Visual Optics

1
Visual Optics

• Chapter 3
• Retinal Image Quality

2
The Net Monochromatic Wavefront Aberration and
its Components
Page 3.34
3
Photorefractive Keratectomy (PRK)
Page 3.34

• PRK for myopia flattens the central cornea
• Patients becomes emmetropic, but some have
  problems

When?
Why?
Figure 3.28 Shape of a myopic patients cornea
before (dashed line) and after photorefractive
keratectomy (PRK). Note the abrupt change in
anterior corneal surface contour at the edge of
the ablated zone.
4
LASIK Procedure
Page 3.37
Figure 3.31

• For LASIK, a flap of anterior cornea is first cut
  and folded back
• Underlying stroma ablated, then the flap is
  folded back into place
• Success rate higher, but some patients still have
  problems

When? At nightLarger pupil
Why? Aberrations due to rapid change in corneal
contour at edge of ablation zone
5
Measuring the Eyes Wavefront Aberration
Page 3.35
6
Shack-Hartmann Aberrometer Object Grid
Page 3.35
Light from the object grid refracts into the eye,
reflects from the retina, and re-refracts out of
the eye (double pass) The emergent wavefront
represents a double sampling of intrinsic
ocular aberrations and diffraction
Figure 3.29 Shack-Hartmann object array
consists of a rectangular grid pattern of point
sources
7
Aberrometer measures the eyes aberrations
Based on Page 3.35
IN (first pass)
OUT (second pass) ideal system
To detector
8
Aberrometer measures the eyes aberrations
Based on Page 3.35
IN (first pass)
OUT (second pass) aberrated system
To detector
9
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10
Aberrometry on PRK Patients
Page 3.34

• Aberrometer detects large wavefront aberration at
  edge of ablation zone
• Translates to significant problems with glare and
  halos at night
• Similar findings in LASIK patients

Figure 3.28 Shape of a myopic patients cornea
before (dashed line) and after photorefractive
keratectomy (PRK). Note the abrupt change in
anterior corneal surface contour at the edge of
the ablated zone.
11
Adaptive Optics Smoothing the Aberrated
Wavefront
Pp 3.36 3.38

• Aberrometers measure the eyes wavefront
  aberration
• Can we change the aberrated wavefront to an ideal
  one?
• Yes. By using an optical device to compensate
• The optical device is a deformable (active)
  mirror that continually adapts to a changing
  wavefront based on computer feedback

12
Page 3.38
Deformable Mirror Adaptive Optics Response
Deformable mirror resides on the input side of an
aberrometer Contains a series of actuators to
change local shape across mirror surface Based on
computer analysis of the unmodified exiting
wavefront, the mirror deforms into a shape that
produces minimal output aberration
Figure 3.32 Basis of a deformable mirror. The
mirror surface consists of multiple segments,
each independently controlled by one or more
underlying actuator(s).
13
Adaptive Optics in Imaging Systems
based on Page 3.39

• Early applications of deformable mirrors in
  astronomy to overcome atmospheric turbulence

Jupiters moon Europa, AO off resolution 0.5
seconds arc
Europa with AO on resolution 0.007 seconds arc
14
Identical Principle used in Ophthalmic Imaging
Page 3.39
Scanning Laser Ophthalmoscope
Mirror changes shape
Figure 3.34 - Schematic of high-resolution
ophthalmic imaging system the micro-deformable
mirror (µDM) compensates for aberrations in the
eye.
15
SLO
Page 3.39
16
Wavefront-Guided Refractive Surgery
Pp. 3.38-39

• Same principles can be applied to refractive
  surgery procedures
• Can change refractive surgery from an
  aberration-inducing liability to a procedure
  that corrects ocular aberrations
• Requirements of the adaptive optics system
• Real-time detection of the eyes wavefront
  aberration during surgery (extremely high
  response rates)
• Real-time compensation for the wavefront
  aberration active (deformable) mirror (extremely
  high response rates)
• What would limit visual acuity in an
  aberration-corrected eye?

