Re: Coherence

To: tesla-at-pupman-dot-com
Subject: Re: Coherence
From: "Malcolm Watts" <MALCOLM-at-directorate.wnp.ac.nz> (by way of Terry Fritz <terryf-at-verinet-dot-com>)
Date: Fri, 25 Sep 1998 02:44:02 -0600
Approved: terryf-at-verinet-dot-com
Dear all,
          I have made an error of my own in this piece which I just 
discovered:

>                Here is the lowdown from the horse's mouth:
> 
> According to the 1990 ITS Notes (p2-7, 2-8) :
> 
> 
>      "How long does it take for caollapsing initially stored energy 
> to build up a standing wave on a distributed resonator? For that 
> matter, how long does it take for light to interfere and form a 
> diffraction pattern? (And what happens if the light is not 
> monochromatic?) The answer is not what is commonly called the 
> resonator fill time, which corresponds to the time for standing waves 
> to build up when forced by a sinewave generator (Tfill = 2Q/w), 
> Furthermore, other things being equal, why is the voltage rise far 
> less on a resonator driven by a bandpass signal than the rise 
> produced on the same resonator by a narrowband source?"
> 
> NOTE: I take the notion of a sinewave generator and a narrowband 
> signal to be the same thing. That leaves us with a clear oxymoron!!
> 
>      "The bottom line on our analyses was that the time taken for the 
> waves to build up, from initial uniform energy storage (at the 
> primary spark quenching instant), is inversely related to the 
> spectral width of the resonator. This is the famous Fourier 
> reciprocity relation:
> 
>     dt.df >= 1/(4.PI)
>     
> where df is related to the cavity Q as Q = df/fo," (NOTE ????????) 
> "and dt is the coherence time. We interpret the latter as the time 
> duration required for coherent oscillations to build up on a 
> distributed resonator and a standing wave pattern to form."
> 
> NOTES:  There is a glaring formulaic error in there. In fact we know 
> that Q = fo/df. 

That is what was said in the piece. However, they have used to correct
formula for Q in deriving: 

> Working things out using their stuff, 
> 
>           df = fo/Q
>       =>  dt.fo/Q >= 1/(4PI)
>       =>  dt >= Q/(4.PI.fo)

this formula since it is obvious df = fo/Q is the correct formula.

In fact had I used Q = df/fo I should have obtained 
dt >= 1/(4.PI.Q.fo)  which would be garbage.

My math in the next bit is wanting as well....
       
> If we take the correct formula for Q, 
> 
>           df = Q/fo
>       =>  dt.Q/fo >= 1/(4PI)
>       =>  dt >= fo/(4.PI.Q)

In fact the first line is completely wrong. 

The upshot is that their coherence time formula is quite correct 
despite the wrong formula for Q being given. Howver, the sight of 
coherence on the scope remains as elusive as ever. My sincere and red-
faced apologies.

Malcolm
<snip>