CHAPTER TWO THEORETICAL BACKGROUND
Optically pumped FIR lasers are molecular gas lasers in which the laser transition
occurs between two rotational energy levels associated with the same vibrational state.
Most FIR lasing gases are symmetric top molecules. The lasing mechanism is described
as a ‘pseudo’ three level system and is relatively simple, (figure 2.3). The output from a
CO2 laser is tuned into resonance with a ro-vibrational transition in the FIR lasing gas,
i.e. v’, (J+1)’, K’v’’, J’’, K’’, exciting the molecules to states that are normally almost
totally empty. Population inversions are therefore created between the rotational energy
levels in both the ground and excited vibrational states of the molecule. Laser emission is
possible from three types of transitions:
1. rotational transitions in the upper vibrational state,
2. cascade transitions in the upper vibrational state,
3. refilling transitions in the lower vibrational state.
A slow vibrational relaxation returns the excited molecules to the ground state [4].
1st excited
Vibrational
State – v’
(J+2)’
(J+1)’
Lasing transition
Cascade transition
J’
(J-1)’
(J-2)’
Ground
Vibrational
State - v’’
(J+2)
CO2 laser
pump  9 - 11m
Vibrational
relaxation
’’
(J+1)’’
J’’
(J-1)’’
(J-2)’’
Re-filling
transition
Figure 2.3: The molecular energy levels involved in the absorption and emission
processes in an optically pumped FIR laser.
31
CHAPTER TWO THEORETICAL BACKGROUND
Relative to their long operating wavelengths, these lasers have very narrow gain
profiles that are homogeneously broadened. Consequently, only one longitudinal mode
lies under the profile. Significant output powers are only obtained when the centre
frequency of the gain profile exactly matches one of the resonant frequencies of the laser
cavity. In practice this is achieved by manually adjusting the cavity length. The laser
tube itself acts as a hollow, cylindrical, dielectric waveguide and is constructed with a
relatively large diameter (>5cm) to minimise diffraction losses. The laser end mirrors are
positioned beyond this waveguided region. The field distributions of the laser radiation
are described as resonator modes, which are linear combinations of the hybrid modes of
a waveguide, (EHnm) [5]. These EHnm modes have E- and H-fields directed tangentially
and radially to the waveguide surface. The lowest order hybrid mode, EH11, is linearly
polarised and doubly degenerate, as the E-field vector can be directed along the x or y
axis for a beam propagating along the z-axis. The mode amplitude is circularly
symmetric and closely resembles a HG TEM00 free space mode. If the laser cavity is
very long and the pump laser is introduced co-axially, higher order hybrid modes are
effectively damped (have low gain) and the EH11 mode predominates. When the EH11
mode is coupled out of the laser, 98% of its energy is converted into the HG TEM00
mode. The output beam profile is then linearly polarised (either parallel or perpendicular
to the polarisation of the pump radiation) and Gaussian, so it is possible to propagate the
FIR radiation efficiently over a few metres in free space. This mode selectivity is the
main advantage of a FIR waveguide laser over more conventional systems.
The main drawback to FIR lasers is their low power output. Since each incoming
IR photon, pump, can only generate a maximum of one FIR photon, FIR, the theoretical
upper limit for the conversion of the pump power, Ppump, to FIR laser power, PFIR, is
given by the Manley-Rowe condition [6]:
 FIR
PFIR  1
Ppump
2
(2.5)
pump
The additional loss factor of ½ arises because the populations of the upper and lower
lasing states can only ever be equalled, and not inverted, by saturated pumping. If the gas
only absorbs a fraction of the pump energy then the effective pump power is given by
[7]:
Ppump  1  exp N ( ) L Ppump
32

CHAPTER TWO THEORETICAL BACKGROUND
where () is the frequency dependant absorption coefficient for the pump transition
(0.01-0.001cm-1Torr-1), L is the cavity length (1.5m), and N is the effective number of
passes the pump radiation makes through the laser cavity before it is dissipated by losses
other than absorption by the lasing gas (1). Harth has shown that the FIR output power
is ultimately limited because the laser gain profile becomes power broadened at high
pump powers, and the absorption coefficient of the pump transition decreases [8].
Experimentally, this means that there is always an optimum pump power level beyond
which the FIR laser output will not noticeably increase. In practice, FIR lasers can only
provide around 1% of the maximum theoretical output power, (between 1 and 100mW
for pump powers around 100W).
In waveguide lasers the radiation is confined within the cavity so the intensity of
the pump radiation is higher at a given pump power [6]. The pump radiation is focused
into the laser through a small hole in one of the end mirrors and expands slowly as it
propagates through the cavity. The beam is usually introduced slightly off-axis to ensure
that radiation is not fed back into the pump laser after each round trip of the FIR cavity.
