Diverging Optics and Barlow Lenses
●
●
“Negative” lenses cause rays to diverge (or to converge less
rapidly).
Placing a negative lens in a converging beam (Barlow
configuration) increases the effective focal length of the
system, increasing magnification for a given eyepiece.
F objective
Magnification =
F eyepiece
Ray Tracing
●
Calculating ray trajectories as they encounter reflective or
refractive surfaces is straightforward.
–
●
angle of incidence = angle of reflection / Snell's Law
In computer graphics/animation once can render realistic
scenes with distributed light sources.
Ray Tracing
●
●
Calculating ray trajectories as they encounter reflective or
refractive surfaces is straightforward.
For astronomical optics one typically traces rays from a point
source at infinity (parallel rays) to the focal plane.
Optics, Ray Tracing, and Optical Design
●
Ray traces can by physical as well as computational.
Optics, Ray Tracing, and Optical Design
Geometrical vs. Physical Optics
A raytrace of a parabolic mirror will
produce an infinitesimally small point
image for an object on the axis of a
parabola at infinite distance.
In reality, the image size is limited by
diffraction to an Airy pattern.
http://library.thinkquest.org/22915/reflection.html
Single Slit Diffraction
●
Each point on a wavefront acts like a source (rock in a pond).
–
For an infinite plane wavefront the superposition of all of these
sources propagates the plane wave without distortion.
–
The sources constituting only a segment of a plane wave will
interfere constructively and destructively.
–
The effect is wavelength dependent.
Single Slit Diffraction
●
Each point on a wavefront acts like a source (rock in a pond).
–
For an infinite plane wavefront the superposition of all of these
sources propagates the plane wave without distortion.
–
The sources constituting a segment of a plane wave will
interfere constructively and destructively.
–
The magnitude of the effect is wavelength dependent.
The Airy Pattern
●
●
●
A circular aperture is the two dimensional analog of
the single slit.
Diffraction blurs light passing through a clear circular
aperture. The larger the aperture the smaller the
angular blur.
The size of the blur is wavelength dependent.
Short wavelengths produce sharper images through
the same aperture.
λ
FWHM = 1.03∗
D
λ
first null = 1.22∗
D
λ
first ring = 1.64∗
D
The Airy Pattern
●
●
●
A circular aperture is the two dimensional analog of
the single slit.
Diffraction blurs light passing through a clear circular
aperture. The larger the aperture the smaller the
angular blur.
The size of the blur is wavelength dependent.
Short wavelengths produce sharper images through
the same aperture.
λ
FWHM = 1.03∗
D
λ
first null = 1.22∗
D
λ
first ring = 1.64∗
D
The Airy Pattern
The mathematical form of the Airy Pattern
is related to the Bessel function of the first
kind of order one.
πD
2J 1
sin θ
λ
I (θ) = I o
πD
sin θ
λ
(
(
J 1 ( x)
I (θ) ∝
x
(
)
2
)
)
2
The Resolution/Rayleigh Criterion
●
●
Two stars are considered to be “resolved” if they are sufficiently
separated to fall beyond each other's first Airy nulls.
Thus, formally, the limiting resolution of a telescope is 1.22 λ/D.
Diffraction Limited Resolving Power
A telescope operating at radio wavelengths (21 centimeters, for
example)must have a huge aperture to achieve good resolution.
Hubble Space Telescope – 2.4 meters
The Atacama Large Millimeter Array
http://www.christophmalin.com/files/BIGchile_cm_117.jpg.jpg
Real “Point Spread Functions”
●
●
Telescope
apertures
are often
clear “circles”
Typically telescope
apertures
are not
not clean
unobstructed circles.
The diffraction pattern is the Fourier
transform of the clear aperture.
Real “Point Spread Functions”
Typically telescope apertures are not clean “circles”
http://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/iracinstrumenthandbook/images/IRAC_Instrum
ent_Handbook091.png
Gaming the Physics
Typically telescope apertures are not clean “circles”
Why Does it Matter - Poisson Statistics
●
Why is a sharp high-resolution image important?
