Macroscopic Maxwell Equations: Linear Response, Kramers-Kronig Relations, and Causality

Intuition Beginner

When an electric field is applied to a material, the material pushes back. The electrons shift, the molecules reorient, and the interior field is different from the applied field. This is the essence of macroscopic electrodynamics: instead of tracking every atom, we describe the material's averaged response using bulk quantities like the permittivity $ϵ$ and the permeability $\mu$.

In a dielectric (insulator), the electrons are bound to their atoms but can shift slightly, creating tiny dipoles. This polarization reduces the internal field. In a conductor, the electrons are free to move and current flows in response to the field. In glass, the material bends light, and the amount of bending depends on the color (frequency) of the light. Blue light bends more than red light in a prism because the material responds differently at different frequencies. This frequency dependence is called dispersion.

The deep principle governing all of this is causality: the material's response at the present moment can only depend on the electric field at past moments. The material cannot anticipate what field will be applied next. This seemingly obvious constraint has a striking mathematical consequence: once you know how much energy the material absorbs at each frequency, you can calculate how it disperses (bends differently by color) at each frequency, and vice versa. These are the Kramers-Kronig relations, and they follow from causality alone.

The real part of the permittivity describes how the material stores energy (the refractive index, the bending of light). The imaginary part describes how the material dissipates energy (absorption, attenuation). Kramers-Kronig says you cannot have one without the other: any material that absorbs at some frequencies must also disperse, and any material that disperses must absorb somewhere.

This connection has practical consequences. Optical fiber designers need to know the refractive index of glass across a wide frequency range, but measuring the refractive index directly at every frequency is difficult. Instead, they measure the absorption spectrum (which is easier) and use the Kramers-Kronig relations to compute the refractive index. Conversely, measuring the refractive index across a broad band allows you to predict where the material absorbs, even if the absorption peaks are in a frequency range (like the deep UV) that is hard to measure directly.

Visual Beginner

Quantity	Symbol	What it describes
Permittivity	$ϵ(\omega )$	How a dielectric responds to an oscillating field at frequency $\omega$
Permeability	$\mu (\omega )$	How a magnetic material responds at frequency $\omega$
Susceptibility	$\chi (\omega )$	The fractional response: $ϵ={ϵ}_0(1+\chi )$
Refractive index	$n(\omega )$	How much the material slows and bends light: $n=\sqrt{ϵ\mu /{ϵ}_0{\mu }_0}$

Kramers-Kronig relation	Connects
Real part of $ϵ(\omega )$	Dispersive response (bending, slowing)
Imaginary part of $ϵ(\omega )$	Absorptive response (energy loss)
Causality	Forces these two parts to be Hilbert transform pairs

Worked example Beginner

A simple model for a dielectric is a single electron bound to a nucleus by a spring with resonant frequency ${\omega }_0$. When an oscillating electric field $E(t)=E_0\cos (\omega t)$ drives this electron, the electron oscillates at the driving frequency $\omega$, not at its natural frequency ${\omega }_0$. The amplitude of the oscillation depends on how close $\omega$ is to ${\omega }_0$.

Far below resonance ($\omega \ll {\omega }_0$), the electron follows the field easily and the polarization is large. Far above resonance ($\omega \gg {\omega }_0$), the field oscillates too fast for the electron to follow and the polarization is small. Near resonance ($\omega \approx {\omega }_0$), the response peaks and energy is efficiently transferred from the field to the material: this is absorption.

The resulting permittivity is:

$$ϵ(\omega )={ϵ}_0\left(1+\frac{{\omega }_p^2}{{\omega }_0^2-{\omega }^2-i\gamma \omega }\right)$$

where ${\omega }_p$ is the plasma frequency (related to the density of electrons) and $\gamma$ is a damping constant. The real part of $ϵ$ gives the refractive index (how much light is bent). The imaginary part of $ϵ$ gives the absorption (how much light is lost). As the driving frequency $\omega$ sweeps through the resonance ${\omega }_0$, the real part changes rapidly and the imaginary part has a peak. Both behaviors are locked together by the Kramers-Kronig relations.

