The Fundamental Radiometer Equation

7/29/2019 The Fundamental Radiometer Equation

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Leslie Green CEng MIEE 1 of 15 18 Jan [email protected]

The Fundamental Radiometer Equation

Leslie Green 1

Digital Barriers (ThruVision)

Abstract

The current radiometer equation in widespread use is 2

f

TT SYS

=

which is a scaled value of a measured system noise temperature.

A new equation is derived which effectively gives a fundamental limit on the system noisetemperature, independent of imperfections within the system components.

fk

TfhT CIDEAL

=

10

The approximation for noise power, fkTPN , is examined and found to be

inappropriate for use in the mm-wave region when operating well below 77K. A newapproximation formula is presented which accurately works down to 0.5K and atmm-wave frequencies,

+

+

+

3

0

2

00

222

11

fh

Tk

fh

Tk

fh

TkfkTP

N

Radiometers are shown to become remarkably insensitive at cryogenic targettemperatures, the sensitivity dropping by half at a (Kelvin) temperature given by

k

fhT

3

0%50 =

1

The author has been designing, building, testing, and characterising room-temperature radiometersoperating around a 250GHz centre frequency for the past 8 years.2

Kraus, J.D. (Tiuri, M.E.), 'System Noise' in Radio Astronomy, (McGraw-Hill Book Company, 1966), p. 244.


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Fundamental Radiometer Equation

Leslie Green 2 of 15 18 Jan 2013

Noise Power

If measuring the noise power from a resistor we should use Nyquists equation,3 modifiedby the inclusion of the zero-point energy.4

+

=

U

L

f

f

Ndf

hf

kT

hf

hfP

21exp

Where h is the Planck constant, 6.610-34 Js

k is the Boltzmann constant, 1.3810-23 J/K

f is the operational frequency, Hz

T is the temperature, K (using the degree symbol to avoid confusion with

k which is usually represented by K in

electronics work)

And using the subscripts U and L to represent the Upper and Lower input frequenciesrespectively, signifying ideally sharp (brickwall ) cutoff frequencies.

It is not very convenient to work with the above integral so we approximate the integral as

fkTPN

(see Appendix 1) noting carefully that this approximation is accurate in a range

f < 1 THz (which could also be expressed as f < 1000 GHz)

T > 77K

but is highly inaccurateat much lower temperatures.

3H Nyquist, 'Thermal Agitation of Electric Charge in Conductors', in Physical Review, 32 (July 1928), pp.

110-113.4Van Der Ziel, A., 'History of Noise Research', in Advances in Electronics and Electron Physics, 50 (1980),

p. 377.


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Power Fluctuations

NowNP is a mean power level, measured in Watts, and measurable (in principle) with a

power meter. But one might then reasonably ask what the noise is on this quantity. If, forexample, a moving coil power meter had a fast responding needle, how noisy would thereading be?

We are used to random variables being modelled by a Gaussian distribution where the

mean and variance 2 are independent variables. Looking first at a binomial

distribution, we see that the mean is given by

PN=

where N is the number of events and P is the probability of a particular outcome. Thevariance is then given by

( )PPN = 12

For low probability events, P is small so that ( ) 11 P and 2 . Expressed in words,the mean value is equal to the variance.

We then look at a Poisson distribution and see that it is modelled based on low probabilityevents from lots and lots of independent sources, which is exactly what we expect whenlooking at the aggregated response from electronic fluctuations at the atomic level. Thefinal step is to realise that the Gaussian distribution is used to model both Binomial andPoisson distributions.

Notice in the examples given above that the mean and variance are both dimensionlessquantities. This is essential since the mean, variance, and standard deviation then all havethe same units (none). Also notice that both the mean and the variance are proportional tothe number of events and hence to the measurement interval.

Shot Noise

If we consider radiant energy (photons) arriving in a unit of time we have a situationanalogous to the arrival of electrons crossing a barrier as used in the derivation of the shotnoise formula.5

For shot noise we consider a mean currentSI which is generated by a number of

electrons narriving within a measurement interval . Then

enI

S = , where the bar over

the n represents a mean value. Re-arranging this we finde

In S

= .

