Run this chapter’s code in Google Colab (no installation).
Slides 40–64. A cylindrical core of radius \(a\) and index \(n_1\), inside a cladding of index \(n_2\). Two kinds of rays appear (slide 42). Meridional rays cross the axis. Skew rays spiral around it. In the wave picture the two families give the LP\(_{lm}\) modes.
When \(\Delta\ll1\) (typically 0.3 % to 2 %), the guided modes are almost transverse and linearly polarised:
\[ E_{LP}=E_{lm}(r,\phi)\,e^{j(\omega t-\beta_{lm}z)},\qquad E_{lm}(r,\phi)=f(r)\cos(l\phi)\ \text{or}\ f(r)\sin(l\phi). \]
Inside the core \(f(r)\propto J_l(ur/a)\) and in the cladding \(f(r)\propto K_l(wr/a)\), with \(u^2+w^2=V^2\). Continuity of \(f\) and \(f'\) at \(r=a\) gives the eigenvalue equation
\[ \frac{u\,J_{l-1}(u)}{J_l(u)}=-\frac{w\,K_{l-1}(w)}{K_l(w)} . \tag{2.1}\]
The mode index \(l\) sets the azimuthal order (\(2l\) maxima around a circle). The index \(m\) counts the radial maxima. The function fl.LP_b(V, l, m) solves Equation 2.1, and you can read it in fiberlib.py: it is a sign-change scan followed by Brent’s method.
for l, m, b in fl.LP_all(5.0):
print(f"LP{l}{m}: b = {b:.4f}")
print("Guided modes incl. polarisation and orientation:", fl.LP_count(5.0), " (V²/2 =", 5**2/2, ")")LP01: b = 0.8409
LP11: b = 0.6024
LP21: b = 0.3015
LP02: b = 0.2154
Guided modes incl. polarisation and orientation: 12 (V²/2 = 12.5 )Vg = np.linspace(0.3, 10, 400)
modes = [(0,1),(1,1),(2,1),(0,2),(3,1),(1,2),(4,1),(2,2),(0,3),(5,1)]
curves = {}
fig, ax = plt.subplots(figsize=(8, 4.6))
for k, (l, m) in enumerate(modes):
b = np.array([fl.LP_b(V, l, m) for V in Vg])
curves[f"LP{l}{m}"] = b
ok = np.isfinite(b)
ax.plot(Vg[ok], b[ok], color=fl.PALETTE[k % 8], lw=1.8)
i = np.where(ok)[0][-1]
ax.text(Vg[i] + 0.05, b[i], f"LP$_{{{l}{m}}}$", fontsize=8.5, va="center")
Va = np.linspace(1.2, 3.5, 100)
ax.plot(Va, fl.b_approx(Va), "--", color=fl.INK, lw=1, label="approximation (LP01)")
for vc in [2.405, 3.832, 5.136, 5.520]:
ax.axvline(vc, color=fl.GRID, lw=1, zorder=0)
ax.set_xlabel("V"); ax.set_ylabel("b"); ax.set_ylim(0, 1); ax.set_xlim(0, 10.8)
ax.legend(loc="lower right")
plt.show()
The cut-offs are zeros of Bessel functions. LP\(_{0m}\) is cut off at the zeros of \(J_1\) (0, 3.832, 7.016 …). LP\(_{lm}\) with \(l\ge1\) is cut off at the zeros of \(J_{l-1}\), which gives 2.405 for LP\(_{11}\).
Below you can sweep \(V\) and see which modes are guided, together with their \(b\) and effective index.
V = 8.0
allm = fl.LP_all(V)
x = np.linspace(-1.6, 1.6, 241)
X, Y = np.meshgrid(x, x)
R, PHI = np.hypot(X, Y), np.arctan2(Y, X)
n = len(allm); cols = 5; rows = int(np.ceil(n / cols))
fig, axs = plt.subplots(rows, cols, figsize=(1.9 * cols, 2.0 * rows))
for ax in axs.flat: ax.axis("off")
for ax, (l, m, b) in zip(axs.flat, allm):
E = fl.LP_field(R, V, b, l) * np.cos(l * PHI)
E /= np.abs(E).max()
ax.imshow(E, cmap="RdBu_r", vmin=-1, vmax=1, extent=[-1.6, 1.6, -1.6, 1.6])
ax.add_patch(plt.Circle((0, 0), 1, fill=False, color="k", lw=0.6))
ax.set_title(f"LP$_{{{l}{m}}}$ b={b:.2f}", fontsize=9)
plt.tight_layout(); plt.show()
r = np.linspace(0, 2, 400)
fig, ax = plt.subplots()
ax.axvspan(0, 1, color=fl.PALETTE[0], alpha=0.07, lw=0)
for k, (l, m, b) in enumerate(allm[:8]):
I = fl.LP_field(r, V, b, l) ** 2
ax.plot(r, I / I.max(), color=fl.PALETTE[k], lw=1.6, label=f"LP{l}{m}")
ax.set_xlabel("r / a"); ax.set_ylabel("normalized irradiance"); ax.legend(ncol=4, fontsize=8)
plt.show()
| quantity | formula |
|---|---|
