CHAPTER TWO THEORETICAL BACKGROUND
Equation 2.15 can be rewritten as:
I ( t )  a1Vo  a 2Vo2
 a1V FIR  2a 2VoV FIR sin2  FIR t 


 a1V   wave  2a 2VoV   wave sin2    wave t 
 a 2V
1
2
2
FIR
1  cos4  FIR t  

1
2
a 2V
1  cos4 
t   cos2 
2
  wave
 a 2V FIR V   wave cos2  FIR     wave
  wave
FIR

t
(2.16)

    wave t 
This shows that the diode current oscillates at the input frequencies, FIR and -wave, the
second harmonics of these input frequencies, 2FIR and 2-wave, and the sum- and
difference frequencies of the input frequencies, (FIR + -wave) and (FIR - -wave). All
these frequencies can be re-radiated from the diode. The sum-frequency signal is
conventionally referred to as the upper sideband: the difference frequency signal is
known as the lower sideband.
Equation 2.16 also suggests that both sidebands have equal intensities that are
directly dependent on the relative intensities of the FIR and microwave radiation at the
diode, but independent of the bias voltage. Bicanic et al conducted extensive studies on
the bias characteristics of FIR Schottky diode mixers to verify this relationship [19]. If
FIR and/or microwave power was incident on the diode, they found that the
characteristic I-V curve was modified, as the incoming signals effectively provided an
external bias. Consequently, a higher current response was obtained at lower values of
the forward dc bias. They also showed that the sideband intensity increased linearly as
the FIR power was increased, but was saturated above a certain microwave power level
as the diode absorbed the microwave power more efficiently than the laser power.
Farhoomand et al obtained identical results on their TuFIR spectrometer [1].
On the Cambridge TuFIR system, the upper sideband was consistently weaker
than the lower sideband: this intensity ratio was frequency dependent. Figure 2.7 shows
the how the intensity of one particular lower sideband varied with the input power of the
FIR and the microwave radiation. The sideband was generated from the 410.712m FIR
laser line in CD3OD and 28GHz microwaves. The microwave signal was 100% AM
modulated at 1.5kHz, so the sideband intensity was taken as the measured peak-to-peak
voltage reading on the LIA, after 50dB amplification. The FIR power was measured
directly in front of the laser output coupler using a calibrated Scientech 36-0001
41
CHAPTER TWO THEORETICAL BACKGROUND
Calorimeter [20]. The microwave power was set in dBm units (i.e. a log based power
scale where 0dB=1mW) and the fraction of this output power impinging on the diode,
Pcube, was estimated (in mW) using:
  0.035    wave 
1  P 
Pcube  log 1 
  log  
(2.17)
10
 10 


