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Ideal Photon Detectors
Astronomy 6525
Lecture 4
A6525 - Lecture 4Ideal Photon Detectors 2
Outline Radiation Transport: Terminology Detectors Attributes Electrical Bandwidth Ideal Photon detection
Poisson Noise Signal-to-Noise Ratio Notes on signal extraction Bose-Einstein corrections
Supplemental material Addition detector performance measures Optimal aperture photometry extraction Bose-Einstein statistics - details
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Radiation Transport: Terminology Iν = specific intensity
energy from a given direction
srHzcm
ergs 111
sec 2
Ω=
d
dnchI ν
ν ν number density of photons perunit frequency interval (#/cm3/Hz)
Fν = Flux density sec
or sec 22
mcm
ergs
Hzcm
ergs
μ
const.) (
sym.) (az. cos2)(cos
cos
=→
→
Ω=
νν
ν
νν
π
θπθ
θ
II
Id
IdF
dAproj = dA cosθ
dΩ = 2πsinθ dθ Iν
n̂
A6525 - Lecture 4Ideal Photon Detectors 4
Flux from a star
fν = monochromatic flux see by an observer observed flux density
f = flux seen by an observer
2
2
22
)(coscos22
d
dR
d
rdr
d
dAd proj φφππ −===Ω
)1cos( ≈Ω= θνν Idfr
Rφ to observer
ν
νν φμμμμπ
Fd
R
dId
Rf
2
2
1
02
2
)cos( ),0(2
=
==
)(2
2
νννν π BIifBd
Rf ==
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Attributes of Detectors Responsivity, Ro [amps/Watt]
ratio of output current to input power (due to photons)
gives no information about the noise properties of a detector, hence does not indicate sensitivity
measure at constant (DC) power
for one electron per photon: Ro = e/hν = 0.81 λ (μm)
Spectral Response, Ro(λ) wavelength response of Ro
Frequency Response, R(λ,f) response to a modulated signal,
e.g. chopped radiation
1.0
0.0
λ (μm)
λc
Ro(λ)
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Attributes (cont’d) Quantum Efficiency, η (QE)
Average number of electrons generated per photon
Fraction of photons absorbed for bolometers
Detective Quantum Efficiency, (DQE) Square of output S/N relative to input S/N ratio
Dark Current, id Detector output with no photons falling on the device
Read Noise, RN (RN) Detector noise w/ no input photons
Usually independent of integration time
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Electrical Bandwidth We wish to characterize the electrical bandwidth of
the system If a detection system responds uniformly to
modulation frequency between f1 and f2 and no response outside, then the bandwidth is:
Otherwise, if the response function, R( f ) varies continuously then the equivalent bandwidth is:
where
( )df
R
fRf
∞
=Δ0
2
max
( )( )fRR maxmax =
12 fff −=Δ
A6525 - Lecture 4Ideal Photon Detectors 8
Frequency Response: RC Circuit One can characterize the
response function of a detector
by specifying the dependence
of its output on the frequency
of a sinusoidally varying input
photon power.
The frequency response can be limited by many factors in the detector system However, most factors can be modeled as RC circuits where the R
and C are in parallel
RC circuits have an exponential time rise response, so that charge deposited on the capacitor bleeds off through the resistor with an exponential time constant, τRC = RC
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A6525 - Lecture 4Ideal Photon Detectors 9
Frequency Response If we input a voltage pulse to the system: vin(t) = voδ(t),
then the output voltage, as observed with an oscilloscope will have the form:
The same event can be viewed in terms of the effect of the circuit on the input frequencies instead of in the time dependence of the voltage
To do so, we take a Fourier transform of the input and output voltages to change to the frequency domain
≥
<= − 0t,
v0t,0
)(v /t0 RCet
RC
out τ
τ
{ } ( ){ }ttfin δ0in vFT)(vFT)(V =≡
A6525 - Lecture 4Ideal Photon Detectors 10
Frequency Response The δ (t) function has contributions from all
frequencies at equal strength, so that:
Since the input, Vin( f ) = constant, any deviations from a flat spectrum in the output must be due to the action of the circuit, i.e. the output spectrum yields the frequency response of the circuit directly:
dteedtetf iftt
RC
iftout
RC∞
−−∞
∞−
− ==0
2/02out
v)(v)(V πτπ
τ
02
0 v)(v)(V == ∞
∞−
− dtetf iftin
πδ
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Frequency Response So that
The imaginary part is just a phase shift.
