Chapter 2. Wave Optics When do we use Wave Optics?

Lih Y. Lin, http://www.ee.washington.edu/people/faculty/lin_lih/EE485/ Optics regimes

LENS DESIGN LIGHT DESIGN PHOTON DESIGN ( d >> λ ) ( d ~ λ ) ( d << λ )

Geometrical Optics EM Wave Optics Photonics Ray tracing Wave propagating Photon tunneling

반사/굴절 회절 / 간섭 유도방출 / 유도투과

렌즈 설계 (lens design) 회절격자 설계 나노구조 설계 (nano-tech) 금형 (metal mastering) Wafer mastering Embedded mastering 사출 (injection molding) 복제 (embossing) Nano imprinting 조립 (assembling) 합체 (packaging) 집적 (integrating) 측정 (MTF monitoring) 측정 (extraction effi) 측정 (quantum effi) 기술 Etendue (Δk Δx)(Δk Δx)>1 Diff. Limit ( k x> 1) Uncertainty ( k x > 1) 한계 x x Δ xΔ Δ xΔ

Light design (extraction) : field profile, polarization

LED

Photon design Lens design (e-h combination) (projection) 2-1. Postulates of Wave Optics

Wave Equation

Intensity, Power, and Energy

The optical energy (units of joules) collected in a given time interval is the time integral of the optical power over the time interval. 2.22.2 MONOCHROMATICMONOCHROMATIC WAVESWAVES

Complex representation

The real function is Harmonic Waves - Period and Frequency - HelmholtzHelmholtz equation equation

( wavenumber )

: Helmholtz equation “The wave equation for monochromatic waves”

The optical intensity

The intensity of a monochromatic wave does not vary with time. Helmholtz sought to synthesize Maxwell's electromagnetic theory of light with the central force theorem. To accomplish this, he formulated an electrodynamic theory of action at a distance in which electric and magnetic forces were propagated instantaneously.

Helmholtz, Hermann von (1821-1894) ElementaryElementary waveswaves ofof HelmholtzHelmholtz eq. eq. Plane Wave :

This is the equation describing parallel planes perpendicular to the wavevector k (hence the name “plane wave”).

k : wavelength A plane wave One-dimensional Traveling Wave

v = 1 m/s, -z

v = 2 m/s, +x Spherical Wave : Fresnel Approximation of the Spherical Wave; Paraboloidal Wave

Fresnel Approximation Æ Paraboloidal Wave Fresnel Approximation is valid when

()x22+=ya 2 ParaxialParaxial waveswaves

A wave is said to be paraxial if its wavefront normals are paraxial rays. ParaxialParaxial HelmholtzHelmholtz equation equation

ÎSlowly varying envelope approximation of the Helmholtz equation

Î Paraxial Helmholtz equation. RelationRelation betweenbetween wavewave opticsoptics andand rayray opticsoptics

Eikonal Equation

: Ray equation can be also derived 2-4.2-4. SimpleSimple opticaloptical componentscomponents Reflection from a Planar Mirror

At the boundary, the wavefronts of the two waves match, i.e., the phase must be equal, Reflection and refraction at a planar dielectric boundary BOUNDARYBOUNDARY CONDITIONSCONDITIONS

Wavelength (phase) matching at the boundary = Snell’s law

Suppose that at a particular instance and at a particular location of the boundary, the oscillation of the incident wave is at its maximum; then both reflected and transmitted waves have to be at their maxima.

In other words, Z the wavelengths along the interface surface must have the same temporal and spatial variation.

λz1 = λz2 = λz3 2π Propagation constant : βi = = constant λzi Called also as β, k, phase, or momentum matching

But all mean the same thing: wavelength matching at the boundary!

Snell’s law : B. Transmission Through Optical Components

DiffractionDiffraction gratingsgratings

: Grating Equation C.C. Graded-IndexGraded-Index OpticalOptical ComponentsComponents 2.52.5 INTERFERENCEINTERFERENCE Interferometers

Mach-Zehnder Michelson

Sagnac

B.B. Multiple-beamMultiple-beam interferenceinterference : Finesse 2.62.6 POLYCHROMATICPOLYCHROMATIC LIGHTLIGHT

A polychromatic wave can be expanded as a sum of monochromatic waves by the use of Fourier methods.

The complex wavefunction (also called the complex analytic signal) is therefore obtained from the wavefunction by a process of three steps: (1) determine its Fourier transform; (2) eliminate negative frequencies and multiply by 2; (3) determine the inverse Fourier transform.