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The problem for photons is that you cannot observe them time to time. If you do measure their state, photons will collapse to some Eigen state and become classical. In fact, a photon can have a lot of eigenstates in theory. That is special compared to EM for photons.


Generally to propagate, one needs a term like $\mathbf{k} \cdot \mathbf{x}$ in the exponent (i.e., a finite phase velocity). The last term you showed should not propagate anywhere. It's envelope is a modulated Gaussian (i.e., from the $e^{-x^{2}}$ term) and its amplitude grows in time (i.e., from the $e^{-t^{2}}$ term) while it oscillates as a standing ...


If you use Lumerical or MEEP to do the FDTD calculation, you can simply add a Gaussian source polarized along $x$-direction but propagating along $z$-direction with the given pulse width and $\tau$ and frequency. Alternatively, you can make both $y$ and $z$ components equal to zero for the source. Is this what you want?


Your question is unclear. If you add transverse dimensions, your problem is no longer 1-dimensional. If you mean adding transverse components of your interested quantities, then there is no change needed to the usual FDTD method. For example, if you are analysing the propagation of an electromagnetic plane wave in one-dimension, say the positive $z$ ...


EM waves don't "stop" they just slowly become weaker as $r^{-2}$, so one could conceivably answer "forever." On the other hand, the wave will quickly become so dissipated/spread out that there isn't much to measure, so you have a practical limit where it won't be detectable. However if this is your intent, you haven't given us enough information to answer ...


Each boundary condition comes from an independent Maxwell equation, so the four boundary conditions are independent. The right-hand side of the third boundary condition should K, the surface current. The first boundary condition could be replaced by $\phi_1=\phi_2$, which is easier to implement


No - assuming they don't hit anything they don't decay. The distance dependant "decay" is the drop in the number of photons per volume as the volume gets bigger - it's not a decay of individual photons. It's the same as a crowd dispersing as it leaves a subway exit - nobody is disappearing. Photons can lose energy as they collide with gas or dust in space ...

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