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Week 7
• “Midterm” exam in class next Thursday, March 23• Evening sessions this week are Python data analysis package
oriented in support of Lab 3 writeups.– Given the interruption for the exam the due date will be March 27.
• Problem Set 4 will be available shortly (and in-‐part intended for midterm exam preparation).
• Lab 4 (making beautiful three-‐color calibrated images with the Fan Mountain RRRT) will be out early next week.– Observing happens the week after midterm.
• Topics through the midterm:– Photon detection/Imaging devices– Poisson statistics/noise/background– Determining detector “gain” with Poisson statistics– Astronomical photometry and filters
• Aperture vs. PSF fit photometry
Analysis in the Frequency Domain
• Any time series signal can be reconstructed from the sum of a continuum of sine waves of different frequencies and phases.
• The “Fourier Transform” provides a means of calculating the frequency spectrum decomposition of a time-‐domain signal.
• |S(f)|2 represents the “power spectrum” of the signal – the amount of power in the time series at every frequency.
The Ideal Imaging DeviceYou tell me…
The Ideal Imaging Device
• Reports light incident on cells • Detects only photon events – no noise
– No random spontaneous counts – No continuous leakage of “fake” signal
• Does not “miss” incident photons– Perfect “quantum efficiency”– 100% “fill factor
• Does not saturate– Infinite “well” capacity
• Uniform sensitivity from pixel to pixel– Perfect “flat field”
Real-‐world Imaging Devices• Reports light incident on cells
– Measures electronic signal ~proportional to counts.
• Detects only photon events – no noise– No random counts – Electronic measurement is susceptible to noise
– fake counts…– No leakage of “fake” signal
• “Dark current” creates counts not originating from photons.
• Does not “miss” incident photons– Perfect “quantum efficiency”– 60-‐80% typical…very wavelength dependent– 100% “fill factor– Nope… but close
• Does not saturate– Infinite “well” capacity– Maximum count capacity in any cell. If the
source is too bright the measurement fails.• Uniform sensitivity pixel to pixel
– Nope… but close again. Pixel to pixel variability of a few percent.
Quantifying Light: Photon Detection
• Three methods for converting photons into “data”1. Direct electromagnetic detection
• Electromagnetic waves drive currents in an electrical conductor. Amplifiers enable direct measurement of amplitude (signal) vs. frequency of oscillation.
à Radio… save that method for a different class2. Photon counting (conversion of photons into free electrons in free space or
within an otherwise electrically “insulating” solid making it more conductive)• The photoelectric effect describes the ability for a photon to liberate an electron from a metal if the photon carries enough energy to overcome the potential barrier binding the electron to the metal.– Direct evidence for the quantization of photon energy.
• Similar behavior can occur in the solid state.3. Integrated photon response
• Instead of liberating electrons, photons can “warm up” a lump of material.– The smaller the lump the bigger the effect.– The altered temperature leads to a change in the lump’s properties, for
example altered electrical conductivity.
Making “Free” Electrons with Photons
• A freed 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” that is 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
.
Making “Free” Electrons with Photons
• A freed 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
Dark Current
• Warm metals will emit free electrons, those with thermal energy in excess of the material's work function
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.
The Photoelectric Effect
• Photomultipliers are based on the cascade amplification of individual electrons liberated from a photocathode by the photoelectric effect
• Work functions for metals are typically a few electron volts– 1 eV 1240 nm
http://hyperphysics.phy-astr.gsu.edu/hbase/tables/photoelec.html
Work Functions of Metals
• These numbers sure don’t look too interesting if you goal is to detect low energy photons.
Photomultiplier Shortcomings
● Poor wavelength coverage due to large work function of materials (limited to visible operating wavelength with some exceptions)
● Poor quantum efficiency (<20% conversion of photons to electrons)● Thermally emitted electrons (known as dark current, requiring cooling to suppress)
● Large single-‐detector area.One big advantage → fast photon counting
One big disadvantage à one tube = one measurement
Solid State Detection: Metals vs. Insulators
• At T=0K, the world contains only conductors and insulators.• Conductivity (or not) depends on how atomic energy levels shift and
spread as interatomic distance decrease going from a gas to a solid.• Materials with energy gaps (as illustrated below) are insulators.
Silicon atoms(energy levels)
Solid silicon(energy bands)
Solid State Detection: Semiconductors
• A photon with energy (hn) greater than the gap energy (Eg) can transform a “stuck” electron in the valence (insulator) band to a mobile electron in the conduction band.
For silicon, specifically, photons with energy greater than 1.1 eV can lift an electron up to the conduction band (wavelength shorter than about 1 micron)
hν = hcλ> Eg
Semiconductor Detectors: Bandgaps• Photoexcitation only occurs if hn > bandgap energy.• Different materials have different “cutoff” wavelengths.