1. Pupil size 2. Photoreceptor mosaic (anatomical
limit of resolution)
17
Deformable Mirrors in LASIK Systems
Page 3.39
18
Deformable Mirrors in LASIK Systems
Page 3.39

• The VISX Wavescan? system incorporates an active
  mirror measuring 8.0 mm ? 9.6 mm across with
• 48,000 separate mirror elements (each 40 ?m ? 40
  ?m)
• each element has four independently operated
  actuators
• each element can readjust 250 times per second

Mirrors based on micro-electro-mechanical system
(MEM) technology are compact, with drive
electronics fabricated directly onto the mirror
substrate
19
Defining and Quantifying Monochromatic Wavefront
Aberrations of the Eye
Page 3.39
20
Perfect refracted wavefront
Lenses and the eye do not produce perfect
wavefronts
Ideal Lens
Spherical Wavefront
Spherical Wavefront
How do we quantify the aberrated
wavefront? Typically use a polynomial function
based on a circular aperture Examples Zernike
polynomial, Seidel function
PointImage
PointObject
PARAXIAL IMAGE PLANE
EXIT PUPIL PLANE
21
Making Sense of a Complex Wavefront Shape
Zernike Approach describe wavefront as a
polynomial function with 0, 1st, 2nd, 3rd..nth
order terms First five orders include 21 separate
terms
22
The First Five Zernike Orders
Zernike polynomials from zero to fifth order.
The zero order (piston), and first order (tilt),
also called prism have no bearing on image
quality. Fifth order aberrations (bottom row)
are pentafoil (far right and far left), secondary
trefoil (second from left and right), and
secondary coma (third from left and right).
23
Seidel (Third Order, Monochromatic) Aberrations
Page 3.40
24
Seidel Approach Wavefront Shape in Exit Pupil
Image Plane

• Paraxial Optics predicts that an axial point
  object produces an axial point image

r
Page 3.40
Figure 31 Relationship between wavefront
coordinates in the (exit) pupil plane (x, y, z)
and image plane (x0, y0, z0). r wavefront
radius of curvature.
25
Seidel Approach Wavefront Shape in Exit Pupil
Image Plane
For the ideal wavefront, all locations in the
exit pupil would converge to (x0 y0 z0 ) at the
paraxial image point
r
Page 3.40
Figure 31 Relationship between wavefront
coordinates in the (exit) pupil plane (x, y, z)
and image plane (x0, y0, z0). r wavefront
radius of curvature.
26
An aberrated wavefront does not converge to x0
y0 z0 (paraxial image point). Different parts
of the wavefront converge to different locations
in image space
27
Defining Wavefront Shape in Exit Pupil Plane
Based on page 3.40
Exit Pupil
Paraxial image plane
Object plane
Most important wavefront attributes to quantify
mono-chromatic aberrations
1. Aperture (?) distance from center of
ExP 2. Meridian (?) orientation in exit
pupil 3. Off-axis position (?)
28
Coordinates in Exit Pupil Wave at Oblique Angle
Page 3.40
Paraxial image plane
?W
z
Object plane
Exit Pupil
Defining wavefront position as a distance (?W)
from the exit pupil plane at aperture height (?)
and meridian (?)
29
Coordinates in Exit Pupil (and displacement in
image plane) Off-axis Object Point
Page 3.40
Paraxial image plane
Object plane
Exit Pupil
For an off-axis object point, how does the image
point vary from the paraxial prediction, x0 ?
30
Ideal vs Aberrated Wavefront
Page 3.41
31
Seidel Aberrations Aperture (?), Angular (?) and
Object Height (?0 ) dependence
Page 3.41
Which aberrations are aperture-dependent?
Spherical aberration and Coma (aperture
dependence gt ?2)
32
Seidel Aberrations Aperture (?), Angular (?) and
Object Height (?0 ) dependence
Page 3.41
Define the off-axis aberrations
Coma, off-axis astigmatism, field curvature and
distortion (all have an ?0 term).
33
Seidel Aberrations Aperture (?), Angular (?) and
Object Height (?0 ) dependence
Page 3.41
Which are the meridionally-dependent aberrations?
Coma, OAA (? cos2 ? greatest meridional
variation) and distortion
Significance of no meridional dependence of SA
and field curvature?
Symmetrical image
34
Point-spread functions Which are the
meridionally-dependent aberrations?
Spherical aberration
Ideal wavefront
Airy disc pattern
35
Spherical Aberration
36
Spherical Aberration Ray Diagram
Page 3.43
Figure 3.36 Spherical aberration
37
Quantifying Spherical Aberration
Page 3.44

• Longitudinal Spherical Aberration (LSA)
• Transverse Spherical Aberration (TSA)

