The fields

This experiment solves a monochromatic, lossless boundary problem. The field is invariant along the vertical axis. A finite-height window exposes its three-dimensional electric and magnetic vectors; fading at the viewing edges is graphical only. The surfaces show dielectric boundaries or a cavity cutaway, rather than an aperture that emits a prescribed field.

Energy and observation

Energy shows the dielectric-weighted mean energy density in the volume. On the observation plane it shows forward axial power flux for the guides and tunnelling experiment, and mean energy density for the circulating cavity. The latter has no single forward direction. The displayed equation identifies the sampled quantity. Signed field shows the real electric component; phase shows its complex angle, weighted by amplitude. The optical cycle advances the monochromatic field, while the detector scan moves the observation plane independently.

The two guide supermodes carry equal integrated power. About 97% of the combined axial power occupies one half-space at each transfer maximum; the residual field in the other guide is part of the exact solution. In the tunnelling model, interference between both evanescent terms transports energy across the gap. A single decaying term alone would carry no normal mean power.

Numerical representation

Analytical fields are evaluated locally in a worker at full resolution, then stored as complex binary16 textures. There is no coarse-to-fine field swap. Spacing is at most transversely and axially; identical vertical rows represent exact vertical invariance. The carrier is restored analytically after interpolation. Material boundaries and the conducting wall have finite display resolution. The tests independently check Maxwell’s curl equations, interface continuity, reflected/transmitted power and the guide transfer length.

The fields

The description above specifies this instrument's boundary model. The Schlieren relay, wavefront sensor, diffuser, coronagraph, microscope and Fabry–Pérot use a two-dimensional angular spectrum, with longitudinal electric field and all three magnetic components. The conveyor sums two cones of plane waves. The microcylinder matches cylindrical waves across the glass boundary. Forward and backward waves have their own vector relations. The Michelson, multilayer, graded-index glass, absorbing film, sound cell and triangular cavity use vertically invariant TE fields, drawn inside a finite-height viewing window.

These are monochromatic steady-state fields. Bright volumes reveal the computed field; they are not scattered light visible in otherwise empty vacuum. Hardware outside the stated optical boundaries is illustrative. Thin-lens transmissions, ideal mirrors and the beam splitter do not simulate their microscopic coatings or edge scattering.

Energy and phase

The volume shows mean energy density in Energy mode. The Schlieren camera, wavefront sensor, diffuser, microscope, sound cell and coronagraph show forward power flux at the observation plane. The cavities and the conveyor show local mean energy density, including electric and magnetic contributions. The conveyor beads follow maxima of the electric contribution. Electric field is signed in the volume and evolves through the optical cycle. Phase is the complex angle of the vertical electric component, with darkness indicating weak amplitude.

Calculation

The field is generated locally at its final resolution. The microscope propagates through each lens and the Fourier-plane phase dot in sequence; no specimen image is pasted onto the detector. The Fabry–Pérot sums the infinite reflection series analytically for every retained angular component. The multilayer matches tangential fields at each dielectric interface. The Michelson conserves power at its ideal scattering junction; its moving mirror changes every angular component’s round-trip path. The graded-index glass sums exact Helmholtz modes; the absorbing film matches the electric field and the magnetic-field jump from its surface current.

The fields

The launch field is prescribed at the optical element. Its tangential components are and , with . The longitudinal electric component and every magnetic component follow from Maxwell’s equations. No paraxial propagator is used.

Superposition and detection

For the double slit, the right-opening field is stored separately and added with its adjustable phase: , with the same operation for . The reference peak uses the in-phase configuration. The fields are summed before intensity is formed. Cross terms create the bright and dark structure. The detector is a virtual sampling plane. Fixed exposure preserves brightness changes with distance; adaptive exposure reveals faint structure. Preview magnification enlarges the region marked by four corners on the three-dimensional plane. In Energy view its image shows nonnegative forward flux, normalized by a fixed volume-wide reference peak or an interpolated per-plane reference peak, and multiplied by display brightness, with a power curve to reveal faint structure; probe values remain linear. Electric field and Phase show the corresponding field quantity on the same observation plane; these are not images directly recorded by an ordinary intensity detector.

For a finite aperture of characteristic radius , the Fresnel number indicates the propagation regime. Aperture structure dominates near the element; the Fraunhofer pattern is approached when . Moving the detector changes the sampled plane, not the propagation equation.

What is drawn

Write the complex electric component as . Its real oscillation is . Phase colour represents the angle; brightness is weighted by the amplitude. At that angle is undefined. Such zeros form the phase-singularity curves traced in the knot and linked-ring setups. A loop around a simple singularity accumulates a phase change of . Other electric or magnetic components can remain nonzero there.

