Foundations

Relate a periodic wave's spatial and temporal repetition.

A local differential equation determines small string disturbances.

Linear disturbances add; their relative phase determines reinforcement.

Boundary conditions select standing-wave frequencies.

Angles at an interface depend on refractive indices.

A paraxial lens relates object and image distances.

An aperture's finite width selects dark far-field directions.

An ideal analyzer transmits the field component along its axis.

In a homogeneous nonmagnetic isotropic dielectric, a monochromatic plane wave has k²=ε_r(ω)ω²/c² and B₀=k×E₀/ω.

At normal incidence on a lossless nonmagnetic dielectric interface, the electric-amplitude reflection coefficient is r=(n₁−n₂)/(n₁+n₂).

A thin spherical lens in a common surrounding medium has 1/f=(n_lens/n_medium−1)(1/R₁−1/R₂).

For scalar co-polarized fields, averaged intensity includes a cross-correlation term.

For equal-phase coherent illumination of narrow slits separated by d, far-field bright directions satisfy d sinθ=mλ and dark directions d sinθ=(m+½)λ.

A monochromatic scalar field in a uniform medium propagates paraxially as E_z(x,y)≈e^(ikz)/(iλz)∫∫E₀(ξ,η)exp[ik((x−ξ)²+(y−η)²)/(2z)]dξdη, with k=2π/λ.

When the input-aperture quadratic phase in the Fresnel integral is negligible, the far-field amplitude is proportional to the two-dimensional Fourier transform of the aperture field evaluated at spatial frequencies x/(λz), y/(λz).

At normal incidence on a periodic grating of pitch d, principal propagating orders satisfy d sinθ_m=mλ.

A uniform ideal circular pupil of diameter D gives an Airy intensity pattern whose first dark ring has small angle θ≈1.22λ/D.

For a linear shift-invariant scalar imaging system, coherent image amplitude is E_image=E_object*h.

For a narrow-band wave packet on a smooth real dispersion branch, its leading envelope velocity is v_g=dω/dk at the carrier, while phase velocity is ω/k.

At an interface between lossless isotropic nonmagnetic dielectrics, normal energy-flux ratios are R_s=[(n₁cosθ_i−n₂cosθ_t)/(n₁cosθ_i+n₂cosθ_t)]² and R_p=[(n₂cosθ_i−n₁cosθ_t)/(n₂cosθ_i+n₁cosθ_t)]², with T_s,p=1−R_s,p.

Reflection angles are measured from the local normal.

Total reflected flux does not mean zero field beyond the interface.

A virtual image is formed by ray extensions, not actual ray convergence.

A solid angle measures a cone of directions rather than a plane rotation.

Optical power, directional power and surface densities are different quantities.

Photometry weights radiation with a specified standard visual response.