.2 If we take the sphere which is through the half diameter H of the oblate spheroid to transit, see the middle picture in figure 5. We will polish largest amount of glass at the edge of mirror and zero in the centre.

Figure 5 Figure 6

2.3 If we take the sphere which is tangent to the oblate spheroid in the center and still through the half diameter H of the oblate spheroid to transit, known as the best fit sphere, see the higher right picture in figure 5. We will polish largest amount of glass at the 0.707 H of the mirror and zero in the center and edge. See the lower right picture in figure 5. The largest polishing depth is 0

maxδ , compared with two former methods, their polishing depth is 4 0maxδ . Using this method the

volume of the fix is only half of the former method, not a quarter.

Figure 6 is drawing for convex oblate spheroid. There are also three methods to take the sphere, not much elaboration. When we take the most closed compare sphere, see right upper and downer picture in the schematic diagram 6. We can



see that we should polish the most amount of glass at 0.707H of the concave oblate spheroid and zero at edge and centre of it. The largest polishing depth is still 0

maxδ , which is contrary to polish ellipsoid, paraboloid and hyperboloid.

3．SEVERAL TEST METHOD OF OBLATE SPHEROID

Now we take the case of the concave oblate spheroid used in domestic 2.16 meters astronomical telescope [1] to explain. The parameters of this concave oblate are: the optical diameter is 403 mm, the vertex curvature radius is 5373.7 mm; e2= -0.2585. Through the parameters we can calculate the biggest asphericity is about 0.089μm. The normal aberration is Δ≈1mm.

According to the specific examples, we can use the following test methods.

3.1Moire fringe technique

For the concave oblate spheroid with small asphericity this method can be used. We can draw its interference fringes comparing with the spherical wave, try to take more stripes, such as 50, then compact it into small plate just like figure 7. We can put this plate on “Laser spherical wave interferometer” [2] and contrast it with the actual interference fringes when test. Then judge the surface shape of the work piece by seeing whether the Moire fringe occurs between them is straight, or whether it has local error or not. Or we can take a photo of the existing stripe like figure 8. Enlarging the photo and comparing with figure7, then estimates that surface shape. Obviously it can be used more conveniently if we made it into a professional instrument. Because we needed only one oblate ellipsoid, so we did not use this method.

Figure 7 Figure 8

3.2 Максутов[3] compensation inspection

This method is to put the point light source on the optical axis within the focus of concave spherical surface. Then the divergent beam will be reflected by the concave spherical and form divergent beam with negative spherical aberration, which is consistent with the normal of the mirror. Now we can make compensation tests. The normal of concave oblate spheroid forms positive spherical aberration. So we should use convex spherical compensation mirror instead of concave spherical compensation mirror, then the problem can be solved. But the calculation results are not very well, so this method cannot be used.

3.3 Reflex compensation test method

We know that concave ellipsoid has two aberration-free point S and S '(or called conjugate point). Put point light source on S, after the ray be reflected by the concave ellipsoid, no matter how high the belt is, it will converge in S 'point with no aberration. See figure 9.

Figure 9 Figure 10

For concave ellipsoid, if we don't put the point light source on aberration-free point but on S1 within S (close to the ellipsoid vertex), see figure 10, the reflected ray will forms positive spherical aberration . And the distance from S1’to the



ellipsoid vertex is far than the distance of S'. For convex hyperboloid this test method has been introduced in the document[4].

If we put the point light source in one point S2 (away from the ellipsoid vertex) outside S, see figure 11. The reflected beam will converge near S2’ but not S’, it also has negative spherical aberration ΔSy’ and the distance between S2’ to the ellipsoid vertex is shorter than distance of S'.

Figure 11 Figure 12

When we test concave oblate spheroid, we can put it on the left side of the figure 10, see figure 12. Choose an ellipsoid parameter and position of S, make an ellipsoid formed aberration as Δ S 'y which is equal to the normal aberration ΔRy of oblate spheroid. Then when we put point light in S, after the ray has been reflected by an ellipsoid, it will form spherical aberration which is equal to the normal aberration of oblate spheroid. By the way, we can only make the normal of half diameter and skew ray to match together, the biggest spherical aberration is in 0.7H. Therefore it requires us to calculate if the remnant surface test error could be accepted when using this method.

The author had calculated the special case before, the result is the test optical path is too long .The oblate spheroid its own vertex radius of curvature has already 4.5 meters, and the distance from the second conjugate point of ellipsoid to the mirror is also too long. We can choose to put the light source on S ', such as figure 13.

