PHYS212 : Lectures

2.4 Bandwidth Limits

• Consider Eq. 2.50, the Fourier transformation of the square pulse of width $L$. Plots of the wavenumber spectra for pulses of widths $L=1$ and $L=2$ are shown in Fig. 2.3. We see that a narrower central peak in the spectrum corresponds to a wider pulse.

• If we define the width $\Delta k$, called the wavenumber bandwidth of the spectrum to be half the width of the central peak, or the distance between $k=0$ and the first zero of the spectrum, we have
  

  $$\Delta k = \frac{2 \pi}{L} = \frac{2 \pi}{\Delta x}$$ (2.75)

  
  
  Hence the spatial and wavenumber widths are inversely proportional. The spatial width $\Delta x$ is called the coherence length.

• The comparing Fig.'s 2.4 and 2.5, we see the same kind of relationship between a Gaussian pulse and its Fourier transformation. Taking the width of a Gaussian to be its standard deviation, we have $\Delta x = \sigma$ and from Eq. 2.62
  

  $$\frac{k^2}{2 \Delta k^2} = \frac{\sigma^2 k^2}{2}$$ (2.76)

  
  
  or
  

  $$\Delta k = \frac{1}{\sigma} = \frac{1}{\Delta x}$$ (2.77)

  
  

• Without a formal proof,^2.5 drawing from our two examples, we arrive at an approximate relation,
  

  $$\Delta k \Delta x \approx 2 \pi$$ (2.78)

  
  

• The direct analogy between wavenumber/position and frequency/time transformations leads to another bandwidth limit,
  

  $$\Delta \omega \Delta t \approx 2 \pi$$ (2.79)

  
  

• Pulse length and duration place lower limits on wavenumber and frequency bandwidths. The reverse is also true.

• Bandwidth limits are of practical importance. For example, they can be used to estimate the minimum length and duration of pulses traveling in a medium with a limited bandwidth. You will work with an example in homework.

• In the context of quantum mechanics, bandwidth limits become uncertainty principles (with $\vec{p} = \hbar \vec{k}$ and $E = \hbar \omega$). However, we do not call them uncertainty principles in the context of classical waves, and we should not. Why not?

Copyright © 2003-2009, Lewis A. Riley

Updated Wed Jan 18 09:51:28 2006