Unidirectional Slow Light Transmission in

applied sciences Article

Unidirectional Slow Light Transmission in Heterostructure Photonic Crystal Waveguide Qiuyue Zhang 1 1 2

*

and Xun Li 1,2, *

Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China; [email protected] Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON L8S 4K1, Canada Correspondence: [email protected]; Tel.: +1-905-525-9140 (ext. 27698)

Received: 9 September 2018; Accepted: 7 October 2018; Published: 9 October 2018







Featured Application: The proposed unidirectional backscattering-immune device may provide a potential way to realize robust optical delay lines and robust on-chip optical buffers. Abstract: In conventional photonic crystal systems, extrinsic scattering resulting from random manufacturing defects or environmental changes is a major source of loss that causes performance degradation, and the backscattering loss is amplified as the group velocity slows down. In order to overcome the limitations in slow light systems, we propose a backscattering-immune slow light waveguide design. The waveguide is based on an interface between a square lattice of magneto-optical photonic crystal with precisely tailored rod radii of the first two rows and a titled 45 degrees square lattice of Alumina photonic crystal with an aligned band gap. High group indices of 77, 68, 64, and 60 with the normalized frequency bandwidths of 0.444%, 0.481%, 0.485%, and 0.491% are obtained, respectively. The corresponding normalized delay-bandwidth products remain around 0.32 for all cases, which are higher than previously reported works based on rod radius adjustment. The robustness for the edge modes against different types of interfacial defects is observed for the lack of backward propagation modes at the same frequencies as the unidirectional edge modes. Furthermore, the transmission direction can be controlled by the sign of the externally applied magnetic field normal to the plane. Keywords: delay-bandwidth products; group velocity dispersion; flat band; slow light; interfacial defects; backscattering-immune

1. Introduction The velocity of light in vacuum is approximately 2.99 × 108 m/s, fast enough to circle the Earth once in 0.3 s. However, faster is not always better, such an ultrahigh speed makes it very difficult to control the optical signals in time domain [1,2]. Slow light with a much slower group velocity makes it possible to delay and temporarily store light in all-optical systems [3–6]. Among all the approaches to realize slow light, photonic crystal waveguide (PCW) has drawn much attention as it supports room temperature operation and is compatible with on-chip integration [7–16]. However, the performance of the conventional photonic crystal (PC) systems is very vulnerable to the backscattering loss induced by disorders or defects [17,18]. The backscattering loss is amplified as the group velocity slows down in the waveguide [19,20]. The development of the photonic chiral edge state (CES) [21–24] may provide an approach to overcome the limitations related to backscattering loss in slow light systems, as it is robust against scattering from interfacial defects. The existence of the photonic CESs at the interface of a two-dimensional (2D) magneto-optical photonic crystal (MOPC) was first theoretically proposed by Appl. Sci. 2018, 8, 1858; doi:10.3390/app8101858

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Haldane [25,26] and was then experimentally verified by Wang et al. [27,28] in microwave regime. Since the CESs are leaky in the free space [21,29], a regular PC with an aligned band gap has to be bounded to the interface of the MOPC to prevent the radiation leakage. The constructed waveguide only supports unidirectional modes as it has only positive (negative) group velocities and the backscattering is completely suppressed. This enabled the light transmission to be immune to fabrication imperfections. The unidirectional transmission feature is of great significance for slow light systems and may provide robust designs for optical delay and light storage applications [30]. There are various approaches to tailor the dispersion curves to realize slow light in PC systems: modification the radii of air holes [31], shifting the air holes to specific directions [32], infiltrating optical fluid into the air holes [33], infiltrating microfluidic selectively [34,35], using asymmetric engineering structure [36], merging coupled cavities [37] and, introducing alternative rows of ellipse-holes [38]. In all the above mentioned ways, adjusting the rod radius is relatively easy to control and have less fabrication issues. In this paper, we propose a unidirectional backscattering-immune slow light waveguide. The waveguide is constructed by a square lattice of MOPC with the rod radii of the first two rows adjacent to the waveguide core are precisely tailored and a titled 45◦ square lattice of Alumina PC as an ancillary cladding. A flat band for slow light is obtained and the slope (group velocities) of the waveguide dispersion curves process only positive (or only negative) values, which is determined by the sign of the externally applied magnetic field. In the waveguide, the backscattering is completely suppressed for the lack of the backward transmission channel. This unique feature ensures that the design can avoid performance degradation induced by impurities or defects, which is difficult to realize in conventional slow light systems. The remaining content of this work is arranged as follows: to better understand the content of the subsequent sections, we briefly introduce the theoretical background of slow light in Section 2. In Section 3, we firstly introduce the design of the unidirectional slow light waveguide and then exhibit the slow light characteristics and the backscattering-immune properties of the waveguide in detail. The followed simulation results are based on the finite element method (FEM). 2. Theoretical Background The group velocity v g (or equivalently the group index n g = c/v g , c denotes the speed of light in vacuum) is not the only key parameter to evaluate the performance of slow light devices. We should also take into account the frequency bandwidth, group velocity dispersion (GVD), and the propagation loss. The frequency bandwidth and the group index are encapsulated by the normalized delay-bandwidth product (NDBP) [16,39]. The NDBP indicates the obtained compromise between a large operating bandwidth and a large group index, as given by eg × NDBP = n