Figure 13

We put the point light source in the second no aberration point S0 of the ellipsoid. Through controlling the ellipsoid parameter and its distance to the oblate spheroid, we can make ΔS’y＝ΔRy to test oblate spheroid.

When the point light source in S0, the spherical aberration produced by the ellipsoid is: 0

2 2 22 0 1 0 11 0 0

21 0 1

( )'(2 )

yS e S RS yR S R

− −Δ ≈

−

………(4) 22 22 0

22y

eR yR

Δ ≈ ………(5)

Make two equal and 0 10

22

' 'y

y

S S yR yR

α− − −

− ≈ = =

S0: 0

10 0

2 2

0

2

1

R

R RS

α

α

=

+

………(6)



And 0 0

22 3 21 2 11 0 0

2 2

1 1( ) ( 1)2

R e ReR Rα α α

= + −−

…………(7)

So if we make sure of the S0 then we can find out 0

1R and 21e which are initial values of the ellipsoid from formula (6),

(7). The skew ray and adaxial ray are all reflected by the ellipsoid and focus near S, but their focus are different. Using these initial values, we can make the distance minimum through changing 2

1e . Then put point light source in S, we can calculate the remaining aberration of all belts and accuracy of this test result.

3.4 Example

3.4.1 Set the initial value: give a series of values to y1 (ellipsoid half diameter), half diameter of the oblate spheroid is 203, the distance between point light source to oblate spheroid vertex is 100, see figure 14.

Figure 14 Figure 15

021 1

20 2 21 1

100 5469.7106y yS R e xtgu tgu

≈ + − + = + (neglect x1)

From formula(6)、(7)we can calculate table1:

y1 S0 0

1R 21e

100 8112．99754 3990.0733 1．158378249

110 8377．32624 4319.96123 1．014582543

125 8773．81927 4803.77962 0．847917346

200 10756.28446 7093.26688 0．432012924

Table 1

From the table 1 we can see when y1 (half diameter of ellipsoid) is small, 21e > 1 it is hyperboloid. Along with the

increase of y1 the 21e gradually decreased to zero. We choose y1=125, which means an ellipsoid with diameter 250, to

calculate the result. Then we take the data into the formula (4), (5) and work out ' 0.99161ySΔ ≈ , 0.99217yRΔ ≈ . These

shows the initial value is right.

3.4.2 Ray tracing. Of course it is a lot more convenient by using ZEMAX calculation program now, but in order to understand the whole process, we provide more details below.

3.4.2.1 First after the half diameter ray and skew ray be reflected by the ellipsoid, work out the difference value of focus on optical axis near S between them. Then change 2

1e to make the difference minimum and put light point source on the focus. See figure 15.



When the slope 'ytgu and the intersection N of linear MN and x axis are known, calculate the intersection 1 1( , )M x y

of MN and the conic, we can solve the simultaneous equation for the conic and the linear.

'( ' )y yy S x tgu= − and 02 2 2

12 (1 )y R x e x= − − then

0 0' 2 ' ' 2 ' 2 2 2 ' 2 2 '

1 11 2 2 '

1

( ) ( ) (1 ) '

(1 )y y y y y y y

y

R S tg u R S tg u e tg u S tg ux

e tg u

⎡ ⎤+ − + − − +⎣ ⎦=⎡ ⎤− +⎣ ⎦

…………(8)

' '1 1( )y yy S x tgu= − ……………(9)

The angle of M point normal and x axis 1 1

02

1 1 1(1 )

ytgR e x

ϕ −⎡ ⎤⎢ ⎥=⎢ ⎥− −⎣ ⎦

The angle of reflected ray and x axis '2y yu uϕ= −

The intercept of reflected ray 1 1yy

y

x tgu yS

tgu+

=

Paraxial ray cut point 0'0 1

0 0'0 12

S RSS R

=−

2' ' 1 1

0

1

3305.7349682

y HH

y yS Stgu R

= = + =

' ' 20 2 2 3304.742796yS S e x= − =

The computed result: when 21 0.8500e = ,

0 0.00039327yS S SΔ = − = −

when 21 0.8501e = ,

0 0.00084346yS S SΔ = − = +

We take 21e = 0.85, all parameter of the ellipsoid using for compensation have been confirmed further.

