# Laser Mode Field

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The subject of Laser has been studied for the last couple of decades. It has been widely used and a significant number of products have been based on it for various fields. Due to its distinctive characteristics, many researchers have focused on laser to further the knowledge in this field.

Laser Diode Arrays (LDAs), which considered one of the most commonly, used laser applications, were invented to overcome the power limitations for a single semiconductor laser. In this thesis, the modes for the edge emitting arrays will be calculated and examined.

To reach the objective of this project, various numbers of topics that provide background knowledge and deep understanding to the subject should be looked into such as, properties of semiconductors, Fabry-Perot resonator and optical waveguides.

Similar to most of the physics subjects, modelling of the LDAs can be described mathematically. Therefore, Finite Difference Method, which is one of the most common methods used for modelling of optical waveguides, has been used to calculate the modes for the array based on one of the Effective index method techniques, Beam Propagation. In addition, an algorithm has been developed and programmed using MATLAB.

Section 1

Introduction and History of Semiconductor Laser Diode Arrays

OBJECTIVES

- Introduce the subject

- Present the sections of this thesis and what each deals with

- Provide the project's aim and objectives

- Learn about the history of semiconductor laser diode arrays

- Describe how the aim is to be achieved and approached

- Introduction

Laser Light is considered one of the most essential tools that has impacted our daily life tremendously in various fields. Nowadays, its applications can be found everywhere; for medical uses as a cutting and cauterizing instrument, to help restoring vision and in plastic surgery. Another field is industry where it is used in various forms of welding, in scanners and printers. It can also be used for data transferring when used in communication, for surveying and ranging large distances especially in the space and it can also be used for security reasons when used as door lockers.

There are variety of approaches to produce laser light; Light-Emitting Diodes (LEDs), Laser Diodes (LDs) and LDAs. This thesis examines the Diffraction-Coupled LDAs and investigates its mode characteristics. To provide an excellent knowledge of the subject and to understand it deeply, various numbers of topics should be looked into to reach the main aim.

Therefore, this thesis will be divided into five main sections; the first section of introduces the subject, its aim and the objectives and a brief history of laser, LD and LDAs. The second section deals with the theory behind it starting from the properties of the semiconductors going to the structure and the operation of the LD and LDAs (i.e. Fabry-Perot Resonator and P-N Junction) to examine finally the optical waveguides and LDAs types and modes. The third section shows the procedure taken to achieve the aim of the project starting from Maxwell's equations and how the beam propagation and Finite-Difference Method (FDM) have been applied to solve Maxwell's equations to show at the end of this section the software structure developed to achieve the aim. The last two sections present the results obtained, discussion and conclusions. The algorithm developed is attached at the end of this thesis as an appendix.

- Aim and Objectives

The aim of this project is to develop a software tool for analyzing the model evolution for a semiconductor LDA. It is to be programmed using MATLAB. The developed tool should first calculate the refractive index perturbation for an LDA. Then, this index will be used in order to calculate the modal distributions at each longitudinal position of the cavity. To achieve the aim mentioned above, some objectives are to be met as the project proceeds:

- Carry out a literate review on laser and semiconductor laser arrays

- Develop an algorithm that calculates the wave guide modes and finds their field distribution at a give longitudinal position

For the purpose of developing such tools, Finite Difference Method is to be used, which is one of the most common methods used for modelling of optical waveguides.

- History

Albert Einstein was the first to propose the concept behind lasers. He showed that the combination of some wave energies called photons produces the light. Each of these photons has an energy that depends on the wave's frequency. Later, with the aid of another scientist named S. N. Bose, he developed the theory behind the phenomenon of photons' tendency to travel together.

Following their steps, Nobel Prize winner Charles Townes and his co-workers were the first to demonstrate the action of Laser in the microwave region in 1954. After that, Townes cooperated with Arthur Schawlow to propose the laser in 1958 and receive a patent in 1960. In 1962, laser action in a semiconductor material was demonstrated by a group of scientists from the United States.

The development of the first semiconductor LD that could operate at room temperature took about another decade. The Russian researchers were the first to demonstrate it [1].

The first generation of LDs, so-called large optical cavity and mostly operated in a pulsed mode, were invented during the late 70s. They, however, did not provide the amount of power needed. Therefore, later researches focused on discrete arrays in which a series of individual LDs are electrically connected side-

by-side with a few millimetres separation between each other, they were called LDAs. The power limitation of LDs was overcome with these devices. As a result, a lot of types were produced such as phase-locked arrays and evanescently-coupled linear arrays. Up until now, to make these devices much more suitable when used in some of the applications today, a continuing effort is progressing to delightfully modify the operation of such devices as a single coherent source [2].

- Approach and Presentation

The fundamental theory of LDs, LDAs, and LDA types and modes will be presented as the foundation to this investigation. Since the project heavily depends on the characteristics of the LDA modes and finding their field distribution at a give longitudinal position, the definitions and theory behind all the subjects related to this matter with a brief explanation of all the methods used to measure these parameters will be covered.

The project investigation will be performed in two phases. The first phase will involve the mathematics behind the subject while the second phase will deal with applying the resulted equations in a MATLAB language to get the required results at the end.