–
A small image minimizes the area contributing contaminating
background and crams the most light into the minimum of detector
pixels.
●
●
Unwanted background = additional noise
Detector pixels also contribute noise (readout noise). Fewer pixels per
star image is better.
Poisson Statistics
●
The uncertainty in a measurement in a counting experiment
(detecting photons in this case) is equal to the square root of the
number of counts.
–
Quantization of light as photons makes astronomical detection a
counting experiment
–
Even with a perfect detection system with no noise and no
interfering light from background, if you detect 100 photons from a
star, the measurement is uncertain by 10 photons, or 10%.
uncertainty =
√ counts
Poisson Statistics
●
The uncertainty in a measurement in a counting experiment
(detecting photons in this case) is equal to the square root of the
number of counts.
–
Quantization of light as photons makes astronomical detection a
counting experiment
–
Even with a perfect detection system with no noise and no
interfering light from background, if you detect 100 photons from a
star, the measurement is uncertain by 10 photons, or 10%.
–
You can't measure a star to a precision of 1% until you have
detected 10,000 photons from that star.
–
Complicating this fact is that detection systems aren't perfect and
there are contaminating sources of light such as the glow of the sky
(and glow of the telescope in the thermal infrared)
●
and extraneous sources of noise (detector “read noise” in particular)
that masquerades as additional unwanted counts
Signal to Noise Ratio
●
Traditionally, astronomers like to express the quality of the
detection of a star or spectral line in terms of the ratio of signal to
noise (signal-to-noise ratio or SNR).
–
In simplest terms take the number of signal counts and divide by the
uncertainty.
–
S/N=10 is a measurement with 10% precision
●
–
S/N=100 is a measurement with 1% precision
●
●
100 photons gets you there if there is no source of contaminating light.
10,000 photons without contamination.
In general, if the star is the only source of counts.
Signal
N
= SNR =
=
Noise
√N
√N
Accounting for Background Contamination
●
●
Sources of background add to the detected photons.
–
These unwanted counts add additional noise.
–
Reducing these backgrounds improve signal-to-noise
●
sharper images (landing on fewer pixels)
●
selecting filter bandpasses to avoid skyglow
●
cooling telescopes used in the thermal infrared
If N is the number of counts from the star and B is the number of
counts from the background.
N
SNR=
√N +B
●
Consider a star which covers 4 pixels, each containing
contaminating background, vs. one which covers 1 pixel.
–
Same “N” but 4 times lower background, B, in the second case....
Sharp images are good:
Seeing, Diffraction, and Resolving Power
A telescope's resolving power is limited by the worst of...
- atmospheric “seeing” - Image blur from atmospheric turbulence
- diffraction - passing light through an aperture blurs the image.
first null = 1.22∗ λ
D
- image quality – blurring, hopefully minimized, by the optical design
Image Quality: Spherical Aberration
An “on-axis” aberration that arises from
different radial zones on a optic
producing a focus at different distances.
By its geometrical definition, a parabola
is free of spherical aberration (but guilty
of others).
Spherical Aberration in Practice
Optical Design and Spherical Aberration
Mitigation
●
The power of optical design is illustrated by the control of spherical aberration
provided by altering lens shape (a.k.a. “bending”).
●
All of the illustrated lenses have the same focal length.
Coma
●
Coma arises when incident rays are not parallel to the optical axis.
●
●
Like spherical aberration, coma is manifested by different radial zones in
the optic
Each pair of symmetric points in each radial zone produces a sharp image,
but since the lateral magnification is different for each pair each ring of
incident rays forms an offset ring producing the classic “comma” image.
Optimizing for Coma via Bending
●
In a simple lens spherical aberration and coma cannot be minimized
simultaneously (but close)
●
The optimal shape is close to plano-convex
●
but not that this is different from convex-plano ... direction matters!
Astigmatism
●
The optical axis and “chief ray” containing
your sourcedefine a plane – the tangential
plane.
