Chapter 2. Wave Optics

When do we use Wave Optics?

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

LENS DESIGN

( d >> λ )

LIGHT DESIGN

( d ~ λ )

PHOTON DESIGN

( d << λ )

Geometrical OpticsRay tracing

반사/굴절

렌즈 설계 (lens design)금형 (metal mastering)사출 (injection molding)조립 (assembling)측정 (MTF monitoring)

Etendue (ΔkxΔx)(ΔkxΔx)>1

EM Wave OpticsWave propagating

회절 / 간섭

회절격자 설계Wafer mastering복제 (embossing)합체 (packaging)측정 (extraction effi)

Diff. Limit (ΔkxΔx > 1)

PhotonicsPhoton tunneling

유도방출 / 유도투과

나노구조 설계 (nano-tech)Embedded masteringNano imprinting집적 (integrating)측정 (quantum effi)

Uncertainty (ΔkxΔx > 1)

Lens design(projection)

Light design(extraction) : field profile, polarization

Photon design(e-h combination)

LED

Optics regimes

기술한계

2-1. Postulates of Wave OpticsWave 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.

2.2 MONOCHROMATIC WAVES2.2 MONOCHROMATIC WAVES

Complex representation

The real function is

Harmonic Waves - Period and Frequency -

Helmholtz equationHelmholtz equation

: Helmholtz equation

( wavenumber )

“The wave equation for monochromatic waves”

The optical intensity

The intensity of a monochromatic wave does not vary with time.

Helmholtz, Hermann von (1821-1894)

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.

Elementary waves of Helmholtz eq.Elementary waves of Helmholtz eq.

Plane Wave :

This is the equation describing parallel planes perpendicular tothe 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 Paraboloidal Wave

Fresnel Approximation is valid when

2 2 2( )x y a+ =

Paraxial wavesParaxial wavesA wave is said to be paraxial if its wavefront normals are paraxial rays.

Paraxial Helmholtz equationParaxial Helmholtz equation

Slowly varying envelope approximation of the Helmholtz equation

Paraxial Helmholtz equation.

Relation between wave optics and ray optics Relation between wave optics and ray optics

Eikonal Equation

: Ray equation can be also derived

2-4. Simple optical components2-4. Simple optical componentsReflection 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

BOUNDARY CONDITIONSBOUNDARY CONDITIONS

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, the wavelengths along the interface surface must have the same temporal and spatial variation.

321 zzz λλλ ==

constant2==

zii λ

πβPropagation constant :

Called also as β, k, phase, or momentum matching

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

Snell’s law :

Z

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

B. Transmission Through Optical Components

Diffraction gratingsDiffraction gratings

: Grating Equation

C. Graded-Index Optical ComponentsC. Graded-Index Optical Components

2.5 INTERFERENCE2.5 INTERFERENCE

Interferometers

Mach-Zehnder Michelson

Sagnac

B. Multiple-beam interferenceB. Multiple-beam interference

: Finesse

2.6 POLYCHROMATIC LIGHT2.6 POLYCHROMATIC LIGHT

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

(1) determine its Fourier transform; (2) eliminate negative frequencies and multiply by 2; (3) determine the inverse Fourier transform.

The complex wavefunction (also called the complex analytic signal) istherefore obtained from the wavefunction by a process of three steps: