Path: utzoo!utgpu!watmath!bmaraldo
From: bmaraldo@watmath.waterloo.edu (Commander Brett Maraldo)
Newsgroups: sci.electronics
Subject: Lasers
Message-ID: <22739@watmath.waterloo.edu>
Date: 15 Dec 88 07:29:27 GMT
Distribution: sci
Organization: Unit 36 Research
Lines: 87


	I think a discussion about lasers would be incomplete without
a little theory.  This is derived from Einstein's Quantum Theory of
Radiation.  It has been simplified and I haven't bothered going through
the derivations step by step.  All you need is a little first year
calculus to get the jist of it.  If you want a more rigorous explanation
I'd suggest books on Thermodynamics (all the Boltzman stuff) and a little
Quantum Mechanics.  For now, just keep in mind that    d(something)/dt
means 'the rate of change of (something) with respect to time'.  Also,
if   d(something)/dt vanishes (=0) then the (something) is a constant.
Other than that the rest id straight forward.  The final conclusions are
what matter, the math just backs them up.



 	We have for spontaneous emission:
	
		 d N2 
(1)		------ = -A21 N2   ---->  sol'n: N2(t) = N20 e^(-A21 t)
		 dt

	... for stimulated emission:

		 d N2
(2)		------ = -B21 N2 p(f)
		 dt
 			     where: p(f) = photon density as func'n of frequency

	... and for absorption:

		 d N1
(3)		------ = -B21 N1 p(f).
		 dt

	Pictorally, for two quantum states E1 and E2 there are corresponding
populations, N1 and N2.  Thus:

	N2 (E2) ---------------------------------------------
		    |	        |	        ^
	      	    |		|		|
	         A21 N2      B21 N2 p(f)     B12 N1 p(f)
		    |		|		|
		    v		v		|
 	N1 (E1) ---------------------------------------------
    	         spont emmit    absorb         stim emmit	

	A21, B21, and B12 are the Einstein coefficeints (The Quantum Theory
of Radiation, 1916).   Einstein assumed that the distribution of N1 and N2
at E1 and E2 followed the Boltzman distribution (at a blackbody temp T
for the spectral density p(f)).  He also assumed that (1) and (2) vanished.
This means: 

		p(f) = R  [e^(hf/kT) - 1] ^ -1    ... assumes quantum thermo-
						  dynamic equilibrium

		N2/N1 = e^-(hf/kT)  		  ... Boltzman distribution

and all this leads to:

		A21/B21 = R  ( = 8 pi h f^3 / c^3)... one of those freaky							  constants.  f= frequency
					But note that A21/B21 is proportional
					to the cube of the frequency.
		B12 = B21

Now, you can see the importance here:  All the coefficeints are dependent
on each other, stim emmit and absorption are negatively equal, for N2 > N1
(look at (2) and (3)) we have amplification and this state is population
inversion, and as f increases the smaller B21 gets compared to A21 and because
B21 contributes to amplification more than A21 the more difficult it is
to amplify the field (ie. xray lasers are damned hard to make and you need
to pump them with killer densities to stimulate emmision - see the latest
Scientific American (... no I don't read it.)). 

	Note this theory was know in 1916.  It took until the 1960 to 
actually apply and come up with the laser.  Also note that the acronym 
LASER makes a hell of a lot of sence... I think its one of the best
acronyms we've come up with.


Brett L Maraldo


-- 
               --------     Unit 36 Research     ---------
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  	 	      bmaraldo@watmath.waterloo.edu 
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