I would like to make a couple additional comments to this discussion which I think mathematically are quite reasonable. There seems to be two different types of "laser" type (or coherent r-f signal) patterns being considered here=one is a plane wave and the other is a narrow pattern that has a finite beam spread/divergence. For both cases, if a given signal amplitude is considered, there needs to be a correlation of the phases of the signal across/throughout the beam (i.e. the 3-D beam pattern)=whether it is considered to be of a photon nature or simply as a completely classical picture. ## \\ ## For both cases, (of a plane wave or a beam pattern with a finite divergence), to double the intensity of the signal, the amplitude at each location only needs to increase by a factor of ## \sqrt{2} ## if the amplitudes are in phase everywhere. Somehow, a factor needs to get introduced where a factor of ## N ## in energy is a factor of ## \sqrt{N} ## in the resultant electric field amplitude everywhere. ## \\ ## One way to achieve this is by overlaying the layers and giving each layer a random phase. Another way, is simply to keep all the superimposed layers in phase, (or some arbitrary phase), with each other, and have a mathematics where ## E_{total}=\sqrt{N} E_o ##, where ## E ## is electric field amplitude, and the usual mathematics with phasor diagrams where ## E ## fields add is no longer applicable. The Q.M. description, (see also post 50), seems to use this latter approach. (edit: see the next paragraph. In this present paragraph, I was treating the photon as a fundamental sinusoidal disturbance with a given amplitude and having some phase. These disturbances would then add linearly, taking into account the phases as they are added. For random phases, the amplitude would then be proportional to ## \sqrt{N} ## . This picture of a photon may or may not be realistic.)## \\ ## After pondering this question further, one additional item occurred to me that perhaps resolves some or most of this: The energy density being proportional to the square of the amplitude is how waves behave in general in linear materials, so that one should not expect to lay a layer of (wave) energy in phase on a linear material and have the electric field amplitude double. This ## \sqrt{N} ## factor is normal behavior for a wave in a linear material.