This would lead to frequency ‘jitters’ in the pump laser and amplitude instabilities in the
FIR output [6]. The energy from the pump beam is fully dissipated after a few trips
through the cavity, either by molecular absorption or heat dissipation on the cavity walls.
A number of studies have shown that temperature gradients induced by the pump
radiation affect the amplitude and frequency stability of the FIR laser [e.g. 9, 10]. This is
overcome by cooling the input coupler, output coupler, and waveguide walls to around
2oC [10]. There are three further advantages to this process:
1. the populations of the vibrational energy levels in the lasing gas are governed
by Boltzmann statistics, so cooling significantly increases the ground
vibrational state population. Consequently more pump power is absorbed,
2. the rate limiting steps in the FIR laser mechanism are the rotational and
vibrational relaxations of the molecules to the ground state. If the walls are
cooled, a wall-molecule collision is more likely to result in one of these
relaxation processes occurring, so the laser efficiency will be increased as the
lower pump level is more rapidly replenished [11],
33
CHAPTER TWO THEORETICAL BACKGROUND
3. line competition is reduced in gases where a single pump frequency can be
used to access a number of FIR laser transitions, as the rotational energy
levels in the ground vibrational state are less highly populated.
All these processes increase the FIR laser output power at a single frequency. Bakos et al
have shown that buffer gases such as He can also be used to increase the vibrational
relaxation rate of the lasing molecules [11] provided that the collisional cross-section for
de-excitation between the molecules is large, and the laser gas is flowing through the
cavity. Most FIR laser gases are sufficiently expensive to preclude flowing
configurations: ideally FIR lasers should always be operated this way, as the ground state
population is then continually replenished.
The output power of the laser can be doubled by using a metallic insert: this is
useful when weak FIR lines are required, but on ‘strong’ FIR laser lines multiple
transverse modes are excited, and the interference between these modes produces
amplitude instabilities in the laser output. Compared to dielectric waveguides, a small
diameter, smoothbore, metallic waveguide exhibits much lower attenuation and
propagation losses at FIR frequencies. In metallic waveguides the radiation is circularly
polarised, usually in TE01 or TE11 non-Gaussian modes. These modes do not couple
efficiently into free-space and give randomly polarised, non-Gaussian output beams.
Jiang et al conducted a thorough investigation into the output characteristics of FIR
lasers with hollow Al or Cu waveguides inserted into the FIR cavity [12]. They found
that a corrugated, over-moded metallic waveguide worked as efficiently as a smoothbore
waveguide provided that the groove corrugations were small in comparison to the laser
wavelength. The advantage to this type of waveguide is that it operates in an HE11 mode
(a hybrid mode similar to EH11) and couples efficiently to the HG TEM00 free space
mode.
FIR radiation is coupled out of the laser by a dichroic mirror. This type of output
coupler is designed to totally reflect the incident pump radiation and partially transmit
FIR radiation [6]. These mirrors are constructed from quartz or Si, with a thin layer of
Ag or Cu coated on one surface, except for a 6-10mm region at the centre, (figure 2.4).
34
CHAPTER TWO THEORETICAL BACKGROUND
un-coated
quartz
CO2 radiation
FIR radiation
FIR radiation
silver backed
quartz with
gold overcoat
TEM00 mode
in free space
EH11 Hybrid
mode
Extra-cavity
Intra-cavity
Figure 2.4: FIR dichroic output coupler. The transverse modes of the laser radiation
both inside and outside the cavity are illustrated for a vertically polarised beam. The
arrow lengths represent the E-field strengths.
The lasers used in the Cambridge TuFIR spectrometer are an Edinburgh
Instruments PL-6 CO2 Laser [13] and Edinburgh Instruments Model 295 FIR Gas Laser
[14]. A clear description of the lasing mechanism in a CO2 laser can be found in Davies
[15] or Hollas [16]. The PL6 CO2 laser is a step-tunable CW CO2 laser operating
between 9 and 11m. The laser cavity is a ‘U’-folded resonator, with parallel, watercooled, discharge tubes that are bevelled to minimise wall reflections and suppress
higher order transverse modes. The discharge tubes are sealed at the cathode end by two
ZnSe Brewster windows, but at the anode end the tubes are open. Two intra-cavity
steering mirrors are positioned at the anode end of the discharge tubes: these are
contained within an ‘O’ ring sealed housing which completes the cavity. The ends of the
cavity are defined by a gold-coated, water-cooled, blazed diffraction grating (90
lines/mm) and an output coupler with a 60% reflective coating on its intra-cavity face.
