Material Builder

Build a passive material from Lorentz oscillators and a Drude contribution. Save the full model, then use it in an optical film.

A material made of several terms

Real materials rarely have a single resonance. Glass absorbs in the ultraviolet and again in the infrared. A metal has free electrons and interband absorption on top of them. This editor builds a permittivity as a sum: a constant background, Lorentz terms and a Drude term, up to 32 terms, one of them added as Drude. It plots the result from $0.15\times10^{15}$ to $6\times10^{15}$ rad/s, which is about 12.6 µm down to 314 nm, and can overlay a measured dataset for comparison. What one oscillator does on its own is covered in Lorentz and Drude Models. This page is about adding them up.

Terms add

Each term describes a separate group of charges driven by the same field. A term's susceptibility $\chi_j$ is how much that group adds to the permittivity. In this linear model, where the groups do not act on each other, their susceptibilities simply add:

\begin{aligned}\varepsilon(\omega)&=\varepsilon_\infty+\sum_j\chi_j(\omega),\\ \chi_j(\omega)&=\frac{S_j}{\omega_j^2-\omega^2-i\gamma_j\omega}\end{aligned}

With the $\exp(-i\omega t)$ convention used in the model section, $\operatorname{Im}\chi_j=S_j\gamma_j\omega/\big[(\omega_j^2-\omega^2)^2+\gamma_j^2\omega^2\big]$ is positive for every term with positive strength and damping. A sum of such terms therefore stays passive: it can absorb energy but not supply it. The second chart shows each term's $\operatorname{Im}\chi_j$ separately.

For a Lorentz term, absorption stays close to its resonance. The real part does not. Far from resonance a Lorentz term still shifts $\operatorname{Re}\varepsilon$ :

\begin{aligned}\chi_j&\approx\frac{S_j}{\omega_j^2} &&(\omega\ll\omega_j),\\ \chi_j&\approx-\frac{S_j}{\omega^2} &&(\omega\gg\omega_j)\end{aligned}

What the background stands for

A resonance far above the plotted range contributes an almost constant $S_j/\omega_j^2$ . Instead of listing every such resonance, the model collects them into $\varepsilon_\infty$ . Starting from vacuum, that gives $\varepsilon_\infty\approx1+\sum_{\text{above}}S_j/\omega_j^2$ . A resonance far below the range is different: its contribution tends to the Drude form $-S_j/(\omega^2+i\gamma_j\omega)$ , which is $-S_j/\omega^2$ when $\gamma_j\ll\omega$ . It fades with frequency and cannot be absorbed into a constant. Over the plotted range it looks like a free-carrier term instead.

A Drude term in the same sum

The Drude term is the case $\omega_j=0$ ; its strength $S$ is the square of the plasma frequency, $\omega_p^2$ (see Lorentz and Drude Models). Its real part is $-S/(\omega^2+\gamma^2)$ , always negative. If everything else adds up to a real, roughly constant $\varepsilon_b$ , the total real permittivity crosses zero at

\omega_z^2=\frac{S}{\varepsilon_b}-\gamma^2

This needs $\varepsilon_b>0$ and $S>\varepsilon_b\gamma^2$ , and holds only while the other terms stay nearly constant near $\omega_z$ . Bound terms and the background do not just sit next to the free carriers: they shift the frequency below which $\operatorname{Re}\varepsilon$ is negative. With weak damping, that is roughly where a metal stops reflecting strongly. A strong Lorentz term can also make $\operatorname{Re}\varepsilon$ negative on its own, just above its resonance, as the default term does.

Derivation from Maxwell's equations

Start from the macroscopic Maxwell equations for a linear, isotropic, nonmagnetic medium with no external charges or currents; the carriers' own current is folded into $\varepsilon$ below. Fields vary as $\exp(-i\omega t)$ , units are SI, $\varepsilon(\omega)$ is the relative permittivity and $k_0=\omega/c$ is the vacuum wave number. After the carriers' current is folded in (below), the material enters only through $\mathbf D=\varepsilon_0\mathbf E+\mathbf P=\varepsilon_0\varepsilon\mathbf E$ , so with $\mathbf P=\varepsilon_0\chi\mathbf E$ the permittivity is $\varepsilon=1+\chi$ .

