Simulations on single-mode waveguides in GaAs
Supervisors Prof. dr. ir. C. H. van der Wal dr. T.A. Schlath¨olter ir. J. P. de Jong Research group Physics of Nanodevices Student
F. A. van Zwol
Date Friday 10th July, 2015 Credits 15 ECTS
Bachelor thesis
Zernike Institute for Advanced Materials
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 cryogenic confocal microscope setup. Making this sample single-mode is challenging because the number of modes increases with core thickness. We solve this problem by bringing the refractive index of the core very close to the refractive index of the cladding of 2D waveguides that confine light in one direction. We looked at the difference between symmetric and asymmetric waveguides. The most asymmetric situation with the refractive index of one cladding close to the core and the refractive index of the other cladding much lower than the core yields the highest possible core thickness. We also looked at 3D waveguide structures that confine light in two directions. We used the Marcatilli method to solve for the rectangular 3D waveguide and found that the same theory as for slab waveguides applies. We also tried to solve more complex structures.
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.
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1
1 Introduction 3
2 Requirements 5
2.1 Materials . . . 5
2.2 Thickness and single-mode . . . 5
3 Planar dielectric waveguides 6 3.1 Symmetric dielectric slab waveguides . . . 6
3.1.1 Polarization . . . 7
3.1.2 Ray approach: Self-consistency condition . . . 7
3.1.3 Single-mode condition . . . 8
3.1.4 Number of modes versus core thickness . . . 9
3.1.5 Refractive index versus thickness . . . 10
3.1.6 Angle of incidence versus refractive index . . . 11
3.2 Asymmetric dielectric waveguides . . . 13
3.2.1 Self-consistency condition . . . 13
3.2.2 Single-mode condition . . . 13
3.2.3 Thickness versus refractive index . . . 14
3.2.4 Number of modes versus thickness . . . 14
3.2.5 Thickness versus refractive index . . . 14
3.2.6 Angle of incidence versus refractive index . . . 16
4 Rectangular dielectric waveguides 18 4.1 Marcatilli’s method . . . 18
4.1.1 Polarization . . . 19
4.1.2 Self-consistency condition . . . 19
4.1.3 Single-mode condition . . . 20
4.2 Rib waveguides . . . 21
Appendices 26
A Matlab scripts 27
2
Introduction
Quantum information systems use quantum bits (qubits) as information units, whereas regular information systems use regular bits. These qubits are quantum mechanical systems consisting of two eigenstates. A qubit can be in a superposition of states, in contrast to a normal bit. Several different quantum systems can be a qubit, like the electron spin or the polarization of light. Solid systems are usually chosen to be a qubit because because they stay in place. Photons are used to move quantum information. Photons can interact with matter and move quantum information over large distances.
Quantum optics is a field that researches the interaction between light and matter. Quantum optics aims to 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 single quantum emitter and a single photon. Another way to improve the interaction is by using waveguides. A waveguide 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 experiments more efficiently [1]. Another application of waveguides is the transport of light inside quantum experiments on 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. This sample has been placed on an AlAs substrate. This thesis will determine the geometry for the GaAs sample such that it is a single-mode waveguide. Single-mode means that only one wave pattern of light will be supported by the waveguide. The waveguide needs to be single-mode because we can be sure about the stability over time of the properties of the light exiting the waveguide. Making a waveguide single-mode can be difficult because the number of modes generally increases with thickness of the waveguide core. We need a 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 condition for the phase of light waves reflecting on a dielectric boundary and the number of modes as a function of refractive index [2, 3]. We applied the equations for the dielectric slab waveguides that can be found in literature to our material. From the resulting equations, we investigated which parameters have to change to get a single-mode waveguide with a core thickness of about a micrometer. The first parameter we changed was the refractive index of the claddings when the refractive index of both claddings was equal, creating a symmetric situation. Another situation we looked at is the asymmetric situation where we varied the refractive 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 our optical setup, we calculate the angle of incidence of the mode at the waveguide entrance. How the theory of waveguides changes when waveguides confine light in two directions instead of one direction, can also be found in literature. We investigated rectangular waveguides, which confine light in two directions, and compared them to slab waveguides, which confine light in only one direction. The same analytic approach and equations apply to this waveguide according to literature when some approximations are made.
3
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 refractive index for both the symmetric waveguide and the asymmetric waveguide versus the maximum single-mode thickness. 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 is impossible.
