# The spectrum of a double pulse

snatcos
Hi everyone,
I was working recently with a Michelson interferometer and measured the spectrum of the two pulses and see how the fringe spacing of the spectrum change as I change the position of one of the two mirrors in the interferometer, that is the time interval between the two pulses.

I first define the double pulse, then Fourier trnasform and finally take the squared absolute value to have an expression for the spectrum.

I do this:
Two gaussian pulses at optical frequency w0 with same FWHM and spaced by t0:

E1(t) = exp(-a·t^2) · exp(j*w0*t) pulse 1
+
E2(t) = exp(-a·(t-t0)^2) · exp(j*w0*(t-t0)) pulse 2

Foruier Transform:

E1(w) = exp(-1/(4a)*(w-w0)^2 FT pulse 1
E2(w) = exp(-1/(4a)*w^2) * exp(-j*t0*w) * delta(w-w0) * exp(-j*t0*w) FT pulse 2

E1(w) + E2(w) = exp(-1/(4a)*(w-w0)^2 * (1 + exp(-j*2*t0*w)

Taht is a gaussian function modulated by (1 + exp(-j*2*t0*w).

The absolute value of a complex number c is : sqrt( Re(c)^2 + Im(c)^2 ), so:

|(1 + exp(-j*2*t0*w)|^2 = (1+cos(2*t0*w)^2 + sin(2*t0*w)^2 =

2 * (1+cos(2*t0*w)) = 4*cos^2(t0*w)

And this gives the fringes in the spectrum.

Does anyone see any errors here ? I am not so sure about the result and appreciate any suggestion, critics.

Thanks a lot