Detection of Electromagnetic Detection of Electromagnetic Radiation IV and V:Radiation IV and V:

Detectors and AmplifiersDetectors and Amplifiers

Phil Mauskopf, University of Phil Mauskopf, University of RomeRome

21/23 January, 200421/23 January, 2004

Noise: Equations

Include Bose-Einstein statistics and obtain the so-called‘Classical’ formulae for noise correlations:

Sij*() = (1-SS)ij kT (I-SS)ij /(exp(/kT)-1)

Seiej*() = 2(Z+Z)ij kT 2(Z+Z)ij /(exp(/kT)-1)

Relations between voltage current and input/output waves:

1/4Z0 (Vi+Z0Ii) = ai1/4Z0 (Vi - Z0Ii) = bi orVi = Z0 (ai + bi) Ii = 1/Z0 (ai - bi)

Noise: Derivation

Quantum Mechanics II: Include zero point energy

Zero point energy of quantum harmonic oscillator = /2

I.e. on the transmission line, Z at temperature, T=0 thereis still energy.

Add this energy to the ‘Semiclassical’ noise correlation matrixand we obtain:

Seiej*() = 2 (Z+Z)ij coth(/2kT) = 2 R (2nth +1)

Sij*() = (1-SS)ij coth(/2kT) = (2nth +1)

Noise: Derivation - Quantum mechanics

This is where the Scattering Matrix formulation is moreconvenient than the impedance method:

Replace wave amplitudes, a, b with creation andannihilation operators, a, a, b, b and impose commutationrelations:

[a, a ] = 1 Normalized so that a a = number of photons[a, a ] = Normalized so that a a = Energy

Quantum scattering matrix: b = a + cSince [b, b ] = [a, a ] = then the commutator of the noise source, c is given by:

[c, c ] = (I - ||2)

Quantum Mechanics III: Calculate Quantum Correlation Matrix

If we replace the noise operators, c, c that representloss in the scattering matrix by a set of additional portsthat have incoming and outgoing waves, a, b:

c i = i a

and:(I - ||2)ij

= i j

Therefore the quantum noise correlation matrix is just:

c i c i = (I - ||2)ij

nth = (I - SS)ijnth

So we have lost the zero point energy term again...

Noise: Quantum Mechanics IV: Detection operators

An ideal photon counter can be represented quantummechanically by the photon number operator for outgoingphotons on port i:

di = b i b

i which is related to the photon number operator forincoming photons on port j by: b

i b i = (n S*

inan)(m Simam) + ci ci = d Bii()

(n S*inan

)(m Simam) = n,m S*in Sim a

n am

an am = nth(m,) nm which is the occupation number of

incoming photons at port m

Noise: Quantum Mechanics IV: Detection operators

Thereforedi = m S*

imSim nth(m,) + ci ci = d Bii()

Where: ci ci = (I - SS)iinth

The noise is given by the variance in the number of photons:

ij2 = di dj - di di = d Bij() ( Bij()+ ij )

Bij() = m S*imSjm nth(m,) + ci cj

= m S*imSim nth(m,) + (I - SS)ijnth(T,)

Assuming that nth(m,) refers to occupation number of incomingwaves, am , and nth(T,) refers to occupation number of internallossy components all at temperature, T

Noise: Example 1 - single mode detector

No loss in system, no noise from detectors, only signal/noiseis from port 0 = input single mode port:Sim = 0 for i, m 0S0i = Si0 0

di = d S*i0Si0 nth(0,) + ci ci = d Bii()

ii2 = di dji - di di = d Bii() ( Bii()+ ii )

For lossless system - ci ci = 0 and

ii2 = d Bii() ( Bii()+ ii ) = d Si0

2 nth() (Si02 nth()+ 1)

Recognizing Si02 = as the optical efficiency of the path from

the input port 0 to port i we have:

ii2 = d nth() (nth()+ 1) express in terms of photon number

Noise: Gain - semiclassical

Minimum voltage noise from an amplifier = zero pointfluctuation - I.e. attach zero temperature to input:

SV() = 2 R coth(/2kT) = 2 R (2nth +1)

when nth = 0 then

SV() = 2 R

Compare to formula in limit of high nth :

