Wave Phenomenon in
 Geometric Optics
       Tom Cuypers
        Se Baek Oh
    Roarke Horstmeyer
      Ramesh Raskar
Part 1
Introduction
Overview
1.   Introduction and Welcome
2.   Relating wave propagation to Light Fields
3.   Augmented Light Fields
4.   App...
Motivation
• Dual representation of light:
  – Photons travelling in a straight line
                 Computational Photog...
Motivation
• Dual representation of light:
  – Photons travelling in a straight line
  – Waves traveling in all directions...
Motivation
• Dual representation of light:
  – Photons travelling in a straight line
  – Waves traveling in all directions...
Wave phenomena in the real world
• Fluid surfaces




           http://4.bp.blogspot.com/_NpINLHeo8rM/Rsl52vjOKII/AAAAAAA...
Wave phenomena in the real world
• Fluid surfaces

• Sound waves




         http://fetch1.com/wp-content/uploads/2009/11...
Wave phenomena in the real world
• Fluid surfaces

• Sound waves

• Electromagnetic waves
  – Microscopic scale



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

• Correlation of two points on w...
Coherence
• throwing stones......




   single point source          many point sources
       ⇒ coherent         ⇒ if th...
Coherence
• Temporal coherence:
  – spectral bandwidth
    • monochromatic: temporally coherent
    • broadband (white lig...
Example
Temporally incoherent;           Temporally &
  spatially coherent           spatially coherent




      Temporal...
What is a wave?
• Types
  – Electromagnetic waves
  – Mechanical Waves




                 http://en.wikipedia.org/wiki/F...
What is a wave?
• Types
  – Electromagnetic waves
  – Mechanical Waves




                http://www.gi.alaska.edu/chapar...
What is a wave?
• Types                                                            λ
• Properties                        A...
What are wave phenomena?
• Huygens principle
What are wave phenomena?
• Huygens principle
What are wave phenomena?
• Huygens principle
What are wave phenomena?
• Huygens principle
What are wave phenomena?
• Huygens principle
• Diffraction
What are wave phenomena?
• Huygens principle
• Diffraction
What are wave phenomena?
• Huygens principle
• Diffraction
What are wave phenomena?
• Huygens principle
• Diffraction
What are wave phenomena?
• Huygens principle
• Diffraction
What are wave phenomena?
• Huygens principle     Wave A

• Diffraction           Wave B

• Interference

                 ...
What are wave phenomena?
• Huygens principle     Wave A

• Diffraction           Wave B

• Interference

                 ...
What are wave phenomena?
•   Huygens principle
•   Diffraction
•   Interference
•   Example                 N


          ...
What are wave phenomena?
•   Huygens principle
•   Diffraction
•   Interference
•   Example


                           R...
What are wave phenomena?
•   Huygens principle
•   Diffraction
•   Interference
•   Example


                           R...
What are wave phenomena?
•   Huygens principle
•   Diffraction
•   Interference
•   Example


                           R...
What are wave phenomena?
•   Huygens principle
•   Diffraction
•   Interference
•   Example


                           R...
What are wave phenomena?
•   Huygens principle
•   Diffraction
•   Interference
•   Example


                           R...
What are wave phenomena?
•   Huygens principle
•   Diffraction
•   Interference
•   Example


                           R...
Part 2
Relating Wave Phenomena
       to Light Fields
Introduction
• Review of Light Fields
• Review of Waves using Fourier optics
  principles ? (intro)
• Introduction to the ...
Plenoptic Function




• Q: What is the set of all things that we can ever see?
• A: The Plenoptic Function (Adelson & Ber...
Gray Snapshot




• P(θ,φ) is intensity of light
– Seen from a single view point
– At a single time
– Averaged over the wa...
Color Snapshot




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




P(θ,φ,λ,t,Vx, Vy, Vz)
• Can reconstruct every possible view, at every moment, from every position,
...
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 Objects
  P(θ,φ,VX,VY,VZ)
• 4D
  2D position
  – 2D direction
• The space of all lines in 3-D space i...
Representation

        (θ,φ)
                                     (u,v)
(x,y)                    (x,y)




