Techreport Slides

Wave Phenomenon in Geometric Optics

Tom CuypersSe Baek Oh

Roarke HorstmeyerRamesh Raskar

Part 1Introduction

Overview

1. Introduction and Welcome

2. Relating wave propagation to Light Fields

3. Augmented Light Fields

4. Applications in Imaging

Motivation

• Dual representation of light:

– Photons travelling in a straight line

Computer Graphics

http://graphics.ucsd.edu/~henrik/images

Computational Photography

http://graphics.stanford.edu/projects/lightfield

Motivation



– Waves traveling in all directionsHolography

http://www.humanproductivitylab.com/images

Optics

Motivation



– Waves traveling in all directions

• Goal of the course:

Provide a gentle introduction of wave phenomenon using ray-based representations

Wave phenomena in the real world

• Fluid surfaces

http://4.bp.blogspot.com/_NpINLHeo8rM/Rsl52vjOKII/AAAAAAAAFMM/WnESejvzq5Y/s400/s

plash-water-waves-4559.JPG


• Fluid surfaces

• Sound waves

http://fetch1.com/wp-content/uploads/2009/11/hd-800_detail_sound-waves1.jpg


• Fluid surfaces

• Sound waves

• Electromagnetic waves

– Microscopic scale

http://upload.wikimedia.org/wikipedia/commons/archive/1/1f/20090127195426!Ggb_in_soap_bubble_1.jpg

1

Coherence

• Degree of making interference– coherent ⇐ partially coherent ⇒ incoherent

• Correlation of two points on wavefront– (≈phase difference)

Coherent: deterministic phase relation

Incoherent: uncorrelated phase

relation

1

Coherence

• throwing stones......

single point source many point sources

⇒ coherent ⇒ if thrown identically, still coherent!

⇒ if thrown randomly, then incoherent!

1

Coherence

• Temporal coherence:

– spectral bandwidth

• monochromatic: temporally coherent

• broadband (white light): temporally incoherent

• Spatial coherence:

– spatial bandwidth (angular span)

• point source: spatially coherent

• extended source: spatially incoherent

1

ExampleTemporally incoherent;

spatially coherent

Temporally &

spatially coherent

Temporally &

spatially incoherentTemporally coherent;

spatially incoherent

laser

rotating diffuser

What is a wave?

• Types

– Electromagnetic waves

– Mechanical Waves

http://en.wikipedia.org/wiki/File:EM_spectrum.svg

What is a wave?

• Types

– Electromagnetic waves

– Mechanical Waves

http://www.gi.alaska.edu/chaparral/acousticspectrum.jpg

What is a wave?

• Types

• Properties

– Wavelength: λ

– Frequency :

– Phase: p

– Amplitude: A

– Polarization

λ

p=0p=π/2

p=πp=3π/2

A

http://www.ccrs.nrcan.gc.ca/glossary/images/3104.gif

What are wave phenomena?

• Huygens principle









• Diffraction



• Diffraction



• Diffraction



• Diffraction



• Diffraction



• Diffraction

• Interference

Wave A

Wave B

Constructive

interference



• Diffraction

• Interference

Wave A

Wave B

Destructive

interference



• Diffraction

• Interference

• Example N

Reflection

Ray-based



• Diffraction

• Interference

• Example

Reflection

Huygens Principle



• Diffraction

• Interference

• Example

Reflection

Huygens Principle



• Diffraction

• Interference

• Example

Reflection

Huygens Principle



• Diffraction

• Interference

• Example

Reflection

Huygens Principle



• Diffraction

• Interference

• Example

Reflection

Huygens Principle



• Diffraction

• Interference

• Example

Reflection

Huygens Principle

Part 2Relating Wave Phenomena

to Light Fields

Introduction

• Review of Light Fields

• Review of Waves using Fourier optics principles ? (intro)

• Introduction to the Wigner Distribution Function

• Augmented Light Fields to represent wave phenomena

Plenoptic Function

• Q: What is the set of all things that we can ever see?

