^{1,a)}

### Abstract

The performance of a laser-window material must be assessed not only in terms of its ability to transmit high-power beams without generating undue optical distortion but also in terms of the constraints imposed by stress-related failure modes. In operational use, the stress field images the superposition of stresses originating from the mechanical load created by the pressure differential and the thermal load created by the laser beam. Here, we provide the tools to carry out an analysis of both pressure- and beam-induced stresses, and illustrate the procedure in the context of assessing the performance of a “model” window made of fusion-cast . The analysis assumes (a) operation on a time scale such that lateral heat diffusion can be ignored, and (b) cylindrically symmetric Gaussian beam shapes, which permit straightforward calculations of stress distributions that should be representative of worst case situations. Pressure-induced stresses strongly depend on the window’s aspect ratio, which suggests increasing the thickness to minimize the stress, but considerations relating to the optical performance require minimum allowable thicknesses based on a Weibull statistical analysis of the fracture probability. Beam-induced stresses are best evaluated in terms of (a) thickness-averaged radial and azimuthal stresses, which increase linearly with exposure time and depend on radial distances through the truncation parameter, and (b) across-the-thickness stress deviations relative to the average stress, which are caused by surface absorption and reach steady-state configurations on a time scale much shorter than the characteristic time for lateral heat transport. The average stress is always compressive and equibiaxial in the central region of the window, but its azimuthal component turns tensile in the rim region, thus threatening the structural integrity through brittlefracture. In addition, the coating-induced stress results in on-axis surface compressions that may exceed the yield strength of the windowpane material. In this light we formulate a figure of merit for stress, which demonstrates that promising laser-window materials must combine a small stress factor (expansion coefficient times elastic modulus) with superior thermal properties in terms of the product of heat capacity and thermal conductivity; and are the only known candidates that exhibit outstanding optical features at chemical laser wavelengths together with acceptable thermomechanical properties.

The author is indebted to John Detrio, University of Dayton Research Institute, for providing valuable information on the nature of beam-induced stresses and relevant properties of laser-window materials.

I. INTRODUCTION

II. WINDOW DESIGN CONSIDERATIONS

A. Beam description

B. Aperture diameter

C. Window thickness

III. PRESSURE-INDUCED STRESSES

IV. BEAM-INDUCED AVERAGE STRESSES

A. Prediffusion regime

B. Thickness-averaged stresses

C. model window

V. PLANAR STRESS AXIAL DEPENDENCE

A. Axial temperature profile

B. Steady-state regime

C. model window

VI. WINDOW PERFORMANCE EVALUATION

A. Combined planar stresses

B. Stress-induced failure

C. Window temperature excursion

VII. FIGURE OF MERIT

VIII. CONCLUSION

### Key Topics

- Materials properties
- 17.0
- Laser materials
- 16.0
- Stress strain relations
- 12.0
- Brittleness
- 9.0
- Optical materials
- 9.0

## Figures

Beam profile factor for the far-field focal intensity and the window hoop stress as a function of the beam truncation parameter.

Beam profile factor for the far-field focal intensity and the window hoop stress as a function of the beam truncation parameter.

Gaussian beam profiles for truncation parameters ranging from 0.3 to 3.0. The case illustrates the profile of an optimally truncated beam of total power and peak intensity. The and shapes refer to a beam passing through an -diameter window.

Gaussian beam profiles for truncation parameters ranging from 0.3 to 3.0. The case illustrates the profile of an optimally truncated beam of total power and peak intensity. The and shapes refer to a beam passing through an -diameter window.

Failure probability of an -diameter window made of fusion-cast and subjected to a uniform biaxial stress. The Weibull statistical parameters are as derived in the Appendix. Note that surface treatment does not alter the failure probability.

Failure probability of an -diameter window made of fusion-cast and subjected to a uniform biaxial stress. The Weibull statistical parameters are as derived in the Appendix. Note that surface treatment does not alter the failure probability.

Schematic description of the window geometry.

Schematic description of the window geometry.

Predicted pressure-induced planar stresses acting in a freely-supported cylindrical window made of fusion-cast , measuring in diameter and in thickness, subjected to a uniform load of .

Predicted pressure-induced planar stresses acting in a freely-supported cylindrical window made of fusion-cast , measuring in diameter and in thickness, subjected to a uniform load of .

Beam-induced thickness-averaged stress dependence on the radial coordinate and the truncation parameter for Gaussian axisymmetric temperature distributions. (a) Nondimensional radial stress; (b) nondimensional azimuthal stress. Note that the radial stress remains compressive throughout the parameter space, whereas the azimuthal stress turns tensile for and reaches peak values at some distance off the edge if .

