Design and Modeling of the Photonic Crystal Waveguide Structure for Heat-Assisted Magnetic Recording

)e application of the photonic crystal (PC) waveguide (WG) as the light delivery system in the heat-assisted magnetic recording (HAMR) system is demonstrated. )e structure consists of a 90° bending PC waveguide and a ridge dielectric waveguide taper coupler. )ree-dimensional (3D) models of structures are built and simulated in order to determine light coupling and transmission efficiencies. Geometric parameters including the taper length (LTP), coupler inlet width (WFW), and PC waveguide width (WWG) are investigated. )e initial simulation shows that the transmission efficiency of over 90% can be achieved with the coupler integrated with the straight PC waveguide. However, the overall transmission efficiency is substantially reduced to 53.8% when the coupler is attached to the 90° bending PC waveguide. Our analysis shows that the wave mode matching and light decay rate in the waveguide cavity are significant contributing factors. )e transmission efficiency increases to around 60.8% after some modification of the bending region.


Introduction
Photonic crystals (PCs) are periodical optical micro-and nanostructures that receive increasing attention due to their ability to manipulate light propagation while maintaining high transmission efficiency.
Coupling is an important issue when it comes to the WG integration with other components of an optical communication device.Several taper PC coupling structures were investigated [20][21][22][23].ese studies agree on one common fact that the light propagation depends largely on the geometries of the coupler and guiding structure.
e capability of the PC to manipulate light as described above suggests the use of the PC structure as a potential candidate for light delivery in the heat-assisted magnetic recording (HAMR) system [24].Several HAMR light delivery systems have been proposed, for example, the structure consisting of grating, slab waveguide, and near-field transducer (NFT) [24], the fiber optic connected with a spot-size converter (SSC) [25], and the butt-grating structure associated with multilayer core and cladding [26].ese proposed light deliveries, however, suffer from power losses at different locations and produce less than 50% transmission efficiency.
In this paper, the photonic crystal waveguide for the light delivery in HAMR is demonstrated.e proposed structure shown in Figure 1 replaces the conventional structure [24] by combining grating, slab waveguide, and fixed curve mirror into a single piece through utilizing of coupling and bending sections.Note that the NFT is not included in the proposed design.e PC waveguide is attached at the front of the slider.e light path can be guided directly to the waveguide via the coupler at the inlet, and the xed curve mirror is thus not required [24,27,28].e light is then guided via a 90 °bend and propagates in the straight guiding section to the output.e 3-dimensional (3D) design of the 2D PC waveguide is carried out using the basic Photonic Band Gap (PBG) software [29].e modeling is performed with the widely used nite element method (FEM) simulator COMSOL Multiphysics [30].e performances of the waveguide including transmission and coupling e ciencies through bending and coupling sections are investigated.

Photonic Band Gap (PBG)
e 2D photonic crystal structure is the dielectric constant periodically modulated in two dimensions.In this paper, our PC waveguide is operated at 1,550 nm [31], and our PC

2
Advances in Materials Science and Engineering waveguide is constructed using a 2D triangular lattice array of air holes embedded in polysilicon (p-Si).At this wavelength, the frequency-dependent refractive index of p-Si is 3.476 [9]. is structure is preferred since its band gap contains a larger overlapping area than other PC lattices and can be associated with both transverse electric (TE) and transverse magnetic (TM) modes.A simple laser source can thus be utilized without the need for a mode selector.Moreover, silicon has high refractive index and exhibits low loss at the wavelength of interest [32].Note that if the NFT is included in the design, the waveguide must possess a certain polarization as most NFTs are polarization speci c.
e PBG diagram is created by calculating each PBG, the forbidden wavelength range for propagation, associated with a speci ed ratio of air hole radius (r) and lattice constant (a).
e consolidation of band gaps can be depicted by the gap map shown in Figure 2(a).e selected ratio of radius and lattice space constant (r/a) is 0.43, a relatively low value in the overlapping area known as the complete band [6].A lower r/a ratio is desirable in order to prevent structure collapse during fabrication.e calculated photonic band diagram of the lattice with r/a 0.43 is shown in Figure 2(b).In this gure, the normalized frequency is plotted versus the wave vector into the lattice (Γ, M, and K). e calculations show that the band gap width is about 49.45% (from 0.276 to 0.458) for the TE mode and 5.12% (from 0.385 to 0.406) for the TM mode, respectively.An overlapping area with the center normalized frequency (ωa/2πc) is chosen at 0.395.e radius of the air hole (r) and lattice space constant (a) are determined by the following relation: where ω is the angular frequency (rad/s), c is the light velocity (m/s), and λ is the wavelength (m).erefore, the air hole radius (r) and the lattice space constant (a) are 263 and 612 nm, respectively.