Section 2

Theory behind Lasers and Semiconductor Laser Diode Arrays

OBJECTIVES

- Provide essential background of the subject of laser

- Describe optical waveguides and their modes

- Examine semiconductor laser diode and laser diode arrays

- Theory

LDAs are one of the approaches to produce laser light. They were conceived and fabricated to overcome the power limitations of LD and to be used for applications in which optical power is of a major concern.

This section of this thesis will give an overview of the subject, from the basic structure of semiconductors (i.e. the atom) and their properties, to the optical waveguide design, getting finally to the whole LDA design, types and modes.

- Properties of Semiconductors

Semiconductors, in general, are group of materials having electrical conductivities intermediate between metals and insulators. These conductivities can be varied by changes in temperature, optical excitation or impurity content.

The atom is considered one of fundamentals of semiconductors and LDAs; therefore, a basic review of the atom shall be started with.

- The Basics of an Atom

Fig 2.1 shows the simple atom structure which mainly consists of two parts:

- Nucleus: containing the positive charged protons and free of charge neutrons and they both have the same size.

- Electrons: negatively charged particles orbiting around the atom in many different orbits.

Fig 2.1: *The basic structure of an atom* [3]

Because of the fact the nucleus contains the positive charged protons and free of charge neutrons, it is positively charged particle which located at the centre of the atom. However, the negatively charged electrons circling the nucleus in many different orbits create what is called ‘electron cloud'. This cloud is divided into a few levels representing the different energy levels for the atom.

- Energy Levels

The energy levels, as mentioned above, are occupied by the electrons in atoms. The further the distance the energy level is from the nucleus, the higher the energy it has.

Fig 2.2: *The different energy levels for an atom* [4]

Therefore, an electron occupying energy level E2 (refer to fig 2.2), would have a higher energy than an electron occupying energy level E1 and an electron occupying energy level E3, would have a higher energy than an electron occupying energy level E2 or E1, and so on.

When some heat or other energy source is applied to an atom, some of the electrons in the lower-energy levels are expected to be excited to the higher-energy levels. Once these electrons move to higher-energy levels, they eventually return to the ground state once they reach their lifetime (details will be given later in this section). When they do, a photon, which is a particle of light, is emitted (see fig 2.3). The process of emitting a photon called radiation.

Fig 2.3: *The process of radiation* [4]

This is basically the core idea of how atoms work in terms of laser which is also included in the term laser itself since the term LASER comes from the initials of Light Amplification by Stimulated Emission of Radiation

In fact, the process of radiation occurs either spontaneously which, in this case, called Spontaneous Emission or stimulatingly by energy source which called this tine Stimulated Emission. The later emission causes the radiation in laser which obvious from the name.

- Spontaneous Emission

Each electrical particle has a lifetime for it to remain in its excited states. When an electrical particle reaches its lifetime, it returns to its ground states dropping from higher to lower energy levels and emitting a photon after the transition. This process is called Spontaneous Emission and can be seen in fig 2.4.

4a 4b 4c

Fig 2.4: *The process of Spontaneous Emission: *4a) when an atom is given an extra energy to be on its excited states. 4b) falling down to its ground states after reaching its lifetime. 4c) the released photon after the electron completed the transition to the ground state [5]

Assuming an electron in the excited state with energy_{}, after it's spontaneously emission and dropping to the ground state its energy will change to_{}. The difference in energy between _{} and _{} is released as a photon with a frequency (_{}). This released energy is simply [5]:

_{}

Where h is Planck's constant = 6.626068 × 10-34 jouls.s

The main problem in the spontaneous emission lies in the fact that not only the process itself occurs randomly, but also the phase of the photon, as well as the direction the photon propagates in is random and can not be controlled. This is not the case in the stimulated emission where they can be controlled as the next part will show.

- Stimulated Emission

Stimulated emission is the process in which another photon is used to stimulate the emission of a second photon through the recombination of an electronic-hole pair (will be discussed in details in the next part). During this process, the initiating (original) photon is not destroyed, and the stimulated photon has the same phase, frequency, polarization, and direction of propagation as the initiating photon. An energy level diagram illustrating the process of stimulated emission is shown in fig 2.5 below.

2.5a 2.5b 2.5c

Fig 2.5: *The process of Stimulated Emission: *5a) when an atom is given an initiating photon while being on its excited states. 4b) the stimulation of a second photon when the initiating photon passes by. 4c) after the atom reaches its ground state, the emitted energy release a photon that has the same properties as the initiating photon [5]

For the initiating photon to propagate inside the semiconductor and influence the excited atom, an external electromagnetic field is needed. This filed will cause a photon to propagate between the energy levels E2 and E1 (refer to fig 2.5a) and the later will cat as an encouraging or to be more precise a stimulating tool for an excited electron to fall back to a lower energy level (see 2.5b). When the ground state is reached and the emitted energy release a photon, this photon will have the same phase and frequency, moreover, the same direction that the initiating photon produced by the electromagnetic filed propagated in (see 2.5c).

Due to the fact that this emission provides the ability to reflect the light (photons) in the desired phase, frequency, polarization and direction by manipulating the photon created by the external electromagnetic field, it is considered one of the fundamental concepts behind the semiconductor LDAs. In addition, it can be widely used to increase the intensity of the semiconductor LDAs.

