整合

org/other/question1_pdms_emissivity.py

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+import os
+from dataclasses import dataclass
+from typing import Dict, List
+
+import numpy as np
+from matplotlib import pyplot as plt
+
+AIR_INDEX = 1.0  # normal incidence, non-absorbing ambient
+WAVELENGTH_MIN = 0.4  # µm
+WAVELENGTH_MAX = 25.0  # µm
+
+
+def cauchy_index(wavelength_um: np.ndarray) -> np.ndarray:
+    """
+    Dispersion model for PDMS based on Cauchy equation fitted to literature data:
+        n(λ) = A + B / λ^2 + C / λ^4
+    The coefficients (A=1.385, B=0.0125, C=0.00045) match n≈1.43 @ 0.55 µm
+    and n≈1.40 beyond 2 µm.
+    """
+    A = 1.385
+    B = 0.0125
+    C = 0.00045
+    lam2 = wavelength_um ** 2
+    return A + B / lam2 + C / (lam2 * wavelength_um ** 2)
+
+
+def extinction_coeff(wavelength_um: np.ndarray) -> np.ndarray:
+    """
+    Construct an approximate extinction coefficient profile by superposing
+    Gaussians centered at the known vibrational bands of PDMS in the mid-IR.
+    Peaks taken from FTIR measurements (Si-O-Si at ~9.2 µm, Si-CH3 rocking at
+    12.7 µm, CH stretches at 3.4 µm, etc.).
+    """
+    peaks = [
+        # (center µm, amplitude, width µm)
+        (3.4, 0.06, 0.25),
+        (8.0, 0.30, 0.50),
+        (9.2, 0.65, 0.60),
+        (10.6, 0.35, 0.45),
+        (12.7, 0.45, 0.35),
+        (14.5, 0.25, 0.60),
+    ]
+    k = np.full_like(wavelength_um, 5e-4)
+    for center, amp, width in peaks:
+        k += amp * np.exp(-0.5 * ((wavelength_um - center) / width) ** 2)
+    # enforce plausible upper bound
+    return np.clip(k, 0, 2.0)
+
+
+def thin_film_emissivity(
+    wavelength_um: np.ndarray, thickness_um: float, n_complex: np.ndarray
+) -> np.ndarray:
+    """
+    Normal-incidence emissivity for a PDMS slab in air computed using
+    the transfer-matrix solution for a single absorbing layer.
+    """
+    n0 = AIR_INDEX
+    ns = AIR_INDEX
+    thickness_m = thickness_um * 1e-6
+    # vacuum wavenumber
+    beta = 2 * np.pi / (wavelength_um * 1e-6)
+    delta = beta * n_complex * thickness_m
+
+    r01 = (n0 - n_complex) / (n0 + n_complex)
+    r12 = (n_complex - ns) / (n_complex + ns)
+    exp_term = np.exp(2j * delta)
+    numerator_r = r01 + r12 * exp_term
+    denominator = 1 + r01 * r12 * exp_term
+    r = numerator_r / denominator
+
+    t01 = 2 * n0 / (n0 + n_complex)
+    t12 = 2 * n_complex / (n_complex + ns)
+    exp_half = np.exp(1j * delta)
+    t = (t01 * t12 * exp_half) / denominator
+
+    R = np.abs(r) ** 2
+    T = (ns.real / n0) * np.abs(t) ** 2
+    A = 1 - R - T
+    return np.clip(A.real, 0, 1)
+
+
+@dataclass
+class EmissivityResult:
+    wavelengths: np.ndarray
+    emissivity_map: Dict[float, np.ndarray]
+    window_avg: Dict[float, float]
+
+
+def compute_emissivity(thicknesses_um: List[float]) -> EmissivityResult:
+    wavelengths = np.linspace(WAVELENGTH_MIN, WAVELENGTH_MAX, 1200)
+    n = cauchy_index(wavelengths)
+    k = extinction_coeff(wavelengths)
+    n_complex = n + 1j * k
+
+    emissivity_map = {}
+    window_avg = {}
+    window_mask = (wavelengths >= 8) & (wavelengths <= 13)
+
+    for d in thicknesses_um:
+        eps = thin_film_emissivity(wavelengths, d, n_complex)
+        emissivity_map[d] = eps
+        window_avg[d] = float(np.trapz(eps[window_mask], wavelengths[window_mask]) /
+                              np.trapz(np.ones_like(wavelengths[window_mask]),
+                                       wavelengths[window_mask]))
+
+    return EmissivityResult(wavelengths, emissivity_map, window_avg)
+
+
+def plot_emissivity(result: EmissivityResult, outdir: str) -> str:
+    plt.figure(figsize=(9, 5))
+    for d, eps in result.emissivity_map.items():
+        label = f"{d:.0f} µm (ε̄₈₋₁₃={result.window_avg[d]:.2f})"
+        plt.plot(result.wavelengths, eps, label=label)
+
+    plt.axvspan(8, 13, color="tab:gray", alpha=0.15, label="Atmospheric window")
+    plt.xlabel("Wavelength (µm)")
+    plt.ylabel("Spectral emissivity")
+    plt.title("PDMS Thin-Film Emissivity vs. Wavelength")
+    plt.ylim(0, 1.05)
+    plt.xlim(WAVELENGTH_MIN, WAVELENGTH_MAX)
+    plt.legend()
+    plt.grid(alpha=0.3)
+
+    os.makedirs(outdir, exist_ok=True)
+    output_path = os.path.join(outdir, "question1_emissivity.png")
+    plt.tight_layout()
+    plt.savefig(output_path, dpi=300)
+    plt.close()
+    return output_path
+
+
+def main():
+    thicknesses = [1, 5, 10, 25, 50, 100]
+    result = compute_emissivity(thicknesses)
+    outdir = os.path.join(os.path.dirname(__file__), "outputs")
+    figure_path = plot_emissivity(result, outdir)
+
+    summary_lines = ["thickness_um,avg_emissivity_8_13um"]
+    for d in thicknesses:
+        summary_lines.append(f"{d},{result.window_avg[d]:.4f}")
+    csv_path = os.path.join(outdir, "question1_emissivity_summary.csv")
+    with open(csv_path, "w", encoding="utf-8") as f:
+        f.write("\n".join(summary_lines))
+
+    print(f"Figure saved to: {figure_path}")
+    print(f"Summary saved to: {csv_path}")
+
+
+if __name__ == "__main__":
+    main()
+