Simulations on single-mode waveguides in GaAs · A dielectric waveguide uses the concept of total...

Simulations on single-modewaveguides in GaAs

SupervisorsProf. dr. ir. C. H. van der Wal

dr. T.A. Schlatholterir. J. P. de Jong

Research groupPhysics of Nanodevices

StudentF. A. van Zwol

DateFriday 10th July, 2015

Credits15 ECTS

Bachelor thesis

Zernike Institute for Advanced Materials

Abstract

Waveguides improve the interaction between light and matter in quantum experiments. We examine the pos-sibility to integrate a single-mode waveguide into the GaAs/AlGaAs sample, for experiments in a cryogenicconfocal microscope setup. Making this sample single-mode is challenging because the number of modesincreases with core thickness. We solve this problem by bringing the refractive index of the core very closeto the refractive index of the cladding of 2D waveguides that confine light in one direction. We looked atthe difference between symmetric and asymmetric waveguides. The most asymmetric situation with therefractive index of one cladding close to the core and the refractive index of the other cladding much lowerthan the core yields the highest possible core thickness. We also looked at 3D waveguide structures thatconfine light in two directions. We used the Marcatilli method to solve for the rectangular 3D waveguide andfound that the same theory as for slab waveguides applies. We also tried to solve more complex structures.

Acknowledgements

Thanks to prof. dr. ir. van der Wal for the research opportunity.Thanks to J. P. de Jong for the clear feedback and daily guidance.Thanks to the quantum devices/FND-research group for the great time during the project.This page has been intentionally left daubed.

1

Contents

1 Introduction 3

2 Requirements 52.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Thickness and single-mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Planar dielectric waveguides 63.1 Symmetric dielectric slab waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.2 Ray approach: Self-consistency condition . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.3 Single-mode condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.4 Number of modes versus core thickness . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.5 Refractive index versus thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.6 Angle of incidence versus refractive index . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Asymmetric dielectric waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.1 Self-consistency condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Single-mode condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.3 Thickness versus refractive index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.4 Number of modes versus thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.5 Thickness versus refractive index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.6 Angle of incidence versus refractive index . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Rectangular dielectric waveguides 184.1 Marcatilli’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.2 Self-consistency condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.3 Single-mode condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Rib waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Appendices 26

A Matlab scripts 27

2

Chapter 1

Introduction

Quantum information systems use quantum bits (qubits) as information units, whereas regular informationsystems use regular bits. These qubits are quantum mechanical systems consisting of two eigenstates. Aqubit can be in a superposition of states, in contrast to a normal bit. Several different quantum systems canbe a qubit, like the electron spin or the polarization of light. Solid systems are usually chosen to be a qubitbecause because they stay in place. Photons are used to move quantum information. Photons can interactwith matter and move quantum information over large distances.

Quantum optics is a field that researches the interaction between light and matter. Quantum optics aimsto improve the interaction between photons and matter because photons interact very weakly with matter.This can be improved by using ensembles of quantum emitters instead of a system consisting of a singlequantum emitter and a single photon. Another way to improve the interaction is by using waveguides. Awaveguide can be used to create highly localized light to improve the interaction between matter and light.A waveguide can also be used to collect photons that are emitted from single photon sources in experimentsmore efficiently [1]. Another application of waveguides is the transport of light inside quantum experimentson a chip. Waveguides can guide light between matter qubits with very little loss.

Our group is currently running a quantum experiment in a cryogenic setup using a GaAs sample. Thissample has been placed on an AlAs substrate. This thesis will determine the geometry for the GaAs samplesuch that it is a single-mode waveguide. Single-mode means that only one wave pattern of light will besupported by the waveguide. The waveguide needs to be single-mode because we can be sure about thestability over time of the properties of the light exiting the waveguide. Making a waveguide single-mode canbe difficult because the number of modes generally increases with thickness of the waveguide core. We needa waveguide that is about a micrometer thick due to the spot size of the laser beam used in our experiment.

The theory and equations of waveguides can be found in literature, like the self-consistency conditionfor the phase of light waves reflecting on a dielectric boundary and the number of modes as a function ofrefractive index [2, 3]. We applied the equations for the dielectric slab waveguides that can be found inliterature to our material. From the resulting equations, we investigated which parameters have to change toget a single-mode waveguide with a core thickness of about a micrometer. The first parameter we changedwas the refractive index of the claddings when the refractive index of both claddings was equal, creatinga symmetric situation. Another situation we looked at is the asymmetric situation where we varied therefractive index of only one cladding and the refractive index of the other cladding was kept constant.To check whether the single-mode waveguides obtained from these parameters are compatible with ouroptical setup, we calculate the angle of incidence of the mode at the waveguide entrance. How the theoryof waveguides changes when waveguides confine light in two directions instead of one direction, can alsobe found in literature. We investigated rectangular waveguides, which confine light in two directions, andcompared them to slab waveguides, which confine light in only one direction. The same analytic approachand equations apply to this waveguide according to literature when some approximations are made.

3

CHAPTER 1. INTRODUCTION 4

Outline

The requirements of the waveguide are laid out in the second chapter.The third chapter explains the theory of planar dielectric waveguides. Analytic equations are derived for

these waveguides. The derived equations are then applied in matlab scripts that simulate a varying refractiveindex for both the symmetric waveguide and the asymmetric waveguide versus the maximum single-modethickness. We also plotted the angle of incidence required to couple to a mode.

Then the Marcatilli method has been used to solve a 3D waveguide and compare it to 2D-slab waveg-uides in the fourth chapter. This is an approximate method because solving 3D waveguides analytically isimpossible.

Chapter 2

Requirements

The goal of this thesis is to determine the parameters of a single-mode waveguide GaAs sample. We derivethe requirements for the waveguide in this chapter. These requirements come from fabricational methodsand specifications of the cryogenic setup.

Figure 2.1: Schematic overview of the simulated sample setup where sample thickness d is varied in thesimulations.

2.1 Materials

The sample studied in this thesis consists of a layer of GaAs with a refractive index n0 of 3.55. This sample isplaced on a substrate with refractive index n2. We need to find an optimal refractive index for the claddingmaterials and the top cladding. The solid materials used for both claddings need to closely match the latticeof the GaAs sample to limit the strain, hence the use of AlGaAs. An AlxGa1−xAs layer could be producedwith a refractive index between 2.8 and 3.55.

2.2 Thickness and single-mode

We know from previous experiments that the focal point of the laser beam is at least two micrometers indiameter and the much smaller diffraction limit is not feasible in our setup [4]. This means that for thepresent study, we aim at realizing a waveguide of several micrometers thick to couple a Gaussian beam intothe waveguide. We want to have a single-mode waveguide because it is not possible to couple light into asingle-mode fiber from a multi-mode waveguide. We can also be sure to which mode our light couples to ina single-mode waveguide. Light entering a multi-mode waveguide could couple to multiple modes and thusmake the characteristics like the intensity profile of the light exiting the waveguide unstable over time. Thechallenge is to keep the waveguide single-mode while achieving a core thickness of several micrometers. Ingeneral, increasing the thickness of the waveguide increases the number of modes, as explained in chapter 3.

