1. Introduction

Coupling power loss arising due to waveguide width mismatches (or discrepancy in modal cross-sectional area) should be addressed in order to successfully convey optical data through different parts of the photonic chip. Connecting waveguides with different widths via butt-coupling method produces large back-reflection of light [1]. Almost perfect transmission of light can be achieved if an adiabatic transition region with a longer taper length is implemented [2]. When the ratio of waveguide widths is large then extremely long intermediate coupler region is required for an adiabatic coupling. Consequently, complete configuration turns out to be a bulky structure.

Dielectric waveguides composed of core and cladding with high-refractive index contrast can guide light waves efficiently and strongly. However, combining waveguides with different sizes causes transmission power losses due to Fresnel reflections. There have been various studies conducted in order to achieve efficient coupling strategies. In Ref. 3, planar version of a parabolic mirror called J-coupler is designed and fabricated to couple light from an input waveguide into a single-line-defect photonic crystal (PC) waveguide. In another study, light transport from PC waveguide to dielectric waveguide was achieved by using tapered coupler [4]. Also, coupling from ridge waveguide to PC waveguide and a compact mode converter have been studied in Refs. 5 and 6. In these studies, authors performed three dimensional (3D) electromagnetic analyses. Moreover, the insertion losses were reduced by etching holes with varying dimensions into the planar waveguides. However, the fabrication issues due to varying dimensions of the etched holes should be precisely controlled. To overcome fabrication difficulties in another study [7], holes are intentionally introduced to obtain a tapered section for the coupler. Besides, alignment tolerances and different type of tapered regions have been analyzed. In order to integrate single-mode optical fiber with PC structure nonuniform-shape PC taper was proposed [8]. The influences of the tapering parameters on coupling efficiency were also explored in the same study.

Recently, an experimental study has been demonstrated to achieve low-loss coupling between fiber and nano-photonic silicon waveguide by using an integrated Luneburg lens [9]. The working principle of Luneburg lens [10] is that all the incoming parallel waves of light that hits the lens on one side are curved by the structure and then, brought to a single point on the other side. This means that when light waves come from one side then they will all be tightly focused by the specified lens at the other side of the structure. Operating in the reverse direction, a point source at the interface turns out to be collimated after traversing the Luneburg lens. However, if there are curved surfaces between the two planar waveguides, the designed structure may be susceptible to alignment problem and coupling losses. In a similar way, due to spherical (3D)/cylindrical (2D) shape of the Luneburg lens, coupling and misalignment losses may occur between input/output waveguide and the lens that may reduce the coupling efficiency at the interfaces.

Graded index (GRIN) medium based upon artificially engineered dielectric materials has gathered great promise due to the variety of promising applications in photonics and optics [11–22]. It is feasible to design GRIN medium if the parameters of the two dimensional PCs are rearranged appropriately. These structures are known as GRIN PCs. The use of GRIN PCs can be an approach for tackling the coupling problem. Considering GRIN PC’s focusing characteristic and mimicking ability to continuous media make such structures a powerful tool for the coupler design. There are some methods that can be followed to generate GRIN PC medium. One method is based on the arrangement of the radii of the dielectric rods or air holes so that filling factor can be varied [23]. Another one is to adjust the size of unit cells such that distances between the adjacent rods are arranged appropriately [24]. Finally, infiltration of air holes with different substances can be implemented to have GRIN medium. We used a novel type of two dimensional (2D) GRIN lens coupler whose dielectric distribution is adapted to cross-section of half of the Maxwell fish-eye (HMFE) lens distribution. The reason for the selection of a semi-circular HMFE lens is that it focuses all the incoming parallel rays on a single point along the semi-circular region just as Luneburg lens does (an example of the generalized Luneburg lens is the Maxwell's fish-eye lens). Moreover, the coupling region can be shortened in HMFE case since the half of Maxwell fish-eye lens is required to collimate wide beam spot size into a narrower one. We also implemented a 2D GRIN PC lens to mimic the designed coupling section by using continuous GRIN medium. For that reason, the locations of dielectric rods with constant radii are modified according to continuous dielectric distribution. Effective medium theory (EMT) that is valid at the long wavelength region has been applied to attain the appropriate index distribution. The moderate coupling performance of GRIN PC medium has been boosted by moving the operating frequency to upper values. The front and back surfaces of GRIN PC are coated by a single column of rods that act as an antireflection coating (ARC) by reducing unwanted back-reflections [25].

