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        riddles >> general problem-solving / chatting / whatever >> Optics: Mirrors for Perfect Transmission
        
(Message started by: william wu on Apr 2^nd, 2003, 5:03am)

Title: Optics: Mirrors for Perfect Transmission
Post by william wu on Apr 2^nd, 2003, 5:03am

I don't know much about this but I'd like to more.

Recently I overheard optoelectricians discussing something vaguely like the following: if you have two mirrors with 99% reflectivity, one behind the other, you can adjust the distance between the mirrors such that 100% of the light entering the 1st mirror comes out of the back side of the 2nd mirror. Essentially something to do with cascading highly reflective surfaces to surprisingly produce an extremely transmissive device.

If anyone knows what I'm talking about, I'd appreciate it if you dropped some good google search parameters for me; I can teach myself the material. Or feel free to explain it if you're so inclined.

Title: Re: Optics: Mirrors for Perfect Transmission
Post by aero_guy on Apr 2^nd, 2003, 5:15am

Sounds like you are talking about polarized lenses. If the the first lens (or just flat plate) is polarized and the second is aligned to it I believe you can get what you describe. You may be able to create the same effect by adjusting the distance to the wavelength you wish to transmit without worrying about alignment.

Title: Re: Optics: Mirrors for Perfect Transmission
Post by BNC on Apr 2^nd, 2003, 5:49am

Hi Willy,

Try looking for "interferometer".
What you're describing sounds to me (EO-guy...) like what's known as a Fabry-Perot interferometer.

Bets luck searching,
BNC

Title: Re: Optics: Mirrors for Perfect Transmission
Post by BNC on Apr 2^nd, 2003, 11:42pm

Still not sure if what you overheard was indeed a reference to Fabry-Perot, but since I think it is, here are a few more comments on that.

But first, as for the possible reference to polarizers: many polarizers will absorb waves in the “wrong” polarization, and won’t reflect them (although that’s kind of dependant on the technology used, and that’s dependant on the spectral range). In any case, the distance between the polarizers is irrelevant.

As for interferometry : most physicists will probably remember the Michelson interferometer (used in the Michelson-Morley experiment). However, the Fabry-Perot interferometer, used as an optical resonator, is probably the most widespread (as it is used in lasers – more on that later). I’ll try to give a few pointers.

Terminology and symbols:
- The Greek letter ni is used for the optical frequency – the wavelength. I will write it as v, for simplicity.
- k is the wave number (a result of solving the Helmhotz equation)
- I will use p instead of pi in equations

An optical resonator is the optical counterpart of an electronic resonant circuit. It confines and stores light at certain resonance frequencies. The simplest approach is a resonator constructed of two parallel, highly reflective, flat mirrors, separated by a distance d (used mostly in classrooms, as this resonator is instable, but we’ll use it here). This simple case is known as a Frbry-Perot etalon. For the following derivation, I will use lossless mirrors (reflectivity of 100%).

A monochromatic wave of frequency v has a wavefunction:
U(r,t) = Re{ U(r)exp(j2pvt) }
which represents the transverse component of the electric field. r, in general, is a 3D vector, but we will solve for the 1D case.
As this wave travels in space, it accumulates phase, according to the distance it travels. If you will now draw for yourself the two parallel mirrors, and allow a wave at the “input” of the system, you will see infinite reflection between the mirrors. Where the waves are reflected, you may sum-up the components. You will see that the waves may only exist if the phase allows constructive interference.
The phase shift imparted by the two mirror reflections is 0, or 2pi (pi at each mirror). The phase shift imparted by a single round trip of propagation (a distance 2d) is:
Phi = k2d = 4pvd/c. Phi must be a multiple of 2pi to maintain the wave, so:
Phi = k2d = q2p, q=1,2,… (it can be shown that q=0 is associated with a mode that carries no energy).
This leads to the relation kd=qp. Thus, we may write:
v_q=qc/(2d), q=1,2,…
which are the resonance frequencies of the resonator. q is the mode number.

To show that only these waves (or a combination of them) can exist within the resonator, consider a monochromatic plane wave of complex amplitude U0 at mirror1 traveling toward mirror2 along the axis of the resonator. The wave is reflected from mirror2 and propagates back to mirror1, where it is again reflected. Its amplitude at mirror1 then becomes U1. Yet another round trip results in a wave of complex amplitude U2, and so on ad infinitum. Because the original wave U0 is monochromatic, it is “eternal”. The partial waves U0, U1… are monochromatic and perpetually coexist. Furthermore, their magnitudes are identical (remember – lossless case). The total wave U is therefore represented by the sum of an infinite number of phasors of equal magnitude: U=U0+U1+U2+…
The phase difference of two consecutive phasors imparted by a round trip of propagation is, as we saw Phi=k2d. If the magnitude of the initial phaseor is infinitesimal, the magnitude of each of these phasors must be infinitesimal. The magnitude of the sum of the infinite number of infinitesimal phasors is itself infinitesimal unless they are aligned, i.e., Phi=q2pi. Thus, an infinitesimal wave can result in the buildup of finite power in the resonator, but only is Phi=q2pi.

If we want some energy out of the resonator, the mirrors (at least one of them) must be imperfect. The analysis in this case in slightly more complex, but follows the same logical path. The result is the “widening” of the ideal case from a “delta-function train” at discreet frequencies to finite width “pulses”, that are very high at the resonator frequencies, and drop low (but no to 0) at frequencies away form resonance. The “width” of these “pulses” defines a parameter of the resonator (similar to the Q factor) called finesse. The finesse is lower (worse) as the losses increase. Thus, in real-life cases, the mirrors should be made are reflective as possible. The reflectivity is sometimes defined by “number of nines” – a mirror of 5-nines is a mirror with reflectivity 99.999%. The mirror in the overheard conversation of 99% are but 2-nines – very low reflectivity.

When used in lasers, an optical amplifying material is placed between the resonator mirrors. The result is that the energy “losses” at the laser output (as well as the losses of realistic systems) are compensated by the energy amplification inside the resonator. At the output, there will be laser light – monochromatic at a constant phase.

The reference to multiple reflectors is probably a reference to Bragg reflectors, which use similar approach to achieve high reflectivity from multiple low-reflectivity surfaces spaces “just-right”. It’s used, for example, to achieve very high-grade anti-reflective coatings for optical elements.

Wow, that was longer than I thought! I hope it helps, at least somewhat.