Equating the mean and variance gives ( )2nne

In S ==

. Note that it is not the mean

current and the variance of the current that are equal, it is the mean number of electrons

5Van der Ziel, A., 'Fluctuations in Cathode Emission', in Noise(New York: Prentice-Hall Inc, 1954), pp. 90-

92.


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arriving and the variance of that number which are equal. Both the mean current and thestandard deviation of the current are dimensionally equal to Amps.

If, at the end of the time interval , we have received a different number of electrons than

the mean number, we have a current deviation, ( )nn

e

i=

which gives rise to a mean

squared current deviation ( )

SS eI

e

Ien

enn

ei =

=

=

=

222

2 which is readily

seen to have the correct dimensions of amps squared, since [coulombs/second] = [amps].

Photon Noise

For a radiometer we consider the arrival of individual photons at the detector analogouslyto electrons crossing a barrier. Instead of [coulombs/second] = [amps] we expect[joules/second] = [watts].

If n is the number of photons arriving at the detector during a measurement interval ,

where each photon has an energy of , then the mean power is

= nP

N.

Therefore ( )2nnPnN

==

If, at the end of the time interval , we have received a different number of photons thanthe mean number, we have a power deviation

( )nnp =

The variance of the power deviation is

( ) NN PPnnnp ====

2

2

2

22

2

22

From which we get the standard deviation (the AC RMS value)

NPP =

Watts

To find the minimum resolvable T of a system 6 we view an ideal 7 cold target and

measure both the standard deviation of the noise, P and the mean noise CP . Then we

measure the mean noise for an ideal hot target, HP . The T is then defined by

( )CH

CHPP

PTTT

K

6which we will call T for brevity and read as delta-Tee.

7a black-body target at a constant temperature across the field of view.


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From our theoretical analysis we can now set a lower bound on the T of an ideal system.

( )NCNH

NC

CHIDEALPP

P

TTT

=

K

Until we use an approximation equation forNP we cant get very far with this.

Using fkTPN we get

( )( ) fk

T

fTTk

fkT

TTT C

CH

C

CHIDEAL

=

=

1K

This equation shows a significant feature in the form of the term. A higher photon

energy results in a worse(larger) T . A moments thought and the reason, apart from themaths, is obvious. The photon arrivals are discrete events. Less of them means a highergranularity and therefore a worse noise performance.

We can then use Plancks relation, 0fh= , with 0f as the centre frequency, to give

fk

TfhT CIDEAL

=

10 K

Although we have to appreciate that this formula is not validfor K77


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Cryogenic Operation

Cryogenic radiometers are not uncommon and it would be helpful to extend the precedinganalysis to liquid Helium temperatures and below. Appendix 2 shows that a relativelysimple full-temperature approximation to the power integral is possible, but for some

reason this approximation, and others for the whole integral, do not yield useful results interms of an ideal radiometer equation.

The problem is seen by plotting the power integral over temperature at mm-wavefrequencies (using Mathcad).

df 1 109

:= h 6.6 1034

:= k 1.38 1023

:= T 0.5 0.6, 200..:=

P F T,( )

Fdf

2

Fdf

2+

fh f

exph f

k T

1

h f

2+

d:= RJ F T,( ) df k T:=

0.1 1 10 100 1 .1030

5 .10 13

1 .1012

1.5 .1012

2.

10

12

2.5 .1012

3 .1012

Integrated Power variation with Temperature for 1GHz bandwidth

Temperature (K)

P 100 109

T,( )

P 500 109

T,( )

P 1 00 0 1 09

T,( )

RJ 5 00 1 09

T,( )

T

Now it is easy to see that below about 10K the available power is relatively unchanging.This is a real problem for a radiometer monitoring targets below the 4K region because itssensitivity to temperature fluctuations is almost zero. We can see this sensitivity more

clearly by plotting dP/dT, evaluated numerically by using a 0.1K temperature differenceand scaling the result by 10.


