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Chapter 5
Geometrical OpticsMirrors and Prisms
Phys 322Lecture 14
Optical bench
http://webphysics.davidson.edu/Applets/optics4/default.html
Mirrors
Ancient bronze mirror
Liquid mercury mirror
Hubble telescope mirror
Planar mirror
i rsi = -so
Sign convention: s on the object side is positive, and negative on the opposite side
also called plane, or flat mirrors
Planar mirror
For a plane mirror, a point source and its image are at the same distance from the mirror on opposite sides; both lie on the same normal line.Image is virtual, up-right, and life-size (MT = +1)
o
i
o
iT s
syyM The equation for lens works:
si = -so
Sign convention: s on the object side is positive, and negative on the opposite side
Exercise: plane mirror height
How high should be the mirror for a person to see a full image ofhim/her-self?
Solution:A
B
C
D
ETriangle ABC is twice as small as ADE
BC is half DE (the height of the guy)
1. Mirror (BC) should be at least half of the guy’s height (DE) 2. Its bottom should 1/2 of the height of guy’s eyes from the ground
‘Mirror image’
Mirror image of left hand is a right hand
Inversion: converting right-handed coordinate system into left-handed one
Even number of mirrors can be used to avoid inversion
Applications: steering light
reflex camera (SLR)
Atomic force microscope
DLP projection TV
http://www.plus-america.com/papers.html
Parabolic aspherical mirror
Make a mirror that will converge plane waves into a pointFermat’s principle:
FAAWFAAWOPL 222111
22221111 DAAWDAAW
111 DAFA 222 DAFA
In general: ADAF
This is the surface of paraboloid: y2 = 4fx (origin at vertext V)
Application:headlights, flashlights,radars,dish antenna,….
V
Aspherical mirrors
Collects light from one point to another
divergingdiverging
convergingconverging
convergingdiverging
divergingconverging
off-axis parabolic
Spherical mirror
222 RRxy
fxy 42
Paraboloid and sphere are similar in paraxial approximation
2222 2 RRxRxy 2 2 2y x xR
small when close to axis x
Spherical mirror formula
SAP is bisected by AC:
PACP
SASC
R<0 in real object spaceso>0 in real object spacesi>0 in real image spacef >0 concave mirror
Sign convention:
RsSC o
isRCP
Paraxial approximation: osSA isPA
i
i
o
o
sRs
sRs
Rss io
211
Focal lengths:lim1/ 1/ 2 /i
o o osf s f R
lim 1/ 1/ 2 /o
i i isf s f R
Mirror Formula
Rfss io
2111
Spherical mirrors
Rfss io
2111
Note: Both mirror and lens equations are the same, except the real image is in front of mirror, but it is behind the lensMagnification equations are the same as well.
Concave mirror: principal axes and image
S
f
1) Parallel to principal axis reflects through F.2) Through F, reflects parallel to principal axis.
3) Through center.
C
#1
#2#3
NOTE: Any other ray from object tip which hits mirror will reflect through image tip
Image is:Real (light rays actually cross)Inverted (Arrow points in opposite direction)Diminished (smaller than object, only if object is further than C)
si
Principal rays for concave mirror:Rfss io
2111
Convex mirror: principal axes and image
Image is:Virtual (light rays don’t really cross)Upright (same direction as object)Diminished (smaller than object)
SP
1) Parallel to principal axis appear to originate from F after reflection.2) Through F, reflects parallel to principal axis.3) Through center.
C
#1
#2#3
**For a real object, image is always virtual, upright and diminished
f
Principal rays for convex mirror:
F
Exercise: can a concave mirror form a virtual image?
fss io
111
Concave mirror: so and f are always positive, want to get negative si
siso
0111
oi sfs
fso
An object must be between mirror and its focal plane
F
so si
virtualimage
Spherical mirrors
Examples
Dispersing prism
ttii nn sinsin
Bending depends on wavelength: dispersing prism, i.e. n=n()
Can we use optical flat for dispersing light?Rays emerge parallel to each other.Practically we don’t see them (focused by the eye at the same spot).
Example
Dispersing prism equation
i1
ttii nn sinsin
t2
nt=nni=1
cossinsinsinarcsin 1122
1 iti n
Total deviation is a function of refraction index:
Minimum deviation min occurs when i1 = t2
2/sin
2/sin min
n can use to determine n
And this arrangementmaps position to angle:
Spectral analyzer
out inx
Prism spectrometers
Drawbacks:() - nonlinear dependenceLow spectral resolutionSmall aperture
Constant-deviation dispersing prisms
min=90o always!
Pellin-Broca prism:
Pellin, Ph. and Broca, André (1899), "A Spectroscope of Fixed Deviation". Astrophysical Journal 10 337
Abbe prism:
Ernst Abbe1840-1905
min=60o always!
Fix input-output at 90o or 60o and rotate prism for different wavelengths
Reflecting prismReflect the beam with no dispersion using total internal reflection
If we make t1 = i2 - like in flat glass plate
= 2i1 + “achromatic” prism
Reflecting prisms
The right-angle prism The Porro prism
The Dove prism
The penta prism