Phase Velocity, Group Velocity, and Pulse Spreading

Derivation

Phase Velocity, Group Velocity, and Pulse Spreading

D-205 Home PU-206 Threads waves · light Depends on Dispersion and Absorption from the Lorentz Oscillator, fourier-transform-pairs

Statement

Given a dispersion relation \(\omega=\omega(k)\) for a linear wave, the phase velocity of a single Fourier mode is \(v_p=\omega/k\) and the velocity of a narrow-band wave packet is the group velocity \(v_g=\dfrac{d\omega}{dk}\). Expanding \(\omega(k)\) to second order about the carrier \(k_0\), the curvature \(\beta\equiv\left.\dfrac{d^2\omega}{dk^2}\right|_{k_0}\) — the group-velocity dispersion — broadens an initially Gaussian packet of width \(\sigma_0\) to \(\sigma(t)=\sigma_0\sqrt{1+\left(\beta t/\sigma_0^{2}\right)^{2}}\).

Why it matters

Energy and information ride on the envelope of a wave, not on its individual crests. In any dispersive medium — glass, an optical fibre, a waveguide, the ocean, or the vacuum for a de Broglie matter wave — the crests (phase velocity) and the envelope (group velocity) travel at different speeds, and generically \(v_p\neq v_g\). Getting this distinction right is the difference between a signal that arrives when you expect and one that does not.

The same second-order term that separates \(v_g\) from \(v_p\) also forces every real pulse to spread: a bit in a fibre smears into its neighbours, a laser pulse loses peak power, a quantum wave packet delocalises. Pulse spreading is therefore the fundamental bandwidth limit of dispersive transmission, and this derivation makes it quantitative.

Assumptions

The medium is linear and time-invariant.If dropped, Fourier modes couple, harmonics are generated, and \(\omega(k)\) is no longer a single-valued relation you can superpose over — the whole spectral decomposition fails. A dispersion relation \(\omega=\omega(k)\) exists and is real.A complex \(\omega(k)\) means gain or absorption; the packet amplitude then changes as it moves and \(v_g\) alone no longer describes envelope transport. The spectrum \(A(k)\) is narrow-band about a carrier \(k_0\).If the bandwidth is not small, the second-order Taylor truncation of \(\omega(k)\) is inaccurate; third-order and higher terms distort the envelope shape (asymmetry, oscillatory tails), not just its width. The packet stays in a regime where \(\omega(k)\) is analytic (no band edge, resonance, or cutoff nearby).Near a resonance \(v_g\) can exceed \(c\), vanish, or turn negative; the naive "signal travels at \(v_g\)" reading breaks and one must return to the full \(\omega(k)\) and causal (Sommerfeld–Brillouin) analysis.

Derivation

\[ \psi(x,t)=e^{i(kx-\omega t)},\qquad kx-\omega t=\text{const}\ \Rightarrow\ v_p=\frac{dx}{dt}=\frac{\omega}{k} \]

A surface of constant phase moves so that \(k\,dx=\omega\,dt\); this defines the phase velocity of one monochromatic mode. A

\[ \psi=\cos\!\big[(k_0+\delta k)x-(\omega_0+\delta\omega)t\big]+\cos\!\big[(k_0-\delta k)x-(\omega_0-\delta\omega)t\big] \]

Superpose two equal-amplitude modes symmetrically placed about \((k_0,\omega_0)\) — the minimal model of a "packet". A

\[ \psi=2\underbrace{\cos(k_0x-\omega_0t)}_{\text{carrier}}\;\underbrace{\cos(\delta k\,x-\delta\omega\,t)}_{\text{envelope}} \]

Apply \(\cos A+\cos B=2\cos\frac{A+B}{2}\cos\frac{A-B}{2}\). The slow envelope factor rides on the fast carrier. A

\[ \delta k\,x-\delta\omega\,t=\text{const}\ \Rightarrow\ v_g=\frac{\delta\omega}{\delta k}\ \xrightarrow[\delta k\to0]{}\ \frac{d\omega}{dk}\bigg|_{k_0} \]

The envelope's constant-phase condition gives its speed; taking the two modes infinitesimally close promotes the ratio to a derivative. B

\[ \psi(x,t)=\int_{-\infty}^{\infty} A(k)\,e^{i\left(kx-\omega(k)t\right)}\,dk \]

