Complex Refractive Index and the Extinction Coefficient

A silver mirror and a windowpane are made of the same physics. aq-optics-refractive-index introduced the complex index $\tilde n=n+i\kappa$ as a promissory note — one number whose real part refracts and whose imaginary part absorbs; aq-optics-lorentz-oscillator-model built the complex dielectric function $\varepsilon(\omega)$ that stands behind it. This article cashes the note. It owns the identity $\tilde n^2=\varepsilon=\varepsilon'+i\varepsilon''$ , the exact inversion between the two descriptions, the propagation factor that splits a wave into phase times decay, and the normal-incidence reflectance $R$ that decides whether a material is a mirror or a window. The central move is small and consequential: let the wavenumber be complex, $\tilde k=\tilde n\,\omega/c$ . Then the same plane wave that aq-optics-plane-waves-isotropic-media built in a lossless medium becomes $E\propto e^{-\kappa\omega z/c}\,e^{i(n\omega z/c-\omega t)}$ — an oscillation at wavelength $\lambda_0/n$ inside an exponentially dying envelope. The extinction coefficient $\kappa$ is dimensionless: it measures absorption per radian of vacuum phase, which is why a metal with $\kappa\approx4$ extinguishes light within nanometres while glass with $\kappa\lesssim10^{-11}$ carries it for kilometres. The numbers make the point brutally. Take a metal-like $\varepsilon=-16+0.5i$ at 500 nm: the inversion gives $n=0.0625$ and $\kappa=4.001$ , hence $R=98.5\%$ — almost everything bounces off — and the 1.5% that enters is gone within a penetration depth $1/\alpha=9.95$ nm. Take glass, $n=1.5$ , $\kappa\to0$ : $R=4.0\%$ per surface, the familiar Fresnel loss of every uncoated lens. One formula spans both, and the whole spread is carried by $\kappa$ .

Key takeaways

The complex refractive index $\tilde n=n+i\kappa$ makes the wavenumber complex, $\tilde k=\tilde n\omega/c$ : the wave becomes $e^{-\kappa\omega z/c}\,e^{i(n\omega z/c-\omega t)}$ — $n$ advances the phase, the extinction coefficient $\kappa$ decays the amplitude.
It is the exact square root of the dielectric function of aq-optics-lorentz-oscillator-model: $\tilde n^2=\varepsilon=\varepsilon'+i\varepsilon''$ , so $\varepsilon'=n^2-\kappa^2$ and $\varepsilon''=2n\kappa$ .
The inversion runs both ways: $n=\sqrt{(|\varepsilon|+\varepsilon')/2}$ , $\kappa=\sqrt{(|\varepsilon|-\varepsilon')/2}$ — from any measured or modelled $\varepsilon$ , the optical constants follow.
Intensity decays as $I=I_0e^{-\alpha z}$ with $\alpha=2\kappa\omega/c=4\pi\kappa/\lambda_0$ : the factor 2 between field and intensity is the most-forgotten factor in the subject.
Normal-incidence reflectance $R=\dfrac{(n-1)^2+\kappa^2}{(n+1)^2+\kappa^2}$ : glass ( $n=1.5$ , $\kappa\to0$ ) reflects 4.0%; a metal with $\varepsilon=-16+0.5i$ ( $n=0.0625$ , $\kappa=4.001$ ) reflects 98.5% and absorbs the rest within $1/\alpha=9.95$ nm.
Key idea: one complex number $\tilde n=n+i\kappa=\sqrt{\varepsilon}$ carries the entire linear propagation story — phase ( $n$ ), absorption ( $\kappa$ ), and reflection ( $R$ ) — and the metal/dielectric divide is just the size of $\kappa$ .

The core insight

Let the index be complex: $\tilde n=n+i\kappa=\sqrt{\varepsilon}$ . The very same plane wave then splits into phase × decay, $e^{i(n\omega z/c-\omega t)}\,e^{-\kappa\omega z/c}$ , with $\varepsilon'=n^2-\kappa^2$ , $\varepsilon''=2n\kappa$ , absorption $\alpha=2\kappa\omega/c$ , and reflectance $R=((n-1)^2+\kappa^2)/((n+1)^2+\kappa^2)$ — refraction, absorption, and reflection from one number.