Our laser has a wavelength in a range around 817 nm. The used wavelength λ everywhere in this thesisis therefore chosen to be 817 nm unless otherwise specified.

5

Chapter 3

Planar dielectric waveguides

We derive analytic equations for planar dielectric slab waveguides in this chapter. We look at the symmetricsituation with equal refractive indices of the cladding as well as the asymmetric situation with unequalrefractive indices of the cladding. We also look the differences between transverse magnetic and transverseelectric polarization. We will also impose a condition for which the waveguide is single-mode. We make plotsof the refractive index versus core thickness using the single-mode condition. We also look at the angle ofincidence of the light that the light coupled to the waveguide should have.

Total internal reflection

Figure 3.1: Total internal reflection and critical angle. Figure taken from [5]

A dielectric waveguide uses the concept of total internal reflection to reflect light. A core material issurrounded by a cladding material of lower refractive index. The main cause of losses in dielectric waveguidesare scattering due to impurities and absorption. We ignore these losses when deriving equations becausethey are very small.

The definition of the angle from where total internal reflection occurs is:

sin θc =ncladncore

(3.1)

This is called the critical angle. Light inciding on a boundary at an angle greater than the critical angle willreflect internally as seen in figure 3.1.

3.1 Symmetric dielectric slab waveguides

We start in this section with the symmetric dielectric slab waveguide because this waveguide gives insightto the concepts of more complicated waveguides. The symmetric dielectric slab waveguide is the waveguide

6

CHAPTER 3. PLANAR DIELECTRIC WAVEGUIDES 7

θcθm

θa

θi

n1

n0

n2

d

n

Figure 3.2: A dielectric slab waveguide where the mode angle θm and critical angle θc are indicated. Theangles θi and θa are angles the light entering the waveguide needs to have to propagate under the modeangle and critical angle respectively. The refractive indices n0 of the core, n1 and n2 of the cladding and nof the medium outside the waveguide are also labeled. The thickness of the waveguide core is labeled by d.

shown in figure 3.2 where the claddings n1 and n2 are equal. This waveguide consists of a core materialof height d. We assume that the height of the cladding regions is infinite. We also assume that the widthof the waveguide is infinite so it only confines light in one direction. We also assume the material of thewaveguide is lossless. This makes it possible to solve this waveguide analytically, which is nearly impossible3D structures that confine light in two directions.

3.1.1 Polarization

In free space, waves can be transverse electromagnetically (TEM) polarized. These waves have no magneticor electric field in the direction of propagation. TEM polarized waves are not supported by a dielectricslab waveguide [3]. Only transverse electric (TE) and transverse magnetic (TM) polarization are supported.TE polarized electromagnetic waves have no electric field in the direction of propagation of the mode andTM polarized waves have no magnetic field in the direction of mode propagation. Note that the directionof propagation of the light is not the same as the direction of propagation of the mode. The propagationconstant of the mode β is the z-component of the propagation vector k of the light as seen in figure 3.3. Themode thus travels in the z-direction.

3.1.2 Ray approach: Self-consistency condition

Waves propagating inside the waveguide at an angle greater than the critical angle will reflect as seen infigure 3.3. When these waves interfere constructively, there will be a pattern that does not change along thedirection of travel as seen in figure 3.4. The condition for this constructive interference is that the wave hasto be equal in phase to itself after two reflections. The amplitude and propagation constant of the wavesare not changed by reflection. The phase change depends on the indices of refraction of the materials atthe reflection boundaries. We can then set a condition consisting of the phase change due to travel of thewave and add it to the phase change due to reflection of the wave. The total phase change must be equal to2πm where m = 0, 1, 2... for the wave to reproduce its pattern. A wave pattern that satisfies this conditionis called a mode. These modes have a bounce angle θm that is bigger than the critical angle θc as seen infigure 3.2. All angles are defined with respect to the normal.


Figure 3.3: An illustration of plane waves in a waveguide. The self-consistency after two reflections is visiblein this ray/wave picture. The lines parallel to the light rays are plane waves. The thickness of the waveguidecore has been indicated as d, the mode angle with θ and the wavelength λ. The line AB depicts the path ofthe original wave, while AC is the actual wave. This image has been adapted from the book Fundamentalsof Photonics [6].

Figure 3.4: The pattern of a mode does not change in the direction of the propagation of the wave. Imagefrom Fundamentals of Photonics [6].

The self-consistency condition for a planar symmetric dielectric waveguide for TE modes is:

tan

(2πd

λcos θm −mπ

)= 2

√sin2 θm − n2

1

n20

cos θm(3.2)

The self-consistency condition for the symmetric for TM modes is:

tan

(2πd

λcos θm −mπ

)= 2

n20n21

√sin2 θm − n2

1

n20

cos θm(3.3)

There is an extra constantn20

n21

added in the transverse magnetic (TM) case caused by the polarization

dependence of the phase shift.

3.1.3 Single-mode condition

Because we want a waveguide that only supports one mode, we impose a cut-off condition. This cutoffhappens when the mode angle θm just equals the critical angle θc [3]. At this angle, the mode transits from aguided mode to an unguided mode called a radiation mode. Note that modes slightly above this cutoff leakaway very slowly because this cutoff condition is not a hard boundary. Internal reflection as well as externalreflection happens past the critical angle. So the light propagating in the waveguide will still be guided fora certain distance. The cutoff condition is:

sin θm = sin θc =n1n0

(3.4)


When we add this cutoff condition to the self-consistency condition (equation 3.2) we get :

tan

(2πd

λcos θm −mπ

)= 0 (3.5)

When tanx = 0 then x = 0, so:2πd

λcos θm = mπ (3.6)

The number of modes is then given by:

M.=

2d

λcos θm (3.7)

Where the dot on the equal sign means that M is rounded up to the nearest integer. This has to be donebecause there is one more mode than the mode index m (first mode has m = 0, second mode has m = 1etc.).

The number of modes can also be represented with respect to the numerical aperture and the normalizedfrequency Vc:

NA =√n20 − n21 (3.8)

Vc =πd

λNA (3.9)

M.=

2Vcπ

=2d

λ

√n20 − n21 (3.10)

3.1.4 Number of modes versus core thickness

100 200 300 400 500 600

0.5

1

1.5

2

2.5

3

Thickness in nm

Num

ber

of m

odes

Number of modes versus thickness

TE polarization

TM polarization

Figure 3.5: The number of modes versus the thickness of the waveguide core for n0=3.55 and n1=2.8. Thenumber of modes have been plotted for both TE and TM polarization for the symmetric case. The lines areequal for both polarizations.