We are only concerned here with the amplitude of the frequency response which is given by:
( )( )( ) 2/12
02/1*outoutout
21
vVV)(V
RCff
τπ+==
RC
tifout if
tdefτπ
π
21
vv)(V 0
0
)21(o +
=′= ∞
′+−
A6525 - Lecture 4Ideal Photon Detectors 12
Frequency Response
The frequency response can be characterized by a cutoff frequency:
At which the amplitude drops to 1/sqrt(2) of its value at f = 0RC
cf πτ2
1=
( ) ( )0V2
1V outout =cf
( )( ) 2/12
0out
21
v)(V
RCff
τπ+=
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A6525 - Lecture 4Ideal Photon Detectors 13
E-BW: Exponential Decay For such a system, with exponential decay time, τ, the
response function, R( f ) is therefore:
τπ fi
RfR
21)( 0
+=
Or using the definition of equivalent electrical bandwidth introduced earlier
R0 is the DC responsivity
∞
+=Δ
02)2(1 τπ f
dff
τ4
1=Δf
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E-BW: Integrator over time T For a system that integrates over a time, T, the
response is:
The frequency response is:
The electrical bandwidth is then:
fTπ
fTπRdte
T
RfR πift
T
T
sin)( 0
22/
2/
0 == −
−
∞
=
0
2sin
dffT
fTΔf
ππ
<<
=otherwise0
2/t2-1)(v
TT/tout
TΔf
2
1=
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A6525 - Lecture 4Ideal Photon Detectors 15
Ideal Photon Detector No output current in the absence of incident power
No noise except that due to the randomness of emission times of photoelectrons
Let P be the power falling onto the detector with quantum efficiency, η, in a small bandwidth, Δν.
Assume emission events are probabilistic in the sense that they are uncorrelated and occur at an average rate r, given by:
νηh
Pr =
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Ideal Photon Detector: Shot Noise
The average number of photoevents occurring in a time T is:
The actual number of events will fluctuate around for any one particular interval of length T.
The probability, P(N ), that in any one such interval exactly N photoevents occur is given by the Poisson probability distribution:
rTN =
NN
eN
NNP −=
!)(
N
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Applicability of Poisson Distribution
Divide the time interval T into n segments. The average number of photons per segment is /n , where is the average number of photon events for the time interval T.
For sufficiently large n, /n << 1, so that /n can be interpreted as the probability that one photoevent occurs in a given segment.
The probability that exactly N events occur in the total interval T is given by the binomial distribution
N
NnN
n n
N
n
N
NnN
nNP
−
−
−
= 1)!(!
!)(
N
N
N
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Poisson Distribution
Rewriting gives
Taking the limit as n → ∞, we arrive at the Poisson distribution
Nn
n
N
nn
n
N
N
N
NPNP
−
∞→
∞→
−=
=
1lim!
)(lim)(N
N
eN
NNP −=
!)(
Nn
Nn
N
nnn n
NN
NNP
−+
−−−−
= 1!
)1()1)(1(1)(
121
NnN
n n
N
n
N
NnN
nNP
−
−
−
= 1)!(!
!)(
Probability that photoevents occur in N specific segments
Probability that photoevents do not occur in the remaining n-N segments
Combination of n things taken N at a time – the total number of ways that N indistinguishable events can occur in n segments
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Poisson Probability Distribution Figure 8.4 in Boyd – the
Poisson probability distribution
Note how has the average number of photons gets large, the width of the distribution (noise) approaches 2/1N
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Properties of Poisson Dist’n Normalization: 1
!)(
00
=== −∞
=
−∞
= NN
N
NN
N
eeN
NeNP
One can show:
Expectation value:
Variance:
RMS noise:
Signal to noise ratio:
NNPNNN
== ∞
=0
)(
( ) ( ) NNNN =−≡Δ 22
( ) 2/12 NNNrms =Δ≡Δ
rTNT
NN
N
N
S
rms
=∝
=Δ
≡
since ,2/1
2/1
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A6525 - Lecture 4Ideal Photon Detectors 21
Noise for a Photodetector The photoevents give rise to a photocurrent (and
likewise noise) in a photon detector.