A general packet is the continuous superposition; \(A(k)\) is the Fourier amplitude of the initial profile (prior result: fourier-transform-pairs). B

\[ \omega(k)\approx\omega_0+v_g\,\kappa+\tfrac12\beta\,\kappa^{2},\qquad \kappa\equiv k-k_0,\quad v_g=\omega'(k_0),\ \beta=\omega''(k_0) \]

Taylor-expand \(\omega(k)\) about the carrier to second order — the leading correction beyond uniform translation. Validity requires the narrow-band assumption. B

\[ \psi=e^{i(k_0x-\omega_0t)}\!\int A(\kappa)\,\exp\!\Big[i\Big(\kappa\,\xi-\tfrac12\beta\kappa^2 t\Big)\Big]d\kappa,\qquad \xi\equiv x-v_g t \]

Factor out the carrier and change to the envelope frame \(\xi=x-v_g t\); the linear term produces pure translation at \(v_g\), leaving the \(\beta\)-term as the only shape-changing piece. C

\[ A(\kappa)\propto e^{-\sigma_0^{2}\kappa^{2}/2}\quad\Longleftrightarrow\quad \psi(x,0)\propto e^{-x^{2}/2\sigma_0^{2}}\,e^{ik_0x} \]

Choose a Gaussian initial envelope of width \(\sigma_0\); its transform is Gaussian (fourier-transform-pairs), which makes the propagation integral exactly doable. B

\[ \psi\propto\int \exp\!\Big[-\tfrac12(\sigma_0^{2}+i\beta t)\kappa^{2}+i\kappa\xi\Big]d\kappa \propto\frac{1}{\sqrt{\sigma_0^{2}+i\beta t}}\exp\!\left[-\frac{\xi^{2}}{2(\sigma_0^{2}+i\beta t)}\right] \]

Complete the square and use \(\int e^{-a\kappa^2+ib\kappa}d\kappa=\sqrt{\pi/a}\,e^{-b^2/4a}\); the width parameter has become complex, \(\sigma_0^2\to\sigma_0^2+i\beta t\). C

\[ |\psi|^{2}\propto\exp\!\left[-\frac{\xi^{2}\sigma_0^{2}}{\sigma_0^{4}+\beta^{2}t^{2}}\right] \equiv\exp\!\left[-\frac{\xi^{2}}{\sigma^{2}(t)}\right],\qquad \sigma^{2}(t)=\sigma_0^{2}+\frac{\beta^{2}t^{2}}{\sigma_0^{2}} \]

Take the squared modulus; the real part \(\mathrm{Re}\,[1/(\sigma_0^2+i\beta t)]=\sigma_0^2/(\sigma_0^4+\beta^2t^2)\) fixes the intensity width. The envelope stays Gaussian but widens. C

Result

\[ v_p=\frac{\omega}{k},\qquad v_g=\frac{d\omega}{dk},\qquad \sigma(t)=\sigma_0\sqrt{1+\left(\frac{\beta t}{\sigma_0^{2}}\right)^{2}},\quad \beta=\frac{d^2\omega}{dk^2}\bigg|_{k_0} \]

Reading. Crests move at \(v_p\); the envelope (energy, information) moves at \(v_g\). The two are equal only when \(\omega\propto k\). The curvature \(\beta\) of the dispersion relation makes the packet spread symmetrically about its centre, the width growing linearly in \(t\) once \(\beta t\gg\sigma_0^2\). A narrower initial packet (small \(\sigma_0\), broad spectrum) spreads faster, because more widely separated \(k\)-components drift apart more quickly.

Units check. \([v_p]=[v_g]=(\mathrm{s^{-1}})/(\mathrm{m^{-1}})=\mathrm{m\,s^{-1}}\). \([\beta]=(\mathrm{s^{-1}})/(\mathrm{m^{-1}})^2=\mathrm{m^{2}\,s^{-1}}\), so \(\beta t/\sigma_0^{2}=(\mathrm{m^2\,s^{-1}})(\mathrm{s})/\mathrm{m^2}\) is dimensionless and \(\sigma(t)\) has units of length. Consistent.