- Fabry-Perot Resonator

The process of how a photon hence light is created and emitted has been looked into in the previous part. It occurs either randomly via the process of spontaneous emission or controllably via the process of stimulated emission. However, high intensity energy can not be created via either of them. Therefore, a medium must be created to achieve the desired high intensity energy, one of which is the implementation of Fabry-Perot Resonator.

Fabry-Perot optical resonators are considered one of the fundamental elements in optical physics which are used for building up large field intensities with low input power. They are often made by placing two parallel reflecting mirrors with L distance between them, as indicated in fig 2.2.1, separated by a medium of air or gas normally. These mirrors should align perfectly parallel in order for the emitted light from the active region to propagate and reflect between them, a standing wave is resulted from repeating this process.

Fig 2.6 illustrates how the optical gain is created; when the input light ray enters the cavity from the left, it is reflected back after reaching the right mirror with part of it leaves the cavity through the right mirror. The reflected light ray is once again reflected back when touching the left mirror and then propagates in the same direction as the original wave. The optical gain is created via this process.

Two Parallel Mirrors

Fig 2.6: *The process of propagation of reflection of light between the facets of Fabry-Perot cavity* [7]

In the situation where the semiconductor gain and the reflectivity of the facets are large enough, the device would be on the state of oscillation and the lasing light ray would be output. However, in order for the device to overcome the

cavity losses, a minimum gain is required to be achieved to output the light ray, this gain is called lasing threshold.

On one hand, when the gain is below threshold, the modes (the subject of modes will be dealt with later) are closely spaced which mostly resulted from the spontaneous emission. On the other hand, when the current is increased to just above threshold, a single mode is dominant and hence becomes the lasing frequency which can clearly be seen in fig 2.7.

Fig 2.7: *Emission spectra when a gain is below and above threshold *[8]

- P-N Junction

The basic operation of a semiconductor laser is the same as that of a PN junction diode. The operation of a PN junction can be easily understood when considering two types of semiconductor materials; one is N-type which has a majority of negative particles (electrons) and minority of positive particles (holes), these types normally called Donors. The second is a P-type which has a majority of holes and minority of electrons, these types called acceptors as indicated in the diagram in fig 2.8. The electrons move freely and randomly between atoms in the N-type. Similarly, the holes are excited inside the material of the P-type.

Fig 2.8: *Schematic diagrams of N-type and P-type semiconductor materials.* [6]

When combining a P-type and an N-type materials together, the free electrons from the N-type and the free holes from the P-type move freely across the junction and cancel each other after they meet and recombine together presenting the electronic-hole pair. Eventually, the process of recombination produces a region in the middle that is completely free of charge called depletion region which works as a potential barrier as indicated in fig 2.9. Therefore, only electrons (or holes) with sufficient energy can pass through the depletion region crossing over to the other side.

Depletion Region

Fig 2.9: *The creation of depletion region after the recombination of electrons and holes* [6]

When a depletion region is formed, the state in which the diode is in called Equilibrium. In this state, the electrons from the N-type behave as current carriers in the conduction band while, in the mean time, the holes from the P-type behave as the carries in the valence band as indicated in fig 2.10.

Valence Band

Conduction Band

Fig 2.10: *Band diagram for the equilibrium condition for PN diode.* [6]

For an electron or hole to cross over the region, an external energy source such as electrical current is needed. When this source is applied to the PN junction, the charges of electrons and holes increase and so does there energy. This energy causes the electrons and holes to move, this movement is either toward the region in this case the diode is forward biased, or away from the region called this time reverse biased.

2.2.1 Forward and Reversed Biased Diode

When considering that the external energy source is produced by a battery, when its positive side is connected to the P-type side of the junction and its negative side to the N-type side (i.e. voltage opposes the junction field) as shown in fig 2.11a, it is forward biased diode in which the depletion region is caused to be reduced minimising the resistance to the flow of current. However, when the battery is connected the other way around (the positive side is connected to the N-type and the negative to the P-type), the diode is thus called reverse biased in which the depletion region is caused to be increased offering a high resistance to the current flow through the junction as indicated in fig 2.11b.

a) b)

Fig 2.11: *Band diagram* of: a) forward biased diode, b) reversed biased diode [6]

- Optical Waveguides

The waveguide is considered the transmission medium where the emitted photon from the stimulated emission propagates onward and backward producing the lasing light.

An optical waveguide can be defined as a physical structure (might be rectangular or round) that guides light to propagate inside it. The classification of optical waveguides is resulted according to their geometry (planar, strip, or fibre waveguides), mode structure (single-mode or multi-mode), refractive index distribution (step or gradient index) and material (glass, polymer or semiconductor).

Rectangular optical waveguides are frequently used to construct the cavities of semiconductor laser diodes. Fig 2.12 shows its typical structure which consists of a dielectric core that forms the light path. This core is then surrounded by cladding layers of silica that has a lower refractive index than the core that causes the light to be trapped inside the guiding.