Suppose a photon detector is characterized by an averaging time T, the average current is:
T
Nei ≡
The noise in the photocurrent is:
( ) ( ) ( )2
22
2
2222
T
NeNN
T
eiiiiN =−=−≡Δ≡
so thatfie
T
ieiN Δ== 22 (using Δf = 1/2T )
White noise - constant noise power per unit frequency interval.
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Signal-Limited Detection Suppose PS is the signal power falling onto an ideal
detector of quantum efficiency, η, the signal current is:
νη
h
fPefiei S
SN
Δ=Δ=22
2νη
h
Pei SS = w/ noise
The signal-to-noise ratio is then
fh
P
i
i
N
S S
N
S
Δ==
νη
2 ην fh
PNEPNSS
Δ=≡=
21/
T
hNEP
ην=
Minimum detectable power (or NEP) will produce on average one photo-detection per measurement time T.
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Background Limited Instrument Performance (BLIP)
In addition to PS there may be an unwanted background power, PB. The signal and noise currents are now:
( )ν
ηh
fPPei BS
N
Δ+=22
νη
h
Pei SS =
The signal-to-noise ratio is then
( )BS
S
PPfh
P
N
S
+Δ=
νη
2
2
ην fPh
NEP BΔ=2
BLIP occurs for PB >> PS , such as looking through the atmosphere in the infrared.
BLIP
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Insight into Signal-to-Noise Ratio
Photoelectron events are collected for an integration time given by t = 1/(2Δf ), so the total number collected for (monochromatic radiation) the source and background are:
th
PN B
B νη=t
h
PN S
S νη=
( )BS
S
PPh
tP
N
S
+=
νη 2
( ) ( ) ( )ν
ηh
tPPNNNNN BS
BSBSrms
+=+=Δ+Δ=Δ 22
t
Ph
N
SP B
S ην=→
PB >> PS
BLIP
Each component has shot noise which add in quadrature
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A6525 - Lecture 4Ideal Photon Detectors 25
Point Source Sensitivity So for background limited instrument performance we
have
Now for a point source and a small bandwidth (Δλ)
iwTS AfP ττλλΔ=
t
Ph
N
SP B
S ην=
fλ = ergs/cm2/s/μmAT = area of telescopeτw,τi = transmissions
t
Ph
AN
Sf B
iwT ην
ττλλ Δ=
1 Flux density to achieve a given S/N ratio in time t for BLIP.
Note: fν Δν = fλ Δλ, so we can equally use fν rather than fλ
BLIP
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Including Other Noise Sources In addition to the intrinsic shot noise associate with photons from the
source and background, there can be other sources of noise, e.g. Shot noise from dark current
“Read noise” associated with the electronic circuitry associated with the detector
In many cases, these noises are uncorrelated so we can added them in quadrature to get the total noise There may be “excess” noise above that expected from pure shot noise for
some processes.
We will use the factor β (> 1) to characterize this excess.
For the present purpose we (ideally!) include read noise and dark current in a simple way in the signal-to-noise ratio estimate.
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Signal-to-Noise Ratio The signal-to-noise ratio is on a source with S collected photoelectrons is:
[ ] 2/12222NdBS RNNN
S
N
S
+++=
Since the (uncorrelated) noises add in quadrature Thus, solving for S and plugging in gives
( ) 2/1
22
+++= Nddd
BSwiSwi RtiGth
PPG
N
St
h
PG βν
τητβνττη
If one of the noise terms dominates the others then the system is called: background, dark current or read noise limited depending on which dominates.
