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

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

  2. 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

  3. Law of Reflection • 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.

  4. 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?

  5. Application - Rear View Mirror

  6. 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)

  7. p 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:

  8. 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 -

  9. 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

  10. 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

  11. Summary: Concave Mirror Imaging http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=65

  12. 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

  13. 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)

  14. Image with spherical aberration Image without spherical aberration

  15. 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

  16. 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

  17. 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

  18. 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

  19. C 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.

  20. 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

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

  22. 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.

  23. Image Formation by Refraction: Lenses • Lens equation: • Magnification: Spherical Lens Double Convex Or Converging Lens Double Concave Or Diverging Lens + Focal Length (Like Concave Mirror) - Focal Length (Like Convex Mirror)

  24. 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)

  25. 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.

  26. 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.

  27. Image with chromatic aberration Image without chromatic aberration

  28. 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

  29. Fiber Optics: Typical Physical Dimensions

  30. 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.

  31. 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”.

  32. 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….

  33. 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!

  34. Attenuation (Contd.) Losses are minimum at 1.5 mm wavelength!

  35. 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

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

  37. Material Dispersion Shorter wavelengths have higher refractive index so they travel slower through the fiber. Solution: Use lasers with high spectral purity.

  38. Different Types of Fibers Local area networks Long distance applications

  39. Comparison of Data Rates

  40. 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

  41. 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

  42. 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

  43. 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.

  44. 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.

  45. 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!

  46. 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

  47. 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.

  48. 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.

  49. Astigmatism Test Pattern

  50. 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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