Download + (Z 2 +Z 1 )

Detection of Electromagnetic
Radiation II:
Optics and Radiation Scattering
Matrix
Phil Mauskopf, University of Rome
16 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) V(x+dx)
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.
How to calculate inductance:
1. Apply 1 V for 1 sec to loop
with area = d  h
 B = 1/ (d  h)
2. Calculate H from , B
H = B/ = 1/ (d  h)
3. Calculate current from
path integral around loop
I =  Hdl
No field outside, so integral
is just:
I = Hw = w / (d  h)
w
d
I
h
+
-
4. Definition of inductance:
LI =  Vdt = 1  L = (d  h)/w
Proceedure also works if include field outside…modifies L
How to calculate capacitance:
1. Apply 1 A for 1 sec to plates
w
with area = d  h
 Develops D field, charge
D = 1/(d  w)
2. Calculate E from , D
h
E = D/ = 1/(d  w)
3. Calculate current from
Integrate between plates to get V
V =  Edl = 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 transmission
line by changing the geometry and material
L
Z
=
C
Resistive elements in transmission line - loss:
R
L
C
G
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
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
Propagation of electromagnetic radiation:
Geometry of conducting boundaries and properties
of material containing E, H fields (, ) determine impedance
Parallel conducting plates
Enclose in conducting walls - waveguide
Coaxial cable
Micro-strip line
Coplanar waveguide
Coplanar striplines
Slotline
etc.
Given that the solution for the propagation of EM waves
is different for each of the above types of boundary conditions,
how do we transform a giant plane wave coming from a distant
source into a wave travelling down a tiny transmission line
without losing information? - Answer: optics
Dielectric materials, index of refraction, impedance mismatch:
I
1
R
T
2
Transmission line analogy
+
-
Z1
Z2
Fields b/c: 1 + R = T
Energy flow: 1 - R2= (Z1/Z2)T2
T = 2Z2/(Z1+Z2) = 2  1 /( 1+  2) = 2n1/(n1+n2)
R = (Z1-Z2)/(Z1+Z2)
Z1=  /1
Z2=  /2
Anti-reflection coating:
d
I
T1
T2
R
Transmission line analogy
T
1
2
3
+
-
Z1
Z2
T = 4Z2Z3e-ikd/[(Z1+Z2)(Z2+Z3) + (Z2-Z1)(Z3-Z2) e-2ikd]
R = (Z2-Z1)(Z3+Z2) + (Z2+Z1)(Z3-Z2)e-i2kd
-i2kd
(Z1+Z2)(Z2+Z3) + (Z2-Z1)(Z3-Z2) e
k = 2/2
depend on d,  - resonance
Z3
Anti-reflection coating:
d
I
T1
T2
R
Transmission line analogy
+
-
T
1
2
3
-ikd
Z1
Off resonance: d >> , e
=1
T = 4Z2Z3/[(Z1+Z2)(Z2+Z3) + (Z2-Z1)(Z3-Z2)]
Without intermediate dielectric:
T = 2Z3/(Z1+Z3)
Z2
Z3
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
+
R=Z0
Z0
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
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
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 thermal source emitting a power, P
in 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 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
Scattering matrix: S-parameters
The components of the scattering matrix are called
S-parameters.
S11 S12
S=
S13 S14 ...
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
Scattering matrix: Examples - two-port networks
Dielectric interface:
S=
T
-R
R
T
This is because R = (Z1-Z2)/(Z1+Z2)
Going from lower to higher impedance Z1  Z2 gives
the opposite sign as going from higher to lower impedance.
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:
2 Z1
Z1
2 Z1
What is the S-matrix for this circuit? What is the optical
analogue?
Scattering matrix: 4-port networks - 90 degree hybrid
A
(A+iB)/2
B
(A-iB)/2
0
0
S= 1
i
0
0
1
-i
1
i
0
0
1
-i
0
0
Scattering matrix: 4-port networks - 180 degree hybrid
A
(A+B)/2
B
(A-B)/2
0
0
S= 1
1
0
0
1
-1
1
1
0
0
1
-1
0
0
=
0 3
3 0
Two types of mm/submm focal plane architectures:
Bare array
Antenna coupled
IR Filter
Filter stack
Bolometer array
Antennas (e.g. horns)
X-misson line
SCUBA2
PACS
SHARC2
Microstrip Filters
Detectors
BOLOCAM
SCUBA
PLANCK
Mm and submm planar antennas:
Quasi-optical (require lens):
Twin-slot
Log periodic
Coupling to waveguide (require horn):
Radial probe
Bow tie