The coupling problem occurring due to connecting waveguides with different widths causes transmission losses. To address that problem, we designed a modified Maxwell fish-eye (MMFE) lens as an intermediate section between different-sized waveguides to achieve efficient and compact coupling. Broadband performance was observed while connecting waveguides with core sizes of 21a and 3a. We also imitated the continuous lens by adjusting the distances between the dielectric rods that compose the photonic crystals. The planar GRIN PC coupler designed in the present work has index variation that is dependent on both the transverse and longitudinal coordinates. Full-width at half-maximum (FWHM) of intensity profiles obtained for continuous medium and GRIN PC cases are compared. High coupling efficiency with a broad bandwidth is obtained. The effects of angular and lateral misalignments to the coupling of light are investigated and the drop in the coupling efficiency is studied. Efficient, compact and planar coupling method based on the circular cross-section of Maxwell fish-eye lens may steer the design of compact couplers in photonic integrated circuits.

This paper is organized as follows: In Section 2, the coupling problem between two planar waveguides is defined and the coupling performances of tapered and butt couplers are compared. Next, a novel type of coupler design based on HMFE lens index distribution is proposed in Section 3. Then, the comparison between continuous MMFE lens and its approximation by GRIN PC configuration is outlined in terms of their coupling efficiencies in Section 4. After, the effects of misalignments due to laterally shifted and tilted output waveguide are investigated and the variations in the coupling efficiency depending on these misalignments are studied in Section 5. Finally, we summarize the results that we found and conclude our study in Section 6.

2. Problem definition: interface between two dielectric waveguides

The objective of the study is to target a coupler problem in optics by merging GRIN concept (modified Maxwell fish eye represents such a case) with photonic crystals. The proposed solution should satisfy the following features: compact, efficient, broadband and lateral/angular misalignment tolerant. The operating frequency interval belongs to higher frequencies that should stay above the homogenization regime.

Efficient coupling of light between different waveguides in integrated optical circuits has been considered as a challenging issue. We have step-index waveguide configurations for both input and output ports. The core is surrounded by air. There should be an intermediate region that efficiently connects two waveguides with different cross-sectional area. However, strong back-reflections occur when light couples from wider waveguide to a narrower one. Therefore, in order to eliminate reflection losses a focusing structure should be designed at the intermediate region. In the opposite case where a narrower waveguide couples to a wider one, a strong diffraction occurs. In order to suppress diffraction losses, the incident beam should be collimated and then, coupled to a wider waveguide properly. Hence, manipulating the incident beam profile either by narrowing or broadening becomes a mandatory task to effectively transfer and distribute optical data within the optical device. Otherwise, substantial power losses may disrupt the inter-channel communication. In literature, in order to overcome the coupling problem that occurs when wider waveguide feeds the narrower one, two fundamental types of coupling strategies, i.e. coherent and direct coupling, have been proposed. To explore coherent coupling in optical devices propagation constant matching method has been applied. On the other hand, in order to directly couple the incident light mode profile matching for different types of optical transporters has been employed [26]. In the design of optical integrated connecting devices, different types of couplers such as butt-, tapered, adiabatic and J-couplers have been widely investigated [3, 11, 27–30]. As we show later on the study, two of these approaches (butt-and tapered coupling) have some serious drawbacks.

