Ideal Photon Detectors - Cornell Universityhosting.astro.cornell.edu/academics/courses/astro... · A6525: Lec. 04 3 Ideal Photon Detectors 5 A6525 - Lecture 4 Attributes of Detectors

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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


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)


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


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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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 =≡


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 +

=′= ∞

′+−


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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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


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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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 =


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


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


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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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.


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


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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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


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


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


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



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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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


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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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


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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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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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)


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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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.


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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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


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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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


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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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


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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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→


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)

A6525: Lec. 04

26


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.


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

A6525: Lec. 04

27


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).


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.

A6525: Lec. 04

28


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


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

A6525: Lec. 04

29


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


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

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Ideal Photon Detectors - Cornell Universityhosting.astro.cornell.edu/academics/courses/astro... · A6525: Lec. 04 3 Ideal Photon Detectors 5 A6525 - Lecture 4 Attributes of Detectors