= Optical path length
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= OPL
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An <optical path length> measures how much propagation <wave phase> a path accumulates, expressed as the distance that would accumulate the same <wave phase> in <vacuum>. A centimetre in a uniform medium with <refractive index> $3/2$ has <optical path length> $1.5\,\mathrm{cm}$, although its geometric length remains $1\,\mathrm{cm}$. This makes <optical path length> useful whenever two contributions must arrive with matching <wave phases>, from an <optical lens> to a <Michelson interferometer>. <RP Photonics Encyclopedia by Rüdiger Paschotta/Optical Path Length>{ref}
For a specified path $\gamma$ at fixed <angular frequency> $\omega$, the <optical path length> is the <line integral>
$$
\operatorname{OPL}_\gamma(\omega)=\int_\gamma n(\mathbf r,\omega)\,ds,\qquad
ds=|d\mathbf r|,\qquad
\operatorname{OPL}=\sum_j n_jL_j\ \text{for uniform segments}.
$$
Here $s$ is geometric <arc length>, and $n$ is the phase <refractive index>, evaluated at the same <frequency> everywhere. The propagation interpretation assumes a stationary linear medium with <isotropy>, a real positive scalar <refractive index> and negligible loss, on a valid <geometrical optics> branch. The path can curve; at sharp interfaces apply the appropriate matching conditions to each segment. A prescribed curve has a mathematical <line integral>, but it predicts a propagation <wave phase> only when it represents the physical branch. <MIT 2.71 and 2.710 Optics, Spring 2014 by Nicholas X. Fang/Lecture Notes on Geometrical Optics>{ref} <Optical fibers> can require a mode <wavenumber> rather than the bulk <refractive index> along a guessed <light ray>. <RP Photonics Encyclopedia by Rüdiger Paschotta/Optical Path Length>{ref}
Fix the <vacuum> <wavelength> $\lambda_0=c/f=2\pi c/\omega$, with <frequency> $f=\omega/(2\pi)$ and <vacuum> <wavenumber> $k_0=2\pi/\lambda_0$. In a locally uniform medium the <wavelength> is $\lambda=\lambda_0/n$, while the <frequency> stays unchanged at a stationary interface. With the convention $\Phi=k_0S-\omega t+\phi_0$, propagation from a specified starting <wave phase> contributes
$$
\phi_{\mathrm{prop}}=k_0\operatorname{OPL},\qquad
\delta=\delta_{\mathrm{ref}}+k_0\bigl(\operatorname{OPL}_2-\operatorname{OPL}_1\bigr)\pmod{2\pi}.
$$
The relative reference $\delta_{\mathrm{ref}}$ includes launch <wave phases> and any extra <wave phase> from mirrors, <beam splitters> or coatings. Thus $k_0\operatorname{OPL}$ is a propagation contribution, not an absolute detector <wave phase> independent of the apparatus. If the reference contributions stay fixed, a change in <optical path difference> $h$ changes the <phase difference> by $2\pi h/\lambda_0$. Using the material <wavelength> again in this formula would count the <refractive index> twice. <Electromagnetism lecture notes by Richard Fitzpatrick/Laws of Geometric Optics>{ref}
\Image[/optical-path-michelson-phase-readout.png]
{title=<Optical path difference> in a <Michelson interferometer>: apparatus schematic and a calculated <phase difference> readout}
{id=optical-path-michelson-phase-readout}
A motivating measurement turns invisible optical oscillations into a slowly changing electrical signal. MIT's teaching experiment uses a <laser>, a <beam splitter>, two returning mirrors, an amplified <photodiode>, an oscilloscope and a calibrated mirror actuator driven by a function generator. The returning beams overlap on the <photodiode>; one channel records detector voltage and another the actuator drive. The measured observable is the changing <light intensity>, including its maxima and minima, rather than direct timing of an optical cycle. A calibrated displacement per drive voltage and the spacing of the extrema relate geometric motion to the <phase difference>. The guide infers the <laser> <wavelength> from known actuator calibration; conversely, with independently known <vacuum> <wavelength> and ambient <refractive index>, the same relation calibrates mirror motion. These two unknowns cannot both be inferred from fringe spacing alone. <MIT 8.13–14 Experimental Physics I & II, Junior Lab, Fall 2016–Spring 2017 by MIT Department of Physics/Optical Interferometry>{ref}
For a proposed change-of-medium measurement, keep the mirrors fixed and put a transparent cell of internal length $L$ in one arm. Compare its contents at <refractive indices> $n_a$ and $n_b$, leaving its windows in place. At normal incidence the two traversals give
$$
\Delta\operatorname{OPL}=2L(n_b-n_a),\qquad
\Delta\delta=\frac{4\pi L(n_b-n_a)}{\lambda_0},\qquad
N=\frac{\Delta\delta}{2\pi}=\frac{2L(n_b-n_a)}{\lambda_0}.
$$
Here $N$ is the signed number of whole <interference fringe> cycles plus any resolved fractional cycle, counted continuously. A mirror displacement $x$ instead gives $\Delta\operatorname{OPL}=2n_{\mathrm{amb}}x$; distinguish this double-pass factor from a single pass in a <Mach-Zehnder interferometer>. OpenStax's gas-cell worked example provides the same double-pass reasoning; it is a teaching calculation rather than a published measurement dataset.<University Physics Volume 3 by Samuel J. Ling and others/3.5 The Michelson Interferometer>[Example 3.6: mirror displacement and a gas cell]{ref}
An exact toy calculation makes the sensitivity visible. Stipulate $\lambda_0=600\,\mathrm{nm}$, $L=10\,\mathrm{mm}$ and $n_b-n_a=3/100000=0.00003$. Then $\Delta\operatorname{OPL}=600\,\mathrm{nm}$ and $\Delta\delta=2\pi$: a full <interference fringe> cycle, despite unchanged geometric lengths. For a continuous change $n_b-n_a=q(3/100000)$, let $0\leq q\leq2$ and choose the initial detector <phase difference> to be zero. Equal coherent detected arm contributions of <light intensity> $I_0$ predict
$$
\frac{I(q)}{4I_0}=\frac{1+\cos(2\pi q)}{2}=\cos^2(\pi q).
$$
The plot is this stipulated model: $q=1/2$ gives a minimum, $q=1$ restores a maximum, and the reference level is chosen by adjustment of the apparatus. It contains no measured values or assigned properties of a real gas. The equivalent mirror travel in <vacuum> is $300\,\mathrm{nm}$ per cycle. A single endpoint <light intensity> cannot distinguish $q=0$ from $q=1$; the continuous fringe count carries the missing information.
For an interpretable cell measurement, establish the initial <phase difference> with a small calibrated mirror scan, then track slow changes through successive <interference fringes>. Measure the dark signal and each arm separately, keep <photodiode> gain within its linear range, align both beams on the same region and match their <linear polarizations>. Monitor <interferometric visibility> and keep arm delays within the useful <optical coherence time>. Record cell length, contents, <pressure> and <temperature>; return to the starting condition to check drift. Changes in window deformation, cell length, external air or mirror position also change <optical path length>, so a reference cell or an independently monitored mechanical path is needed to isolate the contents' <refractive index>. These controls are deductions from the model, not a claim that the illustrated apparatus was operated. Loss of fringe count leaves a whole-cycle ambiguity; at an intensity extremum the first <derivative> with respect to <phase difference> vanishes, so a scan around <phase quadrature> is more sensitive to small changes. Broadband illumination, unstable <optical coherence>, vibration and strong loss can invalidate a simple one-<frequency> inference.
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