Optics 047 · Interference, coherence, cavities, and metrology

Mach–Zehnder Sensor

A single-pass twin-cell 3D Mach–Zehnder bench connects correct mirror geometry, two complementary power outputs, sample absorption and Jones-polarization projection to a four-step phase acquisition.

Physical detector powers

P_+,P_-

$0.50$

Injected sample phase / sweep visibility

\phi_s,V

$50\%$

Four-step principal index / fringe ambiguity period

\widehat{\Delta n}_{\mathrm{pr}},\Delta n_{\mathrm{period}}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Mach–Zehnder Sensor

Sample index change relative to reference cell

\Delta n

2.40\times10^{-6}

Actual single-pass length (both cells resize)

L_c

10\,\mathrm{mm}

Vacuum wavelength

\lambda_0

632.80\,\mathrm{nm}

Calibrated additional phase bias

\beta

90^{\circ}

Sample single-pass power transmission

\tau

1

Sample polarization angle relative to reference

\theta

0^{\circ}

Both output analyzers: 0 removed, 1 inserted

a

0

Shared analyzer direction (inactive when removed)

\alpha

45^{\circ}

Input laser power

P_{\mathrm{in}}

1\,\mathrm{mW}

Change sample index: one detector brightens as the other dims. Lengthen the cells: both vessels resize and the same index change accumulates more phase. Try absorption: total output falls. Orthogonal polarization tags remove interference without absorbing light; insert both 45-degree analyzers to restore contrast at half throughput. Four ideal phase steps recover only the principal index, never its unknown fringe order. Play animates route markers, not photons or changing measurements.

Ideal monochromatic collinear modes, lossless balanced nonpolarizing splitters, compensated cell windows, homogeneous nondispersive index change and prescribed sample transmission. No diffraction, image reconstruction, detector noise, precision claim, absolute fringe order, quantum photon statistics or delayed-choice inference. Hardware enlarged; flow markers slowed.

Physics tutorial

Audit the physics of Mach–Zehnder Sensor

BackgroundA Mach–Zehnder sends each beam through a separate arm only once. With identical cell windows and matched geometric lengths, the sample contributes $\phi_s=2\pi L_c\Delta n/\lambda_0$ . This is a bulk homogeneous cell with full optical overlap, not a waveguide effective-index sensor. The index control is an imposed difference, not a calculated gas pressure, temperature or concentration.

Why it mattersCan two detector outputs tell an index change from absorption and polarization tagging?

Start with the essentials

Focus question: Can two detector outputs tell an index change from absorption and polarization tagging?
One-sentence intuition: Phase moves power between outputs; absorption and analyzer rejection remove power from the measured pair. Always close $P_++P_-+P_{\mathrm{abs}}+P_{\mathrm{rej}}=P_{\mathrm{in}}$ . Loss of contrast alone does not identify absorption, decoherence or which-path detection. Here the polarization effect is classical Jones optics; no quantum measurement or nonlocality is simulated.

Core mathematical model

A single pass, not Michelson double pass

\begin{aligned}\phi&=\beta+\phi_s\\\phi_s&=\frac{2\pi L_c\Delta n}{\lambda_0}\\\Delta n_{\mathrm{period}}&=\frac{\lambda_0}{L_c}\end{aligned}

Vacuum wavelength and cell length are converted to metres. The two cells lengthen together so the matched air path stays compensated. The bias is an ideal calibrated phase element; it is not independently inferred from detector data. Longer cells give more phase per index change and a smaller unambiguous principal interval.

Unitary splitters combine fields, not ray powers

\begin{aligned}B&=\frac1{\sqrt2}\begin{pmatrix}1&i\\i&1\end{pmatrix}\\\mathbf E_+&=\tfrac{i\sqrt{P_{\mathrm{in}}}}2(\mathbf v_r+\sqrt\tau e^{i\phi}\mathbf v_s)\\\mathbf E_-&=\tfrac{\sqrt{P_{\mathrm{in}}}}2(\mathbf v_r-\sqrt\tau e^{i\phi}\mathbf v_s)\end{aligned}

The reference transmits at the first splitter; the sample reflects. Each folding mirror contributes the same common phase. At the combiner the plus direction is constructive at zero phase, the minus direction destructive. Field amplitude scales with the square root of power transmission. Splitters transmit and reflect; the two opaque folding mirrors only reflect.

