2.8: Pulse Shapes and Time-Bandwidth Products
- Page ID
- 48948
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The following table 2.2 shows pulse shape, spectrum and time bandwidth products of some often used pulse forms.
| \(a(t)\) | \(\hat{a} (\omega) = \int_{-\infty}^{\infty} a(t) e^{-j\omega t} dt\) | \(\Delta t\) | \(\Delta t \cdot \Delta f\) |
|---|---|---|---|
| Gauss: \(e^{-\tfrac{t^2}{t\tau^2}}\) | \(\sqrt{2\pi} \tau e^{-\tfrac{1}{t} \tau^2 \omega^2}\) | \(2\sqrt{\ln 2} \tau\) | 0.441 |
|
Hyperbolicsecant: sech (\(\dfrac{t}{\tau}\)) |
\(\dfrac{\tau}{2}\) sech (\(\dfrac{\pi}{2} \tau \omega\)) | \(1.7627 \tau\) | 0.315 |
| Rect-function: \(= \begin{cases} 1, |t| \le \tau/2 \\ 0, |t| > \tau/2 \end{cases}\) |
\(\tau \dfrac{\sin (\tau \omega/2)}{\tau \omega/2}\) | \(\tau\) | 0.886 |
| Lorentzian: \(\dfrac{1}{1 + (t/\tau)^2\) | \(2\pi \tau e^{-|\tau \omega|}\) | \(1.287 \tau\) | 0.142 |
| Double-Exponential: \(e^{-|\tfrac{t}{\tau}|}\) | \(\dfrac{\tau}{1 + (\omega \tau)^2}\) | \(\ln 2 \tau\) | 0.142 |
Figure 2.14: Fourier relationship to table above.
Figure by MIT OCW.
Figure 2.15: Fourier relationships to table above. Figure by MIT OCW.
Bibliography
[1] I. I. Rabi: "Space Quantization in a Gyrating Magnetic Field,". Phys. Rev. 51, 652-654 (1937).
[2] B. R. Mollow, "Power Spectrum of Light Scattered by Two-Level Sys- tems," Phys. Rev 188, 1969-1975 (1969).
[3] P. Meystre, M. Sargent III: Elements of Quantum Optics, Springer Verlag (1990).
[4] L. Allen and J. H. Eberly: Optical Resonance and Two-Level Atoms, Dover Verlag (1987).
[5] G. B. Whitham: "Linear and Nonlinear Waves," John Wiley and Sons, NY (1973).