Optics

Optics studies light, its propagation, and instruments for forming images and measuring optical spectra.

The central question is how light carries information from a source to an observer. A useful description must explain both where the light goes and how its distribution changes. In geometrical optics, light rays locate paths and images; in physical optics, an electromagnetic wave has a wave amplitude and a wave phase, so overlapping contributions can produce wave interference and diffraction. These descriptions address different levels of detail within the same subject. ^[ref].

An image is especially interesting because a single object point emits light in many directions. An optical lens redirects part of that spreading bundle toward a corresponding image point; repeating this construction over the object preserves spatial information. Simply gathering more light is insufficient: light rays from distinct object points must arrive at distinct locations. This is why an optical camera needs both an optical lens and a correctly positioned receiving surface. The thin lens example below makes that correspondence quantitative. ^[ref].

A specified spectral distribution of illumination. A real scene can contain several sources and reflected illumination, so one global illuminant spectrum is a simplification.

Transport in which light enters a material, scatters within it and exits, potentially at another surface location. A purely local surface-reflection model does not describe all such transport.

A photometric measure of directional light output weighted by a specified luminous-efficiency function, conventionally measured in candela per square metre. It is distinct from illuminance, which concerns light incident per unit area, and from perceived brightness. Chromatic-discrimination experiments must control luminance cues when those cues could reveal the answer.

In linear optics, the response is linear in the optical field and propagation or passive mixing maps mode annihilation operators to linear combinations of mode annihilation operators. Wave plates and beam splitters can rotate or mix photonic modes, but do not by themselves supply a deterministic interaction between independent photon-number qubits. Conditional measurements and auxiliary photons change that resource model.

Absorption that decreases with increasing illumination intensity as the absorber's population distribution changes. It can provide an intensity-dependent loss for passive Q-switching.

An element whose transmission increases as its absorption saturates. In a passive Q-switched laser it initially blocks oscillation, then becomes more transparent when the growing intracavity field bleaches the loss.

Distinct spectra can give the same colour response for a specified observer and illumination. In a simplified three-cone cell model, equal values of $\int L(\lambda )R(\lambda )S_i(\lambda )d\lambda$ for $i=1,2,3$ need not imply identical spectral reflectance functions $R$. Altering illumination or observer can break the match.

A stimulus belonging to a pair of physically different optical spectra that match for a specified observer and viewing condition. Such a match exhibits metamerism. Changing the observer, adaptation or illumination can break the match.

Investigation using a light-based optical microscope. Resolution, specimen preparation and interpretation constrain what can be established from an image.

Aligned two-dimensional sections constrain the three-dimensional structure of a specimen. For parallel sections of known spacing $\Delta z$, the Riemann sum ${\sum }_i{A}_i\Delta z$ approximates volume. Distortion, missing sections and alignment errors constrain the reconstruction; the volume estimate alone does not determine the identity or developmental fate of each visible structure.

Motion blur arises when a detector averages changing image positions during a finite exposure rather than recording an instantaneous position. For a stationary harmonic optical trap with position variance $k_{B}T/k$ and relaxation time $\tau$, rectangular exposure of duration $W$ reduces that variance by the factor $2[\alpha -1+e^{-\alpha }]/{\alpha }^2$, where $\alpha =W/\tau$. Integrating the exponential position covariance over both exposure times proves this formula. Ignoring the reduction makes a naive equipartition stiffness estimate too large.^[ref]

A designed spatial arrangement of dielectric or other electromagnetic material controls optical propagation, confinement or emission. Examples include waveguides, distributed Bragg reflectors, photonic crystals and defect cavities. Spatial interference and mode boundary conditions act on electromagnetic states; they are distinct from the electronic band structure of the material itself.

A structure with spatially periodic dielectric response, giving optical bands and, under suitable conditions, forbidden frequency intervals.

The frequencies ${\omega }_n(\mathbf{k})$ of electromagnetic Bloch modes form allowed bands in a periodic dielectric. Their gaps and curvature determine forbidden propagation, confinement and group velocity.

A frequency interval without bulk propagating electromagnetic modes in the specified directions and polarizations. A complete gap excludes all directions and polarizations; a one-dimensional stop band need not do so.