The default phase is : it subtracts the uniform phase advance of a reference plane wave, leaving the diffraction structure visible. Multiplication by this nonzero factor does not move a singularity. Dark areas indicate small component amplitude, not a particular phase angle. The bright dots mark the scanning plane's intersections with stationary nodal curves; no particles travel along those curves.

The volume is sampled in all three spatial dimensions. In the default palette, cyan and violet encode the positive and negative instantaneous , with opacity weighted by its absolute value after brightness and cutoff. Signed-field and optical-phase views follow the electric component. Electric and magnetic arrows can be selected independently. Arrows occupy fixed points throughout the illuminated volume; they are not particle trajectories. Samples concentrate on the field rather than empty space.

The two arrow colours distinguish and . Both share an adjustable graphical length scale. Readable mode divides both vectors at each anchor by the same complex-field envelope ; directions, oscillation and their local amplitude ratio are preserved. Proportional mode uses a single scale everywhere. Initial arrow size is tuned to each setup. Probe components use the physical axes. All spatial axes have the same scale. Each angular-spectrum mode is transverse and has its magnetic partner fixed by Maxwell’s curl equations. Their superposition need not make the total electric and magnetic vectors perpendicular at every point.

The choices above the preview switch the volume and observation plane together. Energy displays mean energy density in the volume and axial flux on the plane. Electric-field sections default to the instantaneous signed component; remains available in the Light settings. The electric-field volume and arrows remain instantaneous. Energy opacity starts from the square root of the normalized time-averaged density, then uses the same display mapping; relative phase colours show , revealing changes relative to the incident plane wave. Instantaneous phase shows . Both are weighted by amplitude.

Before a mask, an analytic incident plane wave makes the direction of illumination visible: and . Its displayed extent is softly cropped to the collimator. The upstream laser, lens and beam envelope indicate the illumination path; fields inside that hardware are not solved. The outgoing field is prescribed independently by the transmission model; reflected fields and the electromagnetic boundary layer are not simulated. The volume fades at the finite data boundaries for display only; detector and probe values are not faded.

Playback defaults to five seconds per optical cycle. The settings panel changes playback speed and optical phase. Pausing or moving the optical-cycle phase control changes the instantaneous vectors while the time-averaged Energy image remains steady. This is a monochromatic steady state: the source is already on. A physical switch-on front requires a time-dependent, broadband calculation. The floating play control scans the observation plane. Scanning starts automatically and pauses when the detector is moved manually. Optical-cycle playback is controlled separately in Vectors settings.

The bunny recording arm shows an illuminated 3D source, object/reference capture, and a connection to the replay element. Its object-wave energy is the reflected, conjugate counterpart of the calculated replay field. Capture fringes illustrate interference with the spherical reference from the drawn source. The object beam illuminates the sensor-facing surface; the displayed object wave is cropped there to avoid suggesting transmission through the bunny. This is an idealized coherent recording/replay diagram; source material scattering and an acquisition algorithm are not simulated. The translucent reconstructed mesh is an optional surface guide, not a replacement for the calculated field.

Numerical model

Each curated experiment is propagated ahead of time on a angular-spectrum grid spanning . The full complex fields are stored throughout a three-dimensional volume. The detector samples their real and imaginary parts directly on the GPU at its current position; there is no queued recalculation while dragging.

The reconstructed transverse grid aligns with the propagation grid at , with axial spacing . Float32 inverse FFT arithmetic, binary16 storage and trilinear interpolation introduce finite numerical error. The common carrier is removed before interpolation and restored analytically. An aligned half-resolution preview appears first and crossfades to the full field; cached full fields open directly. Compact angular spectra are reconstructed in a worker, retaining every propagated mode. The original stored fields provide a compatibility fallback. Full textures upload in small groups of depth layers. Energy ray marching uses a cached scalar density with a local interpolation correction; sampled RMS differences from direct propagation are below across these experiments. Detector flux and probe values are formed from the complex fields.

This is a band-limited aperture/launch-field model. Material scattering, reflections, thickness, edge charges, and evanescent waves are excluded. The optical elements are schematic representations of the prescribed transmission. Helicon, petal, caustic and vortex phase elements show an equivalent wrapped relief with wavelengths at index contrast , with separate attenuation and a pupil stop. The bunny, five-focus and flat-image holograms use thin patterned displays representing ideal complex-amplitude modulation. They are not glass sculptures. Binary apertures are openings in an absorbing screen; the screen’s graphical transparency does not change its transmission or the computed field. The coloured mask preview in this information view encodes phase delay by hue and transmission amplitude by brightness. These are schematic encodings, not a material simulation. The axicon cone illustrates the equivalent thin-phase profile for an index contrast of 0.5. Colour palettes affect the signed-field and energy volumes and both detectors. In optical-phase view, changing palette rotates the cyclic hue reference.