# Dispersion and Pulse Propagation Conventions: $e^{+j\omega t}$, Haus Fourier transform, field envelope $e^{-t^2/2\tau^2}$ (see [[conventions]]). Fourier tools: [[fourier transform]]. Plane-wave basics: [[waves]]. ## Why pulses change shape A pulse is a superposition of plane waves (Fourier components). Propagating a distance $L$ multiplies each component by its own phase: $ E(L,\omega)=E(0,\omega)\,e^{-j\beta(\omega)L},\qquad \beta(\omega)=\frac{n(\omega)\,\omega}{c} $ If $\beta$ is linear in $\omega$, every component has the same delay and the pulse keeps its shape (the wave equation with constant $v$, pset 1 P1a). Any curvature of $\beta(\omega)$ changes the shape. ## Taylor expansion about the carrier $ \beta(\omega)\approx\beta_0+\beta'\,(\omega-\omega_0)+\frac{1}{2}\beta''\,(\omega-\omega_0)^2 $ | Term | Meaning | Effect | |---|---|---| | $\beta_0$ | phase delay, phase velocity $v_p=\dfrac{\omega_0}{\beta_0}$ | carrier phase shift only | | $\beta'=\dfrac{1}{v_g}$ | group delay per length | the envelope moves at $v_g$ | | $\beta''$ | group velocity dispersion (GVD), units s²/m | broadening and chirp | The group delay after length $L$ is $\beta' L$. The group-delay dispersion is $\mathrm{GDD}=\beta''L$, usually in fs². ## Derivatives from $n(\lambda)$ ($\lambda$ = vacuum wavelength) Use $\omega=\dfrac{2\pi c}{\lambda}$, so $\dfrac{d}{d\omega}=-\dfrac{\lambda^2}{2\pi c}\dfrac{d}{d\lambda}$. $ \frac{d\beta}{d\omega}=\frac{1}{c}\left(n-\lambda\frac{dn}{d\lambda}\right)=\frac{n_g}{c} $ $ \frac{d^2\beta}{d\omega^2}=\frac{\lambda^3}{2\pi c^2}\,\frac{d^2n}{d\lambda^2} $ Fiber-optics dispersion parameter (ps/(nm·km)): $ D=\frac{d}{d\lambda}\left(\frac{1}{v_g}\right)=-\frac{2\pi c}{\lambda^2}\,\beta''=-\frac{\lambda}{c}\,\frac{d^2n}{d\lambda^2} $ - **Normal** dispersion: $\beta''>0$, $D<0$. Red travels faster than blue. This is glass in the visible and near-IR. - **Anomalous** dispersion: $\beta''<0$, $D>0$ (silica above about 1.3 µm, grating pairs). ## Gaussian pulse through a dispersive medium Input: $ E(0,t)=e^{-t^2/2\tau^2}\,e^{j\omega_0 t} \quad\longrightarrow\quad E(0,\omega)=\frac{\tau}{\sqrt{2\pi}}\,e^{-\tau^2(\omega-\omega_0)^2/2} $ Multiply by $e^{-j\beta(\omega)L}$, write $\Omega=\omega-\omega_0$, and use the retarded time $T=t-\beta'L$. The spectral envelope becomes $e^{-(\tau^2+j\beta''L)\Omega^2/2}$, i.e. a Gaussian with a **complex** width. Transforming back (complete the square): $ A(L,T)=\frac{\tau}{\sqrt{\tau^2+j\beta''L}}\;\exp\!\left(-\frac{T^2}{2\left(\tau^2+j\beta''L\right)}\right) $ The intensity is still Gaussian, with $ \tau'^2=\tau^2+\left(\frac{\beta''L}{\tau}\right)^2 \qquad\Longrightarrow\qquad \tau'=\tau\sqrt{1+\left(\frac{\tau_c}{\tau}\right)^4},\qquad \tau_c=\sqrt{|\beta''|L} $ - $\tau_c$ is the **critical pulse width**. Pulses much shorter than $\tau_c$ broaden a lot. - Large-dispersion limit: $\tau'\approx\dfrac{|\beta''|L}{\tau}$. A shorter input pulse ends up **longer**, because it has more bandwidth ($\Delta\omega\sim\dfrac{1}{\tau}$) and the medium maps frequency to arrival time ($\Delta t\approx\beta''L\,\Delta\omega$). - The imaginary part of the width is a **linear chirp**: the instantaneous frequency sweeps linearly across the pulse. Its sign follows $\beta''$. - The spectrum magnitude is unchanged; only the spectral phase changes. So the pulse can be recompressed with the opposite GDD. - Intensity FWHM for the field envelope $e^{-t^2/2\tau^2}$: $ T_\text{FWHM}=2\sqrt{\ln 2}\;\tau\approx1.665\,\tau $ > [!warning] Width convention > All of the above is for the field envelope $e^{-t^2/2\tau^2}$. For $e^{-t^2/\tau^2}$, redo the algebra; e.g. $\tau_c$ becomes $\sqrt{2|\beta''|L}$. ### Numerical recipe (pset 2 P1) 1. Build the **envelope** on a time grid (don't sample the carrier; there are too many oscillations). 2. FFT, multiply by $e^{-j[\beta(\omega_0+\Omega)-\beta_0-\beta'\Omega]L}$ (dropping $\beta_0$ and $\beta'$ removes the phase and the group delay), then inverse FFT. 3. QC: with only $\beta''$ kept, the result must match $\tau'$ above. 4. The grid must span many $\tau'$ (otherwise the pulse wraps around), and $\Delta t$ must resolve $\tau$. ## Material models **Lorentz oscillator** (bound charge on a spring, Haus Prob. 1.4). The equation of motion is $ m\ddot{d}+\sigma\dot{d}+\kappa d=-qE,\qquad \omega_r^2=\frac{\kappa}{m} $ and with $P=-qNd=\epsilon_0\chi E$ it gives $ \chi(\omega)=\frac{\omega_p^2}{\omega_r^2-\omega^2+j\omega\dfrac{\sigma}{m}},\qquad \omega_p^2=\frac{Nq^2}{\epsilon_0 m} $ - The lossless dispersion relation is $k^2=\dfrac{\omega^2}{c^2}\left(1+\chi\right)$. - There is no propagation (imaginary $k$) for $\omega_r<\omega<\sqrt{\omega_r^2+\omega_p^2}$. - Below resonance $n$ increases with $\omega$ (normal dispersion). Near resonance it is anomalous and lossy. - Free electrons (plasma) are the case $\omega_r=0$: $ \epsilon=\epsilon_0\left(1-\frac{\omega_p^2}{\omega^2}\right) $ **Sellmeier equation:** a sum of Lorentz terms far from resonance: $ n^2(\lambda)-1=\sum_i\frac{B_i\lambda^2}{\lambda^2-C_i} $ ## Dispersion compensation and CPA - A **grating pair** (Treacy; Haus §6.6; pset 2 P3) sends longer wavelengths over a longer path, giving **anomalous** GDD. With $l$ the slant distance between gratings, $\Lambda$ the period and $\theta_d$ the diffraction angle: $ \frac{d^2\phi}{d\omega^2}=-\frac{\lambda^3\,l}{2\pi c^2\Lambda^2\cos^2\theta_d} $ (If $G$ is the perpendicular separation, then $l=\dfrac{G}{\cos\theta_d}$.) Physical meaning: $\dfrac{d\phi}{d\omega}$ is the group delay $\dfrac{P}{c}$, where $P$ is the ray path length (Brorson–Haus). - Glass ($\beta''>0$) plus a grating pair ($\beta''<0$) cancel. **Chirped-pulse amplification** (2018 Nobel Prize): stretch, amplify at low peak power, then recompress. ## Frequency combs (pset 1 P3) A periodic pulse train with period $T$ has a line spectrum (Fourier series; see [[fourier transform]]): - tooth spacing $f_r=\dfrac{1}{T}$ - the tooth amplitudes follow the single-pulse spectrum - a carrier–envelope phase slip $\Delta\phi_{ce}$ per pulse shifts every tooth by the same offset: $ f_n=n f_r+f_\text{ceo},\qquad f_\text{ceo}=\frac{\Delta\phi_{ce}}{2\pi}\,f_r $ The sign of $f_\text{ceo}$ depends on how the slip is defined. The CEO slip exists because $v_p\neq v_g$ inside the laser.