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The angles from part (i) have sinesBy Snell law for elastic and acoustic waves, $\sin {\theta }_f/{\alpha }_f=\sin {\theta }_P/\alpha$. Both angles lie in $[0,\pi /2)$, so ${\theta }_f={\theta }_z$. The outgoing fluid ray tilts away from the inward-tilted normal by exactly its tilt from vertical; it therefore travels vertically upwards. This is hyperbolic interface collimation of P-waves.

The source-to-interface distance satisfiesOne can verify this by squaring, using $D^2-s^2=d^2{R}^2$. ConsequentlyAdding the vertical fluid travel time from the interface to height $z$ givesThus surfaces of equal arrival time in the fluid are horizontal planes. Their leading wave amplitude may vary with $R$: a planar wavefront does not imply a spatially uniform plane-wave solution at every order.

Write $\mathbf{X}=ϵ\mathbf{x}$, $\mathbf{k}={\nabla }_{\mathbf{X}}\theta$, and denote the slow density and velocity field amplitudes by $a_{\rho }$ and ${\mathbf{a}}_u$. Acting on the rapid phase, ${\partial }_t+\mathbf{U}\cdot \nabla$ gives $i\Omega$, where $\Omega =\omega -\mathbf{U}\cdot \mathbf{k}$. Spatial differentiation gives $-i\mathbf{k}$, while derivatives of the amplitudes and the mean shear flow are $O(ϵ)$. The leading continuity equation and linearized Euler equations therefore becomeEliminating ${\mathbf{a}}_u$ on the nonzero acoustic branch gives the acoustic eikonal equation in a shear flow:The mean-shear term is small here because $\nabla \mathbf{U}=O(ϵ)$; retaining advection does not require retaining that gradient in the leading phase equation. The positive intrinsic-frequency acoustic dispersion relation is $\omega =U{k}_x+c_0|\mathbf{k}|$, with group velocity ${\mathbf{v}}_g=U{\mathbf{e}}_x+c_0\mathbf{k}/|\mathbf{k}|$.

In a stationary wall boundary layer, horizontal homogeneity preserves $\omega$ and $k_x$ along a Hamiltonian ray-tracing equations trajectory. For a two-dimensional ray,Suppose $U$ increases from zero at the wall to a positive free-stream value. For a downstream-directed wave normal, $k_x>0$, increasing $U$ decreases $\Omega$ and $k_y$. The wave normal bends towards the wall, and the ray becomes more nearly parallel to it. If $U$ reaches $\omega /k_x-c_0$, then $k_y=0$: the ray turns and returns towards the wall rather than reaching the free stream. With launch angle ${\beta }_0$ above the wall and free-stream Mach number $M_{\infty }=U_{\infty }/c_0$, the escape condition isFor upstream-directed wave normals, $k_x<0$, $\Omega$ and $|k_y|$ increase through the layer; there is no such downstream turning barrier on this branch. A wavefront normal is parallel to $\mathbf{k}$, while the energy ray follows the group velocity: advection makes their directions different. In particular, a wall-normal wave normal with $k_x=0$ can still have a ray drifting downstream. The ordinary WKB method needs a local turning-point continuation at $k_y=0$; the ray sketch depicts that qualitative continuation.

Write spatial frequency in cycles across the optical pupil as $u=D/\Lambda$. The previous result gives a maximum axial spatial frequency $N/2$. If one counts one representative per positive-quadrant integer lattice point inside an isotropic disk $u_x^2+u_y^2\le (N/2{)}^2$, the continuum area estimate isThis is the geometric counting convention producing the printed formula. Lattice boundaries, the zero-frequency piston and axis points give lower-order corrections, so the formula is not an exact integer count.

The mode definition in the PDF is $D=j\Lambda /2$, hence $u=j/2$ and $j_{\max }=N$. It does not itself specify a two-dimensional lattice, its orientation, its independent wave phase components or a circular rather than square spatial frequency cutoff. Full-cycle integer frequencies correspond only to the even values of $j$ in that definition. A general real wavefront error also needs both sine and cosine quadratures, and distinct oblique orientations cannot all be identified with a single positive quadrant: the Fourier transform of a real wavefront has conjugate coefficients at $\mathbf{u}$ and $-\mathbf{u}$, rather than identifying every independent sign change of a component.

