Status of Modeling of Damage Effects onFinal Optics Mirror Performance

T.K. Mau, M.S. TillackCenter for Energy Research

Fusion Energy DivisionUniversity of California, San Diego

High Average Power Laser Program WorkshopApril 4-5, 2002

General Atomics, San Diego

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.

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 80o from the GIMM surface normal.

Dimensional Defects Compositional Defects

Gross deformations, δ>λ Surface morphology, δ <λ Gross surfacecontamination

Local contamination

CONCERNS

• Fabrication quality• Neutron swelling• Thermal swelling• Gravity loads

• Las -erinduceddamage

• Thermomechanicaldamage

• Transmutations• Bulk redeposition

• Aeros ,ol dus t&debris

MODELL ING TOOLS

Optica l des ignsoftware Scattering by roughsurfaces (Kirchhoff)

Fresne l mult -i layersolver

Scattering by particles

Mirror Defects and Damage Types, and Approaches to Assess their Effects on Beam Quality

√ √( Ray tracing )

Ray Tracing Analysis of Gross Mirror Deformation Limits

• The ZEMAX optical design software was used for the analysis.

• Gross deformation: δ λ due to thermal or gravity load, or fabrication defect. Deformation: δ = am

2/2rc [ surface sag ]

• Analysis assumes that flat GIMM surface acquires a curvature (rc), 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 λ = 248 nm (KrF)Focusing mirror focal length = 30 mGIMM to target distance = 20 mMirror radius am = 0.3 mGrazing incidence angle = 80o

Target radius = 3 mmBeam spot size asp = 0.64 mm

WallTarget

GIMM

Focusing Mirror

LaserBeam

Typical Output from a ZEMAX Run

Ray Trajectories

GIMM

0ne million rays used; = 80o

FocusingMirror

Target

Mirror Surface Sag rc = 5x104 m

(mm)

-0.3m +0.3m

+0.3m

2-D Illumination Profile 1-D Illumination Profiles

+3mm-3mm

+3mm

x

y

Y-scan

X-scan

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 0.2 0.4 0.6 0.8 1

Beam Spot Size [a,b] (mm)

Mirror Surface Sag (μμ)

= 0.3 GIMM Radius m- - = 20 GIMM to Target Distance m

= 0.64 Nominal Spot Size mm

b

a

Mirror Curvature Causes Spot Size to Elongate

The isotropic surface curvature causes the rays to diverge preferentially in the direction of beam propagation.

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 δ = 0.46 μm

• Mirror surface sag limit for grazing incidence is :

δ < 0.2 μm, (for a mirror of 0.3 m radius)

with the criteria: I / I < 1%, and asp / asp < 10%.

y-scan

δ =0.92μm

0.46μm

0μm

δ = 0 μm

δ = 0.46μm

δ = 0.92 μm

-2mm

+2mm

+2mm

Relative Illumination

• In the presence of a scatterer (mirror surface), total field is given by

where is given by

where S0 is surface of scatterer and G(r,r0) 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, (kz) is the characteristic function of the rough surface, given by

p(h) is the statistical height distribution, and kz is a characteristic wavenumber normal to the mean surface.

• Our interest is focused on the specularly reflected coherent intensity Icoh, which is the component that is aimed at the target.

Kirchhoff Theory of Wave Scattering from Rough Surfaces (δ λ)

ψ(r r ) =ψinc(r r )+ψsc(r r )

ψsc(r r )

ψsc(r) = ψ(r r 0

)∂G(

r r ,

r r 0)

∂n0

−G(r r ,

r r 0)

∂ψ(r r 0)

∂n0

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

S0

∫ dr S 0

ψsc =χ(kz)ψ0sc

χ(s) = p(h)eish

−∞

∞

∫ dh

Icoh= ψsc ψsc∗ ≈Ιoe−g

At 1 = 80o, λ = 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

λ

Reflected Beam Intensity can be Degraded by Microscopic Mirror Surface Roughness ( λ)

Isc=I0e−g+Id

1 = 80o

70o

60o

• For cumulative laser-induced and thermomechanical damages, we may assume Gaussian surface statistics with rms height , giving rise to (kz) = exp{-kz

22/2} . - Grazing incidence is less affected by surface roughness. - To avoid loss of laser beam intensity, λ < 0.01.

12

Isc

Iinc

Inte

n sit y

Deg

rada

ti on

Io : reflected intensity from smooth surfaceId : scattered incoherent intensityg : (4 cos1/λ)2

Specularly Reflected Field is Independent of Surface Correlation Lengths

• Phase difference between two rays scattered from different points (x1, h1) and (x2, h2) is given by:

= k [ (h1-h2)(cos1+cos2) + (x2-x1)(sin1-sin2) ]

where 1 is the incident angle, and 2 is an angle of reflection.

• For specular scattering (1 = 2), = 2k (h1-h2) 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.

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 = k2 2 (cos1 + cos2)2, A = sin1 - sin2 , 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(-R2/λc2),

the diffuse scattered field for slightly rough surfaces is given by:

where F = F(1, 2, R). <Id> is a strong function of the surface correlation length λc .

• The total scattered field intensity is: <I> = Icoh + <Id>

Icoh=k2

π2r2(XY)2exp(−g)[cosθ2(1+R)−cosθ1(1−R)]2sinkAX

kAX⎛ ⎝

⎞ ⎠

2Id =

k2F2λc2

πr2gexp(−g)(XY )exp

−k2A2λc2

4

⎛

⎝ ⎜

⎞

⎠ ⎟

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: δ λ - Ray tracing approach - For a simple gross surface deformation shape, δ < 0.2 μm (1) for a 30-cm radius mirror, λ = 248 nm (2) uniform beam illumination I/I < 1% at the target (2) fixed spot size asp/asp < 10%.

• Microscopic deformation: δ λ - Kirchhoff theory - Surface roughness λ < 0.01 for < 1% illumination reduction - Specularly reflected field is independent of surface correlations

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.