Optical instrument

An optical instrument uses light to form an image, measure a property or control a beam. An optical telescope collects and redirects light from distant objects; a spectrometer separates wavelengths; a refractometer infers refractive index from a measured direction or boundary.

Its usefulness depends on more than enlargement or brightness. An image requires controlled optical aberrations and adequate optical resolution, while a measurement requires a known relation between its readout and the desired quantity. Blocking the input beam, measuring a reference sample and changing one optical setting at a time help distinguish specimen signals from stray light and photodetector offsets.

An optical receiver converts an incoming optical signal into an electrical signal that can be measured or processed. A typical direct-detection receiver combines a photodetector with readout electronics; optical filtering, electrical amplification and bandwidth limitation determine which input variations survive. An optical receiver can characterize a modulated beam as well as receive a communications signal.^[ref]

In a simple linear photodiode model, the signal current is $i(t)=\rho P(t)$, where $\rho$ is the spectral responsivity at the operating wavelength and $P$ the incident radiant power. A low-frequency transimpedance amplifier gives $v(t)=-R_{f}i(t)$ for its ideal inverting readout. For an original example, $\rho =0.8\mathrm{A}{\mathrm{W}}^{-1}$, mean power $20\mu \mathrm{W}$, modulation amplitude $4\mu \mathrm{W}$ and $R_f=10\mathrm{k\Omega }$ give mean photocurrent $16\mu \mathrm{A}$ and modulation amplitude $3.2\mu \mathrm{A}$. The corresponding output has mean $-0.16\mathrm{V}$ and modulation amplitude $0.032\mathrm{V}$, before offsets and frequency-response effects.

A motivating experiment measures the response to optical beating. Hale and colleagues combined two single-frequency lasers near $1319\mathrm{nm}$, with attenuators, polarization control and a beam splitter, into a single-mode fibre. Tuning their frequency difference supplied a tunable optical intensity modulation whose electrical response could be measured.^[ref]

For perfectly overlapping fields of equal polarization, the ideal incident power isA frequency difference of $100\mathrm{MHz}$ therefore produces a $10\mathrm{ns}$ intensity period; it is not the optical carrier period. Sweeping this difference probes modulation response. Measurement calibration must account for optical overlap, receiver gain, electrical losses and impedance mismatch. Spectral sensitivity at an optical wavelength and electrical modulation bandwidth are different specifications; a large direct-current output alone establishes neither low noise nor a fast receiver.

A reticle is a reference pattern positioned in an optical instrument's viewing or imaging path. Crosshairs can mark a reference position; repeated lines or dots can provide a scale or comparison pattern. The pattern may be viewed along with an image or illuminated and projected by the instrument.^[ref]

In an autocollimator, an illuminated reticle is placed so the collimator projects its image at infinity. An external reflector returns the light, and the returned pattern's position is compared with a reference in the image plane. The reticle supplies the recognizable pattern; the optical geometry connects its displacement with angle.^[ref]

For an ideal focal-plane example, suppose adjacent marks are $p=10\mu \mathrm{m}$ apart behind optics with focal length $f=100\mathrm{mm}$. Their small angular separation is $p/f=10^{-4}\mathrm{rad}$, about20.6 arcseconds. In a double-pass mirror-angle measurement, the corresponding mirror tilt is half the return-beam angular change. The numbers are chosen to illustrate the distinction between pattern spacing and the physical quantity ultimately measured.

A reticle is therefore not automatically an absolute measuring scale. Scale interpretation depends on its location, the effective focal length or image magnification, and measurement calibration. Crosshair coincidence alone establishes a position comparison relative to a chosen reference.

An autocollimator is an optical instrument that measures small angular changes by sending collimated light toward a reflector and examining the returning image. The outgoing and returning paths share optical components: a beam splitter directs light from an illuminated reticle through a collimator, while the reflected light is imaged onto a detector or viewed through an eyepiece. The measurement uses autocollimation, the return of the instrument's own projected light.^[ref][ref]

A small tilt $\theta$ of a plane mirror changes the reflected beam direction by $2\theta$. In an ideal paraxial model with focal length $f$, the return-image displacement from its zero position iswith the angle in radians. For the independently chosen values $f=200\mathrm{mm}$ and $x=4\mu \mathrm{m}$, the mirror tilt is $10\mu \mathrm{rad}$, about 2.063 arcseconds. The reflected beam turns through twice that angle: confusing mirror tilt with beam deflection introduces a factor-of-two error.

A mechanical application turns angular readings into height differences. A mirror carried on two supports separated by a base $b$ has height difference $h=b\tan \theta$ across that base. With an illustrative $b=100\mathrm{mm}$ and the tilt above, $h$ is about 1 micrometre. A series of such local slopes can constrain straightness, provided the support geometry and reference are accounted for.^[ref]

The instrument therefore senses orientation through image position. It is not directly a distance gauge: lateral translation of an ideal plane mirror without tilt does not encode that translation in the same angular reading. Zero setting, measurement calibration and optical alignment remain part of interpreting the observation.

