Thickness-dependent PMMA spectra#

This example uses the a multilayer FDM to simulate a SNOM spectrum from a thin layer of poly(methyl methacrylate) (PMMA) on silicon, for different thicknesses of PMMA, and normalises the signal to a reference spectrum taken from bulk gold. It also includes the effects of the far-field reflection coefficient of the sample on the observed SNOM spectra.

import matplotlib.pyplot as plt
import numpy as np
from matplotlib.colors import Normalize

import snompy

# Set some experimental parameters
A_tip = 20e-9  # AFM tip tapping amplitude
r_tip = 30e-9  # AFM tip radius of curvature
L_tip = 350e-9  # Semi-major axis length of ellipsoid tip model
n = 3  # Harmonic for demodulation
theta_in = np.deg2rad(60)  # Light angle of incidence
c_r = 0.3  # Experimental weighting factor
nu_vac = np.linspace(1680, 1800, 128) * 1e2  # Vacuum wavenumber
method = "Q_ave"  # The FDM method to use

# Semi-infinite superstrate and substrate
eps_air = 1.0
eps_Si = 11.7  # Si permitivitty in the mid-infrared

# Very simplified model of PMMA dielectric function based on ref [1] below
eps_pmma = snompy.sample.lorentz_perm(
    nu_vac, nu_j=1738e2, gamma_j=20e2, A_j=4.2e8, eps_inf=2
)
t_pmma = np.geomspace(1, 35, 32) * 1e-9  # A range of thicknesses
sample_pmma = snompy.Sample(
    eps_stack=(eps_air, eps_pmma, eps_Si),
    t_stack=(t_pmma[:, np.newaxis],),
    nu_vac=nu_vac,
)

# Model of Au dielectric function from ref [2] below
eps_Au = snompy.sample.drude_perm(nu_vac, nu_plasma=7.25e6, gamma=2.16e4)
sample_Au = snompy.bulk_sample(eps_sub=eps_Au, eps_env=eps_air, nu_vac=nu_vac)

# Measurement
alpha_eff_pmma = snompy.fdm.eff_pol_n(
    sample=sample_pmma, A_tip=A_tip, n=n, r_tip=r_tip, L_tip=L_tip, method=method
)
r_coef_pmma = sample_pmma.refl_coef(theta_in=theta_in)
sigma_pmma = (1 + c_r * r_coef_pmma) ** 2 * alpha_eff_pmma

# Gold reference
alpha_eff_Au = snompy.fdm.eff_pol_n(
    sample=sample_Au, A_tip=A_tip, n=n, r_tip=r_tip, L_tip=L_tip, method=method
)
r_coef_Au = sample_Au.refl_coef(theta_in=theta_in)
sigma_Au = (1 + c_r * r_coef_Au) ** 2 * alpha_eff_Au

# Normalised complex scattering
eta_n = sigma_pmma / sigma_Au

# Plot output
fig, axes = plt.subplots(nrows=2, sharex=True)

# For neater plotting
nu_per_cm = nu_vac * 1e-2
t_nm = t_pmma * 1e9

SM = plt.cm.ScalarMappable(
    cmap=plt.cm.Spectral_r, norm=Normalize(vmin=t_nm.min(), vmax=t_nm.max())
)  # This maps thickness to colour

for t, sigma in zip(t_nm, eta_n):
    c = SM.to_rgba(t)
    axes[0].plot(nu_per_cm, np.abs(sigma), c=c)
    axes[1].plot(nu_per_cm, np.angle(sigma), c=c)

axes[0].set_ylabel(r"$s_{" f"{n}" r"}$ / a.u.")
axes[1].set(
    xlabel=r"$\nu$ / cm$^{-1}$",
    ylabel=r"$\phi_{" f"{n}" r"}$ / radians",
    xlim=(nu_per_cm.max(), nu_per_cm.min()),
)
fig.tight_layout()
cbar = fig.colorbar(SM, ax=axes, label="PMMA thickness / nm")
plt.show(block=False)

(The dielectric function of PMMA in the above example was based roughly on reference [1], and the dielectric function of gold was taken from reference [2]).

Thickness-dependent PMMA spectra#

References#