∆ω , ω0

(1)

R ω +∆ω dω eg = ω 0 n n g (ω ) ∆ω is an average group index in the desired slow light bandwidth. ∆ω/ω0 denotes 0 the normalized frequency bandwidth, where ∆ω represents the difference between the maximum and the minimum frequency in the slow light region and ω0 represents the center frequency. The reduction of v g should be balanced with the bandwidth requirements in practical applications. The frequency bandwidth is considered as the frequency range corresponding to a ±10% group index variation with eg value [16,39,40]. respect to the obtained mean n The group velocity is strongly dependent on the operation frequencies [41,42], as quantified by the second-order derivative of the dispersion relation: β2 =

d2 k 1 dn g = , 2 c dω dω

(2)

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β 2 is the GVD parameter. Large β 2 can lead to pulse distortion, and thereby degrades the performance of the slow light device. Taking the practical application requirements into consideration, the GVD  

a parameter should below the order of 106 2πc [43,44] (a is the lattice constant in PC systems). 2 The propagation loss, mainly due to light scattering, is the most critical limitations to be solved in slow light systems. The scattering loss is mainly due to random fabrication imperfections or environmental changes. It increases quadratically with the reduced v g as proposed theoretically in [19] and certified experimentally in [20]. If the propagation loss penalty is too high, the superiority of slow light might be compromised. To overcome this crucial limitation in slow light systems, we propose a backscattering-immune slow light waveguide as demonstrated in the followed sections.

3. Model Design and Simulation Results 3.1. The Unidirectinal Waveguide and the Edge Modes Our proposed design is based on an interface between two types of PC structures and the schematic configuration is depicted in the inset in Figure 1. The lower half plane is a square lattice of MOPC consisting of Yttrium–Iron–Garnet (YIG) rods (ε = 15ε 0 , r = 0.11a) [28], where a = 38.69 mm is the lattice constant. With an externally applied magnetic field H0 = 1.27 × 105 A/m normal to the plane (along the +z direction), the induced magnetic permeability sensor is given as [45] 

µr  µ =  −iµk 0 µr = 1 +

iµk µr 0

 0  0  µ0 1

ω m ω0 ωm ω , µk = 2 . 2 2 ω0 − ω ω0 − ω 2

(3)

(4)