We made matlab scripts to investigate the behavior of the dielectric slab waveguide. These scripts canbe found appendix A. To investigate how thick the waveguide core should be to support only one mode,we plotted the number of modes versus the core thickness in figure 3.5 for the symmetric case. We used


refractive indices n1=2.8 and n0=3.55 because 2.8 is the minimum refractive index we can achieve with ourmaterial. We can conclude from figure 3.5 that the waveguide core needs to be around 200 nm thick for thewaveguide to be single-mode in the symmetric case. Coupling light into such a thin waveguide core wouldbe very difficult in our setup, because the diameter of the focal point of the laser beam is 2 µm in our case.

Another interesting observation is that the TM and TE modes have the exact same transition thicknesses.A symmetric dielectric slab waveguide will therefore always have at least one TM mode and one TE mode.This is because the phase change part of the self-consistency condition equals zero in both cases. Whenwe insert the cutoff condition 3.4 into equations 3.2 and 3.3 to get the number of modes, we get an equalexpression for the number of modes for both TE and TM.

It is clear from figure 3.5 that we need to adjust the parameters of the waveguide to achieve a muchthicker core.

3.1.5 Refractive index versus thickness

2.8 2.9 3 3.1 3.2 3.3 3.4 3.5

0

500

1000

1500

2000

2500

3000

3500

Thickness versus refractive index

Refractive index of cladding n1

Thic

kness in n

m

maximum thickness

minimum thickness

Figure 3.6: The refractive index of the cladding n1 versus the thickness of the waveguide core for where itis still single-mode in the symmetric case. The fixed refractive index is n0=3.55.


3.4 3.42 3.44 3.46 3.48 3.5 3.52 3.54 3.56

0

500

1000

1500

2000

2500

3000

3500

Thickness versus refractive index


Thic

kness in n

m

maximum thickness

minimum thickness

Figure 3.7: The refractive index n1 between 3.4 and 3.55 for where it is single-mode in the symmetricsituation. The fixed refractive index is n0=3.55.

The refractive index of the cladding has been varied and plotted against the single-mode core thickness infigure 3.6. This has been done to investigate wether a single-mode waveguide with a core thickness of severalmicrometers would be possible with different refractive indices. We define the minimum thickness as thecore thickness for which the waveguide goes from zero to one supported mode. We define the maximumthickness as the core thickness for which the waveguide goes from one to two supported modes. We cansee from this plot that there is no minimum thickness for which the symmetric dielectric waveguide has nomodes, as expected. We can conclude that a symmetric slab waveguide always has at least one TM and TEmode. The maximum thickness increases rapidly when the refractive index of the core approaches (n0) therefractive index of the cladding (n2). Figure 3.7 shows the preferred refractive index range from 3.4 to 3.55.We see that a single-mode waveguide of several micrometers thick would be possible whether the claddingmaterial has a refractive index close to that of the core. A possible disadvantage of a small difference inrefractive index is that the light in the waveguide is less confined. We can conclude that the refractive indexof the cladding needs to be close to that of the core to achieve a core thickness of several micrometers.

3.1.6 Angle of incidence versus refractive index

Figure 3.8 Shows the angle of incidence (see θi in figure 3.2) versus the refractive index of the cladding. Theangle of incidence can be calculated from the mode angle or critical angle using Snell’s law and yields thefollowing expression:

θi = sin−1(n0n

sin (π/2− θm))

(3.11)

To find out at which angle the laser beam should be coupled to the waveguide, we calculated the angle ofincidence from the mode angle and from the critical angle. The angle of incidence from the critical angleis called the acceptance angle and is usually used as an approximation. The acceptance angle can be usedto approximate the angle of incidence because the mode angle arises from the ray picture with very thinrays and we have a Gaussian intensity distribution that is not negligibly thin. The acceptance and incidenceangles are imaginary between n1 = 2.8 and n1 ≈ 3.34 in our situation. This is because this angle of incidenceexceeds 90 degrees because we get total internal reflection at the entrance of our waveguide. The angle ofincidence becomes smaller as the index of refraction n1 of the cladding approaches n0 of the core, so we needto find a suitable angle we can achieve.

The acceptance angle from the critical angle is equal to the angle of incidence calculated from theminimum thickness line. This follows from the self-consistency condition (equation 3.2). If we evaluate theself-consistency condition at the minimum thickness d=0 and mode index m=0 (first mode), the left side of


the self-consistency condition (equation 3.2) will be zero and we get the definition of the critical angle θc:

sin θm = sin θc =n1n0

(3.12)

3.4 3.42 3.44 3.46 3.48 3.5 3.52 3.54 3.56 3.580

10

20

30

40

50

60

70

80

90Angle of incidence versus n1


Angle

of in

cid

ence

From maximum thickness

From minimum thickness

From critical angle

Figure 3.8: The angle of incidence calculated from the mode angle and the acceptance angle calculatedfrom the critical angle are shown in this figure. The refractive index n0=3.55 and n1=nclad has been variedbetween 3.4 and 3.55, our area of interest. The acceptance angle exceeds 90 degrees before n1 = 3.41 andthe angle of incidence from the critical angle exceeds 90 degrees before n1 = 3.37. Note that the angle ofincidence from the minimum thickness line is equal to the acceptance angle. The refractive index n outsidethe waveguide is 1.


3.2 Asymmetric dielectric waveguides

An asymmetric dielectric slab waveguide is identical to the symmetric dielectric slab waveguide with onedifference: the refractive indices of the cladding materials are different, as shown in figure 3.2. This causesan unequal phase change for both boundary reflections for light propagating in the waveguide. Anotherchange is that the bounce angles of the modes in an asymmetric waveguide also need to be greater than bothcritical angles. We choose the biggest critical angle as the limiting critical angle for the single-mode cutoffcondition. An advantage of the asymmetric setup is that one cladding can have a high refractive index sohigher modes leak out and the other cladding can have a low refractive index for higher optical confinement.In general we assume n0 > n1 > n2 for the derivations, which means that n1 is the limiting refractive indexand yields the biggest critical angle.

3.2.1 Self-consistency condition

The self-consistency condition for this waveguide is nearly identical to the self-consistency condition forsymmetric dielectric slab waveguides (see equation 3.2 ). Instead of two identical phase changes for thereflection, we now have a different phase change for each reflection. The self-consistency condition fortransverse electric (TE) polarization is:

2π

λn0d cos θm −mπ = tan−1

√

sin2 θm − n1

n0

2

cos θm

+ tan−1

√

sin2 θm − n2

n0

2

cos θm

(3.13)

The self-consistency condition for transverse magnetic (TM) modes in the asymmetric situation is:

2π

λn0d cos θm −mπ = tan−1

n20n21

√sin2 θm − n1

n0

2

cos θm

+ tan−1

n20n22

√sin2 θm − n2

n0

2

cos θm

(3.14)


To find out when the waveguide is still single-mode, we impose a cutoff condition for the asymmetric situation.Because we now have two different critical angles, we only choose the biggest critical angle as a cutoffangle. Only the phase change of the boundary we choose for our limiting critical angle will disappearwhen we insert the cutoff condition into the self-consistency condition. Therefore, the left side of the self-consistency condition does not disappear as in the symmetric situation. In this case we choose the criticalangle determined by the n0 - n1 boundary as our limiting critical angle.

