My calculated laser pulse duration is too large. Where am I wrong?

Question

I am currently writing a small Python script to estimate the pulse duration from the optical spectrum.

At the end, the idea is to observe the effects of the spectral phase on the pulse duration and shape. But for now, I cannot retrieve a well-known result without taking into account spectral phase.

Indeed, a time-bandwidth limited Gaussian pulse with an optical spectrum of 30 nm width centered at 800 nm has a duration of around 31 fs. This is a basic result for a rule of thumb. You can check the result here and here, for instance.

Sadly, my Python script, using numpy/matplotlib, gives me a pulse duration which is twice too large:

from numpy import power, linspace, sqrt, log, exp, abs, max, conj
from numpy.fft import fft, fftshift, fftfreq
from scipy.constants import c

from matplotlib import pyplot as plt

# Number of points for the computations
n_points = 2**9

# Spectral width and center in wavelength (nm)
wl_center = 800e-9
wl_width = 30e-9

# Conversion of spectral parameters in frequency (s-1)
freq_center = c/wl_center
freq_width = wl_width*c/power(wl_center, 2)

# Building of the x axes
frequencies = linspace(freq_center - freq_width*8, freq_center + freq_width*8, n_points)
wavelengths = c/frequencies

# Building the Gaussian spectrum according to the parameters
diff_freq = frequencies - freq_center
sig_freq = freq_width/(sqrt(8*log(2)))
spectrum_freq_abs = exp(-(power(diff_freq, 2))/(2*power(sig_freq, 2)))

# FFT
step = frequencies[1] - frequencies[0]
time = fftshift(fftfreq(n_points, d=step))
field = fftshift(fft(spectrum_freq_abs))

# Plotting
# The commented line plot the spectrum vs the wavelength. The second line plot the pulse.
# The second line plot the optical pulse envelope vs the time in fs. It is twice too large (60 fs fwhm instead of 30 fs.)
# plt.plot(wavelengths*1e9, spectrum_freq_abs)
plt.plot(time*1e15, abs(field*conj(field)))
plt.show()

I checked the time array created thanks to the fftfreq function of numpy. It seems to be good according to the frequency array size and step. So I don't think it's an x axis problem.

I cannot find where I am wrong in this small and easy Python script, which is very frustrating. Maybe the community could give me an hint somewhere?

PS: This is a repost from physics community, where I've been kindly redirected to scicomp.

Answer 1

Seems like I'm a bit late, but I was just answering a similar question. The first thing here, is that you probably want to pay attention to the definition of the w-axis. The FFT-algorithm will consider your signal to start at w=0, so the way you defined frequencies, your carrier frequency will be wrong.

But back to your question: the websites you used to convert bandwidth to pulse duration and back use the FWHM of the intensity of the field, rather than the amplitude. Since the intensity is the amplitude squared, for a Gaussian this gives a factor of 2 for the variance, i.e. $\sqrt{2}$ for the standard deviation. You need a factor $\sqrt{2}$ in the definition of sig_freq and another factor $\sqrt{2}$ in the definition of step. This gives the missing factor of $\frac{1}{2}$ you were looking for.