The best condition is signal-noise-limited
τw = (warm) transmissionτι = instrument transmissionη = detector responsivityG = Gainβ = Gain dispersion (excess noise)RN = read noiseNx = Noise due to x
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Point Source Sensitivity (cont’d) BLIP (Background Limited Infrared Performance) is defined as the
background dominating all other noise sources. Assuming the background emission is extended and “thermal”
iTB ATBP τλε λλ ΩΔ= )(ε = emissivityAT = telescope areaΩ = solid angle for extracting source
If there are more sources of background, add them to get PB. Putting the above in for PB gives
( ) Tiw At
Bh
N
Sf
λτβηεν
τλλ
λ ΔΩ= 1
BLIP is the best you can do (unless your source is bright enough to reach the signal-noise-limited regime)
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Point Source Extraction How do we “extract” the photo-electrons for a point source?
(determines background in S/N determination)
Options Add-up signal in pixels that contain
the source.
Fit point spread function (PSF) to source (best way – more in a later)
Optimize extraction radius to get best S/N ratio (if not using PSF). Don’t have to get “all” the flux - but must
account for this in calibration and S/N calculations.
Extraction radius
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Point Sources: At the telescope Consider a detector operating at a telescope.
Use the measured parameters to estimate sensitivity (or integration time). We have:
( ) 2/12NB pRtpSStNoise
StSignal
++=
=S = e-/sec from source
SB = e-/sec/pixel from skyp = number of pixels
(seeing dependent)RN = read noise (e-/pixel)
( ) 2/12NB pRtpSSt
St
N
S
++=
sum over p pixels
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Matched Filtering/Optimal Extraction
For point source extraction, A weighted extraction which matches the point spread
function provides the optimal signal-to-noise ratio
Need to determine center and amplitude in fit
Can ignore bad pixels (hard for aperture photometry)
But … At high S/N (> 50-100?) small errors in PSF dominate
S/N ratio => likely better off using aperture photometry
Also … At low S/N (< 5-10?) exact PSF shape doesn’t matter,
sometimes a Gaussian or other simple function will do
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Optical Imaging Palomar Suppose we have a imaging CCD at Palomar with the
following characteristics (the “old” COSMIC instrument).CCD: 2048x2048 RN: 11 e-Scale: 0.2856 arcsec/pixel
20th mag. star: 637 e-/sec at R-band (SR20)sky: 18 e-/sec/pix at R-band (BR)
Ignoring the read noise (okay for t > 25 sec), the time to obtain a give S/N at R-band on a source of magnitude m is
25.1/)20(220
5.2/)20(20
2
10
10−−
−− +
=
mR
Rm
RR S
pBS
N
St
e.g. tR = 430 seconds to get S/N = 5 for m = 25 with an extraction aperture of 10 pixels containing 50% of the flux.
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A6525 - Lecture 4Ideal Photon Detectors 33
Scaling Laws for Point Sources
For diffraction limited performance:
292.0 λ=ΩTA Diffraction limitedbeam
42
2
2
1
1
DfN
St
tDf
λ
λ
∝
∝
For a fixed beam size (e.g. seeing limited):
2
4 Bθπ=Ω Fixed beam size
22
22
DfN
St
tDf
B
B
λ
λ
θ
θ
∝
∝
92.04
22.12
=
π
( ) Tw At
Bh
N
Sf
λτβηεν
τλλ
λ ΔΩ= 1
For BLIP 2
4DAT
π= 2
4 Bθπ=Ω
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Extended Source Sensitivity For an extended source on the sky. Let
Ω= λλ If Iλ = specific intensity(ergs/cm2/sec/sr/μm)
( ) ΩΔ=
Tiw At
Bh
N
SI
λτβηεν
τλλ
λ1
/
1 Then
292.0 λ=ΩTA
Diffraction limited beam
2
21
1
λ
λ
IN
St
tI
∝
∝
2
4 Bθπ=Ω
Fixed beam size
222
21
1
DIN
St
tDI
B
B
λ
λ
θ
θ
∝
∝
Independent of D!
