Short Course. Phase-Space Optics and Modern Imaging Systems. Markus Testorf Thayer School of Dartmouth College Hanover, NH, USA

Transcription

1 Short Course Phase-Space Optics and Modern Imaging Systems Markus Testorf Thayer School of Dartmouth College Hanover, NH, USA Escuela de Óptica Moderna XI, INAOE April 25-29, 2011

2 Lecture 5 Unification of Optical Models Partial Coherence and Discrete Representations Markus Testorf markus.e.testorf@dartmouth.edu Escuela de Óptica Moderna XI, INAOE April 25-29, 2011

3 Review of Phase-Space Reconstruction Incoherent and Partially Coherent Imaging Self-Imaging Notes About Discrete Phase Space Representations Concluding Remarks Lecture 5

4 Review of Phase Space Reconstruction

5 Review of Phase Space Reconstruction 1. Phase-Space Tomography

6 º x Intensity Measurements at Fractional Fourier Planes = Projections of the WDF at a different of Angles. Projection Tomography applicable to WDF Synthesis WDF from Intensity Measurements Linear Phase Retrieval Phase-Space Tomography

7 Review of Phase Space Reconstruction 2. Integral Imaging (Incoherent, Geom. Optics)

8 f x µ x µ z Lateral Position: x x=2 Ray Direction: µ µ=2 tan µ = x µ x f Lens Array Detector Array Measuring Phase Space

9 º x º x Single shot measurement Phase-space resolution determined by instrument function Phase space resolution subject to uncertainty relation º ¼ µ f Phase-Space Detector

10 Review of Phase Space Reconstruction 3. Shack-Hartmann Sensor (Coherent, Geom. Optics)

11 x A L z L D Shack-Hartmann Sensor

12 x u(x) = exp[iá(x)] A L z L D Shack-Hartmann Sensor

13 x A L z u(x) = exp[iá(x)] Smooth Wavefront (Quasi-Geometrical Optics)! 0 L D Shack-Hartmann Sensor

14 x A L z u(x) = exp[iá(x)] Smooth Wavefront (Quasi-Geometrical Optics)! 0 Measurement of Local Tilt (Local Frequency) L D Shack-Hartmann Sensor

15 x A L z u(x) = exp[iá(x)] Smooth Wavefront (Quasi-Geometrical Optics)! 0 Measurement of Local Tilt (Local Frequency) L D Phase Space of Quasi-Geometrical Optics: W (x; º) ¼ ± º 1 2¼ d dx Á(x) Shack-Hartmann Sensor

16 x A L º W c (x; º) z x L D u(x) = exp[iá(x)] W (x; º) ¼ ± º 1 2¼ d dx Á(x) Shack-Hartmann Sensor

17 x A L º W c (x; º) z x L D Coherent Wavefronts!! Shack-Hartmann Sensor

18 Incoherent and Partially Coherent Imaging

19 x 0 º x Input Signal ¾ Output Signal z u in (x) ~u in (º) ~ h(º) = ~u out (º) u out (x) Linear System Coherent Image Formation

20 x 0 º x Point Source ¾ Point Response z ±(x) ~ h(º) = ±(º ¾=2) ±(º ¾=2) h(x) = 2 cos(¼x¾) Coherent Image Formation

21 x 0 º x Point Source ¾ Point Response z ±(x) jh(x)j 2 = 4 cos 2 (¼x¾) Incoherent Image Formation

22 x 0 º x Point Source ¾ Point Response z ±(x) OT F (º) = Z ~h(º 0 ) ~ h(º 0 º)dº 0 = jh(x)j 2 = 4 cos 2 (¼x¾) = 2±(º) ±(º ¾) ±(º ¾) Incoherent Image Formation

23 x 0 º x Input Signal ¾ Output Signal z I in (x) I out (x) ~I in (º)OT F (º) = ~ I out (º) Linear System Incoherent Image Formation

24 x 0 º x Point Source ¾ PSF z OTF(º) ¾ º The Fourier Optics Approach

25 Extended Source? PSF = Image I 0 (x 0 ) 1 2 [1 cos(2¼¾x)] I(x) = 1 2 Z I 0 (x 0 ) cos [2¼¾(x x 0 )] dx 0 Source Spectrum OTF = Image Spectrum ~I 0 (º) 1 4 [2±(º) ±(º ¾) ±(º ¾)] ~ I(º) = OTF(º) ~ I0 (º) The Fourier Optics Approach