This model — the Lorentz oscillator — is the building block for understanding the optical properties of all materials. Real materials have multiple resonances at different frequencies, each corresponding to a different electronic or vibrational transition. The total permittivity is the sum of contributions from all resonances, and the Kramers-Kronig relations apply to the sum as a whole. A glass window, for instance, has UV resonances from electronic transitions and IR resonances from molecular vibrations. In the visible range (between the UV and IR resonances), the glass is transparent (${ϵ}^{\prime \prime }\approx 0$) but dispersive (${ϵ}^{\prime }$ varies with frequency), which is why a prism made of glass separates white light into a spectrum.

Check your understanding Beginner

Formal definition Intermediate+

The macroscopic Maxwell equations are obtained by averaging the microscopic equations over volumes large enough to smooth out atomic-scale fluctuations. The averaging procedure replaces the microscopic charge and current densities with the macroscopic (smoothed) fields $\mathbf{D}$ and $\mathbf{H}$, which absorb the bound-charge and bound-current contributions into the material response functions. In a linear, isotropic medium, the constitutive relations in the frequency domain are:

$$\mathbf{D}(\omega )=ϵ(\omega )\mathbf{E}(\omega ),\mathbf{B}(\omega )=\mu (\omega )\mathbf{H}(\omega )$$

where $ϵ(\omega )$ is the frequency-dependent permittivity and $\mu (\omega )$ is the frequency-dependent permeability. The electric susceptibility ${\chi }_e(\omega )$ is defined by $ϵ(\omega )={ϵ}_0[1+{\chi }_e(\omega )]$ and the magnetic susceptibility by $\mu (\omega )={\mu }_0[1+{\chi }_m(\omega )]$.

In an anisotropic medium (crystal), the response is direction-dependent and $ϵ$ becomes a tensor: $D_i(\omega )={ϵ}_{ij}(\omega )E_j(\omega )$. The Kramers-Kronig relations then apply to each tensor component ${ϵ}_{ij}(\omega )$ separately, because each component is the Fourier transform of a causal response kernel ${\chi }_{ij}(\tau )$.

In the time domain, the linear response is a convolution:

$$\mathbf{P}(t)={ϵ}_0{\int }_{-\infty }^t{\chi }_e(t-t^{\prime })\mathbf{E}(t^{\prime })d{t}^{\prime }$$

The upper limit $t$ (not $+\infty$) enforces causality: the polarization at time $t$ depends on the field at past times only. The response function ${\chi }_e(\tau )$ is zero for $\tau <0$. The convolution can be rewritten with an infinite upper limit:

$$\mathbf{P}(t)={ϵ}_0{\int }_{-\infty }^{\infty }{\chi }_e(t-t^{\prime })\mathbf{E}(t^{\prime })d{t}^{\prime }$$

because ${\chi }_e(\tau )=0$ for $\tau <0$ automatically truncates the integral. This is the causal convolution: the response is the convolution of the input with a one-sided kernel.

Absorption and the imaginary part. The time-averaged power dissipated per unit volume in a medium with permittivity $ϵ(\omega )={ϵ}^{\prime }(\omega )+i{ϵ}^{\prime \prime }(\omega )$ is:

$$⟨P_{\mathrm{abs}}⟩=\frac{1}{2}\omega {ϵ}^{\prime \prime }(\omega )|E_0{|}^2$$

where $E_0$ is the amplitude of the oscillating field. The imaginary part ${ϵ}^{\prime \prime }(\omega )$ controls absorption: ${ϵ}^{\prime \prime }>0$ means the material absorbs energy from the field, while ${ϵ}^{\prime \prime }<0$ would mean the material amplifies the field (a gain medium). For a passive material, ${ϵ}^{\prime \prime }(\omega )\ge 0$ for all $\omega >0$.

The real part ${ϵ}^{\prime }(\omega )$ determines the refractive index $n(\omega )=\sqrt{{ϵ}^{\prime }(\omega )\mu (\omega )/({ϵ}_0{\mu }_0)}$ (for non-absorbing materials where ${ϵ}^{\prime \prime }$ is small) and hence the phase velocity $v_p=c/n$ of electromagnetic waves in the material. The frequency dependence of $n(\omega )$ is what causes a prism to split white light into its component colors: different frequencies travel at different speeds and refract by different amounts.

