Appendix O — Derivation 15: Photon Internal Structure and Polarisation

Overview

In classical electromagnetism, a photon is a quantised oscillation of the EM field. In quantum mechanics, it is a point-like boson with spin-1 and transverse polarisation.

In modal dynamics, the photon is a latency-preserving coherence mode: it has no mass, no anchoring, and no associated field—but it possesses a rich internal structure.

This appendix derives:

The modal geometry of the photon
The origin of polarisation from phase topology
The distinction between linear and circular polarisation as structural states

1. Photon as a Latent Phase Mode

The photon is a coherence function $ψ_{γ} (x, t)$ that satisfies:

No anchoring: $β = 0$
Constant amplitude envelope
Internally wound phase structure:

ψ_{γ} (x, t) = f (x - c t) e^{i ϕ (x, t)}

Where $f$ is a spatially localised envelope and $ϕ (x, t)$ defines a twisting phase surface across transverse dimensions.

2. Phase Surface and Transverse Geometry

Let the photon propagate along $z$ .

Its internal phase $ϕ (x, y, z, t)$ traces out a helical surface around the axis of motion. The structure is:

ϕ (x, y, z, t) = k z - ω t + θ (x, y)

Where:

$θ (x, y)$ defines the transverse polarisation state
The gradient $\nabla_{⊥} ϕ$ determines the direction of polarisation

3. Linear Polarisation

For linear polarisation along $x$ :

θ (x, y) = α x

Then:

\nabla_{⊥} ϕ = α \hat{x}

The phase surface is planar and tilted—resulting in a straight oscillation in coherence gradient aligned with $\hat{x}$ . This is the internal structural origin of linear polarisation.

4. Circular Polarisation

For circular polarisation:

θ (x, y) = m \arctan (\frac{y}{x})

Then the phase surface winds around the propagation axis, producing a spiral coherence gradient. The direction of winding determines handedness:

Left-handed $\to$ $ϕ$ increases counterclockwise
Right-handed $\to$ $ϕ$ increases clockwise

This structural winding replaces spin-1 formalism:

The photon’s “spin” is its internal phase rotation around its direction of travel.

5. Elliptical and Arbitrary States

Any coherent linear combination of $x$ and $y$ gradients:

θ (x, y) = a x + b y

generates elliptical or tilted phase surfaces, all of which are stable internal structures provided the coherence cost $C [ψ]$ remains bounded.

Polarisation is therefore:

Not an abstract quantum state
But a real geometric property of the photon’s phase surface

6. Implications

Polarisation filtering is coherence field alignment, not vector projection
Photon entanglement in polarisation is structural phase alignment between two modes
Spin-1 behaviour emerges from modal winding, not field angular momentum

These predictions are structurally robust and reproduce observed behaviours without spin operators or quantised fields.

Conclusion

The photon is a wound coherence structure propagating with internal transverse phase.
Polarisation is not a property imposed on it—it is the modal geometry it carries.

Appendix N | [Index](./Appendix Master) | Appendix P