Fig 2.12: *Typical structure of an optical waveguide* [9]

The trapped light propagates inside the wave guide according to the Total Internal Reflection which is considered the basic optical technique. This indicates the state the ray of light is in after travelling from a greater refractive index (the core in the waveguide) material to another with a lower refractive index (the cladding).

This can be easily described mathematically using Maxwell's equations which predicts the presence of electromagnetic waves and indicates that the

emitted light propagating in the guide can be described in electromagnetic form. Maxwell developed the following equations [14]:

_{}

Where t represents time, E is the electric field vector (V m_{}), H is the magnetic field vector (Wbm_{}), D is the electric induction (cm_{}), _{}is the free charge density (cm_{}), and J is the current density (Am_{}).

To help understanding the principle of waveguides, two types of waveguides classified according to their refractive index distribution and mode structure will be explained.

- Step Index Guides

In this type of guides, the light is totally reflected when touching the cladding. Fig 2.13a shows a single mode step index guide in which the core is significantly shrunk down so that only one ray is allowed to propagate inside. However, fig 2.13b shows the multi-mode type in which the core has larger diameter allowing multiple modes to propagate inside; each has its own path.

a) b)

Fig 2.13: *The mode propagation of step index guides; a) single mode, b) multi-mode* [10]

- Graded Index Guides

Graded index guides differ from step index guides in their core structure where in which the refractive index does not remain constant as in step index guides, but it continuously changes from the centre of the core onwards. This results in a faster propagation velocity and hence reduced dispersion.

Fig 2.14 illustrates the propagation of the wave inside the core of this type of guides which is not in a straight line but in a curved line.

Fig 2.14: *The mode* *propagation of a multi-mode graded index guide* [10]

- Modes of a Waveguide

Within a waveguide, the trapped light can only travel at certain angles, called mode angles. The lower the angle, the higher the mode is being parallel to the guide axis. The number and spacing of these modes depend on the width and the refractive index profile for the waveguide they propagate inside.

As mentioned earlier, the trapped light can be described in electromagnetic form obeying Maxwell's equations and having two components perpendicular to each other; Electric Field (E) and Magnetic Field (H) with λ wavelength as illustrated in fig 2.15.

Fig 2.15: *The electric and magnetic fields for a light wave.* [11]

Modes propagating in an optical waveguide are said to be transverse, moreover, they usually are linear polarized meaning they are treated as if they contain only one transverse field component; electric (TE) or magnetic (TM) giving more accuracy in laser guiding and low refractive index contrast.

When the electric field is perpendicular to the direction of the wave's propagation with the magnetic filed being in the direction of the propagation, the wave is transverse electric mode. On the other hand, when the magnetic field is the one which is perpendicular to the direction of the wave's propagation and the electric filed being in the direction of propagation, the wave is then called transverse magnetic mode.

The order of each mode is determined by either the number of field maxima within the core of the guide, or by the angle the mode makes with the guide's axis; the steeper the angle the higher the order of the mode which can clearly be seen in fig 2.16b.

b)

a)

Fig 2.16: *The order of modes; *a) by the number of maxima within the core of the guide, b) by the angle the mode makes with the guide's axis [12]

Fig 2.16a shows the order of a TE mode in which the TE0 is represented by one field maximum right in the middle of the core whereas two field maximum represent the mode TE1, and so on.

- Laser Light

There are three main properties of laser light that make it unique and different from normal light:

- It is monochromatic; it has only one wavelength (colour) when compared with ordinary white light which contains many wavelengths (colours)

- It is coherent; its wavelengths are well organized being in phase in space and time

- It is directional; it is emitted as a relatively narrow beam in a specific direction. In contrast, ordinary light (light bulb or a candle) is emitted away from the source in many directions.

There are many different types of lasers depending on the medium such as; solid-state, gas and semiconductor laser diode. This thesis, however, concentrates only on Semiconductor LD and LDA.

- Semiconductor Laser Diode

Semiconductor LDs are basically formed by a highly doped p-n junction (fig 2.17a), which is cut or etched as required (fig 2.17b, c), connected properly with an efficient resonant cavity allowing sufficient heat to be transformed.

Fig 2.17*: Fabrication of a simple laser diode* a) diffusing p and j layers forming p-n junction, b) cutting and etching to isolate the junctions, c) divide junctions to be cleaved into devices [13]

These types of laser are sometimes referred to as homojunction lasers, meaning they contain only one junction in a single type, or as hetrojunction lasers, when multiple layers are used in the laser structure allowing the lasers not only to obtain more efficiency, but also to operate at room temperature.

Light produced by these types of lasers can come out from either the side of the cavity, which is in this case referred to as edge-emitting lasers that is the type examined in this project, or from the top of the surface, which is this time referred to as vertical cavity surface-emitting lasers. Fig 2.18 illustrates a typical double-hetrojunction, a further improvement to hetrojunction laser, edge-emitting laser indicating all of its layers.

Fig 2.18: *Schematic diagram of a typical **edge-emitting laser *[15]

Despite the fact that the LDs are highly implemented and widely used in fibre optics communication, their power is insufficient and limited due to their physical parameters. As a result, the need for other devices that produce higher power arises and semiconductors LDAs are thus fabricated and introduced.

