Kurdistan Iraqi Region Ministry of Higher Education University of Sulaimani College of Science Physics Department

Numerical Simulation of Laser Pulse

Generation

Prepared by

Rebar F. Karem Gashaw O. Abdullah

Zhilan B. Husain

Supervised by

Dr. Omed Ghareb Abdullah

2009 - 2010

ii

Acknowledgments

Praise be to Allah for providing us the willingness and strength

to accomplish this work, we would to express our deepest gratitude

to our supervisor Dr. Omed for his helps and guidance throughout

this work.

True appreciation for Department of Physics in the College of

Science at the University of Sulaimani for giving us an opportunity

to carry out this work.

We wish to extend our sincere thanks to all lecturers who taught

us along our study many other thanks should go to our colleagues

for their encouraging. Lastly thanks and love to our family for their

patience and support during our study.

Rebar, Zhilan, & Gashaw

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Contents

Chapter One: Basic Concepts

1.1 Introduction.

1.2 Construction of a laser

1.3 Absorption and Emissions

1.4 Population Inversion

1.5 Three-level lasers

1.6 Four-level lasers

1.7 The Gain in Laser

1.8 The Loss in Laser

1.9 The ruby laser

Chapter Two: Q-Switching Techniques

2.1 Introduction.

2.2 Principle of Q-switching

2.3 Active Q-switching

2.3.1 Mechanical Q-Switches

3.2.2 Electro-Optical Q-Switches

3.2.3 Acousto-Optic Q-Switches

2.4 Passive Q-switching

2.5 Passive Q-Switch Processes

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Chapter Three: Passive Q-Switching

3.1 Introduction.

3.2 Rate Equations

3.3 Runge-kutta Fehlberg

Chapter Four: Results and Discussion

4.1 Introduction.

4.2 Solution of Rate Equations

4.3 Effect of concentration of the saturable absorber

4.4 Conclusion

References

Appendix

v

Abstract

Q-switching, sometimes known as giant pulse formation, is a technique

by which a laser can be made to produce a pulsed output beam. The

technique allows the production of light pulses with extremely high (Giga

Watt) peak power, much higher than would be produced by the same laser

if it were operating in a continuous wave (constant output) mode.

In this study the passive Q-switched performance of the ruby laser with

slow-relaxing solid-state saturable absorber calcium fluoride crystals doped

with divalent dysprosium Dy2+:CaF2

is numerically investigated by solving

the coupled rate equations with the Runge-Kutta-Fehlberg method. In the

mean time, important factors such as the laser population inversion at

various stages, the peak photon number inside the laser resonator, the

output energy and the pulse width of the Q-switched laser output are

computed. For typical configuration, a Q-switched laser pulse of 150 𝑛𝑛𝑛𝑛 in

duration and 3 𝑚𝑚𝑚𝑚 in energy is obtained. Also the effects of the saturable

absorber doping concentration on output laser characteristics are studied.

The simulated results show reasonable agreement with those obtained

experimentally by other research groups.

1

Chapter One

Basic Concepts

1.1 Introduction:

Light Amplification by Stimulated Emission of Radiation, LASER (laser), is

a mechanism for emitting light within the electromagnetic radiation region of the

spectrum, via the process of stimulated emission. The emitted laser light is

(usually) a spatially coherent, narrow low-divergence beam, that can be

manipulated with lenses. In laser technology, “coherent light” denotes a light

source that produces (emits) light of in-step waves of identical frequency and

phase. The laser’s beam of coherent light differentiates it from light sources that

emit incoherent light beams, of random phase varying with time and position;

whereas the laser light is a narrow-wavelength electromagnetic spectrum

monochromatic light; yet, there are lasers that emit a broad spectrum light, or

simultaneously, at different wavelengths.

A laser emits a thin, intense beam of nearly monochromatic visible or

infrared light that can travel long distances without diffusing. Most light beams

consist of many waves traveling in roughly the same direction, but the phases

and polarizations of each individual wave (or photon) are randomly distributed.

In laser light, the waves are all precisely in step, or in phase, with each other, and

have the same polarization. Such light is called coherent. All of the photons that

make up a laser beam are in the same quantum state. Lasers produce coherent

light through a process called stimulated emission.

The laser contains a chamber in which atoms of a medium such as a synthetic

ruby rod or a gas are excited, bringing their electrons into higher orbits with

higher energy states. When one of these electrons jumps down to a lower energy

state (which can happen spontaneously), it gives off its extra energy as a photon

with a specific frequency. But this photon, upon encountering another atom with

2

an excited electron, will stimulate that electron to jump down as well, emitting

another photon with the same frequency as the first and in phase with it. This

effect cascades through the chamber, constantly stimulating other atoms to emit

yet more coherent photons. Mirrors at both ends of the chamber cause the light to

bounce back and forth in the chamber, sweeping across the entire medium. If a

sufficient number of atoms in the medium are maintained by some external

energy source in the higher energy state a condition called population inversion

then emission is continuously stimulated, and a stream of coherent photons

develops. One of the mirrors is partially transparent, allowing the laser beam to

exit from that end of the chamber. Lasers have many industrial, military, and

scientific uses, including welding, target detection, microscopic photography,

fiber optics, surgery, and optical instrumentation for surveying.

1.2 Construction of a laser:

A laser consists of an active laser material, a source of excitation energy, and

a resonator or feedback mechanism to perform the three stages of laser action.

The general construction of a laser is shown in Figure(1.1).

Laser material: The lasing material can be a solid (Ruby, YAG and glass

lasers), liquid (Dye lasers), gas (Helium-neon, argon and carbon dioxide) or a

semi-conductor (InGaAlP). A material is said to be in "Normal State" if the

number of atoms in the lower energy level is more than the number of atoms in

the higher energy level. The material is said to be in a excited state if population

inversion has been achieved. The laser material is one in which population

inversion is possible. The downward transition from the excited to the normal

state is triggered by stimulated emission. The lasers are classified depending on

the number of energy levels used for the excitation and the stimulated emission

process. Commercial lasers are 3 level and 4 level systems, while the simple 2

3

level system is not used in practice, as it is difficult to achieve population

inversion in a 2 level system.

Figure (3.2): Construction of Laser.

Excitation Source: Population inversion is achieved by "pumping energy"

from an external source. Depending on the external source, the excitation process

is called as optical pumping or electrical pumping. In electrical pumping, an AC

or DC electrical discharge is used for excitation. Gas lasers and semiconductor

lasers are usually excited using electrical pumping. In optical pumping, light is

the source of energy and is used for most of the solid-state and dye lasers.

Resonator: A Fabry-Perot cavity that has a pair of mirrors, one at each end

of the laser is used as a resonator in most lasers. One of the mirrors is completely

reflective while the other mirror is partially transparent. The reflection of the

laser beam between the two mirrors results in increased power. The beam is

reflected back for amplification, until a specific threshold power is reached. The

portion of the laser beam with the necessary power is coupled as output through

the partially transparent mirror.

1.3 Absorption and Emissions:

In any material, during thermal equilibrium the number of particles in the

excited state is very small and is negligible. When the number of particles in the

excited state is greater than the number of particles in the ground state, the

4

material is in a state of "Population Inversion". Population inversion is a

prerequisite for laser action. Energy can be transferred into a laser medium to

achieve population inversion by several mechanisms including absorption of

photon, collision between electrons (or sometimes ions) and species in the active

medium, collisions among atoms and molecules in the active medium,

recombination of free electrons with ionized atoms, recombination of current

carriers in a semiconductor, chemical reactions producing excited species, and

acceleration of electrons.

In photon absorption, the laser material is optically excited to achieve

population inversion based on Planck’s law. According to Planck’s law, the

change of energy level from 𝐸𝐸1 to 𝐸𝐸2 or vice versa, results in the absorption or

emission of photon respectively. Emission can be spontaneous or stimulated.

Spontaneous emission: is the process by which a light source such as an

atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition

to the ground state and emits a photon, see Figure (1.2). Spontaneous emission of

light or luminescence is a fundamental process that plays an essential role in

many phenomena in nature and forms the basis of many applications, such as

fluorescent tubes, older television screens (cathode ray tubes), plasma display

panels, lasers (for startup - normal continuous operation works by stimulated

emission instead) and light emitting diodes.

Stimulated emission: is the process by which an electron, perturbed by a

photon having the correct energy, may drop to a lower energy level resulting in

the creation of another photon. The perturbing photon is seemingly unchanged in

the process, and the second photon is created with the same phase, frequency,

polarization, and direction of travel as the original. If the resultant photons are

reflected so that they traverse the same atoms or gain medium repeatedly, a

cascade effect is produced. Stimulated emission is really a quantum mechanical

phenomenon but it can be understood in terms of a "classical" field and a

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quantum mechanical atom. The process can be thought of as "optical

amplification" and it forms the basis of the maser (including the laser).

In stimulated emission the photon is emitted in the same direction as the light

that is passing by. When the number of particles in one excited state exceeds the

number of particles in some lower-energy state, population inversion is achieved

and the amount of stimulated emission due to light that passes through is larger

than the amount of absorption. Hence, the light is amplified. Strictly speaking,

these are the essential ingredients of a laser. However, usually the term laser is

used for devices where the light that is amplified is produced as spontaneous

emission from the same gain medium as where the amplification takes place.

Devices where light from an external source is amplified are normally called

optical amplifiers.

The light generated by stimulated emission is very similar to the input signal

in terms of wavelength, phase, and polarization. This gives laser light its

characteristic coherence, and allows it to maintain the uniform polarization and

often monochromaticity established by the optical cavity design. Figures (1.2)

and (1.3) illustrate the spontaneous and stimulated emissions, respectively.

Figure(1.2): Diagram of Spontaneous emission.

6

Figure(1.3): Diagram of Stimulated emission.

The optical cavity, a type of cavity resonator, contains a coherent beam of

light between reflective surfaces so that the light passes through the gain medium

more than once before it is emitted from the output aperture or lost to diffraction

or absorption. As light circulates through the cavity, passing through the gain

medium, if the gain (amplification) in the medium is stronger than the resonator

losses, the power of the circulating light can rise exponentially. But each

stimulated emission event returns a particle from its excited state to the ground

state, reducing the capacity of the gain medium for further amplification. When

this effect becomes strong, the gain is said to be saturated. The balance of pump

power against gain saturation and cavity losses produces an equilibrium value of

the laser power inside the cavity; this equilibrium determines the operating point

of the laser. If the chosen pump power is too small, the gain is not sufficient to

overcome the resonator losses, and the laser will emit only very small light

powers. The minimum pump power needed to begin laser action is called the

lasing threshold. The gain medium will amplify any photons passing through it,

regardless of direction; but only the photons aligned with the cavity manage to

pass more than once through the medium and so have significant amplification.

