Final answer:
The diffusion-cooled laser is the most compact and powerful option for CO₂ lasers due to their efficient cooling mechanism and ability to maintain a high beam quality.
Step-by-step explanation:
The most compact and powerful carbon dioxide (CO₂) laser is the diffusion-cooled laser. Of the options listed, diffusion-cooled lasers, which are also known as slab or waveguide lasers, allow for efficient cooling without the need for external gas flow, leading to a more compact design. They have high beam quality with medium to high power output, making them suitable for precise tasks such as surgical applications.
Conversely, fast axial-flow lasers and slow axial-flow lasers rely on gas flowing through the laser to remove heat, which typically involves larger and more complex systems. Transverse lasers can produce high power, but their construction is also not as compact as diffusion-cooled lasers. Therefore, considering the requirement for both compactness and power, diffusion-cooled lasers stand out as the best choice among the given options.