Scaling Laws
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A6525 - Lecture 4Ideal Photon Detectors 35
Spectral Lines
For best sensitivity Narrow spectral bandpass to
remove extraneous flux without reducing line flux
Line flux = area ( f )= (ergs/cm2/sec)fλ
λ
The integrated flux will be roughly: f = fλΔλ so that the sensitivity is now (BLIP case):
Tiw At
Bh
N
Sf
ΩΔ= λτηεν
τλλ1
narrower spectral BW better sensitivity, unless line is resolved
A6525 - Lecture 4Ideal Photon Detectors 36
But Photons are Bosons
Photons follow Bose-Einstein Probability Distribution:
For which:
1
1/ −
=kThe
n ν( ) n
n
n
nnp ++
= 11)(
( ) ( )12 +=Δ=Δ nnnnrms
for hν >> kT, i.e. n << 1 nnrms =Δ
for hν << kT , i.e. n >> 1 nnrms =Δ Photon “bunching” causes increased dispersion
is average occupation number
when thermally generated
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A6525 - Lecture 4Ideal Photon Detectors 37
Boson Effects: Signal-to-Noise Ratio Including Bose-Einstein statistics
The number of signal electrons generated will be
( ) ( )nh
tPNN i
iBBrms ηετ
νηετ +=≡Δ 122
th
PS Swi
ντητ=
The signal to noise is then given by
BN
S
N
S ≡
correction term for the fact that photons are Bosons
τw = 1 - ε
ε = emissivity of warm background (atmosphere and telescope combined)
See supplemental material for more details
Supplemental Material
Reference: Boyd, R.W. “Radiometry and the Detection of Optical
Radiation” 1983 John Wiley & Sons, Inc.
Appendices Measures of Detector Performance
Extraction size for aperture photometry
Bose-Einstein statistics and S/N
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A6525 - Lecture 4Ideal Photon Detectors 39
Measures of Detector Performance NEP - Noise Equivalent Power (W/Hz1/2)
The input power that produces an output signal-to-noise ratio of unity in a specified electrical bandwidth, Δf.
Often Δf = 1 Hz then the NEP is the minimum signal detectable in a 1/2 second measurement.
NEF – Noise Equivalent Flux (W/m2/Hz1/2) NEF = NEP / ( Atelηatmηtel )
D* - Specific Detectivity (cm-Hz1/2/W)NEP
fAD
Δ=*
NEFD - Noise Equivalent Flux Density Input flux that produces a signal-to-noise ratio of unity in 1/2
second measurement
e.g. NEFD = 10 mJy (1 Jy = 10-26 W m-2 Hz-1)
A6525 - Lecture 4Ideal Photon Detectors 40
Choosing the extraction region The size of the extraction region for counting
the source photons is important. Choosing it too large or too small will degrade
the signal-to-noise ratio
If the PSF is known, determine the location and amplitude of best fit solution. PSF could be determined from other (bright
sources) in the field or from separate reference measurement (if it is stable)
Essentially the choice is the number of pixels to sum, with possible weighting for each according to the PSF.
Let’s look at the Gaussian and Airy functions
Extraction radius
How careful do you need to be? – This depends upon your
science requirements!
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A6525 - Lecture 4Ideal Photon Detectors 41
Gaussian vs. Airy Pattern
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5x
g(x
), I(
x)
g(x) I(x)
G(x) A(x)
What is Ω, the extraction beam? This choice of an extraction beam is science dependent. For point sources, in
the seeing limited case we might expect a Gaussian PSF while for diffraction limited observations the Airy diffraction pattern will hold.
We then integrate the flux (either by summing pixels or fitting a PSF)
Plot of Gaussian and Airy diffraction pattern, and the encircled energy (weighted area integral) vs. x, where x is in units of FWHM for the Gaussian and λ/D for the Airy pattern.
The HWHM of the Airy pattern is 0.51λ/D (and of course 0.5 for the Gaussian).
The Airy pattern is for Dobscur/Dtel = 0.12 or 1.44% obscured area.