• A: The Plenoptic Function (Adelson & Bergen)

Let’s start with a stationary person and try to parameterize

everything that he can see…

Gray Snapshot

• P(θ,φ) is intensity of light– Seen from a single view point– At a single time– Averaged over the wavelengths of the visible spectrum• (can also do P(x,y), but spherical coordinate are nicer)

Color Snapshot

P(θ,φ,λ) is intensity of light– Seen from a single view point– At a single time– As a function of wavelength

Movie

P(θ,φ,λ,t) is intensity of light– Seen from a single view point– Over time– As a function of wavelength

Holographic Movie

P(θ,φ,λ,t,Vx, Vy, Vz) is intensity of light

• – Seen from ANY single view point

• – Over time

• – As a function of wavelength

Plenoptic Function

P(θ,φ,λ,t,Vx, Vy, Vz)

• Can reconstruct every possible view, at every moment, from every position, at every wavelength

• Contains every photograph, every movie, everything that anyone has ever seen.

Sampling Plenoptic Function (top view)

Ray

Let’s not worry about time and color:

5D : P(θ,φ,VX,VY,VZ)

• – 3D position

• – 2D direction

Ray

• No Occluding ObjectsP(θ,φ,VX,VY,VZ)

• 4D2D position– 2D direction

• The space of all lines in 3-D space is 4D.

Representation

(x,y)

(θ,φ)

(x,y)

(u,v)

Position-angle

representation

2 plane

representation

Light Field Camera

Point Grey

Mark levoy

Why Study Light Fields Using Wave Optics?

MacroMicro

x

θz=z0

z=0

x

fz=z0

z=0

Light

Field

Wigner

Distribution

Wave Optics

• Waves instead of rays

• Interference & diffraction

• Plane of point emitters

(Huygen’s principle)

• Each emitter has amplitude

and phase

Parallel rays Plane waves

Position and direction in wave optics

• Spatial frequency: f

1

f


• Spatial frequency: f

• Direction of wave: θ

1

f

λ

θSmall θ assumption:


Complex wavefront = parallel wavefronts

Wigner Distribution Function

• Input: one-dimensional function of position

• Output: two-dimensional function of position and spatial frequency

• (some) information about spectrum at each position

Auto correlation of complex wavefront

5

Wigner Distribution Function

...

...

2D Wigner Distribution

• Projection along frequency yields power

• Projection along position yield spectral power

f

x

W(x,f)




f

x

x

W(x,f)

|h(x)|²




f

x

x

W(x,f)

|h(x)|²

f

|f(x)|²




f

x

x

W(x,f)

|h(x)|²

f

|f(x)|²


Remarks:

• Possible negative values

• Uncertainty principle

f

x

W(x,f)

Relationship with Light Fields:Observable Light Fields

• Move aperture across plane

• Look at direction spread

• Continuous form of plenopticcamera

Scene




Scene





Scene





Scene





Scene





Scene





Scene

Aperture

Position x

θ




Wave

Aperture Window

Fourier Transform

Power


Wave

Aperture Window

Fourier Transform

Power


Wave

Aperture Window

Fourier Transform

Power

Wigner Distribution

of wave functionWigner Distribution

of aperture window


Wigner Distribution

of wave functionWigner Distribution

of aperture window

Blur trades off

resolution in position

with direction


Wigner Distribution

of wave function

At zero wavelength limit

(regime of ray optics)


At zero wavelength limit

(regime of ray optics)

Observable light field and Wigner equivalent!

Observable Light Field

• Observable light field is a blurred Wigner distribution with a modified coordinate system

• Blur trades off resolution in position with direction

• Wigner distribution and observable light field equivalent at zero wavelength limit

Light Fields and Wigner

• Observable Light Fields = special case of Wigner

• Ignores wave phenomena

• Can we also introduce wave phenomena in light fields?

