A plate bearing a photosensitive coating that records an exposure, including exposure to X-rays.

An optical isolator transmits light preferentially in one direction. An ideal Faraday effect device combines a $45^{\circ }$ rotator with polarizers at $0^{\circ }$ and $45^{\circ }$: forward light passes, while backward light acquires another $45^{\circ }$ rotation and reaches the entrance polarizer at $90^{\circ }$. See Faraday rotation on a round trip.

Stray light reaches a photodetector by an unintended path or from outside the intended wavelength band. It can bias spectrophotometry by adding received radiant power.

An optical chopper periodically interrupts a beam of light, for example with a rotating slotted wheel. Its timing can provide a reference for a lock-in amplifier measuring the modulated photodiode signal.

A refractometer measures refractive index through the way a sample redirects or reflects light. Some instruments measure the deviation through an optical prism; others locate the critical angle for total internal reflection at an interface with a prism of known refractive index.

In the latter geometry, for prism index $n_p$ greater than sample index $n_s$, the critical-angle relation is $\sin {\theta }_c=n_s/n_p$. Locating the transition between transmitted and totally reflected directions therefore determines $n_s$. Reference liquids check the angular calibration, while controlled temperature and wavelength make comparisons reproducible. Inferring composition from refractive index additionally requires a sample-specific calibration.

A goniometer measures angles by comparing the orientations of a sample, beam or viewing arm. An optical implementation can rotate a specimen and detector about a common center, measuring the directions of reflected, refracted or diffracted light.

For a prism experiment, a collimated input and a viewing telescope determine the angle through which the beam is deviated as the prism rotates. For a reflection experiment, the incident and viewing directions define the scattering geometry. Accurate centering matters: an off-center sample can move out of the measured beam as the arm rotates, changing signal strength for reasons unrelated to angular response.

An optical filter selectively transmits light according to properties such as wavelength or polarization of waves.

A camera lucida superposes the apparent image of an object on a drawing surface, allowing microscope positions to be marked by hand.

A grism combines an optical prism with a diffraction grating. Their angular deviations can be chosen to cancel at a selected wavelength, while neighboring wavelengths emerge in different directions. This lets an instrument introduce spectral dispersion with approximately straight-through propagation at its design wavelength.

Inserting the grism into an imaging beam turns compact object images into dispersed optical spectra. The cancellation is wavelength-specific and order-specific: overlapping spectra and other diffraction orders remain potential sources of confusion. An optical filter can restrict the recorded band.

A mirror redirects light according to the law of reflection. Its orientation determines the reflected beam direction in an interferometer.

An optical cavity supports optical fields whose phases and spatial profiles reproduce themselves after a round trip through its reflecting or guiding elements. Its resonance conditions include propagation and reflection phases. A cavity bounded by two plane mirrors is a simple example; Fabry–Pérot resonance spacing describes the frequency intervals of an ideal two-mirror arrangement.

Circular dichroism is unequal absorption of opposite circular polarizations. It can produce elliptical polarization; it is the absorptive response related to optical rotation, rather than the same measured quantity.

Matrix mechanics represents an observable by an array of matrix elements between chosen quantum states. In an orthonormal basis of energy eigenstates, each a stationary state, the diagonal entries of the Hamiltonian operator are the energy levels; off-diagonal entries of other observables describe couplings between those states. The same array can therefore answer two different questions: which values can a measurement return, and which pairs of energy eigenstates can a specified interaction connect? The eigenvalues answer the first question; the relevant matrix elements, together with the energy differences, answer the second.

This makes linear algebra a practical language for atomic physics. One can calculate energy levels, electric-dipole selection rules and probability amplitudes for transitions without assigning an electron an unobserved classical orbit. The three-state example below separates a missing optical coupling from a missing energy level; the finite oscillator commutator boundary defect explains why a finite array requires care when approximating a quantum harmonic oscillator.

One inspection of an array slot during an open addressing operation. Its array index follows the chosen probing rule; its contents determine whether to continue, return a matching hash table record, or establish absence.

A deleted-slot marker in open addressing. Lookup continues past it because a hash collision may have displaced a match further along. Linear probing can reuse the slot for insertion after checking for an existing equal key. A genuinely empty slot can terminate lookup; this marker cannot.^[ref]

Delete hash table key $8$ from the original eight-slot example. Writing E into array index $1$ makes lookup of $16$ inspect $0,1$ and incorrectly report absence, although its hash table record remains at $2$. Writing D instead makes lookup pass $1$ and reach $2$, returning $C$ after three hash table probes. The deleted slot must remember that the search route once continued through it.

