Run this chapter’s code in Google Colab (no installation).
Slides 101–147: polarised light, boundary conditions, \(r_\perp,r_\parallel,t_\perp,t_\parallel\), internal and external reflection, Brewster angle, phase change on TIR, evanescent wave, Goos–Hänchen shift, frustrated TIR and beam-splitter cubes, reflectance and transmittance.
Resolve the field into components perpendicular (\(\perp\), TE) and parallel (\(\parallel\), TM) to the plane of incidence (slide 103). The boundary conditions (continuity of tangential \(\mathbf E\) and \(\mathbf B\), slides 106–108) give, with \(n=n_2/n_1\), Fresnel’s equations (slides 110–111):
\[ r_\perp=\frac{\cos\theta_i-\sqrt{n^2-\sin^2\theta_i}}{\cos\theta_i+\sqrt{n^2-\sin^2\theta_i}},\qquad r_\parallel=\frac{\sqrt{n^2-\sin^2\theta_i}-n^2\cos\theta_i}{\sqrt{n^2-\sin^2\theta_i}+n^2\cos\theta_i}, \] \[ t_\perp=\frac{2\cos\theta_i}{\cos\theta_i+\sqrt{n^2-\sin^2\theta_i}},\qquad t_\parallel=\frac{2n\cos\theta_i}{n^2\cos\theta_i+\sqrt{n^2-\sin^2\theta_i}},\qquad r_\parallel+nt_\parallel=1,\quad r_\perp+1=t_\perp . \]
| result | formula | slide |
|---|---|---|
| normal incidence | \(r=\dfrac{n_1-n_2}{n_1+n_2}\), \(\ R=\left(\dfrac{n_1-n_2}{n_1+n_2}\right)^2\) (4 % for glass) | 112, 132 |
| Brewster angle (\(r_\parallel=0\)) | \(\tan\theta_p=n_2/n_1\) | 114–116 |
| TIR phase (\(\perp\)) | \(\tan\dfrac{\phi_\perp}{2}=\dfrac{\sqrt{\sin^2\theta_i-n^2}}{\cos\theta_i}\) | 118 |
| TIR phase (\(\parallel\)) | \(\tan\!\left(\dfrac{\phi_\parallel}{2}+\dfrac{\pi}{2}\right)=\dfrac{\sqrt{\sin^2\theta_i-n^2}}{n^2\cos\theta_i}\) | 119 |
| evanescent field | \(E_t\propto e^{-\alpha_2y}\), \(\ \alpha_2=\dfrac{2\pi n_2}{\lambda}\sqrt{\left(\dfrac{n_1}{n_2}\right)^2\sin^2\theta_i-1}\) | 123–124 |
| Goos–Hänchen shift | \(\Delta z\approx2\delta\tan\theta_i\) | 125 |
| reflectance, transmittance | \(R=\lvert r\rvert^2\), \(\ T=\dfrac{n_2\cos\theta_t}{n_1\cos\theta_i}\lvert t\rvert^2\), \(\ R+T=1\) | 131–138 |
The square root \(\sqrt{n^2-\sin^2\theta_i}\) is taken on the branch with non-positive imaginary part. That is the physical choice in the \(e^{j\omega t}\) convention, where the transmitted field decays into medium 2 under TIR. With this choice ol.fresnel reproduces the slide phases.
for n1, n2, kind in ((1.0, 1.5, "air → glass (external)"), (1.5, 1.0, "glass → air (internal)")):
rs, rp, ts, tp = ol.fresnel(n1, n2, 0.0)
print(f"{kind:24s} r = {rs.real:+.2f} ({'180° phase shift' if rs.real < 0 else 'no phase shift'}), R = {abs(rs)**2:.2%}")
print(f"Brewster angle air → glass: θp = {np.degrees(ol.brewster(1, 1.5)):.1f}° (pile-of-plates polariser, Arago 1812)")air → glass (external) r = -0.20 (180° phase shift), R = 4.00%
glass → air (internal) r = +0.20 (no phase shift), R = 4.00%
Brewster angle air → glass: θp = 56.3° (pile-of-plates polariser, Arago 1812)thp = ol.brewster(1.0, 1.5)
tt = np.arcsin(np.sin(thp)/1.5)
rs, rp, ts, tp = ol.fresnel(1.0, 1.5, thp)
fac = 1.5*np.cos(tt)/np.cos(thp)
print(f"θp = {np.degrees(thp):.2f}°, θt = {np.degrees(tt):.2f}°")
print(f"r⊥ = {rs.real:.3f}, r∥ = {abs(rp):.1e}, t⊥ = {ts.real:.3f}, t∥ = {tp.real:.3f}")
print(f"R⊥ = {abs(rs)**2:.3f}, R∥ = {abs(rp)**2:.1f}, unpolarised R = {abs(rs)**2/2:.3f}")
print(f"T⊥ = {fac*abs(ts)**2:.3f}, T∥ = {fac*abs(tp)**2:.3f} checks: r∥ + n t∥ = {(rp + 1.5*tp).real:.3f}, r⊥ + 1 = t⊥: {np.isclose(rs + 1, ts)}")θp = 56.31°, θt = 33.69°
r⊥ = -0.385, r∥ = 0.0e+00, t⊥ = 0.615, t∥ = 0.667
R⊥ = 0.148, R∥ = 0.0, unpolarised R = 0.074
T⊥ = 0.852, T∥ = 1.000 checks: r∥ + n t∥ = 1.000, r⊥ + 1 = t⊥: TrueThe reflected light is purely \(\perp\)-polarised. The transmitted light is only partially polarised, which is why a stack of plates is needed to make a good polariser.
