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An Abbe diagram plots refractive index $n_d$ against Abbe number $V_d$ to show available optical glass families. Many diagrams put increasing $V_d$ to the left. Ordinary crown glasses have larger $V_d$ than ordinary flint glasses, while modern compositions occupy a broader region.

An Abbé refractometer places a small sample between an illuminating prism and a measuring prism, then locates the boundary between bright and dark parts of an angular field. That limiting ray determines the sample's refractive index through Snell's law and the known measuring-prism geometry.

The prism surfaces must be clean and the sample must make good optical contact: a trapped air layer introduces a different interface. Because refractive index varies with wavelength and temperature, a useful reading specifies both; an instrument using white illumination needs compensation for the colored boundary caused by dispersion. A reference liquid checks the reading before an unknown sample is measured.

Consider a lossless optical system in the paraxial approximation with the same refractive index $n_0$ at its input and output. In ray-height/slope coordinates, writeFor $B\ne 0$, its optical path length generating function isThe factor $n_0$ is required for physical optical path length; omitting it defines the index-normalized length $\mathcal{L}/n_0$.

Fermat's principle gives endpoint derivatives ${\partial }_x\mathcal{L}=-n_0\theta$ and ${\partial }_{x^{\prime }}\mathcal{L}=n_0{\theta }^{\prime }$. Substituting $\theta =(x^{\prime }-Ax)/B$ and ${\theta }^{\prime }=(D{x}^{\prime }-x)/B$ and integrating proves the expression. For uniform free space $B=L>0$, the extra optical length is $n_0(x^{\prime }-x{)}^2/(2L)$, fixing both sign and normalization. Its quadratic wave phase generates a linear canonical transform. The case $B=0$ is a separate imaging limit.

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.

The angle of refraction is the angle between the transmitted light ray and the surface normal vector, measured on the transmitted side. With positive refractive indices, a propagating transmitted light ray satisfies $n_i\sin {\theta }_i=n_t\sin {\theta }_t$ in Snell's law. The incident and transmitted angles are measured from the same normal line, not from the surface and not from the emergent light ray at a second surface.

For a film of refractive index $n$ and thickness $d$ in air, its round-trip propagation phase is $\delta =(4\pi d/{\lambda }_0)\sqrt{n^2-{\sin }^2{\theta }_i}$. Two neighbouring same-type interference fringes differ by $2\pi$. At small angular spacing in radians,The relation assumes slowly varying interface reflection coefficients and no rapidly varying reflection phase. At normal incidence the first derivative vanishes, so a local linear angular-spacing formula must be replaced by a quadratic expansion.

Anomalous dispersion is an interval in which the real refractive index decreases with increasing frequency, opposite to normal dispersion. It commonly occurs near a material resonance, where optical absorption also changes rapidly.

The frequency dependence of the complex refractive index determines both propagation phase and attenuation. A rapidly changing real part can give unusual group velocity, but that value alone does not specify the speed of a new signal in an absorbing medium. Measuring transmitted intensity together with phase or deviation across the band is therefore more informative than fitting a real refractive index while ignoring losses.

A concrete comparison used a Biacore T100 with a CM5 chip bearing the target receptor; unused reactive surface groups were blocked. At $25{}^{\circ }\mathrm{C}$ in a pH 7.4 buffer solution, intact IgG and its Fab fragment were injected for $180\mathrm{s}$ at $5\mu \mathrm{L}/\min$; dissociation was monitored for $3600\mathrm{s}$. The optical response tracks changes in refractive index near the surface, rather than counting occupied paratopes directly. A reference surface and blank injections corrected background; association and dissociation responses were fitted with a 1:1 model. The reported intact-IgG detachment was slower than that of the Fab fragment. ^[ref]

This comparison supports format-dependent attachment, not an intrinsic single-site antibody affinity for intact IgG. The reference and blanks address background, not two-arm avidity, surface accessibility or delivery by diffusion. The paper's fitted 1:1 constants alone cannot distinguish those mechanisms. Its Fab fragment comparison therefore motivates specifying the binding format before interpreting a dissociation constant; a good fit does not establish a unique molecular model.

