^{1}and Reza S. Abhari

^{1}

### Abstract

The emission of extreme ultraviolet radiation in a 2% bandwidth centered at 13.5 nm (in-band) is measured in one hemisphere. The targets of the laser are tin droplets. In-band emission is measured at angles from the laser axis larger than 120°. Analytical models representing the physical processes are developed and calibrated with the experimental data. In the models two assumptions are investigated, isentropic and isothermal 2D-axisymmetric expansion. The parameters of the models are the density distribution of the plasma and the location where the EUV emission is centered. The parameters are inferred by the calibration of the models with the experimental data. The predictions of the models are validated with experiments where slab targets were used.

The authors acknowledge Dr. O. Morris for the support in the development of the alignment procedure for the instrument. Furthermore, the authors wish to thank Dr. B. Rollinger and Dr. S. Ellwi for reviewing the manuscript, and Dr. N. Gambino for valuable discussions.

I. INTRODUCTION

II. EXPERIMENTAL SETUP

III. EXPERIMENTAL RESULTS

IV. MODEL

V. MODELS APPLIED TO SLAB TARGET EXPERIMENTS

VI. MODELS APPLIED TO DROPLET TARGET EXPERIMENTS

A. Calibration of the models with the experimental data

B. The continuum assumption

C. EUV transmission in space

VII. CONCLUSIONS

### Key Topics

- Extreme ultraviolet radiation
- 99.0
- Fluid drops
- 20.0
- Plasma expansion
- 19.0
- Plasma temperature
- 18.0
- Opacity
- 13.0

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## Figures

(a) Picture of the robotic arm inside the vacuum chamber. The EUV energy monitor is mounted on the lower right. The arrows show the rotational movements in longitudinal and latitudinal directions. (b) Schematic of the experimental setup looking at the focusing lens. The laser irradiates the dispensed tin droplets, and the EUV energy monitor scans a predetermined range in one hemisphere. (c) Frame of reference for the measurement positions and for the Mercator projection. λ and φ are the longitude and the latitude, respectively. (d) Frame of reference for the two-dimensional representations. β and θ are the angle versus the horizontal plane and the angle versus the laser axis, respectively.

(a) Picture of the robotic arm inside the vacuum chamber. The EUV energy monitor is mounted on the lower right. The arrows show the rotational movements in longitudinal and latitudinal directions. (b) Schematic of the experimental setup looking at the focusing lens. The laser irradiates the dispensed tin droplets, and the EUV energy monitor scans a predetermined range in one hemisphere. (c) Frame of reference for the measurement positions and for the Mercator projection. λ and φ are the longitude and the latitude, respectively. (d) Frame of reference for the two-dimensional representations. β and θ are the angle versus the horizontal plane and the angle versus the laser axis, respectively.

Mercator projection of the EUV mean energy at ±2%BW centered at 13.5 nm versus the detector position. The origin corresponds to the laser beam intersecting the sphere of the mapping. The black points represent the positions of the measurements. The map is obtained by cubic interpolation and shows EUV energy decreasing moving away from the laser beam.

Mercator projection of the EUV mean energy at ±2%BW centered at 13.5 nm versus the detector position. The origin corresponds to the laser beam intersecting the sphere of the mapping. The black points represent the positions of the measurements. The map is obtained by cubic interpolation and shows EUV energy decreasing moving away from the laser beam.

EUV mean energy in the ±2% BW centered at 13.5 nm versus the angle from the laser axis for different cuts. The cuts are lines defined by the intersection between the mapped spherical surface and a plane defined by the laser axis and the angle β from the horizontal plane, which is reported in the plot legend (β is sketched in Figs. 1(b) and 1(d) ). The blue circles refer to the cut at β = 10°, the black squares to the cut at β = 0°, the green triangles to the cut at β = −20°, and the red crosses to the cut at β = −40°. The error bars show the standard deviation of the measurements. The difference between the cuts is smaller than the standard deviation, therefore the emission is assumed to be axisymmetric.