In order to present and compare coupling performances of tapered and butt-couplers, we employ 2D finite difference time domain (FDTD) method [31, 32]. In order to eliminate reflections originating from the ends of the finite computational window, the boundaries are surrounded by perfectly matched layers (PMLs) [33]. The schematics and structural parameters of the butt-coupler and tapered coupler are shown in Figs. 1(a) and 1(b). The time-domain snap shots of the source for two different structures are also indicated in the same figure. The mathematical form of the input source used to excite the waveguide is as follows: $E_z(y,t)=E_0\exp (-{(y-y_0)}^2/2\Delta y^2)\exp (-{(t-t_0)}^2/2\Delta t^2).$The fields are captured after source excites the input waveguide and then beam propagates reaching the front of the output waveguide.

 

Fig. 1 Schematics of (a) butt-coupler and (b) tapered coupler are shown. Important structural parameters are denoted by W_in, W_out and L. (c) Calculated transmission efficiencies of tapered coupler and butt-coupler. Shaded region represents the interested frequency interval.

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We assume only TM polarization so that non-zero electric and magnetic fields are E_z, H_x, and H_y, respectively. For two types of couplers, the widths of input and output waveguides are denoted by W_in and W_out, respectively. As can be deduced from the same figure, back reflections and diffraction at waveguide ends will become inevitable in the butt-coupling case. In addition, a large portion of light leaks into the free-space. In the case of tapered coupler as illustrated in Fig. 1(b), the amount of back-reflection and light leakage at the interface is slightly reduced. The normalized coupling efficiencies of the coupling structures introduced in Figs. 1(a) and 1(b) are calculated in Fig. 1(c). Coupling efficiency of butt-coupler is around 35% whereas the efficiency of tapered coupler reaches up to 55%. In spite of a better coupling performance compared to the butt-coupling case, the length of the tapered coupler region L should be considerably enlarged in order to achieve higher coupling efficiencies [1]. In other words, in order to match coupling to radiation modes, greater taper length is needed. In optical interconnect designs compact solutions are usually preferred. Thus, aforementioned tapered coupling solution is definitely not a desirable option for on-chip optical interconnect.

In this study, a novel type of coupling structure is proposed to enhance the coupling performance in the butt-coupling case. The designed coupler is a dielectric medium whose refractive index gradually varies along both the optical axis (x-direction) the transverse to propagation direction (y-direction). The 2D index distribution has been adapted by inspiring from the inhomogeneous dielectric profile of the Maxwell fish-eye lens that was proposed by James Clerk Maxwell in 1854 [10, 34]. Furthermore, the 2D index profile of the designed continuous medium is imitated by rearranging the positions PC rods. The design details will be explained in the next section.

3. Design of the coupler structure: modified Maxwell fish-eye lens approach

The generalized refractive index distribution for the Maxwell's fish-eye lens is formulated as

We have adapted Maxwell fish-eye profile into designed coupler in two dimensions (along the x- and y-axes) and modified its form to accomplish efficient light coupling. The core refractive index value is fixed to n = 3.46. In our application, the computational domain is concerned only in 2D spatial domain and the third dimension is taken to be uniform. Therefore, a two dimensional index profile is studied to discuss the in-plane coupling. For this purpose, an equation depending on the longitudinal x- and transverse y-directions is derived to design Modified Maxwell fish-eye (MMFE) lens as a coupler such that;

 

Fig. 2 (a) Schematic representation of coupling strategy used as a modified Maxwell fish-eye lens at the intermediate stage. The values of W_in, L_c, and W_out are 21a, 16.5a and 3a, respectively (b) 2D refractive index distribution of the continuous MMF coupler.

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The corresponding dielectric constants of the input and output waveguide are equal to n = 3.46. Region-II in Fig. 2(a) is the intermediate section and represents the “coupling region” with a length of L_c = 16.5a. Regarding 2D index distribution of the intermediate region is shown in Fig. 2(b). We can see in Figs. 2(a) and 2(b) that the gradient of the index distribution increases along the x-direction. That means for smaller values of x, the index difference between maximum and minimum values is smaller and that difference increases as we move along the x-direction. Therefore, incident light having a large beam aperture can be strongly focused in front of the output waveguide and an efficient coupling to the narrower waveguide can be accomplished.