Where this lives today

Spectroscopic ellipsometry — Drude's polarization trick, automated — is the workhorse metrology of the semiconductor industry: every deposited film on a wafer is characterized by fitting its measured $(n,\kappa)$ spectra, at nanometre thickness sensitivity, thousands of times a day in a fab. The tabulated optical constants that photonics designers pull from databases (Johnson–Christy or Palik for metals, Malitson for silica) are $(n,\kappa)$ pairs, i.e. $\sqrt{\varepsilon}$ — and every simulation of a solar cell, plasmonic sensor, or EUV mirror consumes them through exactly the relations of this article. The two locked numbers of this article each anchor a technology. The metallic skin depth of $\sim10$ nm is why a few tens of nanometres of aluminium makes a perfect mirror coating and why your phone's metal shell blocks light utterly — and it sets the confinement scale of plasmonics, which deliberately works inside that evanescent layer. And the dielectric 4% governs optical engineering everywhere: it is the ghost reflection in eyeglasses, the fibre-connector return loss the telecom budget must absorb, and the reason multi-element lenses are unusable without the anti-reflection coatings that impedance-match it away.

The physical picture

Think of the optical field as a rotating phasor carried through the medium. The real part of $\tilde n$ sets how fast the phase rotates per metre; the imaginary part shrinks the phasor's length as it goes. Both live in one exponent because $i\tilde kz=i(n\omega/c)z-(\kappa\omega/c)z$ : multiplying the wavenumber by $i$ turns the imaginary part of the index into a real, negative exponent — an envelope $e^{-\kappa\omega z/c}$ wrapped around an ordinary wave. Nothing about the machinery of aq-optics-plane-waves-isotropic-media changed; the amplitude simply pays rent as it propagates. The scale of the rent is set by $\kappa$ alone, measured per vacuum wavelength: the amplitude falls by $e^{-2\pi\kappa}$ over each $\lambda_0$ . That single statement spans eleven orders of magnitude of materials. A metal with $\kappa\approx4$ loses a factor $e^{-8\pi}\approx10^{-11}$ of amplitude per wavelength — light effectively cannot exist inside it. Window glass in the visible has $\kappa\sim10^{-8}$ or less; telecom silica at 1550 nm reaches $\kappa\approx6\times10^{-12}$ , which is why a fibre can be twenty kilometres long and still deliver most of the light. "Transparent" and "opaque" are not categories but exponents.

Figure

A wave with complex index ñ = n + iκ: the field oscillates at the shortened wavelength λ₀/n inside a decaying envelope e^(−κωz/c); the amplitude falls to 1/e at the field depth δ = c/(κω) = 2/α, the intensity already at 1/α = δ/2

At an interface the same two numbers decide how much light ever gets in. Reflection is an impedance mismatch: the larger $|\tilde n-1|$ is — whether through $n$ far from 1 or through a large $\kappa$ — the harder the boundary bounces the wave back. A strong absorber is therefore a strong reflector: the wave inside a metal is so heavily damped that it cannot accept the incident energy, and the boundary conditions return almost all of it. That is why the punchline figure's metal reflects 98.5% while its interior would annihilate light in ten nanometres — and why the same formula, fed a modest $n=1.5$ with $\kappa\to0$ , yields the gentle 4% of a windowpane.

Figure

The punchline: one reflectance formula, two materials — metal ε = −16 + 0.5i at 500 nm inverts to n = 0.0625, κ = 4.001 giving R = 0.985 with the transmitted 1.5% absorbed within 1/α = 9.95 nm; dielectric glass n = 1.50, κ → 0 gives R = 4.0% per face

The core equations

The definition packages both optical constants into one number, in the chapter's fixed $e^{i(kz-\omega t)}$ convention. Its job appears the moment it multiplies the wavenumber.

