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# chapter 4: geometric opticshow is light collected and focused to form images - PowerPoint PPT Presentation

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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Geometric Optics

How is light collected and focused to form images?

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

• Fermat’s principle of least time.

• Which path takes the least time?

B

B

B

A

A

A

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?

• 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)

f

q

• 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:

• 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 -

• 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

• 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

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

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 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 - 1.33

• Flint glass - 1.66

• Air - 1.0003

• Diamond - 2.4

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

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

q1

n1

n2

q2

• 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:

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

q

A

B

• 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.

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

Sun appears flatter at sunset

Things appear shallower in water

Mirages

Dispersion and rainbows

http://www.ps.missouri.edu/

rickspage/refract/refraction.html

• Optical fibers

• SLR Cameras & binoculars

• Diamonds

• appear

• bright.

Spherical Lens

Double Convex

Or Converging Lens

Double Concave

Or Diverging Lens

+ Focal Length

(Like Concave Mirror)

- Focal Length

(Like Convex Mirror)

• 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)

• 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.

• 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.

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

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.

• 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!

• 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!

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.

Shorter wavelengths

have higher refractive

index so they travel

slower through the fiber.

Solution: Use lasers with

high spectral purity.

Local area networks

Long distance applications

• 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

• 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.)

• 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.

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!

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

• 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.

• Presbyopia (aging sight): Abnormal eyeball shape and weak ciliary muscles.

• Correction – bifocal lenses.

• Astigmatism:

• Sharper curvature of

• the cornea.

• Correction – cylindrical

• lenses.

• 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

• Parts:

• Light proof box

• Shutter with variable speed (Duration of exposure)

• Film (Light sensitive material)

• 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.

F

Short FL Lens

Film

Small Image Size

Large Field of View

Film

F

Long FL Lens

Large Image Size

Small Field of View

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

focal length 138 mm

focal length 276 mm

focal length 432 mm

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

• For the same object,

• Define f# as

• Then

• 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

• 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

• 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 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

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