38
Longitudinal Spherical Aberration (LSA)
Page 3.44
Note in Geometrical Optics, the symbol y is
often used for aperture diameter instead of ?
39
LSA
Figure 3.37 LSA for (a) small, (b) medium, and
(c) large pupil
Page 3.45
40
Spherical Aberration Longitudinal (LSA) and
Transverse (TSA)
Page 3.46
Figure 3.38 LSA and TSA for a reduced eye
(spherical reduced surface) with large pupil
diameter. Object is a distant axial point
source. The waist of least aberration (WOLA) is
well to the left of the paraxial focus.
Spherical aberration greatly exaggerated.
41
LSA and TSA
Page 3.47
Figure 3.39 (a) Relationship between LSA and
TSA. f?m marginal focal length f?p paraxial
focal length. Appearance of screen image at the
paraxial focus also shown. (b) Using similar
triangles to relate LSA to TSA in terms of pupil
diameter (y) ? angle subtended by marginal ray
(at optical axis).
42
SA and Real Eyes
Page 3.48

• Real eyes do not have spherical corneas or
  crystalline lenses
• Experimental results show positive corneal SA and
  negative lenticular SA
• Positive corneal SA lt predicted for spherical
  cornea
• SA also differs between myopes and hyperopes

Type of Ametropia Total LSA (?m) Corneal LSA (?m) Lenticular LSA (?m)
Myopia (n 17) 0.10 ? 0.13 0.29 ? 0.13 ?0.19 ? 0.16
Hyperopia (n 13) 0.38 ? 0.19 0.41 ? 0.11 ?0.04 ? 0.20
Why would hyperopes have higher corneal SA?
Less aspheric corneas (less peripheral
flattening). Reason? Unknown
43
Page 3.49
Figure 3.38 (a) Spherical cornea (b)
Aspheric cornea (c) Aspheric rays (_______)
vs spherical rays (- - - - - - - -)
44
Ocular SA and Refractive Surgery
Page 3.50
If pupil diameter exceeds ablation zone diameter
? SA? Larger ablation zone means deeper
ablation LASIK produces similar problems at the
edge of the ablated zone Replacing a non-ablated
flap over the reshaped stroma is also a problem,
often introducing higher order aberrations
Figure 3.41 Light ray traveling through the
center of the pupil and two rays either side are
shown refracting through the flattened (ablated)
corneal zone. Another ray immediately either
side of the ablated zone refracts at a
significantly sharper angle through the now
steepest corneal curvature. This causes
considerable positive spherical aberration
45
Coma
Page 3.52
46
Coma
Page 3.52
Comatic image ofa point
Most complex monochromatic aberration Off-axis
version of spherical aberration Asymmetric,
comet-shaped, image very detrimental to overall
image quality
47
Coma
Page 3.52
Greater wavefront curvature above axis
converges marginal image rays below the paraxial
image point
Lower wavefront curvature below axis
leaves too little convergence for marginal rays
to reach the paraxial image point
Figure 3.42 (bottom) Seidel positive coma ray
pattern for a spherical refracting surface
48
Reference Planes for Asymmetric Aberrations
Page 3.53
x0
x
Tangential Plane
plane passing through optic axis in direction of
OA point
contains chief ray (pupil ray) from OA point and
optic axis
perpendicular to tangential plane also passing
through optic axis
Sagittal Plane
49
Coma Tangential Sagittal Planes
Page 3.54
Greatest wavefront asymmetry in tangential
plane Object presents the most asymmetric profile
in this plane
Least wavefront asymmetry in sagittal plane All
sagittal rays focus in the tangential plane
Figure 3.44 Coma produced by a spherical
refracting surface (e.g. the cornea) for a
below-axis object point in the tangential plane
(top) and sagittal plane (bottom).
50
Coma ?45? Oblique Planes
Page 3.55
?45? oblique planes focus right and left of the
tangential plane Stronger refraction above axis
weaker below-axis
Figure 3.45 Coma produced by a spherical
refracting surface for a below-axis object point
in oblique incident planes, ? 45? from vertical
(top) 45? from vertical (bottom).
51
Coma Composite
Page 3.56
Filling in all other oblique planes the result is
a series of comatic circles Each circle
corresponds to a given incident height at the
aperture (? value)
Figure 3.46 View looking through the
spherical refracting surface toward the image
plane, showing the parts of the comatic
(comet-shaped) image produced by tangential,
sagittal and ?45? oblique meridians.
52
Coma Aperture Dependence
Page 3.57
Reducing aperture diameter cuts out larger
comatic circles
Figure 3.47 Effect of reduced aperture diameter
on coma produced in the tangential (top) and
sagittal (bottom) planes for a below axis object
point. Aperture diameter is reduced by less than
half (see Figure 3.44), but the comatic image
pattern is reduced by more than half
53
Coma Aperture Dependence
Page 3.58
Reducing aperture diameter cuts out larger
comatic circles
Figure 3.48 Effect of reduced aperture diameter
on coma produced in the incident planes 45?
counterclockwise from vertical (top), and 45?
clockwise from vertical (bottom) for a below axis
object point.
54
Coma Aperture Shells
Page 3.59
Greatest comatic effect in T-plane Least comatic
effect in S-plane ?quantify coma in these planes
(two extremes)
Figure 3.49 Relationship between incident
height on refracting surface and location in
comatic image.
55
Quantifying Coma
Page 3.61
Measure tangential and sagittal coma from the
paraxial image point Measure T and S coma to the
largest comatic circle (coming from the aperture
margin)
Figure 3.50 Tangential (CT ) and sagittal (CS )
coma. Greatest image displacement occurs at the
periphery of the lens/refracting surface/aperture
corresponding to the largest comatic circle in
the diagram on the left. .
56
Quantifying Coma
Page 3.60
Using the tangential coma equation Doubling
aperture diameter increases tangential coma
4? How much does sagittal coma increase? Move the
object point twice as far off-axis ? double
tangential coma. Effect on a sagittal coma?
57
Coma Aperture and Object Height Effects
Page 3.62
Double aperture diameter Increase coma 4?
Double off-axis object height Increase coma 2?
Figure 3.51 Variation in coma with aperture
diameter and distance off-axis of the object
point.
58
Oblique Astigmatism
Page 3.63
Off-axis aberration (varies exponentially with
object/paraxial image height)
Matches ideal ?2 not aperture-dependent (as a
longitudinal error)
Varies exponentially by meridian (around the exit
pupil) tangential, sagittal intermediate
planes
Figure 3.52 Oblique (radial) astigmatism 3D
wavefront profile in the (exit) pupil plane
produced by oblique astigmatism for a below-axis
object point. .
59
Oblique Astigmatism
Page 3.57
Arises because oblique power (effective power for
oblique incidence) of a positive lens or
refracting surface always exceeds paraxial power
For a below-axis object point, greatest wavefront
curvature in tangential meridian
Figure 3.51 Oblique (radial) astigmatism 3D
wavefront profile in the (exit) pupil plane
produced by oblique astigmatism for a below-axis
object point.
60
Focal Lines in Oblique Astigmatism
Vertical
Page 3.58
Optic axis
Horizontal
Chief Ray
Optic axis
Figure 3.52 Oblique (radial) astigmatism. A
point object (B) below the optic axis (A) creates
an image spread along the optic axis direction
from the tangential to sagittal focus.
61
Focal Lines in Oblique Astigmatism
Page 3.58

