A Study of a Silicon Quasi-Periodic Segmented Waveguide

Abstract This paper presents a study of a segmented waveguide where the layers are distributed following the Fibonacci sequence and aims to diminish the scattering losses usually caused by imperfections in the core boundaries of the waveguide. This phenomenon normally happens in structures which present a high contrast between the refractive index of the core and the refractive index of the clad. The analyses were obtained through observations of wave fields that propagates along the waveguide, how the effective refractive index (neff) varies along the z-direction and the study of its normalized power. The simulation of this photonic component was accomplished with the help of the Vectorial Beam Propagation Method (VBPM).


I. INTRODUCTION
Optical waveguide works confining the light through a contrast between the refractive index (n) of the core and the clad of the structure, where first has a n higher than its clad.When there is a high contrast in a continuous waveguide, it usually occurs scattering in the core-cladding boundary imperfections [1].
Over time, several methods of reducing these losses have been developed.One of the ways found to avoid scattering losses was try not to use waveguides with square core sections, thus having the width of the base larger than the side walls, but this resulted in only good effects for the Transverse Electrical (TE) mode, practically canceling the Transverse Magnetic configurations (TM) [2]- [4].
Another structure also used is the photonic crystal, but this structure model usually presents a photon bandgap interval in the operating frequency [5].Another problem associated with photonic crystals is their high losses when curved [6], greatly limiting their use in some systems.
With time was developed a new type of dielectric waveguide, the segmented waveguide.This structure presents an alternation of the material of its core, where the two material have different refractive index, as shown in Figure 1.Normally n 1 > n 2 and n 2 is equal or higher than the refractive index of the clad.
Some applications of the segmented waveguides are to diminish the scattering [2], frequency filters [7], frequency demultiplexer [8], polarization filter and in second harmonic generation [9].
From the study related to the segmentation of the cores of the optical guides, a new type of guide, the subwavelength grating waveguide (SWG), was presented in [2], where its segmentation tries to introduce a grid step smaller than the Bragg condition, which enables the dispersion to decrease and the nucleus to behave as if it were continuous [10].The SWG exhibits a mode delocalization from the core, which leads to less scattering loss in the boundary between the clad and the core, as well as allowing the control of the effective refractive index without the need to add other materials [10].
This work presents a new waveguide configuration, based on the SWG model, in which it has a quasiperiodic grid, in this case following the Fibonacci sequence.For the realization of the simulations carried out in this work, it was used the Vectorial Beam Propagation Method (VBPM) [27].It is important to highlight that this is the first time that this numerical method is used to simulate and analyze this type of segmented waveguide.
This work is organized in the following way: in Section II is presented the VBPM mathematical modeling and the project of the proposed waveguide with its parameters; the results are showed in Section III; and finally in Section IV are presented the conclusions.

A. Vectorial Beam Propagation Method (VBPM)
The VBPM is a numerical method that aggregates the Finite Element Method (FEM) with the Beam Propagation Method (BPM) by performing a vector analysis of the fields, making use of the firstorder Padé approximation [28], and the use of perfectly matched layers (PML) to avoid undesirable reflections.Its formulation is achieved by first combining Maxwell's equations to obtain (1), which is known as the Helmholtz equation.
In ( 1), k 0 is the wave number in free space and k = 1/ ε, with ε being the relative permittivity and, where εT represents an arbitrary transverse tensor, and ûz is associated to the direction z.
Another important information is that the ∇ operator is defined as: where α x , α y and α z are the parameters relative to the PMLs, and are calculated based on the works [29] and [30].
After some algebraic manipulations, and considering that the field varies slowly with each step in the z-direction [27], we develop the Helmholtz equation, using the Garlekin method [28], to arrive at (4): where Ω is the domain, ∂Ω includes all the interfaces and external contours, and the tensors can be defined as: Discretizing (4) using the finite element method, introducing the PMLs parameters directly in the formulation through operator ∇ T , as shown in (3), we achieve the equation ( 6): where { ⃗ h T } represents a column vector containing unknown values of h xj and h yj , 0 is a null column vector, and [M ] and [K] are called global matrices defined in [27].The next step of the formulation is the application of the non-paraxial approximation of the padé type of order (1,1) [28] in (6), producing the matrix (7): where ).Finally, if we apply the θ-finite-difference marching scheme in equation (7), we obtain the equation (8) as follows: where ∆z is the propagation step along the z-direction and θ(0 ≤ θ ≤ 1) is introduced to control the stability of the method.When θ = 0.5, we have a correspondent Crank-Nicholson algorithm [27].
Since the material of waveguide adopt in this paper change along the z-direction, the effective refractive index was calculated in every propagation step (∆z), using the (9): where † represents the conjugated and transposed complex.