Here ω, ω0 = µ0 γH0 and ωm = µ0 γMs indicate the operating frequency of the microwave field, the procession resonance frequency and the characteristic frequency, respectively (γ = 1.759 × 1011 C/kg is the gyromagnetic ratio). The saturation magnetization of the YIG rods is Ms = 1.42 × 105 A/m. If the external magnetic field applied along −z direction, both ω0 and ωm change signs accordingly. According to Equation (4), it can be inferred that µr remains unchanged, but µk becomes its opposite value. As the edge modes at the interface of the MOPC exist inside the light cone and thus will be leaky in the free space. A titled 45◦ square lattice of Alumina PC (the upper half plane) with an aligned band gap (ε = 10ε 0 , R = 0.115a) has to be attached to the interface of the MOPC. Therefore, the excited edges are evanescent in both two types of PC structures. The dashed rectangle in the inset in Figure 1 indicates a supercell of the waveguide. Using a finite element eigenfrequency analysis and imposing period boundary conditions at the supercell’s borders, we can get the band diagram of the waveguide. Theoretically, the supercell must contain infinite rods in y direction, while a supercell of 10 × 1 unite cells of the waveguide can satisfy the accuracy requirements in practical calculations. With an externally applied magnetic field along the +z direction, the projected band diagram of the unidirectional waveguide for transverse magnetic (TM) modes (electric field is polarized along z-axis) are given in Figure 1. The shadow regions are the bulk modes of full MOPC and Alumina PC and the red curve denotes the unidirectional edge mode. As shown in Figure 1, the slope of the red curve (group velocities) has only positive values. It indicates that the waveguide support only forward transmission and the opposite propagation is completely prohibited. This unique feature insures that the edge modes are robust against interfacial disorders or defects. It is of significant importance for slow light systems.

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Figure 1. 1. The The projected projected band band diagram diagram of of the the waveguide. waveguide. The The corresponding corresponding geometry geometry configuration configuration Figure is illustrated in the inset, where the lower plane is the magneto-optical photonic crystal (MOPC) is illustrated in the inset, where the lower plane is the magneto-optical photonic crystal (MOPC) (𝜀 = ε =, 15ε , r = 0.11a , the upper plane is the Alumina photonic crystal (PC) ε = 10ε , R = 0.115a), (15𝜀 ) ( 0 0 0 𝑟 = 0.11𝑎 ), the upper plane is the Alumina photonic crystal (PC) (𝜀 = 10𝜀0 , 𝑅 = 0.115𝑎 ), the the dashed rectangle indicates a supercell of the waveguide. The red curveisisthe theedge edgemode modeat at the the dashed rectangle indicates a supercell of the waveguide. The red curve interface of of the the MOPC. MOPC. interface

3.2. Slow Light Properties 3.2. Slow Light Properties Since the first few rows of YIG rods have the greatest influence on the shape of the dispersion Since the first few rows of YIG rods have the greatest influence on the shape of the dispersion curve, we adjust the radii of the YIG rods to obtain a flat band region for slow light (a large NDBP curve, we adjust the radii of the YIG rods to obtain a flat band region for slow light (a large NDBP value and a low GVD value) with the unidirectional feature. The new configuration of the waveguide value and a low GVD value) with the unidirectional feature. The new configuration of the waveguide is shown in Figure 2, the radii of the YIG rods in the first and second rows adjacent to the waveguide is shown in Figure 2, the radii of the YIG rods in the first and second rows adjacent to the waveguide core are denoted by r1 and r2 , respectively. We perform a serious of numerical simulations for different core are denoted by 𝑟1 and 𝑟2 , respectively. We perform a serious of numerical simulations for r1 values and find that when r1 goes beyond the range of 0.075a to 0.08a, the unique unidirectionality different 𝑟1 values and find that when 𝑟1 goes beyond the range of 0.075𝑎 to 0.08𝑎, the unique of the edge modes vanishes. Therefore, we present the dispersion diagram for parameters r1 = 0.075a, unidirectionality of the edge modes vanishes. Therefore, we present the dispersion diagram for r1 = 0.077a, r1 = 0.08a, and r1 = 0.11a, as shown in Figure 3a. The corresponding group index parameters 𝑟1 = 0.075𝑎 , 𝑟1 = 0.077𝑎 , 𝑟1 = 0.08𝑎 , and 𝑟1 = 0.11𝑎 , as shown in Figure 3a. The characteristics are illustrated in Figure 3b. The dispersion curve becomes flat at r1 = 0.077a and a corresponding group index characteristics are illustrated in Figure 3b. The dispersion curve becomes step-like region (corresponding to the slow light region) appears, which indicates a low GVD feature. In flat at 𝑟1 = 0.077𝑎 and a step-like region (corresponding to the slow light region) appears, which the cases of r1 = 0.075a and r1 = 0.08a, the curve gets very steep and the step-like behavior disappears. indicates a low GVD feature. In the cases of 𝑟1 = 0.075𝑎 and 𝑟1 = 0.08𝑎, the curve gets very steep and the step-like behavior disappears.