Requirements
The goal of this thesis is to determine the parameters of a single-mode waveguide GaAs sample. We derive the requirements for the waveguide in this chapter. These requirements come from fabricational methods and specifications of the cryogenic setup.
Figure 2.1: Schematic overview of the simulated sample setup where sample thickness d is varied in the simulations.
2.1 Materials
The sample studied in this thesis consists of a layer of GaAs with a refractive index n0of 3.55. This sample is placed on a substrate with refractive index n2. We need to find an optimal refractive index for the cladding materials and the top cladding. The solid materials used for both claddings need to closely match the lattice of the GaAs sample to limit the strain, hence the use of AlGaAs. An AlxGa1−xAs layer could be produced with 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 in diameter and the much smaller diffraction limit is not feasible in our setup [4]. This means that for the present study, we aim at realizing a waveguide of several micrometers thick to couple a Gaussian beam into the waveguide. We want to have a single-mode waveguide because it is not possible to couple light into a single-mode fiber from a multi-mode waveguide. We can also be sure to which mode our light couples to in a single-mode waveguide. Light entering a multi-mode waveguide could couple to multiple modes and thus make the characteristics like the intensity profile of the light exiting the waveguide unstable over time. The challenge is to keep the waveguide single-mode while achieving a core thickness of several micrometers. In general, 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 thesis is therefore chosen to be 817 nm unless otherwise specified.
5
Planar dielectric waveguides
We derive analytic equations for planar dielectric slab waveguides in this chapter. We look at the symmetric situation with equal refractive indices of the cladding as well as the asymmetric situation with unequal refractive indices of the cladding. We also look the differences between transverse magnetic and transverse electric polarization. We will also impose a condition for which the waveguide is single-mode. We make plots of the refractive index versus core thickness using the single-mode condition. We also look at the angle of incidence 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 is surrounded by a cladding material of lower refractive index. The main cause of losses in dielectric waveguides are scattering due to impurities and absorption. We ignore these losses when deriving equations because they are very small.
The definition of the angle from where total internal reflection occurs is:
sin θc= nclad
ncore
(3.1) This is called the critical angle. Light inciding on a boundary at an angle greater than the critical angle will reflect 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 insight to the concepts of more complicated waveguides. The symmetric dielectric slab waveguide is the waveguide
6
θ
cθ
mθ
aθ
in 1
n 0
n 2
d n
Figure 3.2: A dielectric slab waveguide where the mode angle θm and critical angle θc are indicated. The angles θi and θa are angles the light entering the waveguide needs to have to propagate under the mode angle and critical angle respectively. The refractive indices n0 of the core, n1 and n2 of the cladding and n of 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 material of height d. We assume that the height of the cladding regions is infinite. We also assume that the width of the waveguide is infinite so it only confines light in one direction. We also assume the material of the waveguide is lossless. This makes it possible to solve this waveguide analytically, which is nearly impossible 3D 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 magnetic or electric field in the direction of propagation. TEM polarized waves are not supported by a dielectric slab 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 and TM polarized waves have no magnetic field in the direction of mode propagation. Note that the direction of propagation of the light is not the same as the direction of propagation of the mode. The propagation constant of the mode β is the z-component of the propagation vector k of the light as seen in figure 3.3. The mode 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 in figure 3.3. When these waves interfere constructively, there will be a pattern that does not change along the direction of travel as seen in figure 3.4. The condition for this constructive interference is that the wave has to be equal in phase to itself after two reflections. The amplitude and propagation constant of the waves are not changed by reflection. The phase change depends on the indices of refraction of the materials at the reflection boundaries. We can then set a condition consisting of the phase change due to travel of the wave and add it to the phase change due to reflection of the wave. The total phase change must be equal to 2πm where m = 0, 1, 2... for the wave to reproduce its pattern. A wave pattern that satisfies this condition is called a mode. These modes have a bounce angle θm that is bigger than the critical angle θc as seen in figure 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 visible in this ray/wave picture. The lines parallel to the light rays are plane waves. The thickness of the waveguide core has been indicated as d, the mode angle with θ and the wavelength λ. The line AB depicts the path of the original wave, while AC is the actual wave. This image has been adapted from the book Fundamentals of Photonics [6].
Figure 3.4: The pattern of a mode does not change in the direction of the propagation of the wave. Image from Fundamentals of Photonics [6].
