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Chapter 2. Wave Optics

Ray

Optics

Wave Optics

Quantum

Optics

Electromagnetic

Optics

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When do we use Wave Optics?

Lih Y. Lin, http://www.ee.washington.edu/people/faculty/lin_lih/EE485/4

L E NS DES IGN

( d >> λ )

L IGHT DE S IGN

( d ~ λ )

PHOTON DES IGN

( d << λ )

Geometrical OpticsRay tracing

( lens design)(metal mastering)

(injection molding)(assembl ing)(MTF monitoring)

Etendue (ΔkxΔx)(ΔkxΔx)>1

EM Wave OpticsWave propagating

Wafer mastering(emboss ing (packaging)(extraction eff i )

Diff. L imit (Δk xΔx > 1)

PhotonicsPhoton tunneling

(nano-tech) Embeddedmastering Nanoimprinting(integrating)(quantum effi)

Uncer ta inty (Δk xΔx > 1)

Lens des ign(projection)

Light des ign(extract ion) :

L E Dfield profi le , polarization

Photon design(e -h combination)

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Opt ics regimes

Ray

Optics

Wave Optics

Quantum

Optics

Electromagnetic

Optics

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.

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2.2 MONOCHROMATIC WAVES

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2.2 MONOCHROMATIC WAVES

Complex representation

The real function is

Harmonic Waves - Period and Frequency -

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Helmholtz equation

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Helmholtz 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, 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.

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

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Plane Wave :

This is the equation describing parallel planes perpendicular to

the wavevector k (hence the name “plane wave”).

: wavelength

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Spherical Wave :

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Fresnel Approximation of the Spherical Wave; Paraboloidal Wave

Fresnel Approximation -7 Paraboloidal Wave

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Fresnel Approximation is valid when

(x2 + y2 ) = a2

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Paraxial waves

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Paraxial waves

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

Paraxial Helmholtz equation

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Paraxial Helmholtz equation

-+Slowly varying envelope approximation of the Helmholtz equation

-+ Paraxial Helmholtz equation.

Relation between wave optics and ray optics

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Relation between wave optics and ray optics

Eikonal Equation

: Ray equation can be also derived

2-4. Simple optical components

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2-4. Simple optical components

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

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BOUNDARY CONDITIONS

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BOUNDARY CONDITIONS

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

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

Snell’s law :

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,

the wavelengths along the interface surface

must have the same temporal and spatial variation.

λz1 = λz2 = λz3

Propagation constant : β =2π

= constantzi

iλ

Z

B. Transmission Through Optical Components

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Thin Lens

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Diffraction gratings

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Diffraction gratings

: Grating Equation

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

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2.5 INTERFERENCE2.5 INTERFERENCE

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Interferometers

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Interference of Two Oblique Plane Waves

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B. Multiple-beam interferenceB. Multiple-beam interference

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: Finesse

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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.

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.32

The Pulsed Plane Wave

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