Light Fields in Ray and Wave Optics Introduction to Light Fields: - PowerPoint PPT Presentation

Space of LF representations Time-frequency representations Phase space representations Quasi light field Other LF representations WDF Augmented Observable LF LF Other LF representations Traditional light field incoherent Rihaczek Distribution Function coherent 34

Property of the Representation Constant along Interference Non-negativity Coherence Wavelength rays Cross term always always positive only Traditional LF zero no constant incoherent Observable LF nearly constant always positive any coherence any yes state Augmented LF only in the paraxial region positive and any any yes negative only in the paraxial region positive and WDF any any yes negative Rihaczek DF no; linear drift complex any any reduced 35

Benefits & Limitations of the Representation Adaptability Ability to Modeling Simplicity of to current Near Field Far Field propagate wave optics computation pipe line Traditional x-shear no very simple high no yes LF Observable not x-shear yes modest low yes yes LF Augmented x-shear yes modest high no yes LF WDF x-shear yes modest low yes yes better than Rihaczek DF x-shear yes WDF, not as low no yes simple as LF 36

Observable Light Field 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � 37

Observable Light Field 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � Fourier transform 37

Observable Light Field 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � wave Fourier transform 37

Observable Light Field aperture window 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � wave Fourier transform 37

Observable Light Field aperture window 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � power obs ( s, u ) = � � � � wave Fourier transform 37

Observable Light Field 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � l ( T ) s, u − s, u � � � � obs ( s, u ) = W U ⊗ W T λ λ 37

Observable Light Field 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � l ( T ) s, u − s, u � � � � obs ( s, u ) = W U ⊗ W T λ λ Wigner distribution of wave function 37

Observable Light Field 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � l ( T ) s, u − s, u � � � � obs ( s, u ) = W U ⊗ W T λ λ Wigner distribution Wigner distribution of wave function of aperture window 37

Observable Light Field 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � blur trades off resolution in position with direction l ( T ) s, u − s, u � � � � obs ( s, u ) = W U ⊗ W T λ λ Wigner distribution Wigner distribution of wave function of aperture window 37

Observable Light Field at zero wavelength limit (regime of ray optics) l ( T ) s, u � � − s, u � � obs ( s, u ) = W U W T ⊗ λ λ Wigner distribution Wigner distribution of wave function of aperture window 38

Observable Light Field at zero wavelength limit (regime of ray optics) l ( T ) s, u � � obs ( s, u ) = W U ⊗ δ ( − s, u ) λ Wigner distribution of wave function 38

Observable Light Field at zero wavelength limit (regime of ray optics) l ( T ) s, u � � obs ( s, u ) = W U λ observable light field and Wigner equivalent! 38

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 39

Application - Refocusing u s light field 40

Application - Refocusing Isaksen et. al u 2000 s light field 40

Application - Refocusing Isaksen et. al u 2000 s light field image at z=0 40

Application - Refocusing Isaksen et. al u 2000 s light field image at z=z 0 40

Application - Refocusing Isaksen et. al u f u 2000 light s f s light Fourier field field spectrum 40

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum 40

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=0 40

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=z 0 40

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum f ξ ξ Wigner ambiguity f x x Fourier distribution function 40

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=0 f ξ ξ Wigner ambiguity f x x Fourier distribution function 40

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=z 0 f ξ ξ Wigner ambiguity f x x Fourier distribution function 40

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=0 f ξ ξ Wigner ambiguity f x x Fourier distribution function 40

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=z 0 f ξ ξ Wigner ambiguity f x x Fourier distribution function 40

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum Papoulis f ξ ξ 1974 Wigner ambiguity f x x Fourier distribution function 40

Application - Wavefront Coding Dowski and Cathey 1995 same aberrant blur regardless of depth of focus 41

Application - Wavefront Coding Dowski and Cathey 1995 point in scene same aberrant blur regardless of depth of focus 41

Application - Wavefront Coding Dowski and Cathey 1995 point cubic in scene phase plate same aberrant blur regardless of depth of focus 41

Application - Wavefront Coding Dowski and Cathey 1995 point small change cubic in scene in blur shape phase plate same aberrant blur regardless of depth of focus 41

Application - Wavefront Coding slices corresponding to various depths ambiguity function 42

Application - Wavefront Coding 43