### Presentation 4

```Basic Detection Techniques
Quasi-Optical techniques
Andrey Baryshev
Lecture on 18 Oct 2011
Outline
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What is quasi – optics (diffraction)
Gaussian beam and its properties
What is far? (confocal distance), far field, radiation pattern
Gaussian beam coupling
• Concept
• Lens/elliptical mirror
Gaussian beam launching
• Corrugated horn
Polarization elements
• Wire grid
• Roof top Mirror
Quasi-optical components and systems
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A to B
A (source)
B (detector)
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A to B
A (source)
B (detector)
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A to B optical
A (source)
B (detector)
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A to B diffraction
A (source)
A (detector)
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Quasi - optics
Geometrical
Optics   D
Quasi - optics
D
D
• Both, Lens and
Antenna
• Simplification of
physical optics
Lens
Antenna
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What is “quasioptics” ?
“Quasi-optics deals with the propagation of a beam of
radiation that is reasonably well collimated but has
relatively small dimensions (measured in wavelenghts)
transverse to the axis of propagation.”
While this may sound very restrictive, it actually applies to
many practical situations, such a submillimeter and laser
optics.
Main difference to geometrical optics:
Geometrical optics:
Quasi-optics:
λ  0, no diffraction
finite λ, diffraction
Quasi-optics was developed in 1960’s as a result of interest in
laser resonators.
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Why quasi-optics is of interest
Task: Propagate submm beams / signals in a suitable way
Could use
- Cables
- Waveguides
- Optics
 high loss, narrow band
 high loss, cut-off freq
 lossless free-space,
But: “Pure” (geometrical) optical systems would require
components much larger than λ.
In sub- /mm range diffraction is important, and quasi-optics
handles this in a theorectical way.
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Gaussian beam - definition
Most often quasi-optics deals with “Gaussian” beams, i.e. beams
which have a Gaussian intensity distribution transverse to the
propagation axis.
Gaussian beams are of great practical
importance:
• Represents fundamental mode TEM00
• Stays Gaussian passing optical
elements
• Laser beams
• Submm beams
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Gaussian beam – properties I
A Gaussian beam begins as a perfect plane wave at waist but –
due to its finite diameter – increases in diameter (diffraction)
and changes into a wave with curved wave front.
Beam waist
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Gaussian beam properties II
Solution of Helmholtz equation

2
 k  E ( x, y , z )  0
k
2

In cylindrical coordinates
2
E (r , z ) 
e
2
  w ( z)


r2
j r 2
 j k  z 
 j 0 ( z ) 
  2
R( z)
 w (z)

 z 
0  ArcTan 
2 
   w0 
Phase
w0
Waist size
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Gaussian beam – properties III
Gaussian beam diameter (= the distance between 1/e points) varies
along the propagation direction as
 z 
w( z )  w0 1  
2 
  w0 
with
2
λ = free space wavelength
z = distance from beam waist (“focus”)
Radius of phase front curvature is given by
   w2  2 
R( z )  z 1   0  
  z  


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Gaussian beam propagation
Beam diameter
2w at distance z
Beam waist with
Beam profile variation of the fundamental Gaussian beam
mode along the propagation direction z
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Gaussian beam - phase front curvature
Beam profile variation of the
fundamental Gaussian beam mode
along the propagation direction z
Curvature of phase front

0 
  w0
Far field divergence angle
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Confocal (Rayleigh) distance
 w
zc 

2
0
Quasi-optics becomes geometrical
Border between far and near field
zc
Far field
of ALMA
Antenna
377 km
Waist
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Launching Gaussian beam from fiber
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Corrugated horn coupling principle
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Quasi-optical components – Feedhorn (cont’d)
Often used in submm:
Corrugated feedhorn
500 GHz horn
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Lorentz’ reciprocity theorem implies that antennas work equally
well as transmitters or receivers, and specifically that an
antenna’s radiation and receiving patterns are identical.
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This allows determining the characteristics of a receiving
antenna by measuring its emission properties.