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dP F T,( ) 10 P F T 0.1+,( ) P F T,( )( ):=

0.1 1 10 100 1 .1030

2 .1015

4 .1015

6 .1015

8 .1015

1 .1014

1.2 .1014

dP/dT variation with Temperature for 1GHz bandwidth

Temperature (K)

dP 100 109

T,( )

dP 300 109

T,( )

dP 1000 109

T,( )

df k

T

Notice that this sensitivity function, dP/dT, is independent of the decision to include thezero point energy since the zero point energy is independent of temperature and itsderivative is therefore zero.

As an aside, the zero point energy part of the power integral is nicely separable and itsintegral produces a simple result, despite the initial apparent complexity.

dffhdf

fdf

fhfh

dffh

dff

dff

dff

dff

=

+=

=

+

+

2224420

2

0

2

0

2

2

22

2

0

0

0

0

The use of the zero-point energy term in radio astronomy is discussed in detail in an ALMAmemo.8

8

Kerr, A.R., Feldman, M.J, and Pan, S.-K., 'Receiver Noise Temperature, the Quantum Noise Limit, and theRole of the Zero-Point Fluctuations', MMA Memo 161 (8th Int. Symp. on Space Terahertz Tech: 1997(1996)).


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Radiometer Sensitivity at Cryogenic Temperatures

Rather than viewing the sensitivity function in absolute power terms, it is convenient to

normalise the power change relative to the fkdT

dP= approximation.

0.1 1 10 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Radiometer Sensitivity at Low Temperatures

Temperature (K)

dP 100 10

9

T,( )df k

dP 300 109

T,( )df k

dP 1000 109

T,( )df k

T

Now we can clearly see that a 300GHz radiometer viewing a 3K source has 5 lowersensitivity than might be expected from calibration using a higher temperature target.Clearly such considerations are significant for those measuring the Cosmic MicrowaveBackground (CMB) and interpreting earlier data from the COBE 9 satellite (COsmic

Background Explorer). The FIRAS instrument in particular was doing a null test betweenthe CMB and a reference target at roughly 2.8K.

These sensitivity drops are understandable in terms of the classical equipartition theoryequated to Plancks relation. If the molecular motion is small, there is not enough energyavailable to create a high frequency photon.

9http://lambda.gsfc.nasa.gov/product/cobe/


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We therefore equate the average translational kinetic energy Tk23 with half the photon

energy2

fh in order to get the 50% point in the sensitivity curve, giving

k

fhT

3

0%50 =

At 300GHz the predicted 50% sensitivity point is at 4.78K whilst the numerical integralgives 4.76K.

We can also give safe guidance for the lowest target temperature of a radiometer as

k

fhTC

02> using 6 the 50% sensitivity point.

Conclusions

Radiometers which are viewing non-cryogenic targets cannot be made arbitrarily sensitive

as they are subject to fundamental measurement limits.

Radiometers viewing cryogenic targets 10 become unworkable (insensitive) at Kelvin

temperatures much lower thank

fh

3

0 , regardless of the measurement method used.

Acknowledgement

I would like to thank Dr. Chris Mann for getting me started in the field of THz radiometricimaging, for useful technical discussions over the years, and for his strenuous personalefforts to raise capital and keep Thruvision going against all the odds. Without his effortsthis work would not have been possible.

10Note carefully that it is notthe temperature of the radiometer itself that is the critical issue here, it is the

temperature of the target or the scene being viewed that is critical.


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Appendix 1: Simple Approximation of the Power Integral

The intractable power integral,

+

=

U

L

f

f

N dffh

Tk

fhfhP

21exp

W

is typically simplified to fTkPN as follows

Put =Tk

fh. For small values of use ( ) + 1exp

In order to use this approximation with less than 1% error we need 148.0


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( )Tk

fhTk

fh

Tk

fh

fh2

0829.02

1exp

+=

+

Now we are now in a position to evaluate the original integral.

( )( )

3

0829.0

21exp

332

LULU

f

f

N

ff

Tk

hffTkdf

fh

Tk

fh

fhP

U

L

+

+

= W

If we use the substitutions2

0

fff

U

+= ,

20

fff

L

=

Then

32

0

2

0

3

0

3

223

23

+

+

+=

fff

ffffU

32

0

2

0

3

0

3

223

23

+

=

fff

ffffL

So that4

33

2

0

33 fffff LU

+=

Putm

ff 0= with 3>m

Then

+=2

2

0

33

12

113

mffff LU

We can therefore neglect the bracket with much less than 1% error overall.

