168 |
Optics |
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8.5 Geometrical optics
Lenses and mirrorsa
r2 |
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v |
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object |
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f |
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x1 |
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object |
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f |
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u |
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v |
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image |
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R |
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x2 |
f |
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r1 |
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u |
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lens |
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mirror |
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sign convention |
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r |
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centred to right |
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centred to left |
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u |
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real object |
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virtual object |
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v |
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real image |
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virtual image |
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f |
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converging lens/ |
diverging lens/ |
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concave mirror |
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convex mirror |
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MT |
erect image |
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inverted image |
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L = η dl |
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L |
optical path length |
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b |
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η |
refractive index |
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Fermat’s principle |
is stationary |
(8.63) |
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dl |
ray path element |
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1 |
+ 1 |
= 1 |
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u |
object distance |
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Gauss’s lens formula |
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(8.64) |
v |
image distance |
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u |
v |
f |
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f |
focal length |
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Newton’s lens |
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x1x2 = f2 |
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(8.65) |
x1 |
= v − f |
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formula |
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x2 |
= u− f |
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Lensmaker’s |
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1 |
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= (η − 1) |
1 |
1 |
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r |
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radii of curvature of |
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formula |
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u |
+ v |
r1 |
− r2 |
(8.66) |
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i |
lens surfaces |
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Mirror formulac |
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1 |
1 |
2 |
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1 |
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R |
mirror radius of |
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u + v = − R = f |
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(8.67) |
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curvature |
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Dioptre number |
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1 |
m−1 |
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D |
dioptre number (f in |
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D = f |
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(8.68) |
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metres) |
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Focal ratiod |
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n = f |
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(8.69) |
n |
focal ratio |
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d |
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d |
lens or mirror diameter |
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Transverse linear |
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v |
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MT |
transverse |
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magnification |
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MT = |
− u |
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(8.70) |
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magnification |
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Longitudinal linear |
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2 |
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ML |
longitudinal |
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magnification |
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ML = |
−MT |
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(8.71) |
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magnification |
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aFormulas assume “Gaussian optics,” i.e., all lenses are thin and all angles small. Light enters from the left. bA stationary optical path length has, to first order, a length identical to that of adjacent paths.
cThe mirror is concave if R < 0, convex if R > 0. dOr “f-number,” written f/2 if n = 2 etc.
8.6 Polarisation |
171 |
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Stokes parametersa
y |
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V |
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E0y |
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χ |
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pI |
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α |
x |
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Q |
2χ |
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E0x |
2α |
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U |
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2b |
Poincare´ sphere |
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2a |
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Ex = E0xei(kz−ωt) |
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(8.86) |
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k |
wavevector |
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Electric fields |
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ωt |
angular frequency × time |
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i(kz |
− |
ωt+δ) |
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Ey = E0ye |
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(8.87) |
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δ |
relative phase of Ey with |
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respect to Ex |
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Axial ratiob |
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b |
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χ |
(see diagram) |
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tanχ = ±r = ± a |
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(8.88) |
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r |
axial ratio |
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I = Ex2 |
+ Ey2 |
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(8.89) |
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Ex |
electric field component x |
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Q = |
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E2 |
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E2 |
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(8.90) |
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x |
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y |
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Ey |
electric field component |
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y |
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= pI cos2χcos2α |
(8.91) |
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E0x |
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Stokes |
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field amplitude in x direction |
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U = 2 ExEy |
cosδ |
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(8.92) |
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E0y |
field amplitude in y direction |
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parameters |
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= pI cos2χsin2α |
(8.93) |
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α |
polarisation angle |
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V = 2 ExEy |
sinδ |
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(8.94) |
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p |
degree of polarisation |
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mean over time |
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= pI sin2χ |
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(8.95) |
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Degree of |
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(Q2 + U2 + V 2)1/2 |
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polarisation |
p = |
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≤ 1 |
(8.96) |
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I |
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8 |
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Q/I |
U/I |
V /I |
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Q/I |
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U/I |
V /I |
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left circular |
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0 |
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0 |
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−1 |
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right circular |
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0 |
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1 |
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linear x |
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1 |
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0 |
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0 |
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linear y |
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−1 |
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0 |
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linear 45◦ to x |
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1 |
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0 |
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linear |
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45◦ to x |
0 |
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1 |
0 |
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unpolarised |
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0 |
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0 |
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0 |
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aUsing the convention that right-handed circular polarisation corresponds to a clockwise rotation of the electric field in a given plane when looking towards the source. The propagation direction in the diagram is out of the plane. The parameters I, Q, U, and V are sometimes denoted s0, s1, s2, and s3, and other nomenclatures exist. There is no generally accepted definition – often the parameters are scaled to be dimensionless, with s0 = 1, or to represent power flux through a plane the beam, i.e., I = ( Ex2 + Ey2 )/Z0 etc., where Z0 is the impedance of free space.
bThe axial ratio is positive for right-handed polarisation and negative for left-handed polarisation using our definitions.