 Position-angl...
Light Field Camera




Point Grey
Mark levoy
Why Study Light Fields Using Wave Optics?
                     z=z0       θ

                                       x
    ...
Wave Optics
• Waves instead of rays        Parallel rays   Plane waves


• Interference & diffraction

• Plane of point em...
Position and direction in wave optics

• Spatial frequency: f




                                 1
                     ...
Position and direction in wave optics

• Spatial frequency: f

• Direction of wave: θ
                            λ

Small...
Position and direction in wave optics




Complex wavefront   =   parallel wavefronts
Wigner Distribution Function

        Auto correlation of complex wavefront

• Input: one-dimensional function of position...
Wigner Distribution Function




    .
    .
    .

    .
    .
    .
2D Wigner Distribution
             • Projection along
               frequency yields power
             • Projection alo...
2D Wigner Distribution
    |h(x)|²     • Projection along
                  frequency yields power
        x       • Proje...
2D Wigner Distribution
    |h(x)|²     • Projection along
                  frequency yields power
        x       • Proje...
2D Wigner Distribution
    |h(x)|²     • Projection along
                  frequency yields power
        x       • Proje...
2D Wigner Distribution
             Remarks:
             • Possible negative values
             • Uncertainty principle
...
Relationship with Light Fields:
      Observable Light Fields
• Move aperture
  across plane
• Look at direction
  spread
...
Relationship with Light Fields:
       Observable Light Fields
• Move aperture
  across plane
• Look at direction
  spread...
Relationship with Light Fields:
      Observable Light Fields
• Move aperture
  across plane
• Look at direction
  spread
...
Relationship with Light Fields:
      Observable Light Fields
• Move aperture
  across plane
• Look at direction
  spread
...
Relationship with Light Fields:
      Observable Light Fields
• Move aperture
  across plane
• Look at direction
  spread
...
Relationship with Light Fields:
      Observable Light Fields
• Move aperture
  across plane
• Look at direction
  spread
...
Relationship with Light Fields:
      Observable Light Fields
• Move aperture
  across plane
• Look at direction
  spread
...
Relationship with Light Fields:
   Observable Light Fields
Relationship with Light Fields:
   Observable Light Fields
           Aperture Window       Power


       Wave      Fouri...
Relationship with Light Fields:
   Observable Light Fields
           Aperture Window       Power


       Wave      Fouri...
Relationship with Light Fields:
   Observable Light Fields
                Aperture Window       Power


           Wave  ...
Relationship with Light Fields:
   Observable Light Fields

             Blur trades off
          resolution in position
...
Relationship with Light Fields:
   Observable Light Fields

         At zero wavelength limit
          (regime of ray opt...
Relationship with Light Fields:
   Observable Light Fields

         At zero wavelength limit
          (regime of ray opt...
Observable Light Field
• Observable light field is a blurred Wigner
  distribution with a modified coordinate
  system
• B...
Light Fields and Wigner
• Observable Light Fields = special case of
  Wigner
• Ignores wave phenomena
• Can we also introd...
Part 3
Augmenting Light Fields
Introduction


                                   light field

                       position                radiance of ...
Introduction


                                   light field
                                   direction
               ...
Introduction




     Traditional
     Light Field


ray optics based
simple and powerful
Introduction
                    rigorous but cumbersome
                    wave optics based

                    Wigner...
Introduction
                    rigorous but cumbersome
                    wave optics based

                    Wigner...
Augmented LF
                    rigorous but cumbersome
                    wave optics based

                    Wigner...
Augmented LF
• Not a new light field
• A new methodology/framework to create,
  modulate, and propagate light fields
  – s...
Augmented LF framework

LF


         (diffractive)
            optical
           element
Augmented LF framework

LF            LF


                   (diffractive)
                      optical
                ...
Augmented LF framework
                    light field
                  transformer

LF            LF                    ...
Augmented LF framework
                    light field
                  transformer

LF            LF                    ...
Outline
• Limitations of Light Field analysis
  – Ignore wave phenomena
  – Only positive ray -> no interference
Outline
• Limitations of Light Field analysis
• Augmented Light Field
  – free-space propagation
Outline
• Limitations of Light Field analysis
• Augmented Light Field
  – free-space propagation
  – virtual light project...
Outline
• Limitations of Light Field analysis
• Augmented Light Field
  – free-space propagation
  – virtual light project...
Assumptions
• Monochromatic (= temporally coherent)
  – can be extended into polychromatic
• Flatland (= 1D observation pl...
Young’s experiment