In the case of TE polarization, we get the following number of modes from the self-consistency conditionwhen we insert the cutoff condition:

M.=

1

π

(Vc − tan−1

√n21 − n22n20 − n21

)(3.15)

Where:

Vc =πd

λ

√n20 − n21 (3.16)

And in the case of TM polarization:

M.=

1

π

(Vc − tan−1

√(n20n22

)n21 − n22n20 − n21

)(3.17)

M has to be between 0 and 1 for the waveguide to be single-mode.Because the tan−1 term needs to be smaller than Vc for even a single-mode to exist, there is a minimum

thickness. This minimum thickness differs for TM and TE polarization, which means there is a thickness


where only one TE mode exists, as opposed to the symmetric waveguide where at least one TM and one TEmode exists.

Another difference between the symmetric slab waveguide and the asymmetric slab is that no analyticsolution for the mode angle in the asymmetric situation exists. The solution also depends on the polariza-tion in the asymmetric situation, while the solution in the symmetric situation was equal for TM and TEpolarization because the right side of the self-consistency condition did disappear.

3.2.3 Thickness versus refractive index

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Thickness in nm

Nu

mb

er

of

mo

de

s

Number of modes versus thickness

TE polarization

TM polarization

Figure 3.9: The number of modes versus the thickness of the core has been plotted. The that TE polarizationhas a lower transition thickness than TM. The used refractive indices are n0 = 3.55, n1 = 2.8 and n2 = 3.4.

3.2.4 Number of modes versus thickness

We will now take a look at asymmetric dielectric waveguides. To investigate whether the transition thicknessfrom single to multi-mode is different in the asymmetric situation we plotted the number of modes versusthickness again in figure 3.9 for the asymmetric situation. There is a visible difference in the transitionthicknesses of TM and TE polarization. This is due to the cutoff condition. The cutoff condition onlycancels out the part of the phase change in the self-consistency conditions 3.13 and 3.14 of the boundarywe choose to set up the cutoff condition. The resulting terms are slightly different for TM and TE asdemonstrated in section 3.2.2. We can see that TM polarization has a slightly higher transition thicknessthan TE, but nowhere near the amount we need to get close to a core thickness of several micrometers.

3.2.5 Thickness versus refractive index

We plotted the thickness versus the refractive index again for the asymmetric situation. The line for bothTE and TM modes is shown in figure 3.10. This has been done to investigate whether a high maximumthickness could still be achieved when only one of the refractive indices was brought close to the refractiveindex of the core while the other was kept fixed. The small difference in thickness between TM and TEmodes is also visible in the plot. This difference becomes smaller when n2 approaches n0 due to the TMconstant n0

n2from equation 3.14 approaching 1.

To investigate the effect of changing the non-limiting refractive index, n2 has been varied in figure 3.11.The effect of changing this refractive index is small and a lower refractive index for the non-limiting indexyields the highest thickness.


3.4 3.42 3.44 3.46 3.48 3.5 3.52 3.54 3.56 3.580

500

1000

1500

2000

2500

3000

3500

4000Thickness versus refractive index n1 for a single mode

n1

Thic

kness in n

m

Maximum thickness line for TE

Minimum thickness line for TE

Maximum thickness line for TM

Minimum thickness line for TM

Figure 3.10: The refractive index of the cladding versus the thickness of the waveguide core for a single-modewaveguide. The dashed lines are for TM polarization and the solid lines are for the TE polarization. Thefixed refractive indices are n0=3.55 and n2=2.8

2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.60

100

200

300

400

500

600Thickness versus refractive index n2 for a single mode

n2

Thic

kness in n

m





Figure 3.11: The refractive index of the non limiting cladding n2 has been varied in this plot. The dashedlines are for the TM polarization and the solid lines are for the TE polarization. The fixed refractive indicesare n0=3.55 and n1=3.4

We can see from figure 3.11 and figure 3.10 that the limiting refractive index n1 needs to approach that ofthe core as closely as possible, while non-limiting the refractive index n2 needs to be as low as possible. Thismeans the waveguide needs to be as asymmetric as possible. Keeping index n2 low would also yield betteroptical confinement as seen in figure 4.5 in chapter 4. Note that the plot has no meaning beyond n2 = 3.4because n2 > n1. Refractive index n2 becomes the limiting refractive index while it is still analyticallytreated as the non-limiting refractive index.


2.8 2.9 3 3.1 3.2 3.3 3.4 3.5

500

1000

1500

2000

Thickness versus refractive index n2 for a single mode

n2

Th

ickn

ess in

nm





Figure 3.12: The refractive index of the non limiting cladding n2 has been varied in this plot while refractiveindex n1 has been kept very close to the refractive index of the cladding. The dashed lines are for the TMpolarization and the solid lines are for the TE polarization. The fixed refractive indices are n0=3.55 andn1=3.54

To investigate whether the situation remains the same if the limiting refractive index n1 is very close tothe refractive index n0 of the core, figure 3.12 has been plotted. We can see that the thicknesses are muchhigher due to the non-limiting refractive index n1 being close to the refractive index of the core. We can stillsee that the maximum thickness collapses when n2 gets very close to n1 and a lower refractive index yieldsa slightly higher maximum thickness. We could use no top cladding for the sample instead of an AlGaAslayer to strengthen this effect.

3.2.6 Angle of incidence versus refractive index

The angle of incidence in this plot has been calculated from the mode angle using Snell’s law:

θi = sin−1(n0n

sin (π/2− θm))

(3.18)

This angle is called the acceptance angle when θm = θc. The acceptance angle approaches the angle ofincidence near the area of n1 where the waveguide is single-mode. This is why most literature neglect thedifference between the acceptance angle and angle of incidence from the mode angle because the mode anglearises from ray optics with very thin rays, compared to the broader Gaussian distribution we have in practice.The angle of incidence that would be required for the thickness of the waveguide core to be well above 2micrometers is very small, which could make coupling light in more difficult. We could look at 3D-waveguidesnext to investigate whether this is still the case.


3.4 3.42 3.44 3.46 3.48 3.5 3.52 3.54 3.56 3.580

10

20

30

40

50

60

70

80

90Asymmetric situation: n1 versus angle of incidence for TE modes

n1

Angle

of in

cid

ence

From maximum thickness line

From critical angle

From minimum thickness

Figure 3.13: The angle of incidence and the angle of acceptance in the asymmetric dielectric waveguide.n2 = 2.8, n0 = 3.55 and noutside = n = 1

Chapter 4

Rectangular dielectric waveguides

We will look at more complex waveguides that confine light in two directions in this chapter. These waveg-uides are 3D waveguides. We will use the Marcatilli method to solve these waveguides. This approximatemethod has been used because analytic solutions are not possible. We will first look at approximate methodsfor the simplest 3D waveguide: the rectangular dielectric waveguide. We can no longer ignore the width ofthe waveguide as we did with slab waveguides. The precision of the approximate methods is therefore moreprecise as the aspect ration of the waveguide becomes wider.