Autocollimation is an optical return-path method: a source pattern is projected through a collimator, reflected back from an external surface, and examined through the same optical system. It is the process used by an autocollimator, rather than the instrument itself. A plane mirror normal to the outgoing collimated light returns the light along the reference direction.^[ref]

A typical apparatus uses an illuminated reticle near a focal plane, an objective and a beam splitter that separates the illumination and viewing paths. The outward projection and the return image are compared, making small changes in reflector orientation observable.^[ref]

For a small mirror tilt $\theta$, reflection changes the return direction by $2\theta$. In the ideal paraxial model, return-image displacement is $x\approx 2f\theta$, where $f$ is the effective focal length. An original example with $f=150\mathrm{mm}$ and $\theta =20\mu \mathrm{rad}$ gives $x\approx 6\mu \mathrm{m}$. This is displacement in the image plane, not a direct measurement of mirror translation.

The angular interpretation explains why simply moving an ideal plane mirror farther away does not multiply the focal-plane displacement by that distance. In practice, finite aperture, beam walk, alignment and diffraction limit the usable return. Measurement calibration and a stable reference remain necessary to distinguish a genuine angular change from movement of the optical system.

An instrument with a polarizer and an optical analyzer, often with a specimen between them, used to reveal changes of polarization through transmitted intensity. Crossed ideal elements give extinction without a polarization-changing sample.

A spectroscope makes an optical spectrum observable, commonly by passing light from an entrance slit through a collimator and an optical prism or diffraction grating. An eyepiece permits direct inspection of separated spectral lines or bands, while a recording detector makes the instrument a spectrograph.

An emission lamp gives bright spectral lines; placing an absorbing sample in front of a continuous source can produce an absorption spectrum. Slit width affects both received radiant power and the ability to distinguish neighboring features. A visible dark band must be checked against the source spectrum and instrument response before being assigned to sample absorption.

An instrument measuring a radiometric quantity as a function of wavelength under a stated collection geometry and calibration. Measurement at the relevant viewpoint helps connect scene or display output to observer input.

An optical element that adds a controlled wave phase to one path. On two path states, an ideal lossless phase shifter has the displayed unitary matrix. It changes their relative phase while leaving individual path probabilities unchanged; subsequent interference can turn that phase into an output-intensity change.

A device uses an applied electric field to change an optical medium's refractive index and hence the phase, amplitude or polarization of light. A phase change followed by suitable polarization analysis can provide an intensity transmission gate for active mode locking.

An electro-optic element whose birefringent retardance changes approximately linearly with applied voltage. Combined with a polarizer, it converts voltage-controlled polarization changes into variable optical loss for active Q-switching.

A pentaprism redirects a beam through a right angle using two reflecting faces whose planes meet at $45^{\circ }$. In the meridional section their combined action is a $90^{\circ }$ rotation, preserving the relevant image-top direction as the beam turns. It does not by itself perform the additional lateral reversal of a roof pentaprism; the latter has an extra reflecting face.

The Hamilton echelle spectrograph is an astronomical cross-dispersed echelle spectrograph at Lick Observatory. An echelle grating provides high spectral dispersion and a prism separates overlapping diffraction orders in the perpendicular direction. The observatory lists feeds from the Shane and Coudé Auxiliary telescopes.^[ref]

The resulting charge-coupled device image contains several strips of optical spectrum, rather than one continuous line. A wavelength reference must locate the spectral lines in these strips before small line displacements can be interpreted as a Doppler shift.

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.

A transverse optical cavity mode is a resonant field pattern across the direction of propagation in an optical cavity. After propagation and reflection through a round trip, the mode reproduces its transverse shape up to a complex amplitude factor. Longitudinal order counts axial phase cycles; transverse order describes the pattern across the beam. Different transverse patterns can resonate at the same frequency, as in confocal transverse-mode degeneracy.

In an ideal paraxial cavity with suitable spherical mirrors, a Cartesian Hermite-Gaussian optical mode at its waist has field proportional towhere $H_m,H_n$ are Hermite polynomials, $m,n$ are nonnegative integers, and $w_0$ is the beam waist radius. The intensity is proportional to $|u_{mn}{|}^2$, not to the signed field. The primary photonics notes give these transverse solutions, their phase advance and lower-order patterns.^[ref]

For $(m,n)=(0,0)$, the intensity is a single Gaussian beam spot. For $(1,0)$, $H_1(s)=2s$, so intensity is proportional to $x^2{e}^{-2(x^2+y^2)/w_0^2}$. It vanishes on $x=0$ and its two maxima occur at $x=\pm w_0/\sqrt{2}$, $y=0$. The two lobes have opposite field signs even though both have positive intensity. The $(1,1)$ pattern has two crossing nodal lines and four intensity lobes.