Kramers-Kronig relations. Because ${\chi }_e(\omega )$ is the Fourier transform of a causal (zero for $\tau <0$) response function, it is analytic in the upper half of the complex $\omega$-plane. The real and imaginary parts of ${\chi }_e(\omega )={\chi }_e^{\prime }(\omega )+i{\chi }_e^{\prime \prime }(\omega )$ are related by:

$${\chi }_e^{\prime }(\omega )=\frac{1}{\pi }P.V.{\int }_{-\infty }^{\infty }\frac{{\chi }_e^{\prime \prime }({\omega }^{\prime })}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$$ $${\chi }_e^{\prime \prime }(\omega )=-\frac{1}{\pi }P.V.{\int }_{-\infty }^{\infty }\frac{{\chi }_e^{\prime }({\omega }^{\prime })}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$$

where $P.V.$ denotes the Cauchy principal value. These are Hilbert transform pairs: the real and imaginary parts of any causal response function are Hilbert transforms of each other. Equivalent relations hold for $ϵ(\omega )$ and $\mu (\omega )$.

Key derivation Intermediate+

Derivation (Kramers-Kronig relations from causality).

Theorem. If the response function $\chi (\tau )$ satisfies $\chi (\tau )=0$ for $\tau <0$ and $\chi \in L^2(\mathbb{R})$ (square-integrable), then its Fourier transform $\hat{\chi }(\omega )={\chi }^{\prime }(\omega )+i{\chi }^{\prime \prime }(\omega )$ is analytic for $\mathrm{Im}(\omega )>0$, and the real and imaginary parts satisfy the Kramers-Kronig (Hilbert transform) relations.

Proof. Define $\hat{\chi }(\omega )={\int }_0^{\infty }\chi (\tau )e^{i\omega \tau }d\tau$ (the lower limit is 0, not $-\infty$, because $\chi (\tau )=0$ for $\tau <0$ by causality). For $\omega ={\omega }_1+i{\omega }_2$ with ${\omega }_2>0$:

$$|\hat{\chi }(\omega )|\le {\int }_0^{\infty }|\chi (\tau )|e^{-{\omega }_2\tau }d\tau \le \Vert \chi {\Vert }_{L^2}\cdot {\left({\int }_0^{\infty }{e}^{-2{\omega }_2\tau }d\tau \right)}^{1/2}=\frac{\Vert \chi {\Vert }_{L^2}}{\sqrt{2{\omega }_2}}$$

so $\hat{\chi }(\omega )$ is bounded in the upper half-plane. The factor $e^{-{\omega }_2\tau }$ provides exponential damping, ensuring the integral converges uniformly for ${\omega }_2\ge \delta >0$. By the Weierstrass M-test, $\hat{\chi }(\omega )$ is analytic for $\mathrm{Im}(\omega )>0$.

Now consider the contour integral:

$$I={\oint }_C\frac{\hat{\chi }({\omega }^{\prime })}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$$

where $C$ is a large semicircle in the upper half-plane indented below the real point $\omega$. Since $\hat{\chi }$ is analytic inside $C$ and $\hat{\chi }({\omega }^{\prime })\to 0$ as $|{\omega }^{\prime }|\to \infty$ (by the Riemann-Lebesgue lemma), the contribution from the semicircular arc at infinity vanishes. The only contribution comes from the residue at ${\omega }^{\prime }=\omega$:

$$I=i\pi \hat{\chi }(\omega )$$

Expanding into real and imaginary parts and equating:

$$\hat{\chi }(\omega )=\frac{1}{i\pi }P.V.{\int }_{-\infty }^{\infty }\frac{\hat{\chi }({\omega }^{\prime })}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$$

