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”

•Image formation: by actual (real image) or apparent (virtual image) intersection of two or more rays of light.

Ray

Wavefront

Law of Reflection

•Fermat’s principle of least time.

Which path takes the least time?

AB

AB

AB

•Incident ray, reflected ray, and the normal are in the same plane.

•Law is valid for any surface.

http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=57

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.

Convex(Outward curvature)

Concave(Inward curvature)

Image Formation by Spherical Mirrors

• How to locate and describe the image?

• Mathematical treatment: (Applicable to concave or convex mirrors).

• Object mirror distance = p

• Image mirror distance = q

• Focal length of mirror = f

• Object size (height) = Ho

• Image size = Hi

• Mirror (or lens) equation:

p

q

f

qpf

111

Spherical Mirrors (Contd.)

• Image location and its nature are given by:

• Magnification is given by:

• Note: Real image: q is + Concave mirror: f is +

Virtual image: q is – Convex mirror: f is -

fp

pfq

p

q

H

HM

o

i

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?

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?

12 ft.

Location: 40 cm to left of mirrorSize: Same as the object (M=1)Nature: RealDirection: 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?

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 sizeNature: VirtualDirection: Upright

Location: At the focal pointSize: Diminished

Summary: Concave Mirror Imaging

ObjectObject ImageImage ApplicationApplication

At infinityAt infinity At F, Smallest, Inverted, At F, Smallest, Inverted, RealReal

Camera, TelescopeCamera, Telescope

Between infinity Between infinity and Cand C

Left of F, Diminished, Left of F, Diminished, Inverted, RealInverted, Real

CameraCamera

At 2fAt 2f At 2f, Same size, Inverted, At 2f, Same size, Inverted, RealReal

Camera, Fax, XeroxCamera, Fax, Xerox

Between 2f and fBetween 2f and f Left of 2f, Magnified, Left of 2f, Magnified, Inverted, RealInverted, Real

Camera, XeroxCamera, Xerox

At fAt f At infinity (no image)At infinity (no image) HeadlightsHeadlights

Between F and VBetween F and V Behind the mirror, Virtual, Behind the mirror, Virtual, Magnified, UprightMagnified, Upright

Beauty mirrorBeauty mirror

http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=65

Summary: Convex Mirror Imaging

• Image is always:

Diminished

Virtual

Upright

• Application: Collects light from a wide area. Used as rear-view mirror.

http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=65

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:

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 - 1.33Flint glass - 1.66Air - 1.0003Diamond - 2.4

v

cn

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.

http://www.control.co.kr/java1/RefractionofLight/LightRefract.html

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.

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:

n1

n2

)()( 2211 SinnSinn

http://www.ps.missouri.edu/rickspage/refract/refraction.html

Trigonometric Ratio

• Consider a right angled triangle ABC.

• Sine of the angle is defined as the ratio of the sides BC to AC.

• 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.

ACLength

BCLength θSin

C

AB

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

Things appear shallower in water

Sun appears flatter at sunset

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

• Lens equation:

• Magnification:

Spherical Lens

Double ConcaveOr Diverging Lens

Double ConvexOr Converging Lens

+ Focal Length(Like Concave Mirror)

- Focal Length(Like Convex Mirror)

fp

pfq

p

qM

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?

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)

q = -100 cm (To the left of the lens, virtual)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:

• 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.

fP

1

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:

Bundles of fibers

Image Intensifiers / Magnifiers /

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 m 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 refractiveindex so they travelslower through the fiber.

Solution: Use lasers withhigh spectral purity.

Different Types of Fibers

Long distance applications

Local area networks

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).

http://micro.magnet.fsu.edu/primer/java/scienceopticsu/eyeball/index.html

Near Point = 25 cm

Far Point = Infinity

The Human Eye: Features (Contd.)

• Adjustable aperture:

• Iris: A muscle that changes size to adapt.

• Pupil: Opening diameter

• 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?

~ 1.5 mm under bright light ~ 6.0 mm under dim light

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

Effective center of cornea + lens

Image size = HiObject size = Ho

od

mm17

P from distanceObject

P from distance Image

H

HM :ionMagnificat

o

i

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)

Note: The size of a single cone is about 5 m!For scotopic vision this acuity is much less.

Hi = 6.8 x 10-6 m

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?

•Someone wearing glasses of – 2.0 diopters?

•Someone with near point of 25 cm and far point of infinity?

•Someone with near point of 150 cm and far point of infinity?

•Someone with near point of 17 cm and far point of 1.0 m?

Farsighted

Nearsighted

Normal vision

Farsighted

Nearsighted

Comparison

EyeEye CameraCamera

AdaptationAdaptationPupil diameter changes Pupil diameter changes

Photopic/scotopic visionPhotopic/scotopic vision

Aperture diameter Aperture diameter changeschanges

Film speed and exposureFilm speed and exposure

AccomodationAccomodation Lens shape changesLens shape changes Lens position changesLens position changes

ImageImage Real, invertedReal, inverted Real, invertedReal, inverted

Light sensitive Light sensitive materialmaterial RetinaRetina FilmFilm

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:

Note: Zoom lenses have variable focal lengths.

Wide Angle 28 mm

Normal50 mm

Telephoto 300 mm

Smaller image Larger image

Field of view: 950 470 80

2f

1 Intensity Image

f Size Image

Effect of Focal Length on Image Size

F

F

Short FL Lens

Long FL Lens

Film

Film

Small Image SizeLarge Field of View

Large Image SizeSmall Field of View

Effect of Focal Length on Image Size (Contd.)

That's Seattle about 2 miles away. focal length 36 mm focal length 138 mm

focal length 432 mm focal length 276 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 3Area 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

• For the same object,

• Define f# as

• Then

2

Length Focal

Diameter Brightness

Ddiameter

flength focal f#

2

f#

1 B

F-Numbers (Contd.)

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.

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

Review Problems

1.What is the aperture diameter of a 50 mm lens set at f# = 4?

2. What is the f# for a lens of 200 mm focal length and the aperture diameter of the previous problem?

3. How many times does the brightness change when you go from f# = 4.0 to f# = 16?

D = 12.5 mm

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

Or constant

# 2 f

t

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.

• Focus distanceThe greater the focus distance from camera to subject, the greater the depth of field.

• Focal length of lensThe shorter the focal length, the greater depth of field.

F# = 2

F# = 8

F# = 22

http://www.dofmaster.com/dofjs.html