For an ideal empty two-mirror cavity of length $L$, set $g_i=1-L/R_i$. On the near-planar stable branch with $g_1,g_2>0$, the resonance frequencies areThe transverse contribution is a Gouy phase correction to the round-trip phase. Other branches require a consistent phase choice; the square root alone should not erase that choice. For an original example, $L=0.10\mathrm{m}$ and $R_1=R_2=0.20\mathrm{m}$ give $g_1=g_2=1/2$ and a one-order transverse spacing $c/(6L)\approx 499.65\mathrm{MHz}$ at fixed $q$. Modes with the same $m+n$ are degenerate in this ideal symmetric model.

A proposed observation scans cavity length while recording both transmission peaks and a camera image of the transmitted beam. Translating or tilting the incident beam changes its overlap with different transverse modes; matching the incident wavefront and width favors the fundamental mode. A transmission peak alone cannot identify the spatial pattern, and an intensity image alone does not recover its phase. Finite apertures, astigmatism and mirror imperfections alter losses and split ideal degeneracies; state the cavity geometry and imaging plane when comparing observations.

A Hermite-Gaussian optical mode is a separable paraxial field pattern whose transverse dependence is a product of Hermite polynomials and a Gaussian beam envelope. In an ideal aligned optical resonator with suitable spherical mirrors, these patterns form a family of transverse optical cavity modes. The two nonnegative integer orders $m,n$ count the straight nodal lines along the chosen transverse axes. The primary photonics notes give the mode fields and their order-dependent phase advance.^[ref]

At a common waist plane, choose a unit-normalized scalar fieldHere $H_j$ uses the physicists' Hermite polynomial convention, with $H_0(s)=1$, $H_1(s)=2s$ and $H_2(s)=4s^2-2$. The normalization follows fromAfter the two coordinate substitutions, $\iint u_{mn}^{\ast }{u}_{pq}dxdy={\delta }_{mp}{\delta }_{nq}$. Thus coefficients in a common-plane expansion can be recovered by inner products; the modes are orthogonal field patterns, not independent bright spots.

For $(m,n)=(1,0)$, intensity is proportional to $x^2{e}^{-2(x^2+y^2)/w_0^2}$, so its maxima lie at $x=\pm w_0/\sqrt{2}$, $y=0$. At $w_0=100\mu \mathrm{m}$, the bright lobes are about $141.4\mu \mathrm{m}$ apart. The field changes sign across $x=0$, although an intensity camera records positive intensity on both sides. For $(2,0)$, the nodal lines instead lie at $x=\pm w_0/2$. These patterns are illustrated in transverse optical cavity mode using individually peak-normalized images; equal peak intensity is different from equal integrated power.

A second-moment width of a Hermite-Gaussian optical mode distinguishes these multi-lobed beams quantitatively. Applying the Hermite polynomial recurrence inside the normalized integrals givesFor order $(1,0)$, the RMS widths are $\sqrt{3}{w}_0/2$ horizontally and $w_0/2$ vertically, despite the same envelope parameter $w_0$. Reporting a fitted single-lobe radius without its definition can obscure this difference.

During propagation the beam radius and wavefront curvature evolve, and the Gouy phase is multiplied by $m+n+1$. Consequently, transverse order contributes to the resonator's frequency condition and to confocal transverse-mode degeneracy. A proposed experiment scans the resonator while imaging a specified plane, then interferes its output with a reference field to test the phase reversal across a node. An intensity fit alone does not establish a pure coherent mode: mixtures, astigmatism, finite apertures and degeneracy can alter what is observed.

A symmetric confocal optical resonator has two concave mirrors of equal curvature $R$, separated by $R$. Its symmetric fundamental Gaussian beam has a central beam waist and Rayleigh range $R/2$. Its ideal transverse spectrum has degeneracies, so transmission at one resonance need not certify exclusive excitation of the fundamental normal mode.

Each mirror’s paraxial focal length is $R/2$, so their focal points coincide halfway between them. It is used for optical feedback and scanned spectral analysis; the manufacturer’s apparatus manual identifies the equal-radius, equal-separation geometry explicitly.^[ref]

The usual symmetric fundamental Gaussian beam has a central beam waist. Its wavefront curvature a distance $z$ from the waist is $R_b(z)=z+z_R^2/z$, where $z_R$ is the Rayleigh range. Requiring $R_b(L/2)=L$ givesHere the cavity is empty, $\lambda$ is its vacuum wavelength, and $w$ is the standard Gaussian intensity radius. For an original example with $L=0.100\mathrm{m}$ and $\lambda =633\mathrm{nm}$, $z_R=50\mathrm{mm}$, $w_0\approx 100.4\mu \mathrm{m}$ and the mirror spot radius is about $142.0\mu \mathrm{m}$. Finite mirror apertures and imperfect wavefronts change the loss and usable mode structure.

A key feature is confocal transverse-mode degeneracy. The one-way Gouy phase change of this symmetric beam is $\pi /2$. For transverse orders $m,n\ge 0$ and longitudinal index $q$, the ideal paraxial resonance frequencies, with a common mirror-phase offset absorbed into the index convention, areThe primary MIT notes give this confocal resonance condition and the two interleaved transverse-order families.^[ref] One way to obtain it is to set the round-trip propagation phase $4\pi \nu L/c$ minus the round-trip transverse phase $(m+n+1)\pi$ equal to $2\pi q$.