where P is the power setting at the synthesiser in dBm, -wave the microwave frequency
in GHz, and the first log term accounts for the power attenuation between the synthesiser
and the diode. The incoming FIR and microwave radiation were usually sufficiently
intense that the diode became current saturated, and the bias voltage had to be reduced to
a few V to prevent any electrical damage. During TuFIR experiments, the FIR power
was maximised as far as possible and the diode microwave power fixed at around
4.5dBm.
at
at
-
no forward
bias required
 FIR power measurements
 Microwave power measurements
at
Figure 2.7: An example showing how the sideband intensity varies with the FIR laser
power and microwave power. The FIR measurements were made with the microwave
power fixed at 4.5dBm. The microwave power measurements were made with the FIR
laser power fixed at 30mW.
42
CHAPTER TWO THEORETICAL BACKGROUND
At FIR frequencies, the most efficient method for coupling the incoming
radiation on to the mixer diode is an open structure design. The radiation is focused onto
the diode’s wire contact, which is placed near the apex of a 90o-corner reflector. In this
configuration, the wire acts as a whisker antenna, and the corner reflector acts as a
ground plane to capture the incoming radiation. Krautle et al were the first to calculate
the radiated power response of the open structure design in different spatial
configurations. Using the antenna reciprocity theorem, which states that ‘the antenna
pattern for reception is identical to that for transmission’, they found that optimum
signal coupling was achieved with a whisker 4 long and placed 1.2 from the reflector
apex [21]. Fetterman et al verified these results with a 100 times scale model and 60GHz
microwaves [22]. With a 90o reflector, the radiation was almost entirely concentrated in a
single lobe, inclined at 25o to the antenna axis, with a 12o divergence angle at the –3dB
points. Mizuno et al proved that an incoming, Gaussian-profiled radiation beam was
efficiently coupled onto the diode if it was sharply focused inside this lobe, and linearly
polarised in the plane of the reflector corner and whisker antenna [23].
More recently, Grossman published his calculations on the antenna patterns and
coupling of Gaussian beams to open structure corner cube mixers. His two dimensional
results contrasted with those of Krautle et al, (who only derived the antenna pattern in
the plane parallel to the antenna). He suggested that the theoretical maximum coupling
efficiency is increased from 55% to 78% if the antenna is 1.35 long and positioned 0.9
from the reflector apex [24]. This configuration also gave a slightly wider main lobe (15o
at the –3dB points) which would couple more effectively to a focused TEM00 Gaussian
beam, and accommodate small mismatches in the polarisation angle of the incoming
radiation. Corner cubes are now designed with a 4 long antenna positioned 0.9 to 1
from the reflector apex, as a compromise between the optimum coupling efficiency and
practical alignment considerations. The TuFIR sidebands and FIR radiation are both reradiated from the diode in the same direction as the incoming laser beam. Sauter et al
found that the bottom reflector plane made a very minor contribution to the power in the
antenna lobe and could be omitted from the structure [25]. This also eliminates retroreflections of the intense incoming FIR radiation, and reduces the total intensity of the
re-radiated beam. For TuFIR spectroscopy, the intense FIR signal must be filtered out of
the re-radiated beam.
43
CHAPTER TWO THEORETICAL BACKGROUND
The Schottky diode mixer on the Cambridge TuFIR spectrometer is mounted in a
custom-made corner cube (Model CDS-3/SP No. FTL 601) and was supplied by Farran
Technology Ltd [26]. At 350m, the whisker antenna is 4 long, and mounted 0.93
from the apex of the corner reflector. These distances are fixed, and consequently the
system is optimised at frequencies between 800GHz and 1THz. The theoretical cut-off
frequency of the diode is 4.23THz but beyond 2.05THz it is not possible to couple FIR
radiation on to the diode as the antenna configuration actually limits the system’s
frequency response. In practice, TuFIR sidebands can be generated from 600GHz to
1.2THz. The diode itself has an Rs value of 18.8, and an ideality factor, , of 1.135
[26]. A Farran technology FDB-F4 Precision dc Bias supply is used to forward bias the
diode [27]. This runs from a rechargeable lead salt diode battery to prevent the diode
junction from being damaged by spurious currents from the mains supply. The whole
corner cube is mounted on a x, y, z translational stage, with an additional vertical rotation
stage added, so the cube position can be optimised to the incoming FIR radiation. This
radiation is focused onto the whisker antenna from a 90o off-axis parabolic mirror, with a
30mm focal length.
The microwave source is a step tunable, Wiltron Swept Frequency Synthesiser,
Model 6740B, that generates K-band microwaves between 26.5 and 40GHz with an
accuracy of a few Hz [28]. The microwave output can be CW or pulsed, AM or FM
modulated, and the power ranges from 0 to 10dBm. A PC controls the microwave
synthesiser via a GPIB link, (see Chapter 3). The microwaves are transmitted to the
diode via a 12” semi-rigid coaxial cable (Wiltron K120-12) with a minimum bend radius
of 6.4mm. The power loss down this cable is given by the first log term in equation 2.17.
The microwaves and the dc bias current are coupled through a bias-T onto a SMA Ktype connector on the base plate of the diode.
2.3 Sideband Selection Elements
A combination of optical elements is used to select a single sideband frequency
from the beam that is re-radiated from the diode:
1. Grid polariser: linear polarisation selective signal filtration,
2. Diplexer: spatially selective signal filtration,
3. Fabry-Perot Interferometer, (FPI): wavelength selective signal filtration.
44
CHAPTER TWO THEORETICAL BACKGROUND
In practice, it is only necessary to discriminate between the intense FIR radiation and the
weaker TuFIR sidebands since the microwave frequencies (-wave, 2-wave) are
attenuated rapidly in free space and the second harmonics of the FIR (2FIR) are even less
intense than the sidebands. The FIR signal is up to 30dB more intense than the
sidebands.
2.3.1. Grid Polariser
At FIR frequencies, a wire grid polariser can be used to divide and combine
linearly polarised beams, or extract a linearly polarised component from an arbitrarily
polarised signal. An array of thin, conducting, metal wires spaced less than /2 apart will
reflect the E-field component of the incident radiation parallel to the wires and transmit
the orthogonal component. The grids are manufactured from 5 to 10m diameter, W wire
that is tautly wound onto a metal frame, at a pitch spacing of around 25m [29]. These
free standing grid polarisers have lower absorption losses than grids fabricated by
depositing the metal on a dielectric substrate, and do not distort the Gaussian profile of
the beam [30]. Costley et al investigated the power transmissivity and reflectivity of a
number of grids as a function of the wire spacing, over a frequency range from 300GHz
to 5THz [31]. They found that the polarising efficiency was dependent on the uniformity
of the wire spacing and the frequency of the incident radiation. Grids can be
manufactured with a cross-polar leakage of –30dB and an absorption-scattering loss
around –0.01dB at 1THz [32]. If the grid is treated as an array of thin wires, the power
reflectivity of the E-field component parallel to the wires is given by [32]:
r||
2
  2S  2  S  2 
 1  
 ln  