7

1.4 Population Inversion:

According to the Boltzmann distribution, in a collection of atoms at thermal

equilibrium there are always fewer atoms in a higher-lying level 𝐸𝐸2 than in a

lower level 𝐸𝐸1. Therefore the population difference 𝑁𝑁1 − 𝑁𝑁2 is always positive,

which means that the absorption coefficient α(νs) is positive and the incident

radiation is absorbed, see Figure (1.4).

Suppose that it were possible to achieve a temporary situation such that

there are more atoms in an upper energy level than in a lower energy level. The

normally positive population difference on that transition then becomes negative,

and the normal stimulated absorption is correspondingly changed to stimulated

emission or amplification of the applied signal. That is, the applied signal gains

energy as it interacts with the atoms and hence is amplified. The energy for this

signal amplification is supplied by the atoms involved in the interaction process.

This situation is characterized by a negative absorption coefficient α(νs).

The essential condition for amplification is that there are more atoms in an

upper energy level than in a lower energy level, i.e., for amplification,

𝑁𝑁2 > 𝑁𝑁1 𝑖𝑖𝑖𝑖 𝐸𝐸2 > 𝐸𝐸1

as illustrated in Figure(1.4). The resulting negative sign of the population

difference (𝑁𝑁2 − 𝑔𝑔2𝑁𝑁1/𝑔𝑔1) on that transition is called a population inversion.

Population inversion is clearly an abnormal situation; it is never observed at

thermal equilibrium. The point at which the population of both states is equal is

called the “inversion threshold.”

Stimulated absorption and emission processes always occur side by side

independent of the population distribution among the levels. So long as the

population of the higher energy level is smaller than that of the lower energy

level, the number of absorption transitions is larger than that of the emission

transitions, so that there is an overall attenuation of the radiation. When the

numbers of atoms in both states are equal, the number of emissions becomes

8

equal to the number of absorptions; the material is then transparent to the

incident radiation. As soon as the population of the higher level becomes larger

than that of the lower level, emission processes predominate and the radiation is

enhanced collectively during passage through the material. To produce an

inversion requires a source of energy to populate a specified energy level; we

call this energy the pump energy.

The total amount of energy which is supplied by the atoms to the light wave

is: 𝐸𝐸 = ∆𝑁𝑁 ℎ𝑣𝑣

where ∆𝑁𝑁 is the total number of atoms which are caused to drop from the

upper to the lower energy level during the time the signal is applied. If laser

action is to be maintained, the pumping process must continually replenish the

supply of upper-state atoms. The size of the inverted population difference is

reduced not only by the amplification process but also by spontaneous emission,

which always tends to return the energy level populations to their thermal

equilibrium values.

Figure(1.4): Relative populations in two energy levels as given by the

Boltzmann relation for thermal equilibrium.

9

1.5 Three-level lasers:

To achieve non-equilibrium conditions, an indirect method of populating the

excited state must be used. To understand how this is done, we may use a

slightly more realistic model, that of a three-level laser. Consider a group of 𝑁𝑁

atoms, this time with each atom able to exist in any of three energy states, levels

1, 2 and 3, with energies 𝐸𝐸1, 𝐸𝐸2 and 𝐸𝐸3, and populations 𝑁𝑁1, 𝑁𝑁2, and 𝑁𝑁3,

respectively.

Note that 𝐸𝐸1 < 𝐸𝐸2 < 𝐸𝐸3; that is, the energy of level 2 lies between that of the

ground state and level 3.

Initially, the system of atoms is at thermal equilibrium, and the majority of

the atoms will be in the ground state; i.e., 𝑁𝑁1 ≈ 𝑁𝑁, 𝑁𝑁2 ≈ 𝑁𝑁3 ≈ 0. If we now

subject the atoms to light of a frequency 𝑣𝑣13 = 1ℎ(𝐸𝐸3 − 𝐸𝐸1), the process of optical

absorption will excite the atoms from the ground state to level 3. This process is

called pumping, and does not necessarily always directly involve light

absorption; other methods of exciting the laser medium, such as electrical

discharge or chemical reactions may be used. The level 3 is sometimes referred

to as the pump level or pump band, and the energy transition 𝐸𝐸1 ⟶ 𝐸𝐸3 as the

pump transition, which is shown as the arrow marked P in the Figure(1.5).

Figure(1.5): Simplified energy level diagram of a three-level laser.

10

If we continue pumping the atoms, we will excite an appreciable number of

them into level 3, such that 𝑁𝑁3 > 0. In a medium suitable for laser operation, we

require these excited atoms to quickly decay to level 2. The energy released in

this transition may be emitted as a photon (spontaneous emission), however in

practice the 3→2 transition (labeled R in the diagram) is usually radiation-less,

with the energy being transferred to vibrational motion (heat) of the host material

surrounding the atoms, without the generation of a photon.

An atom in level 2 may decay by spontaneous emission to the ground state,

releasing a photon of frequency 𝑣𝑣12 (given by 𝐸𝐸2 − 𝐸𝐸1 = ℎ𝑣𝑣12, which is shown

as the transition L, called the laser transition in the diagram. If the lifetime of this

transition, 𝜏𝜏21 is much longer than the lifetime of the radiation-less 3 → 2

transition 𝜏𝜏32 (if 𝜏𝜏21 ≫ 𝜏𝜏32 , known as a favourable lifetime ratio), the population

of the 𝐸𝐸3 will be essentially zero (𝑁𝑁3 ≈ 0) and a population of excited state

atoms will accumulate in level 2 (𝑁𝑁2 > 0). If over half the 𝑁𝑁 atoms can be

accumulated in this state, this will exceed the population of the ground state 𝑁𝑁1.

A population inversion (𝑁𝑁2 > 𝑁𝑁1) has thus been achieved between level 1 and 2,

and optical amplification at the frequency 𝑣𝑣21 can be obtained.

Because at least half the population of atoms must be excited from the

ground state to obtain a population inversion, the laser medium must be very

strongly pumped. This makes three-level lasers rather inefficient, despite being

the first type of laser to be discovered. A three-level system could also have a

radiative transition between level 3 and 2, and a non-radiative transition between

2 and 1. In this case, the pumping requirements are weaker. In practice, most

lasers are four-level lasers, described below.

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1.6 Four-level lasers:

Here, there are four energy levels, energies 𝐸𝐸1, 𝐸𝐸2, 𝐸𝐸3, 𝐸𝐸4, and populations

𝑁𝑁1, 𝑁𝑁2, 𝑁𝑁3, 𝑁𝑁4, respectively. The energies of each level are such that 𝐸𝐸1 < 𝐸𝐸2 <

𝐸𝐸3 < 𝐸𝐸4.

In this system, the pumping transition P excites the atoms in the ground state

(level 1) into the pump band (level 4). From level 4, the atoms again decay by a

fast, non-radiative transition Ra into the level 3. Since the lifetime of the laser

transition L is long compared to that of Ra (𝜏𝜏32 ≫ 𝜏𝜏43), a population

accumulates in level 3 (the upper laser level), which may relax by spontaneous

or stimulated emission into level 2 (the lower laser level). This level likewise has

a fast, non-radiative decay Rb into the ground state (see Figure(1.6)).

As before, the presence of a fast, radiation-less decay transitions result in

population of the pump band being quickly depleted (𝑁𝑁4 ≈ 0). In a four-level

system, any atom in the lower laser level 𝐸𝐸2 is also quickly de-excited, leading to

a negligible population in that state (𝑁𝑁2 ≈ 0). This is important, since any

appreciable population accumulating in level 3, the upper laser level, will form a

population inversion with respect to level 2. That is, as long as 𝑁𝑁3 > 0, then

𝑁𝑁3 > 𝑁𝑁2 and a population inversion is achieved. Thus optical amplification, and

laser operation, can take place at a frequency of 𝑣𝑣32 (𝐸𝐸3 − 𝐸𝐸2 = ℎ𝑣𝑣32).

Figure(1.6): Simplified energy level diagram of a four-level laser.

12

Since only a few atoms must be excited into the upper laser level to form a

population inversion, a four-level laser is much more efficient than a three-level

one, and most practical lasers are of this type. In reality, many more than four

energy levels may be involved in the laser process, with complex excitation and

relaxation processes involved between these levels. In particular, the pump band

may consist of several distinct energy levels, or a continuum of levels, which

allow optical pumping of the medium over a wide range of wavelengths.

Note that in both three- and four-level lasers, the energy of the pumping

transition is greater than that of the laser transition. This means that, if the laser

is optically pumped, the frequency of the pumping light must be greater than that

of the resulting laser light. In other words, the pump wavelength is shorter than

the laser wavelength. It is possible in some media to use multiple photon

absorptions between multiple lower-energy transitions to reach the pump level;

such lasers are called up-conversion lasers.

1.7 The Gain in Laser:

Another fundamental concept in lasers is the idea of gain, which is basically

a short way of referring to the "free" photons described earlier. Suppose we have

just pumped our laser medium so that all of the particles are in their excited state.

One of those particles now spontaneously decays back down to its ground state,

emitting a photon (hv0

). This photon is of the right frequency to stimulate

emission from another excited state particle, which emits another photon which

can stimulate another excited state particle, and so on, as shown in figure(1.7).

13

Figure(1.7): Diagram illustrate the gain in laser.

1.8 The Loss in Laser:

In addition to stimulated emission processes there are also stimulated

absorption processes in which a ground state particle absorbs a photon matching

the energy gap and jumps to the excited state. (represented by the gray arrow in

the above figure). Thus we lose one photon to each stimulated absorption

process. Since the probabilities for stimulated absorption and emission processes

are equal (relative to population of the ground and excited states -- Einstein's

famous result), it is clearly detrimental to the laser to have any particles in the

ground state. For this reason, two level lasers are not practical -- it is not in

general possible to pump more than half of the molecules into the excited state.

1.9 The ruby laser:

The first laser made from a hard material was the ruby laser; it was first build

in 1960. It consists to 99.95% of AlO2 and the remaining 0.05% are Cr. Those

less Cr-Ions are responsible for the laser effect. Cr-Ions have three energetic

levels 𝐸𝐸1 , 𝐸𝐸2 and 𝐸𝐸3. 𝐸𝐸1 is the base energy level and 𝐸𝐸3 is the level when we

14

penetrate the ions with green light which has a short wavelength. To pump the

ions we usually use a xenon lamp.

Now that we have some ions pushed to the highest energy level they will

drop immediately back to 𝐸𝐸2 but without emitting any light only heat. The Cr-

Ions will stay a long time on this medium energy level therefore it is called meta

stable, see Figure(1.8).

If one ion drops back to 𝐸𝐸1 spontaneously it will emit a photon with the

energy ℎ𝑣𝑣 = 𝐸𝐸2 − 𝐸𝐸1 and forces the other Cr-Ions to initialise the laser process.

Figure(1.8): Three level of Ruby laser.

There are of course many other possibilities to build up a laser but the light

emitted by a ruby laser is red and in the visible part of the electro magnetic

waves.

