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