Quasi-monochromatic waves and coherence

real light waves as a superposition of different frequencies

Consider the superposition of several different monochromatic waves
- random distribution of frequency within a bandwidth of δω
- random phase

bandwidth 0.04(Subscript[ω, 0])

field plot

ω0=10;
δω=0.04ω0;
fLst[t_]=Table[Exp[I ((ω0+RandomReal[{-δω/2,δω/2}])t+RandomReal[2π])],{j,1,5}];

fLst[t][[2]]

Plot[Total[Re[fLst[t]]],{t,-30,30},PlotPoints->100]

intensity plot

Plot[((Abs[Total[Re[fLst[t]]]])^2),{t,-30,30},PlotPoints->100]

bandwidth 0.08(Subscript[ω, 0])

ω0=10;
δω=0.08ω0;
fLst[t_]=Table[Exp[I ((ω0+RandomReal[{-δω/2,δω/2}])t+RandomReal[2π])],{j,1,5}];
Plot[Re[Total[fLst[t]]],{t,-30,30},PlotPoints->100]

Plot[((Abs[Total[Re[fLst[t]]]])^2),{t,-30,30},PlotPoints->100]

bandwidth 0.16(Subscript[ω, 0])

ω0=10;
δω=0.16ω0;
fLst[t_]=Table[Exp[I ((ω0+RandomReal[{-δω/2,δω/2}])t+RandomReal[2π])],{j,1,5}];
Plot[Re[Total[fLst[t]]],{t,-30,30},PlotPoints->100]

Plot[((Abs[Total[Re[fLst[t]]]])^2),{t,-30,30},PlotPoints->100]

if we sum up the complex fields and then compute the abs[ ], this eliminates the common carrier

Plot[((Abs[Total[fLst[t]]])^2),{t,-30,30},PlotPoints->100]

random frequencies (bandwidth 0.16(Subscript[ω, 0])), zero phase difference

ω0=10;
δω=0.16ω0;
fLst[t_]=Table[Re[Exp[I ((ω0+RandomReal[{-δω/2,δω/2}])t+0RandomReal[2π])] ],{j,1,5}];
Plot[Total[fLst[t]],{t,-30,30},PlotPoints->100]

equally-spaced frequencies (bandwidth 0.16(Subscript[ω, 0])), random phase difference

ω0=10;
δω=0.16ω0;
dω=δω/4;
dωLst=Table[j dω,{j,-2,2}];
fLst[t_]=Table[Re[Exp[I ((ω0+dωLst[[j]])t+RandomReal[2π])] ],{j,1,5}];
Plot[Total[fLst[t]],{t,-30,30},PlotPoints->100]

Plot[((Abs[Total[fLst[t]]])^2),{t,-30,30},PlotPoints->100]

equally-spaced frequencies (bandwidth 0.16(Subscript[ω, 0])), zero phase difference

ω0=10;
δω=0.16ω0;
dω=δω/4;
dωLst=Table[j dω,{j,-2,2}];
fLst[t_]=Table[Re[Exp[I ((ω0+dωLst[[j]])t+0RandomReal[2π])] ],{j,1,5}];
Plot[Total[fLst[t]],{t,-30,30},PlotPoints->200]

Time averages of intensity for quasi-monochromatic light

<g(Subscript[>, T])=(1/T)(∫_(-T/2)^(T/2))g(t)dt
Consider our light field as a superposition of N waves: 
- equal amplitude a (real)
- frequency randomly distributed within a bandwidth of δω around (Subscript[ω, 0])
-     δω<<(Subscript[ω, 0])
 
 f(t)=a(∑_(n=1)^N)exp[ -i ((Subscript[ω, n])t+(Subscript[ϕ, n]))] 
 f is a complex function, but the physical field is Re[f]
 
 calculate intensity: 
 
  I(t)=(((|f(t)|))^2)=(((a^2)(| (∑_(n=1)^N)exp[ -i ((Subscript[ω, n])t+(Subscript[ϕ, n]))]  |))^2)
  note that this calculation already implies averaging over the very fast fluctuations in I(t) at 2(Subscript[ω, 0])
  
  rewrite the intensity as a double sum over m, n
  I(t)=(((|f(t)|))^2)=(a^2)((∑_(n=1)^N)(∑_(m=1)^N)exp[- i (((Subscript[ω, n])-(Subscript[ω, m]))t+(Subscript[ϕ, n])-(Subscript[ϕ, m]))] )
  this is a sum that includes a number of beat frequencies at the frequency differences