Due to this fact and because of the possibility to run a ruby laser in almost

every surrounding this type of laser is very popular and it will be used for very

many physical experiments.

15

Chapter Two

Q-Switching Techniques

2.1 Introduction:

Q-switching, sometimes known as giant pulse formation, is a technique by

which a laser can be made to produce a pulsed output beam. The technique

allows the production of light pulses with extremely high (giga watt) peak

power, much higher than would be produced by the same laser if it were

operating in a continuous wave (constant output) mode. Q-switching was first

proposed in 1958 by Gordon Gould, and independently discovered and

demonstrated in 1961 or 1962 by R.W. Hellwarth and F.J. McClung using

electrically switched Kerr cell shutters in a ruby laser.

In Q-switching the Q-factor of the cavity is switched between low and high

values, or in other words, the loss in the resonator is switched from a high to a

low value. The component inducing the loss is called the Q-switch, and it can be

driven either actively or passively. In an active Q-switch, fast electronics is

required for modulating the loss in a time scale of a few nanoseconds. In a

passive Q-switch, the transition is driven by light itself through the process of

saturable absorption. Compared to active Q-switching, no electronics is required

but the laser is pulsed without external control. The choice of the switch type

depends on the application. Passive Q-switches are smaller, simpler and more

robust than active Q-switches, but they lack the ability to generate the pulse

accurately at a chosen time.

2.2 Principle of Q-switching:

Q-switching is achieved by putting some type of variable attenuator inside

the laser's optical resonator. When the attenuator is functioning, light which

leaves the gain medium does not return, and lasing cannot begin. This

16

attenuation inside the cavity corresponds to a decrease in the Q factor or quality

factor of the optical resonator. A high Q factor corresponds to low resonator

losses per roundtrip, and vice versa. The variable attenuator is commonly called

a "Q-switch", when used for this purpose.

Initially the laser medium is pumped while the Q-switch is set to prevent

feedback of light into the gain medium (producing an optical resonator with low

Q). This produces a population inversion, but laser operation cannot yet occur

since there is no feedback from the resonator. Since the rate of stimulated

emission is dependent on the amount of light entering the medium, the amount of

energy stored in the gain medium increases as the medium is pumped. Due to

losses from spontaneous emission and other processes, after a certain time the

stored energy will reach some maximum level; the medium is said to be gain

saturated. At this point, the Q-switch device is quickly changed from low to high

Q, allowing feedback and the process of optical amplification by stimulated

emission to begin. Because of the large amount of energy already stored in the

gain medium, the intensity of light in the laser resonator builds up very quickly;

this also causes the energy stored in the medium to be depleted almost as

quickly. The net result is a short pulse of light output from the laser, known as a

giant pulse, which may have a very high peak intensity.

There are two main types of Q-switching: actively Q-switched lasers and

passively Q-switched lasers discussed as follows.

2.3 Active Q-switching:

The most common type is the actively Q-switched solid state bulk laser.

Solid state gain media have a good energy storage capability, and bulk lasers

allow for large mode areas (such higher pulse energies and peak powers) and

shorter laser resonators (e.g. compared with fiber lasers). The laser resonator

contains an active Q-switch - an optical modulator, which is in most cases an

17

acousto-optic modulator, electro-optical modulator and mechanical modulator

are used for Q-switching also, see Figure(2.1). A small actively Q-switched solid

state laser may emit 100 𝑚𝑚𝑚𝑚 of average power in 10 𝑛𝑛𝑛𝑛 pulses with a 1 𝐾𝐾𝐾𝐾𝐾𝐾

repetition rate and 100 𝜇𝜇𝜇𝜇 pulse energy. The peak power is then 9 𝐾𝐾𝑚𝑚. The

highest pulse energies and shortest pulse durations are achieved for low pulse

repetition rates (below the inverse upperstate lifetime), at the expense of

somewhat reduced average output power.

Figure(2.1): Schematic setup of an actively Q-switched laser.

2.3.1 Mechanical Q-Switches:

Q-switches have been designed based upon rotational, oscillatory, or

translational motion of optical components. What these techniques have in

common is that they inhibit laser action during the pump cycle by either

blocking the light path, causing a mirror misalignment, or reducing the

reflectivity of one of the resonator mirrors.

Near the end of the pump pulse, when maximum energy has been stored in

the laser rod, a high Q-condition is established and a Q-switch pulse is emitted

from the laser.

The first mechanical Q-switch consisted of nothing more than a rotating disc

containing an aperture. This method was soon abandoned in favor of rotating

mirrors or prisms, which allow much faster switching times.

The spinning reflector technique for the generation of Q-switched pulses, as

shown in Figure(2.2), involves simply rotating one of the two resonant cavity

18

reflectors so that parallelism of the reflectors occurs for only a brief instant in

time. If a plane mirror is employed as the rotating element, the axis of rotation

must be aligned to within a fraction of a milliradian parallel to the face of the

opposing reflector. This difficulty can be overcome by using a roof prism as the

rotating element. If the roof of the prism is perpendicular to the axis of rotation,

then the retroreflecting nature of the prism assures alignment in one direction,

while the rotation of the prism brings it into alignment in the other direction.

Rotating-mirror devices are simple and inexpensive.

Figure(2.2): Diagram of a ruby laser employing a spinning prism Q-switch.

For most of the time, the alignment of two mirrors will be such that the loss

will be high and hence the Q low. This will allow a large population inversion to

develop. At the instant that the two mirrors are aligned, the Q will be high and a

large output pulse will be developed. Although this method was the first

developed for Q-switching, it does not give the performance in terms of peak

power of some of the newer methods of Q-switching.

19

3.2.2 Electro-Optical Q-Switches:

Very fast electronically controlled optical shutters can be designed by

exploiting the electro-optic effect in crystals or liquids. The key element in such

a shutter is an electro-optic element that becomes birefringent under the

influence of an external field. Birefringence in a medium is characterized by two

orthogonal directions called the “fast” and “slow” axes, which have different

indices of refraction. An optical beam, initially plane-polarized at 45𝑜𝑜 to these

axes and directed normal to their plane, will split into two orthogonal

components, traveling along the same path but at different velocities. Hence, the

electro-optic effect causes a phase difference between the two beams. After

traversing the medium, the combination of the two components results,

depending on the voltage applied, in either an elliptical, circular, or linearly

polarized beam. For Q-switch operation only two particular voltages leading to a

quarter-wave and half-wave retardation are of interest. In the first case, the

incident linearly polarized light is circular polarized after passing the cell, and in

the second case the output beam is linearly polarized; however, the plane of

polarization has been rotated 90𝑜𝑜 .

The two most common arrangements for Q-switching are shown in

Figure(2.3). In Figure(2.3a) the electro-optic cell is located between a polarizer

and the rear mirror. The inclusion of the polarizer is not essential if the laser

radiation is polarized. The sequence of operation is as follows: During the pump

pulse, a voltage 𝑉𝑉1/4 is applied to the electro-optic cell such that the linearly

polarized light passed through the polarizer is circularly polarized. After being

reflected at the mirror, the radiation again passes through the electro-optic cell

and undergoes another 𝜆𝜆/4 retardation, becoming linearly polarized but at 90𝑜𝑜

to its original direction. This radiation is ejected from the laser cavity by the

polarizer, thus preventing optical feedback. Toward the end of the pump pulse

the voltage on the cell is switched off, permitting the polarizer-cell combination

20

to pass a linearly polarized beam without loss. Oscillation within the cavity will

build up, and after a short delay a Q-switch pulse will be emitted from the cavity.

Figure(2.3): Electro-optic Q-switch operated at (a) quarter-wave

and (b) half-wave retardation voltage

In the arrangement of Figure(2.3b) an electric voltage must first be applied to

the cell to transmit the beam. In this so-called pulse-on Q-switch, the cell is

located between two crossed polarizers. As before, polarizer 𝑃𝑃1, located between

the laser rod and the cell, is not required if the active medium emits a polarized

beam. During the pump pulse, with no voltage applied to the cell, the cavity Q is

at a minimum due to the crossed polarizers. At the end of the pump pulse, a

voltage 𝑉𝑉1/2 is applied to the cell, which causes a 90𝑜𝑜 rotation of the incoming

beam. The light is therefore transmitted by the second polarizer 𝑃𝑃2. Upon

21

reflection at the mirror the light passes again through polarizer 𝑃𝑃2 and the cell,

where it experiences another 90𝑜𝑜 rotation. Light traveling toward the polarizer

𝑃𝑃1 has experienced a 180𝑜𝑜 rotation and is therefore transmitted through 𝑃𝑃1.

3.2.3 Acousto-Optic Q-Switches:

In acousto-optic Q-switches, an ultrasonic wave is launched into a block of

transparent optical material, usually fused silica. By switching on the acoustic

field, a fraction of the energy of the main beam is diffracted out of the resonator,

thus introducing a loss mechanism that prevents laser action. When the acoustic

field is switched off, full transmission through the Q-switch cell is restored and a

laser pulse is created. The acousto-optic Q-switch is the device of choice for

repetitively Q-switching cw lasers. The low-gain characteristics of cw-pumped

solid-state lasers do not require very high extinction ratios, but do demand an

exceptionally low insertion loss. Since high optical quality fused silica with

antireflection coatings can be used as the active medium in the acousto-optical

Q-switch, the overall insertion loss of the inactive Q-switch can be reduced to

less than 0.5% per pass. The low-insertion loss of the acousto-optic Q-switch

offers the convenience of converting from Q-switched to cw operation simply by

removing the RF drive power.

2.4 Passive Q-switching:

A more common method of Q-switching is to use a saturable absorber (dye),

with an absorption characteristic similar to that shown in Figure(2.4). The dye is

usually pumped through a small jet, which sprays the liquid through the laser

beam. At the beginning of the pump cycle the incident intensity on the dye is

low. The dye is highly absorbing, which gives a low cavity Q. Later on during

the pump cycle, the irradiance is much higher, which bleaches the dye. Hence

22

the dye becomes transparent, giving a sharp increase in cavity Q. This increase in

Q gives rise to an overall gain and so a high amplitude out put pulse.

Figure(2.4): Schematic setup of a passively Q-switched laser. The saturable

absorber is a crystal within the laser resonator.

In this case, the Q-switch is a saturable absorber, a material whose

transmission increases when the intensity of light exceeds some threshold. The

material may be an ion-doped crystal like Cr:YAG, which is used for Q-

switching of Nd:YAG lasers, a bleachable dye, or a passive semiconductor

device. Initially, the loss of the absorber is high, but still low enough to permit

some lasing once a large amount of energy is stored in the gain medium. As the

laser power increases, it saturates the absorber, i.e., rapidly reduces the resonator

loss, so that the power can increase even faster. Ideally, this brings the absorber

into a state with low losses to allow efficient extraction of the stored energy by

the laser pulse. After the pulse, the absorber recovers to its high-loss state before

the gain recovers, so that the next pulse is delayed until the energy in the gain

medium is fully replenished. The pulse repetition rate can only indirectly be

controlled, e.g. by varying the laser's pump power and the amount of saturable

absorber in the cavity. Direct control of the repetition rate can be achieved by

using a pulsed pump source as well as passive Q-switching, see Figure(2.5).