For comparison, a circular spatial frequency disk has approximately $\pi N^2/4$ signed full-cycle lattice points, or $\pi N^2/8$ conjugate pairs; two real wave phase components per pair restore approximately $\pi N^2/4$ real degrees of freedom. A square spatial frequency support gives a different count. The circular physical optical pupil similarly has only about $\pi N^2/4$ illuminated actuators. The printed $M$ is a quadrant area estimate under an extra counting convention; it is not a uniquely determined count of all independently correctable wavefront modes from the stated one-dimensional mode condition.

Atmospheric extinction changes the amplitude of the incoming light through absorption and scattering. Its wavelength dependence alters the measured optical spectrum and colors; clouds and aerosols introduce additional time dependence. Scattered moonlight and atmospheric emission increase sky brightness.

Atmospheric refraction changes the apparent position of the source. Its wavelength dependence, atmospheric dispersion, spreads a broadband image toward the zenith. The mean refractive index gradient therefore affects direction even in the absence of small-scale turbulence.

Atmospheric turbulence produces rapidly varying optical path lengths, distorting the phase and curvature of a nominally plane wavefront. Different pupil regions acquire different phase delays, producing astronomical seeing, image motion, and short-exposure speckle patterns. Different lines of sight sample different fluctuations, giving anisoplanatism.

Propagation through the fluctuating medium also changes the intensity through atmospheric scintillation, the familiar twinkling of stars. A phase-only adaptive optics correction can reduce atmospheric phase errors but does not remove absorption, all scintillation, or the atmospheric emission background.

The magnetic vector potential is defined by $\mathbf{B}=\nabla \times \mathbf{A}$. The gauge transformationleaves both $\mathbf{B}$ and $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$ unchanged. In magnetostatics choose the Coulomb gauge $\nabla \cdot \mathbf{A}=0$. Then Ampère's law and the curl-of-curl identity giveThe free-space Green function identity ${\nabla }^2|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^{-1}=-4\pi {\delta }^{(3)}(\mathbf{r}-{\mathbf{r}}^{\prime })$ therefore givesThe boundary condition selects the homogeneous solution; the usual decay condition assumes suitably localized sources. An infinitely long static wire needs a reference for its logarithmically divergent potential, although its magnetic field is finite away from the wire.

For time-dependent sources use the Lorenz gauge, $\nabla \cdot \mathbf{A}+c^{-2}{\partial }_t\varphi =0$, and the outgoing retarded potentialThe scalar potential has the analogous retarded charge-density integral. The two gauges should not be conflated for a general time-dependent source.^[ref]

For the prescribed uncharged wire, $\rho =0$ and $\varphi =0$ is consistent. Translation invariance makes $\mathbf{A}=A_z(r,t)\hat{\mathbf{z}}$ with $\nabla \cdot \mathbf{A}=0$, so this particular solution satisfies both gauge conditions. With $s=z^{\prime }-z$,It is zero for $t<r/c$. For $t>r/c$, set $L=\sqrt{c^2{t}^2-r^2}$. The causal interval is $-L<s<L$, giving the switched infinite-wire retarded potentialDifferentiation gives the fields and Poynting vector, for $r>0$, $t>r/c$:Here $\mathbf{B}={\mu }_0\mathbf{H}$ and $B_{\varphi }=-{\partial }_r{A}_z$. All three fields vanish before arrival. For positive $I$, $\mathbf{E}$ opposes the imposed current and the radial energy flow after switching a line current points radially outward: $(-\hat{\mathbf{z}})\times \hat{\mathbf{\varphi }}=\hat{\mathbf{r}}$.

For the graph, let $u=ct/r$. Normalized to $E_{\ast }={\mu }_{0}Ic/(2\pi r)$, $H_{\ast }=I/(2\pi r)$ and $N_{\ast }={\mu }_0{I}^{2}c/(4{\pi }^2{r}^2)$, the curves areAt the wavefront, $|E_z|$ and $H_{\varphi }$ diverge as $(u-1{)}^{-1/2}$ and $N_r$ as $(u-1{)}^{-1}$. This abrupt line-current wavefront singularity reflects the ideal instantaneous turn-on of a line current; a finite rise time smooths the front. At late times,The vector potential grows logarithmically with time in this infinite-wire idealization even as the local magnetic field approaches its static limit. The divergent front should not be interpreted as a finite-energy pulse from a real instantaneous switching experiment.

In time $\Delta t$ the particle advances $\beta c\Delta t$, while a light wavelet travels $c\Delta t/n$. Coherent addition along the direction of emitted photons gives the Cherenkov cone angleThe angle is measured between photon propagation and the particle velocity. It is complementary to the angle between the conical wavefront surface and the trajectory; confusing those two angles exchanges sine and cosine. At threshold the photon cone closes, $\alpha \to 0$.