Separating real and imaginary parts yields:

$${\chi }^{\prime }(\omega )=\frac{1}{\pi }P.V.{\int }_{-\infty }^{\infty }\frac{{\chi }^{\prime \prime }({\omega }^{\prime })}{{\omega }^{\prime }-\omega }d{\omega }^{\prime },{\chi }^{\prime \prime }(\omega )=-\frac{1}{\pi }P.V.{\int }_{-\infty }^{\infty }\frac{{\chi }^{\prime }({\omega }^{\prime })}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$$

These are the Kramers-Kronig relations. $\square$

Sokhotski-Plemelj formula. The Kramers-Kronig relations can be understood through the Sokhotski-Plemelj formula from distribution theory. For a function $f(z)$ analytic in the upper half-plane:

$$\underset{\eta \to 0^{+}}{\lim }\frac{1}{x-(\omega +i\eta )}=P.V.\frac{1}{x-\omega }+i\pi \delta (x-\omega )$$

This identity decomposes the analytic continuation of $1/(x-z)$ at the real axis into a principal-value part and a delta-function part. Applied to the permittivity:

$$ϵ(\omega )-{ϵ}_{\infty }=\frac{1}{i\pi }{\int }_{-\infty }^{\infty }\frac{{ϵ}^{\prime \prime }({\omega }^{\prime })}{{\omega }^{\prime }-\omega -i{0}^{+}}d{\omega }^{\prime }=\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{{ϵ}^{\prime \prime }({\omega }^{\prime })}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }+i{ϵ}^{\prime \prime }(\omega )$$

where the real part of the right side gives the Kramers-Kronig relation for ${ϵ}^{\prime }$ and the imaginary part is the consistency check $\mathrm{Im}(ϵ)={ϵ}^{\prime \prime }$. The Sokhotski-Plemelj formula thus provides the bridge between the analytic function $ϵ(\omega )$ in the upper half-plane and its discontinuity across the real axis (which is ${ϵ}^{\prime \prime }(\omega )$).

Normal and anomalous dispersion. The frequency dependence of ${ϵ}^{\prime }(\omega )$ describes dispersion: how the phase velocity varies with frequency. When $d{ϵ}^{\prime }/d\omega >0$ (the refractive index increases with frequency), the dispersion is normal: blue light bends more than red light in a prism. When $d{ϵ}^{\prime }/d\omega <0$, the dispersion is **anomalous**: red light bends more than blue. Anomalous dispersion occurs near absorption resonances ($\omega \approx {\omega }_0$ in the single-resonance model) and is always accompanied by strong absorption (${ϵ}^{\prime \prime }>0$). The Kramers-Kronig relations explain why: a rapid change in ${ϵ}^{\prime }$ (anomalous dispersion) requires a nonzero ${ϵ}^{\prime \prime }$ (absorption) in the same frequency range. You cannot have anomalous dispersion without absorption, and vice versa.

Bridge. The Kramers-Kronig relations are among the deepest results in classical electrodynamics because they derive a quantitative, experimentally verifiable prediction from causality alone. The foundational insight is that the causality constraint ($\chi (\tau )=0$ for $\tau <0$) forces the frequency-domain response function to be analytic in the upper half-plane, and analyticity implies that the real and imaginary parts are Hilbert transforms of each other. The central message is that dispersion (frequency-dependent refractive index) and absorption (energy dissipation) are two manifestations of the same underlying physical process, linked by causality. Putting these together, the Kramers-Kronig relations provide a powerful experimental tool: measure absorption across all frequencies and predict the refractive index, or measure the refractive index and infer where absorption must occur. The connection to complex analysis [02.10] is essential: the Cauchy integral formula, analytic continuation, and the residue theorem are the mathematical machinery that transforms the physical statement "effects follow causes" into the precise Hilbert-transform pair relating ${ϵ}^{\prime }$ and ${ϵ}^{\prime \prime }$.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has the Cauchy integral formula and the theory of analytic functions but lacks the Titchmarsh theorem, the Paley-Wiener theorem for the upper half-plane, the Hilbert transform, the Kramers-Kronig relations, the Sokhotski-Plemelj formula, and dispersion sum rules. Hardy space theory ($H^2$ of the upper half-plane) and the connection between causality and analyticity are not present. lean_status: none.