26 x 0 º x ¾ z Partial Coherence in Phase Space

27 Extended Incoherent Source W 0 (x 0 ; º) = I 0 (x 0 ) º Optical System W s (x; º) = 2±(º) cos(2¼¾x) ±(º ¾=2) ±(º ¾=2) º x 0 ¾ x System Analysis in Phase Space

28 WDF in Image Plane Z W (x; º) = W 0 (x 0 ; º)W s (x x 0 ; º)dx 0 Z W (x; º) = 2±(º) I 0 (x 0 ) cos [2¼¾(x x 0 )] dx 0 ±(º ¾=2) ±(º ¾=2) º ¾ x System Analysis in Phase Space

29 Z W (x; º) = 2±(º) º I 0 (x 0 ) cos [2¼¾(x x 0 )] dx 0 ±(º ¾=2) ±(º ¾=2) º ¾ x ¾ Intensity Irradiance Interference Term Encoding Coherence Interference Fringes of Variable Visibility Z I(x) = W (x; º)dº = Z = 2 2 I 0 (x 0 ) cos [2¼¾(x x 0 )] dx 0 x System Analysis in Phase Space

30 THE MUTUAL INTENSITY J(x 1 ; x 2 ) = lim T!1 1 T Z T=2 T=2 u(x 1 ; t)u (x 2 ; t)dt = hu(x 1 )u (x 2 )i THE MUTUAL POWER SPECTRUM ~J(º 1 ; º 2 ) = lim T!1 1 T Z T=2 T=2 ~u(º 1 ; t)~u (º 2 ; t)dt = h~u(º 1 ; t)~u (º 2 ; t)i A Taste of Coherence Theory

31 J(x ¹x=2; x ¹x=2) W (x; º) A(¹º; ¹x) ~J(º ¹º=2; º ¹º=2) Fourier Relationships

32 Self-Imaging

33 William Henry Fox Talbot ( ) H. F. Talbot, Facts relating to optical science no. IV, Phil. Mag. 9, (1836). Portrait of William Henry Fox Talbot, Antoine Claudet, 1840s bromide copy of a daguerreotype, made by Herbert Lambert, 1930s. Source: Talbot's Observation

34 Grating Self-Image Incident Plane Wave z Simulated Intensity The Talbot Effect

35 Grating Self-Image Incident Plane Wave z Simulated Intensity Lord Rayleigh, On copying diffraction gratings, and on some phenomena connected therewith, Phil. Mag. 11, (1881). The Talbot Effect

36 u(x) = 1X n= 1 u n exp ³i2¼ n x d W (x; º) = 1X n= 1 1X n 0 = 1 u n u n 0 ± µº n n0 exp 2d h i2¼(n n 0 ) x i d = X n=n 0 ju n j 2 ± ³ º n d 2 X n6=n 0 ju n jju n 0j± µº n n0 cos 2d h 2¼(n n 0 ) x i d Á n Á n 0 WDF of Periodic Functions

37 1 d º Fresnel Diffraction W (x; º; z) = W (x zº; º; 0) x Self-Imaging Condition W (x; º; z T ) = W (x; º; 0) d Talbot Distance z T 1 2d = d z T = 2d2 The Self-Imaging Condition

38 Notes About Discrete Phase Space Representations

39 Sampling of Incoherent Geometrical Optics Phase Space: Apply Nyquist theorem as dictated by the spatial bandwidth of detected signals. Phase-space detector: microlenses matched to the pixel size of detector. Sampling of Wigner Distribution Function Additional sampling points and zero-padding in space and frequency. Sampling of Phase Space

40 Define DFT independent of the Fourier integral transform: ~u m = N 1 X n=0 u n exp(i2¼n m=n) Derive DFT theorems and properties Show that there is a relationship with the Fourier integral transform Use the DFT to compute continuous signals. The DFT Take 1