Huygens' principle treats each point of a wavefront as the source of a secondary wavelet. The later wavefront is the common forward envelope of those wavelets. Consider a plane front reaching a flat interface between media with speeds $v_1$ and $v_2$. In the same interval $\Delta t$, the incident front advances $v_1\Delta t$ while the first transmitted wavelet grows to radius $v_2\Delta t$.

The two right triangles sharing the interface segment of length $d$ give $d\sin i=v_1\Delta t$ and $d\sin r=v_2\Delta t$, where $i$ and $r$ are ray angles to the normal. HenceThis is Snell's law. The frequency is unchanged at a stationary interface, while wavelength changes with the propagation speed.

Huygens' principle treats every point of an advancing wavefront as a source of secondary wavelets; their common envelope forms the next wavefront. Consider two points of an incident plane wave reaching the interface at different times. During the same interval $\Delta t$, the remaining incident portion travels $v_1\Delta t$ while the first transmitted wavelet expands a distance $v_2\Delta t$. The two wavefront right triangles share their interface segment, soThis is Snell's law, with both angles measured from the surface normal. The speeds are phase velocities at the common frequency in stationary isotropic media. If the equation requires $|\sin {\theta }_2|>1$, there is no propagating refracted ray and total internal reflection applies instead.

In Fraunhofer diffraction, the source and observation geometry make the incident and outgoing wavefronts effectively planar across the aperture. For direct propagation over distance $z$, the quadratic variation in propagation phase across a slit of width $w$ must be negligible, approximately $w^2/(\lambda z)\ll 1$. The far-field angular amplitude is then the Fourier transform of the aperture's complex illumination. A converging lens also produces this angular distribution in its rear focal plane for collimated illumination, without requiring a physically enormous observation distance. The approximation concerns the aperture-dependent phase, not the absence of diffraction.

The Huygens-Fresnel principle treats every point of an advancing wavefront as a source of secondary spherical waves; their forward envelope constructs the next wavefront. Consider two points where an oblique plane wavefront meets a plane interface a distance $AB$ apart. After its first point reaches $A$, let time $t$ elapse until the second reaches $B$. In medium 1, the incident travel distance is $v_{1}t=AB\sin i$. Meanwhile, a secondary wave from $A$ has travelled distance $v_{2}t$ in medium 2. The tangent from $B$ to that secondary wavefront gives $v_{2}t=AB\sin r$. ThereforeHere $i,r$ are the angle of incidence and angle of refraction, measured from the interface normal vector, and $n_j=c/v_j$ are the refractive indices at the common frequency. This is Snell's law.

Compare a directly transmitted ray emerging at $A$ with a ray that makes one extra round trip inside the Fabry–Pérot interferometer and emerges at $C$. Both emerge parallel at angle $\theta$ to the plate normal. The extra internal distance is $2d/\cos \theta$, and their emergence points differ laterally by $AC=2d\tan \theta$. The first ray has already travelled $AC\sin \theta$ towards a common outgoing wavefront when the second reaches $C$. The air-gap optical path difference in a Fabry–Pérot interferometer follows because optical path length equals geometric path length in the air-gap approximation:The subtraction is essential: comparing distances only inside the plates would compare different outgoing wavefronts. The corresponding propagation phase difference is $2\pi \Delta /\lambda$. Coating reflection phases can add a separate phase offset; they do not change this geometric optical path difference.

A Michelson interferometer divides a source beam at a partially reflecting beam splitter. The two perpendicular arms terminate in mirrors whose reflected beams return to the splitter and overlap at the detector. The recombined fields must have overlapping wavefronts and compatible polarization of waves. Moving one mirror by $\Delta L$ changes the round-trip optical path difference by $2\Delta L$, so $N$ passing fringes determine $\Delta L=N\lambda /2$. Conversely a known mirror displacement measures wavelength. Scanning the delay can determine temporal optical coherence and, by Fourier transform spectroscopy, spectral structure. A compensating plate may equalize glass traversal, particularly for broadband illumination.

For equal real peak field amplitudes, the superposition isTime averaging the square over many optical cycles gives $⟨E^2⟩=2a^2{\cos }^2(\varphi /2)=a^2(1+\cos \varphi )$. With $\varphi =kx$ after absorbing a constant port-dependent phase into the zero of $x$,Here $I_0$ is the sum of the two separate-arm intensities; maximum intensity is $2I_0$ and the ideal minimum is zero. Equal amplitudes give unit interference visibility.