A short algorithm correctness proof uses this invariant: no never-used slot precedes a stored hash table key along its linear probing route. Initially the claim is vacuous. Insertion places a new hash table record no later than the first E, possibly at an earlier D, and cannot create an E before another stored hash table record. Replacing a value leaves all slots occupied as before. Deletion to D also creates no E. By mathematical induction over operations, the invariant persists, so stopping lookup at E is sound. Deletion directly to E invalidates that argument, as the diagram shows.

To implement the hash table map operation put(k,v) in one bounded scan, remember the first D encountered, but keep checking occupied hash table keys. A later key equality match means replace that hash table record's value and stop. Only after reaching E, or completing $m$ hash table probes, is absence established. Then insert at the remembered D, if any; otherwise use the stopping E. If all $m$ slots are occupied and no match exists, report full or rebuild before trying again. Merely finding D is not permission to create a second equal hash table key.^[ref]

Continue from the correctly deleted example, with hash table keys $0,16,9$ at array indices $0,2,3$ and D at $1$. put(16,Z) examines $0,1,2$ and updates the value at $2$, leaving $1$ deleted and retaining three live hash table records. Inserting immediately at $1$ would instead create two copies of $16$. Now put(24,W) examines $0,1,2,3,4$, proves absence at E, and places the new hash table record at the remembered array index $1$. The hash table load factor rises from $3/8$ to $4/8$; reusing D avoids extending the linear probing cluster to $4$.

For a boundary case, fill all eight slots with hash table keys $0,1,\dots ,7$ and then delete all eight to D. The live hash table load factor is zero, but absent lookup of $16$ still needs eight hash table probes. put(16,C) also checks eight slots before reusing the first D, at array index $0$. A loop that waits indefinitely for E would never finish in this state. Rebuilding from the live hash table records into a fresh array removes each tombstone in open addressing and restores never-used stopping slots.^[ref]

An open addressing probe rule that visits $h_m(k),h_m(k)+1,h_m(k)+2,\dots$ modulo $m$, using modular arithmetic, until a matching record or an appropriate stopping slot is found.^[ref]

For capacity $m>0$, the hash table probe numbered $i$, starting at zero, inspects the array indexThe hash function $h_m$ gives the first array index. Each subsequent hash table probe moves one slot forward, wrapping from $m-1$ to $0$ by modular arithmetic. Keep the hash function and key equality fixed while hash table records are stored.

Use three distinguishable slot states: occupied by a live hash table record; never used since initialization or rebuilding, written E; or deleted, written D, a tombstone in open addressing. Lookup compares occupied hash table keys using key equality, stops at E, and continues past D.^[ref]

For an original example, take an eight-slot array, integer hash table keys, and the hash function $h_8(k)=k\mathrm{mod}8$. Start with all slots E and insert the hash table records $(0,A),(8,B),(16,C),(9,V)$ in that order. The insertion hash table probe traces, expressed as array indices, areThe resulting array holds hash table keys $0,8,16,9$ at array indices $0,1,2,3$, with E in the other four slots. The first three hash table keys have the same initial array index, a hash collision; the fourth begins at $1$ but joins the same linear probing cluster. Lookup of $16$ examines array indices $0,1,2$, rejecting $0,8$ before returning $C$. Lookup of absent $24$ examines $0,1,2,3,4$, including the stopping E: five hash table probes for a hash table load factor $\alpha =4/8=1/2$.

Bound lookup to at most $m$ hash table probes, even if no E remains. For algorithm termination, the first $m$ array indices are distinct: equality of $p_i(k)$ and $p_j(k)$ implies that $m$ divides $i-j$ by modular arithmetic, and $0\le i,j<m$ forces $i=j$. They therefore cover the entire finite set of slots exactly once. If no occupied hash table key matches and no E is encountered, exhausting these $m$ hash table probes establishes absence; wrapping around again supplies no new information.^[ref]

The attraction is a compact array and a simple hash table probe rule. Consecutive slots can benefit from spatial locality, although actual memory cache behavior depends on representation and hardware. Correctness rests on preserving the search route; speed depends on how long that route becomes. The linear probing cluster below explains why even a moderate hash table load factor gives no universal constant lookup bound.

The linear operator $V_T(r)S_{12}$ couples nucleon spin orientations to their separation direction $\hat{\mathbf{r}}$. Here ${\mathbf{\sigma }}_i$ are Pauli matrices. A scalar radial potential energy alone omits this directional dependence.

Radio astronomy studies celestial sources through their radio waves. The long wavelengths make angular resolution expensive: a filled telescope of diameter $D$ has a characteristic diffraction scale of order $\lambda /D$. Aperture synthesis instead combines measurements from separated antennas, using their separations to measure fine structure while their individual collecting areas determine how much radiation they receive. A kilometre-wide arrangement of small antennas does not collect the power intercepted by a filled kilometre-wide dish.