\tilde n=n+i\kappa,\qquad \tilde k=\tilde n\,\frac{\omega}{c} \qquad (1)

where: $n$ — real refractive index, phase velocity $c/n$ , wavelength $\lambda_0/n$ ; $\kappa$ — extinction coefficient, dimensionless, absorption per radian of vacuum phase; $\tilde k$ — complex wavenumber. Both $n$ and $\kappa$ depend on frequency. Insert $\tilde k$ into the plane wave of aq-optics-plane-waves-isotropic-media and the exponent splits into the two jobs. This one line is the entire content of the first figure.

E(z,t)=E_0\,e^{i(\tilde kz-\omega t)}=E_0\,e^{-\kappa\omega z/c}\,e^{i(n\omega z/c-\omega t)} \qquad (2)

where: $e^{-\kappa\omega z/c}$ — amplitude envelope from the imaginary part; $e^{i(n\omega z/c-\omega t)}$ — ordinary phase factor from the real part. The field's $1/e$ depth is $\delta=c/(\kappa\omega)$ . The bridge to the microscopic physics is the wave equation with the complex dielectric function of aq-optics-lorentz-oscillator-model. The index is its exact square root, and squaring the definition gives the forward dictionary.

\tilde n^2=\varepsilon=\varepsilon'+i\varepsilon''\qquad\Longleftrightarrow\qquad \varepsilon'=n^2-\kappa^2,\quad \varepsilon''=2n\kappa \qquad (3)

where: $\varepsilon',\varepsilon''$ — real and imaginary parts of the relative permittivity ( $\varepsilon''>0$ for absorption in this convention). Note $\varepsilon'$ can be negative — that is a metal — whereas $n$ and $\kappa$ are non-negative for a passive medium. Solving (3) for $n$ and $\kappa$ is derivation A. The result is the closed-form inversion used every day on measured or modelled permittivities.

n=\sqrt{\frac{|\varepsilon|+\varepsilon'}{2}},\qquad \kappa=\sqrt{\frac{|\varepsilon|-\varepsilon'}{2}},\qquad |\varepsilon|=\sqrt{\varepsilon'^2+\varepsilon''^2} \qquad (4)

where: the positive roots are the physical branch (passive medium). For $\varepsilon=-16+0.5i$ : $|\varepsilon|=16.008$ , $n=0.0625$ , $\kappa=4.001$ — the locked metal of this article. Intensity is amplitude squared, so the decay exponent doubles. This is the seed of the Beer–Lambert law that aq-optics-absorption-coefficient-beer-lambert develops next.

I(z)=I_0\,e^{-\alpha z},\qquad \alpha=\frac{2\kappa\omega}{c}=\frac{4\pi\kappa}{\lambda_0} \qquad (5)

where: $\alpha$ — intensity absorption coefficient (m⁻¹); $1/\alpha$ — intensity penetration depth, half the field depth $\delta$ . The factor 2 between field and intensity is the classic trap of the warning below. Finally, matching the wave across a vacuum–medium boundary at normal incidence (derivation B) gives the reflectance. This is the number that separates mirrors from windows.

R=\left|\frac{\tilde n-1}{\tilde n+1}\right|^2=\frac{(n-1)^2+\kappa^2}{(n+1)^2+\kappa^2} \qquad (6)

where: $R$ — fraction of incident power reflected. $\kappa\to0$ gives the dielectric $R=((n-1)/(n+1))^2$ — 4.0% for glass; $\kappa$ large forces $R\to1$ — the metallic mirror.

Derivation AFrom the wave equation to the n and kappa inversion

The goal is the exact two-way dictionary between the field description $(n,\kappa)$ and the material description $(\varepsilon',\varepsilon'')$ : show that the wave equation forces $\tilde n=\sqrt{\varepsilon}$ , then solve the resulting pair $\varepsilon'=n^2-\kappa^2$ , $\varepsilon''=2n\kappa$ in closed form for $n$ and $\kappa$ , choosing the physical branch — the inversion (4) that turns every measured permittivity into optical constants.

