Photons Don't Travel in a Direction

This is the single most important thing to understand about photons, and it contradicts almost everything you've been taught.

When an atom emits a photon, the electromagnetic field does not shoot off in one direction like a bullet. It expands outward in all directions simultaneously as a spherical shell, at the speed of light. The shell is hollow — the field has already passed through the interior. It gets weaker as it expands, following the dipole radiation pattern — but it never reaches zero.^[1]

The photon has no trajectory. There is no path it "takes." The field simply exists everywhere on that expanding shell. What we call the "direction" of a photon is decided after the fact — it's the line drawn from the source to wherever the detection event happened to occur.

The Dipole Radiation Pattern

An excited atom is an oscillating electric dipole — a charge distribution oscillating back and forth along an axis. The radiation from this oscillation follows a precise angular pattern given by the far-field solution to Maxwell's equations.^[2]

The electric field at distance $r$ and polar angle $\theta$ from the dipole axis:

The critical factor is $\sin\theta$. This single term determines the shape of the expanding shell:

• Along the dipole axis ($\theta = 0$ or $\pi$): the field is exactly zero — no radiation at all
• Perpendicular to the dipole axis ($\theta = \pi/2$): the field is at its maximum
• At intermediate angles: the field falls off smoothly as $\sin\theta$

The time-averaged power radiated per unit solid angle:

The $\sin^2\theta$ pattern means the shell radiates like a donut around the dipole axis — bright at the equator, dark at the poles. Integrating over all solid angles gives the total radiated power, the Larmor formula:

$$P_\text{total} = \frac{\mu_0 \, p_0^2 \, \omega^4}{12\pi c}$$

Note the $\omega^4$ dependence: a photon of blue light ($\omega$ about twice that of red) radiates 16 times more power from the same dipole moment. This is the origin of Rayleigh scattering — why the sky is blue.

The Expanding Shell in 3D

Every atom the expanding wavefront washes over has some probability of absorbing the quantum — each one gets a roll of the dice weighted by $\sin^2\theta / r^2$. Most rolls are misses (air molecules are poor absorbers of visible light), but the probability is never exactly zero. The photon's life is a gauntlet of absorption probabilities — expanding while constantly rolling dice at every encounter until one finally hits.

Structure of the Shell

The expanding shell has internal structure in two directions:

• Radial direction (outward from source): the field oscillates — this is the wave nature. The shell has a thickness equal to the coherence length $L_c = c\tau_c$, where $\tau_c$ is the coherence time. For a single spontaneous emission event, $\tau_c$ is on the order of nanoseconds, giving a coherence length of roughly $L_c \sim 1$ metre. Within that thickness, the field goes through sinusoidal oscillations at frequency $\omega$.
• Tangential direction (along the shell surface): the field is a smooth, continuous envelope following the $\sin\theta$ dipole pattern — strongest perpendicular to the source atom's oscillation axis, zero along it.

The shell is a wave in the direction it expands, and a smooth surface perpendicular to that.

What We Call "Direction" is a Probability Distribution

If photons have no direction, why does a flashlight make a beam? Why does a laser seem to shoot in a straight line?

The answer: every photon's field extends in all directions, but the probability of detection can be shaped. The detection probability at any point is proportional to the square of the local field strength:

For spontaneous dipole emission, this gives $P \propto \sin^2\theta / r^2$ — a donut-shaped probability distribution. For a laser, it gives a tight Gaussian peaked on the beam axis.

In both cases the field has no direction. The die doesn't know where it will land. The loading just shaped the probability.

How a Laser Shapes Probability

A laser doesn't aim photons. It shapes the probability distribution so that detection is overwhelmingly likely in one direction.^[3] The field still technically exists in all directions, but the amplitude off-axis is something like $10^{-10000}$ — not physically meaningful.

Here's how it works:

1. Excited atoms inside the cavity emit spontaneously in random directions following their $\sin^2\theta$ dipole patterns
2. Most of those expanding shells are lost — only the field component along the cavity axis happens to hit a mirror
3. That field component bounces between two mirrors and passes through excited atoms
4. Each pass triggers stimulated emission — the atom emits a new photon in the exact same mode: same frequency, same phase, same spatial distribution
5. This is natural selection: the cavity doesn't force a direction, it ensures that only the on-axis mode survives long enough to trigger copies of itself
6. One mirror is partially transparent — the accumulated beam leaks through

The output field was never spherical and then "contained." Each stimulated photon was born with its probability concentrated along the beam axis through stimulated emission.

Mirrors: Absorption and Re-emission

When the "bouncing" photon hits a mirror, it doesn't bounce like a ball. The field is absorbed by electrons in the mirror surface, which oscillate and emit a completely new photon — same frequency, same phase (shifted by $\pi$), opposite direction. Every "reflection" is actually destruction and creation. The coherence is preserved because stimulated emission maintains the mode.