Download PHYSICS 2310

Phys 2310 Mon. Aug. 29, 2011
Today’s Topics
– Properties of Light
• Finish Detection of Light
• Chapter 4: Propagation of Light
– Homework Assigned
• Reading for Next Time
1
Supplementary Material: Detection of Light
• Detectors for Infrared Wavelengths
– Similar to CCDs at l < 25 mm
– Largest formats are 2K x 2K
• High Cost ( ~ $500K)
– Use at l > 3 mm requires liquid He cooling (4 oK)
– Longer wavelength detectors (l > 25 mm ) require
use of bolometers
• Electrical resistance depends on temperature below 4 oK
– Measurement of minute temperature changes due to photon
absorbtion
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Supplementary Material: Detection of Light
• Detectors for Microwave and Radio Wavelengths
– Electric field in radio wave excites electrons in antenna
• Amplification produces detectable signal
– Modern electronics technology sufficiently fast to detect and
amplify signals.
– Both phase and amplitude of photons can be detected (why?)
3
Supplementary Material: Detection of Light
• Detectors for Ultraviolet and X-ray
Wavelengths
– High energy of photons results in strong
photoelectric effect
• Electrons emitted from solids when photons absorbed
4
Supplementary Material: Signal Detection
• Photon rate from a source (N) is never constant
– Signal is an average rate over a given time interval
– Can’t detect a fraction of a photon
• Some time intervals contain little more or little less
– Poisson (counting) statistics:
sN = sqrt(N)
– Signal is seldom (never) without some sort of
background
– Almost all detectors has some intrinsic noise
– Uncertainty in any detection is the combination of
all these effects
– Any measurement has an associated uncertainty
5
Supplementary Material: Signal Detection
• Uncertainty in any signal can be estimated
– Without detailed information we assume any source of
errors is distributed as a Gaussian distribution:
 x  m 2 
1

P( x) 
exp  
2

2
s
s 2


Uncertainty(s) can be estimated via a Gaussian fit to a
histogram made from lots of individual measurements.
Compute Mean, Standard Deviation. Alternatively just
estimate. Errors from each source don’t add in same sense
They are random and (usually) uncorrelated. So, add in
quadrature:


s T  s12  s 22  s 32  ...
6
Chapter 4: Propagation of Light
• Photons interact with matter in a variety of ways
– Photons encountering an opaque solid can be absorbed (black
surface) or reflected (metal surface)
– Photons encountering a transparent surface can be scattered if
path length is long enough (no substance is perfectly
transparent)
– Enhanced scattering of bluer light in atmosphere makes sky
blue
• Molecules can be visualized as absorbing photons and then
emitting them in a new direction (physics is complex)
• Huygen’s Principle
– Behavior of light can be understood as the scattering of
“wavelets”.
– A surface (real or imaginary) can be thought of as a number of
scattering centers
– Provides an explanation for the laws of reflection and refraction
7
Chapter 4: Huygen’s Principle
• Huygen’s Principle:
Propagation of light can be
modeled as if scattering off
atoms in such a way that the
spherical “wavelets”
constructively interfere to
produce a wavefront.
“Every point of a propagating
wavefront serves as the source of
spherical secondary wavelets,
such that the wavefront at some
later time is the envelope of these
wavelets.”
8
Chapter 4: Properties of Optical Materials
• We can define an equivalent optical path
length by considering the index of
refraction
– If speed of light is slower in dense material
the equivalent path in a vacuum would be
longer. Hence:
– O.P.L. = nd
where n is the index
of refraction and d is the material
thickness
• The concepts of optical path leads to
Fermat’s Principle (see sec. 4.5 in text):
“A light ray will take the path between two points that minimizes its travel
time.”
This is not strictly true but it is still a useful concept for deriving Snell’s Law (see below).
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Chapter 4: Properties of Optical Materials
• The optical path length is
found to a function of
wavelength
– The index of refraction of
most materials is higher at
shorter wavelengths
• Not strictly true over all
wavelengths but applies over
visible (more about this later)
• Over broader wavelength this
effect (dispersion) is readily
seen
10
Chapter 4: Plane Surfaces
• Light slows as it enters a
transparent medium
– Light path (ray) is deflected
toward normal when
entering higher index
medium
– Light path (ray) is deflected
away from normal when
entering lower index
medium
• Note that dispersion is occurs
such that blue light (higher
index) is deflected more and
vice versa
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Chapter 4: Fermat’s Principle and Snell’s Law
Minimizing the time (optical path length) between points Q and Q’
yields Snell’s Law:
OPL    nd  n' d
d  ( h 2  ( p  x ) 2 )1 / 2
d '  ( h' 2  x 2 )1 / 2
substituting :
t
QA AQ'

v1
V2
(h 2  ( p  x) 2 )1/ 2 (h'2  x 2 )1/ 2

v1
v2
  n[ h  ( p  x) ]  n' (h'  x ])
differentiating :
d
n/2
n' / 2
 2
(
2
p

2
x
)(

1
)

2x  0
dx [ h  ( p  x) 2 ]1 / 2
( h' 2  x 2 )1 / 2
thus,
px
x
n 2

n
'
[ h  ( p  x) 2 ]1 / 2
( h' 2  x 2 )1 / 2
or
px
x
n
 n'
Similarly, the law of reflection can also be derived
d
d'
(homework)
and
n sin   n' sin  '
12
2
2 1/ 2
2
2
1/ 2
t
Chapter 4: Fermat’s Principle continued
• This approach works for a variable index of refraction in which case
the OPL is now an integral over the path:
s2
OPL   n( s)ds
s1
The book discusses an application of this concept to mirages.
Note, that by setting d/dx = 0 the OPL doesn’t strictly have
to be a minimum it could also be a maximum. A modern
description is that we require alternative paths to have
significant differences in phase of the “wavelets”. Thus we
might think of the atoms as scattering the photons and the
path light takes is the one where constructive interference is
maximized. See the discussion on page 109-110 of text.
13
Chapter 4: Total Internal Reflection
The light path of rays is
reversible.
• Thus we can consider
total internal
reflection via
refraction.
• Note that for rays 1-4
all appears fine. Ray 5
is a problem.
• When we consider the
reversed path any
rays with  >c will
not emerge from the
higher index medium.
For glass (n ~ 1.52) c
~ 42 deg. So 45-deg
prisms can be used.
Substituting  = 90 deg, or sin  = 1 into Snell’s Law
gives:
n
sin c 
n'
14
Chapter 4: Plane Surfaces continued
• Light traversing a parallel
plate is deflected
(e.g. a window)
• The deflection angle can be
calculated
d  l sin(    ' )
d  l (sin  cos  ' sin  ' cos  )
fromABC :
l
t
cos  '
 sin  cos  ' sin  ' cos  

d  t 

cos  ' 
 cos  '
since :
n
sin  '  sin 
n'


cos  n
d  t  sin  
sin  
cos  ' n'


or :
 n cos  

d  t sin  1 
n
'
cos

'


15
Homework this Week (HW #1)
Homework this week due Wed., Sept. 8:
Chapter 3: #14, 20, 24
Chapter 4: #6, 8, 9, 15, 17
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Reading this Week
By Wednesday:
Finish Ch. 3,4 (3.6, 4.1 – 4.5, 4.7) Propagation of
Light
Begin Ch. 5 (5.1 – 5.2) Thin lenses
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