Holding $m,n$ fixed gives the longitudinal cavity mode spacing $c/(2L)$. Increasing $m+n$ by two while reducing $q$ by one leaves the ideal frequency unchanged; orders of the other parity give a family halfway between. Thus an analyzer exciting both families can show interleaved resonances spaced by $c/(4L)$. This confocal spacing is not the longitudinal spacing of one fixed transverse mode. For $L=0.100\mathrm{m}$, the two values are approximately $1.499\mathrm{GHz}$ and $749.5\mathrm{MHz}$.

This distinction has a direct apparatus test. Send a narrow-band beam through mode-matching optics into two confocal mirrors, scan one mirror with a piezoelectric drive, and record transmitted light with a photodetector while monitoring the scan. Compare well-matched on-axis excitation with deliberately changed alignment or beam shape. Extra peaks can be cavity transverse modes rather than extra source frequencies. Luhs’s worked scan uses $100\mathrm{mm}$ mirror curvature, a confocal calibration near $750\mathrm{MHz}$, and an estimated finesse near fifty; its observed $15\mathrm{MHz}$ width bounds rather than independently resolves the narrower probe linewidth.^[ref]

A measured peak spacing therefore needs a stated mode-excitation pattern, and one transmission peak does not prove that only the fundamental spatial mode was excited. Departures from confocal geometry can split degeneracies; mirror loss, aberration and scan nonlinearity require separate characterization. The confocal geometry is a particular symmetric concave-mirror resonator, rather than a rule that applies to every pair of curved mirrors.

For two equal mirrors of radius $L$ separated by $L$, a Gaussian waist at the centre is reflected to another waist at the same plane. Putting $a=L/2$, the mirror maps $q=a+i{z}_R$ to $-a+i{a}^2/z_R$, so the reflected Rayleigh range is $a^2/z_R$. Both beams have equal spots on the mirrors although their waist radii differ. Equal waist sizes select $z_R=L/2$.

Confocal transverse-mode degeneracy occurs when distinct spatial modes of an ideal confocal optical cavity have the same resonance frequency. It follows from the rational Gouy phase advance of the equal-radius, equal-separation geometry, rather than from the modes having identical transverse field patterns.

For mirror separation $L=R$, the paraxial resonance frequencies areThe primary MIT notes derive the confocal phase shift and mode-frequency families.^[ref] Let total transverse order be $N=m+n$. Replacing $(q,N)$ by $(q-j,N+2j)$ leaves the frequency unchanged whenever the resulting indices describe permitted modes. Orders that are even integers therefore share the fundamental-order family, while orders that are odd integers form an interleaved family halfway between.

There is also degeneracy among patterns with the same total order. At $N=2$, the pairs $(m,n)=(2,0),(1,1),(0,2)$ have the same ideal frequency while their transverse distributions differ. A transverse optical cavity mode needs its spatial field as well as its resonance frequency to be identified.

An original frequency example uses $L=0.100\mathrm{m}$ and $c=299792458m/s$. Holding one transverse order fixed gives spacing $c/(2L)=1.49896229\mathrm{GHz}$. Relative to a chosen fundamental resonance, the mode with $N=2$ and longitudinal index one smaller has offset zero; the $N=1$ mode at the original longitudinal index has offset $749.481145\mathrm{MHz}$. Thus a spectrum containing both families can show half the fixed-order longitudinal spacing. It does not follow that every visible peak is a separate laser frequency.

The excitation strength depends on spatial coupling. In an ideal orthonormal cavity-mode basis, the incident field can be expanded with coefficients $a_j=\int u_j^{\ast }(x,y)E_{\mathrm{in}}(x,y)dxdy$, with consistent field normalization. Matching the fundamental field suppresses overlap with orthogonal modes, even if those modes share its frequency. Conversely, transmission at the common resonance can contain several patterns when they are excited; frequency selection alone cannot separate a degenerate subspace.

A practical test scans the cavity length while recording both transmitted power and spatial beam shape, then changes alignment or mode matching. Luhs’s primary apparatus discussion identifies unwanted extra peaks as transverse modes of the analyzer and distinguishes them from the probe’s own longitudinal spectrum.^[ref] A photodetector integrating the whole output can conceal spatial differences that a beam image reveals.

Mirror-spacing error, unequal curvatures, aberration and finite apertures perturb the ideal degeneracy. For a general one-way Gouy phase $\psi$, the corresponding phase model gives ${\nu }_{qN}=(c/(2L))[q+(N+1)\psi /\pi ]$. The nominally paired orders $N$ and $N+2$ then differ by $(c/(2L))(2\psi /\pi -1)$ when their longitudinal indices differ by one. This makes near-degenerate peaks a sensitive geometry probe, but attributing a measured splitting requires separating those perturbations from source structure and polarization effects.

