# Experimental Proposal ## Goal Measure whether the center of a bright interference fringe propagates with a different longitudinal delay than one of the incident beams that forms it. ## Rationale The proposal tests a simple effective-advance law: $$ J = u\,c_{\mathrm{eff}}, $$ where $J$ is the carried flux, $u$ is the recovered local density, and $c_{\mathrm{eff}}$ is the effective longitudinal advance speed of the channel. For one incident beam, $$ J_0 = u\,c. $$ For two equal incident beams, the total incoming budget is $$ J_{\mathrm{in}} = 2u\,c. $$ If coherent overlap recovers a bright-fringe center with local density $4u$, then the same incoming budget concentrated into that denser channel gives $$ c_{\mathrm{eff}}=\frac{J_{\mathrm{in}}}{4u} = \frac{2u\,c}{4u} = \frac{c}{2}. $$ So the hypothesis is simple: coherent concentration into a denser bright channel should produce a lower effective longitudinal speed, operationally like a refractive slowdown. ## Complementary Outputs Let the two equal fields arriving at the final recombination region be $$ f_1(x,z,t)=A\,e^{i(kz-\omega t)}e^{+iqx/2}, \qquad f_2(x,z,t)=A\,e^{i(kz-\omega t)}e^{-iqx/2}, $$ with $$ u := |A|^2. $$ At the final 50/50 beam splitter, the two output modes are $$ f_+(x)=\frac{f_1(x)+f_2(x)}{\sqrt2}, \qquad f_-(x)=\frac{f_1(x)-f_2(x)}{\sqrt2}. $$ Therefore $$ u_+(x)=2u\cos^2\!\left(\frac{qx}{2}\right), \qquad u_-(x)=2u\sin^2\!\left(\frac{qx}{2}\right). $$ So the two outputs satisfy $$ u_+(x)+u_-(x)=2u. $$ A bright fringe on one output corresponds to a dark fringe on the other. This is the basic $\cos^2+\sin^2=1$ structure of the two output branches. ## Raw Overlap Peak Before output-mode normalization, the direct coherent overlap is $$ f_{\mathrm{raw}} = f_1 + f_2 = 2A\cos\!\left(\frac{qx}{2}\right)e^{i(kz-\omega t)}, $$ so the raw overlap density is $$ u_{\mathrm{raw}}(x)=4u\cos^2\!\left(\frac{qx}{2}\right). $$ Therefore $$ 0 \le u_{\mathrm{raw}}(x) \le 4u. $$ At a bright-fringe center $$ x_n = \frac{2\pi n}{q}, $$ the loading reaches $$ u_{\mathrm{raw}}(x_n)=4u. $$ This is the local peak used in the $c_{\mathrm{eff}}$ estimate: two incident channels of density $u$ can recover a bright-center loading of $4u$. ## Bright-Core Region To isolate the strongest part of the fringe, require $$ u_{\mathrm{raw}}(x) > 3u. $$ With $$ x=x_n+\Delta x, $$ this becomes $$ 4u\cos^2\!\left(\frac{q\Delta x}{2}\right) > 3u, $$ so $$ \cos^2\!\left(\frac{q\Delta x}{2}\right) > \frac{3}{4}. $$ Hence $$ \left|\frac{q\Delta x}{2}\right| < \frac{\pi}{6}, $$ which gives $$ |\Delta x| < \frac{\pi}{3q}. $$ If the fringe period is $$ \Lambda = \frac{2\pi}{q}, $$ then the bright-core condition is $$ |\Delta x| < \frac{\Lambda}{6}. $$ So the central stripe of full width $$ \frac{\Lambda}{3} $$ stays above $3u$. ## Crossing Angle and Fringe Width If the two beams recombine with total crossing angle $\theta$, then the transverse wave-number difference is $$ q = 2k\sin\!