                                        screen
light from   double
  a laser      slit




           ...
Young’s experiment

                                         screen
light from   double
  a laser      slit
              ...
Young’s experiment




                        Light Field   WDF



ref. plane
Young’s experiment
             projection    projection




             Light Field    WDF



ref. plane
Virtual light projector
                                              projection




        real projector
              ...
Virtual light projector

                                          first null
        real projector                    (O...
Virtual light projector


                      hyperbola    first null
                                   (OPD = λ/2)
   ...
Virtual light projector
in high school physics          destructive interference
                             (need negati...
Question
• Does a virtual light projector also work for
  incoherent light?
• Yes!
Temporal coherence
• Broadband light is incoherent
• ALF (also LF and WDF) can be defined
  for different wavelength and t...
Young’s Exp. w/ white light
Young’s Exp. w/ white light
                     Red




                     Green




                     Blue
Young’s Exp. w/ white light
                     Red




                     Green




                     Blue
Spatial coherence
• ALF w/ virtual light projectors is
  defined for spatially coherent light
• For partially coherent/inc...
Young’s Exp. w/ spatially
    incoherent light
Young’s Exp. w/ spatially
    incoherent light
Young’s Exp. w/ spatially
    incoherent light
Young’s Exp. w/ spatially
             incoherent light


  w/ random
    phase
(uncorrelated)




                 spatia...
Young’s Exp. w/ spatially
       incoherent light


  w/ random
    phase             Addition
(uncorrelated)
Young’s Exp. w/ spatially
       incoherent light


  w/ random
    phase             Addition
(uncorrelated)
Light Field Transformer
• light field interactions w/ optical elements




                        Light field transformer
Light Field Transformer
Dimension      Property                     Note
 8D(4D)     thick, shift variant,    8D reflectan...
8D LF Transformer
• the most generalized case
6D LF Transformer
• For thin optical elements
                              6D Display


                                 ...
4D LF Transformer
• w/ angle shift invariant elements (in the
  paraxial region)
  – e.g. aperture, lens, thin grating, etc
Part 4
Applications in Imaging
Message
• LF is a very powerful tool to understand
  wave-related phenomena
  – and potentially design and develop new sys...
Augmented LF

                                               light field
                                             tran...
Outline
gaussian beam             wavefront coding




rotating PSF                holography
Gaussian Beam
          (from a laser pointer)
  • Beam from a laser
    – a solution of paraxial wave equation



20 mm b...
Gaussian Beam
• ALF (and WDF) of the Gaussian Beam is also Gaussian in x-θ
  space
Gaussian Beam

 x-θ space     z-x space




20 mm beam       20 m
   width       distance
Wavefront coding
 • ALF of a phase mask(slowly varying ϕ(x))


conventional    wavefront coding




                  exte...
Unusual PSF for depth from
                      defocus
                                          standard PSF    DH PSF
...
Rotating PSF

• Rotating beams
  – Superposition along a straight line
  – Rotation rate related to slope of
    line
  – ...
Rotating PSF




               Courtesy of S. R. P. Pavani
Conceptually...
Conceptually...




other modes need to be balanced...
WDF (ALF) of (1,1) order

                              intensity




                 R. Simon and G. S. Agarwal, "Wigner...
WDF in θx- θy




θy                        intensity in x-           WDF in θx- θy
                                y
    ...
Holography
Recording                  Reconstruction
              laser                         virtual
 object
         ...
Holography
• For a point object




    recording

                        reconstruction
Future direction
• Tomography & Inverse problems
• Beam shaping/phase mask design by ray-
  based optimization
• New proce...
Space of LF representations
Time-frequency representations
Phase space representations
Quasi light field

                ...
Property of the Representation
                  Constant        Non-                                  Interference
      ...
Benefits & Limitations of the Representation
                                     Simplicity of Adaptability
             ...
Conclusions
• Wave optics phenomena can be understood with
  geometrical ray based representation
• There are many differe...
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Techreport Slides