The self-consistency condition for a planar symmetric dielectric waveguide for TE modes is:
tan 2πd
λ cos θm− mπ
= 2 q
sin2θm−nn212 0
cos θm
(3.2) The self-consistency condition for the symmetric for TM modes is:
tan 2πd
λ cos θm− mπ
= 2n20 n21
q
sin2θm−nn212 0
cos θm
(3.3)
There is an extra constant nn202 1
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 cutoff happens when the mode angle θmjust equals the critical angle θc[3]. At this angle, the mode transits from a guided mode to an unguided mode called a radiation mode. Note that modes slightly above this cutoff leak away very slowly because this cutoff condition is not a hard boundary. Internal reflection as well as external reflection happens past the critical angle. So the light propagating in the waveguide will still be guided for a certain distance. The cutoff condition is:
sin θm= sin θc= n1 n0
(3.4)
When we add this cutoff condition to the self-consistency condition (equation 3.2) we get : tan 2πd
λ cos θm− mπ
= 0 (3.5)
When tan x = 0 then x = 0, so:
2πd
λ cos θm= mπ (3.6)
The number of modes is then given by:
M .
= 2d
λ cos θm (3.7)
Where the dot on the equal sign means that M is rounded up to the nearest integer. This has to be done because there is one more mode than the mode index m (first mode has m = 0, second mode has m = 1 etc.).
The number of modes can also be represented with respect to the numerical aperture and the normalized frequency Vc:
NA = q
n20− n21 (3.8)
Vc= πd
λNA (3.9)
M .
=2Vc
π = 2d λ
q
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
Number of modes
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. The number of modes have been plotted for both TE and TM polarization for the symmetric case. The lines are equal for both polarizations.
We made matlab scripts to investigate the behavior of the dielectric slab waveguide. These scripts can be 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 our material. We can conclude from figure 3.5 that the waveguide core needs to be around 200 nm thick for the waveguide to be single-mode in the symmetric case. Coupling light into such a thin waveguide core would be very difficult in our setup, because the diameter of the focal point of the laser beam is 2µm 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. When we insert the cutoff condition 3.4 into equations 3.2 and 3.3 to get the number of modes, we get an equal expression 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 much thicker 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
Thickness in nm
maximum thickness minimum thickness
Figure 3.6: The refractive index of the cladding n1 versus the thickness of the waveguide core for where it is 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
Refractive index of cladding n1
Thickness in nm
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 symmetric situation. 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 in figure 3.6. This has been done to investigate wether a single-mode waveguide with a core thickness of several micrometers would be possible with different refractive indices. We define the minimum thickness as the core thickness for which the waveguide goes from zero to one supported mode. We define the maximum thickness as the core thickness for which the waveguide goes from one to two supported modes. We can see from this plot that there is no minimum thickness for which the symmetric dielectric waveguide has no modes, as expected. We can conclude that a symmetric slab waveguide always has at least one TM and TE mode. The maximum thickness increases rapidly when the refractive index of the core approaches (n0) the refractive 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 cladding material has a refractive index close to that of the core. A possible disadvantage of a small difference in refractive index is that the light in the waveguide is less confined. We can conclude that the refractive index of 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 θiin figure 3.2) versus the refractive index of the cladding. The angle of incidence can be calculated from the mode angle or critical angle using Snell’s law and yields the following expression:
θi= sin−1n0
n 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 of incidence from the mode angle and from the critical angle. The angle of incidence from the critical angle is called the acceptance angle and is usually used as an approximation. The acceptance angle can be used to approximate the angle of incidence because the mode angle arises from the ray picture with very thin rays and we have a Gaussian intensity distribution that is not negligibly thin. The acceptance and incidence angles are imaginary between n1= 2.8 and n1≈ 3.34 in our situation. This is because this angle of incidence exceeds 90 degrees because we get total internal reflection at the entrance of our waveguide. The angle of incidence becomes smaller as the index of refraction n1of the cladding approaches n0of the core, so we need to find a suitable angle we can achieve.
The acceptance angle from the critical angle is equal to the angle of incidence calculated from the minimum thickness line. This follows from the self-consistency condition (equation 3.2). If we evaluate the self-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= n1
n0
(3.12)
3.4 3.42 3.44 3.46 3.48 3.5 3.52 3.54 3.56 3.58
0 10 20 30 40 50 60 70 80 90
Angle of incidence versus n1
Refractive index of cladding n1
Angle of incidence
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 calculated from the critical angle are shown in this figure. The refractive index n0=3.55 and n1=nclad has been varied between 3.4 and 3.55, our area of interest. The acceptance angle exceeds 90 degrees before n1 = 3.41 and the angle of incidence from the critical angle exceeds 90 degrees before n1 = 3.37. Note that the angle of incidence from the minimum thickness line is equal to the acceptance angle. The refractive index n outside the waveguide is 1.