A local departure from periodicity can support a discrete mode inside a bulk photonic band gap. An isolated defect makes a cavity; a line of defects may guide light. Finite surrounding crystals allow leakage.

An ordered array of colloidal building blocks whose optical-scale periodicity gives Bragg reflection and photonic stop bands. Particle size and refractive-index contrast control the response; defects and stacking disorder can broaden it.

A multilayer reflector using constructive interference of reflections from alternating refractive-index layers.

At normal incidence, alternating layers with indices $n_H,n_L$ and optical thicknesses $n_H{d}_H=n_L{d}_L={\lambda }_0/4$ give reinforcing reflected waves near design wavelength ${\lambda }_0$. Their periodic transfer matrix has a stop band when its Bloch phase is complex. A finite stack has nonzero transmission and its exterior impedances matter. Introducing a suitably phased central defect can support a transmission resonance within the stop band; a one-dimensional stop band is not a complete three-dimensional photonic gap.

Color vision distinguishes distributions of visible light across wavelengths through the different spectral sensitivities of visual receptors. A perceived colour need not correspond to one wavelength: different optical spectra can produce the same receptor responses. Consequently the broad spectral filtering of thin-film interference need not produce pure monochromatic colours.

Assigning linguistic labels to color stimuli or appearances. Shared naming, discrimination and matching must be measured separately; one observation does not define the others.

Measurement and specification of color-related stimulus relations, especially matches and coordinates for stated observers and conditions. Device values require spectral and radiometric calibration before they can be interpreted as cone stimulation.

Measurement and adjustment of a display's spectral, intensity and temporal output relative to control settings. The emitted primary spectra and their nonlinear control functions matter; RGB numbers alone do not specify retinal stimulation.

A stimulus manipulation designed to vary one receptor class while holding others approximately fixed. With a calibrated primary-to-receptor matrix $B$, solve for primary changes producing a specified catch change, respecting baseline and gamut constraints. Observer variation, calibration error and unintended receptor inputs can limit isolation.

The set of stimuli or coordinates realizable by a specified device or system. If primary intensities $\mathbf{p}$ are bounded and nonnegative and cone catches satisfy $\mathbf{Q}=B\mathbf{p}$, invertibility of $B$ does not ensure feasibility: a requested $B^{-1}\mathbf{Q}$ can lie outside the permitted primary ranges.

Color information considered apart from an overall intensity coordinate under a stated colorimetric convention. It is distinct from accelerator chromaticity and should not be confused with a complete appearance description.

A coordinate representation of color stimuli or responses under stated conventions. Euclidean separation in a chosen space need not equal perceptual discriminability for every observer and task.

Relative stability of perceived surface color across changes in illumination or viewing conditions. The measured degree depends on task, adaptation and available cues. Perfect physical recovery is impossible in unrestricted scenes because illumination and reflectance can be nonunique.

In a diffuse spectral model $N(\lambda )=E(\lambda )R(\lambda )$, choose any positive $g$ for which $R/g$ remains an admissible reflectance. Then $E^{\prime }=Eg$ and $R^{\prime }=R/g$ preserve $N$ exactly. Thus outgoing spectral information alone does not uniquely identify reflectance and illumination; three-channel receptor sampling introduces further ambiguity.

The experienced color of a stimulus under specified viewing, adaptation and contextual conditions. Appearance judgments, discrimination thresholds and naming are related but different observations.

A stimulus setting judged to have no chromatic tint under stated conditions. An observer adjusting a patch to cancel a perceived tint moves the physical setting in a compensating direction; the adjustment must not be confused with the direction of the original appearance shift.

Perceived surface achromatic reflectance, often judged relative to a white reference. The same outgoing luminance can support different lightness interpretations under different illumination or scene context.

Perceived amount or intensity of light from a stimulus. It is distinct from physical luminance and from lightness as a judgment related to surface reflectance.

The color attribute distinguishing appearances such as red, yellow, green or blue and their combinations. A hue need not identify a single wavelength, and cone-class labels do not directly specify it.

