Aberration Theory. Lens. Optical systems convert the shapes of wavefronts

Transcription

1 Aberration Theory Lens Optical systems convert the shapes of wavefronts

2 Aberrations A perfectly spherical wave will converge to a point. Any deviation from the ideal spherical shape is said to be an aberration.

3 Wavefront Error The wavefront error W(x,y) is the difference between a perfect spherical wave and the actual wavefront. It is usually measured in the exit pupil and W(0,0) = 0. The axis for this system is the Line of Sight. W(x,y) / n > 0 In general, there is no symmetry in the eye, so W(x,y) can take on any complex shape.

4 Wavefront Error r R Image Plane A reference sphere of radius R is taken as the ideal spherical wavefront. R is just the distance from the exit pupil to the image plane. δη δη is the transverse ray error and represents the distance between where the ideal and the actual rays strike the image plane. W(x,y) / n δl dl is the longitudinal ray error and represents the distance the actual ray crosses the line of sight from the image plane.

5 Power Error In Visual Optics, longitudinal aberrations are usually given in terms of a power error as opposed to a distance. Φ Φ+dφ δl P P F P P F

6 Power Error The power error is related to the slope of the wavefront dφ = 1 r W(r) r Defocus Example: λs of defocus (λ = 0.5 μm) at edge of 6 mm pupil ( mm) (3 mm) 3 W (r) = r = r dφ = mm -1 = 0. Diopters

7 Power Error The preceding defocus example shows that dφ is constant, which means that dl is the same regardless of where the ray entered the eye. Φ+dφ δl P P F

8 Measuring Defocus in the Eye Ideally, put a test lens at the principal plane. Power changes slightly as lens is moved from the principal plane. Myopia and hyperopia are defocus, where the prescription is the power of the test lens that cancels dφ. φ test

9 Differential Geometry Every point on a continuous surface has two Principal Curvatures. These curvatures represent the maximum and minimum curvature through this point and The principal curvatures are always along orthogonal axes. Calculated from the Fundamental Forms)

10 Fundamental Forms First Fundamental Form x f 1 E + = = y f x f F y f 1 G + = Second Fundamental Form [ ] 1/ F EG x / f L = [ ] 1/ F EG y x / f M = [ ] 1/ F EG y / f N =

11 Curvatures Mean Curvature EN+ GL+ FM 1 H = = κ EG ( F ) Gaussian Curvature ( κ1 + ) Principal Curvatures κ1 = H + κ = H H H K K LN EG M K = F = κ1κ

12 Axial Astigmatism Since the eye is not rotationally symmetric, astigmatism can appear on-axis. This astigmatism is primarily due to the ocular surfaces having a toric or biconic shape. z = R x ( ) R R + R y x x y y Principal Curvatures are 1/R x and 1/R y at the origin

13 Biconic Surfaces Computationally similar to the toric surface, but more versatile since the biconic surface can add asphericity. ( ) ( ) y y x x y x R y K 1 R x K R / y R / x z = R x R y x y Principal Curvatures are 1/R x and 1/R y at the origin

14 Astigmatic Surfaces The axes with the maximum and minimum radii of curvature are called the Principal Meridia (any axis is called a meridian). There is a steep meridian corresponding to the minimum radius of curvature. There is a flat meridian corresponding to the maximum radius of curvature. The principal meridia are always perpendicular to each other.

15 Astigmatic Surfaces In general, and in the eye, the principal meridia do not lie along the x and y axis, but are rotated through some angle θ o. z = 1+ 1 r cos ( θ θ r o ) / R ( ) ( ) o 1+ K x 1+ K y x x + r cos ( θ θ R ) sin ( θ θ o r ) / R y sin ( θ θ R y o )

16 Astigmatic Power Error W(r) = Ax + By = Ar cos θ + Br sin θ dφ = 1 r W(r) r = A cos θ + Bsin θ For a given value of θ, dφ is a constant. Axial astigmatism can be thought of as defocus that depends on meridian.

17 Axial Astigmatism If a refraction is performed through a series of slits rotated at various angles, the refractive error will oscillate between a minimum and maximum value in the presence of axial astigmatism.

18 Jackson Crossed Cylinder Crossed cylinders have a power +D along one meridian and a power -D along the orthogonal meridian. Spherical Equivalent = Average power

19 Chromatic Aberration θ1 Index n(λ) Index n1 θ Snell s Law n1*sin θ1 = n(λ) *sin θ

20 Chromatic Aberration Chromatic Aberration is wavelength dependent defocus W(r) = A ( λ) r dφ( λ) = A( λ) For a given λ, dφ is constant 0.5 Wavelength (nm) LCA (D) New Eye Model Navarro Eye Model Wald & Griffin Bedford & Wyszecki -

21 Duochrome (Bichrome) Test Red Sharper Green Sharper Equal

22 Transverse Chromatic Aberration

23 Wavefront Tilt or Prism Ideal Prism Error Ideal Image Point Real Image Point The wavefront remains the ideal spherical shape, but it is tilted relative to the ideal wavefront.

24 Wavefront Tilt Strabismus or Lazy Eye causes wavefront tilt The brain sees independent images that do not fuse in the brain. The brain does not like this discrepancy, so it will start ignoring one image. If this is not fixed early, one eye will become blind, even though it works fine from an optical standpoint.

25 Hirschberg Test The subject fixates on a point source. The reflection of the source should be slightly nasal of the pupil center. Pupillary Axis κ LOS E E Fovea Asymmetry between the eyes suggests ocular misalignment

26 Spherical Aberration Ideal Image Point With spherical aberration, the wavefront is rotationally symmetric, but is not spherical. The location of the focus varies with the distance from the centerline.

27 Spherical Aberration

28 Spherical Aberration If a refraction is performed through a series of annular rings, the refractive error will vary with the size of the average ring diameter, when spherical aberration is present.

29 Longitudinal Spherical Aberration LSA (D) Pupil Diameter (mm) Van Meeteren Fit Navarro Model Kooijman Model New Eye Model Gullstrand-Le Grand Eye Model

30 Night Myopia There is a myopic shift of the smallest point that is formed on the retina as the pupil dilates.

31 Oblique Astigmatism Off-axis astigmatism Δ given as the difference between the meridional and sagittal foci Δ= α 1.5 x 10 - Diopters where α is the field angle in degrees

32 Aberrations From Raytracing Defocus and Axial Astigmatism n M F P dφ = n' P' F' n' P' M'

33 Aberrations From Raytracing Chromatic Aberration n F λ F λref P dφ = n' P' F' λref n' P' F' λ λ ref is typically nm

34 Aberrations From Raytracing Spherical Aberration n M F dφ = P n' P' M' n' P' F'