3.2 Asymmetric dielectric waveguides
An asymmetric dielectric slab waveguide is identical to the symmetric dielectric slab waveguide with one difference: the refractive indices of the cladding materials are different, as shown in figure 3.2. This causes an unequal phase change for both boundary reflections for light propagating in the waveguide. Another change is that the bounce angles of the modes in an asymmetric waveguide also need to be greater than both critical angles. We choose the biggest critical angle as the limiting critical angle for the single-mode cutoff condition. An advantage of the asymmetric setup is that one cladding can have a high refractive index so higher modes leak out and the other cladding can have a low refractive index for higher optical confinement.
In general we assume n0> n1> n2for the derivations, which means that n1 is the limiting refractive index and 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 for symmetric dielectric slab waveguides (see equation 3.2 ). Instead of two identical phase changes for the reflection, we now have a different phase change for each reflection. The self-consistency condition for transverse electric (TE) polarization is:
2π
λ n0d cos θm− mπ = tan−1
q
sin2θm−nn1
0
2
cos θm
+ tan−1
q
sin2θm−nn2
0
2
cos θm
(3.13)
The self-consistency condition for transverse magnetic (TM) modes in the asymmetric situation is:
2π
λ n0d cos θm− mπ = tan−1
n20 n21
q
sin2θm−nn1
0
2
cos θm
+ tan−1
n20 n22
q
sin2θm−nn2
0
2
cos θm
(3.14)
3.2.2 Single-mode condition
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 cutoff angle. Only the phase change of the boundary we choose for our limiting critical angle will disappear when 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 critical angle 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 condition when we insert the cutoff condition:
M .
= 1
π Vc− tan−1 s
n21− n22 n20− n21
!
(3.15)
Where:
Vc= πd λ
q
n20− n21 (3.16)
And in the case of TM polarization:
M .
= 1
π Vc− tan−1 s n20
n22
n21− n22 n20− 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 TE mode exists.
Another difference between the symmetric slab waveguide and the asymmetric slab is that no analytic solution 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 TE polarization because the right side of the self-consistency condition did disappear.
3.2.3 Thickness versus refractive index
0 100 200 300 400 500 600
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Thickness in nm
Number of modes
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 polarization has 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 thickness from single to multi-mode is different in the asymmetric situation we plotted the number of modes versus thickness again in figure 3.9 for the asymmetric situation. There is a visible difference in the transition thicknesses of TM and TE polarization. This is due to the cutoff condition. The cutoff condition only cancels out the part of the phase change in the self-consistency conditions 3.13 and 3.14 of the boundary we choose to set up the cutoff condition. The resulting terms are slightly different for TM and TE as demonstrated in section 3.2.2. We can see that TM polarization has a slightly higher transition thickness than 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 both TE and TM modes is shown in figure 3.10. This has been done to investigate whether a high maximum thickness could still be achieved when only one of the refractive indices was brought close to the refractive index of the core while the other was kept fixed. The small difference in thickness between TM and TE modes is also visible in the plot. This difference becomes smaller when n2 approaches n0 due to the TM constant nn0
2 from 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 index yields the highest thickness.
