# 7b. Self-Refraction Chapter 7 recovered the source-free wave equation, and chapter 7a resolved the one transporting field into two complementary transverse aspects. That still does not yet say how the same field could bend its own path. The point of this chapter is that no second substrate is needed. Distinct portions of the same electromagnetic flow interact directly when they meet as one common field. Two coherent sources interfere for exactly that reason. The question is what that overlap does to transport. ## Coherent overlap Write those two aspects in their conventional electromagnetic variables $$ \mathbf E, \qquad \mathbf B. $$ Write two coherent portions of the same field as $$ \mathbf E_1,\ \mathbf B_1, \qquad \mathbf E_2,\ \mathbf B_2. $$ When they overlap, the observables are computed from the total field, $$ \mathbf E=\mathbf E_1+\mathbf E_2, \qquad \mathbf B=\mathbf B_1+\mathbf B_2. $$ So the local energy density is $$ u = \frac{\epsilon}{2}|\mathbf E|^2 + \frac{1}{2\mu}|\mathbf B|^2 $$ that is, $$ u = u_1+u_2 + \epsilon\,\mathbf E_1\!\cdot\!\mathbf E_2 + \frac{1}{\mu}\,\mathbf B_1\!\cdot\!\mathbf B_2. $$ Likewise the energy flux is $$ \mathbf S = \frac{1}{\mu}\,\mathbf E\times\mathbf B $$ so $$ \mathbf S = \mathbf S_1+\mathbf S_2 + \frac{1}{\mu}\,\mathbf E_1\times\mathbf B_2 + \frac{1}{\mu}\,\mathbf E_2\times\mathbf B_1. $$ These cross terms are the interaction terms. They are not added by hand. They appear because overlapping portions of the same field are read as one common field. For in-phase coherent overlap, the energy cross terms are non-null. The overlap region is therefore more heavily loaded than either isolated portion. In that sense, one region of the field acts for another as a denser electromagnetic medium. If the exact overlapping fields are kept, nothing further is needed. The interaction is already present in the total-field observables written above. For the simplest local case, take the two portions to be parallel and co-propagating, with the same polarization and a definite relative phase $\Delta$. Then the local loading law can be written as $$ u = u_1+u_2+2\sqrt{u_1u_2}\cos\Delta, $$ and likewise $$ \mathbf S = \mathbf S_1+\mathbf S_2+2\sqrt{|\mathbf S_1||\mathbf S_2|}\cos\Delta\, \hat{\mathbf s} $$ when the two fluxes are parallel to the same unit direction $\hat{\mathbf s}$. So if one wants an addition law for the local load rather than for the field, this is it. For equal in-phase overlap, $\Delta=0$ and $u_1=u_2$, so $$ u=(\sqrt{u_1}+\sqrt{u_2})^2=4u_1, \qquad \mathbf S=4\mathbf S_1. $$ Written in normalized units, the point is simply $$ 1+1+2=4. $$ The extra $2$ is exactly the interference term between $1$ and $1$. In this ontology the square is read geometrically, not just algebraically. A localized pulse may be pictured as an occupied normal extent above the local $\mathbf E\times\mathbf B$ base area. Two separate equal pulses therefore realize two base areas and two normal extents. When those pulses overlap coherently and in phase, the two base areas collapse into one shared base, the two normal extents add, and that doubled extent is compacted into the one surviving base cell. One factor of two comes from merging the two occupied bases into one, and the other comes from doubling the normal extent above that base. That is the geometric reading of the fourfold local density. So the statement is not that four times the total energy has appeared out of nowhere. The statement is that the same total field, when reorganized into one coherent occupied region, can carry four times the local energy density. The result concerns local loading. The total bookkeeping still belongs to the full field through $$ \partial_t u+\nabla\cdot\mathbf S=0, $$ applied to the total field. That result is strange enough to have encouraged the operational dogma that a photon interferes only with itself. But mathematically it is just ordinary wave mechanics: amplitudes add linearly, while the observable load is quadratic in the total field. The overlap itself is the effect. ## Effective-medium summary Sometimes one wants a local summary of that loaded overlap region without tracking each contributing portion separately. Then it is convenient to write the exact overlap phenomenologically as a local effective index $$ n_{\mathrm{eff}}>1, \qquad