Although we expect a high coupling performance by the non-homogeneous MMFE coupler, several challenging issues about its implementation appear that can be described: such as fabrication difficulties during diffusion process, the need of planar faces and limitation of index gradient (usually smaller gradient occurs). In order to overcome these difficulties, the use of PC for the approximation of continuous MMFE index profile is considered. For this purpose, the same coupler structure has been imitated by dielectric PC rods with radii r = 0.20a. The dielectric rods have refractive index value of 3.13, which is the index value of alumina rods. Refractive index profile has been adjusted by changing the distances between each rod along the y-direction. We put some restrictions on the displacement of the dielectric rods. Since diameter of each rod is 0.40a, the minimum separation between the two rods can be the case where they are touching each other. We added an additional value so that the minimum distance becomes ${\Delta }_y=0.44a,$increases up to 1.50a. After setting the distribution of rods for one column, the next one is slightly modified so that minimum index value is further reduced (consequently the index gradient increases).

Figure 3 describes the calculation steps to determine the effective index distribution. The dispersion curves of the first band for different sizes of PC unit cells are shown in Fig. 3(a). While keeping the rods’ radii constant, when the unit cell size increases, the bands move to higher frequencies. The group index of each band$N_g$is calculated by using the slope information of the relevant band present in Fig. 3(a) and obtained group index variation plot, which is illustrated in Fig. 3(b). For longer wavelengths (lower frequencies), group index curves are closely spaced and thus provides slight group indices variation. However, when we move towards the edges (cutoff region) the dispersive effect provides nonlinear behavior. Each curve enters the cutoff region at different frequencies and strong dispersion occurs around these regimes. We denoted three selected intervals. The first two in the low frequency region have linear characteristics and the latter one shows nonlinear variation with respect to the changes in the unit cells size. We decided the required cell sizes for coupler design by inspecting the group index graph, which is represented in Fig. 3(b).

 

Fig. 3 (a) The dispersion curves corresponding to the first band are shown. (b) Group index dependency of each dispersion bands shown in (a). The three colored rectangles demonstrate different frequency regions where group index variations exhibit distinct characteristics.

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The GRIN structure based on PC is first designed for a normalized frequency 0.18. Effective medium theory that is valid at the long wavelength region has been applied to attain the appropriate index profile. After obtaining the desired refractive index distribution, the same medium is subject to broad band spectral analysis.

The intermediate coupling section is divided into eight regions to achieve strong focusing. In each region, index profiles rely on vertical y-dimension and are arranged as follows:

Table 1. The values of the adjustable parameters at each distinct eight regions.

View Table

In Fig. 4(a) , the schematic view of proposed PC based MMFE coupler that placed in the intermediate region is presented. The intermediate coupling section consists of eight distinct regions whose index distributions are arranged according to the design parameter k_i as demonstrated in Table 1. The parameter k_i determines focusing effect of each region. That means spatial separation of the rods in the lateral y-direction becomes larger as we move along the propagation direction. Moreover, as we pass from region 1 to region 8, the gradient of effective index distribution gets larger and hence, strong focusing is observed. The reason why Region 8 is wider than other regions is one of the optimization issues in order to achieve a better coupling performance. Since the gradient of effective index distribution in Region 8 is higher, enlarging the length$L_d$of Region 8 improves the focusing capability of designed coupling structure. Thus, the incident beam can be coupled efficiently from input waveguide to the output waveguide.

 

Fig. 4 (a) Schematic representation of the complete structure, i.e., input waveguide, coupler section, output waveguide. Intermediate region represents approximation of the MMFE lens by dielectric rods. (b) The index profiles of the specified eight regions are presented.