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Figure 2. The configuration slow light photonic crystal waveguide (PCW). Figure 2.geometry The geometry configurationof theone-way one-way slow slow light photonic crystal waveguide (PCW). Figure 2. The geometry configuration ofofthe the one-way light photonic crystal waveguide (PCW). ◦ square upper half plane still titled 45° square lattice lattice of of PC as cladding, and thethe The upper half plane is still a titled 4545° lattice ofAlumina Alumina as an ancillary cladding, and the TheThe upper half plane is is still a atitled square Alumina PCPC asan anancillary ancillary cladding, and lower half plane is a square lattice MOPC with the rodradii radii of first tuned. lowerlower half plane is a is square lattice of MOPC with the rod two rows areprecisely precisely tuned. The half plane a square lattice ofofMOPC with the rod radiiof offirst firsttwo tworows rowsare precisely tuned. The radii of the Yttrium–Iron–Garnet (YIG) rods in the first and the second rows adjacent to the radii of the Yttrium–Iron–Garnet (YIG) in the the second rows adjacent to the radiiThe of the Yttrium–Iron–Garnet (YIG) rods in rods the first andfirst theand second rows adjacent to the waveguide waveguide are given by 𝑟1 and 𝑟2 , respectively. waveguide by 𝑟1 and 𝑟2 , respectively. are given by r1 are andgiven r2 , respectively.

(a)

(b)

(a) curves of the edge modes for different 𝑟1 with an externally (b) Figure 3. (a) Dispersion applied magnetic field along the +z direction. The shadow region denotes the extended modes of a perfect MOPC; (b) Figure 3. Dispersion (a) Dispersion curvesofofthe the edge edge modes 𝑟1 rwith an an externally applied magnetic Figure 3. (a) curves modesfor fordifferent different externally applied magnetic 1 with Group indices as a function of the frequencies, the relevant step-like behavior is indicated.

along direction. The The shadow denotes the extended modesmodes of a perfect fieldfield along thethe +z +z direction. shadowregion region denotes the extended of a MOPC; perfect(b) MOPC; Group indices as a function of the frequencies, the relevant step-like behavior is indicated. (b) Group indices asoptimum a function𝑟1of the frequencies, thefurther relevant step-like indicated. Based on the , we then adjust 𝑟2 to engineer the behavior dispersioniscurve. Figure 4a

illustrates the projected band diagram of TM modes with an external magnetic field applied in the +z