\begin{aligned}\nabla\times\mathbf E&=i\omega\mu_0\mathbf H, & \nabla\cdot(\varepsilon\mathbf E)&=0,\\ \nabla\times\mathbf H&=-i\omega\varepsilon_0\varepsilon\mathbf E, & \nabla\cdot\mathbf H&=0.\end{aligned}

Polarizations add. Split the charges into groups $j$ that each respond linearly to the same macroscopic field, with no direct coupling between groups and no local-field correction. Then $\mathbf P=\sum_j\mathbf P_j$ . Free carriers carry a current rather than a displacement, but at nonzero frequency a current density $\mathbf J$ acts exactly like a polarization $\mathbf P_J=i\mathbf J/\omega$ , since $\partial\mathbf P_J/\partial t=-i\omega\mathbf P_J=\mathbf J$ . So it joins the sum. Each group's driven-oscillator response, derived in Lorentz and Drude Models, gives

\begin{aligned}\varepsilon(\omega)&=1+\sum_j\chi_j(\omega),\\ \chi_j&=\frac{S_j}{\omega_j^2-\omega^2-i\gamma_j\omega}\end{aligned}

Where the constant comes from. For a resonance with $\omega_j\gg\omega$ , expand in $\omega/\omega_j$ . Damping adds only a small imaginary part, of relative size roughly $\gamma_j\omega/\omega_j^2$ , which is dropped here:

\begin{aligned}\chi_j&\approx\frac{S_j}{\omega_j^2}\left(1+\frac{\omega^2}{\omega_j^2}+\dots\right)\\ \varepsilon_\infty&=1+\sum_{\text{above}}\frac{S_j}{\omega_j^2}\end{aligned}

The editor keeps the terms that matter inside the plotted range and replaces the rest with this constant. The constant falls short of that term's own contribution by the fraction $\omega^2/\omega_j^2$ ; the error in $\varepsilon$ is smaller when other terms dominate. For a resonance at $18\times10^{15}$ rad/s (about 105 nm), three times the top of the chart, that is under 0.01% at the low end, 7% at 400 nm and 11% at $6\times10^{15}$ rad/s. A constant background is a good approximation only well below the omitted resonances. In the opposite limit, $\omega_j\ll\omega$ , the same term tends to the Drude form $-S_j/(\omega^2+i\gamma_j\omega)$ , which is $-S_j/\omega^2$ when $\gamma_j\ll\omega$ .

The sum absorbs. The field does work on the charges at the rate $\mathbf J\cdot\mathbf E$ per volume, with $\mathbf J=\partial\mathbf P/\partial t$ (bound and free charges together). For complex amplitudes, $\mathbf J=-i\omega\varepsilon_0\chi\mathbf E$ , and the time average is

\begin{aligned}\langle\mathbf J\cdot\mathbf E\rangle&=\tfrac12\operatorname{Re}\big(\mathbf J\cdot\mathbf E^*\big)\\ &=\tfrac12\,\omega\varepsilon_0\operatorname{Im}\varepsilon\,|\mathbf E|^2\end{aligned}

Here $\operatorname{Im}\varepsilon=\sum_j\operatorname{Im}\chi_j$ (lecture above). Every term with $S_j\ge0$ and $\gamma_j>0$ contributes a nonnegative amount, and a real $\varepsilon_\infty$ contributes nothing. The absorbed power is never negative, whatever the combination.

The Drude zero crossing. Put one Drude term, $\omega_j=0$ , on a background $\varepsilon_b$ , everything except the Drude term, that is real and constant near the crossing. Setting the real part to zero gives

\begin{aligned}\operatorname{Re}\varepsilon&=\varepsilon_b-\frac{S}{\omega^2+\gamma^2}=0\\ \omega_z^2&=\frac{S}{\varepsilon_b}-\gamma^2\end{aligned}

A real crossing exists only when $\varepsilon_b>0$ and $S>\varepsilon_b\gamma^2$ .

From permittivity to films. In a uniform region $\varepsilon$ is constant, so $\nabla\cdot\mathbf E=0$ , and the equations above give the Helmholtz equation:

\nabla^2\mathbf E+\varepsilon k_0^2\mathbf E=0

The complex index is $n+ik=\sqrt\varepsilon$ , taking the root with $k\ge0$ (and $n\ge0$ when $k=0$ ) so waves decay as they travel. The planar solver uses $\varepsilon$ directly through the normal wave number $k_z=k_0q$ , $q=\sqrt{\varepsilon-n_0^2\sin^2\theta}$ , where $n_0$ is the real index of the transparent incidence medium, with the same root choice ( $\operatorname{Im}q\ge0$ , so total internal reflection gives a decaying field); Thin-Film Interference works through that step.

