(14) Coherent radiation

My complaint about claiming that laser radiation is monochromatic and coherent is not that such a claim isn't true, but that it is redundant. Good quality laser radiation (linearly polarized in just one direction) is indeed (almost) monochromatic, i.e., has Fourier components (which are sinusoidal, i.e., sine-wave shaped) of just one frequency, but the additional claim that it is coherent [the meaning of this is commonly said to be that all the radiation's components are in step, i.e., the radiation can be written as b(sin(t + φ))] adds no further information, since all monochromatic waves are coherent in that sense. This is shown by the following: A scalar sinusoidal wave L(t) is completely specified by only three parameters: Its frequency f, its amplitude a, and its phase (wrt a given time or space coordinate system) 𝜃. Monochromatic radiation of frequency f thus is specified, in addition to the frequency's being f, completely by the two other variables, a and 𝜃. Suppose L(t) is monochromatic radiation that is incoherent because it is made up of two waves of different amplitude and phase, L(t) = sin(t) + a(sin(t + 𝜃)). Then there exist b and φ such that L(t) = b(sin(t + φ)), so L(t) is coherent, i.e., is equal to some sine wave. A little trigonometry shows that unless 1 + a(cos(𝜃)) = 0, L(t) = √ [1 + 2a(cos(𝜃)) + a^2](sin(t + arctan[a(sin(𝜃))/(1 + a(cos(𝜃)))])). If 1 + a(cos(𝜃)) = 0, L(t) = a(sin(𝜃)cos(t)) = a(sin(𝜃)sin(t +𝜋/2)). Thus, in either case, L(t) is coherent.

(19) Skolem's Paradox