# Longitudinal Delay of an Isolated Circular Bright Interference Branch ## Abstract This proposal tests whether a spatially isolated bright region of a circular interference pattern propagates with the same delay as an ordinary reference beam, or whether its effective longitudinal advance is reduced by coherent loading. Two coherent laser beams are recombined so that their wavefronts produce fringes. Standard interference gives a spatial redistribution of energy density: bright rings exceed the two-beam mean while neighboring dark rings fall below it, with the full pattern conserving the two-beam energy budget. The experiment isolates a circular bright region, or a narrow annular bright ring, and measures its modulation delay over a known longitudinal distance against a reference beam. The standard expectation is that the selected bright region and the reference beam have the same propagation speed. The loaded-branch expectation is different (in analogy with dielectrics): if the selected bright region carries the available two-beam flux on a higher local energy density, then its effective longitudinal advance must be slower. For equal local beam densities at a bright maximum, the predicted speed is one half of the local available longitudinal transport speed. The experiment is therefore a direct time-of-flight test between ordinary output transport and loaded raw-overlap transport. --- # 1. Goal Measure the longitudinal delay of an isolated bright region in a real circular interference pattern. The experimental question is: $$ \boxed{ \text{Does an isolated circular bright region propagate like ordinary output light, or like a loaded raw-overlap branch?} } $$ The standard expectation is $$ v_{\mathrm{bright}}=v_{\mathrm{ref}} $$ within experimental error. The loaded-branch expectation is $$ v_{\mathrm{bright}}v_{\mathrm{bright}}>\frac{v_{\mathrm{ref}}}{2}, $$ then the selected region may not remain a pure raw bright branch. Partial mixing, diffraction, finite aperture averaging, imperfect visibility, unequal local densities, or incomplete branch isolation may contribute. In that case the measured velocity should be compared against the exact aperture prediction $$ v_{\Omega} = \frac{ \int_{\rho_-}^{\rho_+} \left[ J_{1z}(\rho)+J_{2z}(\rho) \right]\rho\,d\rho } { \int_{\rho_-}^{\rho_+} \left[ u_1(\rho)+u_2(\rho) + 2\sqrt{u_1(\rho)u_2(\rho)} \cos\Delta\phi(\rho) \right]\rho\,d\rho }. $$ --- # 16. Summary This proposal establishes: 1. For real circular fringes, the exact raw density is $$ u_{\mathrm{raw}}(\rho) = u_1(\rho)+u_2(\rho) + 2\sqrt{u_1(\rho)u_2(\rho)} \cos\Delta\phi(\rho). $$ 2. At an equal-beam bright maximum, $$ u_{\mathrm{raw}}=4u_0. $$ 3. The exact available longitudinal flux through a selected circular or annular aperture is $$ \Phi_z(\Omega) = \int_\Omega \left[ J_{1z}+J_{2z} \right]dA. $$ 4. The loaded-branch prediction for the selected region is $$ v_{\Omega} = \frac{\Phi_z(\Omega)} {\int_\Omega u_{\mathrm{raw}}\,dA}. $$ 5. At an equal-beam bright center with equal local longitudinal transport, $$ v_{\mathrm{bright}}=\frac{c_z}{2}. $$ 6. On axis, where $c_z=c$, $$ v_{\mathrm{bright}}=\frac c2. $$ 7. The dielectric-style proportional response gives the same result: $$ E_2=kE_1 \Rightarrow E_{\mathrm{tot}}=(1+k)E_1 \Rightarrow u_{\mathrm{tot}}=(1+k)^2u_1 \Rightarrow v_{\mathrm{eff}}=\frac{c}{1+k}. $$ For $k=1$, $$ v_{\mathrm{eff}}=\frac c2. $$ The decisive experimental signature is $$ \boxed{ m_{\mathrm{bright}}\approx2m_{\mathrm{ref}} } $$ for an equal-beam on-axis bright center, where $m=d\tau/dL$ is the measured delay slope. --- # Appendix A — Exact Spherical-Wave Circular Rings This appendix gives the explicit no-approximation circular-ring formula for two effective spherical wavefronts. Let $$ R_1(\rho)=\sqrt{\rho^2+a_1^2}, $$ and $$ R_2(\rho)=\sqrt{\rho^2+a_2^2}, $$ where $$ a_i=Z-z_i. $$ Then $$ \Delta\phi(\rho) = k[R_1(\rho)-R_2(\rho)]-\phi_0. $$ The raw density is $$ u_{\mathrm{raw}}(\rho) = u_1(\rho)+u_2(\rho) + 2\sqrt{u_1(\rho)u_2(\rho)} \cos\!\left(k[R_1(\rho)-R_2(\rho)]-\phi_0\right). $$ Bright rings obey $$ k[R_1(\rho_b)-R_2(\rho_b)]-\phi_0=2\pi N. $$ Dark rings obey $$ k[R_1(\rho_d)-R_2(\rho_d)]-\phi_0=(2N+1)\pi. $$ The local longitudinal transport factors are $$ \cos\alpha_i(\rho)=\frac{a_i}{R_i(\rho)}. $$ Therefore $$ J_{iz}(\rho)=u_i(\rho)c\frac{a_i}{R_i(\rho)}. $$ The exact loaded-branch aperture prediction is $$ v_{\Omega} = \frac{ \int_{\rho_-}^{\rho_+} \left[ u_1(\rho)c\frac{a_1}{R_1(\rho)} + u_2(\rho)c\frac{a_2}{R_2(\rho)} \right]\rho\,d\rho } { \int_{\rho_-}^{\rho_+} \left[ u_1(\rho)+u_2(\rho) + 2\sqrt{u_1(\rho)u_2(\rho)} \cos\!\left(k[R_1(\rho)-R_2(\rho)]-\phi_0\right) \right]\rho\,d\rho }. $$ No plane-wave approximation is used. No small-angle approximation is used. No constant-density approximation is used. --- # Appendix B — Status of the Claim The interference identity $$ u_{\mathrm{raw}}(\rho) = u_1(\rho)+u_2(\rho) + 2\sqrt{u_1(\rho)u_2(\rho)} \cos\Delta\phi(\rho) $$ is standard electromagnetic interference. The circular-fringe phase condition $$ \Delta\phi(\rho)=2\pi N $$ for bright rings is standard. The reduced speed does not follow from these standard identities alone. The reduced speed follows from the tested loaded-branch transport law $$ v_{\mathrm{eff}}=\frac{J}{u}. $$ Thus the proposal is a direct time-of-flight test between: $$ \boxed{ \text{ordinary output transport} } $$ and $$ \boxed{ \text{loaded raw-overlap transport with }v_{\mathrm{eff}}=J/u. } $$