●
●
Ray fans in and parallel to this plane
behave differently than ray fans lying in
and parallel to the perpendicular
“sagittal” plane
In particular, the two planes focus at
different distances producing sharp
perpendicular “line” images at two
depths with a circle of least confusion in
between.
http://www.microscopyu.com/tutorials/java/aberrations/astigmatism/index.html
Chromatic Aberration
●
●
The “lensmaker's equation” provides the focal length of a lens of a given
refractive index, n.
Since refractive materials have different refractive index at different
wavelength, light comes to a focus in different places.
1
1
1
= (nλ −1)
−
F
r1
r2
(
)
Chromatic Aberration
●
A lens' focal length depends on the
refractive index of the lens material.
●
●
●
●
Refractive index (both fortunately
and unfortunately) is a function of
wavelength.
Only one wavelength can be exactly
in focus at a time.
Imaging systems often function over
broad bandpasses (e.g. K-band spans
2.0 – 2.4 um)
Optical design mixes materials (e.g.
crown and flint glass in a traditional
achromat) to mitigate chromatic
aberration.
Controlling Chromatic Aberration
●
●
Split the lens into two components (use additional surfaces to control
classical aberrations).
Make lenses out of materials with different dispersive properties
●
The Negative lens has a higher refractive index to control spherical
aberration (compensating for its weaker power).
●
●
Zero spherical aberration can be achieved at only one color. The same is
true of chromatic aberration.
The doublet is far from perfect and some are more perfect than others.
Standard doublet
Abbe formula
Spherical (fig 6.3)
Controlling Chromatic Aberration
●
The secondary spectrum in an achromatic doublet can be
minimized by proper pairing of lens materials.
●
Traditionally the working combination is that of crown
(“soda-lime” glass - 10% potassium oxide n=1.52 ) and
flint (30% lead oxide (crystal) n=1.65) for the positive and negative
elements respectively.
9” Clark objective – Univ of Vermont
Clark Telescopes
Chromatic Aberration Correction
●
A single element suffers from severe
chromatic aberration.
●
●
●
There are no “ideal” glasses with
little or no dispersion over broad
bandpasses
A doublet lens can provide perfect
cancellation of chromatic
aberration at two wavelengths –
the “achromatic doublet”
●
Red
the uncorrected dispersion at other
wavelengths is called “secondary
spectrum”.
With more degrees of freedom, a
triplet can substantially reduce
secondary spectrum. This
configuration is known as an
“apochromatic triplet”
Blue
Controlling Chromatic Aberration
Split the lens into two components (use additional surfaces to
control classical aberrations).
:
n
o
i
t
u
l
o
S
r
e
l
p
m
i
S
s
r
o
r
r
i
M
e
Us
Make lenses out of materials with different dispersive properties
The Negative lens has a higher refractive index to control spherical
aberration (compensating for its weaker power).
Zero spherical aberration can be achieved at only one color. The
same is true of chromatic aberration.
The doublet is far from perfect and some are more perfect than
others.
Standard doublet
Abbe formula
Spherical (fig 6.3)
Compound Optics
Virtually all optical systems contain two or more elements.
Most systems can be reduced to an equivalent single thin lens.
The final focus (and focal ratios) can be propagated through the
system one object/image pair at a time.
The Thin Lens Equation
●
For a lens of a give focal length the distance at which an
image is formed depends on the object's distance.
–
In astronomy d0 is typically infinity, so the image is one focal
length away from the lens.
–
In compound optics each image becomes the “object” for the
next element.
1
1
1
+
=
do
di
F
Cassegrain Telescopes as Compound Optics
A Cassegrain telescope is a two-optic system.
The primary forms a real image.
The secondary, which has a negative focal length, relays this real image to
another real image in the focal plane behind the primary mirror.
In a Cassegrain configuration the negative secondary interrupts the
converging beam from the primary before the real image forms, but the
image is there for calculation's sake nonetheless.
Cassegrain Telescopes as Compound Optics
●
A Cassegrain telescope is a two-optic system.