4.1 Marcatilli’s method

n0 n2n4

n3

n1

yx

d

a

Figure 4.1: A frontal view of the rectangular dielectric waveguide. The index of the core is n0 and the otherrefractive indices n1 − n4 are the cladding. The fields outside the corners are neglected or simplified in theapproximate methods (shaded in blue).

The Marcatilli method treats the rectangular 3D dielectric waveguide as two dielectric slab waveguides. Itneglects the fields outside the corners as seen in blue in figure 4.1. This method assumes that light willeither only bounce in the x-direction or only in the y-direction. We look at the symmetric situation wherethe cladding is of the same refractive index (n1 = n2 = n3 = n4) first.

18

CHAPTER 4. RECTANGULAR DIELECTRIC WAVEGUIDES 19

4.1.1 Polarization

We use Ex and Ey polarization instead of TM and TE polarization. Ex modes have no electric field in they-direction. Ey modes have no z-component of the magnetic field. This has been chosen because we cannow split the modes into an x- and -y components, without cross terms.

These modes a characterized by two numbers p and q, where p is the number of peaks in the electricfield in the Y-direction and q is the number of peaks in the electric field in the y-direction. We denote themodes as Epq

x and Epqy . The minimum value for p and q are 1, in contrast to the slab waveguide where the

mode index started at 0. We are interested in the E11y and E11

x modes, which have p = 1 and q = 1. Thesemodes have one electric field peak in both directions and thus yield a Gaussian intensity distribution in thesymmetric case and a near-Gaussian distribution in the asymmetric case.

Figure 4.2: The electric field distribution of modes in the Marcatilli method. Note that the mode notationis Ex

pq instead of Epqx . Image from Okamoto [2].

4.1.2 Self-consistency condition

We can derive a self-consistency condition using the Marcatilli method for a 3D waveguide. This has beendone in several books like the one written by Okamoto [2]. The solution for the Ex modes yields the solutionfor the symmetric dielectric slab waveguide for TE modes for the width of the waveguide and the solution for


TM modes for the height of the waveguide. The difference is that light has to be confined in both the x- and-y-directions, whereas we only needed one self-consistency condition for a slab waveguide. Another differenceis in the notation: The self-consistency conditions have been expressed in terms of the wave number k andthe components kx, ky and kz.

We now need to satisfy both self-consistency conditions to yield a guided mode. The Self-consistencyconditions for Ex modes are:

kxa = (p− 1)π

2+ tan−1

(n21γxn20kx

)(4.1)

kyd = (p− 1)π

2+ tan−1

(γyky

)(4.2)

Where:γ2x = k2(n21 − n20)− k2x (4.3)

γ2y = k2(n21 − n20)− k2y (4.4)

The self-consistency conditions are the same as for TM and TE modes for the slab waveguide. The modepropagation constant is given by:

β = kz = k2n21 − (k2x + k2y) (4.5)

The propagation constant is now also discrete because only certain values for kx and ky are allowed.The self-consistency condition for Ey modes is:

kxa = (p− 1)π

2+ tan−1

(γxkx

)(4.6)

kyd = (p− 1)π

2+ tan−1

(n21γyn20ky

)(4.7)

Where the width is determined by the self-consistency condition of TE modes for slab waveguides and theheight is determined by the self-consistency condition for slab TM waveguides.

We can also see that the self-consistency conditions do not depend on each other. Changing the widthshould not have an effect on the required height as long as the self-consistency conditions are satisfied. Wecan also use the solutions for the asymmetric dielectric slab waveguide for the situation where the refractiveindices of the cladding are not equal (n1 6= n2 6= n3 6= n4) by dividing the rectangular waveguide into twoasymmetric slab waveguides.


We want to find a refractive index n1 for which the wave is single-mode. Because we changed the polarizationsuch that the field components consists of purely horizontal and vertical parts, we can apply the exact sametheory as section 3.2.2 for the same cladding symmetric or section 3.1.3 for unequal claddings. We can usethe same cutoff condition and insert it into the self-consistency condition. We then get equation 3.10 for thenumber of modes in the symmetric situation or equation 3.15 and equation 3.17 for the asymmetric situation.The only difference is that we now have two self-consistency conditions that need to be satisfied. This meansthat both the height and width cannot be arbitrary chosen. We plotted the curves for the maximum widthand height in figure 4.3


2.8 2.9 3 3.1 3.2 3.3 3.4 3.5

500

1000

1500

2000

2500

3000

3500

Maximum width and height for a symmetric rectangular waveguide


thic

kn

ess n

m

Maximum width/height

Minimum width/height

Figure 4.3: The maximum width and height for a 3D rectangular waveguide calculated using the Marcatillimethod. The refractive index of the core n0 = 3.55 and the index n1 of the cladding has been varied.

We can conclude that the required refractive indices for a height and width of several micrometers donot change compared to the dielectric slab waveguide. This is due to the cleverly chosen polarization.

We also investigated the dielectric slab waveguide using the effective index method [2]. This method canalso be used to solve slightly more complicated waveguides, but we did not complete this analysis due totime constraints.

4.2 Rib waveguides

n0

n1

yx

H

a

n2

h

t

Figure 4.4: A rib waveguide. The top and bottom cladding both have refractive index n1. The thicknessof the top cladding is t. The center of the waveguide core has a height H and width a. The sides of thewaveguide core have been etched to a thickness H < h.


After a literature search, we discovered that rib waveguides can be used as single-mode waveguides with acore of several micrometers thick [7]. Because modes higher than the first mode leak out over distance, thedifference in refractive index of the core and the cladding does not have to be as small as in a rectangularwaveguide.

A rib waveguide is made by etching away part of the sides of the waveguide core. A layer can be depositedon top of the core or the top can simply be left open. A substrate with refractive index n2 serves as thelower cladding. Higher order modes leak out over a distance of a few micrometers and only the fundamentalmode is confined when the parameters for this waveguide have been chosen correctly. This is not so trivialto calculate using the Marcatilli or effective index methods. The waveguide core of a rib waveguide couldbe made thicker compared to a rectangular waveguide using the same difference in refractive indices. Therefractive index of the cladding could also be further away from the refractive index of the core when keepingthe core thickness the same as in a rectangular waveguide. The advantage is that either light could becoupled in easier. The waveguide production is also easier because the waveguide can still be single-modeeven if the refractive index is slightly further away than the very small difference required between core andcladding for a rectangular waveguide.

We decided to use the given parameters from the paper [7] to try to simulate such a waveguide inCOMSOL Multiphysics. We started by designing an asymmetric 2D slab waveguide to compare with theplots we made earlier.

Simulating the waveguide in COMSOL could also give us insight into how fast the modes above cutoffactually leak away so we can be sure our waveguide is truly single-mode with our parameters. We can useCOMSOL to test whether the radiation modes are still guided over the length of the waveguide we designed.