  d  

1
(2.18)
and for the E-field component perpendicular to the wires by [32]:
r
2


2
d2


2

   2d 2  2 
 
2S  1  
  2S   
where S is the spacing between the wires and d is the wire diameter.
2
(2.19)
In the TuFIR spectrometer, grid polarisers were used for two different purposes
in each of the configurations, (figure 2.1). In Configuration A, a single grid polariser was
used in conjunction with a roof top mirror to make a double pass of the absorption cell.
45
CHAPTER TWO THEORETICAL BACKGROUND
The grid was rotated so that the wires were set to reflect the sideband beam down the
absorption cell: the roof top mirror rotated the polarisation angle by /2, and the beam
was then transmitted through the polariser after its second pass of the cell. In
configuration B, the polarising properties of the grids were exploited to partially separate
the FIR and TuFIR beams. A grid polariser was placed directly in front of the FIR laser
at 45o to the beam. The wires were set at 10o from vertical for maximum transmission of
the FIR radiation. The beam was linearly polarised in the xy plane rather than along the
x- or y-axis because the Brewster windows in the CO2 laser would only fit the discharge
tubes at a slight angle! Each reflection inverted the E-field component parallel to the
plane of incidence, so that the FIR radiation was actually almost cross-polarised with the
plane of the whisker antenna at the cube. As the polarisation of the re-radiated beam was
dictated by the cube geometry, it was essentially cross-polarised with the incoming beam
and was reflected rather than transmitted by the grid polariser in front of the FIR laser.
However, a small fraction of this signal was transmitted and fed back into the FIR laser.
A second grid polariser, cross-polarised with the first, was placed in one arm of the
diplexer. The ‘handicaps’ of working in the FIR were then exploited: over the frequency
range of this TuFIR spectrometer the reflection and transmission coefficients of the
Melinex beamsplitter were unequal (see next section) so significantly more of the beam
was reflected than transmitted by the beamsplitter. Consequently, the second grid
polariser attenuated the majority of the ‘feedback’ signal, yet the intensity of the
incoming FIR radiation was unaffected. This reduced the feedback to the FIR laser and
improved the sideband’s S:N ratio. In addition, Liu et al have shown that if linearly
polarised light is reflected from a corner cube retro-reflector, it is mostly converted to
elliptically polarised light [33]. Therefore, almost all of the FIR radiation that was
reflected from the corner cube, (rather than coupled onto the diode), was attenuated by
one or other of the grid polarisers, greatly reducing the FIR:TuFIR signal intensity ratio.
2.3.2 Diplexer
A diplexer is an interferometric system that resembles a Michelson interferometer
and acts as a directional filter. It consists of two roof-top mirrors (one of which is
moveable) positioned at 90o to each other and separated by a beamsplitter, (figure 2.8).
Erickson was the first to prove that this type of optical configuration could be used to
separate the input and output frequencies of a mixer from each other [34]. In the TuFIR
spectrometer, each mirror was vertically and horizontally adjustable and the beamsplitter
46
CHAPTER TWO THEORETICAL BACKGROUND
was constructed from a stretched sheet of 50m Melinex (polyethylene tetephthalate or
Mylar) with a reflection coefficient of -0.729 at 850GHz [35]. The whole system was
mounted on a flat sheet of steel plate that was then fixed to the optical bench.
PORT 4 PORT 1
Mirror
movement
D
PORT 2
C
B
PORT 3
A
Mylar
beamsplitter
F
E
Figure 2.8: Schematic of the Diplexer. Red arrows show the paths of the incoming
FIR radiation: green and blue arrows indicate the path of the TuFIR radiation in
configurations A and B respectively.
In each of the spectrometer configurations, the diplexer ports were used for
different purposes. The original diplexer configuration was identical to that described by
Erickson [34]. FIR radiation entered the diplexer from the laser at port 1 and was split
into two beams at A. The beams recombined at D and passed from port 2 towards the
mixer. The re-radiated FIR and TuFIR radiation entered the diplexer at port 2, and were
each split into two beams that followed paths DFEA and DCBA, then re-combined at A.
Both signals exited at port 3.
To calculate the relative intensities of the FIR and TuFIR radiation entering and
leaving the diplexer, the following assumptions are made:
1. the FIR and TuFIR radiation are characterised by:
47
CHAPTER TWO THEORETICAL BACKGROUND
k FIR 
k TuFIR
2
(2.20)
 FIR
2