Two identical concave mirrors of curvature radius R separated by L have half-round-trip ray matrix $A=\left(\begin{array}{cc}1& L\\ -2/R& 1-2L/R\end{array}\right)$. Its determinant is one and its trace is $2(1-L/R)$. The eigenvalues are a conjugate unit-modulus pair for the displayed interval. At the confocal point $L=R$, $A^2=-I$, so it is bounded despite the full-round-trip trace reaching minus two. Matching a Gaussian beam's mirror wavefront gives $z_R^2=(L/2)(R-L/2)$ and $\pi w_0^2=\lambda z_R$. The endpoints are degenerate limiting geometries, not finite confined Gaussian modes.

A normal mode whose longitudinal phase satisfies the resonator's round-trip condition. Holding the transverse mode fixed, an empty nondispersive two-mirror cavity of length L has adjacent optical frequencies separated by $c/(2L)$. An offset from mirror or transverse phases changes the position of the mode sequence without changing this ideal spacing. A dispersive cavity instead uses its round-trip group delay.

Optical mode matching makes the spatial field of an incident beam equal to a desired optical cavity normal mode. For fundamental circular Gaussian beams, matching the complex beam parameter fixes the beam waist size and location; the optical axis, polarization and frequency must also match. A single converging optical lens connects waist Rayleigh ranges $z_A,z_B$ only when $f^2\ge z_A{z}_B$. With $s=\sqrt{f^2/(z_A{z}_B)-1}$, waist-to-lens distances obey $u=f\pm z_{A}s$, $v=f\pm z_{B}s$ with correlated signs, followed by physical distance constraints.

The photon lifetime is the exponential decay time of stored energy in an unpumped optical cavity. If a round trip of duration $T$ retains an energy fraction $\rho$, then ${\tau }_c=-T/\log \rho$ and the quality factor is $Q=\omega {\tau }_c$. Field-amplitude decay takes twice this time. Output coupling contributes to the loss as well as absorption and scattering.

A Lorentz transformation of a resonant optical cavity transforms both the electromagnetic field and the mirror worldlines. Boundary conditions and event-by-event phase matching remain satisfied. For a longitudinal boost, a cavity of rest length $L$ has length $L/\gamma$ and unequal forward/backward frequencies $\gamma {\omega }_0(1\mp \beta )$. The coordinate round-trip time is $2\gamma L/c$, while the phase rate on a moving mirror is ${\omega }_0/\gamma$, so the round-trip phase is still $2{\omega }_{0}L/c$. A stationary-mirror resonance formula with only length contraction inserted describes the wrong boundary-value problem.

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

A parabolic reflector has a paraboloidal reflecting surface. The law of reflection makes all incident light rays parallel to its optical axis pass through one focus in ideal geometrical optics. This removes on-axis spherical aberration and explains its use as a receiving antenna for distant sources.

A concave spherical mirror reflects at the inside of a spherical surface of radius of curvature $R$. For incident light rays near its optical axis, the law of reflection gives paraxial focal length $f=R/2$. Rays farther from the optical axis have different axial crossing positions, causing spherical aberration; one must specify a criterion to define the best full-aperture focus.

A plane mirror has a flat reflecting surface. The law of reflection sends incident unit direction $\mathbf{d}$ to $\mathbf{d}-2(\mathbf{d}\cdot \mathbf{n})\mathbf{n}$, where $\mathbf{n}$ is its unit normal. It forms a virtual image of an object the same perpendicular distance behind the reflecting plane as the object is in front. Two successive plane mirrors whose planes meet at angle $\gamma$ rotate the ray geometry through $2\gamma$ about their intersection.

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

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

An interference filter uses wave interference in a layered optical structure to transmit or reject selected wavelengths. The accumulated optical path length makes reflected contributions reinforce or cancel differently across the optical spectrum.

A narrow-band example can isolate emission before a photodetector or reject an excitation beam in fluorescence microscopy. Tilting the filter changes the propagation angles and optical paths inside its layers, shifting the transmitted band. A filter characterized in a collimated beam may therefore behave differently in a strongly converging beam; temperature and polarization can also affect transmission.

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.

The focal ratio, or f-number, of a converging optical instrument focused at infinity is the dimensionless quantitywhere $f>0$ is effective focal length and $D$ is the on-axis entrance pupil diameter. The notation f/4 means $N=4$, not $f=4$ in a particular unit. A $100\mathrm{mm}$ focal length with $D=25\mathrm{mm}$ gives f/4. Closing the aperture stop until $D=12.5\mathrm{mm}$ gives f/8 while retaining the same focal length. The mechanical opening need not equal $D$. ^[ref].