A submerged object looks shallower because its emerging light rays bend away from the surface normal vector. Their backward extensions give an approximate virtual image nearer the surface. For an object at perpendicular depth $d$ in a medium of refractive index $n_{\mathrm{obj}}$, viewed through a flat boundary from a medium of refractive index $n_{\mathrm{obs}}$, the near-normal paraxial approximation gives $d_{\mathrm{par}}=(n_{\mathrm{obs}}/n_{\mathrm{obj}})d$. Thus the model water depth $12\mathrm{cm}$ appears approximately $9\mathrm{cm}$ from air. This is a bundle-of-light rays statement, not a deflection of the axial light ray.^[ref]

For general transmitted light rays, specify what “depth” means. Put the object at $(0,-d)$ below a horizontal boundary $z=0$. A light ray reaches the boundary at $(x,0)$, with object-side angle ${\theta }_{\mathrm{obj}}$ and observer-side angle ${\theta }_{\mathrm{obs}}$, both measured from the upward normal vector. For $x>0$, its backward extension meets the object's normal line $x=0$ at $z=-d_{\mathrm{axis}}$, whereThis axis-intercept depth is exact for that one transmitted light ray in the stated model. It tends to $d_{\mathrm{par}}$ as the angles tend to zero; at exactly zero the quotient is interpreted as a limit. These elementary tangent, sine and cosine relations show precisely where the small-angle approximation enters.

The original calculations below use $d=12\mathrm{cm}$, $n_{\mathrm{obj}}=4/3$ and $n_{\mathrm{obs}}=1$:

At appreciable obliquity these depths differ, so the full outgoing bundle has no single exact virtual image on the object's normal line. An oblique eye samples a narrow bundle; its local apparent position also depends on that bundle and cannot be inferred by declaring $d_{\mathrm{axis}}$ to be a universal image depth. A plane refracting boundary gives approximate image formation near normal incidence, with departures at larger angles. The construction assumes a flat surface, uniform transparent media with isotropy, and a propagating transmitted light ray; it does not extend to total internal reflection.

When both endpoint bundles have the filled circular-cone geometry of etendue of a circular aperture and cone, conservation of etendue givesFor the same refractive index, quadrupling the area halves the sine of the cone half-angle. In the $10\mathrm{mm}$, $30^{\circ }$ example, an ideal output of diameter $20\mathrm{mm}$ must have $\sin {\theta }_2=1/4$, so ${\theta }_2\approx 14.4775^{\circ }$. The exact result differs from simply halving $30^{\circ }$. Conversely, reducing the diameter to $5\mathrm{mm}$ requires a complete forward hemisphere, ${\theta }_2=90^{\circ }$, as a limiting case.

Equal bounding area and solid angle do not alone prove equal etendue: an actual output can have unoccupied directions or position-direction correlations. The filled-cone equation is an exact model of the specified bundles and a capacity constraint on an optical instrument, not a guarantee of an aberration-free implementation.

Atmospheric optics studies how light is redirected, scattered and absorbed by an atmosphere. Spatial changes of refractive index bend paths, suspended particles produce scattering of light, and moving variations of refractive index distort a distant source's wavefront.

The distinction is experimentally useful. A small source viewed through rising warm air changes shape and position even without an absorbing obstacle, while a cloudy beam path can lose direct radiant power through scattering of light. An optical telescope must therefore contend with both atmospheric transmission and changing optical aberrations; a large aperture alone does not guarantee a sharp image.

Atmospheric refraction bends a ray through the vertical refractive index gradient, shifting its apparent direction. Because of optical dispersion, the shift varies with wavelength.

Atmospheric turbulence produces changing refractive index fluctuations. These alter optical path lengths, distorting wavefronts and causing astronomical seeing and atmospheric scintillation.

Axial resolution describes the ability to distinguish features separated along an imaging system's axis. In optical microscopy, the three-dimensional point spread function is usually longer along that axis than across it, so two layers can overlap even when comparable sideways separations are distinguishable.