41 Find the DFT as a Riemann sum approximation of the Fourier Integral: ~u(º) = Z 1 1 u(x) exp( i2¼xº)dx ¼ ¼ x N 1 X n=0 u(n x) exp( i2¼n xº) Evaluation a discrete points: ~u m = 1 N 1 x ~u(m º) = X n=0 m º = m=n x u(n x) exp( i2¼n m=n ) The DFT Take 2

42 The discretization has to be determined with the additional help of the sampling theorem The DFT is perceived as an approximation for all continuous signals (not true!) Error analysis and error correction not intuitively accessible. The DFT Take 2

43 Consider the class of bandlimited and strictly periodic signals u(x) = u (sp) (x)? 1X n= 1 ±(x np) This corresponds to a discrete finite spectrum The DFT Take 3

44 u(x) p N Samples/Period ~u(º) ±º = 1=p x º = N=p N Samples/Period º The DFT Take 3

45 Sampling of the periodic signal f u(x) = u (sp) (x)? 1X n= 1 f ±(x np) 1X m= 1 ±(x mp=n) Replication of the discrete spectrum The DFT Take 3

46 Signal and spectrum are discrete and periodic The samples of a single period in one space are related to the samples of a single period in the reciprocal space via a DFT This is a rigorous solution of the Fourier Integral The DFT Take 3

47 The sampled Fourier integral transform and the DFT are equivalent for bandlimited and periodic functions: u(x) p N Samples/Period x ~u(º) º = N=p N Samples/Period º The Discrete Fourier Transform

48 1 d n = 2 n = d n = 0 º n = 1 n = n = n = 4 x n 0 = 5 n 0 = 4 n 0 = 3 n 0 = 2 n 0 = 1 n 0 = 0 n 0 = 1 n 0 = 2 n 0 = 3 n 0 = 4 n 0 = 5 W comb (x; º) = 1 2d 1X 1X n= 1 n 0 = 1 ( 1) n n0 ± (x nd=2) ± (º n 0 =2d) WDF of comb-function

49 x º WDF of Discrete Periodic Signal 4N 2 Samples/unit cell p p=n N p 1 p

50 º x WDF of Truncated Bandlimited Signal

51 º x Periodic Extension of Discrete Signal

52 º x Cross-Term Aliasing

53 º x Oversampling and Zeropadding

54 Oversample by factor 2 Zeropadding by factor 2 Use 1/4 of discrete WDF as approximation of non-periodic signal. Sampling Rule for WDF

55 Phase-Space Optics as a Unification of Optical Modelling: Geometrical Optics Radiometry Wave Optics Partial Coherence Discrete Representations Summary

56 Phase-Space Optics as a Unification of Optical Modelling: Phase-Space Measurement as the Central Task of Signal Recovery and Imaging Technology Phase-Space Tomography Integral Photography Shack-Hartmann Sensor Summary

57 Phase-Space Optics as a Unification of Optical Modelling: Phase-Space Measurement as the Central Task of Signal Recovery and Imaging Technology The Phase-Space Toolbox: Signals and System in Phase Space The Phase Space Diagram Geometrical Optical Signal Dynamics Summary

58 Phase-Space Optics as a Unification of Optical Modelling: Phase-Space Measurement as the Central Task of Signal Recovery and Imaging Technology The Phase-Space Toolbox Applications: Integeral Photography/3D Displays/Plenoptic Camera Superresolution and Compressive Sensing The Moiré Effect Extended Depth of Field Wavefront Sensing/Phase Retrieval Summary

59 K. B. Wolf, Geometric Optics on Phase Space, (Springer, Berlin, 2004). A. Torre, Linear Ray and Wave Optics in Phase Space, (Elsevier, Amsterdam, 2005). H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, (Wiley, Chichester, 2001). M. Testorf, J. Ojeda-Castañeda, A. W. Lohmann, eds., Selected Papers on Phase-Space Optics, (SPIE Press, Bellingham, 2006). M. Testorf, B. Hennelly, J. Ojeda-Castañeda, eds., Phase-Space Optics Fundamentals and Applications, (McGraw-Hill, New York, 2010) IEEE International Conference on Computational Photography (ICCP) April , 2011, Robotics Institute, Carnegie Mellon University, Pittsburgh, PA YouTube Account: cmurobotics; Markus Testorf: Phase-Space Tools for Computational Imaging and Photography References