Specifications for an ideal bolometer array using bare pixels or feedhorns at the 30m telescope

S.Leclercq

13/10/2008

4 bands available at the 30mATM opacity model at Pico Veleta, for winter (260K) and summer (300K) with good weather (1mm of water vapour) and bad weather (7mm)

6.2"3450.87

8.8"2401.25

14.5"1462.05

22.6"943.2

Airy HPBW

ν(GHz)

λ(mm)

Centre of the bands for a maximal width, and corresponding size of the FWHM diffraction pattern

Beams and efficienciesSmall angle approximation of the image plane "angular radius":

Relative power:

Pattern from diffraction through a hole (Airy beam) (I = relative intensity)

I t r t,( )1

it0 t( ) 0

1

pt p( ) J0 r p⋅( )⋅ p⋅⌠⌡

d

2

⋅:=

I a r( )2 J1 r( )

r( )

2

:=

rπ θ2

D

λ⋅:=

Effect of feed horn: tapered beam

Effect of a bare pixel size: smoothed detected beam(u·ka is the pixel size in unit of λ/D)

I p r u,( )

u− k a⋅π

2⋅

u k a⋅π

2⋅

ρI a r ρ−( )⌠⌡

d

u− k a⋅π

2⋅

u k a⋅π

2⋅

ρI a ρ−( )⌠⌡

d

:=

0 0.5 1 1.50

0.20.40.60.8

11

0

tg p 10dB,( )

1.50 p

Effective aperture (telescope area) at zero incident angle: Ae=Pcollected(0)/(Pincident/A)=εa·A,

εa is the aperture efficiency.

Conservation of energy

0

4 πΩI

⌠⌡

dλ

2

A e:=

A e

L r( )1

2 0

r

ρρ I ρ( )⋅⌠⌡

d⋅:=

Beam efficiency= relative power at the main beam radius (1stdark ring of the Airy beam): Beff = L(rmb)

Forward efficiency= relative power from the 2π steradian plane in front of the telescope: Feff = L(r2π)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51 .10

4

1 .103

0.01

0.1

11

0.0001

I a q r mba⋅( )I tg q r mba⋅ Te,( )I p q r mba⋅ 0.5,( )ε tsg Te( ) I tg q r mba⋅ Te,( )⋅

50 q

Beams and efficienciesRelative powersBeams

0.4 0.2 0 0.2 0.4 0.6 0.80.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

11

0.5

L a 10q

r mba⋅

L ntg 10q

r mba⋅ Te,

L np 10q

r mba⋅ 0.5,

10.4− q

AiryTapered(feedhorn*)Smoothed (bare pixel)

Beff

Feff

εa

(*on this plots the taper level at dish edge is Te=10dB)

Beams on the 30m: effect of surface errorsDeformation parametersof the telescope main mirror: steepness factor (R), aperture efficiency at long wavelength (ε0), RMS values of the surface deformation height (σh), and their correlation lengths (de)

R 0.85:=σh

0.055

0.055

0.055

mm:= de

2.5

1.7

0.3

m:=ε0 0.62:=

3 components for the 30m: large scale deformations, panel frame misalignments, and panel deformations [A.Greve].

Antenna Tolerance Theory:Theoretical beam without deformation (Imb) summed to gaussian error beams defined by the deformation parameters

I eg r a, r b, i,( )v

ai v, exp

r

2ρ r b( )i v,

2

−

⋅

∑:=

2 models for the beam:Empirical Gaussians:Sum of gaussians, with amplitudes (a) and widths (rb) linked to measures (am) and (θm) with different detectors at the wavelengths λc

I T r n, I mb, eT, a0, ae, ε, σ,( ) a0 n σ,( ) I mb r eT,( )⋅

ι

ae n σ, ε, ι,( ) exp

r d eι⋅

2 D⋅

2

−

⋅∑+:=

ae n σ, ε, ι,( ) 1

ε1 exp σ ϕ n σ,( )2

−−

⋅

d eι

D

2

⋅:=

θ m

27.5

16

10.5

7.2

300

175

125

80

410

280

180

150

2500

1500

950

600

as:= a m

1

1

1

1

0.0004

0.0015

0.003

0.004

0.0002

0.0007

0.001

0.002

0.00003

0.00005

0.00008

0.00025

:=

Ruze :ηa(ν)=ε0 exp(-Σ(σϕ2))

λ c

3.4

2.0

1.3

0.86

mm:=

σ ϕ ν σ h,( ) R4π ν⋅

c⋅ σ h⋅:=

=> can be calculated for any λ

0 20 40 60 80 1001 .10

6

1 .105

1 .104

1 .103

0.01

0.10.1

0.000001

I a q mb⋅( )

