3. Geometrical Optics
Geometric optics—process of light ray through lenses and mirrors
to determine the location and size of the image from a given
object .
Reflection and Mirror
Law of reflection
i   r
i : incidentangle
 r : reflectionangle
Image Formation by Reflection
Application of Double Reflection-Periscope
DIY Periscope
DIY Periscope (Cont’)
Law of reflection (Snell’s law)
n1 sin 1  n2 sin  2
Types of Lenses
Ray Tracing through Thin Lenses
Image Formation by thin Lenses
Lens equation:
1 1 1


(Gaussian form)
d1 d 2 f
f 2  z1 z2
(Newtonianform)
Magnification
h2 d 2
M 

h1 d1
ABCD Matrix
ABCD Matrix (Cont’)
ABCD Matrix (Cont’)
ABCD Matrix (Cont’)
ABCD Matrix (Cont’)
ABCD Matrix (Cont’)
ABCD Matrix (Cont’)
Aberrations of Lenses
• Primary Aberration  image deviate from the original
picture/the first-order approximation
Monochromatic aberrations
 Spherical Aberration
 Coma
 Astigmatism
 Curvature of field
 Distortion
Chromatic aberration
General Method of Reducing Aberration
in Optical Systems-Multiple Lenses
United States Patent 6844972
General Method of Reducing Aberration
in Optical Systems-Multiple Lenses (Cont’)
United States Patent 6995908
Chromatic Aberration
The focal lengths of lights with distinct
wavelengths are different.
Solution of Chromatic Aberration-Using
Doublet, Triplet, or Diffractive Lens
Spherical Aberration (SA)
Spherical Aberration for Different Lenses
(a)
(b)
(c)
(d)
Simple biconvex lens
“Best-form” lens
Two lenses
Aspheric, almost plano-convex lens
Solutions of Spherical AberrationUsing Aspherical Lens or Stop
Coma
Coma (Cont’)
(a) Negative coma (b) Postive coma
Astigmatism
Astigmatism (Cont’)
Solutions of Astigmatism-Using Multiple Lenses
Curvature of field
Solutions of Curvature of field-Using Multiple
Lenses
Distortion
Picture taken by a wide-angle
camera in front of graph paper
with square grids
Solution of Distortion-Using Multiple Lenses
Nearsightedness (Myopia) and
Farsightedness (Hyperopia)
Image Formation  Camera
Camera
F-number
F  num ber
focal length
diameterof aperture
Eg. 50 mm camera lens, aperture stop 6.25mm:
F-number = 8 (f/8)
Exposure
BA Bd 2
E 2 
f
4f 2
E: energy collected by camera lens
B: brightness of object
A: area of aperture
d: diameter of aperture stop
For any given object E 
1
(F - number)2
Camera Lenses
• Wide-angle Lensesthe Aviogon and the
Zeiss Orthometer
lenses
• Standard Lenses-the
Tessar and the Biotar
lenses
• Lens of reducing the
3rd-order aberrationthe Cooke triplet lens
Depth of Field (DOF)
• The distance between the nearest
and farthest objects in a scene that
appear acceptably sharp in an
image.
• In cinematography, a large DOF is
called deep focus, and a small
DOF is often called shallow focus.
• For a given F-number, increasing
the magnification decreases the
DOF; decreasing magnification
increases DOF.
• For a given subject magnification,
increasing the F-number increases
the DOF; decreasing F-number
decreases DOF.
Numerical Aperture (NA)
• The numerical aperture of
an optical system is a
dimensionless number
that characterizes the
range of angles over
which the system can
accept or emit light.
• Generally,
• For a multi-mode optical
fiber,
Telescope
Astronomical (Keplerian) Telescope
Magnification (magnifying power):
General Keplerian telescope: d=fo+fe
'

: angle subtended at input end in front of objective
’: angle subtended at output end behind eyepiece
M
For small angle:
f
'
M   o 0

fe
(inverted image)
Galileo Telescope
 ' fo
M 
0
 fe
General Galileo telescope: d=fo-fe
Terrestrial Telescope
All images are erecting
Optical Microscope
Microscope Theory
Overall magnification:
M  mo me
mo: linear magnification
of objective
me: angular magnification
of eyepiece
Objective
Linear magnification: m o 
y'
x'

y
f
Numerical aperture (NA)
NA 
D
1

(for objective)
f F - number
Microscope Theory (Cont’)
Eyepiece
y
(if   1)
25
Angular magnification:
  tan  
me 
' 25
25

1 
(usually, f  25cm )

f
f
'  tan ' 
y' y

(if '  1)
25 x
Overall magnification of microscope:
M  mo me

x ' 25
f o fe
fo: focal length of objective
fe: focal length of eyepiece
(normal reading distance)
Simple Projection System
Fresnel Lens and Plates
focusing point
(in phase)
• Radius of the concentric
circular: rn = [(n)2+2fn] ½ ,
n=0, 1, 2,….
• Sapce between two adjacent
circular
• zone: rn = rn+1rn