Add chirp to time domain pulse

Question

I have a Gaussian pulse $E(\omega)$ centered at 500 PHz with a bandwidth of 10 PHz. The pulse is transform limited initially.

I am trying to add chirp, through multiplying the FT of $E(\omega)$ by a quadratic phase factor given as

$$E(t)_{chirp} = E(t)\cdot\text{exp}(-ibt^2)$$

however, the exponential factor never seems to change the electric field?

def gaussian(x, mu, sig):
    return np.exp(-np.power(x - mu, 2.) / (2 * np.power(sig, 2.)))

# frequency in Hz
freq = np.linspace(0,2000*10**12,1000)
dt = np.diff(freq)[0]
wl = (3*10**8)/freq;
band = len(freq)


I_stokes = gaussian(freq,500*10**12,10*10**12)

phase = freq*0



E_stokes = np.sqrt(I_stokes)*np.exp(+1j*phase)
#FT:
n = len(E_stokes)
Fs = np.diff(freq)[0]
t = np.arange(0, n/Fs, 1/Fs)

Et = fftshift(ifft(E_stokes))

Phi = 1

E2t = Et**2

E2 = (fft(E2t))



I = E2t*np.exp((-1j*((t)**2)/2*Phi))

p = np.angle(I)

#plt.plot(p)

plt.plot(np.unwrap(np.angle(I))- np.unwrap(np.angle(E2t)))

Here is the difference between the angles of I and E2t:

Answer 1

Couple points that may be helpful:

The command plt.plot(I) is only plotting the real component.

The command phase = freq**0 results in a constant phase of 1 for all values of frequency, rather than $bt^2$ (was this intended)?

That said, E_stokes ends up being E_stokes = np.sqrt(I_stokes)

To see the difference in the waveforms, I recommend looking at the unwrapped phase as determined between I and E2t using

np.unwrap(np.angle(I))- np.unwrap(np.angle(E2t))

Below is the plot of the result in doing that.

Other differences may be in the specific fft ifft used (unknown origin in the OP's code as the imports are not listed). Below is the complete code I used to generate the plot, resulting in the expected quadrature phase at twice the frequency (due to the product done). As far as I am aware, I don't think I changed anything other than fft references and plotting the unwrapped phase:

import numpy as np
import scipy.fftpack as fft
import matplotlib.pyplot as plt
def gaussian(x, mu, sig):
    return np.exp(-np.power(x - mu, 2.) / (2 * np.power(sig, 2.)))

# frequency in Hz
freq = np.linspace(0,2000*10**12,1000)
dt = np.diff(freq)[0]
wl = (3*10**8)/freq;
band = len(freq)       
I_stokes = gaussian(freq,500*10**12,10*10**12)    
phase = freq**0            
E_stokes = np.sqrt(I_stokes)*np.exp(+1j*phase)
#FT:
n = len(E_stokes)
Fs = np.diff(freq)[0]
t = np.arange(0, n/Fs, 1/Fs)    
Et = fft.fftshift(fft.ifft(E_stokes))    
Phi = 1    
E2t = Et**2    
E2 = (fft.fft(E2t))
I = E2t*np.exp((-1j*((t*10**10)**2)/2*Phi))
plt.figure()
plt.plot(freq, np.unwrap(np.angle(I))- np.unwrap(np.angle(E2t)))