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Is it possible to have a monochromatic wave which is incoherent? On the one hand, it could be obtained by irradiating a laser beam on a strongly scattering medium like white paper. But on the other hand, once the light is monochromatic its electric field is a sinusoidal wave of constant frequency and if we sum a series of such sinusoidal waves with random phases and amplitudes we always get a sinusoidal wave of the same frequency same as in coherent light. Is it true that if we want to have incoherent light we need an electric field which fluctuates in time in a more complex way than simple sine and therefore cannot be made of plane waves of single frequency but non-zero bandwidth is crucial?

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  • $\begingroup$ coherent with what? $\endgroup$
    – hyportnex
    Commented Feb 7 at 14:06
  • $\begingroup$ Monochromatic in experimental sense does not really mean perfect sine wave, it means an oscillation with stable intensity over a long time, with concentrated spectrum around single frequency, but not infinitely thin (zero-width) spectral line. The wave may look like a sine wave for a while, but its phase detunes from the perfect sine randomly. $\endgroup$ Commented Feb 7 at 20:28

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As you suggest, it can't be. The wave $A \sin(\omega t - \vec{k} \cdot \vec{r} + \phi(t))$ for random $\phi(t)$ is not a monochromatic wave

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A truly monochromatic wave with a delta function Fourier Transform must exist for infinite time. If its amplitude changes over time, it isn't truly monochromatic. It would have an infinite coherence length.

But many waves with a narrow spectrum are called monochromatic. E.G. lasers are monochromatic. They have a finite coherence length. Those with a long coherence length must have a narrower spectrum.

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You can not produce really monochromatic light waves. But you are right , to have incoherent waves, they are in reality wave packets. So it is not clear, what you want.

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There are different forms of coherence. Leaving aside quantum aspects, we'll focus on first order coherence as seen in classical optics. There one can distinguish between temporal coherence and spatial coherence. Temporal coherence is determined by the temporal frequency spectrum, which is the coherence that the OP seems to be referring to. It is represented by a coherence length which is related to the inverse of the spectral bandwidth. In the monochromatic limit the coherence length grows to infinity, which is obviously just an approximation.

Spatial coherence has to do with the size of the source. Think of the exit pupil of a light source such as a laser and imagine it is a surface consisting of independent point sources. All these point sources contribute to the observed interference. The lack of coherence is thus caused by the different path lengths that are formed by different points on the source to the point where the interference is formed. When these path lengths vary by more than a wavelength then the light would appear incoherent even if the light is monochromatic.

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