“The vast interplanetary and interstellar regions will no longer be regarded as waste places in the universe, which the Creator has not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find them to be already full of this wonderful medium; so full, that no human power can remove it from the smallest portion of space, or produce the slightest flaw in its infinite continuity.”
— James Clerk Maxwell

Maxwell here speaks of the electromagnetic field -the “wonderful medium” that permeates everywhere, even in empty space. Of the four fundamental forces (gravity, electromagnetism, the weak nuclear force, and the strong nuclear force), electromagnetism alone spans both the very small (binding electrons to nuclei) and the very large (governing interactions across galaxies). It is this force that is governed by Maxwell’s Equations.

Introduction

When first approaching Fourier Optics, there are generally two methods of introduction: beginning with the Fourier Transform and Signals & Systems framework, or starting from Maxwell’s Equations and first physical principles. For this post, I choose the latter — a ground-up, physics-first perspective.

This approach risks feeling overwhelming at first, since substantial groundwork must be built before reaching directly applicable techniques like spatial filtering or holography. However, this method pays dividends in the long run: by working from first principles, one gains a deeper, more general intuition. For example, diffraction can then be understood fundamentally — as arising from point sources and the Huygens-Fresnel principle — rather than merely as a far-field pattern predicted by a Fourier transform.

Ultimately, while beginning with Fourier analysis offers faster practical results, grounding Fourier Optics in Maxwell’s Equations builds a foundation that promotes flexibility, insight, and the ability to generalize to new problems beyond textbook cases.

With that said, let’s begin with Maxwell’s Equations — specifically, the four equations as rewritten by Heaviside.

Gauss’s Law

When two point charges $q$ and $Q$ are separated by a distance $r$, they exert an electric force on each other that is attractive if the charges have opposite signs or repulsive if the charges have the same sign. This force is given by Coulomb’s Law:

where $\vec{F}$ is the force vector, $Q$ and $q$ are the magnitudes of the charges, $r$ is the distance between the charges, $\hat{r}$ is the unit vector pointing from the charge $Q$ to $q$, and $\epsilon_0$ is the permittivity of free space.

The electric field $\vec{E}$ at a point in space is defined as the force per unit positive charge that would be experienced by a positive test charge placed at point $r$. Mathematically, this is

For a point charge $Q$, the electric field at a distance $r$ is:

This equation gives us the strength and direction of the electric field due to a point charge $Q$ at any point in space.

The electric flux quantifies how much electric field passes through a given surface area. For an infinitesimal surface element $d\vec{A}$, the electric flux $d\Phi$ is defined as:

where $\theta$ is the angle between the electric field vector and the normal to the surface. Notice that when $\vec{E}$ is perpendicular to the surface ($\theta = 0°$), the flux is maximized whereas when $\vec{E}$ is parallel to the surface ($\theta = 90°$), the flux is zero.

The total flux through a surface is the integral of the electric flux over the entire surface:

Let us now consider a point charge $Q$ at the center of a spherical surface with radius $r$. For this special case, the electric field $\vec{E}$ is radially outward and has the same magnitude at all points on the sphere. The field is perpendicular to the surface at every point ($\theta = 0$). The magnitude of the electric field is $\vec{E} = \frac{Q}{4\pi\epsilon_0 r^2}$ The total flux through the sphere is therefore

Since $\oint dA = 4\pi r^2$ (the surface area of a sphere), we have:

Notice in the resultant equation, the flux does not depend of the radius $r$ of the sphere. Why? While the electric field weakens with distance ($\propto \frac{1}{r^2}$), the surface area increases proportionally ($\propto r^2$), exactly canceling each other out. Additioanlly, notice how the shape of the surface does not matter. For any closed surface enclosing a charge, the total flux depends only on the enclosed charge.

For multiple point charges enclosed within a surface, the principle of superposition applies. The total flux is the sum of the fluxes due to each charge:

where $q_j$ are the individual charges inside the surface.

For a continuous charge distribution with volume charge density $\rho$, we can write:

where the volume integral extends over the entire volume enclosed by the surface.

Gauss’s Law states that the total electric flux through any closed surface is proportional to the enclosed electric charge:

Using the divergence theorem from vector calculus, Gauss’s law can be expressed in differential form:

where $\nabla \cdot \vec{E}$ is the divergence of the electric field and $\rho$ is the volume charge density. In materials, we often use the electric displacement field $\vec{D} = \epsilon_0 \vec{E}$, giving:

No Magnetic Monopoles Exist

In electromagnetism, magnetic fields arise from moving charges (currents) and from changing electric fields. Unlike electric charges, which can exist as isolated positive or negative entities, magnetic sources always appear in pairs—as magnetic dipoles.

Similar to electric flux, magnetic flux measures the amount of magnetic field passing through a surface area. For an infinitesimal surface element $dA$, the magnetic flux $d\Phi_B$ is defined as:

wher $B$ is the magnetic field vector, $dA$ is the surface area element with its normal direction, and $\theta$ is the angle between the magnetic field and the normal to the surface. The total magnetic flux through any surface is the integral of the magnetic flux over the entire surface:

By analogy with Gauss’s law for electric fields, we might attempt to write a similar law for magnetic fields:

where $\Phi_B$ is the total magnetic flux through a closed surface, $Q_j^B$ would represent hypothetical “magnetic charges” or “magnetic monopoles”, and $k$ would be a constant analogous to $\epsilon_0$ in the electric case. However, a critical distinction arises: no magnetic monopoles have ever been observed in nature. While electric charges can exist in isolation (positive or negative charges), magnetic poles always come in north-south pairs.