c_{\mathrm{eff}}=\frac{c}{n_{\mathrm{eff}}}. $$ This does not replace the exact superposition above. It is only a compact summary of the fact that the loaded overlap region advances more slowly than an isolated portion would. ## Local bending Once different sides of a local wavefront are loaded differently, they do not advance at the same speed. The more heavily loaded side moves more slowly, so the transport bends toward it. That is refraction. Approximating the overlap region as a higher-index layer of the same field, with exterior index $1$, and approximating the entering transport as locally tangent to that layer, Snell's law gives $$ \sin\theta_{\mathrm{in}} = n_{\mathrm{eff}}\sin\theta_{\mathrm{tr}}, \qquad \theta_{\mathrm{in}}=\frac{\pi}{2}, $$ so $$ \sin\theta_{\mathrm{tr}}=\frac{1}{n_{\mathrm{eff}}}. $$ If $\beta$ denotes the complementary angle to the local tangent, then $$ \beta=\frac{\pi}{2}-\theta_{\mathrm{tr}}, \qquad \cos\beta=\frac{1}{n_{\mathrm{eff}}}, \qquad \tan\beta=\sqrt{n_{\mathrm{eff}}^2-1}. $$ This is the local self-refraction law: stronger coherent loading means larger $n_{\mathrm{eff}}$, stronger bending, and larger departure from a straight transport line. ## The retarded case For a self-refracting closure, the overlap is not produced by two independent laboratory sources. It is produced when a later portion of the same flow enters a region already shaped by an earlier portion of that same flow. Let $\gamma(s)$ describe a local transport line, parameterized by arclength $s$. A local segment at position $s$ interacts causally with earlier source positions $s_{\mathrm{ret}}$ on the same flow, related by $$ \left|\gamma(s)-\gamma(s_{\mathrm{ret}})\right| = c\,(t-t_{\mathrm{ret}}). $$ For a harmonic mode with period $T=2\pi/\omega$, $$ E(s,t)=\Re\!\left[\widetilde E(s)e^{-i\omega t}\right], \qquad B(s,t)=\Re\!\left[\widetilde B(s)e^{-i\omega t}\right], $$ the retarded contribution can be written as $$ E_{\mathrm{ret}}(s,t) = E(s_{\mathrm{ret}},t_{\mathrm{ret}}) = \Re\!\left[\widetilde E(s_{\mathrm{ret}})e^{-i\omega t}e^{\,i\omega\tau}\right], \qquad \tau=t-t_{\mathrm{ret}}, $$ and likewise for $B_{\mathrm{ret}}$. So the retarded self-action enters as a phase lag $\omega\tau$ carried by earlier portions of the same flow. The retarded case is therefore not the definition of self-interaction. It is the closure-relevant causal specialization of the overlap principle already derived above. In the retarded case one may write the same coarse-grained summary more explicitly as $$ D = \epsilon E_{\mathrm{loc}} + P_{\mathrm{self}}[E_{\mathrm{ret}},B_{\mathrm{ret}}], \qquad H = \frac{1}{\mu}B_{\mathrm{loc}} - M_{\mathrm{self}}[E_{\mathrm{ret}},B_{\mathrm{ret}}]. $$ In the thin, nearly uniform, slowly varying regime, linearize the exact retarded response against the local field: $$ P_{\mathrm{self}}\approx \epsilon\,\chi_{e,\mathrm{eff}}\,E_{\mathrm{loc}}, \qquad M_{\mathrm{self}}\approx \chi_{m,\mathrm{eff}}\,H. $$ Then $$ D \approx \epsilon_{\mathrm{eff}}E_{\mathrm{loc}}, \qquad B_{\mathrm{loc}} \approx \mu_{\mathrm{eff}}H, $$ with $$ \epsilon_{\mathrm{eff}}=\epsilon(1+\chi_{e,\mathrm{eff}}), \qquad \mu_{\mathrm{eff}}=\mu(1+\chi_{m,\mathrm{eff}}). $$ Therefore $$ c_{\mathrm{eff}} = \frac{1}{\sqrt{\mu_{\mathrm{eff}}\epsilon_{\mathrm{eff}}}}, \qquad n_{\mathrm{eff}} = \frac{c}{c_{\mathrm{eff}}} = \sqrt{\frac{\mu_{\mathrm{eff}}\epsilon_{\mathrm{eff}}}{\mu\epsilon}} = \sqrt{(1+\chi_{e,\mathrm{eff}})(1+\chi_{m,\mathrm{eff}})}. $$ This recovers the same local refraction law, now written in the explicit causal form relevant when the field bends back and meets its own earlier transport. ## What This Does and Does Not Yet Give This chapter derives the principle, not yet the global shape. It shows: - how coherent overlap of distinct portions of the same field produces non-null interaction terms in the observables, - how that overlap is summarized phenomenologically by a local effective refractive index, - how the closure-relevant retarded case fits inside that more general self-interaction picture and can be written in dielectric form, - and how that loading bends transport by ordinary refraction. It does **not** yet require closure. The next chapter takes the next step: > if self-refraction becomes strong enough to make the path close on itself, > what global standing organizations are then allowed?