5 Group velocity and group index

Run this chapter’s code in Google Colab (no installation).

Slides 60–69: wave packets, beat of two waves, group velocity $v_g=d\omega/dk$, group index $N_g=n-\lambda\,dn/d\lambda$.

5.1 Key relations

There are no perfectly monochromatic waves (slide 60). Two waves at $\omega\pm\delta\omega$ with wave vectors $k\pm\delta k$ add up to (slides 64–65)

\[ E_x=2E_0\cos(\delta\omega\,t-\delta k\,z)\cos(\omega t-kz), \]

a carrier at $\omega$ modulated by a slow envelope. The envelope, which carries energy and information, travels at the group velocity (slides 61, 66–68):

\[ v_g=\frac{d\omega}{dk}=\frac{c}{N_g},\qquad N_g=n-\lambda\frac{dn}{d\lambda}. \]

In a dispersive medium ($n$ depends on $\lambda$), $v_g\neq v$. For silica in the near IR, $N_g\approx1.46$ while $n\approx1.44$ (slide 69).

5.1.1 Carrier and envelope, live

The red dot rides on a crest of the carrier (phase velocity). The orange dot rides on the maximum of the envelope (group velocity). Change their ratio to see normal ($v_g<v$), no ($v_g=v$) and anomalous ($v_g>v$) dispersion.

Show the Python

viewof gv = Inputs.form({
  t:  Inputs.range([0, 40], {value: 0, step: 0.1, label: "time (carrier periods)"}),
  r:  Inputs.range([0.3, 1.7], {value: 0.7, step: 0.01, label: "v_g / v_phase"}),
  dw: Inputs.range([0.02, 0.2], {value: 0.08, step: 0.005, label: "δω / ω"})
})

Show the Python

{
  const k = 2*Math.PI, w = 2*Math.PI;           // carrier: λ = 1, period = 1, v = 1
  const dw = gv.dw * w, dk = dw / gv.r;         // v_g = dω/dk = r
  const t = gv.t, zs = d3.range(0, 30.0001, 0.01);
  const E = zs.map(z => ({z, E: 2*Math.cos(dw*t - dk*z)*Math.cos(w*t - k*z), part: "field"}));
  const env = zs.map(z => ({z, E: 2*Math.abs(Math.cos(dw*t - dk*z)), part: "envelope"}));
  const Lenv = Math.PI / dk;                      // distance between envelope maxima
  const zg = (gv.r * t) % Lenv, zp = t % 1;
  return Plot.plot({
    width: Math.min(width, 1050), height: 280, x: {label: "z (carrier wavelengths)", domain: [0, 30]}, y: {label: "Eₓ", domain: [-2.3, 2.6]},
    marks: [Plot.line(env, {x: "z", y: "E", stroke: "#eb6834", strokeDasharray: "4,3"}),
            Plot.line(E, {x: "z", y: "E", stroke: "#2a78d6", strokeWidth: 1.2}),
            Plot.dot([{z: zg, E: 2}], {x: "z", y: "E", r: 6, fill: "#eb6834"}),
            Plot.dot([{z: zp, E: 2*Math.cos(dw*t - dk*zp)}], {x: "z", y: "E", r: 6, fill: "#e34948"})]
  });
}

Show the Python

html`<div class="result-box">${gv.r < 0.99 ? "Normal dispersion: the envelope lags behind the carrier crests (v<sub>g</sub> &lt; v)." :
 gv.r > 1.01 ? "Anomalous dispersion: the envelope outruns the carrier crests (v<sub>g</sub> &gt; v), as near an absorption line." :
 "No dispersion: carrier and envelope move together."} After ${gv.t} periods the carrier has moved ${gv.t.toFixed(1)} λ and the envelope ${(gv.r*gv.t).toFixed(1)} λ.</div>`

5.2 Worked example: group delay in a fiber link

Show the Python

z = np.linspace(0, 400, 4000)
k1, k2, w1, w2 = 1.0, 1.02, 1.0, 1.01         # v_g = Δω/Δk = 0.5, v_ph ≈ 1
fig, ax = plt.subplots(figsize=(9, 3))
for k, t in enumerate((0, 60)):
    E = np.cos(w1*t - k1*z) + np.cos(w2*t - k2*z)
    ax.plot(z, E + 4.5*k, color=ol.PALETTE[k], lw=0.8, label=f"t = {t}")
ax.set(xlabel="z", yticks=[], title="envelope moves by v_g·t = 30, carrier by v_ph·t ≈ 60")
ax.legend(loc="upper right"); plt.show()

Figure 5.1: Two waves 1 % apart in frequency in a dispersive medium: the carrier moves at v_ph, the envelope at v_g = v_ph/2.