For a thin lens in air with a circular aperture stop at the thin lens, the converging bundle from a distant on-axis point has half-angle $\alpha$ satisfyingThe last step uses the small-angle approximation. For a compound optical lens, the cone is set by the image-side optical pupil and its distance from the receiving plane; the simple triangle is not an exact high-angle rule. At finite object distance a thin lens instead gives $v=f(1+|m|)$, by its imaging equation and transverse magnification, so its working cone corresponds to $v/D=N(1+|m|)$. Nominal focal ratio therefore does not alone specify close-range image irradiance. ^[ref].

Two useful scalings follow under stated conditions. For a fixed uniform incident beam and unchanged transmission, collected radiant flux scales as $D^2$. For an extended source with fixed radiance, the central image irradiance scales approximately as $N^{-2}$ in the paraxial approximation: the solid angle of the cone scales as $(D/f{)}^2$. For an ideal uniformly illuminated unobscured circular entrance pupil, the first dark ring of the point spread function has angular radius and focal plane radiusHere $\lambda$ is wavelength. These are first-zero radii, not diameters or a universal rule for resolving arbitrary objects. A proof outline evaluates the circular aperture's Fraunhofer diffraction using its Fourier transform, then uses the first zero of the resulting Bessel function. ^[ref].

At $\lambda =0.55\mu \mathrm{m}$, the example's f/4 and f/8 first-ring radii are $2.684$ and $5.368\mu \mathrm{m}$. Closing the aperture stop halves $D$, quarters collected radiant flux and doubles this diffraction radius. It can also reduce optical aberrations by excluding outer light rays; actual sharpness reflects both effects. This explains why a smaller aperture need not produce a sharper image. ^[ref].

A proposed optical bench check puts a fixed collimator before a converging thin lens and a photodetector large enough to collect the entire image behind it. Compare two centred aperture stops of known diameters while fixing source output, detector gain and integration time. The integrated signal tests the $D^2$ collection law; a spatially resolved optical camera at the focal plane can separately measure the point spread function. Repeat a dark reading and the first aperture stop, and check that neither the incoming beam nor detector clips either bundle. A resolved pinhole source, imperfect collimation, optical aberrations and detector pixel size can dominate the apparent spot width; a single peak reading does not measure total collected radiant flux.

An optical camera images incident beams on a photodetector; a spectrograph optical camera maps a diffracted direction into a position in its focal plane.

An optical slit selects a narrow region of a focal plane. In a slit-limited spectrograph, the width of its image divided by the linear spectral dispersion sets the spectral width.

A focal plane is a plane in which an optical system forms images of distant directions. Aberrations can make the best-focus surface curved; a flat detector may then require a field-flattening optical element.

The plane one focal length beyond a lens. In a paraxial imaging system, incident parallel rays of a given direction focus at one point in this plane; it is therefore an angular or diffraction plane distinct from a finite-distance object's image plane.

High-contrast imaging seeks faint objects near a bright source. Residual point spread functions and speckle patterns can dominate photon shot noise; angular differential imaging and simultaneous spectral differential imaging use source motion or spectral diversity to distinguish them.

A companion or extended source contributes to the stellar reference used in differential imaging. Subtracting that reference removes part of the desired signal and distorts its shape, photometry or astrometry. Artificial-source injection and forward modelling can estimate the throughput loss.

Simultaneous spectral differential imaging compares images in nearby bands, rescaled so stellar speckle patterns approximately coincide. A companion has a different positional or spectral response. Simultaneity reduces temporal variation but does not remove chromatic and non-common-path errors.

Angular differential imaging keeps the instrument pupil fixed while the sky rotates. A stellar reference point spread function is subtracted, then the residuals are derotated and combined. Insufficient rotation and imperfect reference selection cause differential-imaging self-subtraction.

A Fabry–Pérot interferometer uses repeated specular reflections between two nearly parallel, partially transmitting surfaces. Its transmitted light is the sum of contributions that have made zero, one, two, or more additional round trips. Their complex amplitudes reinforce only near particular wave phases, producing narrow transmission peaks that select wavelength intervals. Scanning the spacing or angle of incidence can build an imaging spectral cube: an image at each selected wavelength.

The surprising feature is that highly reflecting mirrors can together transmit all the incident power at resonance in the ideal symmetric, lossless model. Each individual contribution is weak, but many contributions arrive in phase. Between resonances the same contributions cancel strongly. Increasing mirror reflectivity sharpens the peaks instead of simply suppressing every transmitted signal; finesse quantifies that selectivity. An optical spectrum with nearby spectral lines can therefore produce distinguishable peaks, provided their separation is large enough relative to the instrument's peak width and their orders are identified.

Charles Fabry and Alfred Perot developed the instrument together; Optica dates its development from 1896 to 1898.^[ref]

For identical mirrors with power reflectivity $\mathcal{R}$, transmission $\mathcal{T}$ and absorption $1-\mathcal{R}-\mathcal{T}$, summing round trips at normal incidence givesHere $\delta$ includes propagation and both internal reflection phases. Optical absorption reduces the resonant peak below unity, while a nonzero reflection phase shifts the resonance condition $\delta =2\pi m$. Frequency-dependent mirror phase and refractive index can also change the free spectral range.

