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
Magnification of Lens • Lateral change in distance equals change in image size • Measures change in apparent image size
y′ s′ m = M = = − 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
s′ − r m = − s + r
• m is + if image virtual, - if real
Angular Magnification • For the eye look at angular magnification
θ ′ m = M = 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
y y tan(θ≈) = = θ Dv 25
• For maximum magnification place object at lens f (in cm)
y θ ′ = f • Thus magnification is (where f in cm)
θ ′ 25 m = = θ f • e.g. What is the magnification of a lens f = 1 inch = 2.5 cm
θ ′ 25 25 m = = = = 10 θ f 2.5
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
n′ − n P = r
• For a thin lens
1 P = 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 fe f1 f2 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
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 micron) • Eyepiece is simple magnifier of that image at g • Magnification of Objective
g mo = fo
• where g = Optical tube length • Eyepiece magnification is
25 me = fe
• Net Microscope Magnification
g25 M = mome = 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
θ f m = e = o θ o 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
f 2 W2 = 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 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 θ 3 θ 5 sin(θ ) = θ − + 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
r + r q = 2 1 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