SV() ~ 4 kTN R where TN Noise temperature

Quantum noise = minimum TN = /2k

Noise: Gain

Ideal amplifier, two ports, zero signal at input port, gain = G:S11 = 0 no reflection at amplifier inputS12 = G gain (amplitude not power)S22 = 0 no reflection at amplifier outputS21 = 0 isolated output

Signal and noise at output port 2:d2 = d S*

12S12 nth(1,) + c2 c2 = d B22()22

2 = d2 d2 - d2 d2 = d B22() ( B22()+ 1 )

c2 c2 = (1 - (SS)22)nth(T,)

What does T, nth mean inside an amplifier that has gain?Gain ~ Negative resistance (or negative temperature)

namp(T,) = -1/ /(exp(-/kT)-1) -1 as T 0

Noise: Gain

0 0 0 G 0 0 G 0 0 0 0 G2

c2 c2 = -(1 - (SS)22) = (G2 - 1)

d2 = d S*12S12 nth(1,) + c2 c2 = d B22()

222 = d2 d2 - d2 d2 = d B22() ( B22()+ 1 )

= d (G2 nth (1,)+ G2 - 1)(G2 nth (1,)+ G2)

If the power gain is = G2 then we have:

222 = d (nth (1,)+ - 1)(nth (1,)+ ) ~ 2(nth (1,)+ 1)2

for >> 1 and expressed in uncertainty in number of photons

In other words, there is an uncertainty of 1 photon per unit

SS = =

Noise: Gain vs. No gain

Noise with gain should be equal to noise without gain for = 1

222 = d (nth (1,)+ - 1)(nth (1,)+ ) = nth(nth + 1)

for = 1

Same as noise without gain:

ii2 = d nth() (nth()+ 1)

Difference - add ( - 1) to first termmultiply ‘zero point’ energy by

Noise: Gain

22 ~ (nth (1,)+ 1)

expressed in power referred to amplifier input, multiply by theenergy per photon and divide by gain,

22 ~ h(nth (1,)+ 1)

Looks like limit of high nth

Amplifier contribution - set nth = 0

22 ~ h = kTn

or Tn = h/k (no factor of 2!)

Noise: Gain

What happens to the photon statistics?

No gain: Pin = n hand in = h n(1+n) /( )

(S/N)0 = Pin /in = n/(1+n)

With gain: Pin = n hand in = h (1+n) /( )

(S/N)G = Pin /in = [n/(1+n)]

(S/N)0/(S/N)G = (1+n)/n

Incoherent and Coherent Sensitivity ComparisonIncoherent and Coherent Sensitivity Comparison

Implementation:Spectroscopy experiment: Front end

Spectroscopy experiment: Back end FTS on chipPhase shifting FTS on a chipDo this in microstrip and divide all path lengths by dielectric, Problem - signal loss in microstripOK in mm-wave - Nb stripline, submm - MgB2?Also - PARADE’s filters work at submm (patterned copper)

180

X N

Power divider

Implementation:Spectroscopy experiment: Front end

Spectroscopy experiment: Back end filter bank on chip

Problem: Size

BPFBPFBSF

BPFBPFBSF

Implementation:Spectroscopy experiment sensitivity: (Zmuidzinas, in preparation)Each detector measures:Total power in band S(n) = d I () cos(2xn/c)/N N = number of lags = number of filter bands

Each detector measures signal to noise ~ d I ()/N

Then take Fourier transform of signals to obtain the frequency spectrum:

R() = i S(n)cos(2ixn/c) cos(2xn/c)

If the noise is uncorrelated• Dominated by photon shot noise (low photon occupation number)• Dominated by detector noise

Then the noise from each detector adds incoherently:

Each band has signal to noise ~ I ()/N

For filter bank (divide signal into frequency bands before detection):Each band has signal to noise ~ I ()/FTS is worse by N !

x

“Butler Combiner”

… X N

Power divider

Solution: Butler combiner (not pairwise)

2x 3x 4x

All lags combined on each detector:

Signals on each detector cancel except in a small bandLike a filter bank but more flexible:• Can modify phases to give different filters• Can add phase chopping to allow “stare modes”• In the correlated noise limit with phase chopping, each detector measures entire band signal - redundancy

Instrumentation:

Imaging interferometer: Front end

OMT 180

Imaging interferometer: Back end

Single moded beam combiner like second part of spectrometer interferometer(e.g. use cascade of magic Tees), n=N

