The Electromagnetic Field of a Photon

In the previous lesson, we saw that a photon expands as a spherical shell of probability. Now let's look at the structure of the field itself — the object that carries the probability.

The Standard Depiction is Wrong

The standard depiction of a photon as a tiny electromagnetic wave traveling along a narrow line is misleading:^[1]

The electric field (E, red arrows) and magnetic field (B, green loops) extend outward into space, decaying with distance from the beam axis. The field is not confined to the propagation axis — it fills all of space, with a Gaussian profile.

Maxwell's Equations in Free Space

The electromagnetic field obeys Maxwell's equations. In free space (no charges, no currents), these reduce to:^[2]

These four equations are entirely self-sustaining: a changing E creates a curling B, which changes and creates a curling E, and so on. The field propagates itself at $c = 1/\sqrt{\mu_0 \varepsilon_0}$, requiring no medium.

Why the Field Cannot Be Localized

The constraint $\nabla \cdot \mathbf{B} = 0$ is the key to understanding why electromagnetic fields cannot be small. Magnetic field lines have no beginning or end — they are always closed loops extending into space. Since E and B are coupled through the curl equations, if B forms extended loops, E must extend equally far. Even a single photon's field is necessarily distributed over space.

The Wave Equation

Taking the curl of Faraday's law and substituting Ampere-Maxwell:

$$\nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t}(\nabla \times \mathbf{B}) = -\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}$$

Using the vector identity $\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}$ and the fact that $\nabla \cdot \mathbf{E} = 0$ in free space:

This equation has many solutions. The spherical wave from a dipole radiator is one. A Gaussian beam from a laser is another. Plane waves, cylindrical waves — all are solutions. What they have in common is that the field extends throughout space; it cannot be confined to a point.

The Gaussian Beam: Full Paraxial Solution

For a laser beam propagating along the $z$-axis, the electric field is described by the paraxial wave equation's fundamental solution:^[3]

Each of these parameters has a precise $z$-dependence:

$$w(z) = w_0 \sqrt{1 + \left(\frac{z}{z_R}\right)^2}$$

$$R(z) = z\left[1 + \left(\frac{z_R}{z}\right)^2\right]$$

$$\zeta(z) = \arctan\!\left(\frac{z}{z_R}\right)$$

where $z_R = \pi w_0^2 / \lambda$ is the Rayleigh range — the distance from the waist at which the beam cross-sectional area doubles.

Understanding Each Term

The beam radius $w(z)$ tells us the beam spreads: at $z = 0$ the radius is $w_0$; at $z = z_R$ it has grown to $w_0\sqrt{2}$; far from the focus it grows linearly as $w(z) \approx w_0 z / z_R = \lambda z / (\pi w_0)$. A tighter focus (smaller $w_0$) leads to faster divergence — this is diffraction, and it is unavoidable.

The wavefront curvature $R(z)$ is infinite at the waist (flat wavefronts), tightens to a minimum at $z = z_R$, then grows linearly as $R \approx z$ (approximating a spherical wave from a point source). The em_wave animation above shows these curving wavefronts.

The Gouy phase $\zeta(z) = \arctan(z/z_R)$ is subtle but physical. As the beam passes through its focus, it accumulates an extra $\pi$ of phase relative to a plane wave. This phase shift is directly connected to the Heisenberg uncertainty principle: confining the beam transversely (small $w_0$) increases the spread of transverse momenta $\Delta p_\perp$, which shifts the longitudinal phase velocity.^[3]

The Gaussian Envelope and Detection Probability

The factor $\exp(-r^2/w^2(z))$ is the Gaussian envelope that shapes detection probability:

Distance from axis	Field amplitude	Detection probability $\propto |E|^2$
$r = 0$ (on-axis)	$E_0 \cdot w_0/w(z)$	Maximum
$r = w(z)$	$\approx 37\%$ of peak	$1/e^2 \approx 13.5\%$
$r = 2w(z)$	$\approx 1.8\%$	$\approx 0.034\%$
$r = 3w(z)$	$\approx 0.01\%$	Effectively zero

But mathematically, the field extends to $r = \infty$. There is no hard edge to a photon.

The Magnetic Field Structure

For a beam with E polarized along $y$, the magnetic field forms closed loops in the propagation plane. The dominant component and its correction from the Gaussian envelope:

$$B_z = -\frac{1}{c} E_y$$

$$B_x \approx \frac{2x}{\omega \, w^2(z)} \, E_y$$

The first equation is the plane-wave relationship. The second is the correction arising from the transverse variation of the Gaussian envelope — it comes directly from Faraday's law $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$ applied to a field with spatial structure.

The ratio of the correction to the main component:

$$\frac{B_x}{B_z} \approx \frac{\lambda}{2\pi w_0} = \frac{1}{k w_0}$$

For beams wider than a wavelength ($w_0 \gg \lambda$), this ratio is tiny, and the B-field loops are highly elongated ellipses. But they still extend into space — reinforcing the point that the field cannot be confined to a narrow line.