Step 1. Recall from aq-optics-maxwell-equations-dielectric-media the source-free wave equation $\nabla^2\mathbf{E}=\mu_0\varepsilon_0\varepsilon\,\partial_t^2\mathbf{E}$ , now with $\varepsilon$ complex (the Lorentz dielectric function). The plane-wave ansatz gives $\tilde k^2=\varepsilon\,\omega^2/c^2$ , so $\tilde k=\sqrt{\varepsilon}\,\omega/c\equiv\tilde n\,\omega/c$ with $\tilde n=\sqrt{\varepsilon}$ . Why: nothing in the derivation of the wave equation required $\varepsilon$ real — absorption enters not as new physics but as the analytic continuation of the machinery already built.

Step 2. Square the definition: $(n+i\kappa)^2=n^2-\kappa^2+2in\kappa$ , and match real and imaginary parts to $\varepsilon'+i\varepsilon''$ : $\varepsilon'=n^2-\kappa^2$ , $\varepsilon''=2n\kappa$ — the forward dictionary (3). Why: these two lines already explain the metal signature — if $\kappa$ exceeds $n$ , then $\varepsilon'<0$ , so a negative real permittivity is not exotic but simply "more extinction than refraction".

Step 3. Invert. From $\kappa=\varepsilon''/2n$ , substitute into $\varepsilon'=n^2-\kappa^2$ : $n^4-\varepsilon'n^2-\varepsilon''^2/4=0$ , a quadratic in $n^2$ with roots $n^2=\tfrac12(\varepsilon'\pm|\varepsilon|)$ , $|\varepsilon|=\sqrt{\varepsilon'^2+\varepsilon''^2}$ . Why: reducing to a quadratic is what makes the inversion closed-form — no iteration is ever needed to go from permittivity to optical constants.

Step 4. Choose the branch: $n^2\ge0$ requires the $+$ root ( $|\varepsilon|\ge|\varepsilon'|$ always), giving $n=\sqrt{(|\varepsilon|+\varepsilon')/2}$ and then $\kappa=\sqrt{(|\varepsilon|-\varepsilon')/2}$ , both real and non-negative — equation (4). Why: the sign choice is physics, not algebra — $n\ge0$ fixes the wave to propagate forward, and $\kappa\ge0$ makes it decay rather than grow, the passivity condition; a gain medium deliberately takes $\kappa<0$ .

Step 5. Round-trip the locked metal: $\varepsilon=-16+0.5i$ gives $|\varepsilon|=16.008$ , $n=\sqrt{0.0039}=0.0625$ , $\kappa=\sqrt{16.004}=4.001$ ; squaring back, $n^2-\kappa^2=-16.00$ and $2n\kappa=0.500$ . Why: the round trip is the correctness proof of the branch choice, and it is executed symbolically and numerically in the verify script. *Verified: $\tilde n^2=\varepsilon$ identities and the closed-form inversion symbolically (sympy), the branch condition, and the metal round-trip in verify_aq-optics-complex-refractive-index-extinction.py.*

Go deeper · Derivation B · 1 minNormal-incidence reflectance from the complex index

Derivation BNormal-incidence reflectance from the complex index

The goal is the reflectance formula (6): match the plane wave across a vacuum–medium interface at normal incidence with the full complex $\tilde n$ , obtain the amplitude coefficient $r=(1-\tilde n)/(1+\tilde n)$ , and square its modulus into $R=((n-1)^2+\kappa^2)/((n+1)^2+\kappa^2)$ — then read off the two limits that make it the mirror-or-window criterion.

Step 1. Set up the three waves along $z$ : incident $E_ie^{i(\omega z/c-\omega t)}$ and reflected $E_re^{i(-\omega z/c-\omega t)}$ in vacuum, transmitted $E_te^{i(\tilde n\omega z/c-\omega t)}$ in the medium. Why: at normal incidence there is no angle bookkeeping and one polarization suffices — the cleanest place to see how a complex index enters reflection.