For a stationary source with a flat intensity spectrum, different spectral components add in time-averaged intensity. Keeping their individual two-arm interference terms and defining $\mathrm{sinc}z=\sin z/z$ givesOne must not coherently add unrelated source frequencies as if they had a fixed common phase. The sinc envelope is the Fourier transform of the rectangular spectral intensity.

Numerically, ${\nu }_0=5.00\times 10^{14}\mathrm{Hz}$ and $\Delta \nu =1.00\times 10^{13}\mathrm{Hz}$. The annotated interferogram usesIt is even in $x$, has central value 2 and background 1, and its carrier-fringe period in optical path difference is $c/{\nu }_0=0.600\mu \mathrm{m}$. The first envelope zeros are at $x=\pm c/\Delta \nu =\pm 30.0\mu \mathrm{m}$, corresponding to mirror shifts of $\pm 15.0\mu \mathrm{m}$ from equal paths. Side lobes persist beyond the zeros; a negative sinc reverses fringe phase. The intensity remains between $1-|\mathrm{sinc}|$ and $1+|\mathrm{sinc}|$, not between the signed sinc curves with an assumed fixed ordering.

Let the first ray reflect at the upper surface and the second traverse the film, reflect below, and emerge at a laterally displaced point. Its internal round-trip geometric length is $2d/\cos \theta$, but the external path to a common outgoing wavefront must also be included. The lateral separation is $2d\tan \theta$, whose projection along the external ray is $2d\tan \theta \sin {\theta }_i$. The net optical path difference is thereforeusing Snell's law $\sin {\theta }_i=n\sin \theta$. Multiplying by the vacuum wavenumber givesFor the usual index ordering $1<n<n_2$, including the stated later value $n=1.2$, both reflections are towards higher refractive index and both reverse the electric-field sign. Their relative reflection phase is therefore zero; $\delta$ is also their total relative phase modulo $2\pi$. A different index ordering or polarization-dependent reflection phase would require that additional term.

Circular apertures and circular obstructions illustrate both the usefulness and the limits of geometric shadow arguments. In scalar diffraction theory, each point of a transmitted wavefront contributes a complex secondary field. Their coherent sum, described by the Huygens-Fresnel principle, determines the intensity. Circular symmetry simplifies the integral, but the answer depends on whether propagation is in the Fraunhofer diffraction or Fresnel diffraction regime.

For aperture radius $a$ and axial propagation distance $z$, the Fresnel number $N_F=a^2/(\lambda z)$ measures the importance of quadratic phase across the aperture. The far-field condition is $N_F\ll 1$; a lens can produce the same angular spectrum in its back focal plane. Fresnel propagation retains that quadratic phase. For an axial point source a distance $s$ before the object and an observer a distance $r$ after it, the phase uses the effective distance $z_{\mathrm{eff}}=sr/(s+r)$, because both incident and outgoing spherical paths contribute curvature. Scalar, paraxial propagation and an opening large compared with wavelength are assumed throughout.

A circular hole in the far field gives an Airy pattern. For uniform plane illumination the field is the two-dimensional Fourier transform of the aperture. In polar coordinates, let $Q=k\sin \theta$; the angular integral gives the Bessel function $J_0$:ThusThe limiting central value is one because $J_1(u)\simeq u/2$ near zero. A bright central Airy disk is surrounded by progressively weaker concentric rings. The first dark ring has $ka\sin \theta =3.8317$, or $\sin \theta \simeq 1.22\lambda /D$ with diameter $D=2a$. This scale gives the Rayleigh criterion for a circular optical instrument. Larger apertures narrow the angular pattern; a lens of focal length $f$ maps the first dark ring to radius $1.22\lambda f/D$ in its focal plane.

In the Fresnel regime, the axial intensity oscillates as aperture size or distance changes. Fresnel half-period zones have boundaries at path excesses $m\lambda /2$ and approximately opposite phases in successive zones. For a plane wave, their radii obey $r_m^2=m\lambda z$. A circular opening can therefore expose alternating constructive and cancelling zone contributions. More precisely, Fresnel diffraction by a circular aperture gives, normalized by the unobstructed field,An odd integer number of zones gives a bright axial point with ideal intensity $4I_0$; an even number gives an axial zero. Intensity can locally exceed the unobstructed value because interference redistributes energy; total transmitted power still obeys conservation of energy. The relative normalisation here differs from the Airy expression, which normalises each far-field pattern by its own central intensity.