Figure 4.5: An asymmetric dielectric slab waveguide. The Z-component of the electric field is shown in thispicture. The refractive index of the core is 3.5. The upper cladding has a refractive index of 2.8 and thelower cladding has a refractive index of 3.49

The z-component of the electric field of an asymmetric dielectric slab waveguide has been plotted infigure 4.5. We can see that it matches our expectations: the field at the lower refractive index cladding isbetter confined in the core and the electric field has one peak in the plane into the paper.


Figure 4.6: A 3D- rectangular waveguide simulated in COMSOL. The Y -component of the electric field isshown in this picture in the zx-plane. Light travels along the Z-axis. The refractive index of the core is 1.5.The cladding has a refractive index of 1.

We then tried to model a 3D rectangular waveguide. The electric field of this waveguide has been plottedin figure. We can see that this is not the E11

x mode nor the E11y mode because the electric field has more

than one peak. We can also see that the electric field is not as strongly confined in the core as we wouldexpect with a large difference in refractive indices of the core and cladding. Due to time constraints andlimited knowledge of how the COMSOL mode solver exactly gets its results, we were unable to simulate athis waveguide successfully. Because we could not successfully simulate the rectangular waveguide, we alsodid not have time to simulate the rib waveguide. Note that refractive indices in figure 4.6 were chosen to bemuch smaller than the ones of our sample due to memory constraints.

The COMSOL example of a H-bend waveguide [8] and waveguide adapter [9] could be used to learnmore about simulating 3D-waveguides in COMSOL for future research. Another program is WMM modesolver. This program is specifically made for solving waveguide properties. Because you have to compile thisprogram yourself from C++ code and the import the date into MATLAB to plot it, the program is not easyto use. This program can be found in reference [10].

Conclusion

We investigated the possibility to integrate a single-mode GaAs waveguide into experiments on opticaltransitions in GaAs in a cryogenic setup. The core needed to be several micrometers thick due to the spotsize of the focal point of our laser beam in the experimental setup. We were limited to an AlGaAs claddingto avoid strain on the sample. The refractive index range that can be achieved with this material is 2.8-3.55.

We found out that the transition thickness, the thickness for which a waveguide transits from single-modeto multi-mode, of the waveguide core was 200 nm thick for a core refractive index of 2.8. Because this wastoo thin to couple light into, we varied the refractive indices of both claddings to find out the requiredrefractive index for a core thickness of several micrometers. We concluded that the refractive index of thecladding should be close to the cladding of the core. Next, we looked at the asymmetric situation, wherethe refractive index of one cladding was varied. We deduced that the waveguide should be as asymmetricas possible to achieve the maximum thickness. Leaving out the top cladding yields the most asymmetricsituation as possible and is possibly the best solution. We then used the Marcatilli method to solve the 3D-rectangular waveguide. Using this approximate method, we concluded that the required refractive indicesfor single-mode operation were the same as for the slab waveguide. The only change was in the polarization.We modeled 3D-rectangular waveguides using COMSOL in a quest to find out more about 3D-rib and ridgewaveguides. We did not complete the mode analysis on the 3D waveguide due to time constraints.

A recommendation for future research would be to investigate more complex 3D-structures to make awaveguide that is easier to integrate into the setup. A possible way to learn more about the COMSOL modesolver could be done by studying the examples and tutorials in references [8] and [9].

24

Bibliography

[1] van der Wal, C. H. & Sladkov, M. Towards quantum optics and entanglement with elec-tron spin ensembles in semiconductors. Solid State Sciences 11, 935 – 941 (2009). URLhttp://www.sciencedirect.com/science/article/pii/S1293255808000757. E-MRS symposium Nand R.

[2] Okamoto, K. Fundamentals of Optical Waveguides (Second Edition) (Academic Press, Burlington,2006), second edition edn.

[3] Yeh, C. & Shimabukuro, F. The essence of dielectric waveguides (section 2.5) (Springer, 2008).

[4] Sladkov, M. et al. Polarization-preserving confocal microscope for optical experiments in a dilu-tion refrigerator with high magnetic field. Review of Scientific Instruments 82, – (2011). URLhttp://scitation.aip.org/content/aip/journal/rsi/82/4/10.1063/1.3574217.

[5] Josell7. Wikimedia commons: Refraction reflection. URLhttps://commons.wikimedia.org/wiki/File:RefractionReflextion.svg.

[6] Saleh, B. & Teich, M. Fundamentals of Photonics. Wiley Series in Pure and Applied Optics (Wiley,2007).

[7] Soref, R. A., Schmidtchen, J. & Petermann, K. Large single-mode rib waveguides in gesi-si and si-on-sio2. Quantum Electronics, IEEE Journal of 27, 1971–1974 (1991).

[8] COMSOL. H-bend waveguides in comsol. URL http://www.comsol.com/model/h-bend-waveguide-3d-1421.

[9] COMSOL. Waveguide adapter application. URL http://www.comsol.com/model/waveguide-adapter-140.

[10] Lohmeyer, M. Wmm mode solver. URL http://wmm.computational-photonics.eu/.

25

Appendices

26

Appendix A

Matlab scripts

Script for the number of modes graphs:

% Acquire initial valuesn2=2.8; % AlAs Claddingn0=3.55; % Samplen1=3.4; % AlAs Substratedmin=1; % min thickness in nmdmax=600; % Max thickness in nmla=817; % Wavelength of light in nm

% Edited initial valuesd=dmin:1:dmax; % Thickness in nmlambda=2*pi/la*(d/2)*sqrt(n0ˆ2-n1ˆ2); % normalized frequency

% Calculate numer of modes for TE and round them to the next integerM1=(1/pi)*(2*lambda-atan(sqrt((n1ˆ2-n2ˆ2)/(n0ˆ2-n1ˆ2))));MTE=ceil(M1);

% Calculate numer of modes for TM and round them to the next integerM=(1/pi)*(2*lambda-atan(sqrt((n0/n2)ˆ2*(n1ˆ2-n2ˆ2)/(n0ˆ2-n1ˆ2))));MTM=ceil(M);mdiff=MTE-MTM;%plot number of TE modesfigure

plot(d,MTE,d,MTM)legend('TE polarization','TM polarization','Location','NorthWest')title('Number of modes versus thickness')xlabel('Thickness in nm')ylabel('Number of modes')

set(gcf, 'PaperPositionMode', 'auto');print -depsc2 plot/as nmodes.eps

Script for calculating the thickness of the waveguide versus the refractive index of the core. First thesymmetric situation for both TM and TE modes:

clear all% Acquire initial valuesstepsize=10ˆ(-3);n0=3.55; % Samplela=817; % Wavelength of light in nmn2min=3.4; %minimum index of refraction of n2n2max=n0-stepsize; %maximum index of refraction of n2% Edited initial valuesdista=[]; % Define data arrays