\left(\frac{\theta}{2}\right), \qquad k=\frac{2\pi}{\lambda}. $$ This is the small recombination-angle triangle: the opening angle between the two rays sets the transverse phase gradient and therefore the fringe spacing. Therefore the fringe period is $$ \Lambda = \frac{2\pi}{q} = \frac{\lambda}{2\sin(\theta/2)} \approx \frac{\lambda}{\theta} \quad (\theta \ll 1). $$ A $1\,\mathrm{m}$ Mach-Zehnder arm is practical, but the fringe width is set by the recombination angle $\theta$, not by the arm length. For a HeNe laser, $$ \lambda = 632.8\,\mathrm{nm}. $$ Choosing a fairly wide fringe with $$ \theta = 0.2\,\mathrm{mrad}, $$ gives $$ \Lambda \approx \frac{632.8\times 10^{-9}}{2\times 10^{-4}} \approx 3.16\,\mathrm{mm}. $$ Then the bright-core region above $3u$ has full width $$ \frac{\Lambda}{3} \approx 1.05\,\mathrm{mm}. $$ That is large enough for straightforward spatial isolation. If the detector or entrance slit is centered on the fringe maximum and has active width $a$, then the worst-case sampled loading is $$ u_{\mathrm{edge}} = 4u\cos^2\!\left(\frac{\pi a}{2\Lambda}\right). $$ For the same HeNe example: - if $a=0.25\,\mathrm{mm}$, then $u_{\mathrm{edge}}\approx 3.94u$; - if $a=0.50\,\mathrm{mm}$, then $u_{\mathrm{edge}}\approx 3.76u$. So a practical target is a sensor or slit width in the range $$ 0.25\,\mathrm{mm} \;\text{to}\; 0.50\,\mathrm{mm}, $$ centered on the bright-fringe maximum. ## Measurement Model 1. Split a coherent laser beam into two equal beams. 2. Recombine them at a small angle in a Mach-Zehnder interferometer. 3. Use one output branch, where stable straight fringes are visible. 4. Spatially isolate the center of one bright fringe ridge. 5. Propagate that bright ridge over a known distance $L$. 6. In parallel, propagate one incident beam as a reference over the same distance. 7. Amplitude-modulate both channels from the same source. 8. For each channel, measure the delay $\tau$ relative to the common modulation signal on an oscilloscope or phase meter. For each channel separately, collect data $$ (L_1,\tau_1),\ (L_2,\tau_2),\ (L_3,\tau_3),\ \ldots $$ and fit $$ \tau(L)=mL+b. $$ With the zero-length reference chosen appropriately, $b$ should be close to zero. The local slope estimates are $$ m_i = \frac{\delta \tau_i}{\delta L_i}, $$ and the regression returns the mean slope $$ \langle m \rangle \approx \frac{d\tau}{dL}. $$ Since $$ \tau = \frac{L}{v}, $$ the speed is $$ v = \frac{1}{\langle m \rangle}. $$ This is done independently for the incident-beam reference and for the isolated bright-fringe channel: $$ v_{\mathrm{ref}} = \frac{1}{\langle m_{\mathrm{ref}} \rangle}, \qquad v_{\mathrm{fringe}} = \frac{1}{\langle m_{\mathrm{fringe}} \rangle}. $$ ## Experimental Question Does the isolated bright-fringe channel yield $$ v_{\mathrm{fringe}} < v_{\mathrm{ref}} \; ? $$ If yes, the bright fringe carries an additional longitudinal delay. If no, the fringe behaves like the reference beam within experimental error. ## Minimum Requirements - stable coherent source - Mach-Zehnder interferometer - controlled small crossing angle at recombination - stable fringe pattern - spatial filter for one bright ridge - common amplitude modulation source - oscilloscope or phase-delay readout - variable path length ## Summary The proposal is simple: 1. create stable fringes in a Mach-Zehnder output, 2. isolate the center of one bright fringe ridge, 3. measure its delay for several lengths, 4. fit $\tau(L)$, 5. recover $v = 1/\langle m \rangle$, 6. compare that speed against one incident beam. This gives a direct experimental test of whether the bright fringe channel acquires an additional longitudinal delay.