●
●
●
The primary forms a real image.
The secondary, which has a negative focal length, relays this real image to
another real image in the focal plane.
In a Cassegrain configuration the secondary interrupts the converging
beam from the primary before the real image forms, but the image is
there for calculation's sake nonetheless.
Schmidt-Cassegrain Telescopes
●
●
●
This configuration uses a spherical(!) primary mirror.
Light enters through a refractive (but weak) “corrector plate”
that compensates for the spherical aberration.
No “spider” since the corrector plate supports the secondary.
Gregorian Telescopes
In the Gregorian configuration the concave secondary mirror
lies beyond the prime focus of the primary. A real image is
formed in space ahead of the secondary.
Refractive Designs in the Infrared
Most classical glasses become opaque at wavelengths longward
of 2 micrometers.
Alternative crystalline materials come to the rescue
ZnS, CaF2, BaF2, ZnSe, InSb, Si, Ge, As2S3, Sapphire, Diamond...
Eyepieces and Exit Pupils
●
●
●
●
●
In a simple telescope the objective
defines the “entrance pupil”
The eyepiece produces an image of
the objective – the “exit pupil”
This exit pupil constrains the bundle of
rays leaving the system and represents
the ideal location for the eye's pupil.
The exit pupil size should be smaller
than the dark adapted pupil of the eye
and it should be located sufficiently far
from the last optic that there is some
space before the eye (eye relief).
Try moving your eye around at various
distances behind the eyepiece and
note how the field of view gets
constrained.
Eyepieces and Exit Pupils
●
●
●
●
●
In a simple telescope the objective
defines the “entrance pupil”
The eyepiece produces an image of
the objective – the “exit pupil”
This exit pupil constrains the bundle of
rays leaving the system and represents
the ideal location for the eye's pupil.
The exit pupil size should be smaller
than the dark adapted pupil of the eye
and it should be located sufficiently far
from the last optic that there is some
space before the eye (eye relief).
Try moving your eye around at various
distances behind the eyepiece and
note how the field of view gets
constrained.
Seeing and Speckles
●
The atmosphere distorts incoming plane waves. The induced
tilts of the wavefronts cause different portions of the wavefront
to be focused in slightly different directions causing image blur
r0, the Fried Parameter
●
Although atmospheric turbulence is a messy business with a
spectrum of turbulent scales, identifying a single size of
atmospheric turbulence cell that is characteristic of the seeing
can yield simple scalings of seeing behavior.
diameter = r0
Locations of Atmospheric Turbulence
stratosphere
tropopause
10-12 km
wind flow over dome
boundary layer
~ 1 km
Heat sources within dome
From Claire Max's Lectures on Adaptive
Optics (click)
Speckles, r0, and Coherence Time
●
●
●
Atmospheric turbulence can be characterized by a single
characteristic scale, r0, also known as the Fried parameter.
–
r0 varies with the seeing and wavelength. Good seeing = large r0
–
In 1” seeing at 500 nm r0 is approximately 10 cm.
Many characteristics then derive from r0
–
The size of seeing disk is just λ / r0 (if your telescope aperture > r0)
–
The number of speckles is the number of r0's in the telescope
aperture, (D / r0)2
–
The “coherence time,” the time for the speckle pattern to change
(thus the time you have to measure and correct it), is the time it takes
an r0 to move its diameter at the wind speed and is typically 10's of
milliseconds. In good seeing (bigger r0) you have more time.
r0 depends on airmass as (airmass)-0.6 so seeing, which scales as
λ/r0, degrades as (airmass)0.6 . Observe targets near to zenith if
possible!
Speckle simulations: 1-meter 2-meter 8-meter telecopes
Speckle Interferometry
●
●
Since each speckle is effectively a diffraction limited image of
the target, each snapshot contains information about the
target at the scale of the diffraction limit of the telescope.
Extraction of this embedded, repeated small-scale pattern can
be accomplished via the Fourier Transform of the image.