Based on the optimum 𝑟 , wethen then adjust r𝑟2 to to further further engineer the dispersion curve. Figure 4a Based on the engineer dispersion curve. 4a 2 direction foroptimum parametersr1𝑟,11we = 0.077𝑎 adjust , 𝑟2 = 0.113𝑎 ; 𝑟1 = 0.078𝑎 , 𝑟2 =the 0.115𝑎 ; 𝑟1 = 0.079𝑎 , 𝑟2 Figure = illustrates the projected band diagram of TM modes with an external magnetic field applied in the +z 0.117𝑎; and 𝑟1 = 0.08𝑎, = 0.118𝑎.of The slope of the curves (group velocities) have only positivein the +z illustrates the projected band𝑟2diagram TM modes with an external magnetic field applied direction for 𝑟1 = 0.077𝑎for , 𝑟slow ; 𝑟1 highlighted = 0.078𝑎 , 𝑟by 0.115𝑎 ; solid 𝑟1 = 0.079𝑎 , 𝑟2 = 2 = 0.113𝑎 2 =the values and parameters the flat rband regions light are thick lines. direction for parameters = 0.077a, r = 0.113a; r = 0.078a, r = 0.115a; r = 0.079a, rThe 2 2 2 = 0.117a; 1 1 0.117𝑎; and 𝑟1 = frequency 0.08𝑎,1𝑟2 =range 0.118𝑎. The slope of region the curves (group velocities)GHz, have4.2744 only GHz– positive corresponding of the slow light are 4.2886 GHz–4.3077 and rvalues rthe 0.118a. The slope of curves velocities)by have positive valuesThe and the 2 = flat 1 = 0.08a, regions forthe slow light(group are highlighted theonly thick solid lines lines. 4.295and GHz, 4.2599 band GHz–4.2806 GHz, and 4.2483 GHz–4.2692 GHz. The thick solid are flat band regions for slow light are highlighted by the thick solid lines. The corresponding frequency corresponding range of the slow region are 4.2886 GHz–4.3077 GHz, 4.2744 GHz– approximate frequency linear, which correspond to thelight step-like regions in Figure 5a. As illustrated in Figure range4.295 of5a,the slow light behavior region are 4.2886 GHz–4.3077 GHz, 4.2599 GHz–4.2806 GHz, 4.2599 GHz–4.2806 GHz, and 4.2483inGHz, GHz–4.2692 GHz. The thick solid lines are the step-like with a small change group4.2744 index GHz–4.295 in a relatively large frequency GHz,approximate and 4.2483linear, GHz–4.2692 GHz.which Theto thick solid lines are value approximate linear, which which the step-like regions in Figure illustrated in correspond Figure bandwidth is found in all correspond cases, indicates a large NDBP and a5a. lowAs GVD characteristics. 5a,step-like the step-like behavior with acriterion small change iningroup index in astep-like relatively largeof frequency Under theregions constant group index [16,40], the normalized bandwidths 0.444%, to the in Figure 5a. As illustrated Figure 5a,frequency the behavior with a small 0.481%, 0.485%, 0.491% obtained for group indices of 77, 68, 64, 60, respectively. bandwidth is index foundand in a all cases,are which indicates a large NDBP value and a low GVD characteristics. change in group in relatively large frequency bandwidth is found inand all cases, which indicates Consequently, thegroup corresponding NDBP are 0.342,the 0.327, 0.311, and 0.295 respectively. ThisofNDBP Under the value constant index criterion [16,40], normalized frequency bandwidths 0.444%, a large NDBP and a low GVD characteristics. Under the constant group index criterion [16,40], values are higher than previous works based on modifying the rod radius [31,42,46]. The results of 0.481%, 0.485%, and 0.491% are obtained for group indices of 77, 68, 64, and 60, respectively. the normalized frequency bandwidths of 0.444%, 0.481%, 0.485%, and 0.491% are obtained for group our design and works are summarized in 0.327, Table 1. The corresponding GVD characteristics Consequently, theprevious corresponding NDBP are 0.342, 0.311, and 0.295 respectively. This NDBP indices are of 77, 68, 64,inand 60, the2 corresponding NDBP are 0.342, 0.327, Figure 5b,respectively. we find thatbased theConsequently, GVD theradius slow light region keep valuesillustrated are higher than previous works on parameters modifying 𝛽 theinrod [31,42,46]. The around results of 𝑎 4 0.295 0.311,our and respectively. This NDBP values are higher than previous works based on modifying 10 ( ) for all four cases. This value is two orders of magnitude smaller than the previous works design 2𝜋𝑐 2 and previous works are summarized in Table 1. The corresponding GVD characteristics the rod radius [31,42,46]. The results of our design and previous works are summarized in Table 1. are illustrated in Figure 5b, we find that the GVD parameters 𝛽2 in the slow light region keep around 𝑎 The corresponding characteristics illustrated Figure 5b,smaller we find that GVD parameters 104 ( ) for allGVD four cases. This valueare is two than thethe previous works orders  ofinmagnitude 2𝜋𝑐 2

a β 2 in the slow light region keep around 104 2πc for all four cases. This value is two orders of 2 magnitude smaller than the previous works [43,44] and it indicates a more superior performance of our design. In the cases of r1 = 0.077a, r2 = 0.113a; r1 = 0.079a, r2 = 0.117a; and r1 = 0.08a, r2 = 0.118a both negative β 2 values and positive β 2 values are observed, it may reveals potential applications in