EUV mean energy in the ±2% BW centered at 13.5 nm versus the angle from the laser axis for different cuts. The cuts are lines defined by the intersection between the mapped spherical surface and a plane defined by the laser axis and the angle β from the horizontal plane, which is reported in the plot legend (β is sketched in Figs. 1(b) and 1(d) ). The blue circles refer to the cut at β = 10°, the black squares to the cut at β = 0°, the green triangles to the cut at β = −20°, and the red crosses to the cut at β = −40°. The error bars show the standard deviation of the measurements. The difference between the cuts is smaller than the standard deviation, therefore the emission is assumed to be axisymmetric.

Schematic of the model setup. The EDR is at the origin of the cylindrical coordinate system. The form of the iso-density lines is ellipsoidal, with a semi-major axis a, and a semi-minor axis b. The distance between the EDR and the center of the ellipsoidal density distribution is z0 . The parameters of the model are a, b and z0 .

Schematic of the model setup. The EDR is at the origin of the cylindrical coordinate system. The form of the iso-density lines is ellipsoidal, with a semi-major axis a, and a semi-minor axis b. The distance between the EDR and the center of the ellipsoidal density distribution is z0 . The parameters of the model are a, b and z0 .

The density profiles for the isentropic (ρs ) and for the isothermal (ρT ) case are shown for a cut along the axis of symmetry together with the temperature for the isentropic assumption (Ts ). The EDR is located at z = 0.

The density profiles for the isentropic (ρs ) and for the isothermal (ρT ) case are shown for a cut along the axis of symmetry together with the temperature for the isentropic assumption (Ts ). The EDR is located at z = 0.

EUV emission versus the angle from the laser axis for the slab target case. The experimental data come from the work of Ando et al. 10 The measurements and the results from the models of the EUV transmission are plotted. The parameters used in the models are b/a = 1.2, and an offset of z0 = 1.55 L.

EUV emission versus the angle from the laser axis for the slab target case. The experimental data come from the work of Ando et al. 10 The measurements and the results from the models of the EUV transmission are plotted. The parameters used in the models are b/a = 1.2, and an offset of z0 = 1.55 L.

EUV emission as a function of the angle from the laser axis. The measurements and the results from the models of the EUV transmission are plotted. The parameters used in the models are b/a = 0.8, and an offset of z0 = 1.55 L. The amount of EUV energy emitted behind the target with respect to the laser (or θ > 90°) is 30% of the overall emitted energy.

EUV emission as a function of the angle from the laser axis. The measurements and the results from the models of the EUV transmission are plotted. The parameters used in the models are b/a = 0.8, and an offset of z0 = 1.55 L. The amount of EUV energy emitted behind the target with respect to the laser (or θ > 90°) is 30% of the overall emitted energy.

Gradient-length local Kn-number versus the axial position. The EDR is located at z = 0. The value stays below 0.05 up to 0.44z0 . The contribution to the EUV absorption of the region outside the continuum is 0.1%. The continuum assumption is valid to obtain the estimation of the overall EUV absorption.

Gradient-length local Kn-number versus the axial position. The EDR is located at z = 0. The value stays below 0.05 up to 0.44z0 . The contribution to the EUV absorption of the region outside the continuum is 0.1%. The continuum assumption is valid to obtain the estimation of the overall EUV absorption.

Spatially integrated EUV transmission. The laser comes from the right and the EDR is at the origin of the coordinate system. The EUV transmission is integrated in the space along lines that propagate outwards, the lines have the EDR as their origin. The isentropic case (above) shows a slower decrease in the transmission than the isothermal case (below) because of the larger temperature between the droplet and the EDR.

Spatially integrated EUV transmission. The laser comes from the right and the EDR is at the origin of the coordinate system. The EUV transmission is integrated in the space along lines that propagate outwards, the lines have the EDR as their origin. The isentropic case (above) shows a slower decrease in the transmission than the isothermal case (below) because of the larger temperature between the droplet and the EDR.

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