  1. 1. Wave Phenomenon in Geometric Optics Tom Cuypers Se Baek Oh Roarke Horstmeyer Ramesh Raskar
  2. 2. Part 1 Introduction
  3. 3. Overview 1. Introduction and Welcome 2. Relating wave propagation to Light Fields 3. Augmented Light Fields 4. Applications in Imaging
  4. 4. Motivation • Dual representation of light: – Photons travelling in a straight line Computational Photography Computer Graphics http://graphics.stanford.edu/projects/lightfield http://graphics.ucsd.edu/~henrik/images
  5. 5. Motivation • Dual representation of light: – Photons travelling in a straight line – Waves traveling in all directions Optics Holography http://www.humanproductivitylab.com/images
  6. 6. Motivation • Dual representation of light: – Photons travelling in a straight line – Waves traveling in all directions • Goal of the course: Provide a gentle introduction of wave phenomenon using ray-based representations
  7. 7. Wave phenomena in the real world • Fluid surfaces http://4.bp.blogspot.com/_NpINLHeo8rM/Rsl52vjOKII/AAAAAAAAFMM/WnESejvzq5Y/s400/s plash-water-waves-4559.JPG
  8. 8. Wave phenomena in the real world • Fluid surfaces • Sound waves http://fetch1.com/wp-content/uploads/2009/11/hd-800_detail_sound-waves1.jpg
  9. 9. Wave phenomena in the real world • Fluid surfaces • Sound waves • Electromagnetic waves – Microscopic scale http://upload.wikimedia.org/wikipedia/commons/archive/1/1f/20090127195426!Ggb_in_soap_bubble_1.jpg
  10. 10. 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
  11. 11. Coherence • throwing stones...... single point source many point sources ⇒ coherent ⇒ if thrown identically, still coherent! ⇒ if thrown randomly, then incoherent!
  12. 12. 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
  13. 13. Example Temporally incoherent; Temporally & spatially coherent spatially coherent Temporally & Temporally coherent; spatially incoherent spatially incoherent rotating diffuser laser
  14. 14. What is a wave? • Types – Electromagnetic waves – Mechanical Waves http://en.wikipedia.org/wiki/File:EM_spectrum.svg
  15. 15. What is a wave? • Types – Electromagnetic waves – Mechanical Waves http://www.gi.alaska.edu/chaparral/acousticspectrum.jpg
  16. 16. What is a wave? • Types λ • Properties A – Wavelength: λ – Frequency : p=0 p=π/2 – Phase: p p=π p=3π/2 – Amplitude: A – Polarization http://www.ccrs.nrcan.gc.ca/glossary/images/3104.gif
  17. 17. What are wave phenomena? • Huygens principle
  18. 18. What are wave phenomena? • Huygens principle
  19. 19. What are wave phenomena? • Huygens principle
  20. 20. What are wave phenomena? • Huygens principle
  21. 21. What are wave phenomena? • Huygens principle • Diffraction
  22. 22. What are wave phenomena? • Huygens principle • Diffraction
  23. 23. What are wave phenomena? • Huygens principle • Diffraction
  24. 24. What are wave phenomena? • Huygens principle • Diffraction
  25. 25. What are wave phenomena? • Huygens principle • Diffraction
  26. 26. What are wave phenomena? • Huygens principle Wave A • Diffraction Wave B • Interference Constructive interference
  27. 27. What are wave phenomena? • Huygens principle Wave A • Diffraction Wave B • Interference Destructive interference
  28. 28. What are wave phenomena? • Huygens principle • Diffraction • Interference • Example N Reflection Ray-based
  29. 29. What are wave phenomena? • Huygens principle • Diffraction • Interference • Example Reflection Huygens Principle
  30. 30. What are wave phenomena? • Huygens principle • Diffraction • Interference • Example Reflection Huygens Principle
  31. 31. What are wave phenomena? • Huygens principle • Diffraction • Interference • Example Reflection Huygens Principle
  32. 32. What are wave phenomena? • Huygens principle • Diffraction • Interference • Example Reflection Huygens Principle
  33. 33. What are wave phenomena? • Huygens principle • Diffraction • Interference • Example Reflection Huygens Principle
  34. 34. What are wave phenomena? • Huygens principle • Diffraction • Interference • Example Reflection Huygens Principle
  35. 35. Part 2 Relating Wave Phenomena to Light Fields
  36. 36. 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
  37. 37. 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…
  38. 38. 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)