3.4.2.2 Distinguishing the oblate ellipsoid for 10 zone, then for each zone use above formula to make optical track, after the normal of these zones has been reflected by the certain ellipsoid, calculate the aberration value near S, the results have been showed in table 2 (0.7H is 143.5 high, with H = 203).

y 0 20 40 60 80 100

ySΔ ±0.00000000 +0.00585083 +0.00206675 -0.00016931 -0.00083410 -0.00105596

y 120 143.5 160 180 203

ySΔ -0.00216000 -0.00180396 -0.00194184 -.00170088 -0.00039274

Table 2

3.4.2.3 Surface test error: From table 2 we know aberration of all zone is not zero. If it is zero, it shows that the point source is in S0, this method has no theoretic test error.

But now the spherical aberration of all zone are not zero. Therefore, we can’t get a batter oblate ellipsoid by using this method to test and polish. Its surface will have aberration. If this aberration is less than the mirror precision requirement, it can be ignored and can be used.

Here we estimate the value of this aberration. From the figure 15 it can be seen, that when a no aberration point has remaining aberration ySΔ , we can calculate '

ySΔ through the following formula.

''

'

sinsin

y yy y

y y

u SS S

u SΔ ≈ −Δ , '

ySΔ is the aberration near the curvature center of the oblate ellipsoid vertex.



It means that when we use this method, all belt normal of oblate spheroid will have such deviation. It can be converted to angular aberration ηy through the following formula. See figure 16.

y yδ ηΔ ≈ ⋅Δ

'2

022

yy

S y

Rη

Δ ⋅≈ −

, _

y yδ ηΔ ≈ ⋅Δ

Figure 16 _

yη is average value for yη ( When the mirror is smooth we divide it into limited belt).In this way integral is turning into sum.

_

0

H

y y d yδ η= ⋅∫ can use _

0

H

y y yδ η≈ ⋅ Δ∑ …………(10) to replace

Calculate the largest deviation value 14000/000038.00max λμδ ≈≈ m (λ=550 nm). The result seems to be available.

But through further analysis, this ellipsoid is very difficult to be tested, because the distance S from the first no aberration point to the ellipsoid vertex is 2499, while the distance S’ from the second no aberration point to the ellipsoid vertex is about 61551,it’s too far. In addition when we test the oblate spheroid, the test point F/# is 1/35(D＝250, S0＝8792), which is too small and inconvenience for test, at the same time the air flow is serious.

From the above consideration, paraboloid can also be used. Choosing y＝111.15 from table1, e21＝1.00004511, then we

use concave paraboloid with D＝250, F＝2402 to instead of concave ellipsoid (because we can use plane auto-collimation method to test the concave paraboloid). It has small remaining surface aberration after calculation. It seems

can be used, but its F/# is still too small. We also can use concave sphere, make e21＝0, H＝222,

0

1R ＝4355, the maximum remaining surface aberration is about λ/2000. This method also has small f/#=1/38 and long S0, so it is not applicable.

These testing method introduced above may be applicable to other oblate ellipsoid, but for the special case it is limited.

The convex oblate spheroid can use concave spherical for compensation test.

3.5 Refraction type compensation inspection:

We can use a positive lens, producing negative spherical aberration, to test conicoid surface with e2＞0. We also can use a negative lens to test oblate spheroid. Now we apply refraction compensation to test the oblate spheroid on above case. This method can also be used for other oblate spheroid test.

3.5.1 Divide the oblate spheroid into a number of belt(we divide it into 11), find out vector height x of all belt, the distance from normal cut point to the vertex is

02

yR R e x= + ⋅ ,and at the same time calculate the angle by the normal and

the optical axis 1y

y

yu tgR x

−=−

. See table 3.

y x 22e x xexeR Ry

22

22

0

7028.53702 −=•+= yu

0 0 0 5370.7028 0°.00000000

20 0．037239292 0.009626357 5370.65593 0°.21336506

40 0.148958998 0.038505901 5370.51533 0°.42673536

60 0.335165037 0.086640162 5370.28099 0°.64011618



DO

DO

OBLATE

-51=-606.839

-U=-2.0328

-R1=-84.15

n= 1.5163

d1= 10

d2= 4770.7

-R2=-90

-R0=-5370.7

en2=-0.2585

Y=

20.2

8.000030

y=

40.5

0.000103

y=

60.7

0.000170

Y=

81.0

0.000161

y=

101.3

0.000013

y=

121.6

)-0.000304

y=

141.9

-0.000745

y=

162.2

-0.001133

y=

182.6

-0.001097

y=

203.0

0.008800

d=

tl=

d=

d=tl=

d=

d=

d=

d=

d=

1/(.000170+.001133

767.4597084

00001982.07.19.

