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Take equal arm lengths, no plate, and coherent illumination. At a coated beam splitter, choose the real-amplitude convention where transmission has amplitude $1/\sqrt{2}$ and reflection has amplitude $+1/\sqrt{2}$ from one side and $-1/\sqrt{2}$ from the other; which reflection carries the minus sign is a basis convention. The two paths interfere constructively at detector 1 and destructively at detector 2 in the diagram's labeling. ThusFor example, writing the beam splitter as $B=2^{-1/2}\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)$ and the two equal-reflection mirrors as $M=-\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, one obtains $BMB(1,0{)}^T=(0,-1{)}^T$. The first output component propagates downwards to detector 2; the nonzero second component propagates upwards to detector 1. The overall minus sign is an unobservable global phase. The redistribution is interference of amplitudes, rather than two successive independent coin tosses.

In the stated ordered propagation basis, a downward input is transmitted downward and reflected upward, giving the first column $(1,1{)}^T/\sqrt{2}$. An upward input is reflected downward with the opposite reflection phase and transmitted upward, giving the second column $(-1,1{)}^T/\sqrt{2}$. ThereforeEqual mirror reflection phases exchange the two directions. Choosing a phase of $\pi$ per reflection givesEach path encounters one mirror, so $M$ describes both arms together, not two successive reflections of each photon. Changing its common phase leaves all detection probabilities unchanged. Direct multiplication gives $B^{†}B=M^{†}M=I$, so both are unitary matrices. They preserve the norm of a quantum state and hence total probability for these lossless optical elements.

Ideal detectors are represented by measurement matrices, but detection itself is not unitary evolution within the two traveling modes. With $e_1=(1,0{)}^T$ and $e_2=(0,1{)}^T$, the projective measurement operators areThey give Born rule probabilities $⟨\psi |P_j|\psi ⟩$ and sum to $I$. An absorbing detector maps the incident photon to an excited detector and vacuum field; a complete physical description includes those additional degrees of freedom.

Positioning repeatability describes the scatter of positions reached when a motion system repeatedly returns to the same commanded position under specified conditions. It is an application of measurement repeatability, a form of measurement precision. A system may return to almost the same wrong position each time, so small scatter does not establish measurement accuracy.^[ref]

The approach direction belongs in the test description. Unidirectional positioning repeatability keeps it fixed; bidirectional positioning repeatability includes returns from both directions. To measure the assembled stage, independently observe its final position rather than only the motor's rotary encoder count. Physik Instrumente describes a static test that lets the platform stop before reading it, using an interferometer for displacement and an autocollimator for angular error, with two reflecting mirrors on the stage.^[ref]

For an original numerical example, repeatedly command $x_{\ast }=10.000\mathrm{mm}$. Suppose an independent position readout with known measurement calibration gives the errors $e_j=x_j-x_{\ast }$, in micrometres,Each direction has the same sample standard deviation,Yet their arithmetic means are 0 and $2.0\mu \mathrm{m}$. The magnitude of their mean difference, a positioning reversal error at this target, is 2.0 micrometres, far larger than the within-direction scatter. Pooling all eight values gives a sample standard deviation of $\sqrt{8.04/7}\approx 1.149\mu \mathrm{m}$; that statistic mixes the directional shift with the smaller return-to-return variation.

The calculation uses a declared sample standard deviation, not an unspecified manufacturer's tolerance or a universal repeatability convention. A positioning specification must state its approach directions, target positions, settling/readout procedure, load, environmental conditions, statistic and measurement uncertainty. A single target does not establish performance over the travel range. The manufacturer's method explicitly separates directional means from scatter and combines information from multiple targets when forming axis-level specifications.^[ref]

Backlash is one possible cause of direction-dependent behavior, while thermal expansion and instrumental drift can move the apparent target over time. A repeatability experiment must distinguish those effects from random scatter. Measuring small returns to one target also answers a different question from minimum incremental motion, which asks whether a small commanded change can actually be executed.

Consider an ideal circular vacuum path of laboratory radius $R$, a limit of closely spaced mirrors constraining light around the circle at laboratory speed $c$. The source and detector rotate uniformly together with signed angular velocity $\Omega$, where $|\Omega |R<c$, in flat spacetime. Let $L=2\pi R$ be the laboratory circumference and $A=\pi R^2$ the positive area. The $+$ pulse travels counterclockwise and the $-$ pulse clockwise, independently of the sign of $\Omega$. Both depart from the detector at $t=0$.

Unwrap the circle into a laboratory arc coordinate. The two first-return conditions areConsequently the exact laboratory arrival times and their difference areThe denominators $c\mp \Omega R$ are closing rates in these laboratory equations; they are not local speed of light measurements by a rotating observer. The two arrival spacetime events lie on the same detector world line. Its constant speed is $|\Omega |R$, so proper time along that world line givesThus laboratory coordinate-time and rotating-detector proper time delays agree only to leading order at small $|\Omega |R/c$. The radius and circumference here are laboratory quantities, not lengths obtained by laying comoving rods around the rim. Finite mirror polygons, transparent media, and optical dispersion require their own path and propagation assumptions before using an exact formula.