Material Bandgap Cutoff WavelengthSilicon 1.1 eV 1.05 micronsGermanium 0.67 eV 1.8 micronsPbS 0.37 eV 3.6 micronsInSb 0.23 eV You Tell Me
Carbon 5.5 eV !
Bandgap varies slightly with temperature because the crystalline lattice spacing changes as the temperature changes.
Q: based on previous slide, would you expect greater or smaller gap in material with more tightly packed atoms?
Semiconductor Detectors: Bandgaps• Photoexcitation only occurs if hn > bandgap energy.• Different materials have different “cutoff” wavelengths.
Q: is there a downside? Why not always use a small gap?
Semiconductors and Cooling
• Semiconductors are insulators with bandgaps so small that thermal energy can maintain some population of electrons in the conduction band at room temperature making them weak (semi) conductors.– Of course these same small bandgaps make a material interesting from a photon
detection perspective.– These materials would suffer from significant dark current if maintained at room
temperature.
• The solution… operate the detectors at low temperature.– Most inexpensive CCD cameras use thermoelectric coolers (another cool but
complicated semiconductor effect) to keep the detectors at -‐20C/253K or cooler.– Professional CCD’s operate at near liquid nitrogen temperature (<100K)– Infrared detectors (small bandgap materials) may have to be cooled to liquid
helium temperature (4K) if the bandgap is small enough.
Cryogenics• Since dark current is the result of thermal excitation, cool the
detector so that kT << bandgap energy.
Longer wavelength = smaller bandgap = lower operating temperature.
Imaging Devices• Because the solid state detector materials are crystalline (e.g. silicon) the
same crystal growth techniques used to make integrated electronic circuits (computer chips) apply to the detectors themselves.– Enables the construction of precision structures on the submicron scale containing
both electronics and detectors – Arrays!
Solid State Photon Detection in Semiconductors
• Technically the electron is not so much “free” in a semiconductor as it is “borrowed”.
• Photon excitation creates a unbound electron and a corresponding “hole” within the crystal lattice. Both are mobile. – If they find one another they recombine = no detection.– As long as they are kept separate (typically via an electric field) they can be
detected…• as a current if they change the resistance of the material.• as a voltage if they are collected on a capacitor.
The MOS Capacitor – The Pixel
• The Charge Coupled Device’s (CCD’s) unit cell, the pixel, is based on a silicon structure that permits the collection of electrons from photon-‐created electron-‐hole pairs at a positively charged “gate”.
• Additional gates permit the dragging of the accumulate charge across the device and ultimately to a readout circuit that converts the electrons into a measurable voltage.
CCD Architecture
Test open shutter
closed shutter
Note that bad things can happen when buckets overflow (saturation).
HST F656N CTE issues
CCD vs. CMOS• CCD’s drag charge to a destination amplifier.
– Good: the few amplifiers on the chip can be engineered to be very sensitive.– Bad: charge can be lost and smeared along the way, each “bucketfull” has a
long journey with potential pitfalls. Also, readout takes a long time
• CMOS arrays “x:y” address each pixel. The charge stays “local”– Good: fast readout, non-‐destructive readout (you can “peek” at the
accumulating image without destroying it).– Bad: millions of amplifiers, but today their sensitivity is comparable to or
better than CCD’s.
“Sandwich” Infrared Arrays
http://gruppo3.ca.infn.it/usai/cmsimple3_0/images/PixelAssembly.png
http://www.flipchips.com/tutorial10.html
• Silicon is a terrific material because it not only makes great detectors, but it is the basis of nearly all integrated circuit electronics. Silicon CCD arrays can be “grown”.
• Infrared detector material (e.g. InSb) must be attached to silicon integrated circuits, typically through mechanical means
• metallic bumps of elemental indium here• differential thermal expansion here is a nightmare!
What Do You Actually Measure?
Photons make electrons, but electronics of some sort must convert that signal into a detectable voltage.
Electrons → Voltage
Analog to DigitalConverter (ADC) Digital “counts”
proportional to thevoltage
For example 5 Volts might correspond to 4096 counts, in which case measuring 1640 counts corresponds to 2 Volts.
What Do You Actually Measure?
Photons make electrons, but electronics of some sort must convert that signal into a detectable voltage.
Electrons → Voltage
Analog to DigitalConverter (ADC) Digital “counts”
proportional to thevoltage
For example 5V might correspond to 4096 countscounts = 4096* actual voltage
5V
Where Do The Volts Come From?
Circuitry converts collected electrons into electronically quantified information.