Limiting cases

Non-dispersive (\(\omega=ck\)): \(v_p=v_g=c\) and \(\beta=0\); the packet propagates rigidly, forever un-spread. This is light in vacuum or a wave on an ideal string.
Weak dispersion, early times (\(\beta t\ll\sigma_0^2\)): \(\sigma(t)\approx\sigma_0\big[1+\tfrac12(\beta t/\sigma_0^2)^2\big]\) — spreading is initially quadratic and negligible; the packet holds shape over a dispersion length \(L_D\sim\sigma_0^2/\beta\cdot v_g\).
Long times (\(\beta t\gg\sigma_0^2\)): \(\sigma(t)\to|\beta|t/\sigma_0\), linear ballistic spreading set by the spectral width \(\sim1/\sigma_0\).
Rayleigh relation: writing \(v_g=v_p+k\,\dfrac{dv_p}{dk}=v_p-\lambda\,\dfrac{dv_p}{d\lambda}\); normal dispersion (\(dv_p/d\lambda>0\)) gives \(v_g<v_p\), anomalous dispersion the reverse.
Optics via refractive index: \(v_g=\dfrac{c}{n+\omega\,dn/d\omega}=\dfrac{c}{n_g}\); the group index \(n_g\) exceeds \(n\) in normal dispersion.

Breaks when

Near a resonance or absorption band (e.g. the Lorentz-oscillator pole, prior result lorentz-oscillator-dispersion): \(\omega(k)\) acquires a large imaginary part and strong curvature, \(v_g\) can exceed \(c\), go to zero, or reverse sign. It then no longer measures signal or energy speed, and the front velocity stays \(\le c\) by causality — the second-order expansion is invalid.
Broadband / ultrashort pulses: when the fractional bandwidth is not small, the truncation at \(\beta\kappa^2\) drops third-order dispersion \(\gamma=d^3\omega/dk^3\); the envelope then develops asymmetry and oscillatory tails, not the symmetric Gaussian broadening derived here.
Nonlinear media (high intensity): if \(n\) depends on amplitude (Kerr effect), self-phase modulation competes with GVD; in the anomalous-dispersion regime the two can balance to form a soliton that does not spread, violating the linear result entirely.
Band edges / cutoffs: where \(d\omega/dk\to0\) (waveguide cutoff, edge of a photonic or electronic band) the group velocity vanishes and \(\beta\) diverges; the packet is not narrow-band relative to the local curvature and spreads anomalously fast.

Failure modes

Confusing \(v_p\) and \(v_g\): quoting \(\omega/k\) as "the speed of the pulse". The envelope moves at \(d\omega/dk\); only for \(\omega\propto k\) do they coincide.
Superluminal panic: finding \(v_g>c\) near a resonance and concluding relativity is violated. There, \(v_g\) is not the signal velocity; the causal front moves at \(\le c\).
Sign error in the Rayleigh relation: mixing up \(v_g=v_p-\lambda\,dv_p/d\lambda\) (in \(\lambda\)) with \(v_g=v_p+k\,dv_p/dk\) (in \(k\)); the sign flips because \(k=2\pi/\lambda\).
Amplitude vs intensity width: forgetting that squaring the Gaussian amplitude changes the \(1/e\) width by \(\sqrt2\); state which width \(\sigma\) refers to.
Assuming narrow packet = slow spreading: the opposite is true — a spatially narrow packet has a broad spectrum and spreads fastest.
Dropping the carrier phase: discarding \(e^{i(k_0x-\omega_0t)}\) and then being surprised the "envelope" oscillates.

Discussion

The deep content of this derivation is a separation of scales. A wave packet carries a fast internal clock — the carrier oscillation at \(\omega_0\) whose crests move at \(v_p\) — and a slow external shape, the envelope whose centroid moves at \(v_g\) and whose width evolves under \(\beta\). Because these are governed by successive derivatives of the same function \(\omega(k)\) evaluated at \(k_0\), the entire near-carrier behaviour of a dispersive medium is encoded in the local Taylor coefficients: \(\omega_0\) (phase), \(v_g=\omega'\) (transport), \(\beta=\omega''\) (spreading), \(\gamma=\omega'''\) (distortion).

Group velocity, not phase velocity, is what carries energy and information in a transparent medium, and it is what a stopwatch measures for a pulse. This is why radio engineers care about waveguide group delay, why fibre-optic links are dispersion-managed, and why the ocean's swell (envelope) outruns or lags its individual crests — for deep-water gravity waves \(v_g=v_p/2\), so crests visibly appear at the back of a group, march forward through it, and vanish at the front.

