A laser generates amplified light through stimulated emission. Its name abbreviates “light amplification by stimulated emission of radiation”. Lasers supply the intense beams used in optical tweezers; optical coherence also makes them useful for wave interference measurements. ^[ref].

In the usual material picture, a gain medium stores energy supplied by pumping, for example by incident light or an electric current. Stimulated emission transfers that stored energy to a selected optical normal mode, while competing photon absorption removes light. Producing net optical gain requires a suitable population distribution, such as population inversion in the simple model below.

A familiar laser oscillator uses mirrors to return light through the gain medium. This optical feedback favours resonant normal modes; a partly transmitting mirror lets useful power escape. Feedback and mode selection have different implementations, including ring arrangements: every laser need not resemble the same two-mirror device. The laser threshold example treats just that device. ^{[ref] [ref]}.

The useful combination is concentrated power and controlled optical normal modes. It gives optical tweezers a localized source of force and wave interference measurements a source with useful optical coherence. A laser can nevertheless emit several frequencies; “laser” alone does not guarantee a single frequency. ^[ref].

An original experimental landmark is T. H. Maiman's 1960 report of stimulated optical radiation from optically pumped ruby at about $694.3\mathrm{nm}$. His paper describes both the pumping path and the observed emission, connecting stored material energy with amplified light. ^[ref].

Spectral pulse shaping changes an optical pulse by applying a complex amplitude-and-phase mask $M(\omega )$ to its frequency components. The output in time is the inverse Fourier transform of the masked spectrum. A passive device cannot amplify unavailable spectral components; finite bandwidth and spectral resolution constrain the realizable waveforms and time window.

A spatial light modulator applies an electronically controlled spatially varying optical phase, amplitude, or polarization change. In a 4f pulse shaper, frequency maps to transverse position, allowing the pixels to implement a calibrated spectral mask. A phase-only modulator generally needs additional polarization optics or another modulation stage for independent amplitude control.

At a smooth mirror with unit normal vector $n$, an incoming unit direction $i$ has outgoing direction $i-2(i\cdot n)n$. Thus the incident and reflected rays make equal angles with the normal.

More precisely, specular reflection redirects each incident light ray into the direction prescribed by the law of reflection, rather than spreading it over many outgoing directions. The incident light ray, reflected light ray, and local normal vector lie in one plane. The surface must be sufficiently smooth on the scale of the wavelength for this model to describe its reflected beam; a visibly flat surface can still scatter light if its microscopic roughness is appreciable. ^[ref].

The displayed formula uses unit vectors $\mathbf{i},\mathbf{n},\mathbf{r}$ and the dot product. It preserves the component parallel to the mirror and reverses the normal component. For a mirror along $y=0$, take $\mathbf{n}=(0,1)$ and $\mathbf{i}=(3/5,-4/5)$. Then $\mathbf{i}\cdot \mathbf{n}=-4/5$ and $\mathbf{r}=(3/5,4/5)$: both directions have vector norm one, and both make an angle $\arccos (4/5)\approx 36.87^{\circ }$ with the relevant normal direction. Replacing $\mathbf{n}$ by $-\mathbf{n}$ leaves the result unchanged.

This same local rule explains a flat mirror's virtual image. For a source at $(0,a)$ above $y=0$, backward extensions of reflected light rays intersect at $(0,-a)$. The light remains above the mirror; its apparent origin lies below it. A curved mirror changes its normal vector from point to point and can instead make reflected light rays converge. The geometry of image formation therefore starts with one simple direction rule. ^[ref].

// Target: optics.bigb

Image formation assigns each object point a corresponding image point: outgoing light rays from that object point either converge there, giving a real image, or diverge as though they originated there, giving a virtual image. The requirement applies separately to each point. Bringing unrelated light rays together makes a bright patch, but does not preserve the object's pattern. This point-by-point correspondence explains how an optical camera records a scene. ^[ref].

Use the same convention as thin lens: place the optical lens at $x=0$, with incident light travelling to the right and a real object at $x=-u$, $u>0$. Write the image coordinate as $x=v$, so $v>0$ for a real image and $v<0$ for a virtual image. Heights $h_o,h_i$ are positive above the optical axis and negative below it. In air, within the paraxial approximation, a thin lens obeysHere $f$ is the signed focal length, positive for a converging optical lens and negative for a diverging optical lens; the expression for $v$ assumes $u\ne f$. Object distance $u$ is positive even though the object's coordinate is negative. These coordinate rules describe transmission through the optical lens; a mirror reverses the outgoing direction, so do not identify its image distance with this same rightward coordinate. ^[ref].