Advanced results Master

Titchmarsh theorem. The equivalence between causality and analyticity is made precise by the Titchmarsh theorem (also called the Paley-Wiener-Titchmarsh theorem). For a function $f\in L^2(\mathbb{R})$, the following are equivalent:

1. $f(t)=0$ for $t<0$ (causality in the time domain).
2. The Fourier transform $\hat{f}(\omega )$ is the boundary value of a function analytic in the upper half-plane with ${\sup }_{y>0}{\int }_{-\infty }^{\infty }|\hat{f}(x+iy){|}^{2}dx<\infty$ (i.e., $\hat{f}\in H^2$, the Hardy space of the upper half-plane).
3. The real and imaginary parts of $\hat{f}$ are Hilbert transforms of each other (Kramers-Kronig relations).

This three-way equivalence — causality, analyticity, and Hilbert transform structure — is the mathematical backbone of dispersion theory. The connection to complex analysis [02.10] is direct: the Kramers-Kronig relations are the dispersion relations of analytic function theory applied to causal response functions. The proof of the Titchmarsh theorem uses the Cauchy integral and the fact that $H^2$ functions are determined by their boundary values on the real axis (a fact that itself depends on the Poisson kernel and harmonic extension).

Passivity and positive-real functions. A passive material absorbs energy but never generates it. In the frequency domain, passivity requires ${ϵ}^{\prime \prime }(\omega )\ge 0$ for $\omega >0$. Combined with causality (analyticity in the upper half-plane), passivity makes $ϵ(\omega )$ a positive-real function: it is analytic in $\mathrm{Im}(\omega )>0$ with $\mathrm{Re}(ϵ(\omega ))\ge 0$ there. Positive-real functions have the property that they map the upper half-plane into the right half-plane. This structure constrains the possible behavior of permittivity and permeability far more than causality alone.

The positive-real condition has an equivalent formulation in terms of the Herglotz property: $\mathrm{Im}(ϵ(\omega ))>0$ whenever $\mathrm{Im}(\omega )>0$. Herglotz functions have an integral representation (the Nevanlinna representation):

$$ϵ(\omega )=a+b\omega +\frac{1}{\pi }{\int }_{-\infty }^{\infty }\left(\frac{1}{{\omega }^{\prime }-\omega }-\frac{{\omega }^{\prime }}{1+{\omega }^{\prime 2}}\right){ϵ}^{\prime \prime }({\omega }^{\prime })d{\omega }^{\prime }$$

where $a=\mathrm{Re}(ϵ(i))$ and $b\ge 0$ (for physical materials, $b=0$ because $ϵ(\omega )$ is bounded at infinity). This representation shows that the entire permittivity function is determined by its imaginary part on the real axis — the absorption spectrum. The Kramers-Kronig relations are the specialization of this representation to the real axis.

The Sellmeier equation and multi-resonance models. Real materials have multiple absorption resonances (electronic transitions in the UV, vibrational modes in the IR). The multi-resonance model gives:

$$ϵ(\omega )={ϵ}_0\left(1+\sum _k\frac{f_k{\omega }_{p,k}^2}{{\omega }_{0,k}^2-{\omega }^2-i{\gamma }_k\omega }\right)$$

where $f_k$ are the oscillator strengths (with the sum rule ${\sum }_k{f}_k=Z$, the number of electrons per atom). The Sellmeier equation is the undamped (${\gamma }_k\to 0$) version, valid far from resonances. Each resonance contributes a Lorentzian peak to ${ϵ}^{\prime \prime }$ and a sign change to ${ϵ}^{\prime }$.

The oscillator strengths $f_k$ satisfy the Thomas-Reiche-Kuhn sum rule ${\sum }_k{f}_k=Z$ (one per electron), which ensures that the high-frequency limit matches the free-electron response. The sum rule is a consequence of the completeness of the set of quantum states and has no classical analogue — it is a uniquely quantum mechanical constraint on the oscillator model.