Design and Modeling of the PC Waveguide for HAMR
e PC waveguide is designed as a slab waveguide with coupling and bending sections.e 3D model and light path are shown in Figures 3 and 4, respectively.e coupling e ciencies associated with di erent coupling structures will be determined rst.After the integration of the bending section, they will be reevaluated along with the transmission e ciencies.
e length of the PC waveguide is about 24.48 (40a) µm, and there are 21 rows in the air hole array.e PC guiding path is created by removing a center row.e slab thickness (t) of the PC waveguide is chosen at 0.4a or around 245 nm.
is thickness value is determined following the relation between the gap size and the PC thickness described in [33].
e taper section is formed by a conventional ridge dielectric slab waveguide whose thickness can be expressed as follows [34,35]: where h is the dielectric taper slab thickness (m) and n 1 is the core material (silicon) refractive index.At the operating wavelength of 1,550 nm, the thickness of the taper slab is around 245 nm (t/a 0.4), similar to the thickness of the PC slab.Advances in Materials Science and Engineering 3

Ridge Dielectric Taper Coupler Integrated with the Straight
PC Waveguide Structure.We perform an initial simulation to observe light coupling into the straight PC waveguide from a ridge dielectric taper coupler shown in Figure 5. e taper coupler is an inlet width converter that is employed to convert the large inlet to the smaller one.e ridge dielectric waveguide taper structure, in particular, is simple and easy to fabricate [36].
e coupler inlet width (W FW ) and the taper length (L TP ) are varied from 5 to 30 µm.Note that the initial coupler inlet width of 5 µm is selected based on the actual size of a typical fiber optic outlet.A parametric study is performed to observe the dependence of the coupling efficiency on the coupler inlet width (W FW ) and taper length (L TP ).Two sizes of the PC waveguide inlet width (W WG ) are considered which are 0.8a√3 and 1.0a√3 (about 0.848 and 1.060 µm).Advances in Materials Science and Engineering e former is selected to demonstrate the extreme case of coupling.e PC waveguide length (L WG ) is 24.48 µm (40a), and the waveguide slab thickness is 0.245 µm (0.4a).However, the computed power distribution is displayed as the net power.In order to accurately determine the coupling e ciency or the transmission e ciency over the coupler, values of incoming and re ected powers are needed.e coupling e ciency can be calculated from power density integration over the cross-sectional plane at the input (P in P source + P re ection ) and output (P out P transmission ) ports of the coupler.e transmission T is determined by T 2P out /(P in + P out ) P transmission /P source , whereby we assume P source P re ection + P transmission [37].P out is the surface integration of power densities over the cross-sectional plane at the output port (point B in Figure 5), and P in is the surface integration of power densities over the cross-sectional plane at the input port (point A in Figure 5).e input power is set at 1 mW for convenience.Note that these formulas can be applied to any kind of waveguiding structures.
Figure 6 shows the coupling e ciency pro les with W WG equal to 0.8a√3 and 1.0a√3, respectively.It can be seen that the coupling e ciency decreases as the taper length increases.At W WG 0.8a√3, the maximum e ciency of 95.3% is achieved with W FW 10 µm and L TP 5 µm, as shown in Figure 6(a).e power density distribution associated with peak coupling e ciency is depicted on the right.e coupling e ciencies obtained from other structures are much lower and stay between 30 and 45%.
e results are however di erent for W WG 1.0a√3 as shown in Figure 6(b).It is observed that there are 3 structures that can achieve over 90% coupling e ciency, that is, W FW 15 µm and L TP 5 µm, W FW 25 µm and L TP 5 µm, and W FW 30 µm and L TP 5 µm, respectively.e smaller the L TP , the better the coupler e ciency.At higher taper lengths, the bell-shaped pro les are observed.In our case, the good matching condition appears at smaller taper lengths.e power density distribution of the W FW 25 µm and L TP 5 µm structure is shown on the right of Figure 6(b).Compared to the W WG 0.8a√3 PC structure, the W WG 1.0a√3 structure provides better overall coupling e ciency.It is believed that a certain amount of power loss occurs from a re ection at the inlet of the PC waveguide.From the PC design standpoint, a bending section with a narrow inlet is di cult to construct.erefore, the W WG 1.0a√3 structure is selected for further study.