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Gaussian vs. Airy Pattern
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5x
Sig
nal-t
o-N
ois
e
Gaussian
Airy
Optimum Signal-to-Noise Ratio For BLIP observations the signal-to-noise ratio will vary as the encircled
energy divided by x (since the background power will scale as x2 and the noise scales as the square root of the power - However at longer wavelengths this is no longer true because Bose-Einstein statistics become important).
Plot of relative signal-to-noise ratio for a Gaussian and Airy diffraction pattern vs. x, where x is in units of FWHM for the Gaussian and λ/D for the Airy pattern.
The maximum S/N occurs at x= 0.673 and 0.660 for the Gaussian and Airy pattern respectively. Choosing x = 0.5 for the extraction radius causes a 6% decrease in S/N.
The Airy pattern is for Dobscur/Dtel = 0.12 or 1.44% obscured area.
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A6525 - Lecture 4Ideal Photon Detectors 43
Choosing Ω Our first guess for an extraction diameter for a diffraction limited beam
might have been 1.22λ/D which would give
2
2
2 92.022.144
λλππ =
=Ω
TTT D
DA
From the results in optimizing S/N a better choice of the extraction diameter is 1.294×FWHM = 1.32 λ/DT. Now we have
2
2
2 08.132.144
λλππ =
=Ω
TTT D
DA
The table below shows how the fractional flux and relative S/N change with extraction aperture diameter for the Airy diffraction pattern
θ (λ/D) Flux Frac. S/N
1.00 0.45 0.90
1.25 0.59 0.92
1.345 0.64 0.96
θ (λ/D) Flux Frac. S/N
1.50 0.71 0.94
1.75 0.78 0.89
2.00 0.81 0.81
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Choosing Ω The table below summarizes the results for Gaussian and diffraction limited
PSFs with the addition of a final column for the fraction of the flux in the extracted beam.
Nicely, the ratio of the optimum extraction radius to HWHM is about the same for both so adopting an average value will result in only a few percent error.
PSF HWHM Opt. S/N radius
Opt S/N over HWHM
Fraction of Flux in Extraction
Gaussian 0.500 0.673 1.346 0.715
Airy 0.510 0.660 1.294 0.632
The units for the HWHM and optimum extraction radius are FWHM for the Gaussian and λ/D for the Air diffraction pattern.
A combined expression for the extraction diameter (when diffraction and Gaussian terms are both present) might look like
( ) ( )( ) ( )22
22
02.132.1
346.1294.1
G
GAext
FWHMD
FWHMFWHMd
+=
+=
λ
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A6525 - Lecture 4Ideal Photon Detectors 45
The Complete Story: Photons are Bosons
The probability distribution function for getting n photons per mode, that is, the probability that n photons are excited in a mode of angular frequency ω is given by:
n is called the mode occupation number, and the energy of each mode is (quantized as):
n = 0, 1, 2, …ωnE =
Summing the series gives:
( ) kTnkT eenp //1)( ωω −−−=
πων 2/=∞
=−
−
=0
/
/
)(n
kTn
kTn
e
enp
ω
ω
Nkc /=ω
A6525 - Lecture 4Ideal Photon Detectors 46
Bose-Einstein Probability Distribution
The average occupation number is:
∞
=
=0
)(n
nnpn
We could then write:
( ) 222 nnn −=Δ
Now
1
1/ −
=kThe
n ν
( ) n
n
n
nnp ++
= 11)(
Bose-Einstein probability distribution
∞
=
=0
22 )(n
npnnwhere
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A6525 - Lecture 4Ideal Photon Detectors 47
Noise in Bose-Einstein Distribution
So we have
nnn += 22 2
The rms dispersion becomes:
( )
( )
( )2
2
2
02
2
0
22
1
1
1
11
1)(
−+=
−−=
−==
−−
∞
=
−−∞
=
x
x
xx
n
nxx
n
e
e
edx
de
edx
denpnn
( ) ( )12 +=Δ=Δ nnnnrms
kT
hx
ν=where
for hν >> kT, i.e. n << 1
nnrms =Δ
for hν << kT , i.e. n >> 1
nnrms =ΔPhoton “bunching” causes increased dispersion
A6525 - Lecture 4Ideal Photon Detectors 48
Noise in Ideal Detector So we have
Now consider the power falling onto a detector
( ) ( )12 +=Δ nnn rms
rnhec
hATBAP
kThB
ν
ννν νν
=−
ΔΩ=ΔΩ=1
12)(
/2
3
where
νλ
ΔΩ=2
2A
r r is the rate at which field modes intersect the detector
Ω = solid angle for a single transverse field, Δν⋅t = number of possible physically independent measurements of the field amplitude in time t
The fluctuations in the photon number occur over the coherence time of the radiation field, τc ~ (Δν)-1. (see Boyd & heterodyne detection)
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A6525 - Lecture 4Ideal Photon Detectors 49
Transmission and QE effects However, the sampling time is much longer than the
coherence time. If, t, is the integration time, the we sample a total of rt modes.