Step 2. Impose continuity of the tangential fields at $z=0$ : $E_i+E_r=E_t$ for the electric field, and — using $H=(\tilde n/\eta_0)\,\hat k\times E$ from the plane-wave triad, with the sign flip for the reversed reflected wave — $E_i-E_r=\tilde n\,E_t$ for the magnetic field. Why: these are the same boundary conditions that give the real Fresnel coefficients; the medium's whole identity enters through the single factor $\tilde n$ in the impedance $\eta_0/\tilde n$ .

Step 3. Eliminate $E_t$ : $r\equiv\dfrac{E_r}{E_i}=\dfrac{1-\tilde n}{1+\tilde n}$ — now a complex number: reflection off an absorbing medium shifts the phase by $\arg r$ , not just 0 or π. Why: that phase is real physics — ellipsometry reads $n$ and $\kappa$ simultaneously precisely because the reflected polarization carries it.

Step 4. Form the power ratio: $R=|r|^2=\dfrac{|1-\tilde n|^2}{|1+\tilde n|^2}=\dfrac{(n-1)^2+\kappa^2}{(n+1)^2+\kappa^2}$ — equation (6). Why: modulus-squared adds the real and imaginary parts of numerator and denominator separately, which is where $\kappa^2$ enters symmetrically in both.

Step 5. Read the limits. Dielectric ( $\kappa\to0$ ): $R\to((n-1)/(n+1))^2$ — glass $n=1.5$ gives $(0.5/2.5)^2=4.00\%$ . Metal ( $\kappa\gg1$ or $\kappa\gg n$ ): numerator and denominator are both dominated by $\kappa^2$ , so $R\to1$ — for the locked metal, $R=16.883/17.133=0.985$ . Why: a large $\kappa$ means the medium cannot sustain the wave, so the boundary conditions can only be satisfied by sending the energy back — strong absorbers are strong reflectors. *Verified: the boundary-condition algebra symbolically, $R(1.5,0)=0.0400$ , $R(0.0625,4.001)=0.985$ , and both limits in verify_aq-optics-complex-refractive-index-extinction.py.*

Limiting cases

Case	Result	Meaning
Transparent dielectric ( $\kappa\to0$ )	$\tilde n\approx n=\sqrt{\varepsilon'}$ , $R=\left(\frac{n-1}{n+1}\right)^2$	real index, Sellmeier territory; glass reflects 4.0% per face
Weak absorber ( $\varepsilon''\ll\varepsilon'$ )	$n\approx\sqrt{\varepsilon'}$ , $\kappa\approx\dfrac{\varepsilon''}{2n}$	the square root linearizes — extinction proportional to the loss
Metal ( $\varepsilon'<0$ , $\vert\varepsilon'\vert\gg\varepsilon''$ )	$\kappa\approx\sqrt{\vert\varepsilon'\vert}$ , $n\approx\dfrac{\varepsilon''}{2\sqrt{\vert\varepsilon'\vert}}$	mostly-imaginary index: evanescent interior, $R\to1$ (locked metal: $\kappa\approx4$ , $n\approx0.0625$ )
Resonant absorber ( $\varepsilon'\approx0$ )	$n\approx\kappa\approx\sqrt{\varepsilon''/2}$	on the Lorentz peak $\varepsilon=1+62.5i$ (recall 1486): $n=5.63$ , $\kappa=5.55$ , $R=70\%$
Perfect conductor ( $\vert\varepsilon\vert\to\infty$ )	$R\to1$ exactly	the ideal mirror; skin depth $\to0$

Common trap"κ and α are the same thing — both just mean absorption."