| normalized index difference | \(\Delta = \dfrac{n_1^2-n_2^2}{2n_1^2}\approx\dfrac{n_1-n_2}{n_1}\) |
| numerical aperture | \(\mathrm{NA}=\sqrt{n_1^2-n_2^2}\approx n_1\sqrt{2\Delta}\) |
| acceptance angle | \(\sin\alpha_\text{max} = \mathrm{NA}/n_0\) |
| \(V\)-number | \(V=\dfrac{2\pi a}{\lambda}\,\mathrm{NA}\) |
| single mode | \(V<2.405\), i.e. \(\lambda>\lambda_c=\dfrac{2\pi a\,\mathrm{NA}}{2.405}\) |
| number of modes (\(V\gg1\)) | \(M\approx V^2/2\) |
Vs = np.linspace(0.5, 25, 120)
Mex = [fl.LP_count(v) for v in Vs]
fig, ax = plt.subplots()
ax.step(Vs, Mex, where="post", label="exact count")
ax.plot(Vs, Vs**2 / 2, "--", color=fl.PALETTE[1], label="V²/2")
ax.set_xlabel("V"); ax.set_ylabel("number of modes M"); ax.legend()
plt.show()
ofiber_test.ipynb)n_clad = 1.45
D = np.linspace(0.0005, 0.02, 300)
n_core = n_clad / (1 - D)
fig, ax = plt.subplots()
for k, a in enumerate(np.array([2, 2.5, 3, 3.5, 4, 4.5, 5]) * 1e-6):
lc = 2 * np.pi * a * fl.NA(n_core, n_clad) / 2.405
ax.plot(D * 100, lc * 1e9, color=fl.PALETTE[k], label=f"a = {a*1e6:.1f} µm")
for lam in (1310, 1550):
ax.axhline(lam, color=fl.INK2, lw=0.8, ls=":")
ax.set_xlabel("Δ (%)"); ax.set_ylabel("λc (nm)"); ax.set_ylim(0, 3000); ax.legend(ncol=2, fontsize=8)
plt.show()
Type the fiber parameters and all the quantities of slides 48–64 update. You can also use this panel to check every worked example below.
def report(title, **kv):
print(f"── {title}")
for k, v in kv.items(): print(f" {k:<28s} {v}")
# Slide 57 — multimode fiber
V = fl.V_number(50e-6, 0.85e-6, 1.468, 1.447)
report("Slide 57: MMF, 2a = 100 µm, λ = 850 nm", V=f"{V:.2f}", **{"M ≈ V²/2": f"{V**2/2:.0f}",
"exact count": fl.LP_count(V)})
# Slide 58 — maximum radius for SMF at 1.3 µm
a_max = 2.405 * 1.3e-6 / (2 * np.pi * fl.NA(1.468, 1.447))
report("Slide 58: SMF core radius at 1.3 µm", **{"a ≤": f"{a_max*1e6:.2f} µm"})
# Slides 59-60 — cut-off and MFD
a, n1, n2 = 4.1e-6, 1.4532, 1.4485
lc = 2 * np.pi * a * fl.NA(n1, n2) / 2.405
V = fl.V_number(a, 1.31e-6, n1, n2)
report("Slides 59–60: 2a = 8.2 µm at 1.31 µm", **{
"λc": f"{lc*1e6:.3f} µm", "V": f"{V:.3f}",
"MFD Marcuse": f"{fl.mfd_marcuse(a, V)*1e6:.2f} µm",
"MFD 2a(2.6/V)": f"{2*a*2.6/V*1e6:.2f} µm",
"MFD exact Gaussian fit (new)": f"{fl.mfd_exact_gaussian(a, V)*1e6:.2f} µm",
"MFD Petermann II (ITU, new)": f"{fl.mfd_petermann2(a, V)*1e6:.2f} µm"})
# Slide 63 — NA and acceptance
n1, n2 = 1.480, 1.460
NAv = fl.NA(n1, n2); V = fl.V_number(50e-6, 0.85e-6, n1, n2)
report("Slide 63: MMF NA", NA=f"{NAv:.4f}", **{"α_max": f"{np.degrees(np.arcsin(NAv)):.1f}°",
"2α_max": f"{2*np.degrees(np.arcsin(NAv)):.1f}°", "V": f"{V:.2f}", "M": f"{V**2/2:.0f}",
"Δ": f"{(n1-n2)/n1*100:.2f} %"})
# Slide 64 — SMF with Δ = 0.3 %
n1, Dl = 1.460, 0.003
NAv = n1 * np.sqrt(2 * Dl)
report("Slide 64: SMF, 2a = 8 µm, Δ = 0.3 %", NA=f"{NAv:.3f}",
**{"2α_max": f"{2*np.degrees(np.arcsin(NAv)):.1f}°", "λc": f"{2*np.pi*4e-6*NAv/2.405*1e6:.2f} µm"})── Slide 57: MMF, 2a = 100 µm, λ = 850 nm
V 91.44
M ≈ V²/2 4181
exact count 4202
── Slide 58: SMF core radius at 1.3 µm
a ≤ 2.01 µm
── Slides 59–60: 2a = 8.2 µm at 1.31 µm
λc 1.251 µm
V 2.297
MFD Marcuse 9.31 µm
MFD 2a(2.6/V) 9.28 µm
MFD exact Gaussian fit (new) 9.23 µm
MFD Petermann II (ITU, new) 9.14 µm
── Slide 63: MMF NA
NA 0.2425
α_max 14.0°
2α_max 28.1°
V 89.62
M 4016
Δ 1.35 %
── Slide 64: SMF, 2a = 8 µm, Δ = 0.3 %
NA 0.113
2α_max 13.0°
λc 1.18 µmSMF with \(n_1=1.4480\), \(n_2=1.4400\), \(a=3\) µm at \(\lambda=1.5\) µm. What are \(v_g\) and the group delay over 1 km? Neglect material dispersion.
