Investigation of the Electronic Discharge Circuit of a N2 Laser
Farooq Kyeyune (farooq@aims.ac.za)
African Institute for Mathematical Sciences (AIMS)
Supervised by: Prof. E. G. Rohwer and Prof. H. M. von Bergmann
Laser Research Institute, Stellenbosch University, South Africa
19 May 2011
Submitted in partial fulfillment of a postgraduate diploma at AIMS
Abstract
A nitrogen laser uses molecular nitrogen as the active medium, which in order to achieve lasing, has
to be excited from the ground state to the excited state as fast as possible. This requires a high
pumping power which can be provided by a fast electronic discharge circuit. However, the difficulty lies
in designing a circuit that can provide these high pump powers. In this work, we present a two coupled
LC circuit equivalent to the famous Blumlein circuit. The circuit is designed to provide extremely fastrising high-voltage required to excite the nitrogen gas through electron impact collisions. It is analysed
and simulations are carried out to understand its behaviour. Results from the simulations show that the
circuit operates at an underdamped discharge mode, and a high potential difference of about twice the
charge-up voltage of the capacitors in the circuit, is created across the electrode gap in the discharge
chamber.
Declaration
I, the undersigned, hereby declare that the work contained in this essay is my original work, and that
any work done by others or by myself previously has been acknowledged and referenced accordingly.
Farooq Kyeyune, 19 May 2011
i
Contents
Abstract
i
1 Introduction
1
1.1
Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.3
Properties of laser light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.4
Principle of laser action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.5
Pumping mechanism for a nitrogen laser . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2 Basic processes in gas discharge
8
2.1
Ionisation processes in gas discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
Breakdown in gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3 The equivalent Blumlein circuit design
13
3.1
Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2
Charge-Discharge RC circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3
Inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4
LC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.5
The equivalent Blumlein circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6
The equivalent Blumein circuit analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Experimental set up of a nitrogen laser
25
4.1
Nitrogen laser construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Conclusion
28
References
30
ii
1. Introduction
In this chapter, we present a brief historical background of nitrogen lasers, and their advances to date.
The unique properties of laser light which distinguishes it from an ordinary source of light are discussed
as a tool for understanding the principle behind the laser action. We then discuss the three-level pumping
mechanism for a nitrogen laser.
1.1
Historical background
The milestones in the history of lasers dates back 100 years when Einstein came up with the idea of
stimulated emissions, which became a platform for the Nobel prize winner T. H. Maiman in inventing
the first ever known laser in 1960. The first gas laser was developed by A. Javan, W. Benneth and D.
Harriott of Bell laboratories in 1961 using a mixture of helium and neon gases before Heard demonstrated
the action of the first nitrogen laser in 1963. Although the average power of this laser was very low, it
captured people’s attention to develop new design methodologies to improve the peak power output.
Research continued until a transverse electrical discharge at atmospheric pressure (TEA) nitrogen laser
was developed which was capable of producing megawatt powers using molecular nitrogen at atmospheric
conditions. This appeared at a time when it was needed in ultraviolet laser development, which led to
excimer lasers which are now of great use on large scale [Sve09].
Since then, lasers have developed enormously with a wide range of applications in industries, military,
medical, and academic research. Today, laser devices have numerous applications in communication
systems, in supermarkets as check–out scanners, in photocopying and scanning machines, in home
compact disc players and CDs, and many others. There are various types of lasers that have been
developed so far with a wide range of physical and operating parameters, characterised by their state of
lasing material such as solid-state, liquid or gas.
1.2
Motivation
Nitrogen molecules are the most abundant molecules in the earth’s atmosphere, constituting 78.08%.
Electron collisions with molecular nitrogen continue to receive interest due to their importance in gaseous
discharge processes, such as those involved in nitrogen lasers. Nitrogen lasers are capable of producing
very high power short pulses of ultraviolet light at a wavelength of 337.1 nm. Therefore their studies
play an important role in laser technology. They have a wide range of applications in dye laser pumping,
lidar (remote sensing), fast speed photography, atomic and lifetime spectroscopy, medical and biological
research, etc. Compared to other gas lasers, nitrogen lasers are very cheap and easy to construct. This
explains why they are sometimes called “home built lasers”. However, their applications are limited by
very poor efficiency, which does not often exceed 1%. A surprising report of higher efficiencies of about
3% was reported by Oliveira et al. [MA93]. As long as nitrogen lasers have been in the field, theoretical
and experimental investigations on various parameters and electronic discharge circuit designs continue
in order to improve their efficiencies. Thus, following the previous work done in [RR96, MA93, LSTT97],
it has been shown that nitrogen lasers operate efficiently, depending on a fast discharge circuit to provide
a high voltage across the laser tube, anterior to the gas breakdown.
1
Section 1.3. Properties of laser light
1.3
Page 2
Properties of laser light
In principle, light is made of photons which have an intrinsic energy determined by Planck’s relation
E = hν, where h is Planck’s constant and ν is the frequency of light. The concept of a laser is not
a magical phenomenon, but rather another source of light with very unique or extreme properties. To
get an understanding of these properties, we can think of an ordinary source of light such as a burning
candle, which radiates light in all directions emitting photons with different energies. In contrast, laser
light is concentrated in only one direction. These special properties are briefly discussed below using
the quantum theory of light [Sil04].
Monochromaticity: Like any other source of light, laser light is produced as a result of the emission
of photons, whose intrinsic energy determines its wavelength through the relationship λ = hc/E, where
h is Plank’s constant, E is the energy of the emitted photons and c is the speed of light. Under ideal
conditions, photons emitted by lasers are all of the same energy, giving rise to a single wavelength (very
pure colours). However, in reality, laser light is not purely monochromatic due to its finite wave train
durations.
Coherence: Based upon the fact that laser light is monochromatic, and the emitted photons have the
same wavelength, this intuitively means that the beam from a laser has the same phase relationship.
Laser beams are highly coherent in the sense that their phase relationship is maintained between any
two points. The coherence property is further divided into two parts: Temporal coherence is where two
waves stay in phase with each other at the same location but at different times, (given by lc = λ2 /∆λ)
λ is the average wavelength of the waves and ∆λ, is the difference between the wavelengths. The
second is spatial coherence where two waves stay in phase with each other at any two locations along
the wave profile, given by ls = λ/θk . Where θk u k/r, k is the distance between the waves and r is
the distance of each wave from the reference point [Sil04].
Directionality: Another unique property of laser light worth noting is its high degree of directionality.
In the previous example of an ordinary source of light (a candle), light is distributed in all directions,
unlike laser light which is concentrated in one direction with low divergence1 .
Ultrashort pulse generation: Unlike an ordinary source of light, lasers can be made to operate in
pulse form or continuously, with a time duration related to the spectral bandwidth through the equation
tp ≥ 1/∆ν, where ∆ν is the spectral bandwidth of the laser light.
Spectral brightness: A combination of the narrow line width due to monochromaticity and insignificant
laser beam spreading, are good enough to summarise the laser properties, but spectral brightness is also
superlative in distinguishing laser light from an ordinary source of light. It is defined as the power
emitted per unit surface area per unit solid angle, which is in this case very high.
After gaining an insight into what lasers are all about, we stand in a position to define a laser as a device
with the capability of transforming energy in the form of electrical, chemical or inherently optical into
coherent optical emissions, with its originality from an acronym for Light Amplification by Stimulated
Emission of Radiation [Sve09].
1
Beam spreading is insignificant.
Section 1.4. Principle of laser action
1.4
Page 3
Principle of laser action
After discussing the properties of laser light, in this section, we intend to discuss the basic principle
through which laser light can be generated. The mechanism of laser action involves three processes:
photon absorption, spontaneous emission and stimulated emission. In the description of these processes,
we consider a simple two level–energy system represented in Figure 1.1, where E1 and E2 are the lower
and higher energy levels with populations N1 and N2 , respectively. Normally, atoms or molecules
depending on the laser material undergo repetitive jumps from one level to the other by either the pump
source 1 or the emission of photons.2
Figure 1.1: Energy conversion processes involved in laser action [Kas99].