B. Quasi-Periodic Segmented Waveguide
For this work it was proposed a silicon waveguide, where the substrate is silica (SiO 2 ) and the upper cladding is SU-8 polymer.This waveguide is based on the one proposed in [2], but instead of the use of a periodic segmentation, it is proposed a quasi-periodic segmentation, in this case using the Fibonacci sequence.The Fig. 2 shows the transversal model of the waveguide.The waveguide structure is based on a sequence of blocks, where the first two blocks of the sequence are A and B, and the next blocks are formed by a superposition of A and B following a rule inspired by the Fibonacci sequence, so that the nth generation of its sequence is obtained by (10).
For the (10) n > 2, besides S 1 = A and S 2 = B.The following are the first generations of the Fibonacci sequence: For the calculation of each width of the layers, the theory shown in [2], in which the authors claim to have achieved better propagation results for periods of less than half the effective wavelength, was take into account.Based on this work, it was developed calculations of the width of the A with B together, as shown in (12).
Next, it was calculated the separated width of the layers A and B with the help of ( 13) and (14).
Where L A and L B are the width of the layers A and B, C A and C B are the number of layers A and B for the number of the chosen generation of the Fibonacci sequence, and C T is the total number of layers.The proportion used in ( 13) and ( 14) were adopted after several tests.
The model used was based on the schematic of Fig. 3, in which A and B are the layers of the core that will follow the Fibonacci sequence.The parameters for this simulation are presented in the Table I.In addition, it was used 2380 finite elements and a propagation step of 0.01µm, during a length of 0.39µm.

III. NUMERICAL RESULTS
For this segmented waveguide, it was used the fourth generation of the Fibonacci sequence, which means that it will have seven layers.It was chosen this generation because it was already possible to see the wanted effect in the model of the fields of the wave.
The structure was excited with the fundamental mode obtained from modal analyses, considering several wavelengths and different layers of refractive index, distributed according to the Fibonacci sequence and quasi-periodic in the propagation direction.
In the Fig. 4, it is presented the distribution of the magnetic field when z = 0µm, while in Fig. 5 it is shown the distribution of the magnetic field in the final stage of the simulation (z = 0.39µm).It is possible to observe in Fig. 5, that the field expands, concentrating less power in the core interface with the clad, reducing the loss for scattering in this region of the waveguide.Other important propriety that can be observed in the analyses of the results, is that in theses types of waveguides, the effective refraction index (n ef f ) tends to decrease as propagation occurs, similar of what happens in the periodic SWG.This can be attributed to the small, but existing, insertion losses that occur along the z-direction, due to constant changes in the refractive index of the layers that form the core.The Fig. 6 shows the evolution of the effective refractive index of the quasi-periodic SWG in comparison with the periodic one.Since the proposed grid is still smaller than the Bragg grating, the segmented waveguide will have a behavior similar of a continuous waveguide.The Fig. 7 shows the constancy of the component H y of the magnetic field propagating along the guide.
Comparing the loss of a wave at λ = 1.55µm, along the waveguide, of the proposed segmented device with the periodic SWG, it can be observed in the Fig. 8 that power of the quasi-periodic waveguide decreases less than last one.Comparing the Fig. 6 with the Fig. 8, it can be noticed that the power along the quasi-periodic waveguide maintain higher than the SWG, because the length that the higher effective index occurs in the proposed device is longer than the usual segmented waveguide.This was already intended when the equations ( 13) and ( 14) were developed.
Since the quasi-periodic waveguide will perpetuate this structural characteristic along its longitudinal extension, it can be affirmed that the power behaviour of the device will continue the same along its all length.Adding that the field of the mode was already expanded in this propagation period, as it shown in Fig. 5, it does not display the need of a longer propagation length.
The Fig. 9 shows the normalized power for different wavelengths for the proposed SWG.It can  An interesting characteristic of this quasi-periodic segmented waveguide is that H x component gains power along the propagation direction.The Fig. 10 shows this behaviour for different wavelengths.The Fig. 11 shown the normalized power, stored in the Hx and Hy fields along the propagation direction.The energy remains confined in the magnetic field polarized in the y direction, proving that

n 3 Fig. 1 .
Fig. 1.Example of a segmented waveguide proposed in this work.

Fig. 2 .
Fig. 2. Used model of the Silicon waveguide

Fig. 4 .
Fig. 4. Mode distribution of the field in the beginning of the waveguide.

Fig. 5 .
Fig. 5. Mode distribution of the field in the end of the waveguide.

Fig. 9 .
Fig. 9. Hy component normalized power for different wavelengths for the proposed SWG.

Fig. 10 .
Fig. 10.Hx component normalized power for different wavelengths for the proposed SWG.

TABLE I .
Parameters of the model to be analyzed.Journal of Microwaves, Optoelectronics and Electromagnetic Applications, Vol.22, No. 2, June 2023 DOI: http://dx.doi.org/10.1590/2179-10742023v22i2267287273 All simulations were developed in program language FORTRAN, and the graphs were plotted with the help of PYTHON program language.
Journal of Microwaves, Optoelectronics and Electromagnetic Applications, Vol.22, No. 2, June 2023 DOI: http://dx.doi.org/10.1590/2179-10742023v22i2267287274 Journal of Microwaves, Optoelectronics and Electromagnetic Applications, Vol.22, No. 2, June 2023 DOI: http://dx.doi.org/10.1590/2179-10742023v22i2267287276 be observed that the smaller the wavelength, greater it will be the losses.This happens because the frequency of the transmitted wave is getting closer to the band gap of the waveguide.