Re: How fast does light dissipate in a 2 way spherical mirror?

Greetings:

Your questions are a very interesting because they address one of the 
classic problems studied in electromagnetic circuit theory. In particular 
the decay time (often called the relaxation time) of electromagnetic energy 
confined within metallic boxes, cylinders and spheres (and many other 
shapes) is of importance in electromagnetic circuits. These circuits are 
often called resonators and they are used in most microwave and laser radar 
and communications systems. They do store energy for a period of time that 
is determined by several parameters. First I’ll present the key factors in 
determining an answer to your question and then I’ll put in some typical 
numbers into the equations. 

First I must point out that a mirror that is silvered so that it reflects a 
given percentage of light, say 80 %, and transmits a given percentage of 
light, say 10%, exhibits the same characteristics for light traveling in 
both directions. They are sometimes called one way mirrors but that is a 
misnomer. When used for viewing one way, the viewing room is kept darkened 
and the room being observed is kept bright. If both rooms are equally 
bright, people in both rooms could see each other by the 10% of transmitted 
light passing both ways through the mirror.

Second, light is modified when passing through solids, liquids or gasses by 
absorption and by scattering. The absorbing and scattering particles could 
be atoms, molecules, and impurities in a material, dust, rain, hale ect. 
Absorption converts the electromagnetic energy to heat with in the material. 
The size of the particles is of particular importance when analyzing 
scattering. The reason we can see light beams from the side is because of 
scattering, usually from dust particles. In a perfect vacuum we cannot see 
light beams except by stopping them.

At first it would seem that following the path of electromagnetic rays 
bouncing around inside a resonator (this is called ray tracing) and adding 
up the loss of energy at each surface reflection would be a straightforward 
approach to answering your question. However, Jenkins and White present 
charts of the attenuation of optical energy reflecting off silver, gold, 
copper and steel surfaces and the amount of reflected and transmitted energy 
is a function of 5 major parameters; wavelength, absorption, scattering, 
polarization and angle of incidence.

Reference: F. A, Jenkins, H. E. White, "Fundamentals of optics", Mc Graw- 
Hill, New York, 1950

The charts show that the amount of energy of visible light reflected from 
each ray incident on a silvered surface ranges from 85% to better then 99% 
depending on the parameters listed above.

First lets consider a beam of laser energy confined between two flat mirrors 
placed in a vacuum chamber. We also will assume that no energy leaks out of 
the open sides of the resonator. This configuration is called a Fabry-Perot 
resonator. Lets assume that the mirrors are silver with a reflection 
coefficient of 95% and absorption of 5% lost in surface heating. If we put a 
one-watt peak pulse of laser power into the resonator, 95% of the energy in 
the beam will be reflected from the first mirror leaving 0.95 watt of 
optical energy in the resonator. At the second mirror 95% of .95 watt = 
.9025 watt will remain in the resonator. The first mirror will then reflect 
95% of .9025 watt = 85.74 watts remaining in the resonator. Continuing this 
process if you plot the power in the resonator, as a function of the number 
of reflections back and for you will find the energy decreases in an 
exponential curve. After 90 reflections only 0.01 watt (1%) of the original 
energy will be left in the resonator. Eventually the energy in the resonator 
will become so small that it will not be detectable from thermal noise.

A typical Fabry-Perot resonator would have a mirror spacing of one meter 
(39.3 inches). Therefore the light pulse would travel 90 meters between the 
mirrors before it was reduced to .01 watt. At the speed of light this would 
take 0.3 microseconds. Resonators as long as 10 meters between mirrors have 
been used in large high power lasers so that the energy in them could be 
stored for about 3 microseconds before being reduced by 99%.

To keep light from leaking out of the sides of a Fabry-Perot resonator 
spherical concave mirrors are used in place of flat mirrors to confine 
energy in the resonator. Typically the mirrors are spaced so that the center 
of the radius of curvature of each mirror is placed in the center of the 
second mirror. This keeps focussing the light energy toward the mirror 
centers. This configuration is called a "confocal resonator" and is commonly 
used to confine light within lasers. Very little light leaks from the open 
sides of a confocal Fabry-Perot resonator; however, scattering and 
absorption from the material within the resonator then become more important 
to consider. 