Photon absorption: The atoms jump from the lower level to the upper level, with a discrete amount
of energy given by hν = E2 − E1 , as shown in Figure 1.1a. Due to the energy transfer from E1 to E2 ,
there is an absorption of photons, with the absorption rate given by
dN1
dN2
=−
= −B12 N1 ρ(ν),
dt
dt
(1.4.1)
where ρ(ν) is the spectral energy density (intensity) at a frequency ν, and B12 is Einstein’s coefficient
of absorption. In the absorption process, a photon is annihilated.
Spontaneous emission: The excited atoms in the upper level are unstable, thus they decay to the lower
level by releasing some of their excess energy in the form of photons. The amount of excess energy
is given by hν = E2 − E1 , as shown in Figure 1.1b. Spontaneous emission processes are random and
isotropic in nature, thus occur without any external influence. The probability of the process to occur
is defined in terms of the decay rate of the upper level population and the increase rate of the lower
level population,
dN2
dN1
=−
= −A12 N2 ,
dt
dt
(1.4.2)
where A12 is Einstein’s coefficient which determines the probability of emission of a photon, the negative
sign signifies the decreasing population of atoms in the upper level.
Stimulated emission: The atoms in the upper level can be brought to the lower level through an
external influence. When a stimulating photon with exactly the same energy as the difference in energy
between the upper and lower levels, in the same direction and polarization is made to strike the excitable
laser material, the atom decays to the lower level and emits a photon [Sil04].
1
2
When the atom gains energy to jump from the ground state to the excited state
When the atom decays from the excited state to the ground state, photons are emitted
Section 1.4. Principle of laser action
Page 4
Both the emitted photon and the stimulating photon have an energy given by hν = E2 − E1 , as shown
in Figure 1.1c, with the transition rate given by
dN2
dN1
=−
= −B21 N2 ρ(ν),
dt
dt
(1.4.3)
where B21 is Einstein’s coefficient for stimulated emission, basically an intrinsic property of the system.
dA
I(z + dz)
I(z)
dz
Figure 1.2: A beam of intensity I travelling in z–direction through a medium.
Consider a beam of light with intensity I(z), entering a medium of thickness dz and cross-sectional
area dA. We would like to investigate the effect3 on the beam, as it propagates through the elemental
length dz, due to the absorption and stimulated emission processes we have discussed above. We shall
neglect the contribution due to the spontaneous emission, since the process is random, thus it offers
very small contribution in the direction of the beam.
The change in energy of the beam after traversing a thickness dz, is given by the product of the intensity
change and the cross-sectional area of the medium.
∆E = (I(z + dz) − I(z))dA = dI(z)dA.
(1.4.4)
This energy change can also be expressed as the energy of the photon per transition rate, multiplied by
the difference between the transition rate in the stimulated emission process and the transition rate in
the absorption process, within the volume dV = dAdz. Equation (1.4.4) can then be written as
dI(z)
hν
=
(N2 B21 I − N1 B12 I).
dz
c
(1.4.5)
Since B21 = B12 , equation (1.4.5) can be simplified to obtain
dI(z)
= σI(N2 − N1 ),
dz
(1.4.6)
where σ is the stimulated emission cross-section, given by σ = hνB21 /c.
Assuming the populations N1 and N2 are constant, and integrating equation (1.4.6), we get
I = I0 exp(σ(N2 − N1 )z) = I0 exp(gz),
(1.4.7)
where g = σ(N2 − N1 ), the gain medium and I0 is the intensity of the beam just before entering the
medium.
3
Elemental change of the intensity dI(z)
Section 1.4. Principle of laser action
Page 5
Analysing equation (1.4.7), we see that when N2 < N1 , a case where the population of the atoms in
the lower level is more than that in the upper level, the equation reduces to I0 exp(−αz), where α is
called the absorption coefficient, given by α = σ(N1 − N2 ). When there are equal populations of atoms
in both levels i.e. N1 = N2 , the equation reduces to I0 , thus the intensity of the beam does not change
as it propagates through the medium. The two processes, the stimulated emission and absorption are
then counterbalanced. Lastly, when there are more population of atoms in the upper level than the
lower level i.e. N1 > N2 , the equation becomes I0 exp(gz), thus a gain.
1.4.1 Population inversion. One of the prerequisites for lasing to take place continuously, or to give
an even-pulsed output, is population inversion. A population inversion takes place when there are more
atoms in the upper level than the lower level. Atoms are excited to the upper level through a process
called “pumping”, which differs for different lasing media. Therefore, the pumping mechanism should
in principle be able to excite atoms or molecules continuously to achieve a uniform output of the laser
pulses.
For any ideal population of atoms, not all of them are excited to the upper level since the probability of
stimulated emission is very small and negligible. Thus at a thermal equilibrium, there will be a smaller
number of atoms in the excited state than the ground state. The population between the two energy
levels is given by Boltzmann’s distribution [ME10]
N2
hν
E2 − E1
= exp −
,
(1.4.8)
= exp −
N1
kT
kT
where N2 and N1 are the number of atoms in the excited state and the ground state, k is Boltzmann’s
constant, and T is the absolute temperature of the material.
Energy
E2 N2
N1
E1
Energy
Thermal equilibrium
N2 < N1
E
exp − kT
Population density
(a)
E2 N2
N1
E1
Population inversion
N2 > N1
E
exp − kT
Population density
(b)
Figure 1.3: Population distribution between levels 1 and 2 for thermal equilibrium and population
inversion conditions.
Under equilibrium conditions, the system acts as an absorber at a frequency ν, since N2 < N1 . However,
if the inversion condition is achieved N2 > N1 , then the system acts as an amplifier [Sil04]. This is
defined as population inversion.
1.4.2 Essential elements of a basic laser. The principle of laser action is based on three main features:
a lasing medium, a pump source to provide excitation energy, and a resonator cavity. The general layout
of a laser is shown in Figure 1.4.
Section 1.5. Pumping mechanism for a nitrogen laser
Page 6
Lasing medium: This is also called an active medium, where the atoms or molecules are pumped from
the ground state to the excited state. The lasing medium can be a solid, liquid or gas. Examples of gas
lasers include, carbon dioxide, argon, Helium-neon and nitrogen lasers.
Pump source: A suitable pump source is required to create a population inversion, a necessary condition
for the spontaneous emission of photons. However, the question remains that how does the pump
power required to acquire the population inversion supplied to the active medium? The different types
of pumping are: Chemical pumping, which is basically a chemical reaction, Optical pumping where
the source of energy is light, basically used in solid state lasers, and Electric pumping where energy is
obtained from electricity. We use the latter for the excitation of gas lasers such as nitrogen, which shall
be discussed in the next section.
Resonator cavity: This is sometimes called “feedback” mechanism. The arrangement consists of two
mirrors. One mirror is completely reflective so that the desired emitted photons get back into the
system for more amplification and the other mirror, is partially transparent for the output of the portion
of the laser beam with the required threshold value. It is important to note here that not all lasers use
a feedback mechanism. A nitrogen laser does not require resonator cavity (lases in air), since it has
extremely high gain [ME10].
Excitation
mechanism
(Pump source)
Lasing medium
High reflector
e.g Molecular nitrogen
Output
Partial reflector
Resonator support structure
Figure 1.4: Schematic layout of the laser.