If we add together rt modes the noise is then
If the detector has quantum efficiency, η, and optical transmission, τ, and the emissivity of the background is, ε,then the number of modes “reaching” the detector and producing photoelectrons is reduced by the product of the factors, i.e.
( ) ( ) ( )
( )nrh
Prt
nnrtnrtN
B
rmsrms
+=
+=Δ=Δ
1
122
ν
nn ηετ→
A6525 - Lecture 4Ideal Photon Detectors 50
Boson Effects: Signal-to-Noise Ratio Including Bose-Einstein statistics
The number of signal electrons generated will be
( ) ( )nh
tPNN i
iBBrms ηετ
νηετ +=≡Δ 122
th
PS Swi
ντητ=
The signal to noise is then given by
BN
S
N
S ≡
correction term for the fact that photons are Bosons
τw = 1 - ε
ε = emissivity of warm background (atmosphere and telescope combined)
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A6525 - Lecture 4Ideal Photon Detectors 51
Signal detection with B-E Stats Thus we have
where
( )nt
hP
N
SP i
i
BS ηετ
ητνε
ε+
−= 1
1
1
Thus we can write:
nhA
TBAP TTB νν
λν ν ΔΩ=ΔΩ=
22)( &
1
1/ −
=kThe
n ν
( )2
211
1
λν
ητν
εΩΔ+
−= T
rri
S
A
tnn
h
N
SP nnr ηετ=
The factor of two enters since we have both polarizations. Calling Np the number of polarization measured (1 or 2), we have for point sources
2p
TS
NfAP νν Δ= We reference to the unpolarized flux.
A6525 - Lecture 4Ideal Photon Detectors 52
Sensitivity with B-E Stats Putting it all together gives
Where for diffraction limited performance
TmkT
hx
)(
14388
μλν =≡
Note that for hν >> kT the Bose-Einstein correction factor is unimportant. Let
for T = 290 K; x = 49.6/λ(μm)
So the dividing line is ~ 50 μm for whether to worry about the extra factor (and depends upon η, ε, and τ.
( )t
A
N
nn
A
h
N
Sf T
p
rr
Ti
112
1
12λνητ
νεν
ΩΔ+
−=
2λ∝ΩTA
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A6525 - Lecture 4Ideal Photon Detectors 53
Additional Complications Multiple background contributions
The derivation on the previous page assumed a single background source with emissivity ε [and transmission (1-ε)].
Emission is also usually contributed by the telescope (and other sources) which may or may not be at the same temperature as the atmosphere.
Lost light In the radio and submm parts of the spectrum, typically the surface
roughness of the “dish” is large enough to scatter light from the source out of the beam (but the background will be unchanged because light from the background is also scattered into the beam)
Point source extraction The optimal extraction of a point source will not include the entire PSF –
so we must account for this. Pixilation intrinsically widens the source PSF. To first order this can be
modeled by increasing the effective extraction size (by adding the pixel angular size in quadrature with the nominal extraction beam size).
