Chapter 4:
Geometric Optics
How is light collected and focused to
form images?
Geometric Optics
Reflection:
Light bouncing back from
a surface.
Refraction:
Light traveling from one
transparent medium to another.
•Two parallel descriptions:
Wave optics – “Wavefronts”
Geometric optics – “Light rays”
Ray
Wavefront
•Image formation: by actual (real image) or apparent (virtual
image) intersection of two or more rays of light.
Law of Reflection
•Fermat’s principle of least time.
B
A
B
A
B
A
Which path takes the least time?
http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=57
•Incident ray, reflected ray,
and the normal are in the
same plane.
•Law is valid for any surface.
Image Formation With Plane Mirrors
•Image is:
•Virtual (Virtual images are formed by divergent rays. Light
appears to originate from there).
•Same size as the object.
•Located as far behind the mirror as the object is in front of it.
•Laterally inverted (Right to Left etc.).
•How tall does a mirror have to be so you can see your entire self?
Application - Rear View Mirror
Image Formation With Curved Mirrors
• Curvature: spherical, cylindrical, parabolic…etc.
• Definitions:
•
Center of curvature (C)
•
Radius of curvature (R) = Distance AC
•
Vertex (A)
•
Principal axis (AFC)
•
Focal point (F)
•
Focal length (f) = Distance FA
• Note: Incoming parallel rays will
converge to or diverge from
the focal point.
Concave
(Inward curvature)
Convex
(Outward curvature)
Image Formation by Spherical Mirrors
• How to locate and describe the image?
• Mathematical treatment: (Applicable to concave
or convex mirrors).
p
• Object mirror distance = p
f
• Image mirror distance = q
q
• Focal length of mirror = f
• Object size (height) = Ho
• Image size = Hi
• Mirror (or lens) equation:
1 1 1
 
f
p q
Spherical Mirrors (Contd.)
• Image location and its nature are given by:
pf
q
p  f 
• Magnification is given by:
Hi q
M 

