1. Detection of Electromagnetic Radiation III: Photon Noise Phil Mauskopf, University of Rome 19 January, 2004

2. Example - impedances of transmission lines Parallel strips - 5  m wide, 0.3  m separation,  = 4.5 (SiO) Z ~ h/w · 377/   = 10  Microstrip line - one plate inifnitely wide therefore impedance is slightly lower Z ~ 5  Coaxial cable - Radius of outer conductor = R, inner = r Z ~ ln(R/r) (1/2  ) · 377/   ~ 60  · ln(R/r) /   For R/r = 2.3,  = 1, Z ~ 50 

3. Scattering matrix: Ports Ports are just points of access to an optical system. Each port has a characteristic impedance Any optical system can be described completely by specifying all of the ports and their impedances and the complex coefficients that give the coupling between each port and every other port. For an optical system with N ports, there are NxN coefficients necessary to specify the system. This NxN set of coefficients is called the Scattering Matrix

4. Scattering matrix: S-parameters The components of the scattering matrix are called S-parameters. S 11 S 12 S 13 S 14... S =

5. Scattering matrix: Lossless networks - unitarity condition, conservation of energy For a network with no loss, the S-matrix is unitary: S S = I This is just the expression of conservation of energy, For a two port network: 1 + R = T

6. Scattering matrix: Examples - two-port networks Dielectric interface: S = This is because R = (Z 1 -Z 2 )/(Z 1 +Z 2 ) Going from lower to higher impedance Z 1  Z 2 gives the opposite sign as going from higher to lower impedance. T R -R T

7. Scattering matrix: Power divider How about 3-port networks? Can we make an optical element that divides the power of an electromagnetic wave in half into two output ports? Guess: What is the S-matrix for this circuit? What is the optical analogue? Z1Z1 2 Z 1

8. Scattering matrix: 4-port networks - 90 degree hybrid A B (A+iB)/  2 (A-iB)/  2 S = 0 0 1 1 0 0 i -i 1 1 0 0 i -i 0 0

9. Scattering matrix: 4-port networks - 90 degree hybrid Optical analogue: Half power beam splitter A B (iA+B)/  2 (A+iB)/  2 S = = = 0 0 1 i 0 0 i 1 1 i 0 0 i 1 0 0 0  2  2 0 0 0 1 1 0 0 i -i 1 1 0 0 i -i 0 0 With a 90  phase shift on port 2

10. Scattering matrix: 4-port networks - 180 degree hybrid A B (A+B)/  2 (A-B)/  2 S = = 0 0 1 1 0 0 1 -1 1 1 0 0 1 -1 0 0 0  3  3 0 In general lossless scattering matrices  SU(n)

11. Resistive elements in transmission line - loss: C R G L 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 +i  L )/( G +i  C ) Z has real part and imaginary part. Imaginary part gives loss

12. Resistive elements in transmission line - loss: You can replace loss terms in the scattering matrix (which makes it non-unitary) with additional ports that account for the lost signal. C R G L C G L Z0Z0 Z0Z0 Z=R 

13. Optics: Direct coupling to detectors (simplest) Need to match detector to free space - 377  One way to do it is with resistive absorber - e.g. thin metal film Transmission line model: Converts radiation into heat - detect with thermometer = the famous bolometer! How about other detection techniques? Impedance mismatch? - Non-destructive sampling - sample voltage - high input Z - sample current - low input Z Both cases the signal is reflected 100% E.g. JFET readout of NTD, SQUID readout of TES +-+- Z0Z0 R=Z 0

14. Optics: Direct coupling to detectors (simplest) Without an antenna connected to a microstrip line, the minimum size of an effective detector absorber is limited by diffraction Single mode - size ~ 2 The number of modes in an optical system is limited by the total throughput: n(modes) = A  / 2 The throughput is limited by the coupling between optical elements

15. Two types of mm/submm focal plane architectures: SCUBA2 PACS SHARC2 BOLOCAM SCUBA PLANCK Filter stack Bolometer array IR Filter Antennas (e.g. horns) X-misson line Detectors Bare array Antenna coupled Microstrip Filters