23

Figure(2.5): Schematic of Ruby Passive Q-switch.

2.5 Passive Q-Switch Processes:

The process of passive Q-switching is illustrated in Figure(2.2). The

repeating cycle can be divided into four phases:

Figure(2.2): Q-switching cycle.

24

I: The population inversion and gain increase due to the pumping process. At

the onset of the pulse buildup, the gain in the laser crystal reaches the non

saturated loss level in the resonator.

II: The intra cavity intensity builds up by the process of stimulated emission,

seeded by a spontaneously emitted photon. The circulating light bleaches

the Q-switch and gives rise to a very large net gain due to the accumulated

population inversion. The pulse builds up rapidly.

III: The population inversion has been extracted to a level at which the gain

equals the saturated loss. The net gain is zero, and the pulse starts to

decay. The circulating light extracts more of the population inversion on

its way out of the resonator.

IV: The pulse has been extracted and light has left the resonator. The bleached

Q-switch relaxes back to its ground-state, increasing the loss back to its

non saturated value. The population inversion and gain build up for the

next cycle.

In an ideal passively Q-switched laser, increasing the pump power would just

shorten the pump phase and increase the repetition rate of the laser, while

keeping the pulse properties unchanged. In reality, the instabilities of the pulse

train increase along with the repetition rate, and at some point the laser output

turns chaotic.

25

Chapter Three

Passive Q-Switching

3.1 Introduction:

The idea of Q-switching is to delay the stimulated emission until a large

number of atoms are excited and a lot of photons are released during

spontaneous emission and the great number of photons starts the stimulated

emission together. Then, a strong beam of laser can be produced. Passive Q-

switching involves the addition of a material whose transmission is dictated by

the intracavity photon density in the laser cavity.

Q-switched lasers are often used in applications which demand high laser

intensities in nanosecond pulses, such as dentistry, metal cutting or pulsed

holography non linear optics often takes advantage of the high peak powers of

these lasers, offering applications such as 3D optical data storages and 3D micro

fabrication. However Q-switched lasers can also be used for measurement

purposes, such as for distance measurements (range finding) by measuring the

time it takes for the pulse to get to some target and the reflected light to get back

to the sender. Q-switched lasers are used to remove tattoos. They are used to

shatter tattoo pigment in to particles that are cleared by the body lymphatic

system. Full removal takes an average of eight treatments, spaced at least a

month apart using different lasers for different colored inks.

26

3.2 Rate Equations:

The following rate equations may be utilized to model a solid-state laser

passively Q-switched by a saturable absorber with excited state absorption.

These equations are based on the evolution of atomic population densities in the

laser and Q-switch materials:

nNNKNKNKdtdn

caaaaagg ])([ 0 γβ −−−−= (3.1)

nNKNRdt

dNggggp

g γγ −−= (3.2)

nNKNNdt

dNaaaaa

a −−= )( 0γ (3.3)

The population reduction factor γ equals to one for a four-level laser and

two for a three-level laser. Other parameters used in these coupled rate equations

are defined as following: n is the photon number inside the cavity; gN is the

population inversion of the laser; aN is the ground state population of the

saturable absorber; 0aN is the initial value of aN ; gγ is the effective decay rate of

the upper laser level; aγ is the effective saturable absorber relaxation rate; pR is

the pumping rate; cγ is the cavity decay rate; 𝐾𝐾𝑔𝑔 = 2𝜎𝜎𝑔𝑔 𝜏𝜏𝑟𝑟𝐴𝐴𝑔𝑔⁄ is a coupling

coefficient, where 𝜎𝜎𝑔𝑔 is the laser emission cross-section, 𝜏𝜏𝑟𝑟 is the cavity round-

trip transit time, and 𝐴𝐴𝑔𝑔 is the effective laser beam area; 𝐾𝐾𝑎𝑎 = 2𝜎𝜎𝑎𝑎 𝜏𝜏𝑟𝑟𝐴𝐴𝑎𝑎⁄ , where

𝜎𝜎𝑎𝑎 is the saturable absorber ground state absorption cross-section and 𝐴𝐴𝑎𝑎 is the

saturable absorber beam area; and 𝛽𝛽 is the ratio of the excited state absorption

cross-section, to the ground state absorption cross-section of the saturable

absorber. Notice that 𝛽𝛽 must be less than unity, otherwise a giant pulse will

never develop.

Equation (3.1) models the cavity photon density as a function of time and

consists of a series of gains and losses that contribute to the overall photon

density. The first term models the contribution of photons from the stimulated

27

emission process within the laser material. The next two terms model losses

associated with the production of excited states within the saturable absorber

while the last term represent losses due to decay. The final term concern the bulk

optical losses within the cavity. Equation (3.2) models the excited state

popluation density within the laser material with respect to time and contains one

gain and two loss terms. The gain term is the excited state production rate or

pump rate. The loss terms describe reductions in the excited state density due to

both stimulated and spontaneous emission respectively. Equation (3.3) models

the saturable absorber ground state population density and consists of a gain and

a loss term. The gain term concerns the repopulation of the ground state due to

spontaneous decay within the absorber excited states. The loss term concerns the

excited state population production within the absorber material.

In order to compare the population inversion of the laser to the loss of the

overall laser system, it is convenient to define a normalized loss factor of the Q-

switched laser system from equation (3.1) as:

g

caaaaa

KNNKNKLoss γβ +−+

=)( 0 (3.4)

The differential equations (3.1), (3.2) and (3.3) need to be solved numerically

to obtain the behavior of a specific Q-switched laser system. However, important

characteristics of a saturable absorber Q-switched laser system can be found

from analyzing these three coupled rate equations.

Since the build-up time of the Q-switched laser pulse is generally very short

compared to pumping and relaxation times of the gain medium, it is reasonable

to neglect pumping and spontaneous decay of the laser population inversion

during pulse generation. With this assumption equation (3.2) becomes:

nNKdt

dNgg

g γ−≅ (3.5)

28

When the light intensity is low almost all the population of the saturable

absorber are in the ground state. Hence, the initial population inversion required

for laser action can be approximated by setting the right hand side of Eq. (3.1) to

zero and assuming 0aa NN ≅ , and 0gg NN ≅ i.e.

g

caag K

NKN γ+≅ 0

0 (3.6)

When the light intensity is high most population in the ground state of the

saturable absorber are promoted to the excited state. Therefore, the threshold

population inversion after the bleaching of the saturable absorber can be taken by

setting the right hand side of equation (3.1) to zero and assuming 0≅aN , and

thg NN ≅ i.e.

g

caath K

NKN γβ +≅ 0 (3.7)

Then, equation (3.1), can be rewrite as:

nNNKdtdn

thgg ][ −≅ (3.8)

Equations (3.5) and (3.8) leads to the following equation which relates n and

gN :

−−≅ )ln(1 0

0g

gthgg N

NNNNn

γ (3.9)

As indicated in equation (3.8) the photon number reaches to a peak value pn

when gN is equivalent to thN . Hence, from equation (3.9):

−−≅ )ln(1 0

0th

gththgp N

NNNNn

γ (3.10)

After the release of the Q-switched laser pulse, the laser population inversion

gN is depleted by the photon flux and reduces to a value below thN . This final

population inversion fN can be derived from equation (3.9) by setting 0≅n ,

29

since the photon number is small after the release of the Q-switched laser pulse.

Let fg NN = and 0=n , then equation (3.9) becomes:

0)ln( 00 ≅−−

f

gthfg N

NNNN (3.11)

Equation (3.11) is transcendental and can be solved numerically. When 0gN

and fN are known, the pulse width pulseτ , and the output energy outE of the Q-

switched pulse can be approximated by the following equations:

)ln( 00

0

th

gththg

fgcpulse

NN

NNN

NN

−−

−≅ ττ (3.12)

cfg

out hvNN

E ηγ

)(0 −≅ (3.13)

where cτ is the cavity lifetime, hv is the photon energy and cη is the output

coupling efficiency. Equations (3.5) to (3.13) may be used to quantitatively

evaluate the optical performance of a Q-switched laser analytically.

3.3 Runge-kutta Fehlberg:

One way to guarantee accuracy in the solution of an initial values problems

(I.V.P.) is to solve the problem twice using step sizes h and h/2 and compare

answers at the mesh points corresponding to the larger step size. But this requires

a significant amount of computation for the smaller step size and must be

repeated if it is determined that the agreement is not good enough.

The Runge-Kutta-Fehlberg method (denoted RKF45) is one way to try to

resolve this problem. It has a procedure to determine if the proper step size h is

being used. At each step, two different approximations for the solution are made

and compared. If the two answers are in close agreement, the approximation is

accepted. If the two answers do not agree to a specified accuracy, the step size is

30

reduced. If the answers agree to more significant digits than required, the step

size is increased.

Thus, the approach yields the error estimate on the basis of only six function

evaluations!

Then an approximation to the solution of the I.V.P. is made using a Runge-

Kutta method of order 4:

𝑦𝑦𝑖𝑖+1 = 𝑦𝑦𝑖𝑖 + �37

378𝑘𝑘1 +

250621

𝑘𝑘3 +125594

𝑘𝑘4 +512

1771𝑘𝑘6� ℎ (3.14)

A better value for the solution is determined using a Runge-Kutta method of

order 5:

𝑦𝑦𝑖𝑖+1 = 𝑦𝑦𝑖𝑖 + �2825

27648𝑘𝑘1 +

1857548384

𝑘𝑘3 +1352555296

𝑘𝑘4 +277

14336𝑘𝑘5 +

14𝑘𝑘6� ℎ

(3.15)

Where:

𝑘𝑘1 = 𝑓𝑓(𝑥𝑥𝑖𝑖 + 𝑦𝑦𝑖𝑖) (3.16)

𝑘𝑘2 = 𝑓𝑓 �𝑥𝑥𝑖𝑖 +15ℎ,𝑦𝑦𝑖𝑖 +

15𝑘𝑘1ℎ� (3.17)

𝑘𝑘3 = 𝑓𝑓 �𝑥𝑥𝑖𝑖 +3

10ℎ,𝑦𝑦𝑖𝑖 +

340

𝑘𝑘1ℎ +9

40𝑘𝑘2ℎ� (3.18)

𝑘𝑘4 = 𝑓𝑓 �𝑥𝑥𝑖𝑖 +35ℎ,𝑦𝑦𝑖𝑖 +

310

𝑘𝑘1ℎ −9

10𝑘𝑘2ℎ +

65𝑘𝑘3ℎ� (3.19)

𝑘𝑘5 = 𝑓𝑓 �𝑥𝑥𝑖𝑖 + ℎ,𝑦𝑦𝑖𝑖 −1154

𝑘𝑘1ℎ +52𝑘𝑘2ℎ −

7027

𝑘𝑘3ℎ +3527

𝑘𝑘4ℎ� (3.20)

𝑘𝑘6 = 𝑓𝑓 �𝑥𝑥𝑖𝑖 +78ℎ,𝑦𝑦𝑖𝑖 +

163155296

𝑘𝑘1ℎ +175512

𝑘𝑘2ℎ +575

13824𝑘𝑘3ℎ +

44275110592

𝑘𝑘4ℎ

+253

4096𝑘𝑘5ℎ� (3.21)

Thus, the ODE can be solved with equation (3.15), and the error estimated as

the difference of the fifth- and fourth- order estimates. It should be noted that the

particular coefficients used above were developed by cash and karp (1990).