1.5
Pumping mechanism for a nitrogen laser
A nitrogen molecule like any other diatomic molecule, possesses vibronic energy levels that constitute
both vibrational and electronic states. These energy states are mainly separated by the change in
electron energy of the atoms, and a small contribution resulting from the vibrations within the nitrogen
molecules themselves. The laser action therefore involves a series of the transitions, resulting from the
changes in vibrational and electronic states of the nitrogen molecules, which are nearly spaced to favour
the emission of ultraviolet radiation at 337.1 nm [Cse04]. The pumping mechanism in a nitrogen laser,
involves collision of high energy electrons with the gas molecules in a laser tube through electron impact
excitation. The accelerated electrons in the laser tube strike the nitrogen molecules, thus exciting them
to a higher electronic energy state. Figure 1.5 shows the energy level scheme for a nitrogen laser, with
each level showing a series of vibrational energy level, which depends on the internuclear separation of
the molecule.
Section 1.5. Pumping mechanism for a nitrogen laser
Page 7
The laser normally starts when the nitrogen molecules become excited by an electric discharge in the
lasing tube from the electronic ground state, labelled X 1 Σ+
g energy band to the upper lasing level,
labelled C 3 Πu energy band. The molecules at the upper lasing level are unstable, thus they decay to
the lower lasing level, labelled as B 3 Πg energy band, by emitting photons of ultraviolet radiation at
337.1 nm. This corresponds to only energy levels with the lowest vibration state, for which ν = 0, and
ν = 1 corresponding to the upper vibrational state. However, other transition states are also possible.
For example in a 1-0 state, where in this case the lower lasing level involved in the vibration state is at
ν = 1 and the upper lasing level is the same as the lowest, at ν = 0 [Cse04]. This results into a shorter
jump during the transition state than before, thus an output with a lower energy at a wavelength of
358 nm is given off as shown in Figure 1.5. When a molecule emits a photon as it falls to the lower
lasing level, it then decays to a metastable state where it stays. Thus a nitrogen laser is effectively a
three-level laser.
Figure 1.5: Energies of the molecular nitrogen, as a function of internuclear separation relevant for the
nitrogen laser [Sil04].
The biggest problem with a nitrogen laser is the lifetime of the upper lasing level, which is their greatest
barrier in the laser development. The transition lifetime of the gas molecules from the upper lasing
level to the lower lasing level is very short (≤ 40 ns) as compared to lifetime for the transition between
the lower lasing level and the metastable state (about 10 µs) [Cse04]. Thus the population of the gas
molecules at the lower level exceeds that at the upper lasing level. This means that the lasing process
will terminate, since the condition for lasing to take place is violated. Creating a population inversion
at the upper lasing level in this case, is only possible if the pump power from the source is sufficient
to have the nitrogen molecules in the upper lasing level as quick as possible. Gas lasers in general, are
pumped directly by electron-impact excitation. The gas molecules are excited from the ground level to
the upper level through collisions with the electrons accelerated by the electric field set-up between the
electrodes of the discharge chamber. In the next chapter, we discuss the various mechanisms involved
in gas discharge.
2. Basic processes in gas discharge
Gas discharges normally occur when an applied voltage to the gas in-between the electrodes of a discharge
chamber reaches a critical value called the breakdown voltage, beyond which the gas suddenly switches
from being a dielectric to a conductor, so that current flows through the electrodes. Electrons normally
called “seed electrons” produced by e.g. cosmic rays collide with the gas molecules, thus exciting them
to higher energy states. In this chapter therefore, we present some of the important processes in gas
discharges and later on, introduce the concepts of Townsend breakdown (the electron avalanche) and
Streamer breakdown. Streamers results into non–uniform discharge which is not desired in gas lasers,
thus we discuss the preionisation mechanism, which suppresses the effect.
2.1
Ionisation processes in gas discharge
Ionisation is a process by which an electron is removed from an atom, leaving an atom with a positive
net charge.
N + e− → N + + 2e− .
Collision processes are responsible for the ionisation of atoms and molecules, which are required for
self-sustained discharges and excitation processes in lasers, to selectively populate the upper laser level.
There are three main types of collisions involved in ionisation: elastic collision, inelastic collision of the
first kind and inelastic collision of the second kind. However, for our study purposes, we discuss the
first two types of collisions, which are only distinguished by energy transfers.
2.1.1 Elastic collision. During the collision, there is negligible exchange of energy between the striking
electron and the nitrogen molecule, since the mass of the electron is smaller as compared to that of the
gas molecule. Thus there is only redistribution of their kinetic energy (kinetic energy of the system is
conserved).
2.1.2 Inelastic collision of the first kind. The kinetic energy of the colliding partner is converted into
internal energy. In this case, the neutral gas molecules do not always get ionised on electron impact.
The molecules can subsequently give out a photon, with an energy E = hν, where h is Planck’s constant
and ν is the frequency of the emitted photon. With this kind of collision, the gas molecules are left
in an excited state with an excitation energy equivalent to the difference in the kinetic energy of the
striking electrons.
N + e− → N ∗ + e− .
Other relevant processes involved in Inelastic collision of the first kind include:
Ionisation by double impact: In the previous collision, the gas molecule is left in an excited state.
Thus the ionisation process can occur through collision of the molecule with a slow moving electron
within the field. This completes the ionisation process, where the electron would require less energy
than the ionisation energy of the molecule, but greater than the addition energy required for ionisation.
N ∗ + e− → N + + e− .
Photo ionisation: As a result of high energy, a molecule or an atom at the ground level can be ionised
by a photon, with an energy given by hν. The end result is emission of a photon, whose energy must
be greater than the ionisation energy of the molecule, for the ionisation to occur.
N + hν → N + + e− .
8
Section 2.2. Breakdown in gases
2.2
Page 9
Breakdown in gases
There are basically two known mechanisms which leads to breakdown of gases: These are the Townsend
breakdown and the Streamer breakdown.
2.2.1 Townsend breakdown mechanism. It is one of the processes considered in breakdown of low
pressure gases, and it involves generation of successive secondary electrons at the cathode by the action
of photons or ions striking the cathode.
When a high voltage is applied to the electrodes of a gas discharge chamber, an electric field is created.
The field accelerates the free electrons as they move from the cathode to the anode under the influence
of an electric force. The electrons acquire kinetic energy which they pass onto the gas molecules through
collision as they traverse the field. When the molecules gain enough energy, necessary for ionisation to
take place, they may ionise, releasing new electrons. The new electrons and the striking electrons are
then accelerated in the field and each will again ionise a gas molecule, liberating other new electrons.
The process repeats itself, resulting in an exponentially growing electron cloud, a state called electron
avalanche, as shown in Figure 2.1. The ionisation process increases rapidly with increasing voltage until,
ultimately gas breakdown occurs, which is normally characterised by sparks.
Figure 2.1: Illustration of the electron avalanche mechanism.
When the applied voltage across the chamber exceeds the breakdown voltage Vbr , we can formulate the
condition for a uniform discharge using the following mathematical analysis.
2.2.2 Townsend breakdown mathematical analysis. Electrons drifting in a gas for a distance dx,
under the influence of an applied electric field will undergo ionising collisions, thereby increasing their
number by
dn = α ndx,
(2.2.1)
where n is the number of electrons moving a distance x from the cathode, and α is Townsend’s first
ionisation coefficient, defined as the average number of ionising collisions caused by a single electron
drifting a distance 1 cm in the direction of the applied uniform electric field [Raz91].
Section 2.2. Breakdown in gases
Page 10
Rearranging, and integrating equation (2.2.1), gives
Z
Z
dn
= α dx,
n
n(x) = n0 eαx ,
(2.2.2)
where n0 is the number of electrons drifting from the cathode.
If the anode is at a distance d from the cathode as shown in Figure 2.1, then the number of electrons
striking the anode per second is given by
nd = n0 eαd .