  39. 39. 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
  40. 40. Movie P(θ,φ,λ,t) is intensity of light – Seen from a single view point – Over time – As a function of wavelength
  41. 41. 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
  42. 42. 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.
  43. 43. Sampling Plenoptic Function (top view)
  44. 44. Ray Let’s not worry about time and color: 5D : P(θ,φ,VX,VY,VZ) • – 3D position • – 2D direction
  45. 45. Ray • No Occluding Objects P(θ,φ,VX,VY,VZ) • 4D 2D position – 2D direction • The space of all lines in 3-D space is 4D.
  46. 46. Representation (θ,φ) (u,v) (x,y) (x,y) Position-angle 2 plane representation representation
  47. 47. Light Field Camera Point Grey Mark levoy
  48. 48. Why Study Light Fields Using Wave Optics? z=z0 θ x Light z=0 Field Macro Micro f z=z0 x Wigner z=0 Distribution
  49. 49. Wave Optics • Waves instead of rays Parallel rays Plane waves • Interference & diffraction • Plane of point emitters (Huygen’s principle) • Each emitter has amplitude and phase
  50. 50. Position and direction in wave optics • Spatial frequency: f 1 f
  51. 51. Position and direction in wave optics • Spatial frequency: f • Direction of wave: θ λ Small θ assumption: θ 1 f
  52. 52. Position and direction in wave optics Complex wavefront = parallel wavefronts
  53. 53. Wigner Distribution Function Auto correlation of complex wavefront • Input: one-dimensional function of position • Output: two-dimensional function of position and spatial frequency • (some) information about spectrum at each position
  54. 54. Wigner Distribution Function . . . . . .
  55. 55. 2D Wigner Distribution • Projection along frequency yields power • Projection along position yield spectral power f W(x,f) x
  56. 56. 2D Wigner Distribution |h(x)|² • Projection along frequency yields power x • Projection along position yield spectral power f W(x,f) x
  57. 57. 2D Wigner Distribution |h(x)|² • Projection along frequency yields power x • Projection along position yield spectral power f W(x,f) f |f(x)|² x
  58. 58. 2D Wigner Distribution |h(x)|² • Projection along frequency yields power x • Projection along position yield spectral power f W(x,f) f |f(x)|² x
  59. 59. 2D Wigner Distribution Remarks: • Possible negative values • Uncertainty principle f W(x,f) x
  60. 60. Relationship with Light Fields: Observable Light Fields • Move aperture across plane • Look at direction spread • Continuous form of plenoptic Scene camera
  61. 61. Relationship with Light Fields: Observable Light Fields • Move aperture across plane • Look at direction spread • Continuous form of plenoptic Scene camera
  62. 62. Relationship with Light Fields: Observable Light Fields • Move aperture across plane • Look at direction spread • Continuous form of plenoptic Scene camera
  63. 63. Relationship with Light Fields: Observable Light Fields • Move aperture across plane • Look at direction spread • Continuous form of plenoptic Scene camera
  64. 64. Relationship with Light Fields: Observable Light Fields • Move aperture across plane • Look at direction spread • Continuous form of plenoptic Scene camera
  65. 65. Relationship with Light Fields: Observable Light Fields • Move aperture across plane • Look at direction spread • Continuous form of plenoptic Scene camera
  66. 66. Relationship with Light Fields: Observable Light Fields • Move aperture across plane • Look at direction spread • Continuous form of plenoptic Scene θ camera Aperture Position x
  67. 67. Relationship with Light Fields: Observable Light Fields
  68. 68. Relationship with Light Fields: Observable Light Fields Aperture Window Power Wave Fourier Transform
  69. 69. Relationship with Light Fields: Observable Light Fields Aperture Window Power Wave Fourier Transform
  70. 70. Relationship with Light Fields: Observable Light Fields Aperture Window Power Wave Fourier Transform Wigner Distribution Wigner Distribution of wave function of aperture window
  71. 71. Relationship with Light Fields: Observable Light Fields Blur trades off resolution in position with direction Wigner Distribution Wigner Distribution of wave function of aperture window
  72. 72. Relationship with Light Fields: Observable Light Fields At zero wavelength limit (regime of ray optics) Wigner Distribution of wave function