Lucky Imaging
●
Every once in a while a large atmospheric cell passes over the
telescope. For an instant the telescope becomes diffraction
limited. It may take thousands of short exposures to find
nearly perfect images, but the results can be spectacular.
0.65” seeing
Hubble
Lucky (Palomar)
The limitation here is exposure time since only 1 in thousand frames might
be diffraction limited.
Lucky Imaging
●
Every once in a while a large atmospheric cell passes over the
telescope. For an instant the telescope becomes diffraction
limited. It may take thousands of short exposures to find
nearly perfect images, but the results can be spectacular.
Seeing in the Infrared
●
r0 scales as λ1.2 so the scale size is larger at longer wavelengths.
●
Larger scale size is beneficial
●
●
●
–
smaller seeing disk λ/r0 → λ-0.2
–
fewer speckles (one speckle = diffraction limited)
–
longer coherence time
(IR seeing is better than visual)
Diffraction for the full aperture becomes worse going to longer
wavelengths so at some point (usually around 3-5 microns for
moderate aperture telescopes) telescopes no longer are affected by
seeing.
All of these factors favor real-time correction of the atmospheric
effects at infrared wavelengths.
“Adaptive optics,” using deformable mirrors to correct for
atmospheric wavefront distortion, becomes practical at wavelengths
longward of 1000 nm (1 micron).
Intensity
Adaptive Optics
x
●
●
Correct for atmospheric
wavefront corruption in real
time with a deformable
mirror that “undoes” the
atmosphere.
Requires correction on
millisecond timescales.
Strehl Ratio
●
Strehl is the ratio between the peak intensity of a point source
(star) image and the peak that the image would have if the
light were concentrated into a perfect diffraction-limited image.
Wavefront Sensors and Deformable Mirrors
Deformable mirror behavior
Switching on adaptive optics
Isoplanatic Angle
●
Different lines of sight, if they are sufficiently separated in
angle, will encounter different atmospheric turbulence.
–
r0 = 20cm at 2km altitude subtends 40 arcseconds.
–
Outside this isoplanatic angle star's speckle patterns, and thus
deformable mirror AO correction are unrelated.
Laser Guide Stars
●
●
If you are observing a faint source you need to have a bright
reference star within the isoplanatic angle.
If no such star exists, you can make one!
One strategy is to excite
neutral sodium atoms in
a layer about 80-100
kilometers above the
ground.
Laser guide star tutorial
Keck laser guide star
observer's page
Atmospheric Transmission vs. Wavelength
●
text
Atmospheric Transmission vs. Wavelength
●
Solution 1 – leave the atmosphere behind
Spitzer Infrared
Hubble – Ultraviolet, Visible, Infrared
Compton Gamma-ray
Observatory
Atmospheric Transmission
●
Molecular absorption, water in particular, contributes
substantial atmospheric opacity in the infrared.
Atmospheric Transmission
●
Since water predominately resides in the troposphere, you just
have to get into the stratosphere to see into space.
Atmospheric Transmission vs. Wavelength
●
Submillimeter transmission at 17000 feet
Atmospheric
Extinction
●
Calibrating stellar
photometry requires
correction for loss of
light passing through
the atmosphere.
Extinction Correction in Practice
●
In each filter measure the star at a variety of airmasses (∆x
below is (airmass – 1)) and determine the extinction in units of
magnitudes per airmass for each observing band.
●
●
●
●
●
Alternatively, have a calibrated star in your field of view (easy
in the era of sky surveys.
Time
Variability of Extinction
Systematics,
I
• Extinction
– Rayleigh scattering
(optical; proportional
to static pressure
and airmass)
– Ozone (optical)
– Water (IR)
– Volcanic aerosols
• Can vary by 0.1-1%
• Episodic problem
• IR impact uncertain
SCTF 1/29/2014
Nabro eruption, 13 June 2011
(Bourassa, et al. (2012))
68
Filter Bandpasses
●
Calibrating observations precisely is dependent upon having
precisely defined bandpasses.
Infrared Bandpasses
●
Atmospheric absorption provides natural boundaries for
defining infrared filter bandpasses.