Group velocity and pulse propagation. In a dispersive medium, the group velocity $v_g=d\omega /dk=c/[n(\omega )+\omega dn/d\omega ]$ determines the speed at which wave packets (pulses) propagate. The Kramers-Kronig relations constrain the group velocity through the frequency dependence of $n(\omega )$: in regions of normal dispersion ($dn/d\omega >0$), $v_g<c$, but in regions of anomalous dispersion ($dn/d\omega <0$), the group velocity can formally exceed $c$ or even become negative. This does not violate causality because the anomalous-dispersion region coincides with strong absorption, and the pulse is severely distorted. The signal velocity — the speed at which the front of a step-modulated signal propagates — never exceeds $c$, as required by causality and guaranteed by the analytic properties of $ϵ(\omega )$ in the upper half-plane (the Brillouin precursor analysis).

Conductors and the Drude model. For a conductor, the free electrons contribute a Drude term to the permittivity:

$$ϵ(\omega )={ϵ}_{\infty }-\frac{{\omega }_p^2}{\omega (\omega +i\gamma )}$$

where ${\omega }_p$ is the plasma frequency and $\gamma =1/\tau$ is the collision rate. In the low-frequency limit ($\omega \ll \gamma$), this reduces to $ϵ(\omega )\approx {ϵ}_{\infty }+i\sigma /\omega$ where $\sigma ={ϵ}_0{\omega }_p^2/\gamma$ is the DC conductivity. The Drude model is causal (its poles are in the lower half-plane) and passive (${ϵ}^{\prime \prime }>0$ for $\omega >0$). The Kramers-Kronig relations applied to the Drude model correctly reproduce the relationship between the conductivity $\sigma$ and the high-frequency behavior of ${ϵ}^{\prime }$.

Sum rules. Beyond the $f$-sum rule, several other sum rules constrain the permittivity:

• The static limit: ${ϵ}^{\prime }(0)-{ϵ}_0=(2/\pi ){\int }_0^{\infty }{ϵ}^{\prime \prime }(\omega )/\omega d\omega$.
• The inertial sum rule: ${\int }_0^{\infty }[{ϵ}^{\prime }(\omega )-{ϵ}_{\infty }]d\omega =0$ (dispersion averages to zero).
• The perfect-conductor sum rule: ${\int }_0^{\infty }[{ϵ}^{\prime }(\omega )-{ϵ}_0]d\omega /{\omega }^2=-\pi \chi (0)/(2{ϵ}_0)$.

These sum rules provide consistency checks for experimental data: measured spectra that violate them contain errors or miss spectral weight. The sum rules also constrain the design of optical materials: for example, a material engineered to have a very high refractive index at one frequency must have compensating dispersion elsewhere, which limits the bandwidth over which the high index can be maintained.

Synthesis. The macroscopic Maxwell equations with frequency-dependent material response unify the description of dielectrics, conductors, and magnetic materials under a single framework. The foundational insight is that causality — the unremarkable requirement that effects follow causes — has the remarkable mathematical consequence that the response function is analytic in the upper half-plane. The central message is that the Kramers-Kronig relations bind absorption and dispersion into an inseparable pair: you cannot have absorption without dispersion, and the quantitative relationship between them is a Hilbert transform. Putting these together, the theory of linear response in electrodynamics demonstrates that the most powerful constraints on physical behavior often come from the most basic principles (causality, passivity, linearity), and that these constraints take the form of exact mathematical relations — the Kramers-Kronig relations, sum rules, and analyticity conditions — that can be tested experimentally and used to extract complete material properties from partial measurements.

Full proof set Master

Proposition (The permittivity is a Herglotz function). For any causal, passive, linear medium, the function $ϵ(\omega )$ is a Herglotz (or Nevalinna-Pick) function: it is analytic in the upper half-plane and maps the upper half-plane into itself ($\mathrm{Im}(ϵ(\omega ))>0$ when $\mathrm{Im}(\omega )>0$).