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[43,44] and and it it indicates indicates aa more more superior superior performance performance of of our our design. design. In In the the cases cases of of 𝑟𝑟11 = = 0.077𝑎, 0.077𝑎, 𝑟𝑟22 = = [43,44] 0.113𝑎 ; 𝑟 = 0.079𝑎 , 𝑟 = 0.117𝑎 ; and 𝑟 = 0.08𝑎 , 𝑟 = 0.118𝑎 both negative 𝛽 values and 0.113𝑎 ; 𝑟11 = 0.079𝑎 , 𝑟22 = 0.117𝑎 ; and 𝑟11 = 0.08𝑎 , 𝑟22 = 0.118𝑎 both negative 𝛽22 values and positive 𝛽 𝛽22 values values are are observed, observed, it may may is reveals potential applications in dispersion-compensation. dispersion-compensation. positive it reveals potential applications in dispersion-compensation. Zero β 2 values observed in the range of 4.2786 GHz to 4.2925 GHz (the Zero 𝛽 values is observed in the range of 4.2786 GHz to 4.2925 GHz The (the dispersion corresponding Zero 𝛽22 values is observed in the range GHz tor24.2925 GHz (the corresponding corresponding bandwidth is 0.324%) in the caseofof4.2786 r1 = 0.078a, = 0.115a. of such bandwidth is 0.324%) in the case of 𝑟 = 0.078𝑎, 𝑟 = 0.115𝑎. The dispersion of such slow light PCW PCW 1 = 0.078𝑎, 𝑟2 2 = 0.115𝑎. The dispersion of such slow light bandwidth is 0.324%) in the case of 𝑟 1 slow light PCW can be pursued to achieve a much slower v g and lower β 2 values simultaneously by can be be pursued pursued to to achieve achieve aa much much slower slower 𝑣 𝑣𝑔 and and lower lower 𝛽2 values simultaneously simultaneously by by further further can further adjusting the YIG rod radii of the third 𝑔row adjacent𝛽2tovalues the waveguide. Notably, when the adjusting the YIG rod radii of the third row adjacent to the waveguide. Notably, when the externally adjusting the YIG rod radii of the third row adjacent to the waveguide. Notably, when the externally externally applied magnetic field is reversed, the slope of the dispersion curves have only negative applied magnetic magnetic field field is is reversed, reversed, the the slope slope of of the the dispersion dispersion curves curves have have only only negative negative values values as applied as valuesshown as shown in Figure It indicates that the direction of the velocity group velocity can be controlled in Figure Figure 3b. It It 3b. indicates that the the direction of the the group group can be be controlled controlled by the the by shown in 3b. indicates that direction of velocity can by the external field. externalmagnetic magnetic field. field. external magnetic

(a) (a)

(b) (b)

Figure 4. Dispersion Dispersion curves ofthe theedge edge modes modes for for parameters = 0.077𝑎, =r20.113𝑎; 0.113𝑎; = r0.078𝑎, 0.078𝑎, Figure 4. Dispersion curves ofof forparameters parameters r1 0.077𝑎, = 0.077a, = 0.113a; Figure 4. curves the edge modes 𝑟𝑟11 = 𝑟𝑟22 = 𝑟𝑟11 = 1 = 0.078a, = 0.115𝑎; 0.115𝑎; = 0.079𝑎, 0.079𝑎, = 0.117𝑎; 0.117𝑎; and 𝑟1 = = 0.08𝑎, 0.08𝑎, 𝑟0.118a. = 0.118𝑎. 0.118𝑎. under the+(a) (a) +𝑧 (b) and−(b) (b) 2 r2 = 0.115a; r1 = 0.079a, r2 = 0.117a; and rand = 0.08a, r = under the (a) z and z direction 𝑟𝑟22 = 𝑟𝑟11 = 𝑟𝑟22 = 𝑟 𝑟 = under the +𝑧 and –– 𝑧𝑧 2 1 1 2 direction applied magnetic field, The thick solid lines represent the flat band region for slow light. ⊙ direction applied magnetic field, The thick solid lines represent the flat band region for slow light. applied magnetic field, The thick solid lines represent the flat band region for slow light. ⊙ and ⊕ and ⊕ ⊕ indicate indicate +𝑧 +𝑧 and and –– 𝑧𝑧 direction direction externally externally applied applied magnetic magnetic field, field, respectively. respectively. and indicate +z and −z direction externally applied magnetic field, respectively.