Therefore, it is sometimes called the cash-karp RK method.

31

Chapter Four

Results and Discussion

4.1 Introduction:

High energy laser pulses of short durations may be obtained by passive Q-

switching techniques, where energy is stored in the gain medium through

pumping source while the quality factor of the laser resonator is decreased to

prevent laser oscillation. The passive Q-switching with a saturable absorber is

economical and simple because it requires less optical elements inside the laser

cavity and no outside driving circuitry. Passive Q-switching is a better choice for

those applications where compactness of the laser is a prime requirement.

The dye cell is highly absorptive at low light intensity and can be bleached

when the light becomes intense. Short laser pulses with high peak power can be

obtained with the use of a dye Q-switch. However, dye Q-switches suffer from

poor chemical stability and inadequate thermal properties. To overcome this

problem, a flow system has been used at the expense of compactness and

simplicity.

4.2 Solution of Rate Equations:

The laser rate equations (3.1) to (3.3) are numerically solved using the

Runge-Kutta Fehlberg method to investigate the performance of the ruby laser

(Cr4+:Al2O3) with Dy2+:CaF2

The parameters used in this simulation, obtained from experiments published

articles, are as following: laser wavelength=

solid-state saturable absorber as a passive Q-

switch.

nm3.694 , 110 sec1022.7 −−×=gK , 18 sec1046.3 −−×=aK , 18 sec1042.3 −×=cγ , 1sec333 −=gγ , 1sec6667 −=aγ , 75.0=β ,

32

121 sec107.1 −×=pR and 150 1018.5 ×=aN . The population reduction factor γ of the

ruby laser has a value of 2 since it is a three-level laser.

The behavior of the laser population inversion 𝑁𝑁𝑔𝑔 , the loss of the Q-switched

laser Loss , and photon number inside the laser resonator 𝑛𝑛 as functions of time,

are shown in Fig (4.1). The giant laser pulse is devloped 463.6𝜇𝜇𝜇𝜇 after pumping

starts. Another three laser pulses are developed 502.7𝜇𝜇𝜇𝜇, 517.0𝜇𝜇𝜇𝜇, and 530.8𝜇𝜇𝜇𝜇,

after the pumping starts.

Evidently, the amplitudes of these three pulses are much smaller than that of

the first giant laser pulse. This is due to the fact that the Dy2+:CaF2

Fig (4.2), is an expanded picture of Fig (4.1), near the occurrence of the first

giant laser pulse. The output energy of the giant laser pulse is 3.04 𝑚𝑚𝑚𝑚 with a

pulse-width of 150.14 𝑛𝑛𝜇𝜇𝑛𝑛𝑛𝑛 (full width at half maximum).

has a very

long lifetime. Moreover, the population inversion dose not decrease to zero after

the release of the first laser pulse. Hence it requires much less time to develop

the second and subsequent laser pulses (about 14𝜇𝜇𝜇𝜇), and the energy contained

in each of the second and subsequent laser pulses is much smaller than that of the

first giant laser pulse.

33

Fig (4.1): The population inversion of the laser𝑁𝑁𝑔𝑔 , the loss of the Q-switched

laser 𝐿𝐿𝐿𝐿𝐿𝐿𝜇𝜇and photon number inside the laser resonator 𝑛𝑛 as functions of time.

Fig (4.2): Expanded picture of Fig (4.1), near the occurrence of the

first giant laser pulse.

The results indicate that when the phonon number is low, aN is close to 0aN

and the loss of the laser system has an initial value of about 171022.7 × . For the

laser action to occur the laser has to be pumped, so that the gain is greater than

the loss, i.e. LossNg > . When this condition is satisfied the photon number starts

to build up from the noise by depleting the laser population inversion and the

saturable absorber starts to saturate. The photon number reaches to the maximum

value of about 14108.5 × when the laser population inversion equals the cavity

loss, i.e., when 171063.6 ×≅= LossNg . Beyond this point the laser gain is smaller

than the total loss of the laser system and the Q-switched laser pulse dies out

34

quickly while the laser population inversion decreases gradually to a minimum

value of about 171024.6 × .

The theoretical predictions of all pulse characteristics, such as output energy

and pulse duration were in very good agreement with experimental results. Thus,

it allows us to assume that the numerical model proposed is correct and can be

applied for different saturable absorbers.

The accuracy of evaluating the output energy and pulse-width of the Q-

switched laser pulse are within %77.0 and %20.0 when compared to the

experimental results ( mJ6.2 and sec150n ) respectively.

4.3 Effect of concentration of the saturable absorber:

To investigate the effect of concentration of the saturable absorber on the

performance of laser, the different values of 𝑁𝑁𝑎𝑎𝐿𝐿 were used from 1 × 1015 to

9 × 1015 , and all the parameters were calculated for each case, like the

maximum photons number, Period life time (FWHM), time of the first pulse

generation, and the energy of Q-Switched pulse.

Numerical simulation of the ruby passive Q-switching performance, i.e. the

population inversion of the laser 𝑁𝑁𝑔𝑔 , the loss of the overall laser system 𝐿𝐿𝐿𝐿𝜇𝜇𝜇𝜇,

and the photon number inside the cavity 𝑛𝑛, as a function of time, and near the

occurrence of the first giant laser pulse, for different values of 𝑁𝑁𝑎𝑎𝐿𝐿 are shown in

Fig (4-3) through to Fig (4.11).

35

Fig (4.3): Numerical simulation of the ruby passive Q-switching

performance, 𝑁𝑁𝑎𝑎𝐿𝐿 = 1 × 1015 (a) 𝑁𝑁𝑔𝑔 , 𝐿𝐿𝐿𝐿𝜇𝜇𝜇𝜇, and 𝑛𝑛 as a function of time. (b) near

the occurrence of the first giant laser pulse.

36

Fig (4.4): Numerical simulation of the ruby passive Q-switching

performance, 𝑁𝑁𝑎𝑎𝐿𝐿 = 2 × 1015 (a) 𝑁𝑁𝑔𝑔 , 𝐿𝐿𝐿𝐿𝜇𝜇𝜇𝜇, and 𝑛𝑛 as a function of time. (b) near

the occurrence of the first giant laser pulse.

37

Fig (4.5): Numerical simulation of the ruby passive Q-switching

performance, 𝑁𝑁𝑎𝑎𝐿𝐿 = 3 × 1015 (a) 𝑁𝑁𝑔𝑔 , 𝐿𝐿𝐿𝐿𝜇𝜇𝜇𝜇, and 𝑛𝑛 as a function of time. (b) near

the occurrence of the first giant laser pulse.

38

Fig (4.6): Numerical simulation of the ruby passive Q-switching

performance, 𝑁𝑁𝑎𝑎𝐿𝐿 = 4 × 1015 (a) 𝑁𝑁𝑔𝑔 , 𝐿𝐿𝐿𝐿𝜇𝜇𝜇𝜇, and 𝑛𝑛 as a function of time. (b) near

the occurrence of the first giant laser pulse.

39

Fig (4.7): Numerical simulation of the ruby passive Q-switching

performance, 𝑁𝑁𝑎𝑎𝐿𝐿 = 5 × 1015 (a) 𝑁𝑁𝑔𝑔 , 𝐿𝐿𝐿𝐿𝜇𝜇𝜇𝜇, and 𝑛𝑛 as a function of time. (b) near

the occurrence of the first giant laser pulse.

40

Fig (4.8): Numerical simulation of the ruby passive Q-switching

performance, 𝑁𝑁𝑎𝑎𝐿𝐿 = 6 × 1015 (a) 𝑁𝑁𝑔𝑔 , 𝐿𝐿𝐿𝐿𝜇𝜇𝜇𝜇, and 𝑛𝑛 as a function of time. (b) near

the occurrence of the first giant laser pulse.

41

Fig (4.9): Numerical simulation of the ruby passive Q-switching

performance, 𝑁𝑁𝑎𝑎𝐿𝐿 = 7 × 1015 (a) 𝑁𝑁𝑔𝑔 , 𝐿𝐿𝐿𝐿𝜇𝜇𝜇𝜇, and 𝑛𝑛 as a function of time. (b) near

the occurrence of the first giant laser pulse.

42

Fig (4.10): Numerical simulation of the ruby passive Q-switching

performance, 𝑁𝑁𝑎𝑎𝐿𝐿 = 8 × 1015 (a) 𝑁𝑁𝑔𝑔 , 𝐿𝐿𝐿𝐿𝜇𝜇𝜇𝜇, and 𝑛𝑛 as a function of time. (b) near

the occurrence of the first giant laser pulse.

43

Fig (4.11): Numerical simulation of the ruby passive Q-switching

performance, 𝑁𝑁𝑎𝑎𝐿𝐿 = 9 × 1015 (a) 𝑁𝑁𝑔𝑔 , 𝐿𝐿𝐿𝐿𝜇𝜇𝜇𝜇, and 𝑛𝑛 as a function of time. (b) near

the occurrence of the first giant laser pulse.

44

The energy of the laser output pulse, the pulse-width (full width at half

maximum), and the peak photon number, are solved numerically as functions of

the initial population in the ground state of the saturable absorber 𝑁𝑁𝑎𝑎0, assuming

other parameters remain unchanged. The results are shown in Fig (4.12) to Fig

(4.14) respectively.

The output energy, and the pulse-width, are shown as a function of the initial

population in the ground state of the saturable absorber 𝑁𝑁𝑎𝑎0 in Fig (4.12) and Fig

(4.13), assuming other parameters remain unchanged. Obviously, the output

energy increases and the pulse-width decreases when 𝑁𝑁𝑎𝑎0 increases. In other

words, the passive Q-switching laser system has a better performance when a

higher doping concentration of the saturable absorber is used.

Fig (4.12): The output energy as a function of initial value of saturable absorber

𝑁𝑁𝑎𝑎0.

0

1

2

3

4

5

6

0.00E+00 2.00E+15 4.00E+15 6.00E+15 8.00E+15 1.00E+16

Out

put E

nerg

y (m

J)

Nao

45

Fig (4.13): The pulse-width as a function of initial value of saturable absorber

𝑁𝑁𝑎𝑎0.