(2.2.3)
Consider an additional ionisation coefficient, Townsend’s second ionisation coefficient γ, defined as the
average number of secondary electrons produced at the cathode per ionising collision in the gap. For nd
electrons traversing a distance d of the gap to the anode, then the number of the secondary electrons
generated at the cathode is given by
Z d
γ
0
γ n(x)dx = n000 (eαd − 1),
n0 =
(2.2.4)
α
0
where n000 is the total number of electrons leaving the cathode per second , and its given by n000 = n0 +n00 .
Substituting equation (2.2.4) into the previous expression for n000 and rearranging, one gets
n000 =
n0
.
γ αd
1 − (e − 1)
α
Therefore, the total number of electrons arriving the anode is nd = n000 eαd , which can finally be expressed
as
nd =
n0 eαd
.
γ
1 − (eαd − 1)
α
(2.2.5)
The discharge will become self sustained if for each electron leaving the cathode and crossing the gap in
form of an avalanche, at least one secondary electron is generated at the cathode. Thus we require the
denominator of equation (2.2.5) to be less or equal to zero, a condition which is known as Townsend’s
criterion for the spark breakdown of a discharge gap.
γ αd
(e − 1) > 1.
(2.2.6)
α
2.2.3 Paschen’s law. Let us suppose that the electrons and ions are moving in a uniform electric field
E in a gas chamber, with a gas pressure p. The mean energies of the electrons attain equilibrium values
that greatly depend on the ratio of E and p, thus the first and second Townsend ionisation coefficients
are then respectively defined by
α/p = g1 (E/p)
and
γ/p = g2 (E/p),
with E = V /d.
Now, if we apply the Townsend’s criterion for a uniform field spark gap breakdown of a discharge gap
given in equation (2.2.6), at the breakdown voltage Vbr , we shall have
o
g2 (Vbr /pd) n pd g1 (Vbr /pd)
e
− 1 = 1.
g1 (Vbr /pd)
Section 2.2. Breakdown in gases
Page 11
The breakdown voltage for any gas is an implicit function, only dependent on the product p × d and not
separately on pressure p and inter-electrode separation d. It is defined as Vbr = g(pd). The breakdown
charateristics defined by g is called the Paschen curve and can be calculated if the functions g1 and g2
are known.
Figure 2.2: Breakdown characteristics (Paschen curve).
The minimum breakdown voltage is called the Paschen’s minimum. When the density of atoms decreases, there are fewer collisions of the electrons with the gas molecules. Therefore the minimum of
the characteristic curve signifies the low density atoms which in turn affects the chances of ionisation
[Luc01]. Hence breakdown can only take place if the probability of the gas molecule ionisation is increased, thus accounting for the the increase in the breakdown voltage to the left of the curve from
point C through A as shown in Figure 2.2.
For most gases, the Paschen curve assumes a minimum at Vbr u 300 V, and at pd = 0.1−0.57 cm Torr.
The atmospheric pressure breakdown value for nitrogen gas is Vbr = 30 kVcm−1 . However, Paschen’s
law fails at very low pressure and very high pressure compared to atmospheric pressure [Raz91].
2.2.4 Streamer breakdown mechanism. The Townsend breakdown mechanism, relying on successive
avalanches crossing the gap, is only valid for slowly rising voltages and low pressures. This results in a
very diffuse form of discharge. However, if the product p × d is large enough and at a fast rising voltage,
the space charge field of an individual avalanche can become comparable to the applied electric field.
This causes a distortion of the electric field, whereby secondary avalanches triggered by photo-ionization
will converge towards the head of the primary, leading to a highly filamentary channel (a streamer)
bridging the gap.
E
CATHODE
+
+
+
+
ions
+
+
+
+
+
+
--
+
+
+
- - --- ANODE
- - - - - - - -- - -- electrons
Figure 2.3: Illustration of the generation of the secondary avalanches on the primary avalanche.
Section 2.2. Breakdown in gases
Page 12
Figure 2.3 illustrates successively generated secondary avalanches along the tail of the primary avalanche
due to the presence of a high space charge field in the avalanche tip. The space charges form a sort of
dipole, with all the electrons at the head of the avalanche [Raz91].
Since the positive ions are heavier than the electrons, the latter advance rapidly in the electric field, and
the former are left behind, thus forming a tail like as shown in Figure 2.3.
2.2.5 Preionised breakdown. As a result of streamer formation discussed in the previous subsection,
this ignites a non-uniform discharge in the space between the electrodes, triggering rapid spark breakdown
according to Raether’s breakdown criterion. This kind of discharge is undesired in gas laser, where a
uniform glow discharge is required. The effect can be suppressed by proper preionisation of the gas
volume, leading to a homogeneous gas breakdown and uniform discharge, even at large enough values
of pd. However, to produce a uniformly distributed discharge at large values of pd, two most important
conditions must be met. Firstly, the number of initiating electrons drifting from the cathode must
satisfy the condition n0 > rc−3 , where rc is the critical size (radius) of the primary avalanche. Thus if a
large number of avalanche initiating electrons is uniformly distributed throughout the gap, the density
of the initiated avalanches can increase, leading to their overlap before reaching a critical size. Hence
the space gradients are then reduced, suppressing streamer development and in turn initiating a uniform
glow discharge.
Secondly, since preionisation mechanism is limited in time, the initial electrons which form the primary
avalanche drifting from the cathode with a thickness x, must satisfy the condition x < rc . Therefore
gas volume preionisation is purposely desired to create a uniform distribution of electrons in the gap,
with high enough density to cause overlapping of the avalanches [PT00].
In summary, ionisation of the gas molecules takes place when an electric field is created between the
electrodes of the discharge chamber. This accelerates the electrons which collide with the gas molecules
within the gap, thereby losing energy to them in form of excitation energy. However, in order to achieve
this, a high rising voltage is needed across the discharge chamber, which is provided by a fast discharge
circuit. In the next chapter, we present a detailed review of the fundamental concepts in circuit analysis,
with emphasis laid on LC circuits, aimed at designing and analysing a two coupled LC circuit which can
be used in the excitation of molecular nitrogen gas in a nitrogen laser.
3. The equivalent Blumlein circuit design
We present in this chapter an overview of the charging and discharging processes of capacitors in RC
circuits. A fast discharge circuit, based on two coupled LC circuits which can be used for the excitation
of nitrogen molecules is designed, and then analysed. In this chapter, the words laser tube and discharge
chamber are used interchangeably.
3.1
Capacitors
Capacitors are devices used to store energy as potential energy in an electrical field. The energy stored
can be availed within very short period of time, which is useful in electronic timing circuits. They have
many applications such as noise filtering and coupling circuits to mention but a few. Capacitors basically
consist of two conductive plates separated by an insulator, called a dielectric such as air or paper. The
magnitude of the charge stored on either plate of the capacitor is defined as
Q = CV,
(3.1.1)
where V is the voltage across the capacitor plates and C is a constant of proportionality called the
capacitance, a measure of the capacitor’s ability to store electrical energy [HRW01]. However, this
measure depends on the geometry of the plates (shape, position, and size), and the insulator between
the plates.
Parallel plate capacitors.
+Q
+
+
+
+
+
+
+
+
+
+
E
-Q -
-
-
d
-
-
-
-
-
-
-
Figure 3.1: Parallel plate capacitor, with a small separation between the plates to avoid edge effects.
If the plates of the capacitor are each of area A, and are at a separation d, with both plates given equal
but opposite charge Q, then from electrostatics, the field between the plates is given as
E=
σ
Q
,
=
ε0
ε0 A
where ε0 = 8.857 × 10−12 Fm−1 , the permittivity of free space, and σ is the charge density defined as
σ = Q/A. Assuming the separation d is very small (to avoid edge effects), then the electric field between
the plates is uniform. Therefore, the potential difference across the plates is defined by V = Ed. The
capacitance C = Q/V is now expressed in the form:
ε0 A
.