Beam-induced thickness-averaged stress dependence on the radial coordinate and the truncation parameter for Gaussian axisymmetric temperature distributions. (a) Nondimensional radial stress; (b) nondimensional azimuthal stress. Note that the radial stress remains compressive throughout the parameter space, whereas the azimuthal stress turns tensile for and reaches peak values at some distance off the edge if .

Predicted thickness-averaged planar stress acting in a window made of fusion-cast , in diameter and in thickness, exposed to an optimally truncated Gaussian chemical-laser beam for durations of 1, 3, and .

Predicted thickness-averaged planar stress acting in a window made of fusion-cast , in diameter and in thickness, exposed to an optimally truncated Gaussian chemical-laser beam for durations of 1, 3, and .

Axial temperature profile of an AR-coated laser window as a function of the nondimensional time parameter . Note that the “zero” axial position identifies the middle plane. The profiles are identical on both sides of this plane.

Axial temperature profile of an AR-coated laser window as a function of the nondimensional time parameter . Note that the “zero” axial position identifies the middle plane. The profiles are identical on both sides of this plane.

On-axis, temperature-gradient-induced stress enhancement as a function of the axial position for beam-exposure times of 1, 3, and . The calculation applies to an AR-coated window made of fusion-cast , thick, exposed to a Gaussian chemical laser beam of peak intensity. Note that, as expected, the thickness-averaged stress enhancement is equal to zero.

On-axis, temperature-gradient-induced stress enhancement as a function of the axial position for beam-exposure times of 1, 3, and . The calculation applies to an AR-coated window made of fusion-cast , thick, exposed to a Gaussian chemical laser beam of peak intensity. Note that, as expected, the thickness-averaged stress enhancement is equal to zero.

Temperature-gradient-induced stress enhancement as a function of radial position and exposure time, for the two surfaces and the middle plane of a model window as in Fig. 9, on assuming a truncation parameter .

Temperature-gradient-induced stress enhancement as a function of radial position and exposure time, for the two surfaces and the middle plane of a model window as in Fig. 9, on assuming a truncation parameter .

Combined pressure- and beam-induced planar stresses as a function of the normalized radial distance. The figure depicts the situation on the front (high-pressure) side, the middle plane, and the back (low-pressure) side of a model window made of fusion-cast (diameter: ; thickness: ) subjected to a pressure differential and exposed to an optimally truncated chemical laser beam. (a) Laser run time: ; (b) laser run time: ; (c) laser run time: .

Combined pressure- and beam-induced planar stresses as a function of the normalized radial distance. The figure depicts the situation on the front (high-pressure) side, the middle plane, and the back (low-pressure) side of a model window made of fusion-cast (diameter: ; thickness: ) subjected to a pressure differential and exposed to an optimally truncated chemical laser beam. (a) Laser run time: ; (b) laser run time: ; (c) laser run time: .

Combined peak stresses in a model window made of fusion-cast , in diameter and in thickness, exposed to optimally truncated Gaussian chemical laser beams for durations of . The symbol refers to the peak irradiance level. (a) Peak tensile stress at the backface edge; (b) peak compressive stress at the frontface center.

Combined peak stresses in a model window made of fusion-cast , in diameter and in thickness, exposed to optimally truncated Gaussian chemical laser beams for durations of . The symbol refers to the peak irradiance level. (a) Peak tensile stress at the backface edge; (b) peak compressive stress at the frontface center.

Predicted on-axis temperature rise experienced by a model window made of fusion-cast , in diameter and in thickness, exposed to an optimally truncated Gaussian chemical laser beam for durations up to .

Predicted on-axis temperature rise experienced by a model window made of fusion-cast , in diameter and in thickness, exposed to an optimally truncated Gaussian chemical laser beam for durations up to .

Figure of merit for stress of the six laser-window material candidates identified in Table III. Note that, for comparative purposes, the of was set equal to 1.

Figure of merit for stress of the six laser-window material candidates identified in Table III. Note that, for comparative purposes, the of was set equal to 1.

Uniaxial fracture-strength Weibull plot for fusion-cast ; the “as-received” and “FCP-treated” data points reflect fracture-strength test results recorded in Refs. 17 and 18, respectively.

Uniaxial fracture-strength Weibull plot for fusion-cast ; the “as-received” and “FCP-treated” data points reflect fracture-strength test results recorded in Refs. 17 and 18, respectively.

## Tables

List of symbols (see also Table II).

List of symbols (see also Table II).

Key properties of fusion-cast at room temperature.

Key properties of fusion-cast at room temperature.

Relevant room-temperature properties and figure of merit for stress, , of key laser-window material candidates.

Relevant room-temperature properties and figure of merit for stress, , of key laser-window material candidates.

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