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Two parts of this answer: (1) a quick sketch of how one gets the result cited and (2) why I believe one can't get a less awkward, more succinct result. Proof Sketch The one thing that the numerical …

Another explanation, further to J.G.'s answer, is simply "owing of symmetry". The answer has to be invariant if we rotate our co-ordianates about the normal, which comes to mean that the only transmis …

The equations you write are not those of a focussing wave (I think you're missing an $i\,\pi$ in your exponent for the $x$ variation for $z>0$). There is no way for the wavefront curvature to change w …

Actually there is one imaging mode where the spherical lens is a perfect lens: converging all rays, no matter how far from paraxial, to a single focal point. This is the case of the aplanatic sphere, …

One can make a large core fiber single moded by adjusting the core-cladding index difference so that the waveguide parameter: $$V = \frac{2\,\pi\,\rho}{\lambda}\sqrt{n_{co}^2-n_{cl}^2}$$ is less tha …

The answer is given as part of my answer to your other similar question here. In a dielectric waveguide, nothing special happens: the field spreads out so that it is significantly different from zer …

Probably nothing too unusual. The modal field will tend to be a little larger than the core. If the core cladding index difference is small, then the modal field spreads out a long way into the claddi …

I think the project assigned is calling on you to work out something fairly significant for yourself, so be patient and prepared to think fairly hard and not expect the answer to come too trivially. …

There are very recent devices discovered which this model may not be adequate for (more later), but the model is nonetheless very general. It works for any axially symmetric optical system made of the …

The paraxial imaging geometry of any axially symmetric optical system can be described by three real parameters: the axial positions of the two principal planes and one focal length (i.e. $f_f=f_b$ in …

It's a bit complicated. There are two competing effects here as one varies the focal length of a spherical lens. Assuming that the beamwidth $w$ is constant, the $f$-number $\frac{q}{w}$ is then propo …

You are correct that the emerging rays are parallel to the incoming ones, but sketch the situation. A ray cone $POQ$ converging to point $O$ with no glass plate converges, instead, in the presence of …

This is a good question: those marks are mystifying when you meet a beamsplitter for the first time, but I understand from manufacturers that the following can all be reasons for the marks (point 1 is …

Snow is simply a random collection of snowflakes and bits of irregularly shapen bits of ice. Each of these is clear, but a small fraction of the light incident on each clear entity is reflected and sc …

I presume your 70km comes from the $30{\rm km\,s^{-1}}$ Earth orbital speed about the Sun multiplied by the 2.5 second return trip time to the Moon for a pulse of light. However, recall that, althoug …