Let’s look at the time averaged intensity: 
<I(t)(Subscript[>, T])=(a^2)(1/T)(∫_(-T/2)^(T/2))((∑_(n=1)^N)(∑_(m=1)^N)exp[ -i (((Subscript[ω, n])-(Subscript[ω, m]))t+(Subscript[ϕ, n])-(Subscript[ϕ, m]))] ) dt

for averaging over a very long time: 

(Subscript[T, 0])>>2π/δω
all of the oscillations should average out, and the only terms that count are for m = n, where the exp[ ] = 1

<I(t)(Subscript[>, (Subscript[T, 0])])=(a^2)(1/(Subscript[T, 0]))(∫_(-(Subscript[T, 0])/2)^((Subscript[T, 0])/2))((∑_(n=1)^N)1 ) dt=(a^2)N

Short term average: 
split out terms where m = n
<I(t)(Subscript[>, (Subscript[T, 1])])=(a^2)N+2(a^2)(1/(Subscript[T, 1]))(∫_(-(Subscript[T, 1])/2)^((Subscript[T, 1])/2))((∑_(n=1)^N)(∑_(m<n))cos[ ((Subscript[ω, n])-(Subscript[ω, m]))t+(Subscript[ϕ, n])-(Subscript[ϕ, m])] ) dt  
our averaging time will be less than the period for any of the beats, 
(Subscript[T, 1])δω<(π/2)
so during that time the cos[ ] term will be approx constant, but with an random value. 

approximate form: 
<I(t)(Subscript[>, (Subscript[T, 1])])=(a^2)N+2(a^2)(∑_(n=1)^N)(∑_(m<n))cos[ (Subscript[ϕ, n])-(Subscript[ϕ, m])] 

These fluctuations are of the same order as the average intensity: we see 100% fluctuations. 

Over a very short time, t<<(Subscript[T, 1]) the wave just looks like a sine wave: 

f(t)=a(∑_(n=1)^N)exp[ i ((Subscript[ω, n])t+(Subscript[ϕ, n]))] = a (e^(-i (Subscript[ω, 0])t))(∑_(n=1)^N)exp[- i (((Subscript[ω, n])-(Subscript[ω, 0]))t+(Subscript[ϕ, n]))] 

again, during the short interval (less than the minimum beat period), we can write approximately 

f(t)= a (e^(-i (Subscript[ω, 0])t))(∑_(n=1)^N)exp[i (Subscript[ϕ, n])] 

Random phase, but still nominally oscillating at the carrier frequency. 

This shorter time period is the coherence time.

Interference with quasi-monochromatic waves

example

ω0=10;
δω=0.16ω0;
fLst[t_]=Table[Exp[I ((ω0+RandomReal[{-δω/2,δω/2}])t+RandomReal[2π])],{j,1,5}];

τDelay=5;

Plot[Re[Total[fLst[t]]],{t,-30,30},PlotPoints->100]
Plot[Re[Total[fLst[t+τDelay]]],{t,-30,30},PlotPoints->100]

tRange=100;
Plot[((Abs[Total[fLst[t]]+Total[fLst[t+τDelay]]])^2),{t,-tRange,tRange},PlotPoints->100]

Clear[τDelay];

Iint[τDelay_,T0_]:=(1/T0)NIntegrate[((Abs[Total[fLst[t]]+Total[fLst[t+τDelay]]])^2),{t,-T0/2,T0/2}]

tCoh=2π/δω;
Print["coherence time: ",tCoh];
τDelayTst=5;
IintLst=Table[Iint[τDelayTst,j 5 tCoh],{j,1,100}];

ListPlot[IintLst]

dτDelay=0.1;
interfometerOutputLst=Table[Iint[j dτDelay,100 tCoh],{j,-400,400}];
ListLinePlot[interfometerOutputLst]