(3.1.2)
d
However, if the space between the capacitor plates is completely filled with a dielectric material, the
capacitance C increases by a factor κ, thus given by
C=
C=κ
13
ε0 A
d
Section 3.2. Charge-Discharge RC circuits
Page 14
where κ > 1, the relative permittivity of the dielectric. Introduction of the dielectric material between
the capacitor plates or any two electrodes limit the potential difference that can be applied between them
to a given value Vmax , called the breakdown voltage. Therefore capacitors or merely two electrodes are
rated to a specific voltage, beyond which the dielectric material will breakdown and form a conducting
path between the plates or electrodes. Often this breakdown is catastrophic, and it results in wastage on
energy (burning of devices). However, in fast discharge processes, it controls the triggering of a spark
gap, leading to a conducting state of the spark gap electrodes. A few such values are listed in the table
below [HRW01].
Table 3.1: Breakdown voltage for different materials.
Material
Air (1 atm)
Paper
Rubber
Glass
Pyrex
Polystyrene
Dielectric constant
(κ)
Breakdown voltage
(kV/mm)
1.00059
3.5
2-3.5
4-6
4.7
2.6
3
16
30
9
14
24
Geometrically, the value of C depend on the quantities A and d, and the dielectric material. In terms of
interpretation, we see that large plates give rise to more charges on the plates, thus a higher capacitance
and the decrease in separation increases the capacitance.
When a capacitor is connected to the battery, an electric field is set-up across the capacitor plates.
Therefore, the work done by the battery in transporting the charges from one capacitor plate to the
other against the electrostatic forces, is converted into energy which is stored in the electrical field
between the plates. This energy is given as
Z
U=
0
3.2
Q
q
1 Q2
dq =
.
C
2 C
(3.1.3)
Charge-Discharge RC circuits
Charging and Discharging processes play an important role in the energy output characteristics in electric
discharge circuits. In this section, we present expressions and analyses for the voltages and currents in
RC circuits, when subjected to sudden excitations e.g. charging and discharging processes of a capacitor.
3.2.1 RC Charging circuit. In an RC charging circuit, a capacitor that is initially uncharged is connected
in series with a charging resistor R, to a constant voltage supply V0 as shown in Figure 3.2(a). When
the switch is open at t = 0, no current flows through the circuit and there will be no charge on the
capacitor plates. With the switch closed, the current begins to flow through the resistor, causing a
voltage to appear across the the capacitor plates. Since electrons flow in the opposite direction to that
of current, the side of the capacitor plate connected to the positive terminal develops positive charges
whilst the other develops negative charges. The voltage build-up across the capacitor increases and that
across the resistor decreases, until the former matches with that of the constant supply voltage. At this
point, there is no more flow of electrons.
Section 3.2. Charge-Discharge RC circuits
R
Page 15
C
I
+ +
V0
V0
K
C
- -
R
K
(a) RC circuit for charging a capac- (b) RC circuit for discharging a caitor.
pacitor.
Figure 3.2:
Therefore the charge on the capacitor at a maximum being equivalent to CV0 . To understand the
behaviour of the voltage and current when the switch is closed, we analyse the circuit diagram in Figure
(3.2(a)) above using Kirchoff’s voltage law, defined by V0 = VC (t) + VR (t).
Using equation (3.1.1) and Ohm’s law, then the total charge traversing all the components in the circuit,
is given by
CV0 = Q + RC
dQ
.
dt
Re arranging this expression and integrating both sides, yields,
ln |CV0 − Q| = −
t
+ A.
RC
Initially at t = 0, the charge on the ideal capacitor plates is zero, thus the constant of integration
A = ln |CV0 |. Finally the charge on the plates at any time is given by
Q = CV0 (1 − exp(−t/RC)).
(3.2.1)
From equation (3.1.1), the voltage across the capacitor at any time is given by
VC (t) = V0 (1 − exp(−t/τ )).
(3.2.2)
Differentiating equation (3.2.1), we obtain the current through the circuit,
I(t) = I0 exp(−t/τ ).
(3.2.3)
where τ is called the time constant, which determines the rate at which the capacitor charges when
connected to a constant voltage supply through a charging resistor.
Analysing equations (3.2.2) and (3.2.3), we see that when the switch is open at t = 0, the voltage across
the capacitor is zero (VC = (1 − e0 )V0 = 0) and the current through the circuit is maximum. When the
switch is closed, the voltage across the capacitor increases exponentially with time as shown in Figure
3.3(a). At t = τ , the voltage across the capacitor is approximately 63% of V0 i.e. (1 − e−1 )V0 u 0.63V0 ,
and it can be shown that the capacitor is said to be almost fully-charged at t = 5τ , that is 99% of V0
[THG11]. Therefore the time frame for charging a capacitor depends on the product C and R. Thus in
a fast discharge circuit like the one under study in our work, a small charging resistance is required to
charge the capacitor as fast as possible (possibly time in nanosecond).
Section 3.2. Charge-Discharge RC circuits
Page 16
3.2.2 RC discharging circuit. We consider a capacitor C and a resistor R connected in parallel with a
constant voltage supply V0 (no internal resistance) as shown in Figure 3.2(b). When the switch is closed
for a time compared to five times the time constant discussed in the previous subsection, the potential
difference across the resistor and the capacitor is equal (fully-charged capacitor). The capacitor begins
to discharge through the resistor when the switch is opened. Applying Kirchoff’s voltage law to the
capacitor-resistor loop in the circuit diagram shown in Figure 3.2(b), and then Ohm’s law, we obtain
Q
− IR = 0.
C
(3.2.4)
Since the charge across the capacitor is decreasing, then I = −dQ/dt. We rearrange equation (3.2.4)
to obtain a first order differential equation
dQ
Q
+
= 0.
dt
RC
(3.2.5)
The solution to this equation is given by Q(t) = Q0 exp(−t/τ ), where τ = RC, the time constant,
which determines the rate at which the capacitor discharges through R. From equation (3.1.1), the
voltage across the capacitor at any time is given by
VC (t) = V0 exp(−t/τ ).
(3.2.6)
The exponential decay characteristics of the voltage across the capacitor as it discharges through the
resistor is shown in Figure 3.3(a). Similar mathematical analysis is made for the voltage across the
capacitor at different times during the discharging as that during the charging process. At t = τ , the
voltage falls to 37% of V0 i.e. V0 e−1 u 0.37V0 , and one can show that at t = 5τ , the voltage across
the capacitor is less than 1% of V0 [THG11]. Again the time scale is determined by the values of R and
C, thus with very small value of R, the rate of discharge is high.
Figure 3.3(b) shows the voltage characteristics in a capacitor which is continuously charged and discharged.
Charging–discharging characteritics
16
14
Charging
Charging
Discharging
12
10
Voltage (kV)
Voltage (kV)
V0
14
12
0.63 V0
8
6
0.37 V0
4
10
8
6
4
Discharging
2
0
Continuous charging–discharging characteristics
16
V0
0
10
20
t=τ
30
Time (ms)
(a)
40
50
2
60
0
0
1
2
3
Time (ms)
4
5
6
(b)
Figure 3.3: Capacitor voltage vs time for the RC charging and discharging circuits.
The results shown in the Figure 3.3 have been calculated from equations (3.2.2) and (3.2.6), using the
following values: V0 = 15 kV, C = 6 µF and R = 3 Ω.
Section 3.3. Inductors
3.3
Page 17
Inductors
Like capacitors are used to produce an electric field, inductors can be used to produce a magnetic field,
thus valuable in electric oscillation systems. Related to equation (3.1.1), the voltage across an inductor
is given by Faraday’s law
VL = L
dI
.
dt
(3.3.1)
The voltage across the inductor is proportional to the inductance L, thus at a constant rate of change
of current (e.g. dc steady state), the voltage across the inductor is zero. Hence the inductor acts as a
conductor in contrast to a capacitor which acts as an open circuit in DC circuits [THG11]. If a current
I is established through an inductor of N turns, it produces a magnetic flux ΦB through the centre of
the inductor. The inductance of the inductor is then given by
L=
N ΦB
.