Proof. Causality gives analyticity in the upper half-plane. For $\omega$ with $\mathrm{Im}(\omega )=\eta >0$:

$$\mathrm{Im}(ϵ(\omega ))=\mathrm{Im}\left({ϵ}_0+{ϵ}_0{\int }_0^{\infty }\chi (\tau )e^{i\omega \tau }d\tau \right)={ϵ}_0{\int }_0^{\infty }\chi (\tau )e^{-\eta \tau }\sin (\xi \tau )d\tau$$

where $\omega =\xi +i\eta$. This is not immediately manifestly positive. Instead, use the Kramers-Kronig representation for a point $\omega =\xi +i\eta$ with $\eta >0$:

$$ϵ(\omega )-{ϵ}_{\infty }=\frac{1}{i\pi }{\int }_{-\infty }^{\infty }\frac{{ϵ}^{\prime \prime }({\omega }^{\prime })}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$$

Taking the imaginary part:

$$\mathrm{Im}(ϵ(\omega ))=\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{\eta {ϵ}^{\prime \prime }({\omega }^{\prime })}{({\omega }^{\prime }-\xi {)}^2+{\eta }^2}d{\omega }^{\prime }$$

Since $\eta >0$ and ${ϵ}^{\prime \prime }({\omega }^{\prime })\ge 0$ (passivity), the integrand is non-negative. Therefore $\mathrm{Im}(ϵ(\omega ))\ge 0$ for $\mathrm{Im}(\omega )>0$. $\square$

Proposition (Superconvergence and sum rules). If $ϵ(\omega )\to {ϵ}_{\infty }$ as $\omega \to \infty$ sufficiently fast, then ${\int }_0^{\infty }[{ϵ}^{\prime }(\omega )-{ϵ}_{\infty }]d\omega =0$.

Proof. Consider the Kramers-Kronig relation for ${ϵ}^{\prime \prime }(\omega )$:

$${ϵ}^{\prime \prime }(\omega )=-\frac{2\omega }{\pi }P.V.{\int }_0^{\infty }\frac{{ϵ}^{\prime }({\omega }^{\prime })-{ϵ}_{\infty }}{{\omega }^{\prime 2}-{\omega }^2}d{\omega }^{\prime }$$

Multiply both sides by $1/\omega$ and integrate over $[0,\Lambda ]$:

$${\int }_0^{\Lambda }\frac{{ϵ}^{\prime \prime }(\omega )}{\omega }d\omega =-\frac{2}{\pi }{\int }_0^{\Lambda }P.V.{\int }_0^{\infty }\frac{{ϵ}^{\prime }({\omega }^{\prime })-{ϵ}_{\infty }}{{\omega }^{\prime 2}-{\omega }^2}d{\omega }^{\prime }d\omega$$

Exchange the order of integration (justified by Fubini under appropriate conditions) and evaluate the $\omega$ integral:

$${\int }_0^{\infty }\frac{{ϵ}^{\prime \prime }(\omega )}{\omega }d\omega =\frac{2}{\pi }{\int }_0^{\infty }\frac{{ϵ}^{\prime }({\omega }^{\prime })-{ϵ}_{\infty }}{{\omega }^{\prime }}{\left[\frac{1}{2}\ln \left|\frac{{\omega }^{\prime }+\Lambda }{{\omega }^{\prime }-\Lambda }\right|\right]}_{\Lambda \to \infty }d{\omega }^{\prime }$$

The logarithmic term diverges unless ${\int }_0^{\infty }[{ϵ}^{\prime }(\omega )-{ϵ}_{\infty }]d\omega =0$. This is the superconvergence result: the dispersive part of the permittivity averages to zero when weighted uniformly. Physically, for every frequency range where ${ϵ}^{\prime }>{ϵ}_{\infty }$ (normal dispersion), there must be a compensating range where ${ϵ}^{\prime }<{ϵ}_{\infty }$ (anomalous dispersion). $\square$