The formal structure is identical to Schrödinger evolution. A free particle has \(\omega=\hbar k^2/2m\), giving \(v_g=\hbar k/m=p/m\) (the classical velocity, exactly de Broglie's point) and \(\beta=\hbar/m\), a constant. The wave-packet spreading \(\sigma(t)=\sigma_0\sqrt{1+(\hbar t/m\sigma_0^2)^2}\) that follows is the same quantum delocalisation one derives from the time-dependent Schrödinger equation — here it is just group-velocity dispersion of a matter wave, with the medium being the vacuum and the "dispersion" being the quadratic kinetic energy. The uncertainty principle is visible in it: the tighter you localise (\(\sigma_0\) small), the broader the momentum spread \(\sim\hbar/\sigma_0\), and the faster the packet disperses.

Common misconceptions. "The wave moves at \(\omega/k\)" — only a single infinite mode does, and it carries no information. "Group velocity is always less than \(c\)" — it can exceed \(c\), be zero, or negative near resonances without any causality violation, because it is not the signal (front) velocity. "Spreading means energy is lost" — in a lossless medium the packet spreads but its total energy is conserved; only the peak amplitude falls.

Worked examples

\[ \omega=\sqrt{gk}\quad\Rightarrow\quad v_p=\frac{\omega}{k}=\sqrt{\frac{g}{k}},\qquad v_g=\frac{d\omega}{dk}=\frac12\sqrt{\frac{g}{k}}=\frac{v_p}{2} \]

Deep-water gravity waves. Differentiate \(\sqrt{gk}\); the group velocity is exactly half the phase velocity. A

\[ \lambda=100\ \mathrm{m}\Rightarrow k=\frac{2\pi}{\lambda}=6.28\times10^{-2}\ \mathrm{m^{-1}},\quad v_p=\sqrt{\frac{9.81}{6.28\times10^{-2}}}=12.5\ \mathrm{m\,s^{-1}} \]

Insert numbers with \(g=9.81\ \mathrm{m\,s^{-2}}\). A

\[ v_p\approx12.5\ \mathrm{m\,s^{-1}},\qquad v_g=\tfrac12 v_p\approx6.3\ \mathrm{m\,s^{-1}} \]

Reading. A 100 m swell has crests travelling at \(\sim25\) knots but the wave group — the energy — advancing at only \(\sim12\) knots. Crests are born at the rear of the group and die at its front.

\[ \omega=\frac{\hbar k^{2}}{2m}\ \Rightarrow\ v_g=\frac{\hbar k}{m},\quad \beta=\frac{d^2\omega}{dk^2}=\frac{\hbar}{m};\qquad \sigma(t)=\sigma_0\sqrt{1+\left(\frac{\hbar t}{m\sigma_0^{2}}\right)^{2}} \]

Free-electron matter wave. The dispersion is quadratic, so \(\beta=\hbar/m\) is a constant. B

\[ \sigma=2\sigma_0\ \Rightarrow\ \frac{\hbar t}{m\sigma_0^{2}}=\sqrt{3}\ \Rightarrow\ t=\frac{\sqrt{3}\,m\sigma_0^{2}}{\hbar} \]

Solve for the doubling time. B

\[ t=\frac{\sqrt{3}\,(9.11\times10^{-31})(1\times10^{-9})^{2}}{1.055\times10^{-34}}\ \mathrm{s} \]

With \(\sigma_0=1\ \mathrm{nm}\), \(m_e=9.11\times10^{-31}\ \mathrm{kg}\), \(\hbar=1.055\times10^{-34}\ \mathrm{J\,s}\). B

\[ t\approx1.5\times10^{-14}\ \mathrm{s}=15\ \mathrm{fs} \]

Reading. A nanometre-localised electron doubles its spatial width in about 15 femtoseconds — dramatic quantum spreading, and a direct manifestation of group-velocity dispersion of the de Broglie wave. A heavier particle (larger \(m\)) or broader initial packet spreads far more slowly.