I eg q mb⋅ a m, r m, 0,( )I eg q mb⋅ a m, r m, 1,( )I eg q mb⋅ a m, r m, 2,( )I eg q mb⋅ a m, r m, 3,( )I nTGb q mb⋅ 0, I t, ε bt,( )I nTGb q mb⋅ 1, I t, ε bt,( )I nTGb q mb⋅ 2, I t, ε bt,( )I nTGb q mb⋅ 3, I t, ε bt,( )

1000 q

0 1 2 3 4 51 .10

4

1 .103

0.01

0.1

11

0.0001

I a q mb⋅( )

I eg q mb⋅ a m, r m, 0,( )I eg q mb⋅ a m, r m, 1,( )I eg q mb⋅ a m, r m, 2,( )I eg q mb⋅ a m, r m, 3,( )I nTGb q mb⋅ 0, I t, ε bt,( )I nTGb q mb⋅ 1, I t, ε bt,( )I nTGb q mb⋅ 2, I t, ε bt,( )I nTGb q mb⋅ 3, I t, ε bt,( )

50 q

1 0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0

L a 10q

mb⋅( )

L neg 10q

mb⋅ am, r m, 0,

L neg 10q

mb⋅ am, r m, 1,

L neg 10q

mb⋅ am, r m, 2,

L neg 10q

mb⋅ am, r m, 3,

L TGo 10q

mb⋅ 0, L t, ε bt, σ h,

L TGo 10q

mb⋅ 1, L t, ε bt, σ h,

L TGo 10q

mb⋅ 2, L t, ε bt, σ h,

L TGo 10q

mb⋅ 3, L t, ε bt, σ h,

21− q

Beams on the 30m: effect of surface errorsRelative powersBeams Airy

Real beam λ=3.4mmReal beam λ=2.0mmReal beam λ=1.3mmReal beam λ=0.86mm

ηa=

Dash lines = EGSolid lines = ATT

61453516

% Beff=

73544219

%

Feff= %92908675

All efficiencies degrading the transmission from the sky to the instrument

Components of the aperture efficiency from measures conducted last year [C.Thum]

Atmospheric transmission (ta) depends on the opacity (τ): ta = exp(-ττττ(νννν,v)) (see ATM curves)

Attention:

ηa affects point sources ; for extended sources many diffraction pattern are superposed, so that the efficiency of the optical chain is closer to Feff.

Ground emissivity: eg = 30% e = 1 - t

Transmission for 6 mirrors 1% ohmic loss + 1 cryostat window: tt = (1-0.01)6 0.95 = 89%

Transmission for 7 cold filters (2@77K,5@<4K, including Thermal blockers, Edge filters and Band Pass), 95% transmissible each: tf = 0.957 = 70%

Detector quantum efficiency and other factors: to = 85%More realist (bad) cold part [Desert]: tfD = 13%

Detectors architectures2Fλ "efficient" round

feedhorns array 0.5Fλ square bare pixels filled array

Pixel aperture efficiency: η b

15

13

9

5

%=

Central ν:

0.48 due to 5% inter pixel gap

η hiε tsg T e 2( )( ) exp

ι

σ ϕ ν i σ hι,

2∑−

⋅:=η biL T 0.48 ν i,( ):=

η h

73

63

42

21

%=

Effective throughput(sees the sky) [Griffin]:

AΩ s υ λ,( ) υ λ2

⋅:=

υ b u( )π4

u2⋅:= υ b 0.48( ) 0.18= υ h u( ) ε sg T e u( )( ):= υ h 2( ) 0.84=

Number of pixelsfor 2 fields of view:

FOV = (4.8' 10')n mb

26

41

67

96

55

85

140

201

= n mh

7

10

17

24

14

21

35

50

=N b

538

1312

3528

7283

2336

5693

15312

31609

= N h

39

95

255

526

169

411

1105

2281

=

Instrument diameter: Instrument diameter:Total square grid: Total hexagonal grid:

Global efficiency: < 50 % for extended sources

~ ηηηηb/2 for point sources

< 65 % for extended sources

~ ηηηηb/1.5 for point sources

2w

36

40

90

24

GHz=

Band width:

ν

94

146

240

345

GHz=

Collected power and dynamicsBlack body brightness

Spectral power

Power on the pixel

B T s ν,( ) 2 h⋅

po c2⋅

ν3

exph ν⋅

k T s⋅

1−⋅:= Pn T s ν, υ,( ) υ

c

ν

2

⋅ eso s≠( )

t o∏⋅

⋅ B T s ν,( )⋅:=

o s≠

P Ts υ,( )ν i wi−

ν i wi+

nPn T s ν, υ,( )⌠⌡

d:=

i

Background sources: CMB, atmosphere, ground, telescope & warm optics, nitrogen stage filters, helium stage filters and cold chamber

