Detection of Electromagnetic Detection of Electromagnetic Radiation II:Radiation II:

Optics and Radiation Optics and Radiation Scattering MatrixScattering Matrix

Phil Mauskopf, University of Phil Mauskopf, University of RomeRome

16 January, 200416 January, 2004

1990s: SuZIE, SCUBA, NTD/composite

1998: 300 mK NTD SiN

PLANCK: 100 mK NTD SiN

Transmission line wave equation:

V(x) I(x) L

C

L dx dI(x)/dt = V(x+dx) - V(x) L dI/dt = dV/dx C dx dV(x)/dt = I(x) - I(x-dx) C dV/dt = dI/dx

L d2I/dxdt = d2V/dx2

C d2V/dt2 = d2I/dxdt = (1/L) d2V/dx2

LC d2V/dt2= d2V/dx2

Same equation for current

Wave solutions have property: V/I = L/C = Z of line v2 = 1/LC = speed of prop.

V(x+dx)

How to calculate inductance:

w

h

I

+-

1. Apply 1 V for 1 sec to loopwith area = d h B = 1/ (d h)2. Calculate H from , BH = B/ = 1/ (d h)3. Calculate current frompath integral around loopI = HdlNo field outside, so integralis just:I = Hw = w / (d h)

4. Definition of inductance:

LI = Vdt = 1 L = (d h)/wProceedure also works if include field outside…modifies L

d

How to calculate capacitance:

w

h

1. Apply 1 A for 1 sec to plateswith area = d h Develops D field, chargeD = 1/(d w)2. Calculate E from , DE = D/ = 1/(d w)3. Calculate current fromIntegrate between plates to get V

V = Edl = h/(d w)

4. Definition of capacitance:

CV = Idt = 1 C = (d w)/h = A/h

Just like we knew...

d

I

Impedance of transmission line:

C = (d w)/h C = (w/h)

L = (d h)/w L = (h/w)

Z = L/C = L/C = (h/w) /

First part depends on geometry, second on materials

Therefore, we can choose the impedance of a transmissionline by changing the geometry and material

Z =

L

C

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+iL)/(G+iC)

Z has real part and imaginary part. Imaginary part givesloss

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 impedanceis slightly lower

Z ~ 5

Propagation of electromagnetic radiation:

Geometry of conducting boundaries and propertiesof material containing E, H fields (, ) determine impedance

Parallel conducting platesEnclose in conducting walls - waveguideCoaxial cableMicro-strip lineCoplanar waveguideCoplanar striplinesSlotlineetc.Given that the solution for the propagation of EM wavesis different for each of the above types of boundary conditions,how do we transform a giant plane wave coming from a distantsource into a wave travelling down a tiny transmission linewithout losing information? - Answer: optics

Dielectric materials, index of refraction, impedance mismatch:

1 2

IT

R

Fields b/c: 1 + R = TEnergy flow: 1 - R2= (Z1/Z2)T

2

T = 2Z2/(Z1+Z2) = 2 1 /( 1+ 2) = 2n1/(n1+n2)

R = (Z1-Z2)/(Z1+Z2)

Z1= /1

Z2= /2

Z1 Z2

Transmission line analogy

+-

Anti-reflection coating:

1 2

I T1

R

T = 4Z2Z3e-ikd/[(Z1+Z2)(Z2+Z3) + (Z2-Z1)(Z3-Z2) e

-2ikd]

R = (Z2-Z1)(Z3+Z2) + (Z2+Z1)(Z3-Z2)e-i2kd

(Z1+Z2)(Z2+Z3) + (Z2-Z1)(Z3-Z2) e-i2kd

k = 2/2

depend on d, - resonance

Z1 Z2

Transmission line analogy

+-

3

T2

Z3

d

T

Anti-reflection coating:

1 2

I T1

R

Off resonance: d >> , e-ikd = 1 T = 4Z2Z3/[(Z1+Z2)(Z2+Z3) + (Z2-Z1)(Z3-Z2)]

Without intermediate dielectric:

T = 2Z3/(Z1+Z3)

Z1 Z2

Transmission line analogy

+-

3

T2

Z3

d

T

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. thinmetal 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

+-

Z0 R=Z0

Optics: Direct coupling to detectors (simplest)

Without an antenna connected to a microstrip line, theminimum size of an effective detector absorber is limitedby diffraction

Single mode - size ~ 2

Optics: Modes and occupation number

A mode is defined by its throughput: A = 2

The occupation number of a mode is the number ofphotons in that mode per unit bandwidth

For a single mode thermal source emitting a power, Pin a bandwidth , with an emissivity,

The occupation number is:

N = (P/2h)(1/ )

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 byspecifying all of the ports and their impedances andthe complex coefficients that give the coupling betweeneach port and every other port.

For an optical system with N ports, there are NxN coefficientsnecessary to specify the system.

This NxN set of coefficients is called the Scattering Matrix

Scattering matrix: S-parameters

The components of the scattering matrix are calledS-parameters.

S11 S12 S13 S14 ...

S =

Scattering matrix: Lossless networks - unitarity condition,conservation of energy

For a network with no loss, the S-matrix is unitary:

SS = I

This is just the expression of conservation of energy,

For a two port network: 1 + R = T

Scattering matrix: Examples - two-port networks

Dielectric interface:

S =

This is because R = (Z1-Z2)/(Z1+Z2)

Going from lower to higher impedance Z1 Z2 givesthe opposite sign as going from higher to lower impedance.

T R-R T

Scattering matrix: Power divider

How about 3-port networks? Can we make an opticalelement that divides the power of an electromagneticwave in half into two output ports?

Guess:

What is the S-matrix for this circuit? What is the opticalanalogue?

Z1

2 Z1

2 Z1

Scattering matrix: 4-port networks - 90 degree hybrid

A

B

(A+iB)/2

(A-iB)/2

S =

0 0 1 10 0 i -i1 1 0 0i -i 0 0

Scattering matrix: 4-port networks - 180 degree hybrid

A

B

(A+B)/2

(A-B)/2

S = =

0 0 1 10 0 1 -11 1 0 01 -1 0 0

0 3

3 0

Two types of mm/submm focal plane architectures:

SCUBA2PACSSHARC2

BOLOCAMSCUBAPLANCK

Filter stack

Bolometer array

IR Filter

Antennas (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