1. Status of Modeling of Damage Effects on Final Optics Mirror Performance T.K. Mau, M.S. Tillack Center for Energy Research Fusion Energy Division University of California, San Diego High Average Power Laser Program Workshop April 4-5, 2002 General Atomics, San Diego

2. Background and Objectives GIMM (Grazing Incidence Metal Mirror) has been proposed as the final optical element that provides uniform illumination of the DT fusion targets by the driver laser beams to achieve a full target implosion. Threats to GIMMs such as X- and  -rays, neutrons, laser, charged particles and condensable target and chamber materials can cause damage to the mirror surface, resulting in increased laser absorption, reduced damage threshold, shorter lifetime, and reduced beam quality (mirror reflectivity, focusing, and illumination profile). The objectives of final optics modeling are threefold: (1)Quantify mirror damage effects on mirror and beam performance. (2)Provide analysis of laser-target experimental results. (3) Provide design windows for the GIMM in an IFE power plant e.g., power density threshold, coating and material selection, etc.

3. Prometheus-L reactor building layout (30 m) (SOMBRERO values in red) (20 m) Final Mirror is a Critical Component in a Laser-Driven IFE Power Plant GIMM Typically there are 60 beamlines all focusing on the target at the center of the chamber. The incident angle is 80 o from the GIMM surface normal.

4. Mirror Defects and Damage Types, and Approaches to Assess their Effects on Beam Quality √√ ( Ray tracing )

5. Ray Tracing Analysis of Gross Mirror Deformation Limits The ZEMAX optical design software was used for the analysis. Gross deformation:  due to thermal or gravity load, or fabrication defect.  Deformation:  = a m 2 /2r c [ surface sag ] Analysis assumes that flat GIMM surface acquires a curvature (r c ), and calculates resultant changes in beam spot sizes on the target, and intensity profiles as the deformation size is varied. Beam propagation between focusing mirror and target is modeled. Prometheus-L final optics system as a reference: Wavelength = 248 nm (KrF) Focusing mirror focal length = 30 m GIMM to target distance = 20 m Mirror radius a m = 0.3 m Grazing incidence angle = 80 o Target radius = 3 mm Beam spot size a sp = 0.64 mm Wall Target GIMM Focusing Mirror Laser Beam

6. Typical Output from a ZEMAX Run Ray Trajectories GIMM 0ne million rays used;  = 80 o Focusing Mirror Target Mirror Surface Sag r c = 5x10 4 m (mm) -0.3m +0.3m 2-D Illumination Profile 1-D Illumination Profiles +3mm -3mm +3mm x y Y-scan X-scan

7. Mirror Curvature Causes Spot Size to Elongate The isotropic surface curvature causes the rays to diverge preferentially in the direction of beam propagation.

8. Spot Size and Illumination Constraints Limit Allowable Gross Mirror Deformation The dominant effect of gross deformation is enlargement (and elongation) of beam spot size, leading to intensity reduction and beam overlap. Secondary effect is non-uniform illumination:  I / I ~ 2% for  = 0.46  m Mirror surface sag limit for grazing incidence is :  < 0.2  m, (for a mirror of 0.3 m radius) with the criteria:  I / I < 1%, and  a sp / a sp < 10%. y-scan  = 0.92  m 0.46  m 0m0m  = 0  m  = 0.46  m  = 0.92  m -2mm +2mm Relative Illumination

9. In the presence of a scatterer (mirror surface), total field is given by where is given by where S 0 is surface of scatterer and G(r,r 0 ) is the full-space Green’s function. With appropriate approximations[Ogilvy], the average scattered field is given as, where  0 sc is field scattered from smooth surface,  (k z ) is the characteristic function of the rough surface, given by p(h) is the statistical height distribution, and k z is a characteristic wavenumber normal to the mean surface. Our interest is focused on the specularly reflected coherent intensity I coh, which is the component that is aimed at the target. Kirchhoff Theory of Wave Scattering from Rough Surfaces (  )

10. At  1 = 80 o,  = 0.1, degradation, e -g = 0.97. 0 0.1 0.2 0.3 0.4 0.5 1.0 0.8 0.6 0.4 0.2 0  Reflected Beam Intensity can be Degraded by Microscopic Mirror Surface Roughness (  )  1 = 80 o 70 o 60 o For cumulative laser-induced and thermomechanical damages, we may assume Gaussian surface statistics with rms height , giving rise to  (k z ) = exp{-k z 2  2 /2}. - Grazing incidence is less affected by surface roughness. - To avoid loss of laser beam intensity,  < 0.01. 11 22 I sc I inc Intensity Degradation I o : reflected intensity from smooth surface I d : scattered incoherent intensity g : (4   cos  1 / ) 2

11. Specularly Reflected Field is Independent of Surface Correlation Lengths Phase difference between two rays scattered from different points (x 1, h 1 ) and (x 2, h 2 ) is given by:  = k [ (h 1 -h 2 )(cos  1 +cos  2 ) + (x 2 -x 1 )(sin  1 -sin  2 ) ] where  1 is the incident angle, and  2 is an angle of reflection. For specular scattering (  1 =  2 ),  = 2k (h 1 -h 2 ) cos  1 Thus, around the specular direction, the relative phases of waves scattered from different points on the surface depend only on the height difference,  h, between these points, but not on the point separations,  x. Correlations along the surface do not affect scattering around the specular direction, in which the coherent field is most strongly reflected. Thus, only the characteristic function  (k) needs to be specified to evaluate the coherent field.

12. Analytic Form of the Scattered Intensities Along the plane of incidence, the coherent scattered field is where the rough surface is assumed rectangular : -X ≤ x ≤ X, -Y ≤ y ≤ Y, g = k 2  2 (cos  1 + cos  2 ) 2, A = sin  1 - sin  2, and R is the reflection coefficient. The factor [sin(kAX)/kAX] 2 is due to diffraction from the surface edge, and as long as kX >> 1, the specular lobe is very narrow around  2 =  1. Assuming a Gaussian surface correlation function : C(R) = exp(-R 2 / c 2 ), the diffuse scattered field for slightly rough surfaces is given by: where F = F(  1,  2, R). is a strong function of the surface correlation length c. The total scattered field intensity is: = I coh +

13. Summary of Results and Conclusions A number of techniques have been used to assess the mirror surface damage limits on GIMM and driver beam performance, depending on the characteristics and size of the damage. Gross deformation:  - Ray tracing approach - For a simple gross surface deformation shape,  < 0.2  m (1) for a 30-cm radius mirror, = 248 nm (2) uniform beam illumination  I/I < 1% at the target (2) fixed spot size  a sp /a sp < 10%. Microscopic deformation:  - Kirchhoff theory - Surface roughness  < 0.01 for < 1% illumination reduction - Specularly reflected field is independent of surface correlations

14. On-Going and Future Work Kirchhoff theory will be extended to evaluate beam wavefront distortion from reflection off a damaged mirror at grazing incidence, for various types of microscopic surface deformation (Gaussian, spatially anisotropic, multiple scale lengths) and for measured data. The effect of self-shadowing and multiple reflections will be investigated. Quantify effect of gross and macroscopic surface damage on mirror and beam performance using the ray tracing technique, among others. The surface damage characteristics should be consistent with the damage source. The effects of local contaminants in the form of aerosol, dust and other debris on mirror reflective properties will be examined.