Ridge Dielectric Taper Coupler Integrated with the
Bending PC Waveguide Structure.
e ridge dielectric taper coupler is integrated with the 90 °bending PC slab waveguide to form the light delivery structure of HAMR as shown in Figure 7. e dimension of the 90 °bending PC slab waveguide includes 50 µm of width (w) and 55 µm of length (l).Unfortunately, we cannot enlarge the waveguide dimension due to processing limitations.A number of periodic structures in the horizontal and vertical planes are 40 and 80 periods, respectively.
e transmission e ciencies are determined at 3 locations, shown as points B, C, and D in Figure 7. ese efciencies are de ned by ratios of power ows at the locations of interest and powers at the inputs.e coupling e ciency, transmission e ciency after the bending section, and the overall waveguide transmission e ciency are expressed by CT (%) Referring to Figure 7, P A is the power over the crosssectional surface at point A (Watt), P B is the power over the cross-sectional surface at point B (Watt), P C is the power over the cross-sectional surface at point C (Watt), and P D is the power over the cross-sectional surface at point D (Watt).
Figure 8(a) shows the plot of the taper coupling e ciency (CT) where W FW is varied from 5 to 30 µm and L TP is varied from 2 to 30 µm.All structures exhibit bell-shaped e ciency pro les as a result of re ection and scattering in the guiding bodies at certain geometries.ese ndings indicate the strong dependence of the light transmission on the geometry.e highest CT of 97.2% is achieved with the W FW 15 µm and L TP 5 µm structure.However, the highest overall e ciency is achieved with the W FW 5 µm and L TP 10 µm structure.Figures 8(b) and 8(c) show the plots of the bending transmission e ciency (CB) and overall transmission e ciency (CO), respectively.We observe that, for every case, the efciency decreases along the waveguide; moreover, it drops substantially after the bending section.e maximum overall e ciency is achieved at around 53.8% with the W FW 5 µm and L TP 10 µm structure.
e coupling behavior of the proposed structure is analyzed as follows.According to the coupled-mode theory [38], the wave mode pro les of the ridge dielectric taper waveguide and PC waveguide must be matched in order to optimize the transmission.Some researchers have reported applications of the taper waveguide as a wave mode converter [39][40][41].We then calculate the wave mode pro les displayed in terms of electric eld distributions of a single piece of the ridge dielectric taper waveguide with W FW and L TP both varying from 5 to 30 µm, with a 5 µm increment, followed by those of a single-piece PC waveguide.Note that the wave mode pro les are determined at the outlet of the ridge taper waveguide and at the input of the PC waveguide.
Our results show that the number of wave mode pro les increases when W FW increases.
e second-order wave mode pro le of the W FW 5 µm structure is observed.In particular, we found that the wave mode at the output of the ridge taper waveguide with the W FW 5 µm and L TP 10 µm structure exhibits a strong match with that at the input of the PC waveguide, as shown in Figure 9. is result supports the highest value of coupling e ciency achieved for this particular structure.6 Advances in Materials Science and Engineering For comparison, an example of mismatched wave mode pro les is illustrated in Figure 10.
is structure has W FW 10 µm and L TP 10 µm.e wave mode pro le at the output of the ridge taper waveguide is in the third order, while that at the input of the PC waveguide is in the second order.is could explain a nearly 0% e ciency obtained in the coupling section.
Although the matching wave modes between the ridge taper waveguide and PC waveguide can be obtained, power loss in the waveguide is still realized.
e well-known concept of temporal coupled-mode theory [33,42] is then applied to analyze the optical phenomena inside the light delivery structure.We construct the equivalent diagram of the light delivery structure as shown in Figure 11.Advances in Materials Science and Engineering e light delivery structure can be separated into two regions.Region one is the ridge dielectric taper coupler waveguide (WG 1 ) connected with PC waveguide (WG 2 ).Region two is the horizontal PC waveguide (WG 2 ) connected with vertical PC waveguide (WG 3 ) to form a 90 °bending.
Region one is considered rst.When the light I 1 reaches WG 2 , some part will be re ected and some will be radiated away due to the di erence between WG 1 and WG 2 .
e re ected and transmitted waves in WG 1 are denoted as R 1 and T 1 , respectively.
According to the temporal coupled-mode theory, the cavity that connects WG 1 and WG 2 is considered as the resonator.Ideally, if the cavity resonates with a frequency (ω 0 ) equal to the operating frequency (ω) and light decays into WG 1 and WG 2 with equal lifetimes τ 1 τ 2 , 100% transmission can be achieved.Note that the resonant frequency and decay rate of the cavity-resonant mode strongly depend on geometry [33,43,44].erefore, the geometry of the cavity should be modi ed to increase the coupling between WG 1 and WG 2 .For region two, another cavity is formed between WG 2 and WG 3 .Again, to achieve 100% transmission e ciency over this section, the abovementioned conditions must be applied.
e improvement of cavity matching is then demonstrated here.Several modifying patterns of the W FW 5 µm and L TP 10 µm structure are created by randomly removing some air holes near the joint and corner.Figure 12(a) shows the location of air hole removal, where h ij is the coordinate of removed hole position at the joint between the taper and PC Resonance Resonance   It can be seen that the coupling and bending transmission efficiencies of the PC waveguide structure can actually be improved when joints and corners are modified.In our 6 best cases, MOD#5 provides the highest efficiency.Specifically, CT, CB, and CO increase from 88.6% to 97.2%, 59.8% to 72.2%, and 53.8% to 60.8%, respectively.Note that the obtained overall transmission efficiency from the MOD#5 case is better than those reported previously [24][25][26].To further improve the transmission efficiency, a systematic approach should be implemented.In addition, deformation of some air holes might be necessary [13].
Finally, the hotspot size at the output facet of the W FW � 5 µm and L TP � 10 µm structure is determined.e power distribution at the output facet of the structure is shown in Figure 14.By considering the full width half maximum (FWHM) of the power distribution profile, we obtain a spot size of about 103 nm × 413 nm. is spot size is still too large for high areal density recording.More alteration of air holes near the output or NFT integration is recommended for further confinement of the spot size.