• Chief ray subtends angle ? from optic axis (?
  angle of obliquity)
• Longitudinal image spread from tangential focus
  to sagittal focus
• Focal lines ? to meridians producing them
• Lateral image spread (length of focal lines and
  diameter of COLC) vary with aperture diameter

Figure 3.52 Oblique (radial) astigmatism. A
point object (B) below the optic axis (A) creates
an image spread along the optic axis direction
from the tangential to sagittal focus.
62
OA Wavefront Retardation in T and S Planes
Page 3.59
Figure 3.53

• On-axis object point subtends zero angle of
  obliquity (? 0)
• Incidence symmetrical for all meridians ? no OA
• Chief ray (CR) incident normal to lens traverses
  minimal lens thickness sees full lens profile
  (full diameter)

63
OA Wavefront Retardation in T and S Planes
Page 3.59
Figure 3.53
Tangential Plane

• CR subtends ? with optic axis
• T rays asymmetric incidence
• CR traverses greater lens thickness
• CR sees reduced effective lens diameter (d')
  due to more oblique T incidence
• ? thickness ? diameter (d') both retard
  wavefronts
• ?T focus (B'T) closer to lens than paraxial focus
  (A').

64
OA Wavefront Retardation in T and S Planes
Page 3.59
Figure 3.53

• Viewing sagittal plane from above (B below axis)
• S rays incident at angle ?
• CR traverses greater lens thickness
• CR sees full lens diameter
• ? thickness alone retards wavefronts
• ?S focus (B'S) between B'T and paraxial focus
  (A').

Sagittal Plane
65
Quantifying Astigmatic Error
Page 3.60

• For an equiconvex spherical lens, distant object
  (and angles of obliquity up to 25?)

