Run this chapter’s code in Google Colab (no installation).
Slides 25–34 and 79–84: intensity profile, Rayleigh range, divergence, \(M^2\), Gaussian beam in a cavity, power and irradiance of a Gaussian beam.
The output of most lasers, and the fundamental mode of a single-mode fiber (slide 27), is a Gaussian beam. Its irradiance across the beam is Gaussian (slides 26, 80):
\[ I(r,z)=\frac{2P}{\pi w^2(z)}\exp\!\left(-\frac{2r^2}{w^2(z)}\right),\qquad w(z)=w_0\sqrt{1+\left(\frac{z}{z_0}\right)^2},\qquad z_0=\frac{\pi w_0^2 n}{\lambda}. \]
| quantity | formula | slide |
|---|---|---|
| Rayleigh range (beam area doubled) | \(z_0=\pi w_0^2/\lambda\) | 28–29 |
| far-field half-angle divergence | \(\theta=\lambda/(\pi w_0)\), total \(2\theta\) | 26 |
| real beams | \(2\theta=M^2\,\dfrac{4\lambda}{\pi(2w_0)}\), \(M^2\ge1\) | 30–31 |
| wavefront radius | \(R(z)=z\left[1+(z_0/z)^2\right]\) | 34 |
| Gouy phase | \(\psi(z)=\arctan(z/z_0)\) | — |
| power inside radius \(w\) | \(1-e^{-2}\approx86\,\%\) | 82 |
The complex beam parameter \(q(z)=z+jz_0\) (engineering convention: \(1/q=1/R-j\lambda/(\pi w^2)\)) packs \(w\) and \(R\) into one number. It transforms through any paraxial optical system with its \(ABCD\) matrix: \(q'=(Aq+B)/(Cq+D)\).
A 5 mW He-Ne laser at 633 nm has a spot size \(2w_0=1\) mm. Find the maximum irradiance, and the axial irradiance 25 m away.
P, w0, lam = 5e-3, 0.5e-3, nm(633)
I0 = 2*P/(np.pi*w0**2)
w25, z0 = ol.gaussian_w(25, w0, lam)
print(f"I0 = {I0/1e4:.3f} W/cm², z0 = {z0:.2f} m")
print(f"at 25 m: 2w = {2*w25*1e3:.1f} mm, axial irradiance = {2*P/(np.pi*w25**2)/1e4*1e3:.2f} mW/cm²")
print(f"fraction of power inside r = w: 1 − e⁻² = {1-np.exp(-2):.3f}")I0 = 1.273 W/cm², z0 = 1.24 m
at 25 m: 2w = 20.2 mm, axial irradiance = 3.13 mW/cm²
fraction of power inside r = w: 1 − e⁻² = 0.865The same formulas size the transmitter of a LiDAR: how large must the output waist be to keep a small footprint at 100 m?
z = np.linspace(0, 200, 400)
fig, ax = plt.subplots()
for k, w0 in enumerate((0.5e-3, 2e-3, 5e-3)):
for M2, ls in ((1, "-"), (1.5, "--")):
ax.plot(z, 2e2*ol.gaussian_w(z, w0, nm(905), M2)[0], ls, color=ol.PALETTE[k],
label=f"2w₀ = {2e3*w0:.0f} mm, M² = {M2}")
ax.set(xlabel="distance z (m)", ylabel="spot diameter 2w (cm)")
ax.legend(fontsize=8, ncol=2); plt.show()
for zt in (50, 100, 200):
print(f"optimum waist for range {zt} m: w0 = √(λz/π) = {np.sqrt(nm(905)*zt/np.pi)*1e3:.2f} mm")
optimum waist for range 50 m: w0 = √(λz/π) = 3.80 mm
optimum waist for range 100 m: w0 = √(λz/π) = 5.37 mm
optimum waist for range 200 m: w0 = √(λz/π) = 7.59 mmThe optimum waist for a given range \(z\) minimises \(w(z)\): \(w_0^\text{opt}=\sqrt{\lambda z/\pi}\), that is \(z=z_0\).