( )

+=

+

2

0

2

0 0829.010829.0TkfhfkTf

TkfhfkTPN W

With all the previous requirements for the approximation formulae, the second term in thesquare bracket is less than 0.032, meaning we can neglect it with less than 3.2% error.

In summary, fTkPN W

provided f < 1 THz

T > 77K


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We might refer to this as the low frequency, high temperature model, the meaning of lowand high being associated with the boundaries shown above.

We can verify all the mathematical manipulations in one go by plotting the ratio of theapproximation and the numerically integrated exact value using Mathcad. We can then

readily see the dominant 3.2% error at 77K and 1000GHz.

df 10 109

:= h 6.6 1034

:= k 1.38 1023

:= T 77 77.2, 350..:=

R F T,( )1

df k T

Fdf

2

Fdf

2+

fh f

exph f

k T

1

h f

2+

d:=

60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 3601

1.005

1.01

1.015

1.02

1.025

1.03

Approximation Ratio - Variation with Temperature

Temperature (K)

R 100 109 T,( )R 500 10

9 T,( )

R 1000 109

T,( )

T


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We can use the same Mathcad worksheet to see how bad the approximation becomes atlow cryogenic temperatures.

To really understand this plot, look at 5K on the 1000GHz (black) line. Theapproximation is wrong by a factor of 5.

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

22

24


Temperature (K)

R 100 109

T,( )

R 500 109

T,( )

R 1000 109

T,( )

T


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Appendix 2: Full Range Approximation for the Power Integral

+

=

U

L

f

f

Ndf

hf

kThf

hfP

21exp

Starting from the same integral we now want an approximation that works down to 0.5K.The numerical integration procedure becomes inaccurate below this level. Now we seek a

multiplier for the fkTPN

value such that the result approximates the correct value

over a much broader range of temperature.

Our new operating parameters are chosen asf > 100 GHz

T < 350K

An exhaustive numerical search for approximations 11 eventually yielded a new formula

+

+

+

2

9

0

9

0

90

101730

105.451

100237.0

f

T

f

T

fTfkP

N

In the process of eliminating the dimensioned constants the final form gradually emerged.

+

+

+

3

0

2

00

222

11

fh

Tk

fh

Tk

fh

TkfkTP

N

This formula has been tested over the range of 1MHz to 5000GHz and 0.5K to 350K andfound to be accurate to better than 3.5%. Marginal improvement is possible on the error

amount by tweaking the constants slightly, but that detracts from the simplicity andsymmetry of the above formula.

NOTE: Do not use the above formula to numerically evaluatedT

dP. Subtracting the two

nearly equal values exaggerates the error in the formula and makes the result horriblyinaccurate.

11

The search consisted of taking the numerical integration data from Mathcad (version 13), exporting it toMS Excel (version 2003), importing the data from Excel into Eureqa Formulize (version 0.98.1) and picking asuitably accurate but simple approximation. This process was iterated until a good solution was found.


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Mathcad worksheet showing the accuracy of the approximation formula

max 150:= n 0 1, max..:= T n( ) 1.04n

0.5:= T max( ) 358.423=

df 10 109

:= h 6.6 1034

:= k 1.38 1023

:=

R F T,( )1

df k T 11

2 k T

h F

2 k T

h F

2

+2 k T

h F

3

+

+

Fdf

2

Fdf

2+

fh f

exph f

k T

1

h f

2+

d:=

0.1 1 10 100 1 .1030.965

0.97

0.975

0.98

0.985

0.99

0.995

1

1.005

1.01


Temperature (K)

R 100 109

T n( ),( )

R 1000 109

T n( ),( )

R 2500 109

T n( ),( )

R 5000 109

T n( ),( )

T n( )

The df value was varied in the range 0.01GHz to 100GHz and the curves were almostunchanged.

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The Fundamental Radiometer Equation