A small fluorescent bead moved through focus provides an experimental response curve: record the detected intensity at successive axial positions and measure a stated width, such as its full width at half maximum. The result depends on numerical aperture, wavelength, specimen refractive index and the detection method. Confocal microscopy can reject much of the signal from outside the focal plane, improving optical sectioning.

Basic radiance is radiance divided by the square of the refractive index. For radiance $L$ at a fixed wavelength,Lossless propagation conserves reduced radiance along each light ray; $L$ itself is conserved only when the endpoint refractive indices agree. A larger irradiance at a smaller image can result from admitting more directions without increasing reduced radiance. Uniform wave absorption can lower radiant flux while leaving the occupied geometric etendue unchanged; an opaque aperture removes part of that occupied set. ^[ref]

Birefringence is the existence of different optical propagation constants for different eigenpolarizations in an anisotropic material. It provides the propagation-mode separation needed for adiabatic optical following in a twisted nematic.

Two different refractive indices can produce two displaced images, as in the calcite double-image experiment, or a phase retardation between overlapping components, as in a waveplate. Spatial separation concerns where energy flux travels; phase retardation concerns the relative wave phase. A specimen need not show two visibly separated images to have birefringence.

The extraordinary wave's Poynting vector can differ in direction from its wavevector. The latter specifies wave phase propagation, while the former specifies energy flux. This angular separation is walk-off.^[ref]

In the lossless, nondispersive model above, take the gradient of $\omega =c\sqrt{k_x^2/n_e^2+k_z^2/n_o^2}$ to obtain the group velocity, parallel to the Poynting vector. If its angle to the optic axis is $\varphi$, this gives $\tan \varphi =(n_o^2/n_e^2)\tan \theta$. Consequently the extraordinary wave generally carries energy away from its wavevector direction. An internal wavevector normal to a slab face therefore does not guarantee a normal energy path. Birefringent walk-off vanishes at $\theta =0$ and $\theta =\pi /2$; the latter direction still has unequal refractive indices. This is a useful waveplate geometry: overlapping components can acquire phase retardation without spatial splitting. A thin specimen, insufficient imaging resolution, or excitation of only one linear polarization can also hide a double image, for different reasons.

For real $h\in C_c^{\infty }(D)$, the displayed real compactly supported contrast has Fourier transform $\hat{V}(\mathbf{\kappa })=[|\mathbf{\kappa }{|}^4-4({\mathbf{k}}_i\cdot \mathbf{\kappa }{)}^2]\hat{h}(\mathbf{\kappa })$. This vanishes on the shifted Ewald sphere because $|\mathbf{\kappa }{|}^2+2{\mathbf{k}}_i\cdot \mathbf{\kappa }=0$ there. Thus nonzero contrasts can have zero fixed-frequency, fixed-incidence Born approximation data. A small amplitude preserves weak contrast and a positive refractive index.

In the sharp planar, nonmagnetic ionosphere model with frequency above the plasma frequency, the refractive index is $n=\sqrt{1-{\omega }_p^2/{\omega }^2}<1$. From air, the Brewster angle satisfies $\tan {\theta }_B=n$. At that angle the reflected p-polarized wave vanishes, while s polarization retains a nonzero reflected component. A ground-to-ground single reflection from height $h$ spans $D=2h\tan {\theta }_B=2hn$. This is below the critical angle for total internal reflection for total internal reflection, which reflects both polarizations in this lossless isotropic model. The distinction diagnoses ordinary Brewster reflection rather than total internal reflection.

A vacuum ray incident normally on a dielectric moving parallel to its surface becomes oblique in the material rest frame, with $\sin i=|\beta |$. For a transparent nonmagnetic dielectric, the rest-frame Brewster angle satisfies $\tan i_B=n$, giving $|{\beta }_B|=n/\sqrt{1+n^2}$. The reflected electric field is perpendicular to the plane containing the ray and the velocity. A real refractive index $n\ne 1$ is assumed; dispersion makes the condition implicit through the rest-frame frequency, and absorption generally removes exact extinction.