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Beam coupling, lens as example
1 1 1
  '
f R R
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QO Lens with antireflection “coating”
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Refractive index for antireflection coating nAR = n1/2, λ/4 thick
Optical lenses: special material with correct nAR
Submillimeter lenses: grooves of width dg « λ
Effect of AR coating if height and width are chosen such that the
“mixed” refractive index between air and material = nAR
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Elliptical mirror
R1
R2
FP1
FP2
Rotation
axis
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Mirror chain
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Quasi-optical components - Mirrors
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Use of flat and curved mirrors
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Curved mirrors (elliptical, parabolic) for focusing
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Material: mostly machined metal (non-optical quality). Surface
roughness ~few micron sufficient for submm
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Quasi-optical components - Grid
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For separating a beam into orthogonal polarizations
For beam combining (reflection/transmission) of orthogonal
polarizations
Polarization parallel to wire is reflected, perpendicular to wire is
transmitted
Material: thins wires over a metal frame
Also used in more complicated setups
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Quasi-optical components – Quarter wave plate
Quarter-wave plate: linear pol.  circular polarisation
If linear pol. wave incident at 45o Path 1: ½ reflected by grid
Path 2: ½ transmitted by grid
and reflected by mirror
Path difference is ΔL = L1 + L2 = 2d cos θ
Phase delay Φ = k ΔL = (4πλ/d) cos θ
For linear  circular pol. we need
ΔL = λ/4  Φ = π/2 , i.e.
D = λ / (8 cos θ)
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Polarization transfer, roof top mirror
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Quasi – optical components
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Quasi optical systems example
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Martin-Puplett (Polarizing) Interferometer
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Low-loss combination of two beams of different frequency and
polarization into one beam of the same polarization
Often used for LO and signal beam coupling
Use of polarization rotation by roof top mirror:
• Input beam reflected by grid
• Polarization rotated by 90o
through rooftop mirror
•Beam transmitted by grid
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Martin-Puplett Diplexer
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Consider two orthogonally polarized input beams: Signal and LO
Central grid P2 at 45o angle  both beams are split equally and
recombined
For proper pathlength difference setting in the diplexer, both
beams leave at port 3 with the same polarization (and no loss)
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QO system characterization
Beam pattern (PSF) measurements
• E(x,y) phase and amplitude for
near field
• E2(x,y) for far field, in two planes
y
Test source
Moves in x,y
System to measure
x
By fitting Gaussian beam distribution one can
locate waist position and waist size, relative to
measurement XY system
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Beam pattern examples, ALMA main beam
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Alma beam – cross polarization
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HIFI FPU (Focal Plane Unit)
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Common Optics Assembly
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Common Optics Assembly
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Mixer Assembly
Contains two Mixer
Subassemblies (MSA)
Accepts LO and signal
in two polarizations
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Michelson interferometer
Transfer function:
Cosine Fourier transfer
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Interferogram
sis25 05 d3 a.dat
0.006
0.005
Power a.u
0.004
0.003
0.002
0.001
0.000
0.005
0.000
Pathlength
0.005
m
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Fourier transform (band pass)
sis25 05 d3 a.fts
0.0020
Power a.u
0.0015
0.0010
0.0005
0.0000
400
500
600
Frequency
700
800
900
GHz
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Planck formula
Per unit square
In all directions
Integral for gaussion beam over surface and
beam angle gives lambda^2 throughput
3
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Literature on Quasi-optics (examples)
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“Quasioptical Systems”, P.F. Goldsmith, IEEE Press 1998
Excellent book for (sub-)mm optics
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“Beam and Fiber Optics”, J.A. Arnaud, Academic Press 1976
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“Light Transmission Optics”, D. Marcuse, Van NostrandReinhold, 1975
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“An Introduciton to Lasers and Masers”, A.E. Siegman, McGrawHill 1971
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Chapter 5 (by P.F. Goldsmith) in Infrared and Millimeter Waves,
Vol. 6, ed. K.J. Button, Academic Press 1982
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