Problems

An electromagnetic wave in vacuum obeys \(\omega=ck\). Show that \(v_p=v_g=c\) and that \(\beta=0\), and state the physical consequence for a pulse.
Solution
\(v_p=\omega/k=c\). \(v_g=d\omega/dk=c\). \(\beta=d^2\omega/dk^2=0\). Since \(\beta=0\), \(\sigma(t)=\sigma_0\) for all \(t\): the pulse propagates without spreading and both crests and envelope move at \(c\). Vacuum is non-dispersive.
A hollow rectangular waveguide has \(\omega=\sqrt{\omega_c^{2}+c^{2}k^{2}}\) with cutoff frequency \(f_c=10\ \mathrm{GHz}\). Find \(v_p\) and \(v_g\) at \(f=15\ \mathrm{GHz}\), and verify \(v_pv_g=c^2\).
Solution
\(v_p=\omega/k=c/\sqrt{1-(f_c/f)^2}\), \(v_g=d\omega/dk=c^2k/\omega=c\sqrt{1-(f_c/f)^2}\). With \(f_c/f=10/15=0.667\): \(\sqrt{1-0.444}=\sqrt{0.556}=0.745\). So \(v_g=c(0.745)=2.24\times10^{8}\ \mathrm{m\,s^{-1}}\) and \(v_p=c/0.745=4.02\times10^{8}\ \mathrm{m\,s^{-1}}>c\). Product \(v_pv_g=c^2=9.0\times10^{16}\ \mathrm{m^2\,s^{-2}}\). The phase velocity exceeds \(c\) but carries no signal; the group velocity, which does, stays below \(c\).
A deep-water swell has wavelength \(\lambda=64\ \mathrm{m}\). Using \(\omega=\sqrt{gk}\), find the group velocity and the time for a wave group to cross \(10\ \mathrm{km}\) of open water. Take \(g=9.81\ \mathrm{m\,s^{-2}}\).
Solution
\(k=2\pi/64=9.82\times10^{-2}\ \mathrm{m^{-1}}\). \(v_p=\sqrt{g/k}=\sqrt{9.81/0.0982}=9.99\ \mathrm{m\,s^{-1}}\). \(v_g=v_p/2=5.0\ \mathrm{m\,s^{-1}}\). Time to cross \(10\ \mathrm{km}\): \(t=10^4/5.0=2.0\times10^{3}\ \mathrm{s}\approx33\ \mathrm{min}\). (Individual crests, at \(10\ \mathrm{m\,s^{-1}}\), take only \(\sim17\ \mathrm{min}\).)
Crown glass near \(\lambda=589\ \mathrm{nm}\) has \(n=1.512\) and \(dn/d\lambda=-4.4\times10^{4}\ \mathrm{m^{-1}}\). Using \(v_g=c/(n-\lambda\,dn/d\lambda)\), find the group velocity and the group index \(n_g\).
Solution
\(-\lambda\,dn/d\lambda=-(589\times10^{-9})(-4.4\times10^{4})=+0.0259\). Group index \(n_g=n-\lambda\,dn/d\lambda=1.512+0.026=1.538\). \(v_g=c/n_g=(3.00\times10^{8})/1.538=1.95\times10^{8}\ \mathrm{m\,s^{-1}}\). The pulse envelope travels slower than the phase speed \(c/n=1.98\times10^{8}\ \mathrm{m\,s^{-1}}\) — normal dispersion, \(n_g>n\).
A Gaussian light pulse of initial width \(\sigma_0=2.0\ \mathrm{\mu m}\) propagates in a medium with \(\beta=d^2\omega/dk^2=8.0\times10^{4}\ \mathrm{m^2\,s^{-1}}\). (a) Find the dispersion time \(t_D=\sigma_0^2/|\beta|\) after which spreading becomes significant. (b) Find \(\sigma\) at \(t=1.0\times10^{-13}\ \mathrm{s}\).
Solution
(a) \(t_D=\sigma_0^2/|\beta|=(2.0\times10^{-6})^2/(8.0\times10^{4})=(4.0\times10^{-12})/(8.0\times10^{4})=5.0\times10^{-17}\ \mathrm{s}=50\ \mathrm{as}\). (b) Dimensionless \(\beta t/\sigma_0^2=t/t_D=(1.0\times10^{-13})/(5.0\times10^{-17})=2.0\times10^{3}\), so \(\sigma=\sigma_0\sqrt{1+(2.0\times10^3)^2}\approx\sigma_0(2.0\times10^3)=2.0\times10^{-6}\times2.0\times10^{3}=4.0\times10^{-3}\ \mathrm{m}=4.0\ \mathrm{mm}\). The packet, initially micron-scale, has spread by a factor \(\sim2000\) — deep in the linear-spreading regime.