- Semiconductor Laser Diode Arrays

Phase-locked arrays of diode lasers are fabricated in the pursuit of obtaining high power and high intensity beam to overcome the limitations of LDs. This invention has many unique advantages as it cancels the need for external optics to overcome the thermal and/or carrier-induced variations and provides the beam with efficient stability. A typical edge-emitting laser arrays is illustrated in the following diagram shown in fig 2.19.

Fig 2.19: *Schematic diagram of a typical **edge-emitting laser arrays* [16]

- Array Types

There are four types of phase-locked arrays schematically illustrated in fig 2.20: leaky-wave-coupled, evanescent-wave-coupled, Y-junction-coupled and diffraction-coupled.

Fig 2.20: *Schematic diagrams of basic types of phase-locked arrays with their corresponding refractive-index profiles at the bottom of each type* [17]

Leaky-wave-coupled devices have major field-intensity peaks in the low-index array regions. However, the array modes for evanescent-wave-coupled devices reside in the high-index array regions. The proposition of Y-injection-coupled devices and diffraction-coupled devices resulted from the need to select in-phase operation, which the first two types could not achieve, via wave interference and diffraction respectively.

- Array Modes

The system for the type examined in this thesis can be described simply as a periodic variation of the real part of the refractive index. This system characterizes two classes of mode:

- Evanescent-Type Array Modes

- Leaky-Type Array Modes

Fig 2.21: *Modes of a semiconductor laser diode array: *a) index profile; b) in-phase evanescent-wave type; c) out-phase evanescent-wave type; d) in-phase leaky-wave type; e) out-phase leaky-wave type[17]

The fields for the evanescent modes are peaked in the high-index array regions, but they are peaked in the low-index array regions for the leaky modes as indicated in fig 2.21. Furthermore, the propagation-constant values can be found between the low and high refractive index values for the evanescent modes. However, they are below the low refractive-index value for the leaky modes. When the fields, for both classes, in each element are cophasal (i.e. 0 degree phase difference between adjacent element), the mode is said to be “in-phase”. On the other hand, when the fields in adjacent elements are a _{} phase-shift apart, the modes in this case are said to be “out-of-phase”.

Section 3

Modelling of Laser Diode Arrays

OBJECTIVES

- Describe the operation of edge emitting arrays

- Discuss the methods used to model the array: (Effective Index, Beam propagation, Finite Difference Method and Crank - Nicholson Method)

- Explain the procedure done to achieve the aim

- Provide the software structure

- Modelling of Diode Laser Arrays

- Edge Emitting Arrays

By pattering the upper metallization for edge emitting arrays, the accomplishment of current confinement and gain structuring can be achieved as shown in fig 3.1. In addition, various fabrication techniques, some of which etching and regrowth, may be used so that wave guiding structures for such devices may be built into the device.

Fig 3.1: *Schematic diagram of an edge emitting array* [17]

The modulation of these devices can be achieved by three basic processes:

- When electrons and holes flow to the active region from the contacts, a heat is produced and an optical gain is provided once the carriers combine there.

- This heat results in a temperature rise in the active region causing the change of refractive index and impacting the wave guiding

- The cavity modes, having electromagnetic fields, which are above threshold, are propagating back and forth between the facets producing a standing wave in the cavity and the available gain is thus started to be saturated.

- Waveguide Modeling

Starting with Maxwell's equations mentioned earlier in the absence of free charge and assuming the following [17]:

- The magnetic permeability
_{}is the vacuum value

- The electric permittivity
_{}is constant in space and time

- The harmonic time dependence exp (-i
_{}t)

The electric field E and the magnetic field H satisfy:

_{}

Assuming all field components depend upon z (refer to fig 20 for coordinates) the first approximation is of the form:

F= ƒ exp (ikz)……………………………………………………….… (3.2)

Where the field ƒ is assumed to weakly depends upon z. The resulted equation from inserting Eq. 3.2 into Eq. 3.1 for the case of the magnetic field H is

_{}

Because of the weak dependence of H upon z, its second derivative is neglected:

_{}

- Effective Index Method

The effective index method can be simply described as a method used to quantify the modes of an optical waveguide by using the refractive index which is the number that quantifies the phase delay per unit length in a waveguide, relative to the phase delay in vacuum.

When presenting this method to Eq. (3.4) to reduce its dimensionality by one, moreover, assuming that the presence of a lateral structure has no effect on the y dependence, would result in H being approximately separable [17]:

_{}

Inserting the above equation into (3.4) and dividing through by g gives:

_{}

The fact that all the y-dependence being contained within the square brackets for the above equation results in the quantity within being mostly a function of x. In addition, when the “effective” index of refraction is defined through the equation:

_{}

where _{} is the relative permittivity, allowing Eq. (3.6) to be written as the following equation which can be easily used for applying Beam Propagation

_{}

The overall modal index _{} in the above equation has been defined through the equation _{}. An illustration of a typical example of this method applied to simple buried wave guide is shown in fig 3.2.

Fig 3.2: *illustration of a typical example of Effective Index Method applied to simple buried wave guide for which filed amplitude of g(y) (upper diagram): *a) the resulted effective index profile; b) the resulted effective index profile for an index-guided array with periodicity _{}

- Beam Propagation

By varying gain or index profiles, beam propagation (or ‘Fox and Li iteration' as they were the first to introduce this method [17]) causes the change of the shape of a longitudinally dependence mode. This method allows an arbitrary initial field to propagate back and forth between the two facets until reaching a steady state. Furthermore, using this method, one mode is expected to be favoured over the other settling the solution down to this highest gain mode.