Stellar Photometry with Filters
●
Differences between magnitudes (which are ratios when you
think about it) measured in different filters are diagnostic of
temperature of blackbodies (stars).
V R
I
Stellar Photometry with Filters
●
These color differences become more diagnostic (for example
of luminosity class) when you account for stellar spectral
features and how they change with stellar surface gravity.
The Electromagnetic Spectrum and Photons
●
Wavelength alone distinguishes types of light
●
At visible wavelengths – short wavelengths are blue; long are red
●
Wavelength, color, and energy of a photon are all the same thing
hc
E=hν=
λ
λ∗ν=c
–
For reference 1um wavelength corresponds to 2x10 -19J = 1.24 eV
Detectors
Goal: Convert photons to an electronic signal (apologies to
photography...)
● with as little accompanying noise as possible
● ideally at the quantum limit enforced by the photons.
● with as much conversion efficiency as possible
●1 photon yields 1 electron (or ideally a bunch of electrons)
●
●
Primary Detection Methods
●Bulk thermal response (bolometry)
●incident radiation chages the temperature of the detector
●electrical resistance changes with temperature
●Conversion of photons to ''free'' electrons
●quantum response
●photoelectric or solid state detection
●Coherent detection
●sense wave nature (phase) of the photons
●primarily through heterodyning to lower frequencies
Detectors
Goal: Convert photons to an electronic signal (apologies to
photography...)
● with as little accompanying noise as possible
● ideally at the quantum limit enforced by the photons.
● with as much conversion efficiency as possible
●1 photon yields 1 electron (or ideally a bunch of electrons)
●
●
Primary Detection Methods
●Bulk thermal response (bolometry)
●incident radiation chages the temperature of the detector
●electrical resistance changes with temperature
●Conversion of photons to ''free'' electrons
●quantum response
●photoelectric or solid state detection
●Coherent detection
●sense wave nature (phase) of the photons
●primarily through heterodyning to lower frequencies
Electron response -- free electrons / carriers
A free electron is a detectable electron (via voltage or current)
an electron can be free in space -- photoelectric effect
or it can be ''free'' within a crystal lattice -- solid state detection
The Photoelectric Effect
Metals are characterized by a work function which
determines the energy difference between the
highest energy state for an electron within the metal
and the energy of an electron in free space.
A photon with energy in excess of this work function will liberate a free,
detectable, electron -- the photoelectric effect
Heated metals will emit free electrons -- those with thermal energy in
excess of the material's work function -- thermionic emission via a
Boltzmann law.
The Photoelectric Effect
●
●
Photomultipliers are based
on the cascade
amplification of individual
electrons liberated by the
photoelectric effect
Work functions for metals
are typically a few electron
volts
● 1 eV = 1240 nm
http://laxmi.nuc.ucla.edu:8248/M248_99/autorad/Scint/pmt.htmll
The Photoelectric Effect
Photomultipliers are based
on the cascade
amplification of
individual electrons
liberated by the
photoelectric effect
Work functions for metals
are typically a few
electron volts
1 eV = 1240 nm
Photocathodes can be
engineered to have
sensitivity out to 1.5 um
(obviously not using pure
elemental metals...)
http://hyperphysics.phy-astr.gsu.edu/hbase/tables/photoelec.html
The Photoelectric Effect
Shortcomings of photomultipliers
poor wavelength coverage (<1.5um)
poor quantum efficiency (<20%
conversion of photons to electrons)
thermally emitted electrons -particularly for long-wavelength
devices.
large single-detector area
One big advantage -- photon counting
Modern detection systems use semiconductor
detectors which mimic the photoelectric effect in the
solid state.
Photons create “free” electrons within the confines of the
crystal lattice.
Electronic Energy Levels in Conductors
An alternative approach:
At large separations, electronic orbitals have “atomic”
characteristics.
As atomic separation decreases these degenerate states must split
under the interacting potential of all of the nuclei in the crystal.