3.4 3.42 3.44 3.46 3.48 3.5 3.52 3.54 3.56 3.58 0
500 1000 1500 2000 2500 3000 3500 4000
Thickness versus refractive index n1 for a single mode
n1
Thickness in nm
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-mode waveguide. The dashed lines are for TM polarization and the solid lines are for the TE polarization. The fixed 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 3.6
0 100 200 300 400 500 600
Thickness versus refractive index n2 for a single mode
n2
Thickness in nm
Maximum thickness line for TE Minimum thickness line for TE Maximum thickness line for TM Minimum thickness line for TM
Figure 3.11: The refractive index of the non limiting cladding n2 has been varied in this plot. The dashed lines are for the TM polarization and the solid lines are for the TE polarization. The fixed refractive indices are n0=3.55 and n1=3.4
We can see from figure 3.11 and figure 3.10 that the limiting refractive index n1needs to approach that of the core as closely as possible, while non-limiting the refractive index n2 needs to be as low as possible. This means the waveguide needs to be as asymmetric as possible. Keeping index n2 low would also yield better optical confinement as seen in figure 4.5 in chapter 4. Note that the plot has no meaning beyond n2 = 3.4 because n2 > n1. Refractive index n2 becomes the limiting refractive index while it is still analytically treated 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
Thickness in nm
Maximum thickness line for TE Minimum thickness line for TE Maximum thickness line for TM Minimum thickness line for TM
Figure 3.12: The refractive index of the non limiting cladding n2has been varied in this plot while refractive index n1has been kept very close to the refractive index of the cladding. The dashed lines are for the TM polarization and the solid lines are for the TE polarization. The fixed refractive indices are n0=3.55 and n1=3.54
To investigate whether the situation remains the same if the limiting refractive index n1 is very close to the refractive index n0 of the core, figure 3.12 has been plotted. We can see that the thicknesses are much higher due to the non-limiting refractive index n1being close to the refractive index of the core. We can still see that the maximum thickness collapses when n2gets very close to n1and a lower refractive index yields a slightly higher maximum thickness. We could use no top cladding for the sample instead of an AlGaAs layer 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−1n0
n sin (π/2 − θm)
(3.18) This angle is called the acceptance angle when θm = θc. The acceptance angle approaches the angle of incidence near the area of n1 where the waveguide is single-mode. This is why most literature neglect the difference between the acceptance angle and angle of incidence from the mode angle because the mode angle arises 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 2 micrometers is very small, which could make coupling light in more difficult. We could look at 3D-waveguides next 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 3.58 0
10 20 30 40 50 60 70 80 90
Asymmetric situation: n1 versus angle of incidence for TE modes
n1
Angle of incidence
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
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 approximate method has been used because analytic solutions are not possible. We will first look at approximate methods for the simplest 3D waveguide: the rectangular dielectric waveguide. We can no longer ignore the width of the waveguide as we did with slab waveguides. The precision of the approximate methods is therefore more precise as the aspect ration of the waveguide becomes wider.
4.1 Marcatilli’s method
n 0 n 2 n 4
n 3 n 1
y x
d
a
Figure 4.1: A frontal view of the rectangular dielectric waveguide. The index of the core is n0and the other refractive indices n1− n4 are the cladding. The fields outside the corners are neglected or simplified in the approximate methods (shaded in blue).
The Marcatilli method treats the rectangular 3D dielectric waveguide as two dielectric slab waveguides. It neglects the fields outside the corners as seen in blue in figure 4.1. This method assumes that light will either only bounce in the x-direction or only in the y-direction. We look at the symmetric situation where the cladding is of the same refractive index (n1= n2= n3= n4) first.
18
4.1.1 Polarization
We use Ex and Ey polarization instead of TM and TE polarization. Ex modes have no electric field in the y-direction. Ey modes have no z-component of the magnetic field. This has been chosen because we can now 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 electric field in the Y-direction and q is the number of peaks in the electric field in the y-direction. We denote the modes as Expq and Eypq. 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 Ey11and E11x modes, which have p = 1 and q = 1. These modes have one electric field peak in both directions and thus yield a Gaussian intensity distribution in the symmetric 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 notation is Epqx instead of Expq. 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 been done in several books like the one written by Okamoto [2]. The solution for the Exmodes yields the solution for 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 difference is in the notation: The self-consistency conditions have been expressed in terms of the wave number k and the components kx, ky and kz.
We now need to satisfy both self-consistency conditions to yield a guided mode. The Self-consistency conditions for Ex modes are:
kxa = (p − 1)π
2 + tan−1 n21γx
n20kx
(4.1)
kyd = (p − 1)π
2 + tan−1 γy ky
(4.2) Where:
γx2= k2(n21− n20) − k2x (4.3) γy2= k2(n21− n20) − ky2 (4.4) The self-consistency conditions are the same as for TM and TE modes for the slab waveguide. The mode propagation constant is given by:
β = kz= k2n21− (k2x+ ky2) (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 γx kx
(4.6)
kyd = (p − 1)π
2 + tan−1 n21γy
n20ky
(4.7) Where the width is determined by the self-consistency condition of TE modes for slab waveguides and the height 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 width should not have an effect on the required height as long as the self-consistency conditions are satisfied. We can also use the solutions for the asymmetric dielectric slab waveguide for the situation where the refractive indices of the cladding are not equal (n1 6= n2 6= n36= n4) by dividing the rectangular waveguide into two asymmetric slab waveguides.