OBLATE

.HARTMANN

TEST"

2.370

3.052

2.359

3.028

3.716

4.394

3.737

4.416

5.071

5.728

5.084

5.786

6.401

7.074

6.461

7.151

7.718

8.416

7.838

8.542

9.059

9.731

9.213

9.919

10.380

11.051

10.600

11.302

X1+82= 301.5

Y1= 42

DAI KUAN D= 12

RH= 5370.7

e,2=-0.2585

Y=

42,d=-0.01143

Y=

54,d=-0.02193

®Y=

66,4=-0.02625

Y.

78,d=-0.02181

Y=

90,4=-0.02326

Y.

102.4=-0.02090

Y=

114,4=-0.01347

Y=

126,4=-0.00967

Y=

138 d=-0.00107

Y=

150:d= 0.00742

Y=

162,4= 0.00986

Y=

174,d= 0.01085,"

Y=

186,4= 0.00874

Y=

198.4= 0.00000

Rf1S= 0.01553A

1/(0.02625 *0.01085

26.9541779

80 0.595867103 0.154031646 5369.95290 0°.85351276

100 0.931078943 0.240683906 5369.53104 1°.06693036

120 1.340818037 0.346601462 5369.01538 1°.28037425

143.5 1.917522646 0.495679604 5368.28960 1°.53121152

160 2.383967898 0.616255701 5367.70257 1°.70736194

180 3.017433373 0.780006526 5366.90536 1°.92091629

203 3.838190544 0.992172255 5365.87244 2°.16656270

Table 3

3.5.2 Given the initial value of curvature radius according to the existing concave spherical templates. See figure 17.

The data from figure is the final result. Lens diameter=50 (The optical diameter is φ 45.12), r2=90, center thickness=10, R1= 84.15. The results of other parameters see figure17.

Figure 17

3.5.3 First calculated focus aberration of ten normal of the oblate spheroid which have been refracted by the negative lens, see table 4.

From table 4 arrange 2 we can see maximum deviation at range (absolute value sum of deviation) of around 0.01.

Table 4 Table 5

y 'ySΔ

8(10 )y arcη −

0 0 0

20 －0.0002704 ＋0.01571

40 －0.001047607 ＋0.1274

60 －0.002222842 ＋0.4084

80 －0.0036097 ＋0.8831

100 －0.004948037 ＋1.5149

120 －0.005898991 ＋2.1642

143.3 －0.0059458 ＋2.6040

160 －0.004864482 ＋2.3754

180 －0.001770135 ＋0.9704

203 ＋0.005074923 －3.1329



3.5.4 Test error: put point source in S, then use the rays which are connected by S and the refraction points on r1 surface of above ten belt normal refracted by crescent negative lens, in turn to find out their interception S2 and the angle u2 on optical axis after these 10 concentric divergent rays have been refracted by negative lens.

Use the equation (8), (9) to seek the intersection coordinate (x, y) of u2, S2 ray and the oblate spheroid, then compare to the normal of the corresponding points on the oblate spheroid, seek the angle aberration ηy, see table 4 range 3 ηy, with 10－8 radian for the unit. Using the equation (10) to seek the maximum deviation value 750/0

max λδ ≈ (λ＝550nm) and 636/0

max λδ ≈ (λ＝632.8nm). That is to say we don't have to change another negative lens when we use He-Ne laser spherical interferometers to make quantitative test.

4. RESULT Due to the polishing errors, resetting error and so on, we should give some error range to calculate, to see how much the test residual error is? We had calculated the special case, which shows that the residual error is less than 1/50 wavelength at all conditions. As long as these parameters are controlled within limits, we can obtain satisfactory results. For above case refraction inspection is more convenient than reflex inspection. The test F/# is 1/14 which is also small but still can be accepted. More important is the oblate spheroid center can also be tested. Finally we use this project, see figure 17. The test result is satisfactory after using it to the optical aperture 2.16 meters telescope. The design result and the Hartmann test results after processing, respectively see table 5, table 6. The oblate ellipsoid was processed by ZengJinZhu Senior experimentalist.

REFERENCES 1. Su.D.Q. and so on. 2.16 meters astronomical telescope project essays, chief editor Su.D.Q, China science press,2001.26-35

2. Li.D.P. and so on. ACTA Astron. Sinica,1975, 16(1): 51-64

3. Yang.S.J. translation. Astronomical optical technology, Science Press,1964:239-214

4. Pan.J.H. ACTA Astron. Sinica,1960, 8(1): 70-79



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