They are related but not interchangeable, and three factor traps live in the gap. First, dimensions: $\kappa$ is dimensionless (decay per radian of vacuum phase) while $\alpha=2\kappa\omega/c=4\pi\kappa/\lambda_0$ carries m⁻¹ — the same $\kappa$ gives a different $\alpha$ at every wavelength. Second, amplitude versus intensity: the field decays as $e^{-\kappa\omega z/c}$ but intensity as $|E|^2\propto e^{-2\kappa\omega z/c}=e^{-\alpha z}$ — forget the 2 and every penetration depth is wrong by a factor of two. Correspondingly there are two depths: the intensity $1/e$ length $1/\alpha$ (9.95 nm for the locked metal) and the field $1/e$ depth $\delta=c/(\kappa\omega)=2/\alpha$ (19.9 nm) — always say which. Third, sign convention: with $e^{-i\omega t}$ time dependence, absorption means $\kappa>0$ and $\varepsilon''>0$ ; engineering texts using $e^{+j\omega t}$ write $\tilde n=n-j\kappa$ for the same physics. Mixing conventions mid-calculation silently turns absorbers into amplifiers.

Worked examples

1 · Optical constants of a metal from its permittivity (confidence builder)

A metal has $\varepsilon=-16+0.5i$ at $\lambda_0=500$ nm. Find $n$ , $\kappa$ , the normal-incidence reflectance, and how far the transmitted light penetrates.

Step 1. Invert with equation (4). $|\varepsilon|=\sqrt{16^2+0.5^2}=16.008$ ; then $n=\sqrt{(16.008-16)/2}=\sqrt{0.0039}=\mathbf{0.0625}$ and $\kappa=\sqrt{(16.008+16)/2}=\sqrt{16.004}=\mathbf{4.001}$ . The index is almost purely imaginary — the fingerprint of a good metal below its plasma edge.

Step 2. Reflectance from (6): $R=\dfrac{(0.0625-1)^2+16.004}{(0.0625+1)^2+16.004}=\dfrac{16.883}{17.133}=\mathbf{0.985}$ . Only 1.5% of the light gets in at all.

Step 3. Absorption of what enters: $\alpha=4\pi\kappa/\lambda_0=4\pi(4.001)/(500\times10^{-9})=\mathbf{1.005\times10^{8}\ m^{-1}}$ , so the intensity penetration depth is $1/\alpha=\mathbf{9.95\ nm}$ (field depth $2/\alpha=19.9$ nm) — about twenty atomic layers.

Step 4. A curiosity worth keeping: the wavelength inside is $\lambda_0/n=500/0.0625=8.0\ \mu$ m, so across the entire penetration depth the phase advances by only $2\pi n(1/\alpha)/\lambda_0=0.0078$ rad — the "wave" in a metal barely oscillates before it dies; it is an evanescent tail, the same mathematics as the forbidden side of total internal reflection. n = 0.0625, κ = 4.001, R = 98.5%, 1/α = 9.95 nm.

2 · The 4% of every uncoated glass surface

Crown glass has $n=1.50$ and $\kappa\approx0$ in the visible. Find the normal-incidence reflectance per surface, and the transmission of a two-surface window (ignoring interference).

Step 1. With $\kappa=0$ , equation (6) collapses to the dielectric Fresnel form: $R=\left(\dfrac{n-1}{n+1}\right)^2=\left(\dfrac{0.5}{2.5}\right)^2=0.04=\mathbf{4.0\%}$ .

Step 2. Each surface transmits $1-R=96.0\%$ ; two surfaces in series pass $0.96^2=\mathbf{92.2\%}$ . This is the loss that anti-reflection coatings exist to recover — a camera lens with ten uncoated elements would pass only $0.96^{20}\approx44\%$ of the light.

Step 3. Note what $\kappa\approx0$ is doing: it removes $\kappa^2$ from (6) but also makes the bulk lossless, so the 8% missing from the window is redirected, not absorbed. For the metal of example 1 the split is reversed: 98.5% redirected, and the remainder truly absorbed within ten nanometres. R = 4.0% per surface; window transmission 92.2%.

Physics of the portrait

The portrait is the imaginary half of the index — the extinction coefficient $\kappa$ , the single dimensionless number that spans the whole transparency scale of matter.