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Si-based waveguide regions are denoted by I in Fig. 4(a) and have the same widths as mentioned in Fig. 2(a). The index value of homogenous input and output waveguides is equal to n = 3.46. GRIN PC based coupling region in Fig. 4(a) has the length of L_d = 22a, whose effective index distribution varies from the maximum value 3.46 (at the center) to 1.98 (at the edges) in Region-1. Then, the minimum effective index value $\min (n_{eff})$ continues to drop down to 1.31 in Region-8, as tabulated in Table 1. Corresponding to each region, the effective index profiles in terms of y-axis are superimposed in Fig. 4(b). It is important to note that two dimensional index profile of PC MMFE coupler shown in Fig. 4(b) resembles the continuous MMFE coupler index distribution shown in Fig. 2(b). The gradient increase of the index distributions along the propagation direction is closely similar for both cases. In order to diminish the Fresnel reflections at the interfaces, additional PC rods with radii r = 0.10a are placed at the front and back sides of PC coupler. These layers act as antireflection coatings (ARCs) [25]. It is important to briefly explain the selections rules for the design parameters of the coupler. The main design parameters are the selection of rods' positions and the rod numbers within a column. The targeted refractive index profile governs the positions of rods. The number of rods determines the accuracy of the approximated continuous index profile. Input waveguide width dictates the transverse size of the discrete medium. The length of the artificially designed GRIN medium is decided by inspecting the focal point of the structure.

In order to compare the coupling effectiveness of continuous and GRIN PC MMFE designs, two-dimensional FDTD analyses have been conducted. A source with a Gaussian profile is launched to the input waveguide and a detector is located at the output waveguide. In numerical simulations we only employ transverse magnetic (TM) guided mode where the concerned non-zero electric and magnetic field components are E_z, H_x, and H_y, respectively. The coupling efficiencies are calculated and normalized. The investigation of transmission spectrum in the inset of Fig. 5 reveals that there are higher transmission regions as well as forbidden gaps for PC based MMFE coupler case. The transmission efficiency of the lower frequencies is around 60% as shown in Fig. 5 as an inset. The transmission band enters a cutoff region at a normalized frequency ωa/2πc = 0.20. The relatively weak focusing (larger spot size at the focal point) and leakage of light in the transverse direction (due to finite boundary of the GRIN PC medium) cause some transmission power losses. Approximating continuous Maxwell fish-eye index distribution with PC provides additional flexibility to our design. Afterwards, when we investigate the spectral characteristic of the structure we realize that local band gap closes at around normalized frequency ωa/2πc = 0.40 and the second transmission window appears. Since the wavelength gets smaller at this region it is expected that strong focusing hence better coupling efficiency can be achieved.

 

Fig. 5 Calculated transmission efficiencies of the discrete graded-index (D-GRIN) and continuous graded-index (C-GRIN) modified Maxwell fish-eye lenses.

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The zoomed version of transmission efficiency at higher frequencies is presented in Fig. 5. While the proposed continuous MMFE coupler couples almost all the incident power into the narrower waveguide, the approximated MMFE coupler by PC exhibits over 75% coupling efficiency in the frequency range of [0.44-0.53](a/λ) which is represented by shaded region in Fig. 5. The maximum efficiency value reaches up to 90%. The PC based coupler operates with a frequency bandwidth of Δω = 18.56% when the coupling efficiency exceeds 75%. We should note that there are no strong oscillations in the transmission spectrum due to ARCs placed at the front and back sides of PC coupler.

Another superiority of PC based MMFE can be remarked such that continuous GRIN medium has no frequency selectivity property. As a result, all the incident light independent of wavelength can be focused albeit the changes that may occur in terms of the focal point location and spot size of the focused beam. When we introduce discrete dielectric distributions, the medium becomes highly frequency dependent. The transmission window is surrounded by local band gaps. That feature can be exploited such that undesired spectrum of wave can be blocked and only certain frequency region of interest is allowed to propagate.

When we make a comparison with the literature in terms of reported efficiency values, we see the following observations. Before providing the comparison though we should note that each approach has its unique cases such as the waveguide type (index-guided vs. photonic crystal waveguide) and waveguide width ratio differences. However, it will be a useful guidance to collect the reported numbers of previously published studies.

