Review of Basic Principles in Optics, Wavefront and Wavefront Error

Transcription

1 Review of Basic Principles in Optics, Wavefront and Wavefront Error Austin Roorda, Ph.D. University of California, Berkeley Google my name to find copies of these slides for free use and distribution

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 Focused behind retina Optics of the eye: Depth of Focus 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 blur[mrad] = D pupilsize[ mm] blur[minutes] = 3.44 D pupilsize[ mm] 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: Defocus mm (superior-inferior) Wavefront Aberration mm (right-left)

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

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

18 Zernike Polynomials

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

20 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

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

22 The Point Spread Function

23 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.

24 The Point Spread Function The PSF for a perfect optical system is the Airy disc, which is the Fraunhofer diffraction pattern for a circular pupil. Airy Disc

25 Airy Disk θ = 1.22 a λ θ λ angle subtended at the nodal point wavelength of the light θ a pupil diameter

26 As the pupil size gets larger, the Airy disc gets smaller. PSF Airy Disk radius (minutes) λ θ = a θ angle subtended at the nodal point λ wavelength of the light a pupil diameter pupil diameter (mm)

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

28 Resolution

29 Unresolved point sources Rayleigh resolution limit Resolved

30 As the pupil size gets larger, the Airy disc gets smaller. PSF Airy Disk radius (minutes) θ θ min min λ a 1.22 λ = a angle subtended at the nodal point wavelength of the light pupil diameter pupil diameter (mm)

31 Keck telescope: (10 m reflector) About 4500 times better than the eye!

32 Convolution

33 Convolution PSFxy (, ) Oxy (, ) = Ixy (, )

34 Simulated Images 20/20 letters 20/40 letters

35 MTF Modulation Transfer Function

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

37 MTF: Cutoff Frequency modulation transfer mm 2 mm 4 mm 6 mm 8 mm f cut-off frequency cutoff spatial frequency (c/deg) = a 57.3 λ Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size

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

39 PTF Phase Transfer Function

40 low medium high object image phase shift spatial frequency

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

42 The Importance of Phase

43 Relationships Between Wave Aberration, PSF and MTF

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

45 The PSF is the Fourier Transform (FT) of the pupil function 2π i W( x, y) (, ) λ PSF xi yi = FT P( x, y) e The MTF is the amplitude component of the FT of the PSF (, ) { (, )} x y i i MTF f f = Amplitude FT PSF x y The PTF is the phase component of the FT of the PSF (, ) { (, )} x y i i PTF f f = Phase FT PSF x y The OTF (MTF and PTF) can also be computed as the autocorrelation of the pupil function

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

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

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

49 Conventional Metrics to Define Imagine Quality

50 Root Mean Square ( ( ) ( )),, 1 2 RMS = W x y W x y dxdy A A pupil area ( ) W x, y wave aberration ( ) W x, y average wave aberration

51 Root Mean Square: Advantage of Using Zernikes to Represent the Wavefront ( 2) ( 0) ( 2) ( 1) RMS = Z2 + Z2 + Z2 + Z3... astigmatism term defocus term astigmatism term trefoil term

52 diffraction-limited PSF Strehl Ratio H dl Strehl Ratio = H H eye dl actual PSF H eye

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

54 Metrics to Define Image Quality Other Metrics Campbell,C.E. (2004). Improving visual function diagnostic metrics with the use of higher-order aberration information from the eye. J.Refract.Surg. 20, S495-S503 Cheng,X., Bradley,A., Hong,X., & Thibos,L. (2003). Relationship between refractive error and monochromatic aberrations of the eye. Optom.Vis.Sci. 80, Cheng,X., Bradley,A., & Thibos,L.N. (2004). Predicting subjective judgment of best focus with objective image quality metrics. J.Vis. 4, Guirao,A. & Williams,D.R. (2003). A method to predict refractive errors from wave aberration data. Optom.Vis.Sci. 80, Marsack,J.D., Thibos,L.N., & Applegate,R.A. (2003). Scalar metrics of optical quality derived from wave aberrations predict visual performanc. J.Vis. 4, Sarver,E.J. & Applegate,R.A. (2004). The importance of the phase transfer function to visual function and visual quality metrics. J.Refract.Surg. 20, S504-S507

55 Typical Values for Wave Aberration Strehl Ratio Strehl ratios are about 5% for a 5 mm pupil that has been corrected for defocus and astigmatism. Strehl ratios for small (~ 1 mm) pupils approach 1, but the image quality is poor due to diffraction.

56 Typical Values for Wave Aberration Population Statistics trefoil coma coma trefoil spherical aberration

57 Typical Values for Wave Aberration Change in aberrations with pupil size rms wave aberration (microns) Shack Hartmann Methods Other Methods pupil size (mm) Iglesias et al, 1998 Navarro et al, 1998 Liang et al, 1994 Liang and Williams, 1997 Liang et al, 1997 Walsh et al, 1984 He et al, 1999 Calver et al, 1999 Calver et al, 1999 Porter et al., 2001 He et al, 2002 He et al, 2002 Xu et al, 2003 Paquin et al, 2002 Paquin et al, 2002 Carkeet et al, 2002 Cheng et al, 2004

58 Typical Values for Wave Aberration Change in aberrations with age Monochromatic Aberrations as a Function of Age, from Childhood to Advanced Age Isabelle Brunette, 1 Juan M. Bueno, 2 Mireille Parent, 1,3 Habib Hamam, 3 and Pierre Simonet 3

59 Other Optical Factors that Degrade Image Quality

60 Retinal Sampling

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

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

63 Nyquist Sampling Theorem

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

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

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

67 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)

68 MTF: Cutoff Frequency modulation transfer Nyquist limit 1 mm 2 mm 4 mm 6 mm 8 mm f cut-off frequency cutoff = a 57.3 λ Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size spatial frequency (c/deg)

69 Thankyou!