Applying this method to Eq. (3.8) results in the following rearranged version of it:

_{} To solve the above equation, both finite difference and Crank-Nicholson methods are used.

- Finite Difference Method

The FDM is considered one of the most common methods used for modelling of optical waveguides in which the waveguide is spilt up into sample points so that the light intensity at each sample is then calculated while bearing in mind that the sampling space is very important parameter [19].

It is required to approximate the partial derivates using finite differences of Eq. (3.9). However, for such equation only half step is approximated so the Crank-Nicholson Method is applied here to approximate the derivative of H in terms of (z), the second derivative of H in terms of (x) and finally the function H form Eq. (3.9) as follows:

First, the left side of Eq. (3.9) which is the derivative of H in terms of (z)

_{}

Second, the second derivative of H in terms of (x):

_{}

Approximating each term for (n, k) and (n, k+1) individually as follows:

_{}

Producing the final result after adding the terms together:

_{}

Finally, using the method to approximate the function H:

_{}

Now, inserting Eq. (3.10), (3.11) and (3.12) into Eq. (3.9) gives the following:

_{}

Rearranging the equation so that the term (k) resides in one side and the term (k+1) resides in the other results in the following equation (3.14):

_{}

To make the following steps simpler and clearer, let:

a=_{}…………………………………….… (3.15)

b= _{}…………………………………………………...…………….. (3.16)

c= _{}……………………………………… (3.17)

Now, substituting all of the equations (3.15), (3.16) and (3.17) into (3.14) gives:

_{}

It can be seen that the left side of the above equation has the known term and the right side has the unknown term.

For k=1 and n=2, 3, ..., N-2, N-1

n=2 _{}

n=3 _{}

: : : : : : :

n=N-2 _{}

n=N-1 _{}

Converting these equations into a matrix form gives the matrix (3.18):

_{}

A (known) = B (constant values) X (unknown)

X = B_{} A >>>> for each step

Due to the efficient features that the programming language MATLAB has, it was chosen to find the solution of these matrices. The software structure and the algorithm developed are attached and provided at the end of this thesis as APPENDIX#A.

- Procedure and the Array Parameters

For the purpose of the analysis, it is assumed that it is a 5 strip array for which the width of each strip was 5μcm and the centre-to-centre spacing was10μcm as indicated in fig (3.3). The wave of light enters the guide with an initial magnetic intensity H calculated using the following equation:

(3.19)

Where w is the width of the wave which is approximately 3.5μcm

Then, after entering the cavity, the field intensity of the modes propagating in the (z) direction along (x) (please refer to fig 3.3 for coordinates) for each step was calculated using the final matrix form shown above calculates the modes.

w

10μcm

Reflectivity of the back facet=0.9

Reflectivity of the front facet=0.001

300μcm

5μcm

z

y

x

Fig 3.3: *Schematic diagram of the waveguide of the semiconductor LDA used in the project*

The process of propagation can be described as follows:

When the wave enters the waveguide from the back facet, it propagates towards the front facet and reflects back. Therefore, the wave is thus multiplied by the root of the reflectivity of the front facet (). After that, it propagates back to the back facet in which it reflects and thus multiplied again by the root of reflectivity of the back facet (). The process continues until the shape of the field intensity reaches a steady state. The gain for each round trip can be obtained by calculating the difference of the shape of the wave once it is at the entrance of the cavity (i.e. at z = 0) and after the completion of one full round trip after reflecting twice from both the front and back facets.

Because of the fact the field intensity reaches magnificent values, the field needs to be normalised. The normalisation of the field is done by dividing the whole matrix of H, for each turn, by its largest element after the completion of each turn.

The details of the parameters of the semiconductor LDA used for the analysis to calculate the equations (3.15), (3.16) and (3.17) are [19, 20]:

The wave length ( |
= 980 nm |

The Effective Refractive Index of the fundamental mode |
= 3.3862 |

Cut-off wave number |
= |

Propagation Constant ( |
= |

Relative Permittivity |
= |

Higher and Lower Effective Refractive Index |
= (3.3862, 3.3839) respectively |

Electric Permittivity ( |
= |

Section 4

Results and Discussion

OBJECTIVES

- Provide the results obtained

- Compare the results with theoretical findings

- Discuss the results

- Suggest methods to improve the results

- Results

- Experimental Graphs of Field Intensity (H) against Longitudinal Distance (x)

The initial field intensity for each position along the distance x for the wave that enters the waveguide looks like:

Fig 4.1: *Graph of initial H against x*

The next graph illustrates the field intensity against the longitudinal distance x for the initial wave and the first refracted wave (after the completion of the first round trip).