Prather and his colleagues in Ref [3]. have been achieved higher coupling efficiency of over 90% by using a designed J-coupler. In another work, a coupling structure by means of a PC Mikaelian lens has been modeled and fabricated on silicon-on-insulator in Ref [36], where simulation results reported that the corresponding coupling efficiency is around 73%. Using on-waveguide diffraction gratings the Gaussian beam has been coupled from single mode fiber in to a PC waveguide is achieved in Ref [37]. Coupling efficiency for cases with and without mirror layer structure experimentally obtained to be 35% and 57%, respectively. On the other hand, the coupling device on similar concept with different grating period and taper was implemented and the reported values were approximately 33% in Ref [38]. Alternative solution using canonical refractive lenses in micro-dimensions has also been demonstrated in Ref [39]. An estimated coupling efficiency was around 90%. Comparing between inverse taper and aforementioned coupler shows that the latter improved the coupling efficiency by 6.50 dB. Finally, gradually alternating the period and radii of the holes in PC creates a lens medium. Ref [40]. achieved a coupling efficiency of 55%. Coupler design by using GRIN concept is also employed in Ref [9], where Luneburg-mediated coupler is designed and fabricated. Cakmak et al. in Ref [12]. use GRIN PC structure in front of a narrow photonic crystal waveguide in order to improve coupling efficiency. They report an improvement of 6.35 dB numerically and 5 dB experimentally in the microwave regime. Considering all these reported values, we have obtained broadband and efficient coupling method by implementing an alternative configuration that merges the two concepts, GRIN PC and Maxwell fish-eye together.

If we convert the normalized parameters of the continuous and discrete coupling structure into the physical quantities the following values appear. The operating wavelength can be fixed at an optical wavelength λ₀ = 1550 nm. Consequently, the lattice constant can be determined by taking the normalized center frequency of f_cen = 0.475(a/λ) that attain maximum transmission efficiency. In this case the lattice constant and input/output waveguides widths become a = 736.25nm, (W_in, W_out) = (15.46μm, 2.21μm), respectively. Similarly, the lengths of the continuous and discrete coupling structure are equal to L_c = 12.14μm and L_d = 16.19μm, respectively. The available bandwidth with efficiency greater than 75% is 284 nm.

4. Performance comparison of continuous and approximated GRIN PC MMFE couplers

There are important steps to be followed if an efficient light coupling is required between two planar dielectric waveguides. The location and spot size of the focused light should be carefully determined if a lens type of optical element is decided to be implemented. Strong focusing in front of the narrower waveguide may induce severe diffraction. Hence, output waveguide should be slightly wider than the spot size of the focused beam. If a narrow beam is created inside the output waveguide, diffraction causes power loss due to unsatisfied total internal reflection condition. Because, when light hits the boundary with a large incident angle there will be no confinement mechanism.

Figure 6 provides spatial intensity profiles of the continuous MMFE coupler for both frequency intervals (lower and upper) in Figs. 6(a) and 6(c) respectively. Field distributions in three different regions (input, coupler, and output) can be observed from the plots at two selected representative frequencies. We have extracted cross-sectional intensity profiles in propagation direction at two different locations, one is taken at the input side and the second one is positioned at the front of the output waveguide. The cross-sectional intensity profiles of continuous MMFE coupler are shown in Figs. 6(b) and 6(d), respectively. The coupler at lower frequency $\omega a/2\pi c=0.10$creates a beam spot size that is equal to 1.12a as can be seen in Figs. 6(a) and 6(b). The spatial intensity profile for upper frequency $\omega a/2\pi c=0.475$ and its cross-section in the propagation direction are illustrated in Figs. 6(c) and 6(d), respectively. The corresponding beam spot size is calculated as 0.40a, which can be observed in Fig. 6(d).

 

Fig. 6 (a) Spatial intensity profile of the continuous MMFE coupling at $\omega a/2\pi c=0.10$. (b) Cross-sectional intensity profiles at the input and output predefined locations (CS_in and CS_out). (c) Spatial intensity profile of the continuous MMFE coupling at $\omega a/2\pi c=0.475.$ (d) Corresponding cross-sectional intensity profiles for upper frequency band at the same locations.