Must be a type of Butler combiner (as spectrometer) to have similarsensitivity to focal plane array

180

Noise: Multiple modes

Case 1: N modes at entrance, N modes at detectorfully filled with incoherent multimode source (I.e. CMB)Noise in each mode is uncorrelated -

ii2 = N d nth() (nth()+ 1)

where nth() is the occupation number of each mode

Case 2: 1 mode at entrance, split into N modes that areall detected by a single multi-mode detector - must getsingle mode noise. Doesn’t work if we set = 1/N

ii2 = N d nth() (nth()+ 1) ~ (1/N) d nth() (nth()+ 1)

Therefore noise in ‘detector’ modes must be correlatedbecause originally we had only 1 mode

Noise: Multiple modes

Resolution: Depending on mode expansion, either noise is fullycorrelated from one mode to another or it is uncorrelated.

General formula: Mode scattering matrix

2 = d Bop (Bpo + op ) where o,p are mode indicesO,p

Two types of mm/submm focal plane architectures:

SCUBA2PACSSHARC2

BOLOCAMSCUBAPLANCK

Filter stackBolometer array

IR FilterAntennas (e.g. horns)X-misson line

Detectors

Bare array Antenna coupled

Microstrip Filters

Mm and submm planar antennas:

Quasi-optical (require lens):

Twin-slotLog periodic

Coupling to waveguide (require horn):

Radial probeBow tie

Pop up bolometers: Also useful as modulating mirrors...

SAFIR BACKGROUNDSAFIR BACKGROUND

Photoconductor(Semiconductoror superconductorbased):

Bolometer(Thermistor issemiconductoror supercondcutorbased):

Excitedelectrons

Photon

Current

+V

EM wave

Change in R

+V, I

I

Metal film

Phonons

Ther

mis

tor

Basic IR Bolometer theory:

S (V/W) ~ IR/G

R=R(T)is 1/R(dR/dT)I~constantG=Thermal conductivity

NEP = 4kT2G + eJ/S

Time constant = C/GC = heat capacity

Fundamentally limited by achievableG, C - material properties, geometry

Silicon nitride “spider web”bolometer:Absorber and thermal isolationfrom a mesh of 1 mx4 mwide strands of Silicon NitrideThermistor = NTD Germaniumor superconducting film

Bolometers at X-ray and IR:

C

ToG INT

G EXTX-ray

To

T

V ,

TBOLO

BOLO

TIME

= C/G

C

ToG INT

G EXTIR

To

TBOLO

TIME

Teq

Detector Audio Z Readout B-field Coupling-----------------------------------------------------------------------------

Absorber and thermometer independent (thermally connected)

Bolo/TES ~ 1 Ohm SQUID No? Antenna orDistributed

Bolo/Silicon ~ 1 Gohm CMOS No Antenna orDistributed

Bolo/KID ~ 50 Ohms HEMT No Antenna orDistributed

Absorber and thermometer the same

HEB ~ 50 Ohms ?? No Antenna

CEB ~ 1 kOhm ?? No Antenna

Bolometer characteristics:

ThermistorsThermistors Semiconductors - NTD GeSemiconductors - NTD Ge Superconductors - single layer or Superconductors - single layer or

bilayersbilayers Junctions (e.g. SIN, SISe)Junctions (e.g. SIN, SISe)

Superconducting thermometers: monolayers, bilayers,multilayers

Some examples -

Material Tc Reference----------------------------------------------------------Ti/Au <500 mK 30 SRON

Mo/Au < 1 K 300 NIST, Wisconsin,Goddard

Al/Ti/Au < 1 K 100 JPL

W 60-100 mK UCSF

PROTOTYPE SINGLE PIXEL - 150 GHzSchematic:

Waveguide

Radial probe

Nb Microstrip

Silicon nitride

Absorber/termination

TESThermal links

Similar to JPL design, Hunt, et al., 2002 but withwaveguide coupled antenna

PROTOTYPE SINGLE PIXEL - 150 GHzDetails:

Radial probe

Absorber - Ti/Au: 0.5 / - t = 20 nmNeed total R = 5-10 w = 5 m d = 50 m Microstrip line: h = 0.3 m, = 4.5 Z ~ 5

TESThermal links

R represents loss along the propagation path can be surface conductivity of waveguide or microstrip lines

G represents loss due to finite conductivity between boundaries = 1/R in a uniform medium like a dielectric