I
To gain more insight into the geometrical dependence of the inductances on various parameters, we
shall consider a solenoid of length l, with a cross-sectional area A. We present the expression for the
inductance of the solenoid as
L = µ0 n2 lA,
(3.3.2)
where n is the number of turns per unit length, and µ0 = 4π × 10−7 Hm−1 , the permeability constant.
Geometrically, large solenoids produce high inductance, this can be achieved by increasing the number
of windings of the solenoid since it varies as a square.
Due to the induced emf in an inductor, this leads to the establishment of an instantaneous current, so
that the power dissipated in the inductor of inductance L is given by
P =L
dI
dU
I=
.
dt
dt
(3.3.3)
Finally by integrating this equation, we obtain the magnitude of the energy stored in the inductor as
Z
1
U = L IdI = LI 2 ,
(3.3.4)
2
where I is the instantaneous current driven through the circuit. Important to note here is that equation
(3.3.4) compares to equation (3.1.3).
Section 3.4. LC circuit
3.4
Page 18
LC circuit
LC circuit commonly known as a resonant circuit consists of a capacitor initially charged, connected to
an ideal inductor (no resistance) via a switch.
I
+ +
C
L
- -
K
Figure 3.4: Schematic diagram for a simple LC circuit.
When the switch is closed at t = 0, all the energy is stored in the electric field. At this point, there is
no current flowing through the circuit, thus no energy stored in the inductor. As the capacitor begins
to discharge, the current flows through the inductor, thus setting up a magnetic field in the inductor.
The current flowing through the circuit increases until it attains a maximum value, when all the energy
is stored in the inductor, thus no energy stored in the capacitor.
The capacitor begins to charge again and the process keeps repeating itself, a phenomenon called
electrical oscillation. To obtain the angular frequency of oscillations, we shall apply Kirchoff’s voltage
rule and Ohm’s law to the circuit diagram in Figure 3.4, to obtain
−
Q
dI
+L
= 0.
C
dt
(3.4.1)
Since the current increases as the charge decreases, then from the definition of current, we shall have
I = −dQ/dt so that the second order differential equation of the circuit is given by
d2 Q
1
+
Q = 0.
2
dt
LC
(3.4.2)
From the mechanics point of view, the oscillations in an LC circuit are√analogous to the mechanical
oscillations in a spring mass system, with an angular frequency, ω = 1/ LC. Then the second-order
differential equation becomes
d2 Q
+ ω 2 Q = 0.
dt2
(3.4.3)
The general solution to (3.4.3) is given by Q(t) = Q0 sin(ω + φ), where φ is the phase relationship.
π
Substituting for the initial conditions t = 0, and Q = Q0 , we obtain the phase relationship φ = , so
2
that the voltage across the capacitor and the current are given by
VC (t) = V0 cos(ωt)
and
I = I0 sin(ωt).
(3.4.4)
The voltage across the capacitor and the current through the circuit oscillate between maximum and
minimum, and they are 90o out of phase (see [THG11] for details).
Section 3.4. LC circuit
Page 19
3.4.1 Two coupled LC circuit. It consists of two capacitors C1 and C2 connected in parallel across
an inductor L2 of relatively high inductance, a switch S1 and a small inductor L1 .
L2
L1
+HV
A
B
C1
S1
C2
Figure 3.5: Schematic diagram for a simple two coupled LC circuit.
Initially, the two capacitors are simultaneously and slowly charged through L1 and L2 , to the same
voltage and polarity by a high rising voltage. At this point, there is no potential difference across AB.
When S1 is closed, the capacitor C1 is short circuited, thus suddenly discharges through the switch
which drains away the charges from it, leading to its upper terminal becoming more negative. In the
previous sections, we learnt that the time scale of the discharging process depends on the time constant
τ , which is shorter for very small resistances. Since the resistance of the switch is very small , then the
discharging process of C1 is very fast (takes a few nanoseconds). The inductor L2 is intentionally made
to be of high impedance so that during a very short period of discharging, it impedes any flux of current
through it, so that the capacitor C2 discharges slowly. Since the two capacitors discharge at different
time scales1 , we can treat the two coupled LC circuit in Figure 3.5 “independently”, as if they are not
coupled, thus the voltages across the capacitors are given by the first equation in (3.4.4). Then, the
voltage across AB is given by VAB = VC2 − VC1 .
Variation of voltages across C1 and C2 with time
15
VC2
25
5
Voltage (kV)
Voltage (kV)
10
0
5
Voltage characteristics across AB
30
VC1
20
Voltage doubling
15
10
5
10
150
5
10
15
20
Time (ms)
25
30
35
40
(a)
00
5
10
15
20
Time (ms)
25
30
35
(b)
Figure 3.6: Time axis plots for the voltages across the capacitors and AB (in figure 3.5).
1
Capacitor C1 discharges through the switch faster than C2
40
Section 3.4. LC circuit
Page 20
Figure 3.6(a) shows the voltage characteristics across the capacitors C1 and C2 which have been calculated from the first equation in (3.4.4), using the following values: C1 = C2 = 6 µF, L1 = 10 nH,
L2 = 1200 nH, and V0 = 15 kV. The build-up of the potential difference across AB, increases until it
attains a maximum value which is approximately twice the voltage V0 across each of the capacitors just
before discharging as shown in Figure 3.6(b). This voltage is refereed to as the breakdown voltage of the
gas we discussed in section 2. Such a high rising voltage across AB is required in an electric discharge
circuit, to create a high electric field between the electrodes in a laser tube, which consequently cause
the ionisation of the gas molecules.
3.4.2 Spark gap in an LC circuit. Unlike a normal switch which has to be controlled manually, a
free running spark gap in a very fast discharging circuit acts as a fast switch, which opens and closes
repetitively when a high rising voltage is applied to the circuit. This kind of switch is required in fast
discharge circuits, for initiating volumetric discharges, uniformly distributed over the electrodes of the
discharge chamber in gas lasers. However, it has restraints at achieving high repetition rates. The
on/off switching of the spark gap, is controlled by the breakdown voltage of the gas or any dielectric in
between the gap’s electrodes. The circuit diagram is shown in Figure 3.7 below.
L
C
A
+HV
Spark
gap
Discharge load
Figure 3.7: Schematic diagram of a spark gap in a resonant charging circuit.
The capacitor is first charged, to a suitable voltage and then discharges by the spark gap, into the
discharge load. The charging of the capacitor stops when a breakdown voltage is reached, with the plates
of the spark gap forming a conducting path, causing sparks. During the discharge process, the spark gap
is still in the conduction state, with a large amount of current flowing through it. Charging is initiated
again when the spark gap goes into the non-conducting state. The time lag between charging and
discharging is determined by how fast the spark gap recovers, which depends on the current generated
in the circuit. This current must be less than a maximum value called “the hold over current” [BNC98],
for the switch to recover. The situation is achieved by increasing the value of the inductance L, which
impedes the current flow, and in turn reduces the repetition rate2 . However, the repetition rate also
depends on the value of the capacitance, through the product LC, which is a measure of the charge-up
time for the capacitor.
The charging voltage across the capacitor is given by,
Vc (t) = V0 (1 − cos(ωt)),
where ω = √
1
.
LC
(3.4.5)
Differentiating equation (3.4.5), we obtain the rate of voltage rise at point A
dVc (t)
= V0 ω sin(ωt).
dt
2
The switch is very fast!