Electrons → Voltage
Analog to DigitalConverter (ADC) Digital “counts”
proportional to thevoltage
Drive a photon-produced current through a resistor (Ohm's Law).
Collect electrons in a (very small) capacitor, “C”.
CCD Gain
• Gain is the number of electrons that yield one analog-‐to-‐digital count.• Gain is an electronics dependent quantity
Volts / electron = 1.6x10−19 coulombs / electron
Creadout
(capacitance measured in Farads)
Volts / count = 5 Volts4096 counts full range
Combine and get electrons/count.
An Image!
17 22 14 19 16 18 21 20 17 15
22 15 15 18 25 26 15 19 21 11
19 18 27 14 13 18 16 20 12 15
12 15 23 17 15 19 22 21 14 18
15 17 11 24 54 30 21 15 14 19
24 20 13 17 15 21 15 18 21 17
19 12 18 24 15 19 14 22 22 18
17 288 11 20 15 13 18 19 21 22
20 19 18 15 22 14 15 17 20 14
20 14 21 32 102 44 25 17 14 21
FITS Format: Behind the Curtain
• Image storage and “representation” are two different things. • A series of numbers represents a two dimensional image if you have
the format (pixel grid x by y) and other “metadata”available.• FITS files consist of a metadata text “header” followed by data values.
– The header consists of an integral number of 2880 character blocks. • Each block contains a series of 80 character “keyword” parameters• The last keyword of the last block is “END” padded out by blanks• The first bytes (how many and what sort depend on the header information) of the next block is the first pixel of the image.
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 (you’ve seen this before – now it’s serious…).– 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%.
Whatever is being counted
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.
– detection systems aren't perfect (dark current) and there are contaminating sources of light such as the glow of the sky (and glow of the telescope in the thermal infrared) • Not to mention 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). – simplest terms: # signal counts / uncertainty.– S/N=10 is a measurement with 10% precision
• 100 electrons gets you there if there is no source of contaminating light.– S/N=100 is a measurement with 1% precision
• 10,000 electrons without contamination.
• In general, if the star is the only source of counts, N:
• Sources of background add to the detected photons.– These unwanted counts thus add additional Poisson noise.– Reducing these backgrounds improve signal-‐to-‐noise
• sharper images (landing on fewer background-‐containing pixels)• selecting filter bandpasses to avoid skyglow and maximize signal• 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 in each pixel in the measurement
• Consider a star that covers four pixels, each containing contaminating background, vs the same star covering only one pixel.– Same “N” but npix is 4 times smaller leading to 4 times lower total
background. If B is large compared with N sensitivity is improved substantially.
Accounting for Background Contamination
SNR =N fromstar
N fromstar + npixBper pixel
Background and Photometry Footprint
• A telescope collects only so many photons from a given star per unit time.
• These star photons, depending on the optical system and the array pixel size, can land on either a few or a whole lot of pixels.– Each pixel carries a background penalty, so your choice of how many pixels to
use when trying to measure “all” of the collected light from a star has signal-‐to-‐noise consequences.
SNR =N fromstar
N fromstar + npixBper pixel
“Read Noise” from a Poisson Perspective
• The act of measuring the counts, in a CCD pixel for example, can be (usually is) inherently noisy.
– This noise tends to be random/Gaussian (i.e. the value being drawn from a Gaussian distribution of probability)
– It is therefore characterized by the “width” of the distribution of these random counts, “s”.
– Recall that the Poisson noise for an actual “count” of N electrons is sqrt(N), which also behaves in a Gaussian manner for large N.
– Although read noise, characterized by an RMS uncertainty, “RN”, is not Poisson noise, one can pretend that the noise RN is caused by the collection of RN2 counts.
Read Noise and SNR
● If source photons are the only source of noise
● unwanted background photons B add to the Poisson noise,
● Read noise is the random fluctuation (measured in units of electrons) in the measurement (readout) of each pixel. To convert the read noise into the equivalent number of electrons that would produce equivalent noise one has to square read noise and, like with background, account for the number of pixels contributing read noise.
SNR = N fromstar
N fromstar
SNR =N fromstar
N fromstar + npixBper pixel
SNR =N fromstar
N fromstar + npixBper pixel + npixRN2
Back to CCD “Gain” – A Poisson Perspective• Electronic “gain” (a.k.a. amplification) accounts for the difference
between measured digital counts and collected electrons.– 10 electrons may end up on the output capacitor, but the analog to digital
converter may read these 10 electrons as 4 digital counts.• In this example the “gain” is 2.5 electrons per analog to digital unit: 2.5 e-‐/ADU
• Poisson noise provides a tool to determine this CCD gain.– Consider a system with a gain of 100 e-‐/ADU that makes multiple
measurements of a signal of 10,000 counts.– Given the gain, 1,000,000 electrons were collected.– The Poisson noise resulting from those million collected electrons is 1000
electrons, but since the gain is such that it take 100 electrons to make one ADU count the measured RMS noise in the counts will be 1000/100 = 10 ADU.