Show the Python

Lf = 1e3
for lam_nm in (1310, 1550):
    l = nm(lam_nm)
    print(f"{lam_nm} nm, 1 km of silica: phase delay L·n/c = {Lf*ol.n_silica(l)/c0*1e6:.4f} µs, "
          f"group delay L·Ng/c = {Lf*ol.group_index_silica(l)/c0*1e6:.4f} µs")
D = 22                                               # ps/(nm km), bulk silica at 1550 nm (see @sec-index)
dlam_sig = ol.dlam_from_dnu(10e9, nm(1550))*1e9      # 10 Gb/s signal ≈ 10 GHz wide, in nm
print(f"10 Gb/s signal (Δλ ≈ {dlam_sig:.2f} nm) over 80 km: spread ≈ {D*dlam_sig*80:.0f} ps")
print(f"walk-off between channels 0.8 nm apart over 80 km ≈ {D*0.8*80:.0f} ps")

1310 nm, 1 km of silica: phase delay L·n/c = 4.8260 µs, group delay L·Ng/c = 4.8755 µs
1550 nm, 1 km of silica: phase delay L·n/c = 4.8167 µs, group delay L·Ng/c = 4.8787 µs
10 Gb/s signal (Δλ ≈ 0.08 nm) over 80 km: spread ≈ 141 ps
walk-off between channels 0.8 nm apart over 80 km ≈ 1408 ps

A 10 Gb/s bit slot is 100 ps. The dispersive spread over a single 80 km span is already comparable to the bit period, which is why dispersion compensation or coherent DSP is needed (Chapter 2).

Going deeper

Group velocity can exceed $c$ near an absorption line (anomalous dispersion). Why does this not violate causality? Look up “front velocity” and Brillouin–Sommerfeld precursors, and see the deep-dive chapter, Chapter 15.
In a microring or Fabry–Perot resonator the free spectral range is set by $N_g$, not $n$: $\Delta\nu=c/(N_gL_{rt})$. Measure $N_g$ of a silicon waveguide from a ring spectrum (typically $N_g\approx4$ versus $n_\text{eff}\approx2.4$).
Slow light: photonic-crystal waveguides reach $N_g>30$. What limits the usable bandwidth? (Links to Chapter 11.)
Write the Taylor expansion of $k(\omega)$ and relate the second-order term $\beta_2$ to the dispersion coefficient $D$.

Complements

G. P. Agrawal, Fiber-Optic Communication Systems: chapter on dispersion in single-mode fibers.
Saleh & Teich: sections on pulse propagation in dispersive media.

5.3 Try it yourself

Derive $N_g=n-\lambda\,dn/d\lambda$ from $v_g=d\omega/dk$ with $\omega=2\pi c/\lambda$ and $k=2\pi n/\lambda$ (slides 66–67).
Using ol.n_silica, find the wavelength where $N_g$ is minimum. Why is that wavelength special?
A GPS-like timing signal travels 100 km in fiber. How much later does it arrive than the same signal in vacuum?