Fig (4.14): The peak photon number as a function of initial value of saturable

absorber 𝑁𝑁𝑎𝑎0.

In general, when 𝑁𝑁𝑎𝑎0 increases the peak photon number of the Q-switched

laser pulse increase, as shown in Fig (4.14). Since the saturable absorber is used

to store or delay the laser oscillation. During this storage, more molecules can be

0

100

200

300

400

500

600

0.00E+00 2.00E+15 4.00E+15 6.00E+15 8.00E+15 1.00E+16

puls

e-w

idth

(nse

c)

Nao

0.00E+00

2.00E+14

4.00E+14

6.00E+14

8.00E+14

1.00E+15

1.20E+15

1.40E+15

1.60E+15

1.80E+15

0.00E+00 2.00E+15 4.00E+15 6.00E+15 8.00E+15 1.00E+16

nmax

(pea

k ph

oton

num

ber)

Nao

46

excited to the upper level. Once the bleaching occurred, the photons can pass

through and strike with many excited molecules, hence produced many

stimulated emission. This condition results in the production of giant pulse. The

pulse will emit within a very short time.

The temporal profile of the photon number has a narrower width and higher

peak for a larger number of 𝑁𝑁𝑎𝑎0, as expected. It indicates that better passive Q-

switching performance, i.e., a shorter pulse with higher output energy, can be

obtained if a saturable absorber of higher concentration is used.

Fig (4.15) shows the time required to develop the first laser pulse as a

function of the initial value of saturable absorber 𝑁𝑁𝑎𝑎0. It is obvious that the time

required to develop the first laser pulse increase when 𝑁𝑁𝑎𝑎0 increases. This is

reasonable since it takes a longer time to bleach the saturable absorber and the

laser population inversion becomes larger when 𝑁𝑁𝑎𝑎0 increases.

Fig (4.15): The time required for developing the first laser pulse as a function of

initial value of saturable absorber 𝑁𝑁𝑎𝑎0.

0.00E+00

1.00E+05

2.00E+05

3.00E+05

4.00E+05

5.00E+05

6.00E+05

7.00E+05

0.00E+00 2.00E+15 4.00E+15 6.00E+15 8.00E+15 1.00E+16

Dev

elop

ed ti

me

of fi

rst p

ulse

(nse

c)

Nao

47

4.4 Conclusion:

Characteristics and performance of ruby laser passive Q-switching with

Dy2+:CaF2

The characteristics of the Q-switched laser parameters such as the pulse

energy, the pulse width, and the pulse repetition frequency have been explored as

function of saturable absorber doping concentration.

solid state saturable absorber was studied numerically using the

Runge-Kutta Fehlberg method. The important factors such as the initial

population inversion required for laser action, the threshold population inversion

after the bleaching of the saturable absorber, the final population inversion after

the release of the Q-switched laser pulse, and the peak photon number inside the

laser resonator are calculated.

The overall results of this study were summarized below:

1- The simulation results indicate that better passive Q-switching

performance, i.e., a shorter pulse with a higher output energy, can be

obtained by using a higher concentration of Dy2+:CaF2

2- The giant laser pulse is developed 463.6 microsecond after pumping starts.

The sequence of other three laser pulses developed were (502.7,517.0 and

530.8) microsecond respectively, after the pumping starts.

inside the laser

cavity.

3- The amplitudes of these three pulses are much smaller than that of the first

giant laser pulse. This is due to the fact that the Dy2+:CaF2 has a very long

life time.

48

4- The energy of the second and subsequent laser pulses much smaller than

that of the first giant laser pulse. The output energy of the giant laser pulse

is 3.04𝑚𝑚𝑚𝑚 with pulse-width of 150.14 𝑛𝑛𝜇𝜇𝑛𝑛𝑛𝑛 (full width at half maximum).

5- The condition at which results in the production of giant pulse is occurring

the bleaching, that makes photons pass through and strike with many

excited molecules, hence produce many stimulated emission.

7- The time required to develop the first laser pulse increase when

concentration of Dy2+:CaF2

8- The very good agreement between experimental and theoretical results

obtained in this study may be very useful in designing of such lasers.

inside the laser cavity increases.

9- Several useful performance curve are provided in this work, which can be

used as an aid for the design and improve a performance of a passive Q-

switch laser system.

49

References

[1] Azzous I.M., A. El-Nozahy, "Numerical Study of Cr4+:YAG Passively Q-switching Nd:GdVO4

Laser", Egypt. J. Solids, Vol. 29, No. 1, pp. 215-225, (2006).

[2] Swiderski J., M. Skorczakowski, A. Zajac, and P. Konieczny, " Numerical analysis of a passively Q-switched Nd-YAG laser with a Cr4+

:YAG exhibiting ESA", Opto-Electronics Review, Vol. 13, No. 1, pp. 43-50, (2005).

[3] McIntyre D.L., "A Laser Spark Plug Ignition System for a Stationary Lean-Burn Natural Gas Reciprocating Engine", PhD Dissertation, West Virginia University, (2007).

[4] Munajat N.F.B., "Study of saturable absorber materials for Q-switching DYE

laser", MSc thesis, University Technology Malaysia, (2005). [5] Kuo Y.K., H.M. Chen, and C.C. Lin, "A Theoretical Study of the

Cr:BeAl2O4

Laser Passively Q-switched with Cr:YSO Solid State Saturable Absorber", Chinese Journal of Physics, Vol. 38, No. 3-I, pp. 443-460, (2000).

[6] Spiekermann S., "Compact diode-pumped solid-state laser", PhD thesis, Royal Institute of Technology, Stockholm, Sweden, (2004).

[7] Longbotham N.W., "Experimental Characterization of Cr4+

:YAG Passively Q-switched Cr:Nd:GSGG Lasers and Comparison with a Simple Rate Equation Model", PhD thesis, University of New Mexico, (2008).

[8] Kuo Y.K., W. Chen, R.D. Stultz, and M. Birnbaum, "Dy2+:CaF2

saturable-absorber Q switch for the ruby laser", Applied Optics, Vol. 33, No. 27, pp. 6348-6351, (1994).

[9] Kuo Y.K., and M. Birnbaum, "Passive Q switching of the alexandrite laser with a Cr4+:Y2SiO5

solid-state saturable absorber", Appl. Phys. Lett. Vol. 67, No. 2, pp. 173-175, (1995).

[10] Kuo Y.K., H.M. Chen, and J.Y. Chang, "Numerical study of the Cr:YSO Q-switched ruby", Opt. Eng., Vol. 40, No. 9, pp. 2031-2035, (2001).

50

[11] Koechner W., “Solid-State Laser Engineering”, Sixth edition, Springer Science-Business Media, Inc., (2006).

[12] Kuo Y.K., Y. Yang, and . Birnbaum, "Cr4+:Gd3Sc2Ga3O12 passive Q-switch

for the Cr3+:LiCaAlF6

laser", Appl. Phys. Lett. Vol. 64, No. 18, pp. 2329-2331, (1994).

[13] Kuo Y.K., and M. Birnbaum, "Characteristic of ruby passive Q switching with a Dy+2:CaF2

solid-state saturable absorber", Applied Optics, Vol. 34, No. 30, pp. 6829-6833, (1995).

[14] Luo Y.K., M.F. Huang, M. Birnbaum, “Tunable Cr4+

:YSO Q-switched Cr:LiCAF laser”, IEEE Journal of Quantum Electronic, Vol. 31, No. 4, (1995).

[15] Chen H.F., S.W. Hsieh, and Y.K. Kuo, "Simulation of tunable Cr:YSO Q-switched Cr:LiSAF laser:, Proceedings of SPIE, Vol. 5627, pp. 488-498, (2005).

[16] Walter Koechner, "Solid-State Laser Engineering", Sixth Revised and

Updated Edition, Springer Science Business Media, Inc., (2006). [17] www.scientific_web.com/en/physics/optics/Q switching.htm [18] www.wikipedia.org/wiki/Q-switching [19] www.unc.edu/~dtmoore/laser_intro.html [20] www.hazemsakeek.com/QandA/Laser/Laser.htm [21] 1TUwww.laserinternational.org/info/laserintroductionU1T

51

Appendix

Matlab program to solve the rate equation of passive Q-switching %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%% Numerical Solution of %%%%%%%%%% %%%%%%%%%% Rate Equation of Passive Q-Switch %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%% Sulaimani University %%%%%%%%%% %%%%%%%%%% College of Science - Physics Department %%%%%%%%%% %%%%%%%%%% (c) Omed Ghareb Abdullah 2009 %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ka: Coupling coefficient for active medium (1/sec) % kg: Coupling coefficient for saturable absorber material (1/sec) % ga: Decay rate of the first exited level of the saturable absorber material % gc: Cavity decay rate % gg: Decay rate of the upper laser level % rt: Relaxation time of the first excited level of the saturable absorber material % rp: Pumpin rate % b: Ratio of the absorption x-secs of the first excited to the G.S. of % the saturable absorber levels % lambda: LASER wavelength (nm) % clight: light speed (cm/sec) % gama: Population reduction factor, is 1 for a four-level laser, % and 2 for a thre-level laser. % imax: Position of maximum number of photons % no: initial photon numbers % ph: Maximum photons numbers % calstep: Number of calculated steps % nstep: number of steps % tio: initial time (nsec) % tfo: final time (nsec) % tt: times from tio to tfo (nsec) % dt: time interval (nsec) % pr: The period of the Q-Switched pulse % tt(lleft): Position of left % tt(lmax): Position of center % tt(right): Position of right % rise: Rising time (nsec) % fall: Falling time (nsec) % e: The energy of Q-Switched pulse (J) % power: The power of Q-Switched pulse (Watt) % imax=position of the maximum number of photons % nao: Initial number of suturable absorber molecules in G.S % R: Output coupler reflectivity % np: Initial number of photons % nth: Threshold population inversion % ph: Photons number as a function of time % ng: Population inversion as a function of time % na: Number of saturable absorber material % loss: Photon losses % loabs: Rate absorption of G.S. % upabs: Rate absorption of first E.S. % totabs: Total Rate absorption

