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6.2 Refraction

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6.2 Refraction 6.2 Refraction • origin of refraction and the derivation of Snell's law • dispersion, wavelength-dependent refraction and prisms • how lenses use refraction to focus light • the thin lens equation and image formation • light gathering power and the lens f-number • lens aberrations and how they affect the focal spot size and image fidelity 6.2 : 1/13 Refraction Refraction is the change in ___________________ which occurs at a refractive index boundary. Two facts are responsible for refraction: (1) the wavelength of light depends upon the ________________, λ = λ0/n; and, (2) at the refractive index boundary the ________ of the wave has to be the same in both materials. The figure shows refraction for an air/glass boundary. The n2 = 1.5 relationship between the two angles can be determined by geometry. The triangle ABC has θ2 sinθ = BC/AC = (λ/n )/AC. The C 1 1 θ1 B triangle CDA has sinθ2 = AD/AC = (λ/n2)/AC. AC and λ can be eliminated to obtain Snell's Law. A D n1 = 1 n1 sinθ1 = n2 sinθ2 6.2 : 2/13 Refraction Example (1) When incident light is parallel to the surface normal, the transmitted light is ____________ to the surface normal (i.e. there is no refraction). nn122×°=×sin 0 sinθ n1 sinθ2 =×= 0 0 n2 n1 = 1 n2 = 1.5 θ2 = 0 6.2 : 3/13 Refraction Example (2) When incident light is perpendicular to the surface normal, the transmitted light travels at what is called the ______________. n2 = 1.5 nn12×°=×sin 90 sinθc n1 sinθc =× 1 θc = 41.8E n2 −1 ⎛⎞n1 θc = sin ⎜⎟ ⎝⎠n2 n1 = 1 θ = 90E 6.2 : 4/13 Refraction Example (3) When propagating from a higher refractive index region into a region with a lower refractive index, the largest angle that will be transmitted is the critical angle. Light impinging on the refractive index boundary at angles greater than the critical angle will undergo ____________________. The wave in the low index region is a __________________________ that extends for a few wavelengths (this is analytically useful). θi = 65E n2 = 1.5 n1 = 1 n2 = 1.5 -300 -150 0 150 300 amplitude n1 = 1 θ = 65E distance (nm) r 6.2 : 5/13 Prism • for a material with normal dispersion, violet light will get bent toward the surface normal to a _________ extent than red light • when the high index material is shaped into prism, light gets refracted twice - with violet bending more than red both times • different wavelengths of light will then exit the prism at _____________ • wavelengths are dispersed by angle, which becomes ________ at a screen removed from the prism air n = 1 white red glass violet n = f(λ) 6.2 : 6/13 Elliptical Lens • when a plane wave impinges on an elliptically-shaped region of higher refractive index, it is converted into a ________________ collapsing toward the focal point • a lens can be formed by making the second surface spherical • the distance from the front lens surface to the focal point is called the _________________ • the focal spot size is given by the following elliptical surface equation, of lens 0.61λ f ray tracing ρ = r focal point where ρ is the spot radius, λ is the wavelength, f is spherical surface the focal length, and r is of lens the incoming beam radius • ρ is called the plane wave _____________________ spherical wave 6.2 : 7/13 Spherical Lens • elliptical surfaces are very difficult to grind • close to the optical axis a spherical surface nicely approximates an elliptical surface • the approximation works only when sinθ = θ (this is called the _________________________________________________) region where the two surfaces are nearly the same optical axis elliptical surface spherical surface 6.2 : 8/13 Thin Lenses biconvex plano-convex A thin lens is one where the physical thickness is small compared to the surface radii of curvature, the focal length, and the object and image distances. Convex lenses cause collimated light ____________, while concave lenses cause collimated light _____________. The focal length of a lens is determined by the lens makers formula. biconcave plano-concave 111⎛ ⎞ =−()nl 1 ⎜ − ⎟ fRR⎝ in out ⎠ If a surface curves inward toward the right, its radius is ______. If a surface curves inward toward the left, its radius is _______. The focal length of a convex lens is then positive, while that for a concave lens is negative. 6.2 : 9/13 Convex Lens Distances f dim dob f The relationship among the distances is given by, where f is the focal length, dob is the distance of the object from the lens, and dim is the distance of the image from the lens. Note that the image is _______ and magnified by M = ________. 6.2 : 10/13 Concave Lens Distances f dob f dim A concave lens creates a ____________. By convention f, dob and dim are all ______________ in sign. 6.2 : 11/13 Lens f-Number • the f-number of a lens is given by the focal length divided by the diameter, f/# = f/D • the f-number is used as a metric for the _____________ that can be gathered from a point source - the lower the value the higher the collection efficiency • referring to the figure it can be seen that the fraction of light collected is the area subtended by the lens divided by the area of the sphere f/# % gathered 1 _____ 21.5 50.25 D 10 0.062 f 20 0.016 light gathering power drops approximately as the _______ of the f-number 6.2 : 12/13 Lens Aberrations • chromatic: Since the refractive index varies with wavelength, the focal length will __________________. When an object is multicolor the image will focus to a variety of distances. Chromatic aberrations are minimized by combining convex and concave lenses into an achromatic doublet. • spherical: A spherical surface does not focus light to a single point. The further a ray is from the optical axis, the closer it will focus ____________. For approximately equal object and image distances, the use of a lens with equally curved surfaces will minimize the aberration. For an object or image at infinity, the use of a plano-convex lens will minimize the aberration (the planar side is toward the focal spot). • off-axis: When the object is off-axis such that the ray bundle does not satisfy sinθ = θ, the rays impinging on the outer edge of the lens are focused ___________ from the optical axis than center rays. This is called coma and gives comet-shaped images. With an off-axis object the rays in the plane of the image have a different focal distance than those perpendicular. This is called astigmatism. 6.2 : 13/13.
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