A6525 - Lecture 4Ideal Photon Detectors 54
Additional Complications (continued) Detector Noise
Detectors may have read noise, generation-recombination noise, or other sources of noise but we will ignore these for now.
Chopping / Nodding In the infrared/submm a source is move rapidly (chopped) between
two sky positions. The difference is taken to remove atmospheric variation, telescope offsets, etc.
This differencing adds the noises, thus decreasing the sensitivity by sqrt(2).
Additionally if the source is moved off the detector (array) for the second sky position, a factor of two in time is also lost resulting in another factor of sqrt(2) change in sensitivity.
These factors are not included in what follows.
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A6525 - Lecture 4Ideal Photon Detectors 55
Sensitivity with B-E Stats The light loss is characterized by the Ruze factor, gR, which is given by
Let the telescope have emissivity, εT and temperature TT, we then have
( )( )
( )t
A
N
nn
A
h
gg
NSf T
p
rr
TiRfTA
1121
1
/2λνητ
ντεν
ΩΔ+
−=
2)/4( λπ rmssR eg −= srms = rms surface roughness
−+
−=
11 // TA kThT
kThA
r een νν
εεητwhere the atmospheric component is now explicitly labeled and attenuation by the telescope is neglected (normally a 2nd order effect here).
Other backgrounds (emissivity sources) can be added in a similar way.
Finally letting gf be the fraction of the source flux in the extraction beam (typically ~ 70%) and τT the telescope transmission, we have
A6525 - Lecture 4Ideal Photon Detectors 56
Point Source Sensitivity In summary, the sensitivity for instrument on a telescope including the
Bose-Einstein contribution is:
S/N, t = signal-to-ratio and integration time
εT , εA = telescope and atmospheric emissivity
τT , τA = telescope and atmospheric transmission (τA = 1 - εA)
TT , TA = telescope and atmospheric temperature
τ i, η, Np = instrument transmission, detector QE, number of polarizations (1 or 2)
AT , gf = Telescope area and fraction of flux in extraction (~ 0.7 typically)
srms = rms surface roughness
( ) ( )t
A
N
nn
A
h
gg
NSf T
p
rr
TiRfTA
1121/2λνητ
νττν
ΩΔ+=
2)/4( λπ rmssR eg −=
−+
−=
11 // TA kThT
kThA
ir een νν
εεητ
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A6525 - Lecture 4Ideal Photon Detectors 57
Chopping and Pixilation We can add a chopping degradation factor fairly easily since this
directly affects the noise though a difference (sqrt(2)) or integration time on source (for a given wall clock time).
As a first order assumption for “pixilation” noise assume that there are p pixels across the diffraction limit. Then
Where Ω is the non-pixilated beam area. p = 2.0, 2.5, and 3.0 yields factors of 1.12, 1.08, and 1.05 change in sensitivity respectively.
Note that this formulation assumes that a set of randomly “dithered” images are combined so the effective beam is widened which is done to eliminate systematic effects associated with pixel position.
This effect is negated with “perfect” pointing so that this is no “smearing” of the beam, however, quantization effects (finite size pixels) will still enter.
+Ω=Ω
2
11
pPB
A6525 - Lecture 4Ideal Photon Detectors 58
Point Source Sensitivity, again Including pixilation and chopping loses the point source sensitivity is
S/N, t = signal-to-ratio and integration time
εT , εA = telescope and atmospheric emissivity
τT , τA = telescope and atmospheric transmission (τA = 1 - εA)
TT , TA = telescope and atmospheric temperature
τι , η, Np = instrument transmission, detector QE, number of polarizations (1 or 2)
AT , gf = Telescope area and fraction of flux in extraction (~ 0.7 typically)
srms , CL = rms surface roughness and chopping loss (1, 1.414, or 2 typically)
p = pixels across beam (= 2 for Nyquist sampling)
( ) ( )22
11
12/
p
A
N
nn
A
h
gg
C
t
NSf T
p
rr
TiTARf
L +ΩΔ+=
λνητττν
ν
2)/4( λπ rmssR eg −=
−+
−=
11 // TA kThT
kThA
ir een νν
εεητ