52

% pr: Period life time (nsec) % rise: Pulse Rising time (nsec) % fall: Pulse Falling time (nsec) % phmax: Maximum number of photons % hphmax: Wave half mean % This program need approximatelly 95 minutes to RUN clc clear all tfo=600000; % number of steps (nsec) tio=0; % initial time (nsec) dt=5; % time interval (nsec) nstep=(tfo-tio)/dt; kg=7.22e-10; ka=3.46e-8; ga=6667; gc=3.42e8; gg=333; b=0.75; rp=1.7e21; % 1/sec lambda=694.3; % nm lambda=lambda*1e-9; % m nao=5.18e15; gama=2; clight=3e8; % m/sec kg=kg*1e-9; % 1/nsec ka=ka*1e-9; % 1/nsec ga=ga*1e-9; % 1/nsec gc=gc*1e-9; % 1/nsec gg=gg*1e-9; % 1/nsec rp=rp*1e-9; % 1/nsec rt=1/ga; tc=1/gc; nth=(b*ka*nao+gc)/kg; ngo=(ka*nao+gc)/kg; npeak=1/gama*(ngo-nth-nth*log(ngo/nth)); R=0.78; ph(1)=3e6; % initial Photon number ---> zero ng(1)=0; % initial Population inversion=zero na(1)=nao; loss(1)=(ka*na(1)+b*ka*(nao-na(1))+gc)/kg; loabs(1)=ka*na(1); upabs(1)=b*ka*(nao-na(1)); totabs(1)=loabs(1)+upabs(1); tt(1)=tio; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Solveing the Rate equation for three-level model % Using Runge-Kutta Fehlberg method %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for j=1:nstep yph=ph(j); yng=ng(j); yna=na(j); tt(j+1)=tt(j)+dt;

53

kna(1)=(ga*(nao-yna)-ka*yna*yph); kng(1)=(rp-gg*yng-gama*kg*yng*yph); kph(1)=(kg*yng-ka*yna-b*ka*(nao-yna)-gc)*yph; yph=ph(j)+1/5*kph(1)*dt; yng=ng(j)+1/5*kng(1)*dt; yna=na(j)+1/5*kna(1)*dt; kna(2)=(ga*(nao-yna)-ka*yna*yph); kng(2)=(rp-gg*yng-gama*kg*yng*yph); kph(2)=(kg*yng-ka*yna-b*ka*(nao-yna)-gc)*yph; yph=ph(j)+3/40*kph(1)*dt+9/40*kph(2)*dt; yng=ng(j)+3/40*kng(1)*dt+9/40*kng(2)*dt; yna=na(j)+3/40*kna(1)*dt+9/40*kna(2)*dt; kna(3)=(ga*(nao-yna)-ka*yna*yph); kng(3)=(rp-gg*yng-gama*kg*yng*yph); kph(3)=(kg*yng-ka*yna-b*ka*(nao-yna)-gc)*yph; yph=ph(j)+3/10*kph(1)*dt-9/10*kph(2)*dt+6/5*kph(3)*dt; yng=ng(j)+3/10*kng(1)*dt-9/10*kng(2)*dt+6/5*kng(3)*dt; yna=na(j)+3/10*kna(1)*dt-9/10*kna(2)*dt+6/5*kna(3)*dt; kna(4)=(ga*(nao-yna)-ka*yna*yph); kng(4)=(rp-gg*yng-gama*kg*yng*yph); kph(4)=(kg*yng-ka*yna-b*ka*(nao-yna)-gc)*yph; yph=ph(j)-11/54*kph(1)*dt+5/2*kph(2)*dt-70/27*kph(3)*dt+35/27*kph(4)*dt; yng=ng(j)-11/54*kng(1)*dt+5/2*kng(2)*dt-70/27*kng(3)*dt+35/27*kng(4)*dt; yna=na(j)-11/54*kna(1)*dt+5/2*kna(2)*dt-70/27*kna(3)*dt+35/27*kna(4)*dt; kna(5)=(ga*(nao-yna)-ka*yna*yph); kng(5)=(rp-gg*yng-gama*kg*yng*yph); kph(5)=(kg*yng-ka*yna-b*ka*(nao-yna)-gc)*yph; yph=ph(j)+1631/55296*kph(1)*dt+175/512*kph(2)*dt+575/13824*kph(3)*dt+44275/110592*kph(4)*dt+253/4096*kph(5)*dt; yng=ng(j)+1631/55296*kng(1)*dt+175/512*kng(2)*dt+575/13824*kng(3)*dt+44275/110592*kng(4)*dt+253/4096*kng(5)*dt; yna=na(j)+1631/55296*kna(1)*dt+175/512*kna(2)*dt+575/13824*kna(3)*dt+44275/110592*kna(4)*dt+253/4096*kna(5)*dt; kna(6)=(ga*(nao-yna)-ka*yna*yph); kng(6)=(rp-gg*yng-gama*kg*yng*yph); kph(6)=(kg*yng-ka*yna-b*ka*(nao-yna)-gc)*yph; ph4(j+1)=ph(j)+dt*(37/378*kph(1)+250/621*kph(3)+125/594*kph(4)+512/1771*kph(6));

54

ng4(j+1)=ng(j)+dt*(37/378*kng(1)+250/621*kng(3)+125/594*kng(4)+512/1771*kng(6)); na4(j+1)=na(j)+dt*(37/378*kna(1)+250/621*kna(3)+125/594*kna(4)+512/1771*kna(6)); ph5(j+1)=ph(j)+dt*(2825/27648*kph(1)+18575/48384*kph(3)+13525/55296*kph(4)+277/14336*kph(5)+1/4*kph(6)); ng5(j+1)=ng(j)+dt*(2825/27648*kng(1)+18575/48384*kng(3)+13525/55296*kng(4)+277/14336*kng(5)+1/4*kng(6)); na5(j+1)=na(j)+dt*(2825/27648*kna(1)+18575/48384*kna(3)+13525/55296*kna(4)+277/14336*kna(5)+1/4*kna(6)); ph(j+1)=2*ph5(j+1)-ph4(j+1); ng(j+1)=2*ng5(j+1)-ng4(j+1); na(j+1)=2*na5(j+1)-na4(j+1); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if ph(j+1)<ph(1) ph(j+1)=ph(1); end loss(j+1)=(ka*na(j+1)+b*ka*(nao-na(j))+gc)/kg; loabs(j+1)=ka*na(j+1); upabs(j+1)=b*ka*(nao-na(j)); totabs(j+1)=loabs(j+1)+upabs(j+1); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%% whitebg %figure(1) plot(tt,ng,'r-'); hold on; plot(tt,loss,'b--'); title(' '); xlabel('Time (nsec)'); ylabel('Ng configuration & Loos'); legend('Ng ', 'Loss '); xtext=tt(1)+max([tt])/15; ytext=max([ng loss]); ytext=max([ytext]); text(xtext,0.9*ytext,['Wave length = ',num2str(lambda),' nm']); text(xtext,0.8*ytext,['Nao = ',num2str(nao),' ']); text(xtext,0.7*ytext,['B = ',num2str(b),' ']); hold off pause %colordef black %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %figure(2) plot(tt,ph,'r.-'); hold on; title(' '); xlabel('Time (nsec)'); ylabel('n configuration'); xtext=tt(1)+max([tt])/15; xtext=max([xtext]); ytext=max([ph]);

55

ytext=max([ytext]); text(xtext,0.9*ytext,['Wave length = ',num2str(lambda),' nm']); text(xtext,0.8*ytext,['Nao = ',num2str(nao),' ']); text(xtext,0.7*ytext,['B = ',num2str(b),' ']); pause hold off %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %figure(3) plot(tt,loss,'r-'); hold on [AX,H1,H2] = plotyy(tt,ng,tt,ph,'plot'); set(get(AX(1),'Ylabel'),'String','Ng configuration & Loos') set(get(AX(2),'Ylabel'),'String','n configuration') set(H1,'LineStyle','--') set(H2,'LineStyle',':') title(' '); xlabel('Time (nsec)'); title(' '); xtext=tt(1)+max([tt])/15; ytext=max([ng loss ph]); ytext=max([ytext]); text(xtext,0.9*ytext,['Wave length = ',num2str(lambda),' nm']); text(xtext,0.8*ytext,['Nao = ',num2str(nao),' ']); text(xtext,0.7*ytext,['B = ',num2str(b),' ']); pause hold off %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Estimating the principal parameters of first pulse %% by Analytical method, and some approximation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k=1; tol=1e3; nf(k)=1000; ff=ngo-nf(k)-nth*log(ngo/nf(k)); fd=-1+nth/nf(k); nf(k+1)=nf(k)-ff/fd; while abs(nf(k+1)-nf(k))>tol; k=k+1; ff=ngo-nf(k)-nth*log(ngo/nf(k)); fd=-1+nth/nf(k); nf(k+1)=nf(k)-ff/fd; end nff=nf(k+1); disp(['****** Analytical Estimate ****** ']); disp(['Initial population inversion = ',num2str(ngo),' ']); disp(['Threshold population inversion = ',num2str(nth),' ']); disp(['Final population inversion = ',num2str(nff),' ']); disp(['Number of iteration = ',num2str(k+1),' ']); disp(['Maximum photons number = ',num2str(npeak),' ']); hv=6.625e-34*clight/(lambda); eout=(ngo-nff)/gama*hv*(1-R); tpulse=tc*(ngo-nff)/(ngo-nth-nth*log(ngo/nth)); power=eout/(tpulse*1e-9); disp(['Period life time (FWHM) = ',num2str(tpulse),' nsec']); disp(['The energy of Q-Switched pulse = ',num2str(eout*1e3),' mJ']);

56

disp(['The power of Q-Switched pulse = ',num2str(power),' Watt']); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Estimating the principal parameters of first pulse %% by Numerical method (Runge-Kutta Fehlberg method) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% disp(['****** Firest Pulse ****** ']); no1=92500; no2=93000; tt1=tt([no1:no2]); ng1=ng([no1:no2]); ph1=ph([no1:no2]); loss1=loss([no1:no2]); na1=na([no1:no2]); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %figure(4) plot(tt1,loss1,'r-'); hold on [AX,H1,H2] = plotyy(tt1,ng1,tt1,ph1,'plot'); set(get(AX(1),'Ylabel'),'String','Ng configuration & Loos') set(get(AX(2),'Ylabel'),'String','n configuration') set(H1,'LineStyle','--') set(H2,'LineStyle',':') title('Firest Pulse'); xlabel('Time (nsec)'); title('Firest Pulse'); xtext=tt1(1)+max([tt1])/15; ytext=max([ng1 loss1 ph1]); ytext=max([ytext]); text(xtext,0.9*ytext,['Wave length = ',num2str(lambda),' nm']); text(xtext,0.8*ytext,['Nao = ',num2str(nao),' ']); text(xtext,0.7*ytext,['B = ',num2str(b),' ']); hold off pause %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% phmax=max([ph1]); nstep1=no2-no1; for i=1:nstep1 if ph1(i)==phmax; imax=i; end end hphmax=(phmax-ph1(1))/2; tmax=tt1(imax); for i=1:imax if ph1(i)<=hphmax; ileft=i+1; end end for i=imax:nstep1 if ph1(i)>=hphmax iright=i+1; end end

