Lesson 1 · Spectra 101

Mission: compute \(R(\lambda)\) and \(T(\lambda)\) for a layered structure, learn to read the specular order, and meet absorption — the legitimate reason the books don't always sum to one.

A thin-film stack

A 120 nm TiO₂ film on glass, swept across the visible. Nothing is patterned, so only the specular lane exists and RCWA reproduces the exact transfer-matrix spectrum — at n_orders=0, which makes it effectively free:

import numpy as np
from ikarus import RCWA

rcwa = RCWA(period_x=400e-9, period_y=400e-9, n_orders=0)  # specular only
rcwa.add_uniform_layer(np.inf, "Air")
rcwa.add_uniform_layer(120e-9, "TiO2")
rcwa.add_uniform_layer(np.inf, "SiO2")

wavelengths = np.linspace(400e-9, 700e-9, 61)
R, T = [], []
for wl in wavelengths:
    rcwa.set_source(wavelength=wl, theta=0, polarization="linear")
    _, _, res = rcwa.simulate()
    R.append(res.R_total)
    T.append(res.T_total)

R, T = np.array(R), np.array(T)
print(f"R+T spans [{(R+T).min():.6f}, {(R+T).max():.6f}]")  # ≈ 1: lossless here

Why n_orders=0 is not cheating

A uniform stack has no pattern to diffract from — one harmonic (the specular order) captures the physics exactly. Save the big harmonic budgets for patterned layers; thin films fly free.

Plot it

import matplotlib.pyplot as plt

fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(wavelengths * 1e9, R * 100, label="Reflectance")
ax.plot(wavelengths * 1e9, T * 100, label="Transmittance")
ax.plot(wavelengths * 1e9, (R + T) * 100, "k--", lw=1, label="R + T")
ax.set_xlabel("wavelength (nm)")
ax.set_ylabel("efficiency (%)")
ax.legend(); ax.grid(alpha=0.3)
fig.savefig("thin_film_spectrum.png", dpi=150, bbox_inches="tight")

Reflectance, transmittance and their sum for a 120 nm TiO₂ film on glass. R + T pins at 100% — TiO₂ is lossless across the visible.

Totals vs. the specular lane

R_total/T_total sum over all propagating orders. For patterned structures you'll often want just the straight-through lane:

i00 = res.order_index(0, 0)
print("specular T(0,0) =", res.T_orders[i00])
print("specular R(0,0) =", res.R_orders[i00])

For this thin film they coincide — there is only the one lane.

Absorption: when R + T honestly isn't 1

Put 50 nm of gold in the path and watch the energy ledger:

rcwa = RCWA(period_x=400e-9, period_y=400e-9, n_orders=0)
rcwa.add_uniform_layer(np.inf, "Air")
rcwa.add_uniform_layer(50e-9, "Au")          # gold: a glutton in the visible
rcwa.add_uniform_layer(np.inf, "SiO2")

rcwa.set_source(wavelength=550e-9, theta=0, polarization="linear")
_, _, res = rcwa.simulate()
A = 1.0 - res.energy_balance
print(f"R={res.R_total:.3f}  T={res.T_total:.3f}  A={A:.3f}")

A is real ohmic loss, not an error. The diagnostic rule:

energy_balance reads…	Verdict
≈ 1 (lossless materials)	converged, healthy
< 1 (lossy materials)	absorption — physics, working as intended
≳ 1.01 (lossless)	unconverged — raise n_orders
≫ 1	numerical trouble — see Troubleshooting

Expected results

• TiO₂/glass: smooth interference fringes; R+T ≈ 1 to ~10⁻⁹.
• Au film: large A, R+T well below 1 — gold doing gold things.

Pilot habits

• n_orders=0 for thin films — exact and instant.
• One RCWA, many set_source(wavelength=...) calls — never rebuild a fixed stack inside a sweep.
• Glance at energy_balance after every new structure. It's the cheapest bug detector in photonics.

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