The beauty of geometrical optics is that two convenient light rays can locate a point, while a whole admitted bundle forms its image. For a converging thin lens, an incident light ray parallel to the optical axis exits through the rear optical focus; a light ray through the centre continues undeviated. Real optical lens elements do not focus every admitted light ray exactly: optical aberrations and diffraction limit the point correspondence. The diagrams below are original ideal-model constructions, not measured data. ^[ref].

Refraction changes the direction of a transmitted wave when it crosses between media with different phase velocities. For light at a planar interface, Snell's law relates the angles measured from the normal vector through the refractive indices. At normal incidence the direction remains unchanged even though the wavelength and phase velocity can change. The worked example under geometrical optics distinguishes transmission from total internal reflection.

For light, refraction explains why a submerged object looks nearer the surface and why an optical lens can form an image. A wavefront crossing a boundary must retain a consistent wave phase along that boundary: its normal spacing changes with phase velocity, and oblique light rays change direction. Normal incidence separates these two effects especially clearly.

Take stationary, transparent, homogeneous media with isotropy, linearity of their optical response, and real positive refractive indices, at one chosen frequency $f$. With $c$ the vacuum speed of light and ${\lambda }_0=c/f$ the vacuum wavelength, medium $j$ hasThe frequency is unchanged at a stationary boundary; the phase velocity, wavelength and wavenumber can change. Here $n_j$ is a phase refractive index, not the group refractive index. If there is optical dispersion, evaluate $n_j$ at this same frequency on both sides.^[ref]

For an original idealized air–water example, stipulate $n_{\mathrm{a}}=1$ and $n_{\mathrm{w}}=4/3$, with vacuum wavelength $600\mathrm{nm}$. Then $f=c/(600\mathrm{nm})\approx 4.99654\times 10^{14}\mathrm{Hz}$, the water phase velocity is $3c/4$, and its wavelength is $450\mathrm{nm}$. At an air angle of incidence $30^{\circ }$, Snell's law gives a water angle of refraction $\arcsin (3/8)\approx 22.0243^{\circ }$. These stipulated values illustrate the relations; actual refractive indices depend on material conditions and frequency. By reversibility of an optical ray, the same path emerging from water has outgoing air angle $30^{\circ }$.

An optical lattice is a periodic optical trap produced by interfering light fields. Its periodic potential gives atomic Bloch states and energy bands. For the displayed one-dimensional lattice, the recoil energy is $E_R={\hslash }^2{\pi }^2/(2m{d}^2)$ and $s$ measures the depth in recoil units.

For an ideal diffuse emitting or reflecting planar surface with uniform direction-independent radiance, the radiant intensity in a direction at angle $\theta$ to its unit normal is $I(\theta )=I(0)\cos \theta$. This follows immediately from $I=LA\cos \theta$ and the projected area formula. The worked radiance example and diagram show the $0^{\circ }$ and $60^{\circ }$ cases. Lambertian reflectance supplies one physical idealization realizing the law; an ideal blackbody provides an emitting example.

The historical setting is Johann Heinrich Lambert's Photometria sive de mensura et gradibus luminis, colorum et umbrae, published in Augsburg in 1760. Its Part I, §139, explicitly compares normal and oblique illumination using the sine of the source's elevation, equivalently the cosine of its angle from the receiving normal vector. This is an early receiving-surface projection calculation; the modern outgoing-patch statement above keeps radiance, radiant intensity and irradiance distinct. ^{[ref] [ref]}.

Lambertian reflectance is the ideal diffuse-reflection model with direction-independent outgoing radiance, the frequency-integrated specific intensity. If a fraction $0\le \rho \le 1$ of incident radiant flux is reflected, a surface receiving irradiance $E$ has reflected radiance $L=\rho E/\pi$. The reflected outward power per area is $\pi L=\rho E$ by the same hemispherical integral used for radiant exitance. Its radiant intensity still acquires the projected area factor in Lambert's cosine law.

For $E=100\mathrm{W}{\mathrm{m}}^{-2}$ and $\rho =0.80$, the reflected radiance is $80/\pi \approx 25.4648\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$. A patch of area $0.020{\mathrm{m}}^2$ reflects $1.6\mathrm{W}$ in total, and its radiant intensities at $0^{\circ }$ and $60^{\circ }$ are respectively $1.6/\pi$ and $0.8/\pi \mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$. The radiance stays fixed as the view changes, provided the illumination stays fixed. Changing the incident angle can change irradiance and therefore the reflected radiance.