Ho p
•
Note:
Real image: q is +
Concave mirror: f is +
Virtual image: q is –
Convex mirror: f is -
Review Problems
1. If you desired to take a photograph of yourself
while standing 6 ft. from a plane mirror, for what
distance would you set the camera focus?
12 ft.
2. Find the image of an object placed 40 cm from a
concave mirror of focal length 20 cm. What are
the characteristics (location, size, direction, and
nature) of the image?
Location: 40 cm to left of mirror
Size: Same as the object (M=1)
Nature: Real
Direction: Inverted
Review Problems (Contd.)
1. Where would the image of an object very distant
from a concave mirror be located? What would
the size of such an image be?
Location: At the focal point
Size: Diminished
2. Describe the image when an object 5 cm tall is
placed 10 cm in front of a concave mirror of focal
length 20 cm.
Location: q = -20 cm (behind the mirror)
Size: M=2, so 10 cm size
Nature: Virtual
Direction: Upright
Summary: Concave Mirror Imaging
Object
Image
Application
At infinity
At F, Smallest, Inverted, Real
Camera, Telescope
Between infinity
and C
Left of F, Diminished,
Inverted, Real
Camera
At 2f
At 2f, Same size, Inverted,
Real
Camera, Fax, Xerox
Between 2f and f
Left of 2f, Magnified,
Inverted, Real
Camera, Xerox
At f
At infinity (no image)
Headlights
Between F and V
Behind the mirror, Virtual,
Magnified, Upright
Beauty mirror
http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=65
Summary: Convex Mirror Imaging
• Image is always:
Diminished
Virtual
Upright
http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=65
• Application: Collects light from a wide area. Used
as rear-view mirror.
Imperfect Mirrors
• Spherical aberration is an inherent defect. Incoming parallel
rays focus at different points!
• Spherical aberration = (F2 – F1)
F1 (Marginal Rays focus here)
F2 (Paraxial rays focus here)
Image with spherical aberration
Image without spherical aberration
Refraction
• Light rays “bend” when they travel from one transparent
medium into another.
• Refraction (or bending) caused by light traveling at a slower
speed in a denser medium.
•Define “Refractive Index” as:
c
n
v
Where c = 3 x 108 m/s is the speed of
light in vacuum, and v is the speed of
light in any other medium.
•Some common refractive indices:
Water
- 1.33
Flint glass
- 1.66
Air
- 1.0003
Diamond
- 2.4
Review Problem
The index of refraction of a certain type of plastic is 1.7. Find
the speed of light in this plastic.
1.765 x 108 m/s
Refraction: Wave Explanation
When light passes into a new medium, its frequency remains
constant and its wavelength changes.
One side of wave front slows down, and
the entire train of fronts twists. Analogy:
right front tire of vehicle enters mud,
twisting vehicle to the right.
http://www.control.co.kr/java1/RefractionofLight/LightRefract.html
Law of Refraction: Snell’s Law
• Rare to dense medium – light bends towards the normal
• Dense to rare medium – light bends away from the normal
• Angles and refractive indices are related by:
q1
n1
n1Sin(q1 )  n2 Sin(q2 )
n2
q2
http://www.ps.missouri.edu/rickspage/refract/refraction.html
Trigonometric Ratio
• Consider a right angled triangle ABC.
• Sine of the angle q is defined as the ratio of the sides BC
to AC.
C
Length BC
Sin θ  
Length AC
B
q
• Sine of any angle can be found from math tables or your
calculator. Examples:
• Find Sin of 200, 300, 450, 900.
• Find the angles whose sines are 0.1, 0.3, 0.6, 0.9.
A
Review Problems
A ray of light traveling in air strikes a glass surface (n = 1.5)
at an angle of 240 from the normal. At what angle will it be
refracted in glass?
Given: Sin(240) = 0.407, Sin(15.70) = 0.2713
15.70
Some Interesting Effects of Refraction
Sun appears flatter at sunset
Things appear shallower in water
Mirages
Dispersion and rainbows
Total Internal Reflection
• Occurs only when light goes from denser to rarer medium.
http://www.ps.missouri.edu/
rickspage/refract/refraction.html
•Optical fibers
•SLR Cameras & binoculars
•Diamonds
appear
bright.
Image Formation by Refraction: Lenses
Spherical Lens
Double Convex
Or Converging Lens
Double Concave
Or Diverging Lens
+ Focal Length
(Like Concave Mirror)
- Focal Length
(Like Convex Mirror)
pf
• Lens equation: q   p  f 
q
• Magnification: M 
p
Review Problems
1. Using a magnifying glass of 25 cm focal length,
you look at an object that is 20 cm from the glass.
Where and how large will you see the image?
q = -100 cm (To the left of the lens, virtual)
M = 5 (Magnified)
2. An object is placed at a distance of 12 cm from a
lens of focal length 10 cm. Where will its image be
formed and how large will it be?
q = 60 cm (To the right of the lens, real)
M = 5 (Magnified)
Power of a Lens
• Measure of how strongly a lens converges or
diverges rays of light.
• Power of a lens of focal length f is defined as:
1
P
f
• Note: P is in Diopters if f is in meters.
• Example: A converging lens of focal length 50 mm
has +20 D power. A diverging lens of -1.0 D power
has a focal length of 1 meter.
Lens Defects
• Spherical aberration: Marginal and paraxial rays
focus at different points.
• Chromatic aberration: Shorter wavelengths refract
more so different colors focus at different points.
Achromatic Doublet
Image with chromatic aberration
Image without chromatic aberration
Fiber Optics & Communication
• 1854: Fountains carry light.
• 1928: First fiber used to carry light.
• Physical principle: Light is carried by way of “total internal
reflection”.
• Typical core index ~ 1.65; Typical cladding index ~ 1.45
• Critical angle ~ 600
Fiber Optics: Typical Physical Dimensions
Fiber Optics: Applications
Image / Light Carriers:
Image Intensifiers / Magnifiers /
Bundles of fibers
Inverters: Tapered fibers.
Fiber Optic Sensors: Special fibers used for sensing
pressure or temperature changes.
Fiber Optic Communication
• Information can be transmitted by sound, electricity, radio
or microwaves, and light.
• Advantages:
• Light weight, less expensive
• Flexible
• Security (no electrical interference)
• Information carrying capacity
• A wave carries information by
“modulation”.
Fiber Optic Communication (Contd.)
• How much information can a wave carry?
Information carrying capacity is proportional to the
frequency “bandwidth”.
• Example:
FM band ranges from 88 MHz – 108 MHz
So available bandwidth is 2 x 107 Hz!
Red light ranges from 5 x 1014 – 4.3 x 1014 Hz
So available bandwidth is about 7 x 1013 Hz!
Which means light can carry ~1 million times more information than
radio waves.
• Comparison:
1 Telephone wire - 20 simultaneous conversations
1 TV channel
- 1300 …..
1 Optical fiber
- 12000….
Problems with Fiber Optics
• Attenuation (Loss of amplitude): Signal strength is lost
due to absorption by impurities or scattering by
imperfections.
• Need amplifiers (repeater stations) every time the
amplitude drops by a factor of 100,000.
Early fiber losses: 1000 dB/km (need 50m repeaters)
Today: Better than 0.2 dB/km (need 100 km repeaters)
• Note: Microwaves need 30 km repeaters!
Attenuation (Contd.)
Losses are minimum at 1.5 mm wavelength!
Problems with Fiber Optics (Contd.)
• Signal distortion: Limits the information carrying capacity
due to “smearing out” of the signal.
• Mechanisms responsible for distortion are “modal” and
“material” dispersion.
Input signal
After several km
through a fiber
Modal Dispersion
• Signals traveling different paths will arrive at different
times. Solution: Use single mode or gradient index fibers.
Material Dispersion
Shorter wavelengths
have higher refractive
index so they travel
slower through the fiber.
Solution: Use lasers with
high spectral purity.
Different Types of Fibers
Local area networks
Long distance applications
Comparison of Data Rates
Vision Optics
• Working of the human eye as an optical instrument.
• Two important processes responsible for vision:
• ACCOMODATION: Process by which the lens adjusts to
form images.
• ADAPTATION: Process by which the intensity of light is
controlled.
Optical Axis
Visual Axis
The Human Eye: Features
• Adjustable lens system:
•
Cornea (43 diopters): Refracts 70% of incident light.
•
Lens (16 - 26 diopters): Changes shape to accommodate.
•
Both have elliptical shape (minimize spherical aberration).
•
Lens has variable refractive index (minimize chromatic aberration).
Near Point = 25 cm
Far Point =
Infinity
http://micro.magnet.fsu.edu/primer/java/scienceopticsu/eyeball/index.html
The Human Eye: Features (Contd.)
• Adjustable aperture:
•
Iris: A muscle that changes size to adapt.
•
Pupil: Opening diameter
~ 1.5 mm under bright light
~ 6.0 mm under dim light
• Note: Pupil size change accounts for adaptation by a factor of
15 only! Light intensity can change by a factor of 10,000 or
more. Where does the rest of the adaptation come from?
The Human Eye: Features (Contd.)
• Light sensitive material:
•
Retina: Translates light into electrochemical signals. Has two light
sensitive bodies.
•
Rods: For “scotopic” (low light) vision. Response is achromatic and
low resolution.
•
Cones: For “photopic” (bright light) vision. Response is colored and
acute.
The Human Eye: Features (Contd.)
• Fovea:
Has high concentration
of cones so it is used for
acute vision.
• Blind Spot:
Region where optic nerves
join the retina.
The Reduced Eye - A Simplified Model
Object size = Ho
Image size = Hi
Effective center of cornea + lens
H i Image distance from P 17 mm
Magnificat ion : M 