T CMB 2.725K:= T atm 270K:= T grd T atm:=

T tel 280K:= T nf 77K:= T cf 4 K:= T dc 0.3K:=

Benchmark sources: Jupiter, 1K Rayleigh-Jeans extended source, 1mJy point source (Jansky = 10-26 W/(m2Hz)

T j 150K:= T 1KRJ 1K:= F m 1mJy:=

Power from a point source emitting a flux F

Ppt F η,( ) F A⋅

ν i wi−

ν i wi+

νη ν( )o

t o ν( )∏⋅⌠⌡

d⋅:=

i

Pjb

12

12

25

5

11

10

14

1

pW=

Pjh

71

76

152

30

67

61

86

4

pW=

PRJb

78

84

175

36

73

68

98

5

fW=

PRJh

482

516

1055

215

453

414

592

31

fW=

Pptob

1.85

1.74

2.48

0.28

1.74

1.40

1.42

0.04

1017−

W=

Pptoh

11.2

10.6

14.9

1.5

10.6

8.5

8.5

0.2

1017−

W=

PTOTb

6

7

19

7

7

12

39

16

pW=

PTOTh

33

40

105

43

41

67

227

91

pW=

Dynb

1

1

2

6

1

2

5

53

106=

Dynh

4

4

7

19

4

6

15

168

106=

0.5 Fλbare pixel

2 Fλfeedhorn

(no)polarization: po=(1)2

PTD2

6

pW=

4 mm water vapor

1 & 7 mm wv

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.61.6

0

∆ a u( )

∆ ib u( )

∆ 1 u( )

50 u

Photon noise and spatial coherenceBose-Einstein statistics: fluctuation of the mean number of photons per mode in a phase space with g cells: σ2 = n + n2/g. First term = poissonian shot noise (gives optical noise), second term = NON-poissonian bunching noise (gives radiometer noise).

One can show 1/g = ∆/(t·dν) [Lamarre]. 1/(t·dν) is the time coherence of the beam and ∆ is the space coherence of the beam. First approach: ∆ = λ2/(AΩ), but works only if AΩ >> λ2 => incoherent beam approximation. Second approach (HBT): ∆ = 1 when the beam is much smaller than λ2 => coherent beam approximation. General expression for for a uniform detector efficiency and circular pixel:

∆ r ν,( ) 1

2r( )4 r−

r

br−

r

ar−

r

yr−

r

xI x a− y b−, ν,( )⌠⌡

d⌠⌡

d⌠⌡

d⌠⌡

d⋅:= I

Coherent branch

=> single mode detector

Incoherent branch

=> multi-mode detector

Intermediate zone

~1/u in this zone

Curves of ∆ as a function of the pixel size (u) in units of Fλ

Noise Equivalent Power

NEPp υ T, t,( ) 2

ν i wi−

ν i wi+

νh ν⋅ Pn ν υ, T, t,( )⋅⌠⌡

d⋅:=

i

NEPb υ T, t, ∆, u,( ) po

ν i wi−

ν i wi+

ν∆ u( ) Pn ν υ, T, t,( )2⋅⌠⌡

d⋅:=

i

nu10

17−W

Hz:=

Shot noise:

Bunching noise:

Approximations:

NEPpa 2 h⋅ ν⋅ P⋅:=

Convenient noise unit:

NEPbai po 2 k⋅ t⋅ T⋅⋅ P⋅:= t

NEPbacpo

2 wi

P⋅:=i

Summing several components:

NEPpn

NEPpn2

∑:= NEPpn NEPbn

NEPbn∑:= NEPbn

NEPpTb

3

4

8

6

3

5

11

8

nu= NEPbTb

3

3

6

4

3

5

12

9

nu=

NEPpTh

6

9

18

14

7

11

27

20

nu= NEPbTh

18

20

35

28

22

33

77

59

nu=

0.5 Fλbare pixel

2 Fλfeedhorn

NEPpTD2

4

nu= NEPcTD1

2

nu=

Instrumental noise per pixel:

NEPTOTD / 2

NEPTOTb / 6

NEPpix ~<1nu

NEPTOTh / 6

NEPpix ~ few nu

Shot noise: Bunching noise: Total:

NEPTOTb

4

5

10

7

5

7

16

12

nu=

NEPTOTh

19

22

40

31

23

35

81

62

nu=

NEPD2

5

nu=

Sum Nb pixels RN=NEPNb/NEP1

NEPp=2NEPc

NEPp~NEPc

RNh Nb:=

RND11 Nb⋅ 2 Nb

1.5⋅+13

:=

RNbNb Nb

1.5+2

:=

Noise Equivalent Temperature & Noise Equivalent Flux Density

NETNEP 1⋅ K

P1KRJ:=

NEPNEFD

NEP 1⋅ mJy

P1mJy:=

NEPBasic formulas:

These formulas are just describing the noise level in other units. But in general people prefer using them as sensitivity parameters. Since the NEP variation with the pixel size (or number of co-added pixel) is nor unique and depends on the pixel architecture and relative importance of shot noise and bunching noise, the only way to use these parameters as universal sensitivity indicators is to define them for a standard detector size and standard type of observed source.

