Human Eye
• Human eye is a simple single lens system
• Crystalline lens provide focus
• Cornea: outer surface protection
• Iris: control light
• Retina: where image is focused
• Note images are inverted
Human Eye Distance
• Crystalline lens to retina distance 24.4 mm
• Eye focuses object up to 25 cm from it
• Called the near point or Dv = 25 cm
• Eye muscles to change focal length of lens over 2.22<f<2.44
cm
• Near sighted: retina to lens distance too long, focused in front
• Infinity object focused in front of retina: out of focus at it
• When bring objects closer focus moves to retina
• Near sighted people can see objects with Dv < 25 cm
• Far sighted: eye is too short, focuses behind retina, Dv > 25 cm
Magnification of Lens
• Lateral change in distance equals
change in image size
• Measures change in apparent image size
m=M =
y′
s′
=−
y
s
Magnification with Index Change
• Many different ways of measuring magnification
• With curved index of refraction surface
measure apparent change in distance to image
• Called Lateral Magnification
m=−
• m is + if image virtual, - if real
s′ − r
s+r
Angular Magnification
• For the eye look at angular magnification
m=M =
θ′
θ
• Represents the change in apparent angular size
Simple Magnifying Glass
• Human eye focuses near point or Dv = 25 cm
• Magnification of object:
ratio of angles at eye between unaided and lens
• Angle of Object with lens
tan( θ ) =
y
y
=
≈θ
Dv 25
• For maximum magnification place object at lens f (in cm)
y
f
• Thus magnification is (where f in cm)
θ′ =
m=
θ ′ 25
=
θ
f
• e.g. What is the magnification of a lens f = 1 inch = 2.5 cm
m=
θ ′ 25 25
=
=
= 10
f
2 .5
θ
Power of a Lens or Surface
• Power: measures the ability to create
converging/diverging light by a lens
• Measured in Diopters (D) or 1/m
• For a simple curved surface
P=
n′ − n
r
• For a thin lens
P=
1
f
• Converging lens have + D, diverging - D
• eg f = 50 cm, D = +2 D
f = -20 cm, D = -5 D
• Recall that for multiple lens touching
1 1 1 1
= + + L
f e f1 f 2 f3
• Hence power in Diopters is additive
D = D1 + D2 L
Eyeglasses
• Use Diopters in glasses
• Farsighted, Hypermetopia: focus light behind retina
Use convex lens, +D to correct
• Nearsighted, Myopia: focus in front of retina
use concave lens, -D to correct
• Normal human eye power is ~58.6 D
• Nearsighted glasses create a reverse Galilean telescope
• Makes objects look smaller.
Classical Compound Microscope
• Classical system has short fo objective lens
object is near focal length when focused
• Objective creates image at distance g from focal point
• Objective working distance typically small (20-1 mm)
• Eyepiece is simple magnifier of that image at g
• Magnification of Objective
mo =
g
fo
• where g = Optical tube length
• Eyepiece magnification is
me =
25
fe
• Net Microscope Magnification
M = mo me =
g 25
fo fe
Classic Microscope
• To change power change objective or eyepiece
Infinite Corrected Microscopes
• Classical Compound Microscope has limited tube length
• New microscope "Infinite Corrected"
• Objective lens creates parallel image
• Tube lens creates converging image
• Magnification now not dependent on distance
to tube lens: thus can make any distance
• Good for putting optics in microscope
• Laser beam focused at microscope focus
Telescope
• Increases magnification by increasing angular size
• Again eyepiece magnifies angle from objective lens
• Simplest "Astronomical Telescope" or Kepler Telescope
two convex lenses focused at the same point
• Distance between lenses:
d = fo + fe
• Magnification is again
m=
θe fo
=
θo fe
Different Types of Telescopes
• Galilean: concave lens at focus of convex
d = fo + fe
• Eyepiece now negative fe
• Most others mirror types
Telescopes as Beam Expanders
• With lasers telescopes used as beam expanders
qc Parallel light in, parallel light out
• Ratio of incoming beam width W1 to output beam W2
W2 =
f2
W1
f1
Telescopes as Beam Expanders
• Can be used either to expand or shrink beam
• Kepler type focuses beam within telescope:
• Advantages: can filter beam
• Disadvantages: high power point in system
• Galilean: no focus of beam in lens
• Advantages: no high power focused beam
more compact
less corrections in lenses
• Disadvantages: Diverging lens setup harder to arrange
Lens Aberrations: Spherical
• Aberrations are failures to focus to a "point"
• Some are failures of paraxial assumption
sin( θ ) = θ −
θ3 θ5
3!
+
5!
L
• Formalism developed by Seidel
• Light through edge of lens at different focus
• Longitudinal Spherical Aberration along axis
• Transverse Spherical Aberration across axis
• Make Aspheric surfaces to compensate
• Or can use combine two or more spherical surfaces
Astigmatism Aberration
• Off axis rays are not focused at
the same plane as the on axis rays
• Called "skew rays"
• Principal ray, from object through optical axis
to focused object
• Tangental rays (horizontal) focused closer
• Sagittal rays (vertical) further away
• Corrected using multiple surfaces
Coma Aberration
• Comes from third order sin correction
• Off axis distortion
• Results in different magnifications at different points
• Single point becomes a comet like flare
• Coma increase with NA
• Corrected with multiple surfaces
Field Curvature Aberration
• All lenses focus better on curved surfaces
• Called Field Curvature
• positive lens, inward curves
• negative lens, outward (convex) curves
• Reduced by combining positive & neg lenses
Distortion Aberration
• Distortion means image not at paraaxial points
• Grid used as common means of projected image
• Pincushion: pulled to corners
• Barrel: Pulled to sides
Lens Shape
• Coddingdon Shape Factor
q=
r2 + r1
r2 − r1
• Shows how aberrations change with shape
Index of Refraction & Wavelength: Chromatic Aberration
• Different wavelengths have different index of refraction
• Often list wavelength by spectral colour lines (letters)
• Index change is what makes prism colour spread
• Typical changes 1-2% over visible range
• Generally higher index at shorter wavelengths
Chromatic Aberration
• Chromatic Aberrations
different wavelength focus to different points
• Due to index of refraction change with wavelength
• Hence focuses rays at different points
• Generally blue closer (higher n)
Red further away (lower index)
• Important for multiline lasers
• Achromatic lenses: combine different n materials
whose index changes at different rates
• Compensate each other
Lateral Colour Aberration
• Blue rays refracted more typically than red
• Blue image focused at different height
than red image
Singlet vs Achromat Lens
• Combining two lens significantly reduces distortion
• Each lens has different glass index
• positive crown glass
• negative meniscus flint
• Give chromatic correction as well
Combined lens: Unit Conjugation
• Biconvex most distortion
• Two planocovex significant improvement
• Two Achromats, best
Materials for Lasers Lenses/Windows
• Standard visible BK 7
• Boro Silicate glass, pyrex
• For UV want quartz, Lithium Fluroide
• For IR different Silicon, Germanium