– So, in this situation you have a signal of 10,000 counts resulting in an RMS noise of 10 counts – clearly not Poisson, but this mismatch is a clue to how to calculate the gain.
Statistically Estimating CCD Gain• If the gain were unknown in the example on the previous page one
could reverse engineer the value under the assumption that the noise was Poisson.– You illuminate your CCD uniformly and make a bunch of measurements of the
scene, each one illuminated to a mean level of 10,000 ADU counts.• At this point you have no idea how many electrons 10,000 ADU counts represents.
– Given that you have a number of exposures you punch the exact ADU value of a given pixel in each of the frames into your calculator and find the standard deviation.• Your calculator spits out that the standard deviation is 10 ADU, so clearly not Poisson (it would have been 100 ADU if it was Poisson).
– You can now ask (in equation form below) what does the gain, g, have to be in order to make the measurements agree with Poisson statistics.
σ observed =g*ADUaverage
gσ observed =
Number of collected electronsgain
Statistically Estimating CCD Gain• If the gain were unknown in the example on the previous page one
could reverse engineer the value under the assumption that the noise was Poisson.– You illuminate your CCD uniformly and make a bunch of measurements of the
scene, each one illuminated to a mean level of 10,000 ADU counts.• At this point you have no idea how many electrons 10,000 ADU counts represents.
– Given that you have a number of exposures you punch the exact ADU value of a given pixel in each of the frames into your calculator and find the standard deviation.• Your calculator spits out that the standard deviation is 10 ADU, so clearly not Poisson (it would have been 100 ADU if it was Poisson).
– You can now ask (in equation form below) what does the gain, g, have to be in order to make the measurements agree with Poisson statistics.
σ observed =g*ADUaverage
gσ observed =
Number of collected electronsgain
g =ADUavg
σ 2
Why Do You Need to Know the Gain??• Assigning a proper uncertainty to the measurement of a star’s flux is
possibly as important as measuring the flux itself. – A measurement is meaningless if it does not have a reliably assigned statistical
significance.– For a stellar flux measurement extracted from a single image frame proper
quantification of the Poisson noise is the only means of assigning an appropriate uncertainty. Knowing the gain you can calculate the Poisson noise.
CCD Quantum Efficiency
• Quantum efficiency is a measure of what fraction of incoming photons actually make a detectable electron (QE = 1 is 100%)
• Just how efficiently vs. wavelength depends on detector structure. – In simplest terms, light must penetrate to and interact in a region where it
can produce electron-‐hole pairs that ultimately survive to yield a collected electron.
• Electrodes may absorb photons on the way in (short wavelengths).
• Photons may penetrate too far before being absorbed (long wavelengths).
• Photons may reflect off the detector surface.
CCD Quantum Efficiency vs. Wavelength
• The active, photon-‐detecting layer in a CCD lies within about 10 microns of the silicon surface where the readout structures are grown (brown at right).
• Typically light shines on this layer through the gate structures used to shuffle the charge.– These structures are transparent at longer
wavelengths but become opaque in the blue and ultraviolet.
– Simple CCD architectures have poor blue/UV response.
http://hamamatsu.magnet.fsu.edu/articles/quantumefficiency.html
Improving Blue Response via “Thinning”
http://hamamatsu.magnet.fsu.edu/articles/quantumefficiency.html
• “Backside illuminated” CCD’s take advantage of mechanical thinning of the original silicon substrate on which the device is grown to permit illumination from the side opposite the electrodes.
Fluorescent coatings that convert ultraviolet photons to longer wavelength photons can also enhance ultraviolet quantum efficiency.
Optimizing CCD Infrared Response
• Detecting infrared photons requires a thicker “active” layer in the CCD. A special class of “deep depletion” devices optimize quantum efficiency for the infrared.
F = front illuminatedB = back illuminated (thinned)DD = deep depletion
Quantum Efficiency and SNR
• For the same photon flux level in a fixed, say 10 second, integration, the total Poisson noise (square root of collected electrons) is lower for a low quantum efficiency detector compared with a high quantum efficiency detector because the incident photons produce fewer electrons.
– BUT the high quantum efficiency detector will make the detection at a higher signal to noise ratio, which is what counts since SNR depends on the square root of N – the total number of electrons collected.
– If the sentences above make sense you “get it”. If you don’t “get it” start asking questions.
