# Presentation: 516564

## Wayne R. McKinney, Steven C. Irick, Jonathan L. Kirschman, Alastair A. MacDowell, Tony Warwick, Valeriy V. Yashchuk

 Lawrence Berkeley National Laboratory, Berkeley, California, 94720

-  Lawrence Berkeley National Laboratory, Berkeley, California, 94720

- wrmckinney@lbl.gov

**New Procedures for the Adjustment of Elliptically Bent Mirrors with the Long Trace Profiler**

*SPIE Optics and Photonics 2007*
*“**Advances in Metrology for X-Ray and EUV Optics II”*
*San Diego, August 30, 2007*

*Proceedings of SPIE 6704-15*

## Outline

- Micro-focusing often accomplished with KB pairs
- KB Pairs most economically implemented with bendable mirrors
- Systematic method of adjustment needs described
**Use LTP to adjust bendable mirrors**
- Review theory of bendable mirrors—demonstrate linearity
- Show complete method based on regression analysis
- Show 4 radii method
- Derive wave front ray trace as a check
**Give detailed example of adjustment of a bendable mirror**

**Outline**

## Theory of Bendable Mirrors

*X* is the coordinate along the beam,
- the tangential direction.
*Y* is in the direction of the deflection of the beam,
- the surface normal.
*E* is Young’s modulus of the mirror material.
-  *I(x)* is the moment of inertia as a function of position
-  along the beam, or mirror.
*M(x)* is the bending moment.

## Geometry of Tangential Elliptical Mirror

## Geometry of Mirror Bending

## General Theory I

**Add effects of couples**
**Make definitions**
**Add definitions**

## General Theory II

**Integrate to**
**slope**
**Using new definitions**
**            ****The slope is a linear function of f1 and f2**

## Ellipse Theory I

**Let superscript 0**
**indicate  ellipse**
**Let ellipse be**
**imperfect**
**Measure the optic**
**with the LTP**
**Move one bender**
**adjustment, C1, **
**and re-measure**
**Subtract**

## Ellipse Theory II

**Solve for f1,**
**neglecting errors**
**Repeat process,**
**moving second**
**bender adjustment**
**Apply regression**
**analysis**

## Ellipse Theory III

**Solve for best **_**CHANGE**_** in bender**
**adjustments**
**With dispersion**
**Where sigma is:**
**and we differ from the reference below by C****0 ****and using the LTP, not beamline measurements**

**O. Hignette, A. Freund and E. Chinchio, "Incoherent X-ray mirror surface**
** ****metrology," in *****Materials, Manufacturing, and Measurement for Synchrotron***
*** ******Radiation Mirrors. San Diego, CA***** (1997).**

## Now average over the previous method—compute 4 radii

## Why do this?

**Left hand adjuster**
**affects left curvature**
**Right hand adjuster**
**affects right curvature**
**Center curvature and**
**total curvature monitor**
**bending**

## Why do this?  II

**Left hand adjuster**
**affects left curvature**
**Right hand adjuster**
**affects right curvature**
**Center curvature and**
**total curvature monitor**
**bending**

## 4 Radii details

**Fit noisy LTP data to 5****th**** order polynomial, a compromise between low pass filtering, and available adjustments**
**Integrate this fit term by term, which spreads errors out, and is better than using Simpson’s rule, for example**
**Divide into three regions; left, center, and right**
**Do regression fit exactly like the preceding analysis, except we are averaging the curvatures over many points, instead of computing the curvatures for every LTP data point.**

## 4 Radii math I

**Trick from literature,**
**use sum of differences of areas of circles**
**differentiate**
**Divide out**
**Expand**
**Define variables**
**Recast**

## 4 Radii math II

**Write in matrix form:**
**(Solve with IDL)**
**Go back to original**
**variables**
**Retrieve dispersion**
**Figure complicated, but straightforward**
**propagation of error**

## As an overall check, we use a wave front ray trace

## Wave front ray trace details

**Optical path is distance**
**where:**
**Make Mac Laurin series**
**Repeat for other side**
**Compute ray intercept errors**

## Now Let’s adjust

**We will MONITOR with the 4-radii method, but**
**ADJUST by the full regression method based on each point of the LTP data**

## 4-radii regression-based 1st measurement of a mirror 

|  | original r' ≠ 3 m | ellipse r' = 3 m | after 1 adjustment |
| --- | --- | --- | --- |
| Rtotal | 211.0 +/- 0.10 m | 102.8 +/- 0.05 m | 104.6 +/- 0.18 m |
| Rleft | 220.9 +/- 0.26 m | 105.7 +/- 0.03 m | 117.0 +/- 0.19 m |
| Rcenter | 209.6 +/- 0.05 m | 102.9 +/- 0.03 m | 104.6 +/- 0.11 m |
| Rright | 210.6 +/- 0.11 m | 100.0 +/- 0.03 m | 96.6 +/- 0.06 m |

## Summary of the data taken for the 1st adjustment 

|  | upstream encoder | downstream encoder |
| --- | --- | --- |
| original | 105.825 | 126.294 |
| add 10 up | 115.825 | 126.294 |
| add 10 down | 105.825 | 136.294 |
| δC1, δC2 calculated | +83.448 | +152.125 |
| new settings | 189.273 | 278.419 |

## 4-radii regression-based 2nd measurement of a mirror 

|  | original 1st adj | ellipse r' =3 m | after 2nd adjustment |
| --- | --- | --- | --- |
| Rtotal| | 104.6 +/- 0.18 | 102.8 +/- 0.05 m | 102.49 +/- 0.06 |
| Rleft | 117.0 +/- 0.19 | 105.7 +/- 0.03 m | 107.00 +/- 0.08 |
| Rcenter | 104.6 +/- 0.11 | 102.9 +/- 0.03 m | 102.28 +/- 0.04 |
| Rright |  96.6 +/- 0.06 | 100.0 +/- 0.03 m | 100.12 +/- 0.01 |

## Summary of the data taken for the 2nd adjustment 

|  | upstream encoder | downstream encoder |
| --- | --- | --- |
| original | 189.273 | 278.419 |
| add 10 up | 199.273 | 278.419 |
| add 10 down | 189.273 | 288.419 |
| δC1, δC2 calculated | +28.390  | -20.557 |
| new settings | 217.663 | 257.862 |

## One more iteration gave delta values of δC1 = +1.035 and δC2= ‐1.154. 

**One more iteration gave delta values of *****δC1***** = +1.035 and *****δC2*****= ‐1.154. **
**These small numbers confirm that we have reached the best possible adjustment with this method **

## Now a final check by wave front ray trace

## Conclusions

**Basic linearity of two couple bending demonstrated.**
**Two methods given explicitly which are approximately 10 times faster than fitting ellipses in the lab.**
**Method gives complete calibration curve for the bender allowing changes in conjugate distances**
**without removal from the beamline.**

## Observations for the future

**It is our observation that ad hoc adjustments to bendable mirrors at the beam may not be indicated, and can often ruin the careful adjustment of the mirror accomplished in metrology. **
**The next step:  more tightly control *****r*****, *****r*****' and *****θ***** so that the metrology/adjustment process may be completely integrated **

## THANKS!

**L****AWRENCE**** B****ERKELEY**** N****ATIONAL**** L****ABORATORY**

**O****PTICAL**** M****ETROLOGY**** L****ABORATORY**
**E****XPERIMENTAL**** S****YSTEMS**** G****ROUP **
**A****DVANCED**** L****IGHT**** S****OURCE**

*June 16, 2005*

**THANKS!**