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Transcript
The Physics 431 Final Exam
W
Wed,
DECEMBER 16,
16 2009
3:00 -- 5:00 p.m. Ÿ
BPS 1308
•
Calculators, 2 pages “handwritten notes”
•
Graded lab reports ================Î OK
•
Books, old HW, laptops
OK
NO
The exam includes topics covered throughout the semester
Greater emphasis will be placed on the 2nd half of the course
The exam consists of problems totaling 250 pts. Show all work on exam pages — circle your answers
Grades will be posted at BPS 4238 by G
d
ill b
t d t BPS 4238 b 5 pm Friday, December 18. Remember F id D
b 18 R
b
your “pass code” from the final exam.
Check “Midterm
Check
Midterm Review Slides
Review Slides” for topics covered in Midterm I.
for topics covered in Midterm I.
Review “Final Exam Topics” posted/handed out in class.
Telescope
•
•
•
Objectt iis att iinfinity
Obj
fi it so iimage iis att f
Measure angular magnification
Length of telescope light path is sum of
focal lengths of objective and eyepiece
A Magnifying Lens
g y g
fo
M =−
fe
CA0 s θ '
= = = M.
CAe s ' θ
The exit pupil is the image of the aperture stop (AS).
Define CA0 = entrance pupil clear aperture
CAe= exit pupil clear aperture
From the diagram, it is clear that
Microscope
• x‘ is the tube length:
s2 − x '
h'
M0 = = − =
standard x’ ranging160mm to 250mm
h
s1
f0
25
• Magnification is product of lateral
Me =
fe
magnification of objective and angular
magnification of eyepiece
− x ' 25
M total = M 0 × M e =
⋅
g is viewed at infinity
y
• Note: Image
f0 fe
Eye (Hecht 5.7.1 and Notes)
Topics/Keywords: Eye model, Visual Acuity, Cones/Rods
accomodation, eyeglasses, nearsightedness/myopia, /
farsightedness/hyperopia
Monochromatic plane waves
Plane waves have straight wave fronts
– As opposed to spherical waves, etc.
– Suppose
E ( r ) = E 0 e ik r
Î
− iωt
E ( r, t ) = Re{E(r )e
λ
}
= Re{
R {E0 eik r e − iωt }
Ex
By
= Re{E0 ei ( k r −ωt ) }
– E0 still contains: amplitude, polarization, phase
– Direction of propagation given by wavevector:
Di ti
f
ti
i
b
t
k = (k x , k y , k z ) where |k|=2π /λ =ω /c
– Can also define E = ( Ex , E y , Ez )
– Plane wave propagating in z‐direction
E ( z , t ) = Re{{E0 ei( kz −ωt ) } = 12 {E0 ei( kz −ωt ) + E*0 e − i( kz −ωt ) }
Key words: energy, momentum, wavelength, frequency, phase, amplitude…
vz
Poynting vector & Intensity of Light S = E× H
•Poynting vector describes flows of E‐M power
•Power flow is directed along this vector
•Power flow is directed along this vector
•Usually parallel to k
•Intensity is equal to the magnitude of the time averaged Poyning vector: I=<S>
cε 0 2 cε 0
S = I ≡| E ( t ) × H ( t ) |=
E =
Ex 2 + E y 2 )
(
2
2
cε 0 ≈ 2.654 ×10−3 A / V
example E = 1V / m
I = ? W / m2
hω[eV ] =
1239.85
1239
85
λ[nm]
h = 1.05457266
1 05457266 × 10 Js
J
−34
Wave equations in a medium
The induced polarization in Maxwell’s Equations yields another term in Th
i d d l i ti i M
ll’ E
ti
i ld
th t
i
the wave equation:
∂2 E 1 ∂2 E
− 2 2 =0
2
∂z
v ∂t
∂ E
∂ E
− με 2 = 0
2
∂z
∂t
2
2
This is the Inhomogeneous Wave Equation.
The polarization is the driving term for a new solution to this equation.
