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In the context of wave phenomena, such as light, "angular width" refers to the angle over which the majority of the beam's intensity is spread. When light passes through a small opening, it spreads out and this spread is quantified by the angular width. This can often be seen in phenomena such as diffraction.
Consider a nuclear-pumped x-ray laser, which is designed to destroy missiles over great distances. Due to diffraction, the laser beam spreads out as it travels. For a circular aperture, the angular width \( \theta \) of the diffracted beam can be calculated using the formula:

• \( \theta = \frac{1.22 \lambda}{D} \)

where \( \theta \) is in radians, \( \lambda \) is the wavelength of light, and \( D \) is the diameter of the aperture.
In this exercise, the angular width helps determine how much the beam spreads from its starting point to a target 2000 km away. It is essential in understanding how diffraction affects the beam size over a long distance.

The intensity of a beam of light or, in this case, an x-ray laser, is essentially the power per unit area. As the beam travels and spreads due to diffraction, its intensity at a distant point decreases. This happens because the same amount of energy is distributed over a larger surface area.
Initial intensity can be computed near the laser's source, using:

• The formula for area of a circle \( A_0 = \pi r_0^2 \)

where \( r_0 \) is the initial radius of the beam. At a 2000 km distance, the area's radius will be larger, more than 17 meters in this problem.
To find the beam intensity at such a range, the relationship between initial and target intensities can be expressed as a ratio. This ratio indicates that the intensity at the target \( I_t \) becomes a fraction of the initial intensity \( I_0 \), essentially: \[ \frac{I_t}{I_0} = \frac{A_0}{A_t} \] where \( A_t \) is the area where the beam reaches at the target. As beam spread causes significant reduction in intensity, it poses limitations on the effectiveness of such weaponry for distant targets.

X-ray lasers are unique and advanced forms of lasers that use x-rays to generate beams of photons rather than visible light. They emit light at extremely short wavelengths, measured in nanometers, which is suitable for high-precision and high-energy applications, making them potential tools or weapons in advanced technology.
These lasers rely on a compact and powerful energy source, such as a nuclear reaction, to pump the energy levels required for emission in the x-ray spectrum.
Features and applications include:

• Highly focused x-ray beams.
• Potential for cutting-edge medical and technological uses, like imaging and delicate material fabrication.
• Possible military use in targeting through its high energy output and minimal atmospheric interaction.

Despite their potential, x-ray lasers like those described in the exercise are yet to overcome significant challenges, including controlling beam spread and ensuring sustained beam intensity over long distances.