As the diffraction grating angle is changed, the laser is forced to oscillate on one of over
100 discrete frequencies that correspond to ro-vibrational transitions in the CO2 gas. The
output coupler is constructed from ZnSe and is wedged to prevent standing waves
building up within the coupler itself. The intra-cavity face is concave, with a 10m radius
35
CHAPTER TWO THEORETICAL BACKGROUND
of curvature: the outer face is planar, and coated with an anti-reflective material. The
output coupler is mounted on a piezo-electric transducer, (PZT), which can be adjusted
to finely tune the cavity length to match the laser output frequency, (the actual tuning
range is c/2L where L=3.88m). The output beam is vertically polarised, >90% TEM00,
with a 1/e2 beam-waist around 10.5mm. The laser is capable of operating at output
powers up to 200W, but in this experiment it was necessary to achieve a balance between
the output power and beam quality. The laser was usually operated with a 19mA
discharge current (40kV discharge voltage) and a residual gas pressure of 20mBar. In
this configuration the output power ranged from 40W on the weakest laser lines to 130W
on the strongest lines. The laser gas was supplied pre-mixed by MG Gas products at 9%
CO2, 13.5% N2, remainder He.
In the original spectrometer configuration (figure 2.1) the lasers were orientated
at 180o to each other, with a pathlength of around 2m between the CO2 laser output and
FIR laser input. The lasers were originally positioned on separate tables at different
heights, so each laser would mechanically vibrate independently of the other.
Consequently the FIR laser output was very unstable. In configuration B the lasers were
orientated at 90o to each other, the pathlength between them was reduced to 55cm, and
one mirror was completely removed from the system. The whole spectrometer was
positioned on a single optical bench that was isolated from mechanical vibrations below
9Hz. There were three main advantages to this changed orientation:
1. the mechanical vibrations of the lasers were synchronised, so the FIR laser
output was more stable,
2. with one less mirror, fewer distortions could be introduced to the ‘wavefront’, improving the mode quality of the CO2 laser when it entered the FIR
laser cavity,
3. the CO2 laser beam was less severely attenuated over the shorter path length,
so for identical operating conditions in the CO2 laser, more pump power
could be coupled into the FIR laser.
The Model 295 FIR Gas laser is a 36mm large-bore dielectric waveguide
resonator that is pumped co-axially by the CO2 laser. The cavity ends are defined by the
input and output couplers. The input coupler is a planar, gold-coated, stainless steel
mirror with a 1.5mm hole at its centre. The CO2 radiation is focused through this hole
36
CHAPTER TWO THEORETICAL BACKGROUND
and into the laser cavity by a 300mm focal length ZnSe lens. The output coupler is a zcut quartz, multi-layer dielectric mirror with a gold coating on the intra-cavity surface,
(as in figure 2.4). It is mounted on a micrometer translation stage that can be manually
adjusted to match the cavity length with the lasing frequency, (4mm travel with 2.5m
accuracy). The cavity is approximately 1.45m long: the waveguided region is 1.42m. The
input coupler, output coupler and laser cavity are cooled at 2 to 3oC by a ethylene glycolmethanol mixture from a re-circulating chiller. This was added to the system in 1995 to
improve the output power and stability of the FIR laser. The FIR output is >90% TEM00
and linearly polarised. The laser is usually operated statically, with 100 to 200mT of
lasing gas in the cavity. The gases are either obtained directly from lecture bottles or
enter the cavity under their own vapour pressure from a liquid reservoir. In the latter
case, the liquid is pre-purified by a freeze-thaw process. A set of needle values was
added to the gas system in 1996 so that the laser could be operated in a flowing
configuration, but they are only used with in-expensive FIR laser gases. Output powers
between 30 and 60mW are feasible from the strongest FIR lines, e.g. 118m CH3OH,
382.6m CH2F2, 393.6m HCOOH.
2.2 TuFIR Sideband Generation
A Schottky Barrier is a rectifying junction that is formed between a metal contact
(anode) and a semiconductor (cathode) due to the difference in work function between
the two materials [17]. When Schottky diodes are used as mixers at FIR frequencies,
they are designed with a low junction capacitance, Cj, and a low series resistance, Rs, to
minimise the signal conversion losses. The maximum cut-off frequency of the diode, fc,
can be estimated from the dc values of these two parasitics [18]:
fc 
1
2Rsdc C jdc
(2.7)
The cut-off frequency ultimately depends on the carrier mobility across the barrier, so
high frequency mixer diodes are fabricated from n-type semiconductor materials, (figure
2.5). This type of ‘dot matrix’ diode must have a wire attached to the surface to complete
the circuit. When a bias is applied across the junction, the true value of Rs is much
greater than the dc value. Consequently fc is around 2THz: much lower than the value
estimated from equation 2.7.