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It is possible to manipulate PC based GRIN structure such that rearranged rods’ position may provide larger gradient in the index distribution. Consequently, higher coupling efficiency can be obtained. Another way to further improve the coupling is to engineer interfaces that act as an anti-reflection coating and it has been implemented by aligning rods with radii r = 0.1a in y-direction in front of the output and at the back of the input waveguides. Rod locations of the input and output ARC structures have the same locations of rods with Region-1 and Region-8 in Fig. 4(a), respectively.

The focusing and coupling performance of PC MMFE medium is tested first in the low frequency region. A pulse with a center frequency at 0.10 is launched and it is traced while propagating. Figure 7(a) shows spatial intensity profile of the light. The field’s transverse profile in front of the output waveguide is plotted and compared with the input source depicted in Fig. 7(b). The proposed graded PC MMFE medium transforms input beam whose FWHM is 6.42a into a narrower beam with a spot size of 3.32a. Even though, there is a focusing characteristic due to graded medium, the spot size of the field is still large. Therefore, some amount of light escapes without being guided in the output channel as seen in Fig. 7(a). Considering the fact that output waveguide is narrower than 3.32a, then, it is desired to focus light even in a narrower spot. As mentioned before EMT is valid at the long wavelength region, i.e. at the first band in dispersion diagram. However, we expect that increasing the operating frequency by moving outside of the first band in the dispersion diagram depicted in Fig. 3(a), it is possible to obtain a spectral window such that light can penetrate the PC MMFE medium and still focusing behavior can be realized. Smaller wavelength will sense the structural modification even more and effective index gradient of the GRIN PC medium may increase. When we calculate the spatial intensity distribution at a normalized frequency ωa/2πc = 0.475, we immediately realize strong light focusing and efficient coupling of light to output waveguide. Due to increase in the frequency, light pulse is strongly modulated by the GRIN PC as can be seen in Fig. 7(c). The cross-sectional views of electric field intensities at two positions (input and in front of the output waveguide) points out that strong spot size conversion is accomplished as shown in Fig. 7(d). Moreover, corresponding FWHM is equal to 1.35a which is 2.46 times narrower than that one in lower frequency. As a result, we can claim that our expectation for working at higher frequencies is consistent with the obtained results.

 

Fig. 7 (a) Spatial intensity profile of the GRIN PC MMFE coupling at $\omega a/2\pi c=0.10.$ (b) Cross-sectional intensity profiles at the input and output predefined locations. (c) Spatial intensity profile of the GRIN PC MMF coupling at $\omega a/2\pi c=0.475.$ (d) Cross-sectional intensity profiles at the input and output predefined locations.

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The continuous MMFE medium should have the best performance in terms of light coupling (providing larger spot-size conversion ratio and minimizing of coupling loss). This can be seen in Fig. 6(c). Smooth transitions between each interface allow light efficiently coupling to output channel. The cross-sectional views at the input and output locations of the coupler imply that the largest spot size conversion ratio (6.42a/0.41a) is the reminiscent of strong focusing capability of continuous MMFE presented in Fig. 6(d). When we move from continuous MMFE to GRIN PC MMFE it is expected that frequency region may contain both allowed and blocked intervals due to wavelength selective refractive index modulation of the medium.

Even though the waveguide is assumed to be Si, the artificial coupler region can be made of any type of dielectric material such as Si rods as long as it is transparent at the operating frequency interval. Artificial sequencing of one type of dielectric rods gives rise to effective refractive index profile that accommodates both smaller and larger index values than the rods' refractive indices. As a result, one may implement same type of GRIN profile by selecting another type of dielectric material. If an experimental effort is planned to perform for the coupler in the microwave domain then alumina rods with refractive indices of 3.13 can be used.

Three different naming are used throughout the paper, “refractive index” for continuous GRIN medium, “effective refractive index” for an approximated GRIN medium, and “group index” for photonic crystals. The ultimate goal is to finely approximate continuous GRIN medium with a discrete version that is represented by effective refractive index profile.