H o Object distance from P
do
Resolving power (Limit of visual acuity):
Two points must be separated by at least 1/60th of 1 degree.
This means a separation of 0.1 cm at near point!
Limit of Visual Acuity
What is the smallest separation between two points on
the retina so the two points are seen as separate points?
(Hint: Take Ho = 0.1 mm, and do = 25 cm)
Hi = 6.8 x 10-6 m
Note: The size of a single cone is about 5 mm!
For scotopic vision this acuity is much less.
Defects of Vision
•Myopia (nearsightedness):
Abnormal elongation of the
eyeball or too much refracting
power. Far point is closer than
infinity. Correction – diverging
lens.
•Hyperopia (farsightedness):
Abnormal flattening of the
eyeball or not enough refracting
power. Near point is farther than
25 cm. Correction – converging
lens.
Defects of Vision (Contd.)
•Presbyopia (aging sight): Abnormal eyeball shape and
weak ciliary muscles.
Correction – bifocal lenses.
•Astigmatism:
Sharper curvature of
the cornea.
Correction – cylindrical
lenses.
Astigmatism Test Pattern
Review – What kind of vision?
• Someone wearing glasses of +3.5 diopters?
Farsighted
•Someone wearing glasses of – 2.0 diopters?
Nearsighted
•Someone with near point of 25 cm and far point of infinity?
Normal vision
•Someone with near point of 150 cm and far point of infinity?
Farsighted
•Someone with near point of 17 cm and far point of 1.0 m?
Nearsighted
Comparison
Eye
Camera
Adaptation
Pupil diameter changes
Photopic/scotopic vision
Aperture diameter changes
Film speed and exposure
Accomodation
Lens shape changes
Lens position changes
Image
Real, inverted
Real, inverted
Light sensitive
material
Retina
Film
The Camera
Parts:
•Light proof box
•Adjustable lens system
(Accomodation)
•Adjustable aperture
(Adaptation)
•Shutter with variable speed
(Duration of exposure)
•Film (Light sensitive material)
Camera Lens
•Several “coated” elements to reduce aberrations and back
reflections.
•Lens is movable (for accomodation).
•Relationship between focal length, image size, and field of
view:
Wide Angle
28 mm
Normal
50 mm
Smaller image
Telephoto
300 mm
Larger image
Field of view: 950
470
80
Image Size  f
1
Image Intensity  2
f
Note: Zoom lenses have variable focal lengths.
Effect of Focal Length on Image Size
F
Short FL Lens
Film
Small Image Size
Large Field of View
Film
F
Long FL Lens
Large Image Size
Small Field of View
Effect of Focal Length on Image Size (Contd.)
That's Seattle about 2 miles away.
focal length 36 mm
focal length 276 mm
focal length 138 mm
focal length 432 mm
Review Problem
A photographer uses a camera with 50 mm
focal length lens to photograph a distant object.
He then uses a 150 mm lens to photograph the
same object. How will the height of the object
compare on the two resulting photographs?
How do the areas compare?
Image size increases by a factor of 3
Area decreases by a factor of 9
F-Numbers (Brightness)
• Image brightness depends on:
• Focal length of the lens
• Diameter of the aperture (area)
• Intensity of light from the object
 Diameter
• For the same object, Brightness  
 Focal Length
• Define f# as
 focal length f 
f#  