A hue judged to contain no perceptual mixture of a specified opponent pair, such as a yellow judged neither reddish nor greenish. Its measured setting depends on observer and conditions and should not simply be identified with one retinal cardinal axis.

A yellow judged neither reddish nor greenish. A wavelength or matching setting can operationalize the judgment under controlled conditions; shifts in that setting do not by themselves identify the neural locus of a change.

A receptor catch change expressed relative to a positive adapted-background catch. The same absolute increment can represent different contrasts on different backgrounds. These coordinates are useful for isolating signals without asserting that all neurons explicitly divide in this way.

A class of post-receptoral computations comparing signals from different cone populations, for example a gain-weighted L–M difference and an S versus L+M comparison. Such comparisons distinguish spectral differences from shared changes in illumination intensity. Receptor counts, signal gains and later appearance judgments need not vary proportionally.

Stimulus directions distinguished by selective adaptation or physiological response nulls under specified conditions, commonly associated with an achromatic-like combination and two cone-opponent combinations. Their usefulness near a particular background does not make them identical to unique-hue directions or sufficient for every suprathreshold appearance judgment.

If a nonzero weight vector combines cone contrasts as $r={\mathbf{w}}^{\mathsf{T}}\mathbf{c}$, its zero-response contrasts satisfy one linear constraint and form a two-dimensional plane through the adapted origin in three-dimensional contrast space. Any two independent vectors in that plane constrain the direction of the normal $\mathbf{w}$, up to scale. Nonlinearities and response noise limit empirical identification.

The illustrative transform $y_0=c_L+c_M$, $y_1=c_L-c_M$, $y_2=c_S-(c_L+c_M)/2$ has determinant $-2$. Solving gives $c_L=(y_0+y_1)/2$, $c_M=(y_0-y_1)/2$ and $c_S=y_2+y_0/2$. Thus it reorganizes three cone coordinates without adding independent spectral information. Biological response weights and nonlinearities are further empirical questions.

The ability to distinguish stimuli using their chromatic differences under stated conditions. Controlled tests suppress luminance and other unintended cues and estimate a psychometric function. Color matching, color naming and discrimination are different observations.

Color vision using four functionally independent spectral channels. Four pigment variants or four potential cone cell types do not alone demonstrate four-dimensional behavioral discrimination. Tests must establish information use beyond the matches available to a specified trichromat.^[ref]

An observer adjusts lights until they appear to match a comparison under specified viewing and adaptation conditions. The settings and range of accepted matches constrain the available spectral signals. Matching differs from detecting a difference and from assigning a color name.

A match between a mixture of two long/middle-wave lights and a comparison light, with mixture ratio and brightness separately adjustable. The settings and acceptance range help characterize red–green color-vision variation. A satisfactory brightness match alone is not evidence for normal chromatic discrimination.

A model relating a stimulus to the inputs and discrimination capacity of a visual system. Receiver spectral sensitivity, illumination and background matter; human colour distances are not automatically perceptual distances for another animal.

Variability in a sensory receptor signal that limits discrimination. A difference in average receptor quantum catch does not alone establish a perceptually discriminable difference.

Expected photon absorption by a receptor class. A simple relative model is $Q_i=\int L(\lambda )R(\lambda )S_i(\lambda )d\lambda$, with illumination $L$, object reflectance $R$ and effective spectral sensitivity $S_i$. Absolute catch also requires geometric and temporal factors.

Colour vision based on two classes of photoreceptor input.

Colour vision based on three classes of photoreceptor input. Functional discrimination must be tested rather than inferred solely from gene count.

Color vision with three independent receptor classes whose spectral sensitivities differ from a reference trichromat's, often reducing separation of long- and middle-wave signals. A shifted sensitivity, fewer receptors and loss of an entire independent class are different mechanisms and should not be conflated.

The perceived color of a transmitting sample depends on the illuminating spectrum and the wavelength-dependent transmission. Removing a blue band can make a sample appear yellow; absorption maximum alone does not determine a unique visual shade.

This section concerns specialists in optical instruments.

An optician at the Observatoire de la Côte d’Azur.^[ref]

Imaging with X-rays, combining suitable focusing or image-forming optics with contrast from absorption, phase or emitted signals. Photon-energy selection can make core-edge chemical contrast accessible, while achievable resolution depends on the optics, detector and measurement geometry.