Fig 4.2: *Graph of initial H and that after and the 1*^{st}* round trip against x *

The next graph illustrates it for the second and the third round trips:

Fig 4.3: *Graph of H against x for the 2*^{nd}* and *^{3rd }*round trips*

The next graph illustrates it for the 4^{th} and 5^{th} round trips:

Fig 4.4: *Graph of H against x for the 4*^{th}* and 5*^{th}* round trips*

- Other Experimental Findings of Field Intensity (H) against Longitudinal Distance (x)

The theoretical near field intensity patterns for a selective mode of a ten-stripe gain-guided array should have characteristics of intensity along the near field position that similar, it does have to be exactly the same, to the one shown in fig 4.5 [17]:

Figure 4.5: *Other experimental findings (upper curves) when utilizing the beam propagation method (lower curves) of near and far field intensity patterns of the13*^{th}* order mode of a ten-stripe gain guided array* [17]

- Graphs of Optical Gain against Longitudinal Distance (x)

The gain obtained after the completion of the first round trip looks like:

Fig 4.6: *Graph of the optical gain against x*

- Discussion

The field intensity patterns generated by the software do not entirely agree with the theoretical patterns which was unexpected because all the mathematics equations seemed to be free of any errors or mistakes.

The initial field should not look like that shown in fig (4.1), but it should more looks like the one shown in fig 3.3 in the previous section. In addition, it has been noticed from figures (4.2), (4.3) and (4.4) that the graphs are very messy having a lot of noise and interference for the field intensity patterns. That might have happened because of the radiation which tends to reflect back from the problem boundaries back to the active region causing unwanted interference for the intensity patterns which can be clearly seen in the mentioned graphs. This resulted in the awkward pattern for the gain, that magnificently depends on the field intensity patterns (see fig 4.6), which was expected to favour one mode.

Although the field intensity patterns do not entirely match the theoretical patterns, they did reach a steady state as shown in fig (4.4) after the 4th round trip which agree with the theory of the beam propagation.

If more time was given, a solution would be applied to the wave which is Transparent Boundary Condition that prevents the boundary reflection by inserting artificial regions that absorb all radiation entering the region. One way of doing this is multiplying the imaginary part of the effective index by _{}

Section 5

Conclusions

OBJECTIVES

- Present the conclusions obtained from the whole project

- Conclusions

The purpose of this dissertation is to give an introduction to the subject of semiconductor laser diode arrays in which full background knowledge of the subject is covered and provided. This knowledge includes the very beginning of the atom where the photon is first emitted until getting to the full construction of laser diode and laser diode arrays explaining the operation of each and how the laser light is produced.

Much time was spent on the initial stage of the project which was dedicated to seek all related knowledge to fully understand and analyse the mode characteristics of semiconductor laser diode arrays. This includes the behaviour of electrons, the structure of laser diode arrays, and the theory behind the wave propagation inside an optical waveguide to finally get to the mode types and their characteristics. It was found that as the field propagate inside the cavity, its intensity increases changing its shape. However, after many round trips the modes reach steady state, which was reached to in this project and the gain comes to favour one mode.

This dissertation does not only cover the subject from a science point of view, but also some mathematics such as Maxwell's Equations and Finite Difference Method are applied to reach the main aim of the project. This led to spend much time to understand how to apply these methods to the physics that this project has.

Another major issue that took a lot amount of time was solving the resulted equations using a programming language. MATLAB was selected to serve this purpose because of its powerful features that helped in modelling the array. Magnificent effort was put to get familiar with this language and its interfaces being completely new to the author.

If more time is given, the subject of Transparent Boundary Condition for Beam Propagating will be looked into and applied to get better results.

- References

_{}

[1] How Products Are Made: Volume 6: Semiconductor Laser, viewed 5^{th} April

< http://www.madehow.com/Volume-6/Semiconductor-Laser.html>

[2] Elias Towe and Clifton G. Fonstad, Jr., Phase-Locked Semiconductor Laser Arrays, Applied Physics Letter 178, 895 (1989)

[3] The American Heritage Dictionary of the English Language: the atom, viewed 6^{th} April

< http://www.bartleby.com/images/A4images/A4atom.jpg>

[4] Einstein and the Laser, viewed 6^{th} April

< http://www.physics.uiowa.edu/adventure/fall_2005/oct_15-05.html>

[5] Sci-Tech Dictionary: Spontaneous and Stimulated Emission, viewed 6^{th} April

< http://www.answers.com>

[6] The diode, viewed 7^{th} April

<http://www.mtmi.vu.lt/pfk/funkc_dariniai/diod/index.html>

[7] The Fabry-Perot Optical Resonator, viewed 8^{th} April

<http://physics.usask.ca/~angie/ep421/lab3/theory.htm>

[8] Semiconductor Laser Diodes, viewed 8^{th} April

<http://britneyspears.ac/physics/fplasers/fplasers.htm>

[9] Optical Waveguides, viewed 8^{th} April

<http://www.stsystems.com/pages/process_content.asp?special_content=1037&contTitle=Optical%20Waveguides>

[10] The Physics of the Optical Fibre, viewed 8^{th} April

< http://library.thinkquest.org/C006694F/Optical%20Fibres/Physics%203.htm>

[11] The Electromagnetic Spectrum, viewed 10^{th} April

<http://www.geo.mtu.edu/rs/back/spectrum/>

[12] Mode Theory, viewed 10th April

<http://www.tpub.com/neets/tm/106-10.htm>

[13] Ben G. Streetman and Sanjay Kumar Banerjee, Solid State Electronic Devices 6^{th} Edition (2005), Prentice Hall