\(n_1=1.460\), \(n_2=1.440\), \(\lambda=1300\) nm. Find the critical angle, the phase changes at 87° and 90°, and the penetration depth of the evanescent wave.
n1, n2, lam = 1.460, 1.440, nm(1300)
print(f"critical angle = {np.degrees(ol.critical_angle(n1, n2)):.2f}°")
for thi in (87, 89.999):
t = np.radians(thi)
rs, rp, ts, tp = ol.fresnel(n1, n2, t)
d = ol.penetration_depth(n1, n2, t, lam)
print(f"θi = {round(thi)}°: |r⊥| = {abs(rs):.3f}, φ⊥ = {np.degrees(np.angle(rs)):6.1f}°, φ∥ = {np.degrees(np.angle(rp)):6.1f}°, "
f"α2 = {1/d:.3e} m⁻¹, δ = {d*1e6:.3f} µm" + (f", Goos–Hänchen Δz = 2δ tanθ = {2*d*np.tan(t)*1e6:.1f} µm" if thi < 89 else " (grazing: Δz diverges)"))critical angle = 80.51°
θi = 87°: |r⊥| = 1.000, φ⊥ = 143.0°, φ∥ = -36.1°, α2 = 1.104e+06 m⁻¹, δ = 0.906 µm, Goos–Hänchen Δz = 2δ tanθ = 34.6 µm
θi = 90°: |r⊥| = 1.000, φ⊥ = 180.0°, φ∥ = -0.0°, α2 = 1.164e+06 m⁻¹, δ = 0.859 µm (grazing: Δz diverges)The slides quote \(\phi_\perp=143°\), \(\phi_\parallel=-36.05°\) (written as \(143.95°-180°\)), \(\delta=0.906\) µm at 87° and 0.859 µm at 90°. The penetration is larger for smaller incidence angles, and it diverges at the critical angle.
y = np.linspace(-18, 4, 2000)*1e-6
t = np.radians(87)
rs = ol.fresnel(n1, n2, t)[0]
ky = 2*np.pi*n1/lam*np.cos(t)
E1 = np.abs(np.exp(1j*ky*y) + rs*np.exp(-1j*ky*y)) # y < 0: incident + reflected
E2 = np.abs(1 + rs)*np.exp(-y/ol.penetration_depth(n1, n2, t, lam))
fig, ax = plt.subplots(figsize=(8, 3.4))
ax.axvspan(0, 4, color=ol.PALETTE[0], alpha=0.06, lw=0)
ax.plot(y[y < 0]*1e6, E1[y < 0], label="medium 1 (incident + reflected)")
ax.plot(y[y >= 0]*1e6, E2[y >= 0], label="medium 2 (evanescent)")
ax.set(xlabel="distance from interface y (µm)", ylabel="|E⊥| (a.u.)"); ax.legend(fontsize=8)
plt.show()
The evanescent tail is exactly what an evanescent-wave sensor (ATR spectroscopy, integrated refractive-index sensors) probes, and what couples two waveguides in a directional coupler (Chapter 2).
Two glass prisms (\(n=1.5\)) separated by an air gap, light at 45° (above \(\theta_c=41.8°\)). When the gap is comparable to the penetration depth, the evanescent field reaches the second prism and light tunnels through. We compute it with the transfer-matrix solver of Chapter 8.
lam = nm(633); gaps = np.linspace(0, 1.5e-6, 300)
fig, ax = plt.subplots()
for pol in ("s", "p"):
T = [ol.tmm([1.5, 1.0, 1.5], [g], lam, np.radians(45), pol)[1][0] for g in gaps]
ax.plot(gaps*1e9, T, label=f"{pol}-pol")
ax.set(xlabel="air gap d (nm)", ylabel="transmittance"); ax.legend()
plt.show()
A 50/50 splitting ratio requires a gap of a fraction of a wavelength, and the ratio depends on polarisation. This is how the beam-splitter cube of slide 129 works, and why “polarisation-insensitive” cubes need extra layers. The same tunnelling is used to couple a laser beam into a thin-film waveguide with a prism (slide 128).
ol.fresnel.ol.tmm for s and p at oblique incidence, and explain why \(T\ne|t|^2\).ol.tmm.