TuFIR
(2.21)
2. the incoming FIR radiation has unit amplitude, A
3. the diplexer is a loss-less system, i.e. the beam is not diffracted or attenuated
in the system,
4. a path difference exists between the two ‘arms’ of the diplexer such that:
ABCD  AEFD  
(2.22)
5. the beamsplitter is infinitely thin,
6. r and t are the reflection and transmission coefficients of the beamsplitter
where:
2
2
r  t 1
(2.23)
7. the diode mixer is simulated by a mirror with an attenuation factor C in the
FIR, and a TuFIR source.
Consequently, the amplitude of the FIR radiation directed towards the mixer at
port 2 is given by:
A2FIR
 Ae
OUT
 i
2
rt  rte
ikFIR

(2.24)
where the exponential term e-i/2 accounts for the phase retardation between the FIR
radiation at ports 1 and 2, and the exponential component of the second term is
attributable to the path difference, . The FIR radiation re-enters the diplexer 180o out of
phase with the incoming beam, and attenuated by a factor C. The amplitude of the FIR
signal exiting the diplexer at port 3 is given by:
A 3FIR
 CAe
OUT
 i 3
2
rt  rte
ikFIR
t
2
 r 2 e ikFIR

(2.25)
where the exponential term e-i3/2 accounts for the phase retardation between the FIR
radiation at ports 1 and 3, and the final term accounts for the path difference, , and the
phase difference, , between the two beams when they recombine at A. The amplitude of
the TuFIR radiation emitted from the diode is dependent on the amplitude of the
incoming FIR radiation and the efficiency of the diode mixer itself. Consequently the
amplitude of the TuFIR radiation at port 3 is given by:

 i  




(2.26)
A3TuFIR
 DAe 2 rt  rte ikFIR t 2  r 2 e ikTuFIR
OUT
where the pre-exponential factor D accounts for the efficiency of the second order
mixing process in the diode, and the exponential term e-i(/2+) accounts for the phase
48
CHAPTER TWO THEORETICAL BACKGROUND
retardation between the FIR radiation at port 1 and the TuFIR radiation at port 3. The
FIR and TuFIR signal intensities, exiting the diplexer at ports 2 and 3, are given by:


I 2FIR
 A 2 2rt  1  cos k FIR  
OUT
I
FIR
3OUT
I 3TuFIR
OUT
2

  AD  2rt  1  cos k

  t

  AC  2rt  1  cos k FIR   t  r  2rt  cosk FIR    
2
2
2
2
FIR
4
4
4
2
(2.27)
(2.28)