 diameter D 
1
B

• Then
 
 f# 
2



2
F-Numbers (Contd.)
f#
1.4
2.0
2.8
4.0
5.6
8.0
16
(f#)2
2
4
8
16
32
64
256
1 / (f#)2
1/2
1/4
1/8
1/16
1/32
1/64
1/256
Note: Brightness changes by a factor of 2 between adjacent f#’s.
Lenses with the same f# produce the same intensity on the film plane.
Review Problems
1. What is the aperture diameter of a 50 mm lens
set at f# = 4?
D = 12.5 mm
2. What is the f# for a lens of 200 mm focal length
and the aperture diameter of the previous
problem?
f# = 16
3. How many times does the brightness change
when you go from f# = 4.0 to f# = 16?
Brightness decreases by a factor of 16
Exposure
• Correct exposure of the film is determined by
• Image brightness (f#)
• Film speed (ASA)
• Shutter speed
• For a given film speed,
Brightness x Exposure Time = Constant
t
Or
 constant
2
 f #
Review Problem
Suppose a proper exposure of a film could be
achieved by taking a picture at 1/50 s with f# = 8.
If under the same light conditions, we wished to
change the exposure time to 1/200 s, what f#
should we choose?
f# = 4
Depth of Field
• Lens opening (f-stop)
Smaller the aperture, the
greater the depth of field.
F# = 2
• Focus distance
The greater the focus distance
from camera to subject, the
greater the depth of field.
F# = 8
• Focal length of lens
The shorter the focal length,
the greater depth of field.
http://www.dofmaster.com/dofjs.html
F# = 22