4.1.3 Single-mode condition
We want to find a refractive index n1for which the wave is single-mode. Because we changed the polarization such that the field components consists of purely horizontal and vertical parts, we can apply the exact same theory as section 3.2.2 for the same cladding symmetric or section 3.1.3 for unequal claddings. We can use the same cutoff condition and insert it into the self-consistency condition. We then get equation 3.10 for the number 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 means that both the height and width cannot be arbitrary chosen. We plotted the curves for the maximum width and 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
Refractive index of cladding n1
thickness nm
Maximum width/height Minimum width/height
Figure 4.3: The maximum width and height for a 3D rectangular waveguide calculated using the Marcatilli method. The refractive index of the core n0= 3.55 and the index n1of the cladding has been varied.
We can conclude that the required refractive indices for a height and width of several micrometers do not 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 can also be used to solve slightly more complicated waveguides, but we did not complete this analysis due to time constraints.
4.2 Rib waveguides
n 0 n 1
y x
H a
n 2
h t
Figure 4.4: A rib waveguide. The top and bottom cladding both have refractive index n1. The thickness of the top cladding is t. The center of the waveguide core has a height H and width a. The sides of the waveguide 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 a core of several micrometers thick [7]. Because modes higher than the first mode leak out over distance, the difference in refractive index of the core and the cladding does not have to be as small as in a rectangular waveguide.
A rib waveguide is made by etching away part of the sides of the waveguide core. A layer can be deposited on top of the core or the top can simply be left open. A substrate with refractive index n2 serves as the lower cladding. Higher order modes leak out over a distance of a few micrometers and only the fundamental mode is confined when the parameters for this waveguide have been chosen correctly. This is not so trivial to calculate using the Marcatilli or effective index methods. The waveguide core of a rib waveguide could be made thicker compared to a rectangular waveguide using the same difference in refractive indices. The refractive index of the cladding could also be further away from the refractive index of the core when keeping the core thickness the same as in a rectangular waveguide. The advantage is that either light could be coupled in easier. The waveguide production is also easier because the waveguide can still be single-mode even if the refractive index is slightly further away than the very small difference required between core and cladding for a rectangular waveguide.
We decided to use the given parameters from the paper [7] to try to simulate such a waveguide in COMSOL Multiphysics. We started by designing an asymmetric 2D slab waveguide to compare with the plots we made earlier.
Simulating the waveguide in COMSOL could also give us insight into how fast the modes above cutoff actually leak away so we can be sure our waveguide is truly single-mode with our parameters. We can use COMSOL 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 this picture. The refractive index of the core is 3.5. The upper cladding has a refractive index of 2.8 and the lower cladding has a refractive index of 3.49
The z-component of the electric field of an asymmetric dielectric slab waveguide has been plotted in figure 4.5. We can see that it matches our expectations: the field at the lower refractive index cladding is better 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 is shown 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 plotted in figure. We can see that this is not the Ex11 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 would expect with a large difference in refractive indices of the core and cladding. Due to time constraints and limited knowledge of how the COMSOL mode solver exactly gets its results, we were unable to simulate a this waveguide successfully. Because we could not successfully simulate the rectangular waveguide, we also did not have time to simulate the rib waveguide. Note that refractive indices in figure 4.6 were chosen to be much 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 learn more about simulating 3D-waveguides in COMSOL for future research. Another program is WMM mode solver. This program is specifically made for solving waveguide properties. Because you have to compile this program yourself from C++ code and the import the date into MATLAB to plot it, the program is not easy to use. This program can be found in reference [10].
We investigated the possibility to integrate a single-mode GaAs waveguide into experiments on optical transitions in GaAs in a cryogenic setup. The core needed to be several micrometers thick due to the spot size of the focal point of our laser beam in the experimental setup. We were limited to an AlGaAs cladding to 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-mode to multi-mode, of the waveguide core was 200 nm thick for a core refractive index of 2.8. Because this was too thin to couple light into, we varied the refractive indices of both claddings to find out the required refractive index for a core thickness of several micrometers. We concluded that the refractive index of the cladding should be close to the cladding of the core. Next, we looked at the asymmetric situation, where the refractive index of one cladding was varied. We deduced that the waveguide should be as asymmetric as possible to achieve the maximum thickness. Leaving out the top cladding yields the most asymmetric situation 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 indices for 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 ridge waveguides. 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 a waveguide that is easier to integrate into the setup. A possible way to learn more about the COMSOL mode solver could be done by studying the examples and tutorials in references [8] and [9].