(3.4.6)
Section 3.5. The equivalent Blumlein circuit
Page 21
For a very short lived discharge,
in the case where the discharge load resistance is very small, and a high
√
rising impedance ω = 1/ LC very small, then from equation (3.4.6), we have sin(ωt) u ωt.
dVc (t)
V0 t
= V0 ω 2 t =
.
dt
LC
(3.4.7)
Equation (3.4.7) shows the dependence of the accelerating field on the residue charge, on the capacitance. Thus increase in the charging capacitance, implies a low accelerating field, which in turn for
a given value of the charging inductance, favours the recovery of the switch. Figure 3.8 shows the
charging–discharging characteristics when the switch recovers, and when it latches.
30
30
2V0
25
2V0
25
Charging
Discharging
20
Voltage (kV)
Voltage (kV)
20
15
15
10
10
5
5
0
Charging
Discharging
0
10
20
30
40
Time (ms)
50
60
70
80
(a) Recovery of the switch
0
0
10
20
Time (ms)
30
40
50
(b) When the switch latches
Figure 3.8: Charging–discharging characteristics, calculated from equation (3.4.5) using L = 10 nH,
C = 6 nF and V0 = 13 kV.
From equation (3.4.7), the maximum achievable repetition rate depends on the product LC. Therefore,
increasing the capacitance C, implies reducing L in order to maintain the same charge-up time of the
capacitor, but now without jeopardizing the chance of recovery of the switch.
A spark gap as a switch is preferred in a very fast discharging circuits over a manual switch, since the
time scale of such electrical circuits is very small (of order in nanoseconds or less). However, designing
and implementing an effective circuit, with higher repetition rates in practical situations require expertise
and extra care especially in the geometrical construction of the electrodes and their orientation.
3.5
The equivalent Blumlein circuit
The principle of operation of a nitrogen laser can be explained by referring to the electrical schematics
of the circuit shown in Figure 3.9. There are various electrical configurations used in exciting the gas.
However, according to Fitszimmons et al. [MA93], a Blumlein configuration has been reported to provide
higher efficiencies, through generation of a high voltage across the laser tube within a short period of
time (in nanoseconds). In our work, we present an equivalent Blumlein circuit, which is based on two
coupled LC circuits.
The circuit consists of two parallel plate capacitors C1 and C2 of extremely small inductances, connected
through charging inductors L1 and L2 , a laser tube or discharge chamber, and a spark gap.
Section 3.6. The equivalent Blumein circuit analysis
Page 22
L2
L1
+HV
A
Spark gap
B
Laser Tube
C1
C2
Figure 3.9: Schematic diagram of an equivalent Blumlein nitrogen laser.
The main principle behind the operation of this circuit is similar to that we have discussed in subsection
3.4.1, with the switch replaced by a spark gap, and a laser tube where the discharge of the gas takes
place introduced across AB, see Figure 3.5. The discharging process in this case terminates when the
breakdown voltage across the spark gap is reached, where the gas between the gap becomes conducting.
Typically, this state is attained for small scale lasers, when the applied voltage is about 15 kV.
3.6
The equivalent Blumein circuit analysis
L
A
+HV
Ll
S2 Rl
B
S1
Rg
Ig
C1
Il
C2
Lg
Figure 3.10: Circuit analysis for the equivalent Blumlein circuit
We can simulate each of the discharges taking place at the two non–linear elements; the spark gap and
the laser tube in the circuit diagram in Figure 3.9, as a series connection of an inductor and a resistor.
We let Rl and Ll to be the resistance and inductance associated with the laser tube, and Rg and Lg
similarly associated with the spark gap as shown in Figure 3.10. Switches S1 and S2 have been included
to indicate the on/off-set discharges in either the spark gap or laser tube, respectively.
Before firing of the spark gap, S1 and S2 remain open. When the spark gap fires (just before S2 closes),
the gas in the gap is in a conducting state, thus a current Ig flows through the circuit composed of
C1 , Lg and Rg , as indicated by the loop Ig . Thus there is a voltage build-up across the laser tube until
a threshold value is reached when the gas breakdown in the tube occurs, so that S2 closes. A current
Il will now flow through the circuit composed of C1 , C2 , Rl and Ll as indicated by the loop Il .
Section 3.6. The equivalent Blumein circuit analysis
Page 23
This therefore means that two currents simultaneously flow through the circuit after the main discharge
in the laser tube has been initiated. In our analysis, we will consider a stage before the main discharge
of the gas in the laser tube takes place, at the moment when S2 is still open. This implies that no
current flows through the laser tube yet (Il = 0). The circuit diagram when S2 is open, is shown in
Figure (3.11(a)) below.
Rg
Ig
L
L
A
C1
B
C2
I1
A
Rg
Ig
C1
B
I1 C2
Lg
Lg
(a)
(b)
(c)
Figure 3.11: The equivalent Blumlein circuits before the laser discharge, with S1 in Figure 3.10 closed.
As we discussed before, due to different time scales, we can treat the behaviour of the two capacitors
independently, as shown in Figures 3.11(b) and 3.11(c).
Considering Figure 3.11(b) first, we derive the second order differential equation governing the flow of
the current Ig in the RLC circuit, by applying Kirchoff’s voltage rule to the loop Ig .
VLg + VRg − VC1 = 0.
(3.6.1)
Using Ohm’s law and equations (3.1.1) and (3.3.1), we have
Lg
dIg
1
+ Ig Rg −
Q = 0.
dt
C1
(3.6.2)
Substituting for Ig = −dQ/dt, one obtains
Lg
d2 Q
dQ
1
+ Rg
+
Q = 0.
2
dt
dt
C1
(3.6.3)
Equation (3.6.3) is analogues to the general equation for the damped harmonic motion, see [HRW01].
Therefore, the oscillatory solution with an exponentially decaying amplitude (underdamped) to the
equation, is written as
Q = Q0 exp(−t/τ ) cos(ωt),
(3.6.4)
p
where τ u 2Lg /Rg and ω u 1/C1 Lg . The voltage appearing across the capacitor C1 in the RLC
circuit composed of Rg , Lg and C1 itself is given by,
VC1 (t) = V0 exp(−t/τ ) cos(ωt).
(3.6.5)
Next, we consider the LC circuit in Figure 3.11(c). The voltage across C2 , is given by the first equation
of (3.4.4), and the potential difference across the points A and B in Figure 3.11(a) is given by the
difference in the voltages across C2 and C1 , as it has been discussed in subsection 3.4.1.
Section 3.6. The equivalent Blumein circuit analysis
Page 24
Using equation (3.6.5), and the first equation in (3.4.4), we have simulated the behaviour of the voltages
across the capacitors C1 and C2 , and the laser tube, at the time before the breakdown of the gas in the
laser tube takes place. The results obtained in Figure 3.6 are almost similar to those obtained in Figure
3.5. In this case, the voltage across the laser tube rises in a short time (t ≤ 28 ns) until it reaches
a high value close to twice the charge-up voltage of the capacitors. After time t, the voltage then
collapses, following an underdamped mode of oscillation as shown in Figure 3.12(c). The calculations
have been done using the following values: the spark gap inductance Lg = 15 nH, spark gap resistance
Rg = 0.35 Ω, C1 = C2 = 6 nF, L = 120 nH and V0 = 10 kV. Almost similar results were reported in
[KNlR97, MA93, RR96] using experimental data and analytical models.
10
10
8
8
VC1
4
Voltage (kV)
Voltage (kV)
6
2
0
2
4
4
VC2
2
0
6
80
6
20
40
60
80
100
Time (ns)
120
20
140
20
40
(a)
60
80
Time (ns)
100
120
140
(b)
20
Voltage (kV)
15
10
Vl = VC2 −VC1
5
0
50
20
40
60
80
Time (ns)
100
120
140
(c)
Figure 3.12: Voltage waveforms across the capacitors C1 and C2 , and the laser tube.