Three things to try

The default term. The page opens with one Lorentz term at $3\times10^{15}$ rad/s (628 nm), strength $9\times10^{30}\,\mathrm{s^{-2}}$ , damping $3\times10^{14}$ rad/s and background 1. At the low end of the chart $\operatorname{Re}\varepsilon=2.00$ : the background plus $S/\omega_0^2=1$ . At the resonance it returns to exactly 1, and $\operatorname{Im}\varepsilon=S/(\gamma\omega_0)=10$ . The real part is negative from about 3.02 to $4.22\times10^{15}$ rad/s. At the top of the chart it is 0.67, below the background, because the term now pulls downward.

Add a Drude term. Press Add Drude term. The added term has strength $10^{30}\,\mathrm{s^{-2}}$ and damping $3\times10^{14}$ rad/s. On a background of 1 alone, its real part would cross zero at $0.95\times10^{15}$ rad/s. The Lorentz term raises the background below its resonance to about 2.05, and the crossing moves down to $0.63\times10^{15}$ rad/s, about 3 µm. The formula above gives the same value.

Compare with fused silica. Choose the fused silica overlay and switch to n / k; that dataset has no loss data, so it does not appear in the permittivity view. Disable the default term and set the background to 2.1. The model gives a flat $n=1.449$ , while the dataset rises from 1.452 at 900 nm to 1.470 at 400 nm. Now select Lorentz 1, tick Enabled again and type resonance 6 ( $10^{15}$ rad/s), strength 1.32 ( $10^{30}\,\mathrm{s^{-2}}$ ), damping 0.01 and background 2.07. The resonance sits on the right edge of the chart, so the curve shoots up there and flattens everything else; hover the chart or open Sample values to read $n$ . Between 2.09 and $4.71\times10^{15}$ rad/s (900 to 400 nm) the model is within 0.003 of the dataset. At $1.22\times10^{15}$ rad/s (1550 nm) the dataset has dropped to 1.444 and the model is still at 1.452, because silica's infrared absorption is missing from it. The ultraviolet term is a stand-in at the highest resonance the editor allows, not silica's actual absorption. It fits well in the visible and says little about the physics, which is a common combination.

When the material looks right, Use in thin film sends the whole model to the thin-film page. There it is the 250 nm film between air and an ideal $n=1.5$ substrate, calculated from 400 to 900 nm. Use in Layer-Stack Builder sends it there instead.

Oscillator model

Material name

Background permittivity

Selected term

Enabled

Resonance (10¹⁵ rad/s)

Strength (10³⁰ s⁻²)

Damping (10¹⁵ rad/s)

Terms add linearly. Positive damping and nonnegative strength keep the supported model passive. No fitting or effective-medium mixing is performed.

Combined material response

Reference dataset overlay

Spectrum representation

Sum of enabled terms and background

Real Imaginary

Sample values

Angular frequency (10¹⁵ rad/s)	Real (Dimensionless)	Imaginary (Dimensionless)
0.15000	2.0025	0.0050250
0.34500	2.0133	0.011809
0.54000	2.0331	0.019219
0.73500	2.0631	0.027710
0.93000	2.1050	0.037897
1.1250	2.1614	0.050680
1.3200	2.2364	0.067462
1.5150	2.3362	0.090577
1.7100	2.4708	0.12418
1.9050	2.6569	0.17630
2.1000	2.9245	0.26415
2.2950	3.3316	0.43004
2.4900	4.0008	0.80060
2.6850	5.1800	1.8802
2.8800	6.1033	6.2490
3.0750	-2.8736	7.8429
3.2700	-2.9799	2.3063
3.4650	-1.6741	0.92464
3.6600	-0.92725	0.48142
3.8550	-0.47802	0.29164
4.0500	-0.18391	0.19432
4.2450	0.021721	0.13812
4.4400	0.17273	0.10285
4.6350	0.28787	0.079324
4.8300	0.37826	0.062873
5.0250	0.45090	0.050938
5.2200	0.51041	0.042014
5.4150	0.55995	0.035177
5.6100	0.60174	0.029827
5.8050	0.63740	0.025567
6.0000	0.66814	0.022124