Origins · 1 minextinction coefficient and how this was discovered

extinction coefficient

κ, the imaginary part of ñ — absorption per radian of optical phase · electromagnetic optics of media

The extinction coefficient began as an embarrassment. Fresnel's real-index formulas, triumphant for glass, simply failed on metals — the reflected polarization came out wrong. Cauchy's remedy (1849) was formally audacious for its time: let the index be complex, and carry the "coefficient of extinction" through the same algebra. Measuring it took another generation — Kundt (1888) pressed metals into micro-thin prisms and found real parts below unity, while Drude (1890) extracted both constants from the polarization of reflected light, inventing the method that became ellipsometry, and in 1900 his free-electron theory explained why metallic $\kappa$ is large. The modern table of $\kappa$ is the transparency ledger of the universe: $\sim4$ for silver in the visible (mirrors), $\sim10^{-2}$ for silicon at its band edge (solar-cell design), $\sim10^{-8}$ for window glass, down to $\sim6\times10^{-12}$ for telecom silica at 1550 nm — eleven orders of magnitude read by one dimensionless number. And $\kappa$ is not free: causality ties it to $n$ at every frequency (Kramers–Kronig, 1926–27), so the extinction spectrum of a material secretly contains its entire refractive index — the promise aq-optics-kramers-kronig-relations will keep.

History

1849

— Cauchy extends the refractive index into the complex plane to describe metallic reflection

1888

— Kundt measures refraction through micro-thin metal prisms: real indices below 1

1890

— Drude extracts n and κ of metals from reflected polarization — the birth of ellipsometry

1900

— Drude's free-electron theory explains the large κ and high reflectance of metals

1926

— Kronig derives the causality dispersion relations; Kramers completes the optical pair in 1927

1972

— Johnson and Christy publish the standard measured n, κ tables for the noble metals

Going further

Texts. Hecht, Optics, 5th ed., §4.8, §7.4 · Foot, Atomic Physics, §7.5 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1–2 · Saleh & Teich, Fundamentals of Photonics, §5.5, §8.2 · Demtröder, Laser Spectroscopy, §3.1 · Jackson, Classical Electrodynamics, 3rd ed., §7.5.

Provenance. A verifier should re-derive $\tilde k=\sqrt{\varepsilon_r}\,\omega/c$ from Maxwell's equations (Jackson §7.5; Hecht §4.8), confirm $n^2-\kappa^2=\varepsilon'$ , $2n\kappa=\varepsilon''$ and the inversion formulas by squaring, verify $\alpha=4\pi\kappa/\lambda_0$ via $I\propto|E|^2$ (Saleh & Teich §5.5), and reproduce the Lorentz $\varepsilon'(\omega),\varepsilon''(\omega)$ and their dilute-limit $n,\kappa$ (Foot §7.5; Demtröder §3.1). Numerical checks: silica fibre loss $\approx0.2$ dB/km at 1.55 µm ( $\kappa\approx6\times10^{-12}$ , i.e. order $10^{-11}$ ) is a standard tabulated value; the normal-incidence reflectance formula is standard (Hecht §4.6). No fabricated constants used. The locked anchors ( $\varepsilon=-16+0.5i\Rightarrow n=0.0625$ , $\kappa=4.001$ , $R=0.985$ , $\alpha=1.005\times10^8$ m⁻¹, $1/\alpha=9.95$ nm; glass $R=4.00\%$ ) are recomputed in verify_aq-optics-complex-refractive-index-extinction.py.

Prerequisites. Maxwell's Equations in Dielectric Media · Dielectric Susceptibility and Permittivity · Plane Waves in Homogeneous Isotropic Media · The Refractive Index. Leads to. Absorption Coefficient and the Beer–Lambert Law · Kramers–Kronig Relations · Dispersion: Normal and Anomalous · The Lorentz Oscillator Model.

How can a refractive index be 0.0625 — doesn't n measure how much light slows down?

For a monochromatic crest, yes: $n=0.0625$ means a phase velocity of $16c$ . Nothing physical moves that fast — a crest of an infinite wave carries no information, and in a metal the point is nearly moot anyway: the wave is extinguished within ten nanometres, barely 0.008 radian of phase, so there is hardly a "wave" whose speed you could clock. The real content of a small $n$ is impedance and reflection: it makes $|\tilde n-1|$ large, hence $R$ close to 1. The full story of superluminal phase velocities — routine, harmless, and signal-free — is owned by aq-optics-phase-velocity later in the chapter.