27

APPENDIX A. MATLAB SCRIPTS 28

distamin=[]; % min data arraynair=[];

%loop through indices nfor n2=n2min:stepsize:n2max

%Then loop through every distance for every index ndista1=[]; %

for d=0:4000vn=2*pi/la*(d/2)*sqrt(n0ˆ2-n2ˆ2); % normalized frequency% Calculate numer of modes for TE and round them to the next integerM1=(1/pi)*(2*vn);MTE=ceil(M1);

% And then check if it is single modeif MTE==1%if so, hold datadista1(end+1,:)=[n2,d];end

end%Check maximum value and minimum of d of this loop through d for fixed N%valueif isempty(dista1)==0dista(end+1,:)=max(dista1);distamin(end+1,:)=min(dista1);

endend%plot number of TE modesf=figure;set(f,'windowstyle','docked');set(gca,'FontSize',18);plot(dista(:,1),dista(:,2),distamin(:,1),distamin(:,2))title('Thickness versus refractive index')xlabel('Refractive index of cladding n1')ylabel('Thickness in nm')legend('maximum thickness','minimum thickness','Location','northwest')set(gcf, 'PaperPositionMode', 'auto');print -depsc2 plot/n vs d symmetric 34.eps

And the asymmetric version for TM modes:

clear all% Acquire initial valuesstepsize=10ˆ-3; %stepsize of refractive indicesn0=3.55; % Samplen1=2.8; % AlAs claddingla=817; % Wavelength of light in nmn2min=3.4;n2max=n0-stepsize;% Edited initial valuesdista=[]; %nair=[];


%Then loop through every distance for every index nfor d=0:4000vn=2*pi/la*(d/2)*sqrt(n0ˆ2-n2ˆ2); % normalized frequency% Calculate numer of modes for TE and round them to the next integer% Calculate numer of modes for TM and round them to the next integerM1=(1/pi)*(2*vn-atan(sqrt((n0/n1)ˆ2*(n2ˆ2-n1ˆ2)/(n0ˆ2-n2ˆ2))));MTE=ceil(M1);


% And then check if it is single modeif MTE==1%if so, record datadista(end+1)=d;nair(end+1)=n2;end

endend

% Join array nair and dista to edit themM=[nair;dista]';%Split matrix M in matching nair values in array of cell arrays AA = arrayfun(@(x) M(M(:,1) == x, :), unique(M(:,1)), 'uniformoutput', false);% Get length of Alengtha=size(A);% Creat empty data matricesdata=[];datamin=[];

% extract useful data from each submatrix from Afor i=1:lengtha% Extract every matrix from AL=cell2mat(A(i));% Add the max values of every submatrix L of A to datadata(end+1,:)=max(L);% And do the same for the min valuesdatamin(end+1,:)=min(L);

end

%plot for TM modes%f=figure;%set(f,'windowstyle','docked');%set(gca,'FontSize',18);plot(data(:,1),data(:,2),'--',datamin(:,1),datamin(:,2),'--')title('Thickness versus refractive index n1 for a single mode')xlabel('n1')ylabel('Thickness in nm')legend('Maximum thickness line for TE','Minimum thickness line for TE','Maximum thickness ...

line for TM','Minimum thickness line for TM','Location','NorthWest')set(gcf, 'PaperPositionMode', 'auto');print -depsc2 plot/non limit tm.eps

And for TE modes:

clear all%n2 limiting cladding% Acquire initial valuesstepsize=10ˆ-3;n0=3.55; % Samplen1=2.80; % AlAs claddingla=817; % Wavelength of light in nmn2min=3.4; %minimum index of refraction of n2n2max=n0-stepsize; %maximum index of refraction of n2% Edited initial valuesdista=[]; % Define data arraysdistamin=[]; % min data arraynair=[];


%Then loop through every distance for every index n


dista1=[]; %

for d=0:4000vn=2*pi/la*(d/2)*sqrt(n0ˆ2-n2ˆ2); % normalized frequency% Calculate numer of modes for TE and round them to the next integerM1=(1/pi)*(2*vn-atan(sqrt(abs((n2ˆ2-n1ˆ2)/(n0ˆ2-n2ˆ2)))));MTE=ceil(M1);



endend%plot number of TE modesf=figure;set(f,'windowstyle','docked');set(gca,'FontSize',18);plot(dista(:,1),dista(:,2),distamin(:,1),distamin(:,2))title('Distance versus refractive index for a single TE mode')xlabel('n1')ylabel('Thickness in nm')legend('Maximum thickness line','Minium thickness line','Location','NorthWest')set(gcf, 'PaperPositionMode', 'auto');print -depsc2 plot/dist vs n as 34.epshold on

And the script for calculating the mode angle versus the refractive index of the core in the symmetricsituation:

%Edited 7-5-15clear all%n2 limiting cladding% Acquire initial valuesn0=3.55; % Samplela=817; % Wavelength of light in nmn2min=3.4; %minimum index of refraction of n2n2max=n0; %maximum index of refraction of n2% Edited initial valuesdista=[]; % Define data arraysdistamin=[]; % min data arraynair=[];angle max=[];angle min=[];bounce angles=[];y=[];ymin=[];ac=[];%loop through indices nfor n2=n2min:0.001:n2max

%Then loop through every distance for every index ndista1=[]; %for d=0:10000vn=2/la*d*sqrt(n0ˆ2-n2ˆ2); % normalized frequency% Calculate numer of modes for TE and round them to the next integerMTE=ceil(vn);




endend

% Check self consistency condition to find the angle y of the mode% Loops through every index of dfor d=1:length(dista(:,2))

syms x;

%Retrieve mode parametersD = dista(d,2); %max thicknessDmin = distamin(d,2); %min thicknessn2=dista(d,1); %refractive indextc=asin(n2/n0); %critical anglex0=[tc pi/2-0.01]; %inital values between 90 degrees and the critical angle%Use vpasolve to find the solution numerically%Self consistency for minimum and maximum distance lineF = tan(pi*D/817*cos(x))-sqrt(n0ˆ2*sin(x)ˆ2-n2ˆ2)/(n2*cos(x));Fmin = tan(pi*Dmin/817*cos(x))-sqrt(n0ˆ2*sin(x)ˆ2-n2ˆ2)/(n2*cos(x));

% Solve the maximum line

y = vpasolve(F,x,x0);% Solve for the minimum lineymin = vpasolve(Fmin,x,x0);

bounce angles(end+1,:)=[Dmin,ymin];

n=1; %index of refraction of material in front of sample/corey=asin(n0/n*sin(pi/2-y)); %Get the angle of incidence from the mode angleymin=asin(n0/n*sin(pi/2-ymin)); %Get the angle of incidence from the mode angle

%Fill the arrays with this dataangle max(end+1,:)=[n2,y];angle min(end+1,:)=[n2,ymin];