The ensemble of split energy levels is a “band” which may be
full, partially filled and/or overlapping with other bands
Electrons which have immediately adjacent energy states can
change state and thus “conduct”
Electronic Energy Levels in Conductors
Sodium, a metal, has a single 3s valence band electron.
the 3s state is only half filled since any atom can have a second electron in th
state with opposite spin.
In crystalline sodium (recall any chunk of metal is an assemblage of
crystalline domains) the 3s state is shared by the entire crystal.
There are 2N (twice the number of atoms) 3s states in the crystal and only N
electrons.
Conduction is “easy” since a valence electron sees a variety of nearby open
energy/momentum states.
Each atom brings a fixed number
of states and a fixed number of
electrons. It is natural that
bands sometimes end up exactly
full.
Electronic Energy Levels in Conductors
Copper has a single 4s electron making it a natural conductor.
It also has an electronic configuration where the 4s band
overlaps the 3d band at the crystalline interatomic spacing.
Overlap between bands also provides access to infinitesimally
different energy states, permitting conduction (e.g. Magnesium –
which has a filled 3s state is still an electrical conductor).
Note that there are about 10²²
atoms in a fist sized
chunk of metal.
Each atom contributes a
couple of 4s energy
states.
The energy states span a
couple of eV.
The degenerate
conduction band energy
levels are about 10¯²² eV
apart.
Electronic Energy Levels in Insulators
Insulators have filled energy bands which do not overlap with
adjacent energy bands for the interatomic equilibrium spacing.
There is no such thing as a
“semiconductor”!
At T=0 a material will either have
overlapping energy states and is a
conductor, or it will have a “bandgap”
above a completely filled energy state
and be an insulator.
Semiconductivity (if that is a word) is a
manifestation of Boltzmann factors at
finite temperature – kT vs. the “bandgap”.
Band Filling and Band Interactions
Semiconductors
Semiconductors can consist of pure
elemental materials or alloys of different
elements.
In either case, materials with complementary
filled valence shells are likely
semiconductors.
Carbon (diamond) is a “semiconductor”
with an energy gap of 5.33 eV (0.23um)l
http://pearl1.lanl.gov/periodic/default.htm
Semiconductors
At T=0K, the world contains only conductors and insulators.
Above 0K, electrons at the top of the Fermi sea can be excited to
higher energy states if the states are sufficiently ( ~kT ) close.
Small bandgap materials are thus semiconductors with marginal
electrical conductivity at room temperature due to thermally excited
carriers.
The conductivity of metals improves at low
temperatures. The conductivity of semiconductors
declines.
Semiconductor Conductivity
The population of electrons (holes) in the conduction (valence)
band in an intrinsic semiconductor depends on temperature.
The electrons obey Fermi-Dirac statistics, but given the large size
of the bandgap vs. kT, occupancy in the conduction band
follows an apparent Boltzmann law.
Recall 1eV/k = 12000K
E bandgap= 2∗ E−E f 
Semiconductor Detectors
While the photoelectric effect creates free electrons semiconductors
provide an analog in the solid state.
Photoexcitation across the material's “insulating” bandgap produces
free carries.
cutoff =
1.24  m
E gap eV 
Resulting carriers produce a change in bulk material resistance
(photoconductors)
Carriers can also be directly detected as an electrical current in a diode
configuration (photovoltaics)
Photons can also change the bulk temperature of a small piece of
semiconductor changing the electrical resistance (bolometers)
Silicon
PbS
GaAs
InSb
Bandgap Cutoff (um)
1.11
1.12
0.37
3.35
1.43
0.87
0.18
6.89
Note cutoff is for room temperature.
Cutoffs change at cryogenic temperature
due to changing lattice spacing (e.g. InSb
detectors have a 5.5um cutoff at 77K).
The Ideal Imaging Device
The Ideal Imaging Device
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Indium Bump Bonds
http://gruppo3.ca.infn.it/usai/cmsimple3_0/images/PixelAssembly.png
http://www.flipchips.com/tutorial10.html