57

tl=((hphmax-ph1(ileft-1))/(ph1(ileft)-ph1(ileft-1)))*(tt1(ileft)-tt1(ileft-1))+tt1(ileft-1); tr=((hphmax-ph1(iright))/(ph1(iright-1)-ph1(iright)))*(tt1(iright-1)-tt1(iright))+tt1(iright); pr=tr-tl; rise=tt1(imax)-tl; fall=tr-tt1(imax); disp(['Position of left = ',num2str(tt1(ileft)),' ']); disp(['Position of center = ',num2str(tt1(imax)),' ']); disp(['Position of right = ',num2str(tt1(iright)),' ']); disp(['Position of maximum number of photons = ',num2str(imax),' ']); disp(['Maximum photons number = ',num2str(ph1(imax)),' ']); disp(['Period life time (FWHM) = ',num2str(pr),' nsec']); disp(['Pulse Rising time = ',num2str(rise),' nsec']); disp(['Pulse Falling time = ',num2str(fall),' nsec']); for m=2:nstep1 if ph1(m)>(ph1(1)); li=m; end end disp(['Final population inversion = ',num2str(ng1(li)),' ']); disp(['Final number of absorber molecules in the G.S. = ',num2str(na1(li)),' ']); disp(['Final number of absorber molecules in the first excited state = ',num2str(nao-na1(li)),' ']); hv=6.625e-34* clight/(lambda); eout=(ngo-ng1(li))/gama*hv*(1-R); power=eout/(pr*1e-9); disp(['The energy of Q-Switched pulse = ',num2str(eout*1e3),' mJ']); disp(['The power of Q-Switched pulse = ',num2str(power),' Watt']); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Estimating the principal parameters of first pulse %% by Numerical method (Runge-Kutta Fehlberg method) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% disp(['****** Second Pulse ****** ']); no1=100000; no2=101200; tt1=tt([no1:no2]); ng1=ng([no1:no2]); ph1=ph([no1:no2]); loss1=loss([no1:no2]); na1=na([no1:no2]); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %figure(5) plot(tt1,loss1,'r-'); hold on [AX,H1,H2] = plotyy(tt1,ng1,tt1,ph1,'plot'); set(get(AX(1),'Ylabel'),'String','Ng configuration & Loos') set(get(AX(2),'Ylabel'),'String','n configuration') set(H1,'LineStyle','--')

58

set(H2,'LineStyle',':') title(' '); xlabel('Time (nsec)'); title('Second Pulse'); xtext=tt1(1)+max([tt1])/15; ytext=max([ng1 loss1 ph1]); ytext=max([ytext]); text(xtext,0.9*ytext,['Wave length = ',num2str(lambda),' nm']); text(xtext,0.8*ytext,['Nao = ',num2str(nao),' ']); text(xtext,0.7*ytext,['B = ',num2str(b),' ']); hold off pause %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% phmax=max([ph1]); nstep1=no2-no1; for i=1:nstep1 if ph1(i)==phmax; imax=i; end end hphmax=(phmax-ph1(1))/2; tmax=tt1(imax); for i=1:imax if ph1(i)<=hphmax; ileft=i+1; end end for i=imax:nstep1 if ph1(i)>=hphmax iright=i+1; end end tl=((hphmax-ph1(ileft-1))/(ph1(ileft)-ph1(ileft-1)))*(tt1(ileft)-tt1(ileft-1))+tt1(ileft-1); tr=((hphmax-ph1(iright))/(ph1(iright-1)-ph1(iright)))*(tt1(iright-1)-tt1(iright))+tt1(iright); pr=tr-tl; rise=tt1(imax)-tl; fall=tr-tt1(imax); disp(['Position of left = ',num2str(tt1(ileft)),' ']); disp(['Position of center = ',num2str(tt1(imax)),' ']); disp(['Position of right = ',num2str(tt1(iright)),' ']); disp(['Position of maximum number of photons = ',num2str(imax),' ']); disp(['Maximum photons number = ',num2str(ph1(imax)),' ']); disp(['Period life time (FWHM) = ',num2str(pr),' nsec']); disp(['Pulse Rising time = ',num2str(rise),' nsec']); disp(['Pulse Falling time = ',num2str(fall),' nsec']); for m=2:nstep1 if ph1(m)>(ph1(1)); li=m; end end

59

disp(['Final population inversion = ',num2str(ng1(li)),' ']); disp(['Final number of absorber molecules in the G.S. = ',num2str(na1(li)),' ']); disp(['Final number of absorber molecules in the first excited state = ',num2str(nao-na1(li)),' ']); hv=6.625e-34*clight/(lambda); eout=(ng1(1)-ng1(li))/gama*hv*(1-R); power=eout/(pr*1e-9); disp(['The energy of Q-Switched pulse = ',num2str(eout*1e3),' mJ']); disp(['The power of Q-Switched pulse = ',num2str(power),' Watt']); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Estimating the principal parameters of last pulse %% by Numerical method (Runge-Kutta Fehlberg method) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% disp(['****** Third Pulse ****** ']); no1=102800; no2=104000; tt1=tt([no1:no2]); ng1=ng([no1:no2]); ph1=ph([no1:no2]); loss1=loss([no1:no2]); na1=na([no1:no2]); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %figure(6) plot(tt1,loss1,'r-'); hold on [AX,H1,H2] = plotyy(tt1,ng1,tt1,ph1,'plot'); set(get(AX(1),'Ylabel'),'String','Ng configuration & Loos') set(get(AX(2),'Ylabel'),'String','n configuration') set(H1,'LineStyle','--') set(H2,'LineStyle',':') title(' '); xlabel('Time (nsec)'); title('Third Pulse'); xtext=tt1(1)+max([tt1])/15; ytext=max([ng1 loss1 ph1]); ytext=max([ytext]); text(xtext,0.9*ytext,['Wave length = ',num2str(lambda),' nm']); text(xtext,0.8*ytext,['Nao = ',num2str(nao),' ']); text(xtext,0.7*ytext,['B = ',num2str(b),' ']); hold off pause %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% phmax=max([ph1]); nstep1=no2-no1; for i=1:nstep1 if ph1(i)==phmax; imax=i; end end hphmax=(phmax-ph1(1))/2; tmax=tt1(imax);

60

for i=1:imax if ph1(i)<=hphmax; ileft=i+1; end end for i=imax:nstep1 if ph1(i)>=hphmax iright=i+1; end end tl=((hphmax-ph1(ileft-1))/(ph1(ileft)-ph1(ileft-1)))*(tt1(ileft)-tt1(ileft-1))+tt1(ileft-1); tr=((hphmax-ph1(iright))/(ph1(iright-1)-ph1(iright)))*(tt1(iright-1)-tt1(iright))+tt1(iright); pr=tr-tl; rise=tt1(imax)-tl; fall=tr-tt1(imax); disp(['Position of left = ',num2str(tt1(ileft)),' ']); disp(['Position of center = ',num2str(tt1(imax)),' ']); disp(['Position of right = ',num2str(tt1(iright)),' ']); disp(['Position of maximum number of photons = ',num2str(imax),' ']); disp(['Maximum photons number = ',num2str(ph1(imax)),' ']); disp(['Period life time (FWHM) = ',num2str(pr),' nsec']); disp(['Pulse Rising time = ',num2str(rise),' nsec']); disp(['Pulse Falling time = ',num2str(fall),' nsec']); for m=2:nstep1 if ph1(m)>(ph1(1)); li=m; end end disp(['Final population inversion = ',num2str(ng1(li)),' ']); disp(['Final number of absorber molecules in the G.S. = ',num2str(na1(li)),' ']); disp(['Final number of absorber molecules in the first excited state = ',num2str(nao-na1(li)),' ']); hv=6.625e-34*clight/(lambda); eout=(ng1(1)-ng1(li))/gama*hv*(1-R); power=eout/(pr*1e-9); disp(['The energy of Q-Switched pulse = ',num2str(eout*1e3),' mJ']); disp(['The power of Q-Switched pulse = ',num2str(power),' Watt']); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Estimating the principal parameters of last pulse %% by Numerical method (Runge-Kutta Fehlberg method) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% disp(['****** Fourth Pulse ****** ']); no1=105500; no2=106600; tt1=tt([no1:no2]);

61

ng1=ng([no1:no2]); ph1=ph([no1:no2]); loss1=loss([no1:no2]); na1=na([no1:no2]); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %figure(7) plot(tt1,loss1,'r-'); hold on [AX,H1,H2] = plotyy(tt1,ng1,tt1,ph1,'plot'); set(get(AX(1),'Ylabel'),'String','Ng configuration & Loos') set(get(AX(2),'Ylabel'),'String','n configuration') set(H1,'LineStyle','--') set(H2,'LineStyle',':') title(' '); xlabel('Time (nsec)'); title('Fourth Pulse'); xtext=tt1(1)+max([tt1])/15; ytext=max([ng1 loss1 ph1]); ytext=max([ytext]); text(xtext,0.9*ytext,['Wave length = ',num2str(lambda),' nm']); text(xtext,0.8*ytext,['Nao = ',num2str(nao),' ']); text(xtext,0.7*ytext,['B = ',num2str(b),' ']); hold off pause %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% phmax=max([ph1]); nstep1=no2-no1; for i=1:nstep1 if ph1(i)==phmax; imax=i; end end hphmax=(phmax-ph1(1))/2; tmax=tt1(imax); for i=1:imax if ph1(i)<=hphmax; ileft=i+1; end end for i=imax:nstep1 if ph1(i)>=hphmax iright=i+1; end end tl=((hphmax-ph1(ileft-1))/(ph1(ileft)-ph1(ileft-1)))*(tt1(ileft)-tt1(ileft-1))+tt1(ileft-1); tr=((hphmax-ph1(iright))/(ph1(iright-1)-ph1(iright)))*(tt1(iright-1)-tt1(iright))+tt1(iright); pr=tr-tl; rise=tt1(imax)-tl; fall=tr-tt1(imax); disp(['Position of left = ',num2str(tt1(ileft)),' ']); disp(['Position of center = ',num2str(tt1(imax)),' ']); disp(['Position of right = ',num2str(tt1(iright)),' ']); disp(['Position of maximum number of photons = ',num2str(imax),' ']);

62

disp(['Maximum photons number = ',num2str(ph1(imax)),' ']); disp(['Period life time (FWHM) = ',num2str(pr),' nsec']); disp(['Pulse Rising time = ',num2str(rise),' nsec']); disp(['Pulse Falling time = ',num2str(fall),' nsec']); for m=2:nstep1 if ph1(m)>(ph1(1)); li=m; end end disp(['Final population inversion = ',num2str(ng1(li)),' ']); disp(['Final number of absorber molecules in the G.S. = ',num2str(na1(li)),' ']); disp(['Final number of absorber molecules in the first excited state = ',num2str(nao-na1(li)),' ']); hv=6.625e-34*clight/(lambda); eout=(ng1(1)-ng1(li))/gama*hv*(1-R); power=eout/(pr*1e-9); disp(['The energy of Q-Switched pulse = ',num2str(eout*1e3),' mJ']); disp(['The power of Q-Switched pulse = ',num2str(power),' Watt']); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%