Metrology of trapezoidal shape mirror benders
Josep Nicolas
CELLS-ALBA, 08290 Cerdanyola del Vallès, Spain
Abstract
Slope error decoupling
When a high demagnification focusing is required for a plano-elliptical mirror, the required ellipse
cannot be approximated by the 3rd order polynomial that can be reached by introducing two
bending moments onto a solid, elastic parallelepiped. A typical approach to improve this is to cut
the mirror bulk into a trapezoidal shape, in a way that the required ellipse can be approximated
also to higher polynomial orders. In this case, the introduced deformation is described by an infinite
series, which depends on three parameters: the two bending moments, and the width variation
ratio. In this case, the optimization of the mirror cannot be done by simply fitting the measured
profile to a 3rd degree polynomial, and identifying the ellipse parameters from the polynomial
coefficients, since this leads to biased parameters. In this work we propose a method to accurately
fit the ellipse parameters in terms of the elastic deformation of the beam and the residual slope
error. This permits to fit the ellipse parameters accurately, independently of the polishing slope
error.
The width ratio of the mirror 1/η is normally a source of error. It can be fit by
minimizing the differences between residuals for different bending conditions.
Once this is done, the residual slope error is independent of the bending
condition.
Error due to wrong trapezoid constant
200
residual profile (nm)
150
1.145
1.14
160
100
140
50
0
120
-50
1.135
-100
1.13
-100
1.125
-50
0
position (mm)
50
100
Error due to wrong trapezoid constant
200
-0.96
-0.95
η (m)
-0.94
100
80
60
-0.93
150
residual profile (nm)
Elastic deformation of a trapezoid
Differences of slope
error
for
different
bending conditions
40
100
50
20
0
0
-50
-100
-100
-50
0
position (mm)
50
Bending torques
100
-100
-50
0
x (mm)
50
100
Residual profile for different
bending
conditions.
Blue
corresponds to mirror in a
kinematic mount
z
Reflective surface
residual profile (nm)
match error (nm RMS)
1.15
-0.97
M0-∆M
180
y
x
Ellipse parameters
M0+∆M
Since the mirror deformation is described with high accuracy, the estimation of
the deformation parameters is accurate, which allows to fit the right ellipse and
minimize the error.
Error to the best fit ellipse
2.5
b0
Measurement of the profile
Fit measurement to a function, to
obtain coefficients
T2,T3
Obtain ellipse parameters from fit
coefficients
p, q
Generate the ideal ellipse from obtained
parameters
zp,q(x)
Total Slope Error (µrad)
η
zLTP(x)
h
 x
b(x ) = b0 1 + 
 η
The width of the mirror is given by a linear function:
The elastic deformation of the mirror is ruled by the Euler-Bernouilli equation:
Subtract the ideal ellipse from
measurement to obtain residual error
h b0  x  d z ( x )
2x
1 + 
= M 0 + ∆M
E
2
L
12  η  dx
3
2
zLTP(x)-zp,q(x)
200
3
25
30
2
2
E
∆
E
∆
ζ E2 = 2 L2 + 3 L4
3
20
100
50
0
-50
-100
Nominal ellipse fitting
-150
-100
-50
0
x (mm)
50
100
Alternatively we propose the following expression of the solution:
 2
 x 
3
2
zT ( x ) = T2 x + T3 3ηx − 6 η + η x log1 + 
 η 

(
15
20
Sample Nr.
150
The error term is small, and often can be neglected.
Then, one fits the measured profile to a 4th order
polynomial, to determine the error.
2
10
If the measured profile is fit to a polynomial, the resulting ellipse parameters are
not accurately determined, what contributes to the residual slope error.
Error for η = 0.95 m
3
η G(x) (nm)
4
5
Slope error with respect to the best fit ellipse, for
different bending conditions, and using different
deformation functions
Slope Error
 x
zT (x ) = T2 x + T3 x + T4 x + T3η G 
η 
3
1.5
1
The solution is typically written as a 4th order polynomial plus an error term
2
Polynomial fit (3)
Polynomial fit (4)
Elastic fn. fit
2
)
By using the proposed fitting, one can accurately adjust the bending torques so
as to obtain the required nominal ellipse, still preserving the slope error
measured at the mirror being flat.
The propose fitting is used in an automated procedure to adjust the mirror
bending torques.
30
Optimization of the shape of the ellipse
• It allows to decouple the elastic deformation from the polishing error.
p=4.827 m, q=1.245 m, η=0.88 µrad RMS
2
20
• It provides self-consistent ellipse coefficients, allowing exact optimization of
the mirror figure.
• All the coefficients to be fit, are independent each other.
• Allows to predict the slope error at the nominal ellipse.
6M 0
T2 =
Eh 3b0
2M 0
4 ∆M
− 3
T3 =
3
Eh b0 L Eh ηb0
 1 1  cos α
E2 =  + 
 p q 4
 1
1  sin 2α
E3 =  2 − 2 
q  16
p
1 rad
1.5
∆ E3/E3µ
• It is a linear combination of functions that do not depend on the bending
torques, therefore it allows a simple fitting by the LSM method.
Height (nm)
10
1
-10
-20
Polynomial fit (3)
Polynomial fit (4)
Elastic fn. fit
0.5
-30
-40
-30
-20
-100
-50
0
Position (mm)
50
100
p=4.825 m q=1.250 m
0
-0.5
-50
0
-10
0
∆ E2/E2µ
1 rad
Residual error with respect to a best
fit ellipse obtained by simplex
optimiztion of p,q.
The resulting ellipse is virtually
identical to the nominal one.
Path of the bender in the E2,E3 configuration space
during automatic optimization. To use polynomial fit
would lead to a biased ellipse.
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