  73. 73. Relationship with Light Fields: Observable Light Fields At zero wavelength limit (regime of ray optics) Observable light field and Wigner equivalent!
  74. 74. 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
  75. 75. 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
  76. 76. Part 3 Augmenting Light Fields
  77. 77. Introduction light field position radiance of ray Traditional Light Field ray optics based simple and powerful ref. plane
  78. 78. Introduction light field direction position radiance of ray Traditional Light Field ray optics based simple and powerful ref. plane
  79. 79. Introduction Traditional Light Field ray optics based simple and powerful
  80. 80. Introduction rigorous but cumbersome wave optics based Wigner Distribution Function Traditional Light Field ray optics based simple and powerful limited in diffraction & interference
  81. 81. Introduction rigorous but cumbersome wave optics based Wigner Distribution Function holograms beam shaping Traditional Light Field ray optics based rotational PSF simple and powerful limited in diffraction & interference
  82. 82. Augmented LF rigorous but cumbersome wave optics based Wigner WDF Distribution Function Augmented LF Traditional Traditional Light Field Light Field ray optics based simple and powerful Interference & Diffraction limited in diffraction & interference Interaction w/ optical elements Non-paraxial propagation
  83. 83. 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
  84. 84. Augmented LF framework LF (diffractive) optical element
  85. 85. Augmented LF framework LF LF (diffractive) optical element LF propagation
  86. 86. Augmented LF framework light field transformer LF LF LF negative radiance (diffractive) optical element LF propagation
  87. 87. Augmented LF framework light field transformer LF LF LF LF negative radiance (diffractive) optical element LF propagation LF propagation Tech report, S. B. Oh et al.
  88. 88. Outline • Limitations of Light Field analysis – Ignore wave phenomena – Only positive ray -> no interference
  89. 89. Outline • Limitations of Light Field analysis • Augmented Light Field – free-space propagation
  90. 90. Outline • Limitations of Light Field analysis • Augmented Light Field – free-space propagation – virtual light projector in the ALF • Possible negative • Coherence
  91. 91. Outline • Limitations of Light Field analysis • Augmented Light Field – free-space propagation – virtual light projector in the ALF • Possible negative • Coherence – light field transformer
  92. 92. 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)
  93. 93. Young’s experiment screen light from double a laser slit constructive interference
  94. 94. Young’s experiment screen light from double a laser slit destructive interference
  95. 95. Young’s experiment Light Field WDF ref. plane
  96. 96. Young’s experiment projection projection Light Field WDF ref. plane
  97. 97. Virtual light projector projection real projector negative virtual light projector positive at the mid point real projector Augmented LF intensity=0 Not conflict with physics
  98. 98. Virtual light projector first null real projector (OPD = λ/2) virtual light projector real projector
  99. 99. Virtual light projector hyperbola first null (OPD = λ/2) asymptote of λ/2 hyperbola valid in Fresnel regime (or paraxial)
  100. 100. Virtual light projector in high school physics destructive interference (need negative radiance from class, virtual light projector) Video waves
  101. 101. Question • Does a virtual light projector also work for incoherent light? • Yes!
  102. 102. Temporal coherence • Broadband light is incoherent • ALF (also LF and WDF) can be defined for different wavelength and treated independently
  103. 103. Young’s Exp. w/ white light
  104. 104. Young’s Exp. w/ white light Red Green Blue
  105. 105. Young’s Exp. w/ white light Red Green Blue
  106. 106. 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!
  107. 107. Young’s Exp. w/ spatially incoherent light
  108. 108. Young’s Exp. w/ spatially incoherent light
  109. 109. Young’s Exp. w/ spatially incoherent light
  110. 110. Young’s Exp. w/ spatially incoherent light w/ random phase (uncorrelated) spatially incoherent light: infinite number of waves propagating along all the direction with random phase delay