Spectral resolution and order identification are separate design constraints. An instrumental width narrower than a line separation does not prevent a different interference order from placing another wavelength at the same angle. Choose optical thickness so the source range fits a known order interval, sufficient finesse to resolve neighbouring lines, and an angular window or order-sorting filter to exclude unwanted orders. Material dispersion and the internal/external angle conversion must be calibrated.

For an air gap of thickness $d$, an extra internal round trip has length $2d/\cos \theta$, but consecutive transmitted rays emerge at different positions. Comparing them at the same outgoing wavefront subtracts $2d\tan \theta \sin \theta$, leaving optical path difference $2d\cos \theta$. Reflection-coating wave phases are separate from this geometric path calculation.

Consider two identical, lossless parallel mirrors with intensity reflectivity $0\le R<1$, separation $d$, and a uniform transparent medium of refractive index $n$. Fix one polarization of an electromagnetic wave and an internal angle of incidence $\theta$ measured from the mirror normal vector. Illuminate with a monochromatic wave having planar wavefronts; the emerging contributions must overlap coherently in the detected direction. Define $T=I_{\mathrm{transmitted}}/I_{\mathrm{incident}}$ using the same incident and exit media. Each mirror transmits a fraction $1-R$ of incident power. Its reflected wave amplitude has magnitude $\sqrt{R}$, so two internal specular reflections multiply the circulating complex amplitude by magnitude $R$, not $R^2$.

Let $\lambda$ be the vacuum wavelength and choose the reflection wave phase convention in which the product of the two internal amplitude reflection coefficients is real and positive. The phase difference between successive transmitted contributions, evaluated on a common outgoing wavefront, isWith other mirror coatings or phase conventions, use the total $\delta =4\pi nd\cos \theta /\lambda +{\varphi }_{\mathrm{rt}}$, where ${\varphi }_{\mathrm{rt}}$ is the sum of the two internal reflection wave phases. The stated resonance positions presume ${\varphi }_{\mathrm{rt}}=0$ modulo $2\pi$; a constant nonzero ${\varphi }_{\mathrm{rt}}$ shifts them, while a frequency-dependent one can also change their spacing.

Why does the cosine appear rather than its reciprocal? Inside the gap, the extra two legs have optical path length $2nd/\cos \theta$. But the later emerging contribution starts $2d\tan \theta$ farther along the mirror. Comparing both on one outgoing wavefront subtracts $2nd\tan \theta \sin \theta$ from that path difference, leaving $2nd\cos \theta$. Equivalently, a plane wave's mirror-normal wavevector component is $k_z=(2\pi n/\lambda )\cos \theta$, so the round-trip wave phase is $2k_{z}d$. The diagram uses the same medium outside and inside to show the common-wavefront subtraction directly; at a refracting exit interface, tangential wavevector matching in Snell's law supplies the same result.

Absorb the common first-pass wave phase into the incident complex amplitude $E_{\mathrm{in}}$. Successive transmitted contributions are $(1-R)E_{\mathrm{in}}$, $(1-R)E_{\mathrm{in}}R{e}^{i\delta }$, and so on. The superposition principle and the convergent geometric series giveThe last step expands the squared complex modulus as $1+R^2-2R\cos \delta$. This transmission profile is often called the Airy transmission function; it is distinct from the Airy disk, a diffraction image of a circular aperture. The University of Washington's lab handout gives the same transmission profile and distinguishes mirror reflectivity from other sources of peak broadening.^[ref]

For an original example, set $R=3/4$. At $\delta =0$, the amplitudes add to $(1/4)/(1-3/4)=1$, so $T=1$. At $\delta =\pi$, they alternate in sign and add to $(1/4)/(1+3/4)=1/7$, so $T=1/49$. At $\delta =\pi /2$, the amplitude is $4/25+3i/25$ and $T=1/25$. Adding the individual intensities instead would give $(1-R{)}^2{\sum }_j{R}^{2j}=(1-R)/(1+R)=1/7$, independent of wave phase; that discards the wave interference responsible for the peaks.

In general the maximum is $T_{\max }=1$ at $\delta =2\pi m$, with integer $m$, and the minimum is $T_{\min }=[(1-R)/(1+R){]}^2$ halfway between peaks. Perfect resonant transmission requires identical lossless mirrors, coherent illumination, and sufficient time to establish the steady field. Absorption, unequal mirror transmission, finite beams that fail to overlap, and fluctuations of $d$ can reduce it. The exactly reflecting case $R=1$ is excluded: no incident power enters through a perfectly reflecting mirror, and the resonant limit $R\to 1^{-}$ is singular.

At normal incidence, with no optical dispersion and zero round-trip reflection wave phase modulo $2\pi$, the resonance condition in Fabry–Pérot transmission from multiple reflections isHere $\nu$ is frequency and $c$ is the speed of light in vacuum. A change of $\nu$ by one free spectral range advances the round-trip wave phase by $2\pi$. At fixed internal angle of incidence $\theta$, the same nondispersive model gives $c/(2nd\cos \theta )$. Holding the external angle of incidence fixed in a dispersive medium need not hold the internal $\theta$ fixed, because of Snell's law.

For stipulated $n=1$ and $d=10\mathrm{mm}$, the free spectral range is $14.9896229\mathrm{GHz}$ using $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$. A frequency difference of $0.10\Delta {\nu }_{\mathrm{FSR}}$ is therefore $1.49896229\mathrm{GHz}$. Transmission repeats every free spectral range: one isolated scan cannot distinguish that separation from $1.10\Delta {\nu }_{\mathrm{FSR}}$. Independent knowledge of the optical spectrum, an additional filter, or a changed mirror separation is needed to assign orders.

Equal spacing in frequency does not imply equal spacing in wavelength. At normal incidence the exact adjacent resonance wavelengths obeyFor example, choosing $2nd=10{\lambda }_{10}$ makes the exact gap to order 11 equal to ${\lambda }_{10}/11$, while the high-order estimate gives ${\lambda }_{10}/10$. The approximation is already ten percent larger at order 10. With optical dispersion or frequency-dependent reflection wave phase, obtain local spacings from the total phase difference instead: $\Delta \nu \simeq 2\pi /(d\delta /d\nu )$ when its derivative varies little over one order.

The finesse of a Fabry–Pérot interferometer is its free spectral range divided by a transmission peak's full width at half maximum. Large finesse means a narrow accepted frequency interval compared with the spacing between repeated orders. Here half maximum means $T=1/2$ of the ideal unit peak, with zero detector background subtracted; it does not mean halfway between the transmission minimum and maximum.

Write $A=4R/(1-R{)}^2$ for the coefficient multiplying ${\sin }^2(\delta /2)$ in Fabry–Pérot transmission from multiple reflections. This coefficient is sometimes called the coefficient of finesse; it is not the finesse $\mathcal{F}$. About a resonance $\delta =2\pi m$, put $ϵ=\delta -2\pi m$. Solving $T=1/2$ givesand, since one free spectral range advances the wave phase by $2\pi$,These are exact for the ideal nondispersive model with constant $R$ and linear frequency dependence of $\delta$. The inverse sine requires $(1-R)/(2\sqrt{R})\le 1$, or $R\ge 3-2\sqrt{2}\simeq 0.171573$. At equality, neighbouring peaks merely touch half maximum at their shared minima: the nominal width is one full free spectral range and $\mathcal{F}=1$. Isolated half-maximum peaks require the strict inequality. Below this threshold the transmission never falls to $1/2$, so this peak-width definition supplies no finesse; at $R=0$ the transmission is flat.

For large reflectivity, the small-angle approximation $\arcsin x\simeq x$ givesThis is a high-reflectivity approximation, not an identity for every $R$. The original dimensionless examples below have exact full width at half maximum values obtained from the inverse sine formula.

The beauty of the narrowing is visible directly in the geometric series: large $R$ preserves contributions from many round trips, so a small phase difference accumulates across many terms and spoils their addition. For the stipulated $10\mathrm{mm}$ vacuum gap, $R=0.95$ gives an ideal full width at half maximum of about $244.792\mathrm{MHz}$. This instrumental width is not a measured spectral line width. A real scan also depends on the source profile, accepted angles, mirror shape, alignment, and scan stability. The University of Washington handout separately discusses mirror-reflectivity, mirror-flatness, and detector-pinhole contributions; increasing $R$ alone cannot remove the others.^[ref]

For an ideal air-gap Fabry–Pérot etalon at normal incidence, the free spectral range is ${\nu }_{\mathrm{FSR}}=c/(2d)$ and the peak full width at half maximum is $\delta \nu ={\nu }_{\mathrm{FSR}}/\mathcal{F}$. To separate a pair of lines spaced by $\Delta \nu$ by at least $M$ peak widths, require $d\mathcal{F}\ge Mc/(2\Delta \nu )$. Putting both lines inside one unambiguous free spectral range additionally requires $d<c/(2\Delta \nu )$, so $\mathcal{F}>M$. In terms of their vacuum wavelengths, $c/\Delta \nu ={\lambda }_1{\lambda }_2/|{\lambda }_2-{\lambda }_1|$.

The strict order condition needs a margin if tails from adjacent periodic peaks are to be negligible. If every neighbouring peak in a scanned two-line periodic spectrum must be separated by $M$ widths, the stronger condition is $\min (\Delta \nu ,{\nu }_{\mathrm{FSR}}-\Delta \nu )\ge M{\nu }_{\mathrm{FSR}}/\mathcal{F}$. Ordinary resolving power alone does not prevent ambiguous repeated orders. A fixed etalon's transmission as a function of wavelength is a periodic-in-frequency filter; it is not automatically a plot of the incident source's spectral lines.

A Fabry–Pérot etalon fixes mirror separation in a Fabry–Pérot interferometer, selecting wavelengths through its transmission resonances.^[ref]