16. Mm and submm planar antennas: You can have single mode and multi-mode antennas - e.g. scalar feed vs. winston Quasi-optical (require lens): Twin-slot - small number of modes Log periodic - multimode Coupling to waveguide (require horn): Radial probe Bow tie

17. Optics: Modes and occupation number A mode is defined by its throughput: A  = 2 The occupation number of a mode is the number of photons in that mode per unit bandwidth For a single mode source emitting a power, P in a bandwidth , with an emissivity,  The occupation number is: N = (P/2h )(1/   ) For a blackbody source at temperature, T, this is just the Bose-Einstein term: N = 1/(exp(h /kT)-1)

18. Optics: Modes and occupation number N = (P/2h )(1/   ) = 1/(exp(h /kT)-1) Low frequencies (R-J limit): h /kT << 1 N  kT/h >> 1 = High photon occupation number Wave noise dominated = Zero point fluctuations High frequencies (Wein limit): h /kT >> 1 N  exp(-h /kT) << 1 = Low photon occupation number Shot noise dominated = Johnson-Nyquist noise CMB at millimetre-wavelengths: h /kT ~ 1 so it is in between low and high occupation number!

19. 1990s: SuZIE, SCUBA, NTD/composite 1998: 300 mK NTD SiN PLANCK: 100 mK NTD SiN Wave noise Shot noise

20. Noise: Formulae The 1  uncertainty in the optical power is:  p = h   N(1+  N) /(  ) N = mode occupation number  = efficiency  = integration time = central frequency  = bandwidth Limits:  N >> 1   p = h N  /  = P d /(   )  N << 1   p = h  N  /  =  P d h /  / 

21. Noise: Derivation Take an N-port optical system with an NxN scattering matrix, S ij ( ): Port labels are i = 1…N Incoming wave amplitudes are given by: a i ( ) Outgoing wave amplitudes are given by: b i ( ) Considering only linear systems (for which the S-matrix method applies): b i ( ) =  j S ij ( ) a j ( ) or b = Sa S = probability amplitude for photon entering port j to exit at port i

22. Noise: Derivation Start with simplest network - single port = two terminals Z R Port, P has characteristic impedence = Z = R. Therefore there are no reflections. We can think of this as a transmission line terminated at infinity with another resistor, R P

23. Noise: Derivation Start with simplest network - single port = two terminals Z R P Z RRpRp = Where R p is the port impedence In fact, if you use simulation packages such as ADS, they require that you terminate all ports with a characteristic impedance. If R p is infinite = open circuit then we have voltage noise If R p = 0 = short circuit then we have current noise

24. Noise: Derivation Formula for noise can be derived in (at least) two ways: 1. Brownian motion or random walk of electrons 2. Transmission line model and thermodynamics Both methods give classical solutions that are modified by quantum effects We’ll consider only the transmission line model - from Nyquist Z R2R2 R1R1 Based on the principle that in thermal equilibrium there is no average power flow

25. Noise: Derivation R2R2 R1R1 If the voltage noise from R 1 is given by V 1 then the power generated by R 1 and dissipated in R 2 is given by:  V 1 2  R 2 /(R 1 + R 2 ) 2 and the power generated by R 2 and dissipated in R 1 is given by:  V 2 2  R 1 /(R 1 + R 2 ) 2 Thermodynamics says these must be equal at all frequencies so:  V i 2   R i and  V i 2   T. Define power spectrum, S V (  ) =  V i 2   V 1 2  R 2 /(R 1 + R 2 ) 2  V 2 2  R 1 /(R 1 + R 2 ) 2

26. Noise: Derivation l Z RR Suppose R 1 = R 2 = Z Z is a lossless transmission line =  L/C Wave velocity in the transmission line v = 1/  LC The thermal power delivered to the transmission line from either R 1 or R 2 in a frequency interval d  /2  is: dP = (1/4R) S V (  ) d  /2  For a transmission line of length, l the energy stored in the transmission line is equal to the power emitted x travel time = l /v: dE = dP  t  2 = ( l /2Rv) S V (  ) d  /2 