∂2 E
∂2 E
− μ 0ε 0 2 = 0
2
∂z
∂t
∂2 E 1 ∂2 E
− 2 2 =0
2
∂z
c ∂t
Homogeneous (Vacuum) Wave Equation
E ( z , t ) = Re{E0 ei( kz −ωt ) }
= 12 {E0 e
i ( kz −ωt )
+E e
*
0
− i ( kz −ωt )
=| E0 | cos ( kz − ωt )
}
c
=n
v
Phase velocity
Phase velocity
Interference [Hecht 9.1‐9.4, 9.7.2; Fowles 3.1‐3.1; Notes]
E ( r ) = E0 e
Mi h l
Michelson Interferometer
I t f
t
ik r
E ( r, t ) = Re{E(r )e − iωt }
= Re{E0 eik r e − iωt }
= Re{E0 ei ( k r −ωt ) }
Consider the Optical Path Difference (OPD)
Or simply the superpositon of two plane waves
E ( r ) = E1eik •r + E 2 eik •r
1
1
2
2
I =| E ( r ) |2 = E × E*
Key words/Topics: Michelson Interferometer, Dielectric thin film, Anti‐reflection coating, Fringes of equal thickness, Newton rings.
Interference Fringes and Newton Rings
Phase shift on reflection at an interface
Near‐normal
Near
normal incidence
incidence
π phase shift if ni < nt
0 (or 2π
(
phase shift) if n
p
)
i > nt
Young’s double slit interference experiment
order m maxima occur at:
mλ ≈ a sin
i θm ≈ a
ym
s
Diffraction
Diffraction: single, double, multiple slits Study Guide: Hecht Ch. 10.2.1
Study
Guide: Hecht Ch 10 2 1‐10
10.2.6 (detailed but lengthy discussions), 2 6 (detailed but lengthy discussions)
Fowles Ch. 5 (short but clear presentation), or Class Notes
2
⎛ sin β ⎞
I ( β ) = I ( 0) ⎜
⎟
⎝ β ⎠
kb
b
β = sin θ = π sin θ
2
λ
Java applet – Single Slit Diffraction
http://www.walter‐fendt.de/ph14e/singleslit.htm
Diffraction: Double and Multiple Slits
2
2
1
1
⎛ sin
i β ⎞ ⎛ sin
i Nγ ⎞
I (θ ) = I ( 0 ) ⎜
⎟ ⎜
⎟ β = kb sin θ ; γ = ka sin θ
2
2
⎝ β ⎠ ⎝ N sin γ ⎠
See also See
also
http://demonstrations.wolfram.com/MultipleSlitDiffractionPattern/ and
http://wyant.optics.arizona.edu/multipleSlits/multipleSlits.htm
The Diffraction Grating
H ht 10 2 8 F l Ch. 5 p.123 (handout)
Hecht 10.2.8 or Fowles
Ch 5 123 (h d t)
Grating Equation (Optical Path Difference OPD= m λ)
a ( sin
i θ m − sin
i θi ) = mλ
a sin θ m = mλ
Normal incidence θi =0
The chromatic/spectral resolving power of a grating R≡
λ
= mN
Δλ
m is the order number, and is the order number, and
N is the total number of gratings.
Uniform Rectangular Aperture
a
b
2
1
⎛ sin α ⎞ ⎛ sin β ⎞
I (θ ) = I ( 0 ) ⎜
⎟ α = ka sin θ ;
⎟ ⎜
2
⎝ α ⎠ ⎝ β ⎠
2
1
β = kb sin θ
2
Uniform Circular Aperture
R
R
⎛ 2 J1 ( ρ ) ⎞
I (θ ) = I ( 0 ) ⎜
⎟
ρ
⎝
⎠
ρ = kR sin θ ; k =
2π
λ
2
Wave optics of a lens
The spot diameter is
d = 1.22
λf
w
= 1.22
λ
θ
The resolution of the lens as defined by the “Rayleigh” criterion is
d / 2 = 0.61λ / θ
For a small angle θ,
d / 2 = 0.61λ / sin θ = 0.61
λ
NA
Gaussian Beam Optics
Fibers
and
d
.
NA =
(
n 2f
− nc2
)
1/2
The number of modes in a stepped-index fiber is
Nm ≈ 12 (π D× NA / λ0 )
2