24
[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). URL http://www.sciencedirect.com/science/article/pii/S1293255808000757. E-MRS symposium N and 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). URL http://scitation.aip.org/content/aip/journal/rsi/82/4/10.1063/1.3574217.
[5] Josell7. Wikimedia commons: Refraction reflection. URL
https://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
26
Matlab scripts
Script for the number of modes graphs:
% Acquire initial values n2=2.8; % AlAs Cladding n0=3.55; % Sample n1=3.4; % AlAs Substrate dmin=1; % min thickness in nm dmax=600; % Max thickness in nm la=817; % Wavelength of light in nm
% Edited initial values
d=dmin:1:dmax; % Thickness in nm
lambda=2*pi/la*(d/2)*sqrt(n0ˆ2-n1ˆ2); % normalized frequency
% Calculate numer of modes for TE and round them to the next integer M1=(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 integer M=(1/pi)*(2*lambda-atan(sqrt((n0/n2)ˆ2*(n1ˆ2-n2ˆ2)/(n0ˆ2-n1ˆ2))));
MTM=ceil(M);
mdiff=MTE-MTM;
%plot number of TE modes figure
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 the symmetric situation for both TM and TE modes:
clear all
% Acquire initial values stepsize=10ˆ(-3);
n0=3.55; % Sample
la=817; % Wavelength of light in nm
n2min=3.4; %minimum index of refraction of n2
n2max=n0-stepsize; %maximum index of refraction of n2
% Edited initial values dista=[]; % Define data arrays
27
distamin=[]; % min data array nair=[];
%loop through indices n for n2=n2min:stepsize:n2max
%Then loop through every distance for every index n dista1=[]; %
for d=0:4000
vn=2*pi/la*(d/2)*sqrt(n0ˆ2-n2ˆ2); % normalized frequency
% Calculate numer of modes for TE and round them to the next integer M1=(1/pi)*(2*vn);
MTE=ceil(M1);
% And then check if it is single mode if MTE==1
%if so, hold data dista1(end+1,:)=[n2,d];
end
end
%Check maximum value and minimum of d of this loop through d for fixed N
%value
if isempty(dista1)==0 dista(end+1,:)=max(dista1);
distamin(end+1,:)=min(dista1);
end end
%plot number of TE modes f=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 values
stepsize=10ˆ-3; %stepsize of refractive indices n0=3.55; % Sample
n1=2.8; % AlAs cladding
la=817; % Wavelength of light in nm n2min=3.4;
n2max=n0-stepsize;
% Edited initial values dista=[]; %
nair=[];
%loop through indices n for n2=n2min:stepsize:n2max
%Then loop through every distance for every index n for d=0:4000
vn=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 integer M1=(1/pi)*(2*vn-atan(sqrt((n0/n1)ˆ2*(n2ˆ2-n1ˆ2)/(n0ˆ2-n2ˆ2))));
MTE=ceil(M1);
% And then check if it is single mode if MTE==1
%if so, record data dista(end+1)=d;
nair(end+1)=n2;
end
end end
% Join array nair and dista to edit them M=[nair;dista]';
%Split matrix M in matching nair values in array of cell arrays A
A = arrayfun(@(x) M(M(:,1) == x, :), unique(M(:,1)), 'uniformoutput', false);
% Get length of A lengtha=size(A);
% Creat empty data matrices data=[];
datamin=[];
% extract useful data from each submatrix from A for i=1:lengtha
% Extract every matrix from A L=cell2mat(A(i));
% Add the max values of every submatrix L of A to data data(end+1,:)=max(L);
% And do the same for the min values datamin(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 values stepsize=10ˆ-3;
n0=3.55; % Sample
n1=2.80; % AlAs cladding
la=817; % Wavelength of light in nm
n2min=3.4; %minimum index of refraction of n2
n2max=n0-stepsize; %maximum index of refraction of n2
% Edited initial values dista=[]; % Define data arrays distamin=[]; % min data array nair=[];
%loop through indices n for n2=n2min:stepsize:n2max
%Then loop through every distance for every index n
dista1=[]; %
for d=0:4000
vn=2*pi/la*(d/2)*sqrt(n0ˆ2-n2ˆ2); % normalized frequency
% Calculate numer of modes for TE and round them to the next integer M1=(1/pi)*(2*vn-atan(sqrt(abs((n2ˆ2-n1ˆ2)/(n0ˆ2-n2ˆ2)))));