Off axis, the circular Fresnel field isIts concentric structure evolves with $N_F$ and includes fringes near the geometrical edge. The axial zone argument does not determine the complete off-axis pattern.

Complementary holes and opaque disks are related at the field level. Babinet's principle states $U_a+U_d=U_0$ for complementary aperture and disk transmissions under the same illumination. In the Fraunhofer region away from the direct transmitted beam, the background field is absent and the diffracted fields are opposite in sign. Their scattered intensity profiles therefore have the same Airy angular dependence. Exactly in the forward direction the unobstructed field must be retained; subtracting intensities is incorrect. Finite incident beams also give a finite-width direct beam rather than an ideal angular delta function.

For Fresnel diffraction by an opaque disk, the axial complement is particularly striking:The ideal circular disk has an Arago spot at the centre of its geometric shadow. Circular edge contributions reach the axial point with the same phase. This does not mean the whole shadow is illuminated: off-axis cancellation produces the surrounding shadow and rings, and the axial spot becomes narrower for larger obstruction or shorter propagation distance. An aperture's axial intensity oscillates, while its complementary disk's axial intensity stays constant in this ideal paraxial model.

A practical observation uses a compact coherent source, an accurately circular hole or disk, and an aligned screen or camera. Varying the propagation distance or aperture radius reveals the transition between regimes. A point source, or spatial filtering, prevents different source positions from washing out the fine structure. Finite source size, incomplete coherence, edge roughness, misalignment, vector effects for wavelength-sized objects and lens aberrations reduce the ideal contrast. These experimental limitations explain why a geometrically unexpected axial bright point requires a suitable setup even though its scalar derivation is simple.

Both the incoming and outgoing spherical wavefronts contribute to the path excess. At aperture radius $u$, with axial distances $s$ before and $r$ after the aperture,The outer edge of the $n$th Fresnel half-period zone has $\Delta \ell =n\lambda /2$. HenceTherefore the outer radius is $\boxed{u_2=4.47\times 10^{-4}\mathrm{m}=0.447\mathrm{mm}}$. The paraxial approximation is excellent because $u_2\ll s,r$. Treating the illumination as a plane wave would incorrectly omit the source-side path curvature.

In division of amplitude, one incident beam is split into two or more parts at a partially reflecting interface; they follow distinct paths and are recombined. The Michelson interferometer and thin-film interference use this principle. In division of wavefront, spatially distinct parts of the same wavefront are selected and allowed to overlap, as in double-slit interference. Both require adequate optical coherence; wavefront division particularly exposes the need for spatial optical coherence between the selected regions.

In thin-film interference, coherent beams reflected at the two faces of a transparent film overlap. Let the film have refractive index $n_f$, thickness $t$, and internal ray angle ${\theta }_f$, determined by Snell's law. For vacuum wavelength ${\lambda }_0$, the propagation part of the relative phase isAlthough the internal round-trip ray length is $2t/\cos {\theta }_f$, the two emergent rays must be compared on the same outgoing wavefront. Subtracting their external lateral path contribution gives the effective optical path difference $2n_{f}t\cos {\theta }_f$, explaining the cosine rather than its reciprocal.

Reflection also changes phase. At normal incidence in lossless positive-index media, the electric-field Fresnel equations give $r_{ij}=(n_i-n_j)/(n_i+n_j)$: reflection towards higher index reverses sign, whereas reflection towards lower index does not. If the two interfering reflections differ by one such phase reversal, their total relative phase is ${\delta }_{\mathrm{prop}}+\pi$ modulo $2\pi$. Then reflected maxima and minima obeyIf there are zero or two reversals, the conditions interchange. These bright/dark conditions assume overlapping beams with adequate temporal optical coherence; complete cancellation further requires equal amplitudes. In transmitted light, a lossless complete film has complementary intensity variations because reflected plus transmitted power is conserved.

For multiple internal reflections at normal incidence, summing the geometric series gives the exact film reflection amplitudeThis includes both the amplitude mismatch and repeated-beam interference. A normal-incidence antireflection coating on substrate index $n_s>n_0$ can cancel reflection at one wavelength when $n_f=\sqrt{n_0{n}_s}$ and $t={\lambda }_0/(4n_f)$: both interface reflections reverse sign and the film's round-trip propagation adds $\pi$, with equal interface amplitudes. A varying thickness or angle selects different maxima for different wavelengths, producing the colours of transparent films in white light. Broad bandwidth and large incidence angles generally prevent perfect cancellation across all wavelengths.

