Phase Velocity & Group Velocity — Expanded Explanation
By Mohan Dangi Gold medalist
1. Intuition recap
Imagine a set of ripples (many sinusoidal waves) traveling together. Each ripple crest moves at its own phase velocity v_p. But if the ripples are slightly different in wavelength, groups of crests appear and disappear — those groups move at the group velocity v_g. The group carries the energy and the signal, while the phase rides inside the group.
2. Phase velocity (mathematical and geometric)
Plane wave:
y(x,t) = A cos(ω t − k x)
Set the phase constant: φ = ω t − k x = const. Differentiating w.r.t. t:
dφ/dt = ω − k (dx/dt) = 0 ⇒ dx/dt = ω/k = v_p
So geometrically a phase point (e.g., a crest) moves at v_p = ω/k. Note: v_p depends on k when the medium is dispersive (i.e., ω=ω(k)), so different spectral components move at different v_p.
3. Group velocity — derivation & physical meaning
Consider two close frequencies ω₁, ω₂ (carrier near ω̄) and wave numbers k₁, k₂. The superposition becomes:
y = 2A cos(Δω t / 2 − Δk x / 2) · sin(ω̄ t − k̄ x)
Maxima of the envelope satisfy Δω t/2 − Δk x/2 = const, so the envelope velocity is
v_g = (Δω/2) / (Δk/2) = Δω / Δk
In differential limit (narrowband around k₀):
v_g = dω/dk |_{k₀}
Physical meaning: For narrowband signals, v_g is the velocity of the packet center and usually the velocity of energy and information transfer.
4. Dispersion relation, Taylor expansion & pulse evolution
Start from the dispersion relation ω = ω(k) (exact for the medium). Expand around a central wave number k₀:
ω(k) ≈ ω₀ + ω'(k₀)(k−k₀) + 1/2 ω''(k₀)(k−k₀)² + ...
When building a wave packet from spectral components A(k), the field is
Ψ(x,t) = ∫ A(k) e^{i[k x − ω(k) t]} dk.
Inserting the Taylor expansion and pulling out carrier oscillation e^{i(k₀ x − ω₀ t)}:
Ψ(x,t) ≈ e^{i(k₀ x − ω₀ t)} ∫ A(k) e^{i[(k−k₀)(x − ω' t) − 1/2 ω'' (k−k₀)² t + ...]} dk.
The integral shows three effects:
- Carrier motion: the factor
x − ω' tshows the envelope moves atv_g = ω'. - Dispersion-induced spreading: the quadratic term
−½ ω'' (k−k₀)² tcauses the integral to broaden with time — this is group velocity dispersion (GVD). - Higher-order distortion: higher derivatives of
ω(k)cause asymmetry, pulse chirp, or shape changes.
Definition: GVD parameter is β₂ = ω''(k₀). If β₂ > 0 (normal dispersion) pulses broaden; if β₂ < 0 (anomalous) pulses can compress or form solitons in nonlinear media.
D(λ) (ps/(nm·km)) but it’s derived from ω'' via change of variables between k and λ.5. Gaussian wave packet — exact spreading formula (quantum)
Consider a free particle of mass m with an initial Gaussian position wavefunction centered at x=0:
Ψ(x,0) = (1/(2π σ₀²))^{1/4} exp(−x²/(4 σ₀²) + i k₀ x)
Time evolution (using free-particle Schrödinger equation) gives a Gaussian at later times with width
σ_x(t) = σ₀ √{1 + ( (ħ t) / (2 m σ₀²) )² }
This formula shows:
- At small
t,σ_x(t) ≈ σ₀. - At large
t,σ_x(t) ≈ (ħ t) / (2 m σ₀)— i.e., linear growth with time.
Interpretation: A sharply localized particle (small σ₀) has a large spread in momentum and thus spreads quickly. A wide initial packet spreads slowly.
ω(k)=ħ k²/(2m) has ω''≠0, so packets spread.6. de Broglie waves: vp, vg and their relation
Using ω = E/ħ and k = p/ħ:
v_p = ω/k = E/p
For relativistic particle, E = γ m c² and p = γ m v, so
v_p = c² / v
Group velocity for the matter-wave packet equals the particle velocity:
v_g = dω/dk = d(E/ħ)/d(p/ħ) = dE/dp = v
Together they satisfy the elegant relation:
v_p · v_g = c²
So when the particle is slow (v_g ≪ c), the phase velocity is superluminal (v_p ≫ c) — but the product remains c². Superluminal v_p carries no information.
7. Signal, front and information velocities (causality)
There are multiple velocities one might consider:
- Phase velocity
v_p: movement of phase surfaces; not an information carrier. - Group velocity
v_g: usually the velocity of energy/information for narrowband signals. - Front velocity (or signal-front velocity): speed of the very first non-zero part of a signal — this obeys causality and cannot exceed the universal limit (e.g.,
cin vacuum).
In media with anomalous dispersion, v_g can exceed c or be negative for certain narrowband pulses — but the front velocity and causal signal velocity remain ≤ c. The apparent superluminal effects are due to reshaping of the pulse by the medium, not real superluminal transport of new information.
8. Worked numeric examples
Example 1 — de Broglie velocities
Take an electron with classical speed v = 0.01 c (~3×10⁶ m/s):
v_p = c² / v = (3×10⁸ m/s)² / (3×10⁶ m/s) = 3×10¹⁰ m/s ≈ 100 c v_g = v = 0.01 c
Phase velocity is enormous, but group velocity (the particle speed) is subluminal.
Example 2 — optical pulse in normal dispersion
For an optical fiber with β₂ = +20 ps²/km and an initial Gaussian pulse of 1 ps width, the dispersive broadening after L = 10 km roughly scales as
Δτ(L) ≈ τ₀ √{1 + ( (β₂ L) / τ₀² )² } ≈ τ₀ √{1 + ( (20 ps²/km × 10 km) / (1 ps)² )² } ≈ large
This shows why long-distance pulse propagation requires dispersion compensation.
9. Applications and experiments
- Optical communications: Dispersion management (fiber Bragg gratings, dispersion compensating fibers) is essential to prevent pulse overlap and bit errors.
- Ultrafast optics: Chirped pulse amplification and pulse compression exploit GVD and nonlinearities.
- Electron microscopy: Electron wave packet spreading limits temporal resolution in ultrafast electron diffraction.
- Quantum wave-packet experiments: Direct measurements of packet spreading and dispersion relations verify these theoretical predictions.
10. FAQ and common confusions (expanded)
Q: If phase velocity can exceed c, why can't we send messages faster?
A: Because phase velocity describes motion of constant-phase surfaces which carry no new information — only the modulation (envelope) and the signal front can carry information. The front is constrained by causality.
Q: Is group velocity always equal to energy velocity?
A: Usually in weakly dispersive, lossless media, yes — energy propagates at vg. But in strongly absorbing or highly dispersive media the relation can be more complicated; one must compute the Poynting vector or energy flux directly.
Q: What about negative group velocity?
A: Negative vg means the peak of a reshaped pulse appears to exit a sample before it enters — again caused by pulse reshaping and not superluminal information transfer.
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