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Chapter 4: Geometric Optics How is light collected and focused to form images?

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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

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Law of Reflection Fermat’s principle of least time. Which path takes the least time? A B A B A B 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

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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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Application - Rear View Mirror

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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. Convex (Outward curvature) Concave (Inward 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) = H o Image size = H i Mirror (or lens) equation: p q f

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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: q = -20 cm (behind the mirror) Size: M=2, so 10 cm size Nature: Virtual Direction: Upright Location: At the focal point Size: Diminished

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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

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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

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Imperfect Mirrors Spherical aberration is an inherent defect. Incoming parallel rays focus at different points! Spherical aberration = (F 2 – F 1 ) F 1 (Marginal Rays focus here) F 2 (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 10 8 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 10 8 m/s

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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.

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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: q 1 q 2 n 1 n 2 http://www.ps.missouri.edu/rickspage/refract/refraction.html

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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 20 0 , 30 0 , 45 0 , 90 0 . Find the angles whose sines are 0.1, 0.3, 0.6, 0.9. q C A B

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Review Problems A ray of light traveling in air strikes a glass surface (n = 1.5) at an angle of 24 0 from the normal. At what angle will it be refracted in glass? Given: Sin(24 0 ) = 0.407, Sin(15.7 0 ) = 0.2713 15.7 0

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Some Interesting Effects of Refraction Things appear shallower in water Sun appears flatter at sunset Mirages Dispersion and rainbows

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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.

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Image Formation by Refraction: Lenses Lens equation: Magnification: Spherical Lens Double Concave Or Diverging Lens Double Convex Or Converging 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 = 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)

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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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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

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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 ~ 60 0

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Fiber Optics: Typical Physical Dimensions

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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 10 7 Hz! Red light ranges from 5 x 10 14 – 4.3 x 10 14 Hz So available bandwidth is about 7 x 10 13 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 m m 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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Material Dispersion 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 Long distance applications Local area networks

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Comparison of Data Rates

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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). http://micro.magnet.fsu.edu/primer/java/scienceopticsu/eyeball/index.html Near Point = 25 cm Far Point = Infinity

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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 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.

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The Reduced Eye - A Simplified Model Effective center of cornea + lens Image size = H i Object size = H o Resolving power (Limit of visual acuity): Two points must be separated by at least 1/60 th 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 H o = 0.1 mm, and d o = 25 cm) Note: The size of a single cone is about 5 m m! For scotopic vision this acuity is much less. H i = 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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Astigmatism Test Pattern

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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? Farsighted Nearsighted Normal vision Farsighted Nearsighted

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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

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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. Wide Angle 28 mm Normal 50 mm Telephoto 300 mm Smaller image Larger image Field of view: 95 0 47 0 8 0

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Effect of Focal Length on Image Size F F Short FL Lens Long FL Lens Film Film Small Image Size Large Field of View 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 432 mm focal length 276 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. 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

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

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