– -> Augmented Light Fields

Part 3Augmenting Light Fields

Introduction

Traditional

Light Field

light field

radiance of ray

ref. plane

position

ray optics based

simple and powerful

Introduction

Traditional

Light Field

light field

radiance of ray

ref. plane

position

direction

ray optics based

simple and powerful

Introduction

Traditional

Light Field

ray optics based

simple and powerful

Introduction

Wigner

Distribution

Function

Traditional

Light Field

ray optics based

simple and powerful

rigorous but cumbersome

wave optics based

limited in diffraction & interference

Introduction

Wigner

Distribution

Function

Traditional

Light Field

ray optics based

simple and powerful


wave optics based


holograms beam shaping

rotational PSF

Augmented LF

Wigner

Distribution

Function

Traditional

Light Field

WDF

Traditional

Light Field

Augmented LF

Interference & Diffraction

Interaction w/ optical elements

ray optics based

simple and powerful



wave optics based

Non-paraxial propagation

Augmented LF

• Not a new light field

• A new methodology/framework to create, modulate, and propagate light fields

– stay purely in position-angle space

• Wave optics phenomena can be understood with the light field

Augmented LF framework

(diffractive)

optical

element

LF


LF propagation

(diffractive)

optical

element

LF LF


LF propagation

(diffractive)

optical

element

LF LF LF

light field transformer

negative radiance


LF propagation

(diffractive)

optical

element

LF LF LF LF

LF propagation


negative radiance

Tech report, S. B. Oh et al.

Outline

• Limitations of Light Field analysis

– Ignore wave phenomena

– Only positive ray -> no interference

Outline


• Augmented Light Field

– free-space propagation

Outline




– virtual light projector in the ALF

• Possible negative

• Coherence

Outline




– virtual light projector in the ALF

• Possible negative

• Coherence

– light field transformer

Assumptions

• Monochromatic (= temporally coherent)

– can be extended into polychromatic

• Flatland (= 1D observation plane)

– can be extended to the real world

• Scalar field and diffraction (= one polarization)

– can be extended into polarized light

• No non-linear effect (two-photon, SHG, loss,

absorption, etc)

Young’s experiment

light from

a laser

screendouble

slit

constructive interference


light from

a laser

screendouble

slitdestructive interference


ref. plane

Light Field WDF


Light Field WDF

ref. plane

projection projection

9

Virtual light projector

virtual light projector

Augmented

LF

positive

negative

intensity=0

at the mid point

projection

Not conflict with physics

real projector

real projector

9


real projector

real projector

first null

(OPD = λ/2)

virtual light projector

9


first null

(OPD = λ/2)hyperbola

λ/2asymptote of

hyperbola

valid in Fresnel regime

(or paraxial)

1

Virtual light projectorin high school physics

class,

destructive interference(need negative radiance from

virtual light projector)

Video

waves

1

Question

• Does a virtual light projector also work for incoherent light?

• Yes!

1

Temporal coherence

• Broadband light is incoherent

• ALF (also LF and WDF) can be defined for different wavelength and treated independently

1

Young’s Exp. w/ white light

1

Young’s Exp. w/ white lightRed

Green

Blue

1

Young’s Exp. w/ white lightRed

Green

Blue

1

Spatial coherence

• ALF w/ virtual light projectors is defined for spatially coherent light

• For partially coherent/incoherent light, adding the defined ALF still gives valid results!

1

Young’s Exp. w/ spatially incoherent light

1

Young’s Exp. w/ spatially

incoherent light

1


1

w/ random

phase

(uncorrelated)

spatially incoherent light:

infinite number of waves propagating along all

the direction with random phase delay


1

w/ random

phase

(uncorrelated)

Addition


incoherent light

1

w/ random

phase

(uncorrelated)

Addition


incoherent light

1

Light Field Transformer

• light field interactions w/ optical elements

Light field transformer

1

Light Field Transformer

Dimension Property Note8D reflectance field,

volume hologram

6D display,

BTF

many optical elements

shield field

8D(4D) thick, shift variant,

angular variant

thin, shift variant,

angular variant

thin, shift variant,

angular invariant

attenuation

6D(3D)