A focused X-ray probe scans a thin specimen while a detector records transmission. Scanning photon energy across an absorption edge adds local spectroscopic contrast, with spatial resolution set by the probe and scanning performance. Thickness and composition both affect transmission, so an energy stack and suitable references help distinguish them.

A conducting scanning-probe tip acts as a movable electrostatic gate. Mapping conductance changes as the tip perturbs a quantum point contact or surrounding electron gas probes quantum transport. The image is a response to the tip, not automatically a noninvasive map of the unperturbed local current density.

The spatial height and shape of a surface. In atomic force microscopy, apparent lateral shape also depends on AFM tip convolution and the controller response.

Transiently binding fluorescent probes provide spatially separated single-emitter observations whose fitted positions accumulate into a structural map. Probe exchange can replenish bleached emitters. Precision, specificity, event density, optical background and drift all affect the usable resolution; accumulating events does not itself remove systematic labelling error.

Reversible complementary-strand recognition supplies transient fluorescent docking events on a DNA nanostructure. Individual events are localized and accumulated as in PAINT.^[ref]

PALM reconstructs a spatial map by activating sparse subsets of labels, estimating each isolated fluorophore's position and combining many frames. Optical resolution depends on localization uncertainty, labeling density, drift and linkage; repeatedly observing one label cannot recover unlabeled structures.

A small subset of photoactivatable labels is fluorescent at any one time so that their overlapping point spread functions can usually be avoided. Repetition supplies many spatial localizations without requiring all labels to emit simultaneously.

Sparse photoactivation allows successive positions of individual labelled molecules to be linked into trajectories; activation of fresh subsets yields many tracks in living cells. This extends photoactivated localization microscopy from accumulated positions to mobility. For free two-dimensional diffusion with independent per-coordinate fluorescence localization precision $\sigma$, instantaneous frames give mean squared displacement $4D\Delta t+4{\sigma }^2$. Motion blur, tracking errors, confinement and photobleaching must be accounted for before assigning mobility classes.^[ref]

For incoherent imaging with an unobstructed circular aperture of diameter $D$, the intensity-image spatial-frequency cutoff is $D/(\lambda f)$ in the focal plane. The Nyquist–Shannon sampling theorem therefore gives the displayed pitch limit. The Rayleigh criterion instead gives a point-image separation $1.22\lambda f/D$. Consequently two samples per Rayleigh separation and sampling the full optical bandwidth differ by the factor $1.22$. Neither sampling estimate guarantees useful contrast at arbitrarily low signal-to-noise ratio.

The resolution scale imposed by a finite-aperture wave-imaging system under a stated criterion. For conventional optical microscopy, a common lateral scale is $\lambda /(2\mathrm{NA})$. Changing the illumination, measured signal or reconstruction assumptions can change the applicable criterion.

Measuring how specular reflection changes the polarization of an electromagnetic wave to infer optical properties.

The ability of an imaging or measurement system to distinguish nearby spatial features. A resolution criterion must specify the features and measurement conditions: diffraction, noise and sampling can all limit the distinction. A small pixel spacing alone does not establish high spatial resolution; electron microscope resolution illustrates the role of wavelength and numerical aperture.

Light emission from excited matter that is not simply its thermal radiation at the local temperature. Fluorescence is one form of luminescence.

Light emission driven by an electric current or electric field.

Emission of light following the absorption of photons.

A measured emission intensity as excitation photon energy is varied, with emitted energy also selected or resolved. Excitation-emission maps can separate transitions belonging to different semiconducting carbon nanotubes.

A luminescent material, for example one that converts some of a blue light-emitting diode's light into longer wavelengths. The name does not require long-lived phosphorescence.

Producing a perceived color by superposing light with different spectra.

Electron optics studies how electric fields and magnetic fields guide electrons and form images. The Lorentz force supplies a trajectory description, while the de Broglie wavelength supplies the wave scale needed for diffraction. Both descriptions matter: steering a narrow beam and resolving two nearby specimen features are different achievements.