37
CHAPTER TWO THEORETICAL BACKGROUND
Whisker Contact
Antenna
SiO2
Pt/ Au anode
 100m
depletion layer
 5m
n-type GaAs
epitaxial layer
N=2x1017cm-3
 0.1-0.2m
n+ GaAs
Substrate
N=2x1018cm-3
 125m
ohmic contact
Au-Ge
Au
Figure 2.5: The structure of a dot matrix Schottky Barrier Diode, and whisker contact
antenna. N is the doping density in each layer.
The band structure of a Schottky Barrier diode is illustrated in figure 2.6. Under
forward bias conditions, current is conducted across the junction from the semiconductor
to the metal, by two different electronic processes [17]:
1. thermionic emission,
2. field emission.
At room temperature and high dosing levels the current transport mechanism is
dominated by thermionic emission. The I-V characteristics of the diode are derived from
the population densities of the electrons migrating forwards and backwards across the
barrier, and are given by [17]:
  qV  
  1
I (V )  I o exp

kT
(2.8)

 

where q is the electronic charge, V is the bias voltage, T is the diode temperature,  is the
ideality constant, and Io is the saturation current, which, assuming all current conduction
is due to thermionic emission, is given by [17]:
38
CHAPTER TWO THEORETICAL BACKGROUND
a). Before the metal and semiconductor are in contact the electrons in the metal
possess significantly less energy than the electrons in the semiconductor. The
work function of the semiconductor, s varies with its doping density.
metal
semiconductor
E
Eo-free space
energy level
Eo
qs
qm
Efm – Fermi
level in metal
Ec- bottom of
conduction band
Efs-Fermi level in
semiconductor
Ev- top of
valence band
b). Once the metal and semiconductor are in contact mobile electrons collect on
the metal surface leaving a depletion region in the semiconductor. A capacitance
exists between the –ve metal surface and the +ve charges in the depletion region.
The potential difference of this region is given by qb. Thermionic emission
occurs equally in both directions across the barrier.
depletion
zone
metal
semiconductor
E
Eo
Eo
Ef
qb =q(m s)
Ec
Ef
Ev
c). When forward bias is applied across the diode the electron energy increases
relative to the barrier height and electron emission from the semiconductor to the
metal increases.
metal
E
Eo
depletion
zone
+
V
semiconductor
q(b V)
Ef
Eo
Ec
Ef
qV
V0.9V for Au
Ev
Figure 2.6: The energy level band structure of a Schottky barrier diode.
q is the electronic charge.
39
CHAPTER TWO THEORETICAL BACKGROUND
 q 
I o  A * *T 2W exp  b 
 kT 
(2.9)
where A** is the modified Richardson Constant (4.4Acm-2K-2 in GaAs), W is the
junction area, and b is the barrier height, (see figure 2.6). Non-ideal diode behaviour
arises because:
1. imperfections in the fabrication process introduce impurities and crystal defects
to the semiconductor surface,
2. the Schottky Barrier height is lowered when the applied forward bias increases,
3. some current is always conducted by field emission,
4. the electric field is distorted at the edges of the diode.
Frequency mixing occurs on a Schottky diode if, in addition to the bias voltage,
time-dependant voltages are also present. To a first approximation, the Schottky barrier
diode is a square law device, i.e. if equation 2.8 is rewritten as a power series [18]:
I (V )  a1V  a 2V 2  a 3V 3  ...
(2.10)
where:
I  q 
an  o 
n! kT 
n
(2.11)
then the current response of the diode is adequately described by the first and second
order terms in equation 2.10. In the TuFIR spectrometer, tunable FIR radiation is
generated on the diode from non-linear mixing of FIR laser radiation and microwave
radiation. The oscillating E-fields of these signals are converted to alternating voltage
signals on the diode, given by:
Va  VFIR sin2  FIR t 
(2.12)
Vb  V  wave sin2    wave t 
(2.13)
The total voltage at the diode is then:
V  Vo  Va  Vb
(2.14)
and, assuming that the third and higher order terms in equation 2.10 are negligible, the
total current is given by:

 a V

t 
I ( t )  a1 Vo  VFIR sin2  FIR t   V  wave sin2    wave t 
2
o
 VFIR sin2  FIR t   V  wave sin2    wave
40
2
(2.15)