Astronomy 80 B: Light
Lecture 9: curved mirrors, lenses, aberrations 29 April 2003 Jerry Nelson Sensitive Countries LLNL field trip
2003 April 29 80B-Light 2 Topics for Today
¥ Optical illusion ¥ Reflections from curved mirrors Ð Convex mirrors Ð anamorphic systems Ð Concave mirrors ¥ Refraction from curved surfaces Ð Entering and exiting curved surfaces Ð Converging lenses Ð Diverging lenses ¥ Aberrations
2003 April 29 80B-Light 3 2003 April 29 80B-Light 4
2003 April 29 80B-Light 8 2003 April 29 80B-Light 9 Images from convex mirror
2003 April 29 80B-Light 10 2003 April 29 80B-Light 11 Reflection from sphere
¥ Escher drawing of images from convex sphere
2003 April 29 80B-Light 12 ¥ Anamorphic mirror and image
2003 April 29 80B-Light 13 ¥ Anamorphic mirror (conical)
2003 April 29 80B-Light 14 ¥ The artist Hans Holbein made anamorphic paintings
2003 April 29 80B-Light 15 Ray rules for concave mirrors
2003 April 29 80B-Light 16 Image from concave mirror
2003 April 29 80B-Light 17 Reflections get complex
2003 April 29 80B-Light 18 Mirror eyes in a plankton
2003 April 29 80B-Light 19 Constructing images with rays and mirrors
¥ Paraxial rays are used Ð These rays may only yield approximate results Ð The focal point for a spherical mirror is half way to the center of the sphere.
Ð Rule 1: All rays incident parallel to the axis are reflected so that they appear to be coming from the focal point F. Ð Rule 2: All rays that (when extended) pass through C (the center of the sphere) are reflected back on themselves. Ð Rule 3: All rays that (when extended) pass through F are reflected back parallel to the axis.
Ð Parallel Rays Rule: Rays parallel to each other are imaged to the same place on the focal plane
2003 April 29 80B-Light 20 Spherical mirror images ¥ Convex spherical mirrors Ð image is virtual Ð focal length is half the radius of the mirror Ð image is closer to mirror Ð image is erect Ð makes a wide angle mirror ¥ Concave spherical mirrors Ð focal length is half the radius of the mirror Ð If object is further than the center of curvature: ¥ image is real ¥ image is closer to mirror ¥ image is inverted ¥ image is de magnified Ð If object is between the center of curvature and the focus: ¥ image is real ¥ image is further from mirror ¥ image is inverted ¥ image is magnified Ð If object is between the focus and mirror: ¥ image is virtual ¥ image is erect ¥ image is magnified Mirror Equation
11+=− 1 xxoi f f is positive for convex mirror, negative for concave mirror. This equation allows us to calculate the location of the image. xi, xo are positive as shown. xo
so
si f x Ð Magnification: i
si =−xi so xo Ð This equation allows us to calculate the size of the image
2003 April 29 80B-Light 22 Example: convex mirror
11+=− 1 xxoi f B B’ x A o f A’ c xi
If xo = 5 cm f = 2 cm
xi = Ð1.43 cm
2003 April 29 80B-Light 23 Example: Concave mirror
xo = 10 cm f = Ð3 cm x = +4.29 cm i xo
f 11 1 +=− xi xxoi f
2003 April 29 80B-Light 24 Mirror Equation Examples
¥ Given f and xo find xi
1/xi = Ð (1/f) - (1/xo) = [(xo+f)/xof]
xi = Ð (xof)/(xo + f)
¥ Given xo and xi find f
f = Ð (xoxi)/(xo+xi)
¥ Given f and xi find xo
xo = Ð (xif)/(xi + f) ¥ Example
xo = 49 mm, f = Ð 30 mm (neg for concave mirror)
xi = 78 mm
so = 20 mm
si = Ð 32 mm (negative for inverted image)
2003 April 29 80B-Light 25 Refraction at spherical surface
¥ Refracting properties of spherical lens surfaces ¥ (remember, light is “pulled” towards normal as it enters higher index material) ¥ Light is “pushed” away from normal as it enters lower index material
2003 April 29 80B-Light 26 Converging Lenses
¥ Converging lenses can focus parallel light ¥ Converging lenses generate parallel light from a point source placed at the lens focal point ¥ Focal length f related to index n and surface radii r 1 n −1 = 2() f r 2003 April 29 80B-Light 27 Diverging lenses
¥ Often, doubly concave lenses ¥ Make parallel light diverge
2003 April 29 80B-Light 28 Pinholes and eye