In our analysis of the equivalent Blumlein circuit, we only considered the time before the main discharge
in the laser tube takes place, because of its nonlinear properties which leads to a complex system.
Therefore the analysis does not explain the behaviour of the circuit after the main discharge has been
initiated. However, L. Kawecki et al. analysed the Blumlein circuit by considering all the elements of
the circuit, using a Runge-Kutta method to solve the more complex integral-differential equations of
the system and Gauss-Seidel algorithm for parameter identification in the system [KNlR97].
Furthermore, we simplified the circuit by using the idea of decoupling due the different time scales
involved, which lead to a conclusion that the voltage across the laser tube rises upto a maximum value
2V0 . However, this is not always the case. The gas in the laser tube generally breaks before attaining
this value. Thus the voltage across the laser tube can never rise upto that value.
4. Experimental set up of a nitrogen laser
A nitrogen laser based on an equivalent Blumleim configuration which has been discussed in the previous
chapter, is comparatively easy and cheap to build. This is very useful especially in the scaling of various
parameters for higher power outputs, because of its flexibility favouring easy modifications upon need.
In this chapter, we present a nitrogen laser assembly and its view, which was constructed by the Laser
Research Institute, Stellenbosch University. The output beam and the laser pulse obtained in the
experiments, and their discussions are also presented.
4.1
Nitrogen laser construction
The geometrical construction of the laser shown in Figure 4.1 consists of two capacitor plates made
from aluminium plates, a grounded capacitor plate1 , an assembled spark gap, a ribbon inductor made of
aluminium foil, a diode chain for high voltage rectification, a car induction coil which acts as a step-up
transformer (from 12 V to 18000 V) and a power supply. The laser channel, where the nitrogen gas
is supplied via a gas connector, is made of two electrodes placed on top of the capacitor plates. The
electrodes of the laser channel must be carefully designed, with smooth edges and aligned in parallel,
so as to obtain a uniformly distributed discharge along the entire length of the laser channel. This is
a bit tricky, but with reasonable adjustments, it can get done. The whole set-up is firmly fixed on a 6
mm thick perspex base plate.
Figure 4.1: Schematic diagram of the built nitrogen laser, Laser Research Institute, Stellenbosch University.
1
Two sheets of Mylar are placed between the grounded capacitor plate and the top two capacitor plates to act as a
dielectric.
25
Section 4.2. Experimental results
Page 26
Here is a view of the practical nitrogen laser based on the schematic diagram shown in Figure 4.1 above,
housed in a perspex enclosure.
Figure 4.2: Nitrogen laser
A nitrogen laser can operate normally with consistent and evenly distributed discharges, when the
nitrogen gas is fed to the laser channel at a low flow rate. However, this limits the number of gas
molecules available in the laser channel for ionisation, resulting into low power output. The laser shown
in Figure 4.2 operates at atmospheric pressure, and the uniformly distributed discharges are achievable
through preionisation by automatically timed corona discharge. Corona discharge is as a result of anterior
discharge of the gas, brought about by small sparks before complete ionisation of gas molecules in the
gap by the main discharge. The main disadvantage of preionisation is that, it leads to frequent firing in
the laser channel. This takes advantage of the meandering ions from the previous pulses, resulting into
an increase in temperature, which in turn decreases the power output of the pulses.
4.2
Experimental results
Since air is 78.08% nitrogen, the laser could operate within open air as the lasing medium, but with very
poor output, characterised by violent sparking along the laser channel. However, with pure nitrogen gas
from a cylinder, its operation was normal, giving a laser beam along the channel.
4.2.1 Laser output. As indicated in Figure 1.5, the output of a nitrogen laser is in the ultraviolet range,
at a wavelength of 337.1 nm, thus invisible. However, its beam can be observed as a blue spot with
a fluorescent material such a pieces of paper or cardboard placed at distance along the laser channel.
Figure 4.3 shows the output of the nitrogen laser above.
Figure 4.3: Output of the nitrogen laser in Figure 4.2
Section 4.2. Experimental results
Page 27
4.2.2 Laser pulse shape and width. With the nitrogen laser working, a set of pulses where displayed
on a Tektronix TDS 3032 digital storage oscilloscope rated at 300 MHz. The output signal was detected
using a Thorlabs DET210 output detector, rated at 500 MHz. As shown in Figure 4.4, the peak voltage
of the pulse signal is about 4V and the FWHM is about 2.4 ns/div. However, the normal FWHM value
of the nitrogen laser overleaf is 2 ns/div. To avoid the interference of the noise and the high light
intensity from the spark gap, the oscilloscope and the detector where placed at a distance from the laser
itself, which could have lead to the loss of some fine structures of the laser pulse. Aside from other
factors, the former accounts for the difference between the two values.
4.0
3.5
3.0
Voltage (V)
2.5
2.0
1.5
1.0
0.5
0.0
−0.5
−0.4
−0.2
0.0
0.2
0.4
Time (s)
0.6
0.8
Figure 4.4: Nitrogen laser pulse
1.0
1.2
×10−8
5. Conclusion
In the frame of the present work, a detailed review of the principles behind laser action have been
presented. This provided a background to the understanding of the necessary condition for lasing to
take place, the population inversion. Unlike other gas lasers, nitrogen lasers have a shorter transition
lifetime for the upper lasing level than the lower lasing level. Thus to create a higher population of
the gas molecules at the upper lasing level, requires sufficient power to excite the gas molecules to the
upper level as fast as possible. This power can be provided by a fast electronic discharge circuit.
Understanding the processes involved in gas discharges was a basic tool in our study. Gas discharges
normally occur when an applied voltage across a laser tube exceeds the breakdown voltage, so that
the gas transition state changes from non–conducting to conducting. Two important mechanisms in
gas breakdown have been discussed: the electron avalanche and the streamer breakdown mechanisms.
Streamer development, resulting in non–uniform discharges in the electrode gap, which can be suppressed
by preionisation of the gas volume has been discussed.
Capacitors and inductors play an important role in electrical oscillations, thus a review of their behaviour
in circuits was important. An equivalent Blumlein circuit based on two coupled LC circuits required to
provide a high voltage output has been designed. The fast electric discharge circuit was analysed, and
the simulations show that it operates on an underdamped mode. Furthermore, the results from the
analysis of the design were found to be almost similar to those we obtained by treating the two coupled
LC circuits as decoupled circuits, where the voltage across the laser tube doubles the charge-up voltage
of the capacitors.
We have presented the general assembly of a nitrogen laser and its view. The output beam and the
laser pulse obtained in the experiments, and their discussions have also been presented.
Electric pumping mechanism is commonly used in excitation of gas lasers, thus investigations for the
circuit designs required to generate fast rising high–voltage pulses receive interest in gas laser technology.
Therefore, ideas in this work can be extended to the development of atmospheric pressure nitrogen and
TEA carbon dioxide lasers to mention but a few.
28
Acknowledgements
First and foremost, all praise and thanks to the Almighty Allah for his endless blessings, love and
care bestowed upon me. Surely without his mercy, the completion of this work would have not been
possible. I am also greatly indebted to my supervisor Prof. Erich G. Rohwer not for only suggesting this
exciting essay topic, but also his effort in providing the necessary support in the preparation of this work.
Certainly, I would like to express my sincere gratitude to my co-supervisor Prof. H. M. von Bergmann, for
his tremendous guidance, patience and numerous open discussions, which contributed to the success my
work. I further wish to extend my acknowledgements to the AIMS staff and management, in particular
the director AIMS Prof. Barry Green, the AIMS founder Prof. Neil G. Turok, the IT manager Jan
Groenewald, the English teacher Frances Aron and all the tutors. Special thanks to Nagla Numan Ali
Numan for her advice and encouragements. Finally, to all my colleagues who had a hand in this work
in one way or the other, I owe you my sincerest and most heartfelt thanks.
29
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