Z = (R+iL)/(G+iC)

For a section of transmission line shorted at the end: G= 1/R

Z = (R+iL)/(1/R+iC) = (R2+iRL)/(1+iRC) Z = (R2+iLR)/(1+iRC) = (R2+ZLR)/(1+R/ZC)

Example - Think of it as a lossy transmission line:

CR

GL

Example - impedances of transmission lines

Z = (R2+iLR)/(1+iRC) = (R2+ZLR)/(1+R/ZC)So we want ZL < R and ZC > R for good matching

Calculate impedance of C, L for 50 m section of microstrip w = 5 m, h = 0.3 m, = 4.5 Z ~ (h/2w) 377/ ~ 5 0 is magnetic permeability: free space = 4 10-7 H m-1

0 is the dielectric constant: free space = 8.84 10-12 F m-1

d = 50 mL ~ 0(d h)/2w ~ 1.5 m × ~ 2 × 10-12 H C ~ (d 2w)/h ~ 9 mm × 0 ~ 8 × 10-14 F

ZL = L = 2(150 GHz) 2 10-12 H ~ 2 ZC = 1/C = 1/2(150 GHz) 8 10-14 F ~ 13

MULTIPLEXED READ-OUTMULTIPLEXED READ-OUTTDM and FDMTDM and FDM

Why TES are good:

1. Durability - TES devices are made and tested for X-ray to last years without degradation2. Sensitivity - Have achieved few x10-18 W/Hz at 100 mK good enough for CMB and ground based spectroscopy3. Speed is theoretically few s, for optimum bias still less than 1 ms - good enough4. Ease of fabrication - Only need photolithography, no e-beam, no glue5. Multiplexing with SQUIDs either TDM or FDM, impedances are well matched to SQUID readout6. 1/f noise is measured to be low7. Not so easy to integrate into receiver - SQUIDs are difficult part8. Coupling to microwaves with antenna and matched heaterthermally connected to TES - able to optimize absorption and readout separately

Problems:Problems: Saturation - for satellite and balloons.Saturation - for satellite and balloons. Excess noise - thermal and phase transition?Excess noise - thermal and phase transition? High sensitivity (NEP<10High sensitivity (NEP<10-18-18) requires temperatures < 100 mK) requires temperatures < 100 mK Solutions:Solutions: Overcome saturation by varying the thermal conductivity of detector - superconducting heat linkOvercome saturation by varying the thermal conductivity of detector - superconducting heat link Thermal modelling and optimisationThermal modelling and optimisation Reduce slope of superconducting transitionReduce slope of superconducting transition Better sensitivity requires reduced G - HEBs?Better sensitivity requires reduced G - HEBs?

Problems: Excess Noise - PhysicsProblems: Excess Noise - Physics

Width of supercondcuting transition dependsWidth of supercondcuting transition dependson mean free path of Cooper pair and geometry of TESon mean free path of Cooper pair and geometry of TES

Centre of transition = RCentre of transition = RNN/2 = 1 Cooper pair with MFP = D/2/2 = 1 Cooper pair with MFP = D/2Derive equivalent of Johnson noise using microscopic approach with random variation in mean free path of Derive equivalent of Johnson noise using microscopic approach with random variation in mean free path of

Cooper pairCooper pairGives a noise term proportional to dR/dTGives a noise term proportional to dR/dT

Problems: Sensitivity - Requires very low temperatureProblems: Sensitivity - Requires very low temperature

Fundamentally - a bolometer is a square-law detectorFundamentally - a bolometer is a square-law detectorTherefore, it is a linear device with respect to photon fluxTherefore, it is a linear device with respect to photon flux

Response (dR) is proportional to change in input power (dP)Response (dR) is proportional to change in input power (dP)

In order to count photons, it is better to have a non-linearIn order to count photons, it is better to have a non-lineardevice (I.e. digital) - photoconductordevice (I.e. digital) - photoconductor

Hot Electron Bolometer(HEB)

-Tiny superconducting stripacross an antenna(sub micron)- DC voltage biases the stripat the superconductingtransition-RF radiation heats electronsin the strip and creates a normalhot spot-Can be used as a mixer oras a direct detector

Minimum C (electrons only)Sensitivity limited by achievable G

Detector Audio Z Readout B-field Coupling-----------------------------------------------------------------------------

