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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”

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Chapter 4:

Geometric Optics

How is light collected and focused to form images?

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Geometric Optics


Light bouncing back from

a surface.


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.



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Law of Reflection

  • Fermat’s principle of least time.

  • Which path takes the least time?







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

  • Law is valid for any surface.

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  • 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?

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  • 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.


(Inward curvature)


(Outward curvature)

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  • 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:

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  • 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 -

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  • Review Problems

  • 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?

  • 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 mirror

Size: Same as the object (M=1)

Nature: Real

Direction: Inverted

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  • Review Problems (Contd.)

  • Where would the image of an object very distant from a concave mirror be located? What would the size of such an image be?

  • 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: At the focal point

Size: Diminished

Location: q = -20 cm (behind the mirror)

Size: M=2, so 10 cm size

Nature: Virtual

Direction: Upright

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Summary: Concave Mirror Imaging

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  • 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)

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Image with spherical aberration

Image without spherical aberration

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  • 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 - 1.33

  • Flint glass - 1.66

  • Air - 1.0003

  • Diamond - 2.4

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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

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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.

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  • 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:

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  • Trigonometric Ratio

  • Consider a right angled triangle ABC.

  • Sine of the angle q 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.

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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


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Some Interesting Effects of Refraction

Sun appears flatter at sunset

Things appear shallower in water


Dispersion and rainbows

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  • Optical fibers

  • SLR Cameras & binoculars

  • Diamonds

  • appear

  • bright.

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Spherical Lens

Double Convex

Or Converging Lens

Double Concave

Or Diverging Lens

+ Focal Length

(Like Concave Mirror)

- Focal Length

(Like Convex Mirror)

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  • Review Problems

  • 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?

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

M = 5 (Magnified)

q = 60 cm (To the right of the lens, real)

M = 5 (Magnified)

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  • 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.

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Achromatic Doublet

  • 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.

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Image with chromatic aberration

Image without chromatic aberration

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  • 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

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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.

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  • 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”.

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  • 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….

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  • 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!

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Attenuation (Contd.)

Losses are minimum at 1.5 mm wavelength!

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  • 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

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  • Modal Dispersion

  • Signals traveling different paths will arrive at different times. Solution: Use single mode or gradient index fibers.

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Shorter wavelengths

have higher refractive

index so they travel

slower through the fiber.

Solution: Use lasers with

high spectral purity.

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Different Types of Fibers

Local area networks

Long distance applications

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  • 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

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  • 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 =


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  • 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

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  • 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.

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The Reduced Eye - A Simplified Model

Image size = Hi

Object size = Ho

Effective center of cornea + lens

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!

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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 mm!

For scotopic vision this acuity is much less.

Hi = 6.8 x 10-6 m

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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.

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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.

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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?



Normal vision



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The Camera

  • Parts:

  • Light proof box

  • Adjustable lens system (Accomodation)

  • Adjustable aperture (Adaptation)

  • Shutter with variable speed (Duration of exposure)

  • Film (Light sensitive material)

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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.

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Effect of Focal Length on Image Size


Short FL Lens


Small Image Size

Large Field of View



Long FL Lens

Large Image Size

Small Field of View

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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 276 mm

focal length 432 mm

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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

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  • 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

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    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.

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    • Review Problems

    • 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

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    • 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

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    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

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    • Depth of Field

    • Lens opening (f-stop)

    • Smaller the aperture, the

    • greater the depth of field.

    • Focus distance

    • The greater the focus distance

    • from camera to subject, the

    • greater the depth of field.

    • Focal length of lens

    • The shorter the focal length,

    • the greater depth of field.

    F# = 2

    F# = 8

    F# = 22