As your question suggests, the diameter of the end mirrors could be 
increased to the point where the resonator is completely enclosed by 
mirrored surfaces; however, this would not significantly change the energy 
in the resonator. What are needed are more efficient reflecting surfaces. I 
should also point out that making measurements of the power within the 
resonator also absorb some of the energy. However, to get around this we can 
use the increased temperature of the mirror surface as a way to measure the 
energy inside. Optical scientists use this technique of measuring heat 
dissipation in the surface of an enclosure. This is called calorimetery.

Because the absorption and reflection from reflecting surfaces depends on at 
least four parameters, calculating the rate of energy loss by following rays 
bouncing around inside of more complex resonators as we did in the Fabry-
Perot configuration becomes extremely difficult. Adding curvature to the 
surfaces vastly complicates the problem. However, an elegant solution to 
this type of problem and many other problems in physics has been obtained by 
using energy considerations.

It turns out that the following energy related concepts are so broad that 
they can also be applied to conventional electronic circuits using 
inductors, capacitors and resistors and can also be adapted to analyze 
musical instruments.

Reference: A classic text book for analyzing both microwave and optical 
circuits is S. Ramo, J. R. Whinnery and T. Van Duzer, "Fields and waves in 
communication electronics", John Wiley and Sons, New York, 1967

Resonator circuits can be characterized by a "quality factor" which is 
commonly called the "Q" of the circuit. Q is a ratio of the energy stored 
within the volume of a resonator to the energy lost from passing through or 
being converted to heat (absorption) in the surface of the resonator. The 
surface need not be metal, it can also be the interface between two 
dielectric materials such as the different glasses used to confine light 
energy within the core of fiber optic transmission lines. Today, fiber optic 
lines are being fabricated into cylindrical resonators to precisely control 
the frequency of laser diode transmitters. 

The result of the energy analysis in the Ramo text is that
within the limits of the assumption of relatively small
losses, the exponential energy decay, alpha (a), within the circuit is:

           a = (pi * f) / Q

a (alpha) = attenuation constant in nepers/meter
f = the frequency of the electromagnetic energy
pi = 3.14159
Q = quality factor

* symbol is multiply, / symbol is divide

The exponential (exp) rate of energy decay in the circuit then becomes:

        U(t) = Uo * exp –(a * t)

  Where Uo = the initial energy in the circuit
      U(t) = energy in the circuit at time (t) later.

Therefore the higher the value of Q the longer it will take for energy to 
decay within the resonator. It turns out that Q is a relatively easy 
parameter to measure experimentally and from the Q the average values for 
alpha for any shaped resonator (the answer to your question) can be compared 
to the Q values calculated from energy considerations and the agreement 
turns out to be excellent. 

To make optical mirrors more efficient multi-layer thin film dielectric 
coatings are applied to metal and glass surfaces. These coatings are often 
tuned to one exact wavelength (color) of light just as a radio or TV 
resonator is tuned to one specific channel frequency. Mirrors of this type 
can have reflectivity of 99.9%  If we use mirrors of this type in a one 
meter cofocal resonator it would take 4600 passes back and forth before the 
energy is reduced by 99%. That would be 15.3 microseconds with a one-meter 
mirror spacing and 153 microseconds for a large 10-meter mirror spacing. 
With this many passes through the volume, scattering and absorption would 
then become more important parameters to analyze.

Our current microwave communications satellites, such as those used for 
DirecTV, each use more than 48 ellipsoidal (football) shaped resonators to 
control the transmitted energy, one for each broadcast transmitter. Each 
transmitter then  broadcasts several channels of digital TV programming.

It is always good to find out that ones questions and proposed solutions 
have already been studied in depth because it shows that our thinking is on 
the right track!

Best regards, Your Mad Scientist
Adrian Popa