The slides use \(\beta \approx n_2k[1+b\Delta]\) with the approximate \(b(V)\), and a 0.1 % finite difference. We do the same and also use the exact \(b(V)\) and an exact derivative.
n1, n2, a = 1.4480, 1.4400, 3e-6
Dl = (n1 - n2) / n1
def beta(wl, exact):
V = fl.V_number(a, wl, n1, n2)
b = fl.LP_b(V, 0, 1) if exact else fl.b_approx(V)
return n2 * 2 * np.pi / wl * (1 + b * Dl)
for exact, h in [(False, 1.5e-9), (True, 1e-12)]:
w1, w2 = 1.5e-6, 1.5e-6 + h
om = lambda wl: 2 * np.pi * c0 / wl
vg = (om(w1) - om(w2)) / (beta(w1, exact) - beta(w2, exact))
print(f"{'exact b(V), dλ→0' if exact else 'approx. b(V), dλ = 0.1%':<26s}: "
f"b = {fl.LP_b(fl.V_number(a,w1,n1,n2),0,1) if exact else fl.b_approx(fl.V_number(a,w1,n1,n2)):.5f}, "
f"v_g = {vg:.5e} m/s, τ_g(1 km) = {1e3/vg*1e6:.4f} µs")approx. b(V), dλ = 0.1% : b = 0.38609, v_g = 2.07007e+08 m/s, τ_g(1 km) = 4.8308 µs
exact b(V), dλ→0 : b = 0.38614, v_g = 2.06997e+08 m/s, τ_g(1 km) = 4.8310 µsThe LP\(_{01}\) field is very close to a Gaussian \(E\propto\exp(-r^2/w^2)\). The MFD is \(2w\). Because the intensity is Gaussian, the fraction of power inside \(r<w\) is \(1-e^{-2}\approx 86\,\%\), just like a Gaussian beam (Chapter 1).
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 3.9))
V = 2.30; b = fl.LP_b(V, 0, 1)
r = np.linspace(0, 3, 400)
E = fl.LP_field(r, V, b, 0); E /= E[0]
w = fl.mfd_exact_gaussian(1, V) / 2
ax1.plot(r, E**2, label="exact LP₀₁ intensity")
ax1.plot(r, np.exp(-2 * (r / w) ** 2), "--", label=f"Gaussian, w = {w:.3f} a")
ax1.axvline(1, color=fl.INK2, lw=0.8, ls=":"); ax1.text(1.02, 0.9, "core edge", color=fl.INK2)
ax1.set_xlabel("r / a"); ax1.set_ylabel("intensity"); ax1.legend()
Vs = np.linspace(1.2, 3.0, 40)
ax2.plot(Vs, [fl.mfd_exact_gaussian(1, v) / 2 for v in Vs], label="exact (Gaussian fit)")
ax2.plot(Vs, [fl.mfd_petermann2(1, v) / 2 for v in Vs], label="Petermann II")
ax2.plot(Vs, fl.mfd_marcuse(1, Vs) / 2, "--", label="Marcuse")
ax2.plot(Vs, 2.6 / Vs, ":", label="2.6/V")
ax2.axvspan(1.5, 2.5, color=fl.GRID, alpha=0.5, lw=0)
ax2.set_xlabel("V"); ax2.set_ylabel("MFD / 2a"); ax2.legend(fontsize=8)
plt.tight_layout(); plt.show()
ofiber_test.ipynb)The indices themselves depend on \(\lambda\) (Sellmeier, see Chapter 3). Here is \(V(\lambda)\) for a 9 µm core of SiO\(_2\) doped with 2 % GeO\(_2\) inside a pure-silica cladding:
lam = np.linspace(1000, 1650, 200) * 1e-9
V = fl.V_number(4.5e-6, lam, fl.n_doped(lam, 0.02), fl.n_silica(lam))
fig, ax = plt.subplots()
ax.plot(lam * 1e9, V)
ax.axhspan(2.0, 2.405, color=fl.PALETTE[2], alpha=0.12, lw=0)
ax.axhline(2.405, color=fl.INK2, lw=0.8)
ax.text(1610, 2.2, "practical SM window\n2.0 < V < 2.405", ha="right", va="center", fontsize=9)
ax.set_xlabel("λ (nm)"); ax.set_ylabel("V")
plt.show()