Individual imaginary susceptibility contributions

Term 1

Sample values

Angular frequency (10¹⁵ rad/s)	Term 1 (Im χ)
0.15000	0.0050250
0.34500	0.011809
0.54000	0.019219
0.73500	0.027710
0.93000	0.037897
1.1250	0.050680
1.3200	0.067462
1.5150	0.090577
1.7100	0.12418
1.9050	0.17630
2.1000	0.26415
2.2950	0.43004
2.4900	0.80060
2.6850	1.8802
2.8800	6.2490
3.0750	7.8429
3.2700	2.3063
3.4650	0.92464
3.6600	0.48142
3.8550	0.29164
4.0500	0.19432
4.2450	0.13812
4.4400	0.10285
4.6350	0.079324
4.8300	0.062873
5.0250	0.050938
5.2200	0.042014
5.4150	0.035177
5.6100	0.029827
5.8050	0.025567
6.0000	0.022124

Import model JSON

Model and limits

With time dependence $\exp(-i\omega t)$ , the relative permittivity is

\varepsilon(\omega)=\varepsilon_\infty+\sum_j\frac{S_j}{\omega_j^2-\omega^2-i\gamma_j\omega}

Strength $S_j$ has units $\mathrm{s^{-2}}$ ; resonance $\omega_j$ and damping $\gamma_j$ use rad/s. Nonnegative strengths and positive damping describe passive terms. Setting $\omega_j=0$ gives Drude; $\omega=0$ and undamped poles are excluded. $n+ik$ is the passive square root of $\varepsilon$ .

This is a classical, linear, isotropic response model, not a fit to a named material. The background $\varepsilon_\infty$ represents omitted higher-frequency response over the modelled range. A finite number of oscillators does not supply a complete microscopic description of a solid.

History of the method

Paul Drude's Lehrbuch der Optik (1900; English translation The Theory of Optics, 1902) explains dispersion by assuming that the particles of a body have natural periods of vibration. He writes the complex dielectric constant as a sum over every kind of vibrating charge, each with its own period and friction. In a footnote he credits Sellmeier, Helmholtz and Ketteler with using this idea independently, and reports Rayleigh's observation that Maxwell had set out the same basis in an 1869 Cambridge examination.

The same chapter gives the background term its meaning. For transparent substances Drude shows that the constant term of an empirical dispersion formula equals one plus the contributions of resonances in the ultraviolet, while the static dielectric constant also includes those in the infrared. For quartz he quotes a fit with one ultraviolet and two infrared terms, and attributes the remaining difference in the constant to molecules whose periods lie so far in the ultraviolet that they act as a constant. For metals he adds a term for conducting charges with no restoring force to the same expression.

Such sums are still fitted to measured data. For example, Rakić, Djurišić, Elazar and Majewski (1998) modelled the optical functions of 11 metals from 0.1 to 6 eV with Lorentz–Drude and Brendel–Bormann models. They reported that the noble metals needed the Brendel–Bormann form, which this editor does not include, to describe interband absorption.

Method context and references

The bound/free distinction explains why these models remain useful: restoring forces create resonances, while free carriers contribute a low-frequency conductivity. Gang Chen, Radiative Transfer, MIT OpenCourseWare (Spring 2006), chapter 6 reading develops Lorentz and Drude material response. M. S. Dresselhaus, Solid State Physics, Part II: Optical Properties of Solids, MIT course notes, chapter 2, derives the free-carrier contribution. These teaching sources support the formulas, not the history. The history follows Drude's account and his footnote; the original works of Maxwell, Sellmeier, Helmholtz and Ketteler were not checked. The editor and calculations are a later browser implementation.

Paul Drude, The Theory of Optics, translated from the German by C. Riborg Mann and Robert A. Millikan, Longmans, Green and Co. (1902; copyright 1901), chapter “Dispersion”, pp. 382–399. Chapter read in full in the Internet Archive scan of a 1933 impression; the history above follows this chapter and its footnotes. The earlier works by Sellmeier, Helmholtz, Ketteler and Maxwell were not consulted.

Aleksandar D. Rakić, Aleksandra B. Djurišić, Jovan M. Elazar and Marian L. Majewski, Optical properties of metallic films for vertical-cavity optoelectronic devices, Applied Optics 37(22), 5271 (1998). Only the abstract was read.

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