The standard detector size is the beam (FWHM or main (1st dark ring)).

The source temperature is proportional to the brightness = flux/steradian => NET is well adapted to describe the sensitivity to extended sources.

The power received from a point source is proportional to the source flux => NEFD is well adapted to describe the sensitivity to point sources.

In addition people like to define these quantities in a manner that makes them directly proportional to the integration timeneeded to detect a given source at a given signal to noise ratio. This implies the introduction of 2 other factors:

The Nyquist factorfrom the relation linking the integrator bandwidth to the integration time (∆f=1/2t).

A modulation efficiencycounting for the observing mode: ηobs2=2/0.8for On The Fly with background suppression

using a pixel of reference, ηobs2=2/0.45for On-Off.

K

Hz

Jy

Hz

Noise Equivalent Temperature & Noise Equivalent Flux Density

"Sensivity" formulas:(Nb = number of pixels per beam)

K s⋅ Jy s⋅

(no pixels efficiency ηNb in P1KRJ) (pixels efficiency ηNb inside P1mJy)

NETη obs

2

NEPNb 1⋅ K

P1KRJ.Nb⋅:=NEPNb

NEFDη obs

2

NEPNb 1⋅ mJy

P1mJy.Nb⋅:=NEPNb

4 × 0.5 Fλbare pixels

2 Fλ feedhorn

NEFDb1Fλ

2.1

2.6

3.4

20.0

2.5

5.0

10.9

251.5

mJy s⋅=

NEFDh2Fλ

2.4

3.0

3.9

26.0

3.2

6.0

13.9

355.7

mJy s⋅=

NEFDD7.7

20.1

mJy s⋅=

NETb1Fλ

0.37

0.40

0.37

1.35

0.46

0.75

1.17

16.70

mK s⋅=

NETh2Fλ

0.59

0.63

0.57

2.18

0.76

1.28

2.05

29.74

mK s⋅=

NETD0.71

0.92

mK s⋅=

Integration time and mapping speedIntegration time to detect a source with a signal to noise level σ:

t1

2σ

η obs

2⋅

NEPbkg.inst.source

Psource⋅

2

⋅:=NEPbkg.inst.source

t σNETbkg.inst.source

T source⋅

2

:=NETbkg.inst.source

t σNEFDbkg.inst.source

F source⋅

2

:=NEFDbkg.inst.source

Signal to noise ratio

1 bare pixel vs 1 feedhorn:

Extended source T=100K:

Mapping speed comparison: 0.5Fλ bare pixels filled array versus 2Fλ feedhorns array:

σr ebh

80

75

69

69

82

80

81

85

%=

sre1bh

0.65

0.56

0.47

0.48

0.67

0.63

0.65

0.72

=Speed ratio

1 bare pixel vs 1 feedhorn:

Speed ratio

multipixel:

Filling ratio bare square grid vsfeed hexagon grid: Nb/Nh=13.9

sreNbh

9.0

7.8

6.6

6.6

9.2

8.8

9.1

10.0

=

σr pbh

77

73

70

77

82

83

84

91

%=

Point source (Ps<< background):

srp1bh

0.6

0.5

0.5

0.6

0.7

0.7

0.7

0.8

=

srp4bh

1.4

1.2

1.1

1.4

1.5

1.5

1.6

1.9

=

Coaddition of 4 bare pixels

Conclusion• 4 bands from 3mm to 850 µm available at the 30m

• Huge effect of the surface deformation on the beam efficiency

• 2 types of detector architecture possible: filled array of bare pixels or feedhorns

• Atmosphere and telescope dominate the background => 10s pW per pixel

• Ideal dynamics: fraction of fW to tens of pW=> ~26 bits

• Shot noise and bunching noise comparable => need to use a correct model for

the spatial coherence of the beam

• Ideal pixel instrumental noise NEP < 10-17W/Hz1/2

• Ideal sensitivities: NET < mK·s1/2 ; NEFD < few mJy ·s1/2

• Mapping speed better for filled arrays but not dramatically

⇒ Ideal performances comparable for both architectures, other technical aspects

may intervene: stray light problem for filled array ; effect of sky noise on

measures