27. Noise: Derivation Z RR If we suddenly cut the lines at the end of the transmission line, a certain amount of energy is trapped in standing waves: dE = dP  t  2 = ( l /2Rv) S V (  ) d  /2  Expanding the standing waves in modes gives: m = (d  /2  )/(v/2 l ) Equipartition theorem: average energy per mode = kT dE = mkT = (d  l /  v)kT = ( l /2Rv) S V (  ) d  /2   S V (  ) = 4kTR

28. Noise: Derivation Quantum Mechanics I: Include Bose-Einstein statistics Quantum mechanically, the average thermal energy per mode is given by the energy per photon times the photon occupation number: dE = m  n th = (d  l /  v)  /(exp(  /kT)-1) Setting this equal to the energy stored in the transmission line: dE = ( l /2Rv) S V (  ) d  /2  gives, S V (  ) = 4  R/(exp(  /kT)-1) = 4  R n th

29. Noise: Derivation: 4-terminals = 2 ports Impedence representation and S-matrix representation: ZS V 1 (t) I 2 (t)I 1 (t) V 2 (t) Impedance matrix, Z:Scattering matrix, S: V 1 Z 11 Z 12 I 1 b 1 S 11 S 12 a 1 V 2 Z 21 Z 22 I 2 b 2 S 21 S 22 a 2 = a1a1 b1b1 a2a2 b2b2 = Where a i represents the amplitude of incoming waves and b i represents the amplitude of outgoing waves

30. Noise: Derivation: 4-terminals = 2 ports Generalize to multiple ports: Obtain noise correlation matrix Z ij e i S ij  i V 1 (t) I 1 (t) a1a1 b1b1 V 2 (t) I 2 (t) V n (t) I n (t) a2a2 b2b2 anan bnbn S  i  j * (  ) = (1-S S) ij kT S e i e j * (  ) = 2(Z+Z ) ij kT

31. Noise: Equations Include Bose-Einstein statistics and obtain the so-called ‘Classical’ formulae for noise correlations: S  i  j * (  ) = (1-S S) ij kT  (1-S S) ij  /(exp(  /kT)-1) S e i e j * (  ) = 2(Z+Z ) ij kT  2(Z+Z ) ij  /(exp(  /kT)-1) Relations between voltage current and input/output waves:  1/4Z 0 (V i +Z 0 I i ) = a i  1/4Z 0 (V i - Z 0 I i ) = b i or V i =  Z 0 (a i + b i ) I i = 1/  Z 0 (a i - b i )

32. 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 there is still energy. Add this energy to the ‘Semiclassical’ noise correlation matrix and we obtain: S e i e j * (  ) = 2  (Z+Z ) ij coth(  /2kT) = 2  R (2n th +1) S  i  j * (  ) =  (1-S S) ij coth(  /2kT) =  (2n th +1)

33. Noise: Derivation - Quantum mechanics This is where the Scattering Matrix formulation is more convenient than the impedance method: Replace wave amplitudes, a, b with creation and annihilation operators, a, a, b, b and impose commutation relations: [a, a ] = 1Normalized so that  a a  = number of photons [a, a ] =  Normalized so that  a a  = Energy Quantum scattering matrix:b =  a + c Since [b, b ] = [a, a ] =  then the commutator of the noise source, c is given by: [c, c ] =  (I - |  | 2 )

34. Noise: Quantum Mechanics III: Calculate Quantum Correlation Matrix If we replace the noise operators, c, c that represent loss in the scattering matrix by a set of additional ports that 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 n th = (I - S S) ij n th So we have lost the zero point energy term again...

35. Noise: Quantum Mechanics IV: Detection operators An ideal photon counter can be represented quantum mechanically by the photon number operator for outgoing photons on port i: d i =  b i b i  which is related to the photon number operator for incoming photons on port j by:  b i b i  =  (  n S * in a n )(  m S im a m )  +  c i c i  =  d B ii ( )  (  n S * in a n )(  m S im a m )  =  n,m S * in S im  a n a m   a n a m  = n th (m, )  nm which is the occupation number of incoming photons at port m