We designed the structure based on effective medium theory and tested it both in lower and higher frequency regions (see Fig. 5 and its inset) and realized that the designed structure appropriately operates as a GRIN medium for the second transmission window as well. We have strong indication of such a behavior. By just comparing Figs. 6 and 7 one can see that the coupler acts as a GRIN medium for two different normalized frequencies, 0.10 vs. 0.475, respectively.

5. Exceptional cases in coupling design: lateral and angular misalignments

We have explored the consequences of two most probable imperfections that may be encountered while performing waveguide coupling. The misalignment tolerance of the coupler is studied by laterally shifting output waveguide along the y-direction. The shift amount represented by $\delta$ is increased step by step and in each case the power flux is calculated. The result is shown in Fig. 8 . When the shift amount is small ($\delta =0.25a=184$nm) the coupling efficiency slightly drops and Fabry-Perot oscillations appear in the transmission spectra. Moreover, as shift amount increases oscillation amplitudes become larger. If Fabry-Perot oscillations are strong then the “power uniformity” can be deteriorated and may affect negatively coupling performance of the structure. The slight decrease in the amount of power continues until the shift increases up to $\delta$ = 1.0a = 736.25 nm. There is a large drop when lateral shift is increased beyond that value. 50% efficiency level occurs when the lateral movement becomes 1.50a (~1.1$\mu m$). Interestingly, that value is half of the output waveguide width.

 

Fig. 8 Coupling efficiency variation curves of the designed discrete GRIN coupler under the misalignment occurring in lateral direction are presented. Lateral offset is schematically shown as an inset.

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Another possible scenario that can be considered as a deviation from the ideal set up is the angular misalignment. If there is a tilt angle between the optical axes of designed PC MMF coupler and output waveguide, then it is expected that power transmission efficiency will drop from the maximum value. To explore and quantify the coupling efficiency under the angular misalignment, we prepare Fig. 9 . For small values of the deviated angles, coupling efficiency slightly drops. That trend continues until the angular misalignment reaches 4°. After that (when angle value becomes 5° and larger) we witness large drop in coupling efficiency. Even at this 5° angular misalignment, the transmission efficiency is at around 70%. We can conclude that the designed structure is fairly resistant to angularly misaligned output waveguide. It is observed that both lateral and angular misalignments show resistance against coupling loss up to a certain value. Moving beyond these critical values deteriorates the efficiency of coupling.

 

Fig. 9 Coupling efficiency changes of the designed discrete GRIN coupler under the angularly misaligned cases. Angular misalignment is schematically shown as an inset.

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We have studied optical power coupling efficiency between two planar dielectric waveguides. However, similar procedures can be followed to directly couple a source that emits spatially wide beam to a narrow waveguide. Graded PC coupler is created by spatially varying the positions of dielectric rods and keeping the other parameters (radius, refractive index etc.) constant. In order to improve coupling efficiency we have used index distribution of the modified version of Maxwell fish-eye lens as a coupler. It transforms spatially wide optical mode into a tightly focused one at the output channel. The proposed solution does not employ any type of resonant structure. As a result, high coupling is achieved for broad bandwidth. There is a strong field focusing for both cases. However, graded PC coupler produces spot size that is slightly bigger than continuous GRIN case. There are small side lobes as well. That in turn gives rise to reduction on the overall transmission efficiency.

6. Conclusion

The coupling problem occurring due to connecting waveguides with different width causes power loss. To address that problem, we designed a modified Maxwell fish-eye lens as an intermediate section between different sized waveguides to achieve efficient and compact coupling. Broadband performance was observed while connecting waveguides with core sizes of 21a and 3a. We also imitated the continuous lens by adjusting the distances between the dielectric rods that compose the photonic crystals. The planar GRIN lens designed in the present work has index variation that is dependent on both the transverse and longitudinal coordinates. Full-width at half-maximum of intensity profiles obtained with different cases are compared. High transmission efficiency has been obtained with a broad bandwidth. The effect of misalignment and tilted output waveguide case are investigated and the drop in the coupling efficiency is studied. Efficient, compact and planar coupling method based on the idea of Maxwell fish-eye may provide some guidance to design other photonic circuits.