Review of Optics. Austin Roorda, Ph.D. University of California, Berkeley. victorhorvath.com

Transcription

1 Review of Optics Austin Roorda, Ph.D. University of California, Berkeley victorhorvath.com

2 Geometrical Optics Relationships between pupil size, refractive error and blur

3 Optics of the eye: Depth of Focus 2 mm 4 mm 6 mm

4 Optics of the eye: Depth of Focus Focused behind retina In focus Focused in front of retina 2 mm 4 mm 6 mm

5 Demonstration Role of Pupil Size and Defocus on Retinal Blur Draw a cross like this one on a page. Hold it so close that is it completely out of focus, then squint. You should see the horizontal line become clear. The line becomes clear because you have used your eyelids to make your effective pupil size smaller, thereby reducing the blur due to defocus on the retina image. Only the horizontal line appears clear because you have only reduced the blur in the horizontal direction.

6 Computation of Geometrical Blur Size where D is the defocus in diopters

7 Application of Blur Equation 1 D defocus, 8 mm pupil produces minute blur size ~ 0.5 degrees

8 Physical Optics The Wavefront

9 What is the Wavefront? parallel beam = plane wavefront converging beam = spherical wavefront

10 What is the Wavefront? parallel beam = plane wavefront ideal wavefront defocused wavefront

11 What is the Wavefront? parallel beam = plane wavefront ideal wavefront aberrated beam = irregular wavefront

12 What is the Wavefront? diverging beam = spherical wavefront aberrated beam = irregular wavefront ideal wavefront

13 The Wave Aberration

14 What is the Wave Aberration? diverging beam = spherical wavefront wave aberration

15 Wave Aberration Contour Map mm (superior-inferior) mm (right-left)

16 Wave Aberration: Defocus mm (superior-inferior) Wavefront Aberration mm (right-left)

17 Wave Aberration: Coma mm (superior-inferior) Wavefront Aberration mm (right-left)

18 Wave Aberration: Complex Aberration mm (superior-inferior) Wavefront Aberration mm (right-left)

19 Zernike Polynomials It s convenient to have a mathematical expression to describe the wave aberration allows us to compute metrics allows to display data in different forms allows us to breakdown data into different components (remove defocus, for example) Zernike polynomials are a convenient equation to use to fit wave aberration data for the eye

20 Zernike Polynomials are not a big deal! they are just an equation used to fit data y=mx+b is used to fit linear data y=ax 2 +bx+c is used to fit parabolic data Z(x,y) are useful for fitting ocular wave aberration data

21 Zernike polynomial components what magnitude of each of these shapes is present in my wave aberration data?

22 Zernike term Breakdown of Zernike Terms Coefficient value (microns) astig. defocus astig. trefoil coma coma trefoil spherical aberration 2 nd order 3 rd order 4 th order 5 th order

23 The Reason we Measure the Wave Aberration PSF (point spread function) OTF (optical transfer function) PTF (phase) MTF (contrast) Image Quality Metrics

24 The Point Spread Function

25 The Point Spread Function, or PSF, is the image that an optical system forms of a point source. The point source is the most fundamental object, and forms the basis for any complex object. The PSF is analogous to the Impulse Response Function in electronics.

26 The Point Spread Function The PSF for a perfect optical system is not a point, but is made up a core surrounded by concentric rings of diminishing intensity It is called the Airy disc. Airy Disc

27 Airy Disk θ

28 As the pupil size gets larger, the Airy disc gets smaller. PSF Airy Disk radius (minutes) pupil diameter (mm)

29 Point Spread Function vs. Pupil Size 1 mm 2 mm 3 mm 4 mm 5 mm 6 mm 7 mm Perfect Eye Typical Eye

30 Resolution

31 Unresolved point sources Rayleigh resolution limit Resolved

32 As the pupil size gets larger, the potential resolution improves resolution in minutes of arc pupil diameter (mm)

33 Minutes of arc 20/20 20/10 5 arcmin 2.5 arcmin 1 arcmin

34 Keck telescope (10 m pupil) About 4500 times better than the eye! (0.022 arcseconds resolution)

35 Convolution

36 Convolution

37 Simulated Images 20/20 letters 20/40 letters

38 MTF Modulation Transfer Function

39 low medium high object: 100% contrast image contrast 1 0 spatial frequency

40 modulation transfer MTF: Cutoff Frequency 1 mm 2 mm 4 mm 6 mm 8 mm spatial frequency (c/deg) cut-off frequency Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size

41 Spatial frequency 20/20 20/10 5 arcmin 2.5 arcmin 60 cyc / deg 30 cyc / deg

42 Modulation Transfer Function vertical spatial frequency (c/d) horizontal spatial frequency (c/d) c/deg

43 PTF Phase Transfer Function

44 low medium high object image phase shift spatial frequency

45 Phase Transfer Function Contains information about asymmetry in the PSF Contains information about contrast reversals (spurious resolution)

46 The Importance of Phase

47 Relationships Between Wave Aberration, PSF and MTF

48 The Reason we Measure the Wave Aberration PSF (point spread function) OTF (optical transfer function) PTF (phase) MTF (contrast) Image Quality Metrics

49 The PSF is the Fourier Transform (FT) of the pupil function The MTF is the amplitude component of the FT of the PSF The PTF is the phase component of the FT of the PSF The OTF (MTF and PTF) can also be computed as the autocorrelation of the pupil function

50 Wavefront Aberration Point Spread Function mm (right-left) Modulation Transfer Function Phase Transfer Function arcsec c/deg c/deg

51 Wavefront Aberration 0.5 Point Spread Function mm (right-left) -0.5 Modulation Transfer Function Phase Transfer Function arcsec c/deg c/deg

52 Wavefront Aberration Point Spread Function mm (right-left) Modulation Transfer Function Phase Transfer Function arcsec c/deg c/deg

53 Conventional Metrics to Define Imagine Quality

54 Root Mean Square

55 Root Mean Square: Advantage of Using Zernikes to Represent the Wavefront

56 diffraction-limited PSF Strehl Ratio H dl actual PSF H eye

57 Modulation Transfer Function contrast /20 20/10 Area under the MTF spatial frequency (c/deg)

58 Retinal Sampling

59 Sampling by Foveal Cones Projected Image Sampled Image 20/20 letter 5 arc minutes

60 Sampling by Foveal Cones Projected Image Sampled Image 20/5 letter 5 arc minutes

61 Nyquist Sampling Theorem

62 Photoreceptor Sampling >> Spatial Frequency 1 I 0 1 I 0 nearly 100% transmitted

63 Photoreceptor Sampling = 2 x Spatial Frequency 1 I 0 I 1 0 nearly 100% transmitted

64 1 Photoreceptor Sampling = Spatial Frequency I 0 1 I 0 nothing transmitted

65 1 Photoreceptor Sampling < Spatial Frequency I 0 1 I 0 alias transmitted

66 Nyquist theorem: The maximum spatial frequency that can be detected is equal to ½ of the sampling frequency. foveal cone spacing ~ 120 samples/deg maximum spatial frequency: 60 cycles/deg (20/10 or 6/3 acuity)

67 MTF: Cutoff Frequency Nyquist limit cut-off frequency modulation transfer mm 2 mm 4 mm 6 mm 8 mm spatial frequency (c/deg) Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size