2003 April 29 80B-Light 29 2003 April 29 80B-Light 30 lenses
¥ The three ray rules for constructing an image from a lens
2003 April 29 80B-Light 31 Constructing images with rays and lenses
¥ Paraxial rays are used Ð These rays may only yield approximate results Ð Thin lenses only ¥ focal length is positive for converging lens ¥ focal length is negative for diverging lens
Ð Rule 1: A ray parallel to the axis is deflected through F' (or as if it came from F') Ð Rule 2: A ray through the center of the lens continues undeviated Ð Rule 3: A ray to the lens that (when extended, if necessary) passes through F is deflected parallel to the axis
Ð Parallel Rays Rule: Rays parallel to each other are imaged onto 2003 April 29the same point in the focal80B-Light plane 32 Ray tracing in a converging lens
P’ P
Here, image P’ is virtual, erect and larger than the object P
2003 April 29 80B-Light 33 Visibility in a converging lens
P
Virtual image only visible from shaded area 2003 April 29 80B-Light 34 visibility
¥ a) showing how lens is extended for construction and region of visibility
¥ b) parallel light focussed onto focal plane and region of visibility
2003 April 29 80B-Light 35 Ray tracing a diverging lens
¥ Draw three “standard” rays
¥ These will intersect at image
2003 April 29 80B-Light 36 Converging lens and sign conventions
2003 April 29 80B-Light 37 Lens equation
111+= xxfoi ¥ f is positive for converging lens, negative for diverging lens. This allows us to calculate location of the image
si =−xi so xo ¥ This equation allows us to calculate the size of the image (magnification)
2003 April 29 80B-Light 38 Lens Equation Examples
¥ Given f and xo find xi
1/xi = (1/f) - (1/xo) = [(xoÐ f)/xof]
xi = (xof)/(xo Ð f)
¥ Given xo and xi find f
f = (xoxi)/(xo+xi)
¥ Given f and xi find xo
xo = (xif)/(xi Ð f) ¥ Example
xo = 49 mm, f = 30 mm (positive for converging lens)
xi = 77 mm
so = 20 mm
si = Ð 32 mm (negative for inverted image)
2003 April 29 80B-Light 39 ¥ Constructing imaging from multiple lenses Ð Construct image from 1st lens Ð Add rays that will be useful for the 2nd lens construction (the 3 rays ) Ð Complete ray tracing with these 3 rays through the 2nd lens to find final image
2003 April 29 80B-Light 40 Power of lenses
¥ The power of a lens is its inverse focal length (how strongly it can focus parallel light) Ð P = 1/f ¥ By convention, the units of power are measured in diopters Ð It is numerically equal to the inverse of the focal length of the lens, measured in meters. Ð Examples: ¥ A converging lens with a 1m focal length has a power of 1D ¥ A lens with a power of -5D is a diverging lens with a focal length of -0.2m ¥ A doubly concave lens (diverging) with a focal length of - 2m has a power of -0.5D ¥ The powers of adjacent lenses add to form net power
2003 April 29 80B-Light 41 Fresnel lens principle
2003 April 29 80B-Light 42 Fresnel lens applications
2003 April 29 80B-Light 43 Traffic light ¥ Fresnel lens images scene onto ground glass screen at ~ focal distance ¥ Light source illuminates ground glass ¥ Mask the screen to block light that would go to undesired locations
2003 April 29 80B-Light 44 Aberrations
¥ Chromatic Aberration ¥ Spherical Aberration ¥ Field angle effects (off-axis aberrations) Ð Field curvature Ð Coma Ð Astigmatism Ð Distortion
2003 April 29 80B-Light 45 Chromatic aberration and doublets
2003 April 29 80B-Light 46 Spherical aberration from a lens
2003 April 29 80B-Light 47 Spherical aberration from a concave mirror
2003 April 29 80B-Light 48 Parabolic mirrors have no spherical aberration (on axis)
2003 April 29 80B-Light 49 Ellipsoidal reflector
2003 April 29 80B-Light 50 Spherical aberration in glass of water
2003 April 29 80B-Light 51 Field curvature from lens
2003 April 29 80B-Light 52 Comatic aberration
2003 April 29 80B-Light 53 Astigmatism aberration
2003 April 29 80B-Light 54 Distortion
¥ Upper image shows barrel distortions
¥ Lower image shows pincushion distortion
2003 April 29 80B-Light 55