<<

Astronomy 80 B: Light

Lecture 9: curved , , 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 ¥ 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

2003 April 29 80B-Light 10 2003 April 29 80B-Light 11 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 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 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

¥ ¥ Spherical Aberration ¥ Field angle effects (off-axis aberrations) Ð Field curvature Ð Ð Astigmatism Ð

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