Why does a strong absorber reflect instead of absorbing everything?

Because absorbing requires entering, and a large $\kappa$ makes the medium unable to host the wave. The boundary conditions must join the incident field to an interior wave that decays within nanometres; the only consistent solution sends almost all the energy back out. Formally, $\kappa$ enters the reflectance (6) exactly like a huge impedance mismatch. The distinction shows up in the energy budget: glass reflects 4% and transmits the rest; the locked metal reflects 98.5% and truly absorbs the remaining 1.5% within $1/\alpha\approx10$ nm. A material that absorbed strongly and reflected weakly would need $\kappa$ large in the bulk but $\tilde n\approx1$ at the surface — which is precisely how graded-index "moth-eye" absorbers and camera-sensor black coatings are engineered.

Is κ absorption, or does it include scattering too?

$\kappa$ — the extinction coefficient — quantifies whatever removes energy from the coherent forward wave. In the ideal homogeneous medium of this article that removal is true absorption: field energy converted through the Lorentz damping $\gamma$ into heat. In real materials, inhomogeneities (bubbles, grain boundaries, density fluctuations) scatter light out of the beam, and a transmission measurement cannot tell the difference — both show up as attenuation. That is why careful spectroscopy distinguishes the absorption coefficient from the extinction (attenuation) coefficient, and why fibre engineers separate Rayleigh-scattering loss from absorption loss even though both contribute to the 0.2 dB/km budget. The bookkeeping meets the Beer–Lambert law in aq-optics-absorption-coefficient-beer-lambert.

Some books write ñ = n − iκ. Which sign is right?

Both — the sign follows the time convention. This chapter writes waves as $e^{i(kz-\omega t)}$ (physics convention), and then decay requires $\tilde n=n+i\kappa$ with $\kappa>0$ and $\varepsilon''>0$ . Engineering texts write $e^{j(\omega t-kz)}$ , and the same decaying wave needs $\tilde n=n-j\kappa$ with the loss in $-j\varepsilon''$ . The physics is identical; only the bookkeeping flips. The danger is mixing them: import a permittivity table built on the opposite convention and your absorber becomes an amplifier. The safe habit is to check one limit — a passive medium must decay forward — before trusting any borrowed $\varepsilon$ or $\tilde n$ .

The square root of a complex ε has two branches. Why is the choice not ambiguous?

Algebraically $\sqrt{\varepsilon}$ has two values, $\pm(n+i\kappa)$ . Physics removes the ambiguity: for a wave launched toward $+z$ in a passive medium, the amplitude must decay ( $\kappa\ge0$ ), and choosing the overall sign so that $n\ge0$ makes phase fronts advance forward. That is the branch equation (4) encodes, with $|\varepsilon|+\varepsilon'\ge0$ guaranteeing real outputs for any physical $\varepsilon$ . The "wrong" branch is not meaningless — $\kappa<0$ describes gain, the working condition of a pumped laser medium — but it must be chosen deliberately, as a statement that the medium holds an energy source, never by accident of a numerical square-root routine (which is why verify scripts round-trip $(n+i\kappa)^2=\varepsilon$ explicitly).

How are n and κ measured at the same time?

Three standard routes. Ellipsometry reflects polarized light and measures the ratio and phase difference of the two Fresnel coefficients — two real numbers per wavelength, exactly enough to invert for $(n,\kappa)$ ; this is Drude's 1890 method industrialized. Transmission plus reflection on a thin film gives $\alpha$ (hence $\kappa$ ) from the decay and $n$ from interference fringes. And when only one quantity is measurable over a broad band — say reflectance of an opaque crystal — causality supplies the other: a Kramers–Kronig transform reconstructs the missing phase, the machinery of aq-optics-kramers-kronig-relations. All three converge on the same $(n,\kappa)$ because both numbers are projections of the single response function $\varepsilon(\omega)$ .