Photometry weights visible electromagnetic radiation by standardized visual response functions; radiometry measures its energy and power without visual weighting.^[ref]

The question is how much light a source supplies under a stated visual weighting. A beam containing $1\mathrm{mW}$ at the defining frequency $540\mathrm{THz}$ has luminous flux $0.683\mathrm{lm}$. Moving the same radiant flux to a region of smaller spectral luminous efficacy reduces its luminous flux. Changing the optical spectrum changes this conversion even when its total power stays fixed. This makes photometry useful for comparing lighting, while radiometry remains necessary for an energy budget. The lumen counts luminous flux; the candela additionally specifies its distribution over solid angle.

Two measurement routes implement the weighting: measure an optical spectrum and evaluate a weighted integral, or use a photometer whose spectral responsivity approximates the prescribed weighting. Both require a specified collection geometry. Neither is a complete prediction of an individual's perception of a scene.^[ref]

Radiometry measures the energy and power carried by electromagnetic radiation, including light. It asks how much radiation a source supplies, where it travels, and how much reaches a receiving surface. The measurements are physical rather than weighted by the sensitivity of the human eye: equal radiant fluxes at different wavelengths need not look equally bright. ^[ref].

The five quantities below answer different questions about the same radiation. The International System of Units makes the distinctions visible: a watt is a joule per second, while a steradian measures solid angle. ^[ref].

Resolving wavelength or frequency adds a further “per unit interval”, as in spectral radiance. The notation is consistent with NIST's table of spectral quantities. In particular, radiant flux is power, whereas the existing astronomical radiative flux is power per area. ^[ref].

This bookkeeping makes an optical camera's collected signal calculable from a source's radiance and the geometry of the receiving aperture. It also lets radiation measurements from different instruments be compared. For example, NIST describes separate calibrations of spectral radiance and spectral irradiance for instruments used to observe Earth. ^[ref].

Online text: pages.cs.wisc.edu/~remzi/OSTEP/file-implementation.pdf

Cited passages: Chapter 40, section 40.7, printed pp. 14–16: caching and buffering.

Online text: pages.cs.wisc.edu/~remzi/OSTEP/file-journaling.pdf

Online text: pages.cs.wisc.edu/~remzi/OSTEP/vm-beyondphys-policy.pdf

Cited passages: Arpaci-Dusseau and Arpaci-Dusseau, OSTEP chapter 22, equation 22.1: average memory access time; Arpaci-Dusseau and Arpaci-Dusseau, OSTEP chapter 22, section 22.11: Thrashing; Remzi H. Arpaci-Dusseau and Andrea C. Arpaci-Dusseau, Operating Systems: Three Easy Pieces, chapter 22, section 22.5 and the Types of Locality aside.

Online text: pages.cs.wisc.edu/~remzi/OSTEP/threads-bugs.pdf

Cited passages: Chapter 32, section 32.3, Prevention: Circular Wait, printed pp. 7–8: consistent acquisition ordering and hidden calls; Operating Systems: Three Easy Pieces, chapter 32, section 32.3, printed pp. 5–7, figures 32.6–32.7: opposing acquisition orders and circular wait.

Online text: pages.cs.wisc.edu/~remzi/OSTEP/threads-locks.pdf

Cited passages: Chapter 28, sections 28.1–28.2 and 28.7–28.8: ownership, multiple lock objects, and spinning; Chapter 28, sections 28.6–28.7, figures 28.1–28.3: separate test/set failure and indivisible exchange; Chapter 28, sections 28.7–28.8, printed pp. 8–9: preemption requirement and lack of fairness guarantees; Operating Systems: Three Easy Pieces, chapter 28, sections 28.1–28.2, printed pp. 1–2: lock ownership and critical sections.

Online text: pages.cs.wisc.edu/~remzi/OSTEP/vm-complete.pdf

Cited passages: Chapter 23, section 23.1, printed pp. 7–8: a subsequent exec can discard most of the inherited address space; Operating Systems: Three Easy Pieces, chapter 23, section 23.1, Other Neat Tricks, printed pp. 7–8: protected sharing, copying on a write, and fork/exec motivation.

Online text: opendatastructures.org/ods-python/11_2_Counting_Sort_Radix_So.html

Cited passages: Open Data Structures, section 11.2.1: associated data, backwards placement, Theorem 11.7 and stability; Open Data Structures, section 11.2.2: least-significant-digit passes, stability argument, and Theorem 11.8.

Online text: opendatastructures.org/ods-python/11_1_Comparison_Based_Sorti.html

Cited passages: Open Data Structures, section 11.1.4, Theorem 11.5: deterministic comparison-tree lower bound.

Online text: opendatastructures.org/ods-python/5_1_ChainedHashTable_Hashin.html

Cited passages: Open Data Structures, section 5.1, the array of lists and the find/remove operations.

Additional cited passages: §5.1.1, multiplicative hashing construction and Lemma 5.1; indicator-variable proof of Lemma 5.2. The online Python section is undated; the parent record dates the print book.