Polarization projection and output powers

\begin{aligned}\mathbf v_r&=A\binom10,\quad\mathbf v_s=A\binom{\cos\theta}{\sin\theta}\\R&=\|\mathbf v_r\|^2,\ S=\tau\|\mathbf v_s\|^2\\C&=\sqrt\tau\,\mathbf v_r\!\cdot\!\mathbf v_s\\P_\pm&=\tfrac14P_{\mathrm{in}}(R+S\pm2C\cos\phi)\end{aligned}

Without analyzers A is the identity. With both inserted, A projects onto their shared linear direction. The tag is an ideal lossless half-wave plate: its fast-axis angle is half the selected polarization rotation. Projection overlap is signed, so minus 45 degrees reverses the eraser fringes. At a 90-degree tag with 45-degree analyzers, contrast returns but half the power is rejected. The cited paper uses lossy arm polarizers and a different input normalization; its factor of one eighth is not copied into this lossless-tag bench.

Power accounting and local response

\begin{aligned}P_{\mathrm{abs}}&=\tfrac12P_{\mathrm{in}}(1-\tau)\\P_{\mathrm{rej}}&=\tfrac12P_{\mathrm{in}}(1+\tau-R-S)\\V&=\frac{2|C|}{R+S}\\\frac{\partial(P_+-P_-)}{\partial\Delta n}&=-P_{\mathrm{in}}C\sin\phi\frac{2\pi L_c}{\lambda_0}\end{aligned}

Visibility is zero if no measured power survives. A blocked sample leaves half the input in the reference, then a quarter at each output: neither detector becomes totally dark. Quadrature is locally most sensitive, while a bright/dark extremum has zero first-order intensity slope. Transmission is a prescribed independent parameter; no absorption-index dispersion law or instrument detection limit is claimed.

Recover from four acquired detector pairs

\begin{aligned}\delta_j&=j\pi/2,\quad D_j=P_{+,j}-P_{-,j}\\\widehat\phi&=\operatorname{atan2}[s(D_3-D_1),s(D_0-D_2)]\\s&=\operatorname{sgn}C\\\widehat{\Delta n}_{\mathrm{pr}}&=\frac{\lambda_0}{2\pi L_c}\operatorname{wrap}_{(-\pi,\pi]}(\widehat\phi-\beta)\end{aligned}

The inverse consumes four simulated power pairs at known extra phase steps plus the calibrated overlap sign. It never reads the injected index as its answer. No modulation means no recoverable phase and the readout is withheld. Otherwise it still reports only a principal value; any integer multiple of the displayed index period produces identical data. The principal branch excludes its endpoints. These noiseless ideal acquisitions do not establish real precision or absolute fringe order.

Common difficulties

A recovered number is not an absolute index

Typical misconceptionFour phase steps reveal the full index change no matter how long the cell is.

Better mental modelThey remove the single-intensity sign ambiguity but not the integer fringe order. Increase the cell length and index until the principal readout wraps. The supplied index is known ground truth, not additional evidence available to the inverse. Orthogonally tagged data without analyzers contain no recoverable phase modulation.

Run the experiment

01
Scene 1: Compensated index sensing
Sweep the phase bias and sample index; compare complementary powers and the four acquired pairs below the apparatus.
What to observe: Two opaque mirrors fold the arms into the second splitter; both outputs end on real photodiode entrances. No diagnostic chart floats in the optical path.
02
Scene 2: Phase versus absorption
Switch to absorption. Block the sample, then restore it; compare detected, absorbed and rejected fractions on one fixed scale.
What to observe: The side and cell views show matched variable-length vessels and open mount apertures. Beam width and hardware size are schematic.
03
Scene 3: Polarization tag and eraser
Try a 90-degree polarization tag. Insert analyzers at 45 degrees, then minus 45 degrees; contrast returns with opposite fringe signs.
What to observe: All plots are simulated detector power. The phase sweep is a virtual calibrated scan, not time data. Flow markers indicate routes only.