Connections Master

• Complex analysis [02.10] provides the analytic continuation and Cauchy integral machinery that underpins the Kramers-Kronig relations.
• Green's functions 10.08.01 are themselves causal response functions; the retarded Green's function satisfies the same analyticity conditions as $ϵ(\omega )$.
• Dielectrics 10.04.01 introduce the polarization $\mathbf{P}$ and displacement $\mathbf{D}$ that are the macroscopic quantities described by $ϵ(\omega )$.
• Wave propagation in media 10.04.03 uses $n(\omega )=\sqrt{ϵ\mu /{ϵ}_0{\mu }_0}$ to determine phase velocity, group velocity, and attenuation in dispersive media.
• Conductors 10.04.02 have a complex permittivity $ϵ(\omega )={ϵ}_0+i\sigma /\omega$ that is a special case of the general frequency-dependent response.
• Dipole radiation 10.07.02 and antenna theory use the permittivity to determine the radiation pattern in material media.

Historical & philosophical context Master

The connection between causality and analyticity was first made explicit by Kramers (1927) and independently by Kronig (1926). Kramers, in his paper "La diffusion de la lumiere par les atomes" (Att. Congr. Intern. Fisici, Como, Vol. 2, 1927), derived the dispersion relations connecting the real and imaginary parts of the refractive index from the requirement that the scattered wave is retarded (causal). Kronig, in "On the theory of dispersion of X-rays" (J. Opt. Soc. Am. 12, 547, 1926), independently derived similar relations for the refractive index. The modern formulation in terms of general response functions is due to Toll (1956), who showed that the Kramers-Kronig relations follow from the most general requirements of causality and linearity.

The mathematical foundation was laid by Titchmarsh (Introduction to the Theory of Fourier Integrals, 1937), who proved the equivalence of causality, analyticity, and Hilbert transform structure (the Titchmarsh theorem). The Paley-Wiener theorem (1934) provides the equivalent result in terms of Hardy spaces.

The $f$-sum rule (Thomas-Reiche-Kuhn sum rule) was derived independently by Thomas (1925), Kuhn (1925), and Reiche (1925) in the context of quantum mechanics. Its connection to the Kramers-Kronig relations was recognized later, providing one of the earliest examples of a sum rule derived from analyticity.

The extension to optical constants was carried out by Altarelli, Dexter, Nussenzveig, and Smith (1972), who derived the complete set of sum rules for $n(\omega )$ and the reflectance $R(\omega )$. The use of multiply-subtracted Kramers-Kronig relations for extracting optical constants from reflectance data (where only one-sided spectral data is available) was developed by Toll, Maradudin, and others in the 1960s.

The Sellmeier equation predates the Kramers-Kronig relations. It was introduced by W. Sellmeier in 1871 as an empirical formula for the dispersion of glass. Sellmeier's insight was that dispersion could be modeled as a sum of resonant contributions, anticipating the Lorentz oscillator model by two decades. The equation remains the standard model for the refractive index of optical glasses, and the Sellmeier coefficients for hundreds of materials are tabulated in the Schott and Ohara glass catalogs.

The philosophical significance is that the Kramers-Kronig relations demonstrate how a physical principle as basic as causality — the future cannot influence the past — can generate precise, quantitative, experimentally verifiable constraints on material properties. This pattern (basic physical principles generating exact mathematical constraints) recurs throughout theoretical physics: the CPT theorem in quantum field theory, the fluctuation-dissipation theorem in statistical mechanics, and the optical theorem in scattering theory all follow the same logic. In each case, the mathematical machinery of complex analysis transforms a physical constraint into a pair of integral equations that relate different measurable quantities.

Bibliography Master

• Kramers, H. A., "La diffusion de la lumiere par les atomes," Atti. Congr. Intern. Fisici, Como 2, 545 (1927).
• Kronig, R. de L., "On the theory of dispersion of X-rays," J. Opt. Soc. Am. 12, 547 (1926).
• Toll, J. S., "Causality and the dispersion relation: logical foundations," Phys. Rev. 104, 1760 (1956).
• Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals (Oxford, 1937).
• Paley, R. E. A. C. and Wiener, N., Fourier Transforms in the Complex Domain (AMS, 1934).
• Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
• Landau, L. D. and Lifshitz, E. M., Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).
• Zangwill, A., Modern Electrodynamics (Cambridge, 2013).
• Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics, 3rd ed. (Wiley, 2019).
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