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Chapter 26 Geometrical Optics Snell’s Law Thin Lens Equation

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1) Index of Refraction, n Speed of light is reduced in a medium Air Water4/3 Glass1.5 Diamond2.4

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2) Snell’s Law a) Reflection and Transmission Transmitted ray light splits at an interface

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Transmitted ray

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(b) Refraction: Transmitted ray is bent at interface toward normal if n increases

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away from normal if n decreases

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toward normal if n increases c) Derivation of Snell’s Law

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Example: Rear-view mirror

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Example: Apparent Depth

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x d For small angles, d’

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3) Total internal reflection a) The concept For small values of 1, light splits at an interface

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For larger values of 1, 2 > 90º and refraction is not possible Then all light is reflected internally Note: this is only possible if n 1 > n 2

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b) Critical incident angle Snell’s law:

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Some critical angles Water-air: 49º Glass - air: 42º Diamond - water: 33º Diamond - air: 24º Why diamonds sparkle

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c) Prisms (glass-air critical angle = 45º)

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Prisms in binoculars –Longer light path –Image erect

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d) Fibre optics Low loss transmission of light, encoded signals.

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Fibre optic bundles, coherent bundles Imaging applications: endoscopy

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4) Dispersion Index of refraction depends on wavelength

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Rainbow

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Sun Dogs (parhelia)

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5) Image Formation a) Seeing an object Diffuse reflection

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b) Image formation with a pinhole Diffuse reflection Diffuse reflection screen

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Characteristics of pinhole imaging –Infinite depth of field (everything in focus) –Arbitrary magnification –Low light (increasing size produces blurring) Diffuse reflection screen Diffuse reflection

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c) Ideal lens

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Characteristics of the ideal lens –All rays leaving a point on object meet at one point on image –Only one perfect object distance for selected image distance (limited depth of field -- better for smaller lens)

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6) Thin lenses a) Converging - thicker in the middle

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(i) Parallel coaxial rays converge at focus Reversible

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(ii) Symmetric - rays leaving focal point emerge parallel (f’ = f)

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(iii) Ray through centre undeviated

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Summary of ray tracing rules for converging lens

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b) Diverging - thinner in the middle

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(i) parallel, coaxial rays diverge as if from focus Reversible

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(ii) symmetric - rays converging toward focus emerge parallel

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(iii) ray through centre undeviated

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Summary of ray-tracing rules for diverging lens

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c) Real lenses: - usually spherical surfaces - approximate ideal lens for small angles (paraxial approximation)

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7) Image Formation with thin lenses (ray tracing) (a)Converging lens - real image Use 2 of 3 rays:

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camera /CCD sensor

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(b) Converging lens - virtual image

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(c) Diverging lens - virtual image

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8) Thin Lens Equation a) The equation

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b) Sign Convention (left to right) (i)Focal Length: f > 0 converging f < 0 diverging (ii) Object distance d o > 0 left of lens (real; same side as incident light) d o < 0 right of lens (virtual; opposite incident light) (iii) Image distance d i > 0 right of lens (real; opposite incident light) d i < 0 left of lens (virtual; same side as incident light) (iv) Image size h i > 0 erect h i < 0 inverted

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c) Lateral magnification Definition: From geometry (and sign convention):

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9) Compound Lenses Image of first lens is object for the second lens. Apply thin lens equation in sequentially.

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Overall magnification is the product:

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Example: Problem Find final image and magnification.

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10) Vision and corrective lenses a) Anatomy of the eye

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120 x 10 6 rods - detect intensity: slow, mono, sensitive 6 x 10 6 cones - detect frequency: R nm, G nm, B nm

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b) Optics - Accomodation: focal length changes with object distance - near point: nearest point that can be accomodated - normally < 25 cm - far point: furthest point that can be accomodated - normally ∞

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c) Myopia - far point < ∞ - near-sighted (far-blind) - correction: object at ∞ --> image at far point

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Correction: object at ∞ --> image at far point (ignoring the eye-lens distance)

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d) Refractive Power For a far point of 50 cm, f = -50 cm, Lens prescription: -2

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e) hyperopia (hypermetropia) - near point > 25 cm - far-sighted (near-blind) - correction: object at 25 cm --> image at near point

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Correction: object at 25 cm --> image at near point (ignoring the eye-lens distance) For near point of 40 cm, f = 66 cm Power = (reading glasses)

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Examples: Problem Age 40: f = 65.0 cm --> NP’ = 25.0cm Age 45: NP’ --> 29.0 cm (a) How much has NP (without glasses) changed? (b) What new f is needed? Problem FP = 6.0 m corrected by contact lenses. (Find f) An object (h = 2.0 m) is d = 18.0 m away. (a)Find image distance with lenses. (b)Find image height with lenses.

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11) Angular Magnification a) Angular size

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b) Angular magnification

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12) Magnifier With magnifier: (Magnifier allows object to be close to the eye)

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Without magnifier: Highest magnification (d i = -N): Lowest magnification (d i = -∞): (tense eye) (relaxed eye) (Magnification quoted with N = 25 cm, for relaxed eye)

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Example: Problem Farsighted person has corrective lenses with f = 45.4 cm. Maximum magnification of a magnifier is 7.50 (normal vision). What is the maximum magnification of the magnifier for the farsighted person without lenses?

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13) Compound Microscope Simple magnifier: M = N/f –to increase M, decrease f –practical limits to decreasing f (and therefore size): small lens difficult to manufacture and use increases aberrations Microscope introduces an additional lens to form a larger intermediate image, which can be viewed with a magnifier

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L Magnification: For image at ∞, d i2 = f e For max M, d o1 f o For d i2 = ∞, d i1 + f e = L

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Example: Problem Microscope with f o = 3.50 cm, f e = 6.50 cm, and L = 26.0 cm. (a) Find M for N = 35.0 cm. (b) Find d o1 (if first image at F e ) (c) Find lateral magnification of the objective.

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14) The Astronomical Telescope Magnifier requires d o ∞ for stars Introduce objective to form nearby image, then use magnifier on the image

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Magnification: Long telescope, small eyepiece

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Example: Problem Yerkes Observatory: f o = 19.4 m, f e = 10.0 cm. (a) Find angular magnification. (b) If h o = 1500 m (crater), find h i, given d o = 3.77 x 10 8 m (c) How close does the crater appear to be.

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Galilean Telescope (Opera glasses)

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Reflecting Telescope

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