# Group velocity and group index {#sec-group} [![Open in Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/{{< var github_user >}}/{{< var repo >}}/blob/main/{{< var subdir >}}/notebooks/group.ipynb){.colab-link}  Run this chapter's code in Google Colab (no installation). ```{python} #| include: false import numpy as np import matplotlib.pyplot as plt import optolib as ol ol.use_style() c0, nm = ol.c0, ol.nm ``` *Slides 60–69: wave packets, beat of two waves, group velocity $v_g=d\omega/dk$, group index $N_g=n-\lambda\,dn/d\lambda$.* ## Key relations There are no perfectly monochromatic waves (slide 60). Two waves at $\omega\pm\delta\omega$ with wave vectors $k\pm\delta k$ add up to (slides 64–65) $$ E_x=2E_0\cos(\delta\omega\,t-\delta k\,z)\cos(\omega t-kz), $$ a carrier at $\omega$ modulated by a slow envelope. The envelope, which carries energy and information, travels at the **group velocity** (slides 61, 66–68): $$ v_g=\frac{d\omega}{dk}=\frac{c}{N_g},\qquad N_g=n-\lambda\frac{dn}{d\lambda}. $$ In a dispersive medium ($n$ depends on $\lambda$), $v_g\neq v$. For silica in the near IR, $N_g\approx1.46$ while $n\approx1.44$ (slide 69). ### Carrier and envelope, live The red dot rides on a crest of the carrier (phase velocity). The orange dot rides on the maximum of the envelope (group velocity). Change their ratio to see normal ($v_g<v$), no ($v_g=v$) and anomalous ($v_g>v$) dispersion. ```{ojs} //| echo: false viewof gv = Inputs.form({ t: Inputs.range([0, 40], {value: 0, step: 0.1, label: "time (carrier periods)"}), r: Inputs.range([0.3, 1.7], {value: 0.7, step: 0.01, label: "v_g / v_phase"}), dw: Inputs.range([0.02, 0.2], {value: 0.08, step: 0.005, label: "δω / ω"}) }) ``` ```{ojs} //| echo: false { const k = 2*Math.PI, w = 2*Math.PI; // carrier: λ = 1, period = 1, v = 1 const dw = gv.dw * w, dk = dw / gv.r; // v_g = dω/dk = r const t = gv.t, zs = d3.range(0, 30.0001, 0.01); const E = zs.map(z => ({z, E: 2*Math.cos(dw*t - dk*z)*Math.cos(w*t - k*z), part: "field"})); const env = zs.map(z => ({z, E: 2*Math.abs(Math.cos(dw*t - dk*z)), part: "envelope"})); const Lenv = Math.PI / dk; // distance between envelope maxima const zg = (gv.r * t) % Lenv, zp = t % 1; return Plot.plot({ width: Math.min(width, 1050), height: 280, x: {label: "z (carrier wavelengths)", domain: [0, 30]}, y: {label: "Eₓ", domain: [-2.3, 2.6]}, marks: [Plot.line(env, {x: "z", y: "E", stroke: "#eb6834", strokeDasharray: "4,3"}), Plot.line(E, {x: "z", y: "E", stroke: "#2a78d6", strokeWidth: 1.2}), Plot.dot([{z: zg, E: 2}], {x: "z", y: "E", r: 6, fill: "#eb6834"}), Plot.dot([{z: zp, E: 2*Math.cos(dw*t - dk*zp)}], {x: "z", y: "E", r: 6, fill: "#e34948"})] }); } ``` ```{ojs} //| echo: false html`<div class="result-box">${gv.r < 0.99 ? "Normal dispersion: the envelope lags behind the carrier crests (v<sub>g</sub> < v)." : gv.r > 1.01 ? "Anomalous dispersion: the envelope outruns the carrier crests (v<sub>g</sub> > v), as near an absorption line." : "No dispersion: carrier and envelope move together."} After ${gv.t} periods the carrier has moved ${gv.t.toFixed(1)} λ and the envelope ${(gv.r*gv.t).toFixed(1)} λ.</div>` ``` ## Worked example: group delay in a fiber link ```{python} #| label: fig-beat #| fig-cap: "Two waves 1 % apart in frequency in a dispersive medium: the carrier moves at v_ph, the envelope at v_g = v_ph/2." z = np.linspace(0, 400, 4000) k1, k2, w1, w2 = 1.0, 1.02, 1.0, 1.01 # v_g = Δω/Δk = 0.5, v_ph ≈ 1 fig, ax = plt.subplots(figsize=(9, 3)) for k, t in enumerate((0, 60)): E = np.cos(w1*t - k1*z) + np.cos(w2*t - k2*z) ax.plot(z, E + 4.5*k, color=ol.PALETTE[k], lw=0.8, label=f"t = {t}") ax.set(xlabel="z", yticks=[], title="envelope moves by v_g·t = 30, carrier by v_ph·t ≈ 60") ax.legend(loc="upper right"); plt.show() ``` ```{python} Lf = 1e3 for lam_nm in (1310, 1550): l = nm(lam_nm) print(f"{lam_nm} nm, 1 km of silica: phase delay L·n/c = {Lf*ol.n_silica(l)/c0*1e6:.4f} µs, " f"group delay L·Ng/c = {Lf*ol.group_index_silica(l)/c0*1e6:.4f} µs") D = 22 # ps/(nm km), bulk silica at 1550 nm (see @sec-index) dlam_sig = ol.dlam_from_dnu(10e9, nm(1550))*1e9 # 10 Gb/s signal ≈ 10 GHz wide, in nm print(f"10 Gb/s signal (Δλ ≈ {dlam_sig:.2f} nm) over 80 km: spread ≈ {D*dlam_sig*80:.0f} ps") print(f"walk-off between channels 0.8 nm apart over 80 km ≈ {D*0.8*80:.0f} ps") ``` A 10 Gb/s bit slot is 100 ps. The dispersive spread over a single 80 km span is already comparable to the bit period, which is why dispersion compensation or coherent DSP is needed (Chapter 2). ::: {.callout-tip collapse="true"} ## Going deeper - Group velocity can exceed $c$ near an absorption line (anomalous dispersion). Why does this not violate causality? Look up "front velocity" and Brillouin–Sommerfeld precursors, and see the deep-dive chapter, @sec-poynting. - In a microring or Fabry–Perot resonator the free spectral range is set by $N_g$, not $n$: $\Delta\nu=c/(N_gL_{rt})$. Measure $N_g$ of a silicon waveguide from a ring spectrum (typically $N_g\approx4$ versus $n_\text{eff}\approx2.4$). - Slow light: photonic-crystal waveguides reach $N_g>30$. What limits the usable bandwidth? (Links to @sec-bragg.) - Write the Taylor expansion of $k(\omega)$ and relate the second-order term $\beta_2$ to the dispersion coefficient $D$. ::: ::: {.callout-note collapse="true"} ## Complements - G. P. Agrawal, *Fiber-Optic Communication Systems*: chapter on dispersion in single-mode fibers. - Saleh & Teich: sections on pulse propagation in dispersive media. ::: ## Try it yourself 1. Derive $N_g=n-\lambda\,dn/d\lambda$ from $v_g=d\omega/dk$ with $\omega=2\pi c/\lambda$ and $k=2\pi n/\lambda$ (slides 66–67). 2. Using `ol.n_silica`, find the wavelength where $N_g$ is minimum. Why is that wavelength special? 3. A GPS-like timing signal travels 100 km in fiber. How much later does it arrive than the same signal in vacuum?