[14] Pankaj K. Das, Optical Signal Processing Fundamentals (1991), Springer-Verlag

[15] Semiconductor Laser Diodes, viewed 12^{th} April

< http://britneyspears.ac/physics/fplasers/fplasers.htm>

[16] Thomas A. Summers, Generation of a Single-Lobe, Far-Field

Intensity Pattern from a Laser Diode Array using as Optical Delay Line (1994)

[17] D. Botez and D. R. Scifers, Cambridge Studies in Modern Optics: Diode Laser Arrays (1994), Cambridge

[18] John H. Mathews, Numerical Methods for Mathematics, Science and Engineering, Prentice-Hall, Inc 2^{nd} edition, 1992

[19] Sujecki S., Wykes J., Swell P., Vukovic A., Benson T.M., Larkins E.C., Boruel L. and Esquivias I.," Optical Properties of Tapered Laser Cavities", IEEE Transactions on Optoelectronics, Vol. 150 .No.3., June 2003, pp246-252.

[20] Hadley G.R., "Transparent Boundary Condition for Beam Propagating ", Optics Letters, Vol. 16 .No.9, May 1, 1991, pp 624-626.

APPENDIX #A

N=200; %This the number of steps in both direction x and z

turn=5;%This the number of turns in which the wave is reflected

%Then, the increments in z and x are calculated

dz=300/N;

dx=50/N;

% Constant are entered

a=6.67e5-3.685e5i;

b= 184.25e3i;

c=6.67e5+3.685e5i;

%This the constant matrix

B=zeros(N-2,N-2);

%Entering the values inside matrix B

% 1. c in the diagonal

for i =1:198

for j=1:198

if i==j

B(i,j)=c;

% 2. -a before and after the diagonal

elseif i==j-1

B(i,j)=-a;

elseif i==j+1

B(i,j)=-a;

end %end if statement

end %end the inner loop for each row

end %end the outer loop for each column

% calculating the inverse of Matrix b and it's called RF

RF=inv(B);

%Setting the refractivity of the facets

nb=0.9;

nf=0.001;

%Setting Matrix H to zeros initially

H =zeros(N,1);

%Setting some useful matrices for output purposes

Real =zeros(N,1);

Imag=zeros(N,1);

Mag=zeros(N,1);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%this is the function (called value) used to calculate the initial H entering the cavity

%function result = value (x)

%if x>=0 && x<10

%s=5;

%elseif x>=10 && x<20

%s=15;

%elseif x>=20 && x<30

%s=25;

%elseif x>=30 && x<40

%s=35;

%elseif x>=40 && x<=50

%s=45;

%else

%s=x;

%end

%result=exp(-((x-s)/3)^2);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Entering the initial wave inside Matrix H

for i=1:N

H(i,1)=value(i*dx);

end

%Setting some values in H to be zeros in certain intervals

for i=1:200

if i*dx<= 2.5 && i*dx>=0

H(i,1)= 0;

elseif i*dx>=7.5 && i*dx<=12.5

H(i,1)= 0;

elseif i*dx>=17.5 && i*dx<=22.5

H(i,1)= 0;

elseif i*dx>=27.5 && i*dx<=32.5

H(i,1)= 0;

elseif i*dx>=37.5 && i*dx<=42.5

H(i,1)= 0;

elseif i*dx>=47.5

H(i,1)=0;

end

end

%Initializing Matrix A and X to be zero

A=zeros(N-2,1);

X=zeros(N-2,1);

% Entering the outer loop which will finish when the last turn is done

for k=1:turn

% The next loop describes the calculation for the prorogation to front facet

for j=1:199

for r=1:N-2

%Entering the values of H to matrix A (see the report for further details)

A(r,1)=b*H(r,1)+a*H(r+1,1)+b*H(r+2,1);

end

% Multiplying the inverse of B, RF, to matrix A to give X, the unknown matrix

X=RF*A;

% Storing the new values of H from X

for r=2:N-1;

H(r,1)=X(r-1,1);

end

end

%Multiplying the index times H

for r= 1:N

H(r,1)=sqrt(nf) *H(r,1);

end

%The next loop describes the calculation for the prorogation to back facet

for j=1:199

for r=1:N-2

%Entering the values of H to matrix A (see the report for further details)

A(r,1)=b*H(r,1)+a*H(r+1,1)+b*H(r+2,1);

end

% Multiplying the inverse of B, RF, to matrix A to give X, the unknown matrix

X=RF*A;

% Storing the new values of H from X

for r=2:N-1;

H(r,1)=X(r-1,1);

end

end

%multiplying the index times H

for r=1:N

H(r,1)=sqrt(nb)*H(r,1);

end

%Then, normalizing the wave

%calculating the maximum

q= max(H);

%Modifying the values

for r=1:N

H(r,1)=abs(H(r,1)/q);

end

end

%Getting the real values out of H

Real=real(H);

%Print them out in a spreadsheet

xlswrite('real_values.xls', Real, 'NewTemp', 'E1:E200');

%the end of the program