 r 4  2rt  cosk TuFIR       (2.29)
2
Since the intensity of the TuFIR radiation at port 3 depends on the intensity of the FIR
radiation at port 2, the TuFIR output intensity is actually slowly modulating, with
maxima at:
  m  FIR  
2n  1 
2
  wave
(2.30)
where mn and both are integers.
If the diplexer were an ideal spatial filter it would be possible to:
1. maximise the FIR transmission from ports 1 to 2 to optimise the FIR signal at
the diode,
2. maximise the TuFIR transmission from ports 2 to 3 for maximum signal
output to the absorption cell,
3. have no FIR transmission from ports 2 to 3 at all.
This requires an ‘ideal’ beamsplitter with |r|2=|t|2=0.5. Figure 2.9 shows the normalised
relative intensities, i.e. A=D=C=1, of the FIR and TuFIR radiation for the Melinex
beamsplitter in this experiment, with LSB set at +15o and USB at -14o. In practice, C was
around A/10, and D around C/100. A path difference could always be found to satisfy
conditions 1 and 2 above. As the two sidebands were at different frequencies their
maxima were not coincident. However condition 3 was never completely satisfied. In
this configuration, around 75% of the re-radiated FIR radiation was transmitted toward
the absorption cell. In addition, a portion of the returning FIR signal leaked out of port 1
and was fed directly back into the FIR laser, which in turn produced more noise in the
laser output and the sidebands.
49
CHAPTER TWO THEORETICAL BACKGROUND
FIR INPUT
FIR CUBE OUTPUT
UPPER SIDEBAND
LOWER SIDEBAND
Normalised Intensity
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
Path difference,  (mm)
Normalised Intensity
1.0
0.8
0.6
0.4
0.2
0.0
3.0
3.5
4.0
4.5
5.0
Path difference,  (mm)
Figure 2.9: The normalised relative intensities of the FIR and TuFIR signals as the
path length in the diplexer is altered. Calculated from equations 2.27 to 2.29 with
FIR=850GHz and TuFIR=885GHz or 815GHz.
50
CHAPTER TWO THEORETICAL BACKGROUND
In configuration B the incoming radiation path was unchanged, i.e. the FIR signal
was directed from port 1 to port 2, and the re-radiated signals entered the diplexer at port
2 as before, but exited at port 1. The relative intensity of the FIR at port 2 is therefore
still given by equation 2.27, but the relative intensities of the FIR and TuFIR radiation
exiting the diplexer at port 1 are now given by:
  1  cos k   1  cosk    
  AD  2rt  1  cos k   2rt  1  cosk
     
I 1FIR
  AC  2rt 
OUT
2 2
2
I 1TuFIR
OUT
2
FIR
2
FIR
2
FIR
TuFIR
(2.31)
(2.32)
In figure 2.10, the normalised relative intensities, i.e. A=D=C=1, of the FIR and TuFIR
radiation are plotted as before. There are two main advantages to configuration B:
1. at each sideband maximum, significantly less FIR radiation exits the diplexer
at port 1 so less FIR radiation can propagate through the system to the
absorption cell,
2. the diplexer is able to discriminate more effectively between the two
sidebands, i.e. when one sideband’s intensity is maximised, the other
sideband is only around 5% as intense.
In addition, this configuration could easily be converted into a Martin-Pulpet polarising
interferometer [36], which is much more efficient over a wider range of frequencies than
the system described above, (see Chapter 6). As it was not possible in practice to
completely discriminate between the TuFIR and FIR signals in either configuration,
further filtration elements were required.
2.3.3. Fabry-Perot Interferometer (FPI)
Fabry-Perot Interferometers are multiple-beam interference devices that can be
used to discriminate between signals of different frequencies. They are constructed from
two plane, parallel, highly reflecting surfaces separated by a distance d. The beam is
successively reflected within the FPI cavity, and interference occurs between the
reflected components of the incident wave. The transmitted intensity is described by
Airy’s formula [37]:


1


 1  F sin 2  
(2.33)
2

where A is the absorbtivity and R the reflectivity of the plane parallel surfaces, F is the
It 
A 
 1 
I i  1  R 
2
 
coefficient of finesse, given by:
F
(2.34)
4R
1  R 2
51
CHAPTER TWO THEORETICAL BACKGROUND
FIR INPUT
FIR CUBE OUTPUT
UPPER SIDEBAND
LOWER SIDEBAND
Normalised Intensity
1.0
0.8
0.6
0.4
0.2
0.0
4.0
4.5
5.0
5.5
6.0
Path difference,  (mm)
Normalised Intensity
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
Path difference,  (mm)
Figure 2.10: The normalised relative intensities of the FIR and TuFIR signals as the
path length in the diplexer is altered in configuration B. Calculated from equations
2.27 and 2.31 to 2.32 with FIR=850GHz and TuFIR=885GHz or 815GHz.
52