4D(2D)

2D(1D)

1

• the most generalized case

8D LF Transformer

1

• For thin optical elements

6D LF Transformer

6D Display

Bidirectional

Texture Function

Courtesy of Paul Debevec

Courtesy of Martin Fuchs

1

4D LF Transformer

• w/ angle shift invariant elements (in the paraxial region)– e.g. aperture, lens, thin grating, etc

Part 4Applications in Imaging

1

Message

• LF is a very powerful tool to understand wave-related phenomena

– and potentially design and develop new systems and applications

1

Augmented LF

WDF

Light

Field

Augmented LF

LF

propagation

(diffractive)

optical

element

LF LF LF LF

LF

propagation


negative radiance

Outline

wavefront coding

holography

gaussian beam

rotating PSF

1

Gaussian Beam (from a laser pointer)

• Beam from a laser

– a solution of paraxial wave equation

20 mm beam

width

20 m distance

1

• ALF (and WDF) of the Gaussian Beam is also Gaussian in x-θspace

Gaussian Beam

1

Gaussian Beam

x-θ space z-x space

20 mm beam

width

20 m

distance

1

Wavefront coding

• ALF of a phase mask(slowly varying ϕ(x))

conventional wavefront coding

extended DOF

(w/ deconvolution)

1

Unusual PSF for depth from defocus

standard PSF DH PSF

Courtesy of S. R. P. Pavani

U. of Colorado@Boulder

Prof. Rafael Piestun’s group

Univ. of Colorado@Boulder

Defocus circle with distance

1

Rotating PSF

• Rotating beams

– Superposition along a straight line

– Rotation rate related to slope of line

– Both intensity and phase rotate

– Maximum rotation rate in Rayleigh range

intensity


1


Rotating PSF

1

Conceptually...

1

Conceptually...

other modes need to be balanced...

1

WDF (ALF) of (1,1) order

intensity

R. Simon and G. S. Agarwal, "Wigner

representation of Laguerre-Gaussian beams", Opt.

Lett., 25(18), (2000)

1

θx

θy

WDF in θx- θy

intensity in x-

y

y

x θx

θy

WDF in θx- θy

θx

θy

WDF in θx- θy

1

Holography

laser

object wave

reference wave

Recording

hologram

Reconstruction

reference wave

hologram

observer

real image

virtual image

object

1

Holography

recording

reconstruction

• For a point object

1

Future direction

• Tomography & Inverse problems

• Beam shaping/phase mask design by ray-based optimization

• New processing w/ virtual light source

Other LF

representations

WDF

Traditional

light field

Augmented

LF

Observable

LF

Rihaczek

Distribution

Function

Space of LF representations

Time-frequency representations

Phase space representations

Quasi light field

incoherent

coherent

Other LF

representations

1

Property of the RepresentationConstant

along rays

Non-

negativityCoherence Wavelength

Interference

Cross term

Traditional LFalways

constantalways positive

only incoherent zero no

Observable

LF

nearly constant

always positive

any coherence

stateany yes

Augmented

LF

only in the paraxial region

positive and negative any any yes

WDFonly in the paraxial region

positive and negative any any yes

Rihaczek DFno; linear

driftcomplex any any reduced

1

Benefits & Limitations of the Representation

Ability to

propagate

Modeling

wave optics

Simplicity of

computatio

n

Adaptability

to current

pipe line

Near Field Far Field

Traditional

Light Fieldsx-shear no

very simple

high no yes

Observable

Light Fields

not x-shear

yes modest low yes yes

Augmented

Light Fieldsx-shear yes modest high no yes

WDF x-shear yes modest low yes yes

Rihaczek

DFx-shear yes

better than WDF, not as simple

as LF

low no yes

1

Conclusions

• Wave optics phenomena can be understood with geometrical ray based representation

• There are many different phase-space representations

• We hope to inspire researchers in computer vision/graphics as well as in optics graphics to develop new tools and algorithms based on joint exploration of geometric and wave optics concepts

Documents

Techreport Slides