Draw the Fabry–Pérot etalon with parallel mirrors and an incident beam making internal angle $\theta$ with their normal. Every emerging transmitted contribution has undergone one more internal round trip than its predecessor.

The extra internal path is $2d/\cos \theta$, but successive contributions leave at laterally separated points, a distance $2d\tan \theta$ apart. Comparing their phases on the same outgoing wavefront subtracts the path projection $2d\tan \theta \sin \theta$. Their net optical path difference in air isHence, in the reflection-phase convention implicit in the supplied transmission formula,For a gap of refractive index $n$, replace $d$ by $nd$ and use the vacuum wavelength. Coating reflection phases can add a round-trip offset; the expression above assumes that offset is absorbed in the resonance order, or is zero modulo $2\pi$ as in the stated ideal model.

Snell's law is $n_1\sin {\theta }_1=n_2\sin {\theta }_2$, with angles measured from the interface normal. The Huygens-Fresnel principle explains it by constructing the transmitted wavefront from secondary wavelets: during the same time interval, distances travelled in the two media are proportional to $c/n_1$ and $c/n_2$. The geometry of the advancing plane wavefront then gives $\sin {\theta }_1/\sin {\theta }_2=(c/n_1)/(c/n_2)=n_2/n_1$.

For incidence from index $n$ into vacuum, the transmitted angle is $\alpha =\arcsin (n\sin \theta )$. The ray bends away from the normal, so its positive deviation isFor $0<\theta <{\theta }_c=\arcsin (1/n)$,Thus the graph rises monotonically from zero to a finite endpoint with a vertical tangent:At the endpoint the transmitted ray grazes the interface. Beyond the critical angle for total internal reflection, no propagating transmitted ray exists and total internal reflection occurs.

Inside a prism, the two ray angles with the face normals sum to its apex angle. For the equilateral cross-section, they are $\theta$ and $\pi /3-\theta$. The entry deviation equals the corresponding exit deviation under ray reversal. Adding the two changes of direction therefore givesThis is symmetric about $\theta =\pi /6$. Symmetry alone gives a stationary point; to establish that it is a minimum, differentiate once more:within the transmitted-ray interval. The sum is therefore strictly convex, andThese expressions require $1<n<2$ for a non-grazing path through both faces; $n=2$ is a grazing limit, and $n>2$ admits no such two-face transmitted path. More explicitly, both internal angles must be at most ${\theta }_c$, which is possible only if $2{\theta }_c\ge \pi /3$.

Atmospheric hexagonal ice crystals can present pairs of faces with a $60^{\circ }$ angle, acting as small prisms. Their orientations turn the minimum-deviation cone around the Moon into an optical halo. No transmitted ray of this family is deflected less than the minimum deviation of a prism. Moreover, near the minimum, $\Delta -{\Delta }_{\min }$ is quadratic in the departure from the symmetric ray, so a range of incident directions is compressed into a small outgoing angular range. This caustic makes the inner rim sharp and bright, with the ray density decreasing farther outside. The common 22-degree halo has an angular radius near $22^{\circ }$ about either the Sun or Moon, hence a diameter near $44^{\circ }$.^[ref]

The Huygens-Fresnel principle constructs a new wavefront as the envelope of secondary wavelets emitted by points on an earlier wavefront. Let the phase velocities in the two media be $v_1=c/n_1$ and $v_2=c/n_2$, with refractive indices evaluated at the light's fixed frequency.

For refraction from secondary wavelets, let point $A$ of the incident wavefront reach the interface first. A second point $B$ reaches the interface at $C$ after time $\Delta t$, so $BC=v_1\Delta t$. During the same interval, the secondary wavelet centered on $A$ advances radius $AD=v_2\Delta t$ into the second medium. The refracted wavefront $CD$ is tangent to this wavelet, hence $AD\perp CD$; also $BC\perp AB$ because a ray is normal to its wavefront in these isotropic media.

Let ${\theta }_1,{\theta }_2$ be the incident and refracted ray angles from the interface normal. They also equal the corresponding acute angles between the wavefronts and the interface. The two right triangles have common hypotenuse $AC$, givingTheir ratio eliminates both $AC$ and $\Delta t$:This is Snell's law. In particular, entering a medium of larger refractive index reduces the ray's angle to the normal. The construction assumes a stationary interface and homogeneous isotropic media with a propagating refracted wave.