For a magnetic electron lens,Here $e>0$ is the elementary charge, $\mathbf{v}$ the electron velocity, and $\mathbf{B}$ the magnetic field. The second equality follows because a cross product is perpendicular to each factor: a static magnetic field bends the trajectory without increasing the kinetic energy. Acceleration instead requires work by an electric field. This distinction lets an electron microscope adjust imaging with magnetic fields while an accelerating voltage sets the electron momentum.

The electric current delivered per transverse probe area per solid angle. For a uniform circular probe of diameter $d$ and a small convergence semiangle $\alpha$, one convention is $\beta =4I/({\pi }^2{d}^2{\alpha }^2)$. State the accelerating voltage, angular convention and whether brightness is reduced by a voltage factor before comparing sources. A smaller aperture changes current as well as convergence; finite electron beam brightness prevents arbitrary current in an arbitrarily small probe.

The principal specimen-adjacent focusing lens of an electron microscope. Its role depends on the mode: forming a small incident probe in scanned imaging differs from forming an enlarged transmitted image in conventional imaging.^[ref]

An illumination lens controlling how an electron beam reaches the probe-forming optics or specimen. Its settings affect current, convergence and illumination geometry.

An electron biprism uses the electric field around a biased fine wire to overlap coherent electron waves for interference. It provides object–reference overlap in electron holography.^[ref]

An electron microscope forms a spatially enlarged representation of a specimen using electrons, electron lenses, and a detector. Its appeal is access to structure too fine for an ordinary optical microscope, but a short de Broglie wavelength alone does not guarantee fine electron microscope resolution. Increasing magnification can enlarge an existing blur without revealing any additional structure.

In transmission electron microscopy, an electron gun illuminates a thin specimen; magnetic electron lenses form and enlarge its transmitted image on a fluorescent screen or detector. A vacuum reduces collisions with gas. Lens electric currents control focusing and magnification.^[ref]

The contrast transfer function describes how an imaging system transfers object information at different spatial frequencies. In phase-contrast electron microscopy, defocus and aberrations can suppress or reverse particular components, requiring calibrated correction and complementary data.

An underfocus setting that balances objective-lens spherical aberration against defocus to provide a broad first phase-contrast band for a weak phase object. A common approximate convention is $\Delta f_S\simeq -1.2\sqrt{C_S\lambda }$ for positive $C_S$; numerical optimization conventions differ slightly.

An energy filter selects electrons according to energy. In cryo-electron microscopy, rejecting energy-loss electrons can reduce background, but filtering cannot reverse chemical damage or remove every source of scattering noise.

A cold field emission gun produces an electron beam by field emission without the high emitter temperature of a thermal source. A narrow energy spread can improve temporal coherence and high-frequency contrast in cryo-electron microscopy.

detective quantum efficiency is the ratio of output to input squared signal-to-noise ratio for a specified measurement convention, often evaluated by spatial frequency. It describes information preservation by a detector rather than just its pixel size.

direct electron detection records incident electrons in a sensor without first converting them through a scintillator into an optical image. Fast readout permits counting and movies that support motion correction.

Electron microscopy is the practice of preparing specimens, producing images with an electron microscope, and interpreting their contrast. A dark region can reflect electron deflection out of the admitted beam rather than complete absorption. A useful image therefore needs a stated preparation, detection signal and length calibration; brightness by itself does not identify a material. Transmission electron microscopy probes thin specimens, whereas a scanning electron microscope can examine the exposed surface of a bulk specimen.^[ref]

Electron optics forms a magnified spatial image of photoelectrons emitted from an illuminated surface. Emission yield and electron-energy selection can supply work-function, electronic or chemical contrast; brightness is not necessarily geometrical height.

A specimen change caused by electron irradiation, including atomic displacement, ionization-driven chemical changes, heating or contamination. Compare low-dose repeated observations and unexposed regions to test whether measured nanostructure is being altered by its measurement.

Deposition or presence of carbonaceous material that changes imaging and spectroscopy.

Orientation-dependent propagation and scattering of electrons along crystal planes or columns.

Reconstruction of three-dimensional structure from electron-microscopy views at different tilts. Preparation, sampling and angular coverage affect the reconstruction.