For a near object (as in figure)
5.0 D spherical lens, ? 25?
Astigmatic error varies with the square of the
angle of obliquity of the chief ray (tan2 ?)
66
OA in Lens Design
Page 3.60
Spread of T and S foci increases exponentially
with ? Equiconvex lens produces maximum OA (equal
plus power distribution between front and back
surfaces)
1.09 D
0.25 D
Figure 3.54 oblique astigmatism produced by an
equiconvex positive lens for angles of obliquity
? and 2 ?. In dioptric terms, the astigmatic
error is the difference in image vergence
between tangential and sagittal focus. Note that
the astigmatic error increases exponentially
between ? and 2?.
67
Oblique Astigmatism of the Human Eye
68
Ocular OA
Page 3.64
Figure 3.57 The human eye has significant
intrinsic oblique astigmatism for larger retinal
eccentricities.
69
Ocular OA
Page 3.64

• Oblique rays travel to more peripheral retinal
  locations
• Resolving power decreases rapidly with retinal
  eccentricity
• Rapid increase in ocular OA with eccentricity not
  a problem because peripheral retina has low
  resolution potential
• Exception patients with central retinal disease
  and VA loss

70
Curvature of Field
71
Curvature of Field
Page 3.62

• Removing SA, coma and OA mono-? point object ?
  point image
• Plane objects do not necessarily form plane
  images
• Common example of curvature of field seen with 35
  mm projector image (parts of image clear parts
  blurred)

Figure 3.56 Curvature of field 3D wavefront
profile in the (exit) pupil plane produced by
curvature of field for a plane object
72
Origin of Curvature of Field
Page 3.62
Figure 3.56 refracting surface (devoid of
spherical aberration, coma, or oblique
astigmatism) with focal length, f?, images
parallel on-axis light rays at its second focus
(F). Oblique incident rays also refract at the
curved surface and focus the same distance away.
73
Origin of Curvature of Field
Page 3.62

• Parallel incident rays focus at F' (distance f '
  from surface)
• Oblique parallel incident rays also focus a
  distance f ' from surface
• Oblique direction of f ' places the focal point
  short of the axial position of F ' (error p)
• Resulting image surface Petzvals Surface

74
Curvature of Field Petzvals Surface
Page 3.63
Positive meniscus lens produces a concave image
of a plane object
Figure 3.57 - The image of a plane object
produced by an ophthalmic lens is curved. Here a
positive lens produces negative (concave) image
curvature for the plane object.
75
Origin of Curvature of Field
Page 3.63
76
Origin of Curvature of Field
Page 3.63

• Radius of curvature of Petzvals surface varies
  with focal length and index of lens (or image
  space medium of spherical refracting surface)
• This limits design options to reduce curvature of
  field in any optical system

77
Eliminating Curvature of Field
Page 3.63
Petzval Condition for lens system

• To produce a flat-field image with a camera lens
  system or microscope, the Petzval condition must
  be met
• Anastigmatic camera lenses have zero OA and also
  satisfy the Petzval condition (flat image)

78
Page 3.79
Distortion
79
Distortion
Page 3.79
Strongly dependent on paraxial image height
Depends on aperture position in the optical
system Distortion produces NO image blur
Figure 3.69 Distortion 3D wavefront profile in
the (exit) pupil plane produced by distortion of
an object.
80
Types of Distortion
Page 3.79
Object
Pincushion distortion
Barrel distortion
Take grid-shaped object and image through a plus
lens-aperture stop combination. Resulting image
With aperture AT the plus lens
With aperture to right of plus lens
With aperture to left of plus lens
81
With aperture AT the plus lens chief ray and
nodal ray equivalent
Page 3.80
?
??
Figure 3.71
Could correctly define lateral magnification (m
??/?) using either ? and ?? measured along the
optic axis, or along the nodal ray (or chief ray)
path
82
With aperture to right of plus lens
Page 3.81
Figure 3.72
83
With aperture to left of plus lens
Page 3.82
Figure 3.73
84
Quantifying Distortion
Page 3.54
D ? (h?)3

• D measured relative to paraxial image point
• Negative lens distortion effects opposite to plus
  lens
• Negative lens and aperture to right
• Negative lens and aperture to left
• Systems affected system with asymmetric stop
  eye with high-power spectacle lens
• Correcting/reducing distortion orthoscopic
  (distortion-free) lens aspheric spectacle lens