An exact dimensionless check takes $\Omega R/c=1/5$. Relative to $L/c$, the laboratory arrival times are $5/4$ and $5/6$, giving $\Delta t/(L/c)=5/12$. The detector instead records $\Delta \tau /(L/c)=\sqrt{6}/6\simeq 0.408248$. The leading Sagnac area law would give $\Delta t/(L/c)\simeq 2/5$. These are model values chosen to make the relativistic distinction visible, not attainable parameters of the historical apparatus.

The rotating loop does not constitute a universal measurement of the one-way speed of light. It compares completed circuits at a single readout; assigning a velocity to each remote leg additionally requires a clock synchronization convention. Local Einstein synchronisation around a rotating rim does not close into a single globally synchronized clock network: the accumulated mismatch is another expression of the rotating-path asymmetry. Local comoving vacuum measurements still give $c$ in both directions. Special relativity predicts a nonzero closed-loop delay because the receiver moves during propagation, while no single comoving inertial frame covers the whole circuit.

For a rigidly rotating prescribed vacuum path, use spatial coordinates attached to the apparatus and laboratory time $t$. At a path point $\mathbf{r}$, the local apparatus velocity is $\mathbf{v}=\mathbf{\Omega }\times \mathbf{r}$. A light displacement has instantaneous laboratory displacement $d\mathbf{r}+\mathbf{v}dt$, up to the common rotation of the spatial axes. The null condition is thereforeWrite $d\ell =|d\mathbf{r}|$. To first order in $|\mathbf{v}|/c$, the positive root isReversing the path reverses the dot product term but leaves the leading path-length term unchanged. Subtract the two return times and use the scalar triple product identity:This is a line integral version of the Sagnac area law; the vector area identity also follows from Stokes theorem. For a mirror-defined polygon the actual ray paths move slightly as the mirrors rotate; the stationary optical path at zero rotation gives the same first-order result. The prescribed-path calculation is an idealization, not an exact finite-speed formula for every moving-mirror layout.

The dot product explains the geometry: only the projection of the oriented vector area along the angular velocity contributes. A planar loop whose normal vector is perpendicular to $\mathbf{\Omega }$ has zero leading delay. For repeated traversals with the same orientation, the signed vector areas add; oppositely oriented lobes can cancel even when the total unsigned area is large. Equal perimeter alone does not imply equal response. A constant translational velocity contributes nothing to this leading closed-path line integral, since ${\oint }_{C}d\mathbf{r}=0$.

A Sagnac interferometer uses a beam splitter to send light with optical coherence around a common closed optical path in opposite directions, then recombines the beams to read their phase difference. Mirrors or a guiding path define the loop; the source, beam splitter, loop and readout rotate together in the basic experiment. A small relative beam tilt makes spatial interference fringes whose displacement reveals a change of phase difference. At one fixed output, a photodetector can instead record the corresponding change in light intensity.

In 1913 Georges Sagnac mounted a lamp, collimator, beam splitter, four mirrors and photographic plate on a horizontal rotating platform. The mirrors defined a closed polygon; the two beams traversed it in opposite senses. A slight beam splitter adjustment produced visible interference fringes. A motor brought the platform to steady rotation, and switching the lamp exposed the plate. Comparing opposite senses of rotation revealed a reversing displacement. Changing the beam splitter adjustment supplied a further check against optical deformation. The report compared measured and calculated displacements. Georges Sagnac interpreted this as evidence for a luminiferous aether; the relativistic derivation below reproduces the delay without that hypothesis.^[ref]

The readout measures a relative wave phase, not two stopwatch readings separated by an optical transit time. Reversal isolates a contribution odd in angular velocity: if a signed interference fringe position is $q(\Omega )=q_0+k\Omega$ to leading order, then $[q(\Omega )-q(-\Omega )]/2=k\Omega$ cancels the fixed offset $q_0$. Changes of alignment or temperature correlated with reversal can survive this subtraction. Thus the experimental inference concerns a reproducible rotating-loop phase difference, with controls for disturbances; it is not by itself a unique test of an underlying propagation hypothesis.

A telescope collects electromagnetic radiation from distant objects and forms an image or supplies a signal to a detector. An optical telescope can use lenses or mirrors; instruments for other wavelengths can require different collecting structures. Collection increases the detectable signal, while the aperture and wavelength constrain the angular resolution.

For an ideal unobscured circular aperture of diameter $D$, the Rayleigh criterion gives an angular scale $\theta \simeq 1.22\lambda /D$. Increasing the magnification of an already blurred image cannot recover angular detail lost through diffraction, optical aberrations or atmospheric seeing. This makes a double star a useful observing test: a larger apparent image is different from two genuinely resolved intensity peaks.^[ref]