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How do lasers work?

To give all the precise details would require going into advanced quantum mechanics, which I'm assuming was not the goal here. But the basic principle of lasers is relatively straightforward.

The simplest kind of laser is just three components: A dark tube, a pair of mirrors, and a fluorescent gas.

Energy is pumped into the fluorescent gas (usually by electric current), which causes electrons in the gas to rise briefly to higher-energy states. They then drop back down to their ground states (the lowest-energy states in which atoms are normally stable), and in doing so emit photons. But not just any photons; these photons are always of the same precise wavelength. These photons in turn hit other atoms and stimulate them to produce even more photons, making the gas light up.

If we stopped there, we'd have a fluorescent lamp. But to make it a laser, the mirrors are positioned carefully at the ends of the dark tube containing the gas. One mirror is "full-silvered", meaning it reflects basically all the light that hits it, while the other is "half-silvered", meaning that it reflects about half the light and transmits the other half. The half-silvered mirror is the place from which the laser beam will emerge.

These mirrors are spaced in such a way that the length of the whole tube is a whole number of wavelengths (for visible light this is spectacularly easy to do, as the wavelength is only about half a micron, so almost any macroscopic distance will be approximately a whole number of wavelengths; for longer waves such as microwaves and radio waves, it is more of a challenge because the wavelengths can get into centimeters or even meters). The reason we do this is to ensure constructive interference; we want the light bouncing back and forth between the two mirrors to line up properly so that the peaks of all the waves line up. This makes the resulting light beam coherent---very powerful, very precise, and all with exactly the same wavelength. The result is a laser.

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