BIB Ge > 1012 OhmCIA No Distributed

QD phot. ~ 1 Gohm QD SET Yes/No Antenna

QWIP ~ 1 Gohm CIA No Not normalincidence

SIS/STJ ~ 10 kOhm FET? Yes Antenna

SQPT ~ 1 kOhm RF-SET Yes Antenna

KID ~ 50 Ohm HEMT No Distributedor antenna

Photoconductor characteristics:

Detectors: Semiconductor Photoconductor

Pure crystal - Si, Ge, HgCdTe, etc.Low impuritiesLow level of even doping

Achieve - ‘Freeze out’ of dopantsIncoming radiation excites dopants into conduction bandThey are then accelerated by electric field and create morequasiparticles measure current

e

V,I

Detectors: Semiconductor BIB Photoconductor

Method of controlling dark current while increasing dopinglevels to increase number of potential interactionsTake standard photoconductor and add undoped part on end

Achieve - ‘Freeze out’ of dopantsIncoming radiation excites dopants into conduction bandThey are then accelerated by electric field and create morequasiparticles measure current

V,I

e

Detectors: Quantum Well Infrared Photoconductor

Easier method of controlling dark current and increasingthe number of potential absorbers - use potential barriersThin sandwich of amorphous semiconductor materialwith low band gapCreate 2-d electron gasEnergy levels are continuous in x, y but have steps in z

AlGaAs

GaAs

AlGaAs

Detectors: Quantum Well Infrared Photoconductor

Solve for energy levels using Schrodinger:

Particle in a box -

H = E, H = p/2m + VV = 0 x, y and for 0<z<a (I.e. within well)V = V x, y and for z<a or z<0 (I.e. outside well)

Solve for wavefunctions within well:

Simple solution:

= A ei(kxx+ kyy) sin(nz/a)

Has continuous momentum in x, y, discrete levels in z

Detectors: Quantum Well Infrared Photoconductor

Advantages over standard bulk photoconductor -

1. Can have large carrier density within quantum wellwith low dark current due to well barriers - high quantumefficiency

2. Can engineer energy levels within well to suit wavelengthof photons - geometry determined rather than material

Detectors: Quantum Dots

Confinement in 3 dimensions gives atomic-like energy levelstructure:

= A sin(lx/a) sin(my/b) sin(nz/c)

E2 = (22/2m*)(l2/a2 + m2/b2 + n2/c2)

Useful for generation of light in a very narrow frequencyband - I.e. quantum dot lasers

Also could be useful for absorption of light in narrowfrequency band

Superconducting Tunnel Junctions: X-ray-IR

Two slabs of superconductor separatedby an insulator

photons excite quaiparticles that tunnelthrough the junction

n(e-)/ ~ h/E

Superconducting photoconductor!With band gap = 1 meV vs. 1 eVfor semiconductors (or 100 meV fordonor level)

Sensitivity limited by:

1. Quantum efficiency2. Dark current

Speed generally not a concern

Readout for superconducting junctions: SETs? RF-SET (e.g. Schoelkopf)

Work for -SIS and SINIS - Antenna coupled photodetectorsSQPT - Antenna coupled photoconductors read out with SETs> 1 e-/photonbut are delicate and require e-beam lithography

Types of antennas/absorbers:

1. Twin-slot - planar quasi optical - JPL, Berkeley

2. Finline - wide band coupling to waveguide - Cam

3. Radial probe - wideband coupling to waveguide - Cam, JPL

4. Spider-web - Low cosmic ray cross section, large areaabsorber - JPL

5. Silicon PUDs - Filled area arrays - SCUBA2, NIST/Goddard

The readout problem - low noise multiplexing technologies:

1. SQUIDs - noise temperature < 1 nK Inductively coupled amplifier 10s of MHz bandwidth

2. FETs - noise temperature < 0.1 K Capacitively coupled amplifier 10s of kHz bandwidth

3. SETs - noise temperature < 1 uK Capacitively coupled amplifier GHz bandwidth

4. HEMTs - noise temperatures < 1 K Capacitively coupled amplifier 10s GHz bandwidth

Conclusions:

Many possible new technologies around

Multiplexable bolometers already satisfy criteria for imagingmissions

New photoconductors (semiconductor or superconductor)or HEBs probably needed for higher sensitivity instruments,probably antenna coupled