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Before the advent of the laser, it was quite difficult to produce monochromatic light. One possibility was to use certain gas discharge lamps and metal vapor lamps (e.g. mercury vapor lamps and sodium vapor lamps), emitting light dominantly in certain narrow spectral lines, and isolating one such line with a suitable monochromator. The achieved optical powers and intensities were quite low.
If you define optical power as energy delivered by unit time, then it indeed oscillates in the way you described. However, that oscillation is by far too fast to be measured with any electronic detector. Therefore, optical power or optical intensity (as the magnitude of the Poynting vector) is often defined to be without that oscillation because that is usually of no practical relevance.
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Monochromatic light is light (optical radiation) where the optical spectrum contains only a single optical frequency. The associated electric field strength at a certain point in space, for example, exhibits a purely sinusoidal oscillation, having a constant instantaneous frequency and a zero bandwidth. Light sources can also be called monochromatic, if they emit monochromatic light.
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The term monochromatic originally means having only a single color. Different optical wavelengths of visible light are associated with different perceived colors. However, light colors are rarely a criterion for monochromaticity in practice, and non-monochromatic light can also have specific colors. Also, the term is applied to infrared and ultraviolet light as well as to visible light.
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Lasers are the primary sources of quasi-monochromatic light. In contrast to narrow-band light obtained by bandpass filtering light from a broadband source (see below), lasers can generate quasi-monochromatic light with high optical powers. Some lasers even exhibit extreme degrees of monochromaticity, i.e., an extremely small optical bandwidth. The highest degree of monochromaticity is achieved with carefully stabilized single-frequency lasers (sometimes with a bandwidth well below 1 Hz).
If the electric field from a monochromatic light source varies sinusoidally, shouldn't its power detected on a square law detector that is sufficiently fast give a squared sinusoidal signal? The detectors for optical frequencies would be too slow for this, but I have been told that what is recorded at lower frequencies is a constant power level instead of a squared sinusoid. Is this somehow related to that practical light sources are never perfectly monochromatic, or how is this reconciled?
Many calculations in optics and photonics are performed for monochromatic light. For example, the evolution of laser beams is usually calculated that way; there is just one given optical wavelength or frequency.
Monochromaticity is not a quantity, but you may judge it by considering the spectral bandwidth. You may measure that with and optical spectrum analyzer or with an interferometer, for example.
Real light sources can of course never be exactly monochromatic, i.e., have a zero optical bandwidth. However, particularly laser sources are often quasi-monochromatic, i.e., the optical bandwidth is small enough that certain behavior of the light can hardly be distinguished from that of truly monochromatic light. Some examples:
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A monochromator is essentially a kind of optical filter which allows one to isolate light in a narrow spectral interval from other light. Its output will therefore be quasi-monochromatic. However, light at all other wavelengths is then lost.
There are very different types of interferometers, some requiring quasi-monochromatic light, while others work only with polychromatic light. See, for example, the article on white light interferometers.
The antonym of monochromatic is polychromatic. A typical example for polychromatic light is light created as thermal radiation, e.g. in an incandescent lamp; such light exhibits a broad range of optical frequencies.
It depends on what you mean was “truly monochromatic”. From a theoretical standpoint, it might mean and exactly constant instantaneous optical frequency, which is never achieved in reality. In practice, one determines what level of optical linewidth is still relevant for a specific application.