%And now use the critical angleyc=asin(n0/n*sin(pi/2-tc)); %Get the angle of incidence from the critical angleac(end+1,:)=[n2,yc];

end%plot

f=figure;set(f,'windowstyle','docked');set(gca,'FontSize',18);plot(angle max(:,1),radtodeg(angle max(:,2)),angle min(:,1),radtodeg(angle min(:,2)),ac(:,1),radtodeg(ac(:,2)),'+')title('Angle of incidence versus n1')xlabel('Refractive index of cladding n1')ylabel(' Angle of incidence')legend('From maximum thickness',' From minimum thickness','From critical ...

angle','Location','northeast')set(gcf, 'PaperPositionMode', 'auto');print -depsc2 plot/angle vs n 34.eps


And the script for calculating the mode angle versus the refractive index of the core in the asymmetricsituation:

%Edited 8-6-15clear all%n2 limiting cladding% Acquire initial valuesn0=3.55; % Samplela=817; % Wavelength of light in nmn1=2.80; % AlAs Substraten=1; %index of refraction of material in front of sample/core

n2min=3.40; %minimum index of refraction of n2n2max=n0; %maximum index of refraction of n2% Edited initial valuesdista=[]; % Define data arraysdistamin=[]; % min data arraynair=[]; % Declare empty arraysangle max=[];angle min=[];bounce angles=[];y=[];ymin=[];ac=[];%loop through indices nfor n2=n2min:0.001:n2max

%Then loop through every distance for every index ndista1=[]; %for d=0:10000vn=2*pi/la*(d/2)*sqrt(n0ˆ2-n2ˆ2); % normalized frequency% Calculate numer of modes for TE and round them to the next integerM1=(1/pi)*(2*vn-atan(sqrt(abs((n2ˆ2-n1ˆ2)/(n0ˆ2-n2ˆ2)))));MTE=ceil(M1);



endend

% Check self consistency condition to find the angle y of the mode% Loops through every index of dfor d=1:length(dista(:,2))

syms x;

%Retrieve mode parametersD = dista(d,2); %max thicknessDmin = distamin(d,2); %min thicknessn2=dista(d,1); %refractive indextc=asin(n2/n0); %critical anglex0=[tc pi/2-0.01]; %inital values between 90 degrees and the critical angle%Use vpasolve to find the solution numerically%Self consistency for minimum and maximum distance line and Fmin for%minimum


F = 2*pi/la*n0*D*cos(x) - atan(sqrt(sin(x)ˆ2 - (n2/n0)ˆ2)/cos(x)) - atan(sqrt(sin(x)ˆ2 - ...(n1/n0)ˆ2)/cos(x));

Fmin = 2*pi/la*n0*Dmin*cos(x) - atan(sqrt(sin(x)ˆ2 - (n2/n0)ˆ2)/cos(x)) - atan(sqrt(sin(x)ˆ2 ...- (n1/n0)ˆ2)/cos(x));

% Solve the maximum line

y = vpasolve(F,x,x0);% Solve for the minimum line

ymin = vpasolve(Fmin,x,x0);

y=asin(n0/n*sin(pi/2-y)); %Get the angle of incidence from the mode angleymin=asin(n0/n*sin(pi/2-ymin)); %Get the angle of incidence from the mode angle

%Fill the arrays with this dataangle max(end+1,:)=[n2,y];angle min(end+1,:)=[n2,ymin];

%And now use the critical angleyc=asin(n0/n*sin(pi/2-tc)); %Get the angle of incidence from the critical angleac(end+1,:)=[n2,yc];

end%plot

f=figure;set(f,'windowstyle','docked');set(gca,'FontSize',18);plot(angle max(:,1),radtodeg(angle max(:,2)),ac(:,1),radtodeg(ac(:,2)),'+',angle min(:,1),radtodeg(angle min(:,2)))title('Asymmetric situation: n1 versus angle of incidence for TE modes')xlabel('n1')ylabel(' Angle of incidence')legend('From maximum thickness line','From critical angle',' From minimum thickness')set(gcf, 'PaperPositionMode', 'auto');print -depsc2 plot/angle vs n 34 as.eps

And a script for plotting the non-limiting refractive index vs thickness: First for TE modes:

clear all% Acquire initial values%n1 limitingn0=3.55; % Samplen1=3.4; % AlAs Substratela=817; % Wavelength of light in nmn2min=2.8;n2max=n0;% Edited initial valuesdista=[]; %nair=[];

%loop through indices nfor n2=n2min:0.001:n2max

%Then loop through every distance for every index nfor d=0:10000vn=2*pi/la*(d/2)*sqrt(n0ˆ2-n1ˆ2); % normalized frequency% Calculate numer of modes for TE and round them to the next integer% Calculate numer of modes for TM and round them to the next integerM1=(1/pi)*(2*vn-atan(sqrt((n1ˆ2-n2ˆ2)/(n0ˆ2-n1ˆ2))));MTE=ceil(M1);

% And then check if it is single modeif MTE==1%if so, record data


dista(end+1)=d;nair(end+1)=n2;end

endend



end

%plot for TM modesf=figure;set(f,'windowstyle','docked');set(gca,'FontSize',18);plot(data(:,1),data(:,2),datamin(:,1),datamin(:,2))title('Thickness versus refractive index n2 for a single mode')xlabel('n2')ylabel('Thickness in nm')legend('Maximum thickness line for TE','Minimum thickness line for TE','Location','NorthWest')set(gcf, 'PaperPositionMode', 'auto');print -depsc2 plot/non limit tm.epshold on

And for TM modes:

clear all% Acquire initial values%n1 limitingn0=3.55; % Samplen1=3.4; % AlAs Substratela=817; % Wavelength of light in nmn2min=2.8;n2max=n0;% Edited initial valuesdista=[]; %nair=[];

%loop through indices nfor n2=n2min:0.001:n2max

%Then loop through every distance for every index nfor d=0:10000vn=2*pi/la*(d/2)*sqrt(n0ˆ2-n1ˆ2); % normalized frequency% Calculate numer of modes for TE and round them to the next integer% Calculate numer of modes for TM and round them to the next integer


M1=(1/pi)*(2*vn-atan(sqrt((n0/n1)ˆ2*(n1ˆ2-n2ˆ2)/(n0ˆ2-n1ˆ2))));MTE=ceil(M1);

% And then check if it is single modeif MTE==1%if so, record datadista(end+1)=d;nair(end+1)=n2;end

endend



end

%plot for TM modes%f=figure;%set(f,'windowstyle','docked');%set(gca,'FontSize',18);plot(data(:,1),data(:,2),'--',datamin(:,1),datamin(:,2),'--')title('Thickness versus refractive index n2 for a single mode')xlabel('n2')ylabel('Thickness in nm')legend('Maximum thickness line for TE','Minimum thickness line for TE','Maximum thickness ...

line for TM','Minimum thickness line for TM','Location','NorthWest')set(gcf, 'PaperPositionMode', 'auto');print -depsc2 plot/non limit tm.eps

Date post:	18-Oct-2020
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Simulations on single-mode waveguides in GaAs · A dielectric waveguide uses the concept of total...

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