Run this chapter’s code in Google Colab (no installation).
Slides 163–171: temporal and spatial coherence, coherence time, coherence length, spectral width, Gaussian pulses.
A perfect sine wave is perfectly coherent and has a single frequency. A finite wave train of duration \(\Delta t\) has a spectrum of width \(\Delta\nu\approx1/\Delta t\) (slides 164–167). Two parts of a beam can only interfere if their path difference is shorter than the coherence length:
\[ \Delta t\approx\frac{1}{\Delta\nu},\qquad l_c=c\,\Delta t=\frac{c}{\Delta\nu}=\frac{\lambda^2}{\Delta\lambda}. \]
The slides use \(\Delta t\approx1/\Delta\nu\), hence \(l_c=c/\Delta\nu=\lambda^2/\Delta\lambda\). Other definitions include a line-shape factor. For a Gaussian spectrum with FWHM \(\Delta\lambda\), the FWHM of the interferogram envelope (in path difference) is \((4\ln2/\pi)\,\lambda^2/\Delta\lambda\approx0.88\,\lambda^2/\Delta\lambda\), and the OCT literature quotes half of it, \((2\ln2/\pi)\,\lambda^2/\Delta\lambda\), as axial resolution. For a Lorentzian line Saleh & Teich use \(\tau_c=1/(\pi\Delta\nu)\). When comparing numbers from different sources, check which definition is used.
A Michelson interferometer splits the beam, delays one arm and recombines. Fringes only appear when the optical path difference (OPD) is shorter than about \(l_c\). Broaden the spectrum and watch the fringe packet shrink.
If the OPD range is much larger than the fringe packet, the fringes are too dense to draw: zoom in with a smaller range.
for name, dnu in (("Na lamp, 589 nm, Δν ≈ 5×10¹¹ Hz", 5e11), ("multimode He-Ne, Δν ≈ 1.5×10⁹ Hz", 1.5e9)):
lc, dt, _ = ol.coherence_length(dnu=dnu)
print(f"{name:36s} Δt = {dt:.2e} s, l_c = {lc*1e3:.2f} mm")Na lamp, 589 nm, Δν ≈ 5×10¹¹ Hz Δt = 2.00e-12 s, l_c = 0.60 mm
multimode He-Ne, Δν ≈ 1.5×10⁹ Hz Δt = 6.67e-10 s, l_c = 199.86 mmsrc = [("LED, 1550 nm, Δλ = 150 nm", 1550, ("lam", 150)),
("Laser diode, 1550 nm, Δλ = 0.1 nm", 1550, ("lam", 0.1)),
("Multimode He-Ne, Δν = 1.5 GHz", 633, ("nu", 1.5e9)),
("Single-mode He-Ne, Δν = 10 MHz", 633, ("nu", 10e6)),
("Sunlight, 550 nm, Δν ≈ 100 THz", 550, ("nu", 100e12)),
("Sodium lamp, 589 nm, Δν ≈ 1 GHz", 589, ("nu", 1e9))]
rows = []
for name, l, (kind, val) in src:
lc, dt, dnu = ol.coherence_length(wl=nm(l), dlam=nm(val)) if kind == "lam" else ol.coherence_length(dnu=val)
rows.append({"source": name, "Δν": f"{dnu:.3g} Hz", "Δt": f"{dt:.3g} s",
"l_c": f"{lc*1e6:.3g} µm" if lc < 1e-3 else f"{lc:.3g} m"})
pd.DataFrame(rows).set_index("source")| Δν | Δt | l_c | |
|---|---|---|---|
| source | |||
| LED, 1550 nm, Δλ = 150 nm | 1.87e+13 Hz | 5.34e-14 s | 16 µm |
| Laser diode, 1550 nm, Δλ = 0.1 nm | 1.25e+10 Hz | 8.01e-11 s | 0.024 m |
| Multimode He-Ne, Δν = 1.5 GHz | 1.5e+09 Hz | 6.67e-10 s | 0.2 m |
| Single-mode He-Ne, Δν = 10 MHz | 1e+07 Hz | 1e-07 s | 30 m |
| Sunlight, 550 nm, Δν ≈ 100 THz | 1e+14 Hz | 1e-14 s | 3 µm |
| Sodium lamp, 589 nm, Δν ≈ 1 GHz | 1e+09 Hz | 1e-09 s | 0.3 m |
The sodium lamp appears twice in the slides with very different widths. Slide 168 uses the whole, pressure-broadened emission (≈ 5×10¹¹ Hz, 0.6 mm), slide 169 the width of a single Doppler-broadened line (≈ 1 GHz, 30 cm). Both are legitimate. The coherence length depends on what the instrument actually sees.
Optical coherence tomography (OCT) exploits low coherence: reflections in tissue are located with a precision of about \(l_c/2\).
opd = np.linspace(-40e-6, 40e-6, 4000)
fig, ax = plt.subplots(2, 1, figsize=(9, 4.4), sharex=True)
for a, (lbl, l0, dl) in zip(ax, (("SLD 1310 nm, Δλ = 100 nm", nm(1310), nm(100)),
("LED 1550 nm, Δλ = 150 nm", nm(1550), nm(150)))):
a.plot(opd*1e6, ol.interferogram(opd, l0, dl), lw=0.7); a.set_title(lbl, fontsize=9, loc="left")
print(f"{lbl}: OCT axial resolution (Gaussian FWHM) = {2*np.log(2)/np.pi*l0**2/dl*1e6:.1f} µm in air")
ax[1].set_xlabel("optical path difference (µm)"); plt.tight_layout(); plt.show()SLD 1310 nm, Δλ = 100 nm: OCT axial resolution (Gaussian FWHM) = 7.6 µm in air
LED 1550 nm, Δλ = 150 nm: OCT axial resolution (Gaussian FWHM) = 7.1 µm in air
ol.interferogram.