A common laboratory task: couple a collimated 1550 nm beam into an SMF-28 fiber (mode-field diameter ≈ 10.4 µm, so \(w_f\approx5.2\) µm). The fiber mode is well approximated by a Gaussian (slide 27), so the power coupling efficiency between two Gaussians with waists \(w_1, w_2\) and lateral offset \(d\) at a common waist plane is
\[ \eta=\left(\frac{2w_1w_2}{w_1^2+w_2^2}\right)^2\exp\!\left(-\frac{2d^2}{w_1^2+w_2^2}\right). \]
lam, wf, w_in = nm(1550), um(5.2), 1.0e-3
f = np.linspace(2e-3, 20e-3, 300)
w_focus = []
for fi in f:
q = ol.abcd(ol.q_from_w0(w_in, lam), ol.lens(fi)) # just after the lens
w_focus.append(ol.w_from_q(ol.abcd(q, ol.free(-np.real(q))), lam)) # new waist where Re(q) = 0
w_focus = np.array(w_focus)
eta = (2*w_focus*wf/(w_focus**2 + wf**2))**2
fig, ax = plt.subplots(1, 2, figsize=(10, 3.8))
ax[0].plot(f*1e3, 100*eta)
f_opt = np.pi*w_in*wf/lam
ax[0].axvline(f_opt*1e3, ls=":", c=ol.INK2)
ax[0].text(f_opt*1e3 + 0.3, 20, f"f = πw·w_f/λ = {f_opt*1e3:.1f} mm", color=ol.INK2)
ax[0].set(xlabel="lens focal length (mm)", ylabel="η (%)", title="mode matching")
d = np.linspace(0, 8e-6, 200)
ax[1].plot(d*1e6, -ol.dB(np.exp(-2*d**2/(2*wf**2))))
ax[1].set(xlabel="lateral offset (µm)", ylabel="excess loss (dB)", title="alignment tolerance")
plt.tight_layout(); plt.show()
An offset of 1.5 µm already costs about 0.35 dB. This is why fiber-to-chip packaging is a precision-mechanics problem, and why silicon photonic chips use spot-size converters (inverse tapers) or grating couplers to enlarge their ~0.5 µm mode.
Between two spherical mirrors, the Gaussian beam is a self-consistent solution: the wavefront curvature \(R(z)\) matches the mirror curvature at both ends. For a symmetric confocal cavity (\(R_\text{mirror}=L\)), the waist is \(w_0=\sqrt{\lambda L/2\pi}\).
L, lam = 0.30, nm(633)
w0 = np.sqrt(lam*L/(2*np.pi)); z0 = np.pi*w0**2/lam
z = np.linspace(-L/2, L/2, 400)
w = w0*np.sqrt(1 + (z/z0)**2)
fig, ax = plt.subplots(figsize=(8, 3))
ax.fill_between(z*100, -w*1e3, w*1e3, color=ol.PALETTE[0], alpha=0.25, lw=0)
ax.plot(z*100, w*1e3, color=ol.PALETTE[0]); ax.plot(z*100, -w*1e3, color=ol.PALETTE[0])
yy = np.linspace(-1.6*w.max()*1e3, 1.6*w.max()*1e3, 50)
for zm, s in ((L/2, 1), (-L/2, -1)):
ax.plot((zm - s*(L - np.sqrt(L**2 - (yy*1e-3)**2)))*100, yy, color=ol.PALETTE[1], lw=3)
ax.set(xlabel="z (cm)", ylabel="w (mm)", title=f"confocal cavity: w₀ = {w0*1e3:.3f} mm, z₀ = L/2 = {z0*100:.1f} cm")
plt.show()
# pip install laserbeamsize (ISO 11146 beam-width and M² fitting)
import laserbeamsize as lbs
# images: list of camera frames recorded at positions z (m) along the caustic
d = np.array([lbs.beam_size(img)[2] for img in images]) * pixel_size
params, errors, used = lbs.M2_fit(z, d, nm(633))
d0, z0_pos, theta, M2, zR = params # waist diameter, waist position, full divergence, M², Rayleigh distance