  111. 111. Young’s Exp. w/ spatially incoherent light w/ random phase Addition (uncorrelated)
  112. 112. Young’s Exp. w/ spatially incoherent light w/ random phase Addition (uncorrelated)
  113. 113. Light Field Transformer • light field interactions w/ optical elements Light field transformer
  114. 114. Light Field Transformer Dimension Property Note 8D(4D) thick, shift variant, 8D reflectance field, angular variant volume hologram 6D(3D) thin, shift variant, 6D display, angular variant BTF 4D(2D) thin, shift variant, many optical elements angular invariant 2D(1D) attenuation shield field
  115. 115. 8D LF Transformer • the most generalized case
  116. 116. 6D LF Transformer • For thin optical elements 6D Display Courtesy of Martin Fuchs Bidirectional Texture Function Courtesy of Paul Debevec
  117. 117. 4D LF Transformer • w/ angle shift invariant elements (in the paraxial region) – e.g. aperture, lens, thin grating, etc
  118. 118. Part 4 Applications in Imaging
  119. 119. Message • LF is a very powerful tool to understand wave-related phenomena – and potentially design and develop new systems and applications
  120. 120. Augmented LF light field transformer WDF LF LF LF LF negative radiance Augmented LF (diffractive) optical element Light Field LF LF propagation propagation
  121. 121. Outline gaussian beam wavefront coding rotating PSF holography
  122. 122. Gaussian Beam (from a laser pointer) • Beam from a laser – a solution of paraxial wave equation 20 mm beam width 20 m distance
  123. 123. Gaussian Beam • ALF (and WDF) of the Gaussian Beam is also Gaussian in x-θ space
  124. 124. Gaussian Beam x-θ space z-x space 20 mm beam 20 m width distance
  125. 125. Wavefront coding • ALF of a phase mask(slowly varying ϕ(x)) conventional wavefront coding extended DOF (w/ deconvolution)
  126. 126. Unusual PSF for depth from defocus standard PSF DH PSF Defocus circle with distance Prof. Rafael Piestun’s group Courtesy of S. R. P. Pavani Univ. of Colorado@Boulder U. of Colorado@Boulder
  127. 127. 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 Courtesy of S. R. P. Pavani
  128. 128. Rotating PSF Courtesy of S. R. P. Pavani
  129. 129. Conceptually...
  130. 130. Conceptually... other modes need to be balanced...
  131. 131. WDF (ALF) of (1,1) order intensity R. Simon and G. S. Agarwal, "Wigner representation of Laguerre-Gaussian beams", Opt. Lett., 25(18), (2000)
  132. 132. WDF in θx- θy θy intensity in x- WDF in θx- θy y θx y θy x θx WDF in θx- θy θy θx
  133. 133. Holography Recording Reconstruction laser virtual object image object wave real image reference reference wave hologram wave hologram observer
  134. 134. Holography • For a point object recording reconstruction
  135. 135. Future direction • Tomography & Inverse problems • Beam shaping/phase mask design by ray- based optimization • New processing w/ virtual light source
  136. 136. Space of LF representations Time-frequency representations Phase space representations Quasi light field Other LF representations Observable LF WDF Augmented LF Other LF Traditional representations light field incoherent Rihaczek Distribution Function coherent
  137. 137. Property of the Representation Constant Non- Interference Coherence Wavelength along rays negativity Cross term always always only Traditional LF constant positive incoherent zero no Observable nearly always any constant positive coherence any yes LF state Augmented only in the positive and paraxial negative any any yes LF region only in the positive and WDF paraxial negative any any yes region no; linear Rihaczek DF complex any any reduced drift
  138. 138. Benefits & Limitations of the Representation Simplicity of Adaptability Ability to Modeling computatio to current Near Field Far Field propagate wave optics n pipe line Traditional very Light Fields x-shear no simple high no yes Observable not x- yes modest low yes yes Light Fields shear Augmented Light Fields x-shear yes modest high no yes WDF x-shear yes modest low yes yes better than Rihaczek WDF, not DF x-shear yes as simple low no yes as LF
  139. 139. 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

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