36. Noise: Quantum Mechanics IV: Detection operators Therefore d i =  m S * im S im n th (m, ) +  c i c i  =  d B ii ( ) Where:  c i c i  = (I - S S) ii n th The noise is given by the variance in the number of photons:  ij 2 =  d i d j  -  d i  d i  =  d B ij ( ) ( B ij ( )+  ij ) B ij ( ) =  m S * im S jm n th (m, ) +  c i c j  =  m S * im S im n th (m, ) + (I - S S) ij n th (T, ) Assuming that n th (m, ) refers to occupation number of incoming waves, a m, and n th (T, ) refers to occupation number of internal lossy components all at temperature, T

37. Noise: Example 1 - single mode detector No loss in system, no noise from detectors, only signal/noise is from port 0 = input single mode port: S im = 0 for i, m  0 S 0i = S i0  0 d i =  d S * i0 S i0 n th (0, ) +  c i c i  =  d B ii ( )  ii 2 =  d i d ji  -  d i  d i  =  d B ii ( ) ( B ii ( )+  ii ) For lossless system -  c i c i  = 0 and  ii 2 =  d B ii ( ) ( B ii ( )+  ii ) =  d S i0 2 n th ( ) (S i0 2 n th ( )+ 1) Recognizing S i0 2 =  as the optical efficiency of the path from the input port 0 to port i we have:  ii 2 =  d  n th ( ) (  n th ( )+ 1) express in terms of photon number

38. Noise: Gain - semiclassical Minimum voltage noise from an amplifier = zero point fluctuation - I.e. attach zero temperature to input: S V (  ) = 2  R coth(  /2kT) = 2  R (2n th +1) when n th = 0 then S V (  ) = 2  R Compare to formula in limit of high n th : S V (  ) ~ 4 kT N Rwhere T N  Noise temperature  Quantum noise = minimum T N =  /2k

39. Noise: Gain Ideal amplifier, two ports, zero signal at input port, gain = G: S 11 = 0 no reflection at amplifier input S 12 = G gain (amplitude not power) S 22 = 0 no reflection at amplifier output S 21 = 0 isolated output Signal and noise at output port 2: d 2 =  d S * 12 S 12 n th (1, ) +  c 2 c 2  =  d B 22 ( )  22 2 =  d 2 d 2  -  d 2  d 2  =  d B 22 ( ) ( B 22 ( )+ 1 )  c 2 c 2  = (1 - (S S) 22 )n th (T, ) What does T, n th mean inside an amplifier that has gain? Gain ~ Negative resistance (or negative temperature) n amp (T, ) = -1/ /(exp(-  /kT)-1)  -1 as T  0

40. Noise: Gain 0 0 0 G 0 0 G 0 0 0 0 G 2  c 2 c 2  = -(1 - (S S) 22 ) = (G 2 - 1) d 2 =  d S * 12 S 12 n th (1, ) +  c 2 c 2  =  d B 22 ( )  22 2 =  d 2 d 2  -  d 2  d 2  =  d B 22 ( ) ( B 22 ( )+ 1 ) =  d (G 2 n th (1, )+ G 2 - 1)(G 2 n th (1, )+ G 2 ) If the power gain is  = G 2 then we have:  22 2 =  d (  n th (1, )+  - 1)(  n th (1, )+  ) ~  2 (n th (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 S S = =

41. Noise: Gain  22 ~  (n th (1, )+ 1)  expressed in power referred to amplifier input, multiply by the energy per photon and divide by gain,  22 ~ h (n th (1, )+ 1)  Looks like limit of high n th Amplifier contribution - set n th = 0  22 ~ h  = kT n  or T n = h /k (no factor of 2!)

42. Noise: Gain What happens to the photon statistics? No gain: P in = n h  and  in = h   n(1+n) /(  ) (S/N) 0 = P in /  in =  n  /(1+n) With gain: P in = n h  and  in = h  (1+n) /(  ) (S/N) G = P in /  in =  [n/(1+n)] (S/N) 0 /(S/N) G =  (1+n)/n

43. Noise: Interferometry What if we want to measure the spectrum of incoming radiation? Two ways: 1. Divide signal into N frequency bands using filters and detect the photons with N detectors 2. Divide signal power by N and detect autocorrelation of the input signal with N lags in N detectors