MTE=ceil(M1);
% And then check if it is single mode if MTE==1
%if so, hold data dista1(end+1,:)=[n2,d];
end
end
%Check maximum value and minimum of d of this loop through d for fixed N
%value
if isempty(dista1)==0 dista(end+1,:)=max(dista1);
distamin(end+1,:)=min(dista1);
end end
%plot number of TE modes f=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.eps hold on
And the script for calculating the mode angle versus the refractive index of the core in the symmetric situation:
%Edited 7-5-15 clear all
%n2 limiting cladding
% Acquire initial values n0=3.55; % Sample
la=817; % Wavelength of light in nm
n2min=3.4; %minimum index of refraction of n2 n2max=n0; %maximum index of refraction of n2
% Edited initial values dista=[]; % Define data arrays distamin=[]; % min data array nair=[];
angle max=[];
angle min=[];
bounce angles=[];
y=[];
ymin=[];
ac=[];
%loop through indices n for n2=n2min:0.001:n2max
%Then loop through every distance for every index n dista1=[]; %
for d=0:10000
vn=2/la*d*sqrt(n0ˆ2-n2ˆ2); % normalized frequency
% Calculate numer of modes for TE and round them to the next integer MTE=ceil(vn);
% And then check if it is single mode if MTE==1
%if so, hold data dista1(end+1,:)=[n2,d];
end
end
%Check maximum value and minimum of d of this loop through d for fixed N
%value
if isempty(dista1)==0 dista(end+1,:)=max(dista1);
distamin(end+1,:)=min(dista1);
end end
% Check self consistency condition to find the angle y of the mode
% Loops through every index of d for d=1:length(dista(:,2))
syms x;
%Retrieve mode parameters D = dista(d,2); %max thickness Dmin = distamin(d,2); %min thickness n2=dista(d,1); %refractive index tc=asin(n2/n0); %critical angle
x0=[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 F = 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 line ymin = vpasolve(Fmin,x,x0);
bounce angles(end+1,:)=[Dmin,ymin];
n=1; %index of refraction of material in front of sample/core
y=asin(n0/n*sin(pi/2-y)); %Get the angle of incidence from the mode angle ymin=asin(n0/n*sin(pi/2-ymin)); %Get the angle of incidence from the mode angle
%Fill the arrays with this data angle max(end+1,:)=[n2,y];
angle min(end+1,:)=[n2,ymin];
%And now use the critical angle
yc=asin(n0/n*sin(pi/2-tc)); %Get the angle of incidence from the critical angle ac(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 asymmetric situation:
%Edited 8-6-15 clear all
%n2 limiting cladding
% Acquire initial values n0=3.55; % Sample
la=817; % Wavelength of light in nm n1=2.80; % AlAs Substrate
n=1; %index of refraction of material in front of sample/core
n2min=3.40; %minimum index of refraction of n2 n2max=n0; %maximum index of refraction of n2
% Edited initial values dista=[]; % Define data arrays distamin=[]; % min data array nair=[]; % Declare empty arrays angle max=[];
angle min=[];
bounce angles=[];
y=[];
ymin=[];
ac=[];
%loop through indices n for n2=n2min:0.001:n2max
%Then loop through every distance for every index n dista1=[]; %
for d=0:10000
vn=2*pi/la*(d/2)*sqrt(n0ˆ2-n2ˆ2); % normalized frequency
% Calculate numer of modes for TE and round them to the next integer M1=(1/pi)*(2*vn-atan(sqrt(abs((n2ˆ2-n1ˆ2)/(n0ˆ2-n2ˆ2)))));
MTE=ceil(M1);
% And then check if it is single mode if MTE==1
%if so, hold data dista1(end+1,:)=[n2,d];
end
end
%Check maximum value and minimum of d of this loop through d for fixed N
%value
if isempty(dista1)==0 dista(end+1,:)=max(dista1);
distamin(end+1,:)=min(dista1);
end end
% Check self consistency condition to find the angle y of the mode
% Loops through every index of d for d=1:length(dista(:,2))
syms x;
%Retrieve mode parameters D = dista(d,2); %max thickness Dmin = distamin(d,2); %min thickness n2=dista(d,1); %refractive index tc=asin(n2/n0); %critical angle
x0=[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