Conclusions
is paper investigates a PC waveguide-based light delivery system for HAMR.We examined taper coupling and bending structures.e PC waveguide model is created by removing a single row of photonic crystals in a triangular lattice array of air holes in the silicon substrate.
e PC waveguide operates at 1,550 nm and can accommodate both TM-and TE-polarized sources.e lattice space (a) and air hole radius (r) are 612 and 263 nm, respectively.e dielectric coupler model is designed by varying the inlet waveguide width (W FW ) and the taper length (L TP ) from 5 to 30 μm.
e PC waveguide widths of interest are 0.8a√3 or 0.848 µm and 1.0a√3 or 1.060 μm. e dielectric taper coupler integrated with the straight PC waveguide structure shows the highest coupling efficiency of 95.3% with the W FW � 10 µm, L TP � 5 µm, and W WG � 0.8a√3 µm structure.For W WG � 1.0a√3 µm, there are 3 structures with L TP � 5 µm that achieve over 90% coupling efficiency, that is, 94.6% for W FW � 15 µm, 97.4% for W FW � 25 µm, and 90.8% for W FW � 30 µm.e 1.0a√3 µm width is chosen for integration with the 90 °bending structure to form the proposed light delivery structure since it provides larger guiding area at the bending section.
e coupling efficiencies of most structures drop slightly after attaching the bending structures.e overall highest CT is achieved at L TP � 5 µm.e bell-shaped profiles are observed in every case of W FW .ese results indicate that the effects of light reflection and scattering are altered in different guiding geometries, and the optimization of waveguide geometry is necessary.e transmission efficiency apparently decreases along the waveguide.e maximum overall efficiency of around CO � 53.8% is achieved with the W FW � 5 µm and L TP � 10 µm structure.

Advances in Materials Science and Engineering
Di erent dimensions of the ridge dielectric coupler taper waveguides produce varied output wave modes that a ect the matching to the PC waveguide.Modi cation of air holes at the joint and corner of the PC waveguide of the W FW 5 µm and L TP 10 µm structure appears to improve the e ciencies at the taper coupler section and the overall structure.e optimal coupling and overall transmission efciencies increase to 97.2% and 60.8%, respectively.ese e ciency levels are de nitely su cient, considering our simple design and fabrication [36].In addition, the value of the overall transmission e ciency is better than those reported previously.
e PC waveguide is therefore a promising candidate for the HAMR light delivery system.However, more alteration of air holes near the output or NFT integration is recommended to further con ne the spot size.

Figure 3 :Figure 4 :Figure 5 :
Figure 3: e 3D model of the PC waveguide as the HAMR light delivery.

WWW
FW = 5 m W WG = 0.8a√3 W FW = 10 m W FW = 15 m W FW = 20 m W FW = 25 m W FW = 30 m FW = 10 m and L TP = 5 m FW = 25 m and L TP = 5 m x z y Taper length (m) W FW = 5 m W FW = 10 m W FW = 15 m W FW = 20 m W FW = 25 m W FW = 30 m W WG = 1.0a√3(b)

Figure 7 :
Figure 7: e PC light delivery model with locations de ned for e ciency calculation.

WWWFigure 8 :
Figure 8: Coupling e ciency of (a) the coupler section, (b) transmission e ciency of the bending section, and (c) transmission e ciency of the entire structure.

Figure 9 :Figure 10 :
Figure 9: e wave mode pro les (E z and E y ) at (a) the output of the ridge taper waveguide and (b) the input of the PC waveguide of the W FW 5 µm and L TP 10 µm structure.

Figure 11 :
Figure 11: e equivalent diagram of the proposed light delivery structure.

Figure 12 :
Figure 12: e air hole pattern removal at the (a) joint and (b) corner.