The spectrum of a double pulse

In summary, the spectrum of a double pulse is the distribution of frequencies present in a signal created by combining two pulses with different frequencies. It can be measured using a spectrum analyzer and is affected by the characteristics of the individual pulses and external factors. Studying this spectrum is important in fields such as telecommunications, radar systems, and medical imaging. The spectrum can be manipulated or controlled by adjusting the characteristics of the individual pulses through techniques like pulse shaping and frequency mixing.
  • #1
snatcos
4
0
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
 
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  • #2


Hi there,

As a fellow scientist, I can see that your approach to calculating the spectrum of the two pulses in the Michelson interferometer is correct. However, I do have a few suggestions and comments that may help improve your understanding and results.

Firstly, when defining the two pulses, it may be helpful to include the amplitude of each pulse, as this can affect the resulting spectrum. Additionally, it may be useful to specify the units of the parameters (e.g. time in seconds, frequency in hertz) to ensure consistency in the calculations.

Secondly, when taking the Fourier transform, it may be helpful to explicitly state that the transform is being taken with respect to time. This can help avoid confusion when dealing with multiple variables.

Thirdly, in the final step where you take the squared absolute value of the complex number, it may be helpful to show the full calculation, including the real and imaginary parts. This can help demonstrate the trigonometric identities being used and provide a better understanding of the result.

Lastly, I would recommend checking your calculations and results with a simulation or experimental data to confirm their accuracy. This will also help identify any potential errors or discrepancies.

Overall, your approach seems sound and I don't see any major errors. However, it's always good to double-check and seek feedback from others in the scientific community. Keep up the good work!
 

1. What is the spectrum of a double pulse?

The spectrum of a double pulse refers to the distribution of frequencies present in a signal that has been created by the combination of two pulses with different frequencies.

2. How is the spectrum of a double pulse measured?

The spectrum of a double pulse can be measured using a spectrum analyzer, which displays the amplitude of each frequency component present in the signal.

3. What factors affect the spectrum of a double pulse?

The spectrum of a double pulse is affected by the characteristics of the two individual pulses, such as their frequencies, amplitudes, and durations. It can also be influenced by external factors such as noise and interference.

4. What are the applications of studying the spectrum of a double pulse?

Studying the spectrum of a double pulse is important in fields such as telecommunications, radar systems, and medical imaging, where understanding the frequency components of a signal is crucial for signal processing and analysis.

5. How can the spectrum of a double pulse be manipulated or controlled?

The spectrum of a double pulse can be manipulated or controlled by adjusting the characteristics of the two individual pulses, such as their frequencies and amplitudes. This can be achieved through various techniques such as pulse shaping and frequency mixing.

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