This means that for any closed surface: If there’s a magnetic north pole inside the surface, there must also be a south pole. Additionally, the net “magnetic charge” inside any closed surface is always zero: $Q_j^B = 0$

Since there are no magnetic monopoles, the magnetic flux through any closed surface must be zero:

This equation is Maxwell’s second equation in integral form. It states that the magnetic field lines that enter a closed surface must also exit it—there can be no sources or sinks of magnetic field lines. Using the divergence theorem from vector calculus, we can convert the integral form to a differential form:

Since the left side equals zero for any arbitrary volume, the integrand on the right must be zero everywhere:

This is Maxwell’s second equation in differential form. It states that the divergence of the magnetic field is zero everywhere, which is a mathematical expression of the fact that magnetic monopoles do not exist.

Faraday’s Law

While the first two Maxwell equations describe static electric and magnetic fields, the third and fourth equations address the dynamic nature of electromagnetic fields - what happens when fields change with time.

Michael Faraday discovered that a changing magnetic field induces an electric current in a nearby conductor. Specifically, moving a magnet near a conducting loop generates an electric current in the loop, even without any physical contact. This phenomenon is known as electromagnetic induction.

Faraday’s key insight was that the induced current is proportional to the rate of change of the magnetic flux through the loop. The electromotive force (EMF), denoted by $\mathcal{E}$, is the work done per unit charge, effectively acting as a voltage that drives the induced current. Faraday found that:

where $\Phi_B = \int_S B \cdot dA$ is the magnetic flux through the surface $S$ bounded by the loop. The negative sign (known as Lenz’s law) indicates that the induced EMF opposes the change that created it

The EMF around a closed loop can also be expressed in terms of the electric field:

where $C$ is the closed path (the conducting loop), $E$ is the electric field, and $dl$ is an infinitesimal element of the path

For a circular loop of radius $r$ with a uniform electric field tangent to the loop, this simplifies to:

Equating the two expressions for EMF:

Since the time derivative can be moved inside the integral (assuming the surface doesn’t change with time):

This is Maxwell’s third equation in integral form:

Stokes’ theorem relates a line integral around a closed loop to the curl of the vector field over the surface bounded by that loop:

Comparing with our previous equation:

Since this must be true for any surface $S$, the integrands must be equal:

Ampère’s Law

Ampère discovered that electric currents create magnetic fields that encircle the current. For a long, straight wire carrying current $I$, the magnetic field at a distance $r$ is:

where $\mu_0$ is the permeability of free space.

Since the magnetic field forms concentric circles around the wire, the line integral around the wire gives:

This is Ampère’s Law. It states that the line integral of the magnetic field around any closed loop equals $\mu_0$ times the current passing through any surface bounded by that loop.

For more complex current distributions, we can express the current $I$ in terms of the current density $J$:

where $J$ is the current density vector (current per unit area).

This allows us to rewrite Ampère’s Law as:

Ampère’s Law works well for steady currents, but Maxwell identified a critical inconsistency when applied to time-varying situations. Consider a charging capacitor. Current flows in the wires leading to the capacitor plates, but No current actually flows through the space between the plates; yet, a magnetic field is observed around this region

To resolve this paradox, Maxwell proposed that a changing electric field must also contribute to the magnetic field, just as a changing magnetic field creates an electric field.

In a parallel plate capacitor with plate area $A$, the electric field is:

where $Q$ is the charge on the plates.

The rate of change of this field is:

where $\frac{\partial Q}{\partial t} = I$ is the current flowing into the capacitor.

Maxwell defined the displacement current density as:

The total displacement current through the capacitor is:

For a uniform electric field, this equals $I = \epsilon_0 A \frac{\partial E}{\partial t}$, which is exactly the current flowing in the wires. Adding this displacement current to Ampère’s Law gives us Maxwell’s fourth equation in integral form.

Applying Stokes’ theorem:

Comparing with the integral form:

Since this must be true for any surface $S$, the integrands must be equal.

Summary of Maxwell’s Equations

Interactions with Matter

While Maxwell’s equations describe the fundamental relationships between electric and magnetic fields, they must be supplemented with material-specific equations that describe how these fields behave inside matter. These are known as the constitutive relations:

Where:

• $\vec{D}$: Electric displacement field (C/m²)
• $\vec{E}$: Electric field (V/m)
• $\epsilon$: Permittivity of the material (F/m)
• $\vec{B}$: Magnetic flux density (T)
• $\vec{H}$: Magnetic field intensity (A/m)
• $\mu$: Permeability of the material (H/m)

Permittivity measures a material’s ability to permit (or support) electric field formation. It quantifies how easily a material becomes polarized when exposed to an electric field. A higher $\epsilon$ means electric dipoles in the material align more easily with the external field, reducing the net field inside. This “screening” effect causes a slower propagation of electromagnetic waves, since more energy is stored in the medium.

In vacuum, this becomes:

where $\epsilon_0 \approx 8.854 \times 10^{-12} \, \text{F/m}$ is the vacuum permittivity.

We often define a dimensionless quantity called the relative permittivity:

$\epsilon_r > 1$ for dielectrics like glass, water, or ceramics, and for a vacuum, $\epsilon_r = 1$

Permeability characterizes how a material responds to magnetic fields. It represents the degree to which magnetic field lines can penetrate or be supported by a material. A higher $\mu$ means stronger magnetization for a given applied field. Materials like iron have high $\mu$ due to the alignment of atomic magnetic moments (domains).

In vacuum, we have

where $\mu_0 = 4\pi \times 10^{-7} \, \text{H/m}$ is the vacuum permeability.

Analogous to permittivity, we define the relative permeability:

$\mu_r > 1$ for ferromagnetic materials like iron or cobalt, and for a vacuum, $\mu_r = 1$