(a) (a)

(b) (b)

Figure 5. (a) Calculated Calculated group indices 𝑛𝑔(b) and (b) the corresponding corresponding 𝛽2function as a function of the the Figure (a) indices 𝑛 the of Figure 5. (a) 5. Calculated groupgroup indices n g and the(b) corresponding β 2 as a𝛽 of the frequencies 𝑔 and 2 as a function frequencies for the four optimized PCWs. frequencies for the four optimized PCWs. for the four optimized PCWs.

The dispersion curve can be tailored further to purse larger NDBP values by adjusting the

The dispersion curve can tailoredfurther further to values by by adjusting the the The dispersion curve can bebetailored to purse purselarger largerNDBP NDBP values adjusting magnitude of of the the externally externally applied applied magnetic magnetic field field 𝐻 𝐻00.. Based Based on on the the results results of of Figure Figure 5, 5, we we keep keep the the magnitude magnitude of the externally applied magnetic field H0 . Based on the results of Figure 5, we keep the parameters 𝑟𝑟11,, 𝑟𝑟22 and and carry carry out out the the optimizations optimizations for for the the 𝐻 𝐻00.. As As shown shown in in Figure Figure 6a, 6a, corresponding corresponding parameters parameters r1 , r2PCW and carry out the optimizations the H shown Figure 6a, and corresponding 0 . As of to the the same same structures in Figure Figure 5, higher higherfor NDBP values 0.394,in 0.347, 0.329, 0.326 are are to to PCW structures in 5, NDBP values of 0.394, 0.347, 0.329, and 0.326 5 5 5 the same PCW structures in Figure 5, higher NDBP values of 0.394, 0.347, 0.329, and 0.326 are obtained obtained for for 𝐻 𝐻00 = = 1.277 1.277 × × 10 105 A/m A/m ,, 𝐻 𝐻00 = = 1.279 1.279 × × 10 105 A/m A/m ,, 𝐻 𝐻00 = = 1.274 1.274 × × 10 105 A/m A/m ,, and and 𝐻 𝐻00 = = obtained 5 105 A/m, H = 1.279 × 105 A/m, H = 1.274 × 105 A/m, and H = 1.277 × 105 A/m, for H01.277 = 1.277 5 1.277 × 10 10× A/m, respectively. respectively. The corresponding corresponding GVD GVD characteristics are are illustrated illustrated in Figure Figure 6b. 6b. It It 0 0 0 in × A/m, The characteristics respectively. The corresponding GVD characteristics are illustrated in Figure 6b. It is shown that the  

GVD parameters β 2 in the slow light region keep around 104 are smaller than the values in Figure 5b.

a 2πc2

for all four cases, and these values

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cases, and these values are smaller than the values in Figure 5b.

(a)

(b)

Figure (a) Normalized delay-bandwidthproduct product (NDBP) ofof thethe four optimized PCWs as a as a Figure 6. (a)6.Normalized delay-bandwidth (NDBP)values values four optimized PCWs function ofexternally the externally applied magnetic fieldH0𝐻;0(b) ; (b)The Theβ𝛽22asasaafunction function of of the the frequencies function of the applied magnetic field frequenciesfor for the the four optimized PCWs in Figure with different four optimized PCWs in Figure 5 with5 different H0 . 𝐻0 . 1. Group index, frequency bandwidth,and and NDBP NDBP values different optimized parameters index, frequency bandwidth, valuesunder under different optimized parameters TableTable 1. Group andcomparison the comparison with previous works. and the with previous works. Current Work

Current Work

Previous Work

Previous Work

Average Group Optimized ParametersAverage Group Optimized Index Parameters Index r1  0.077a , r2  0.113a 77

= 0.077a, r1 r1 0.078 a , r2  0.115a r2 = 0.113a r1 r 0.079 a , r2  0.117a 1 = 0.078a, r1r2 = 0.08 a , r2  0.118a 0.115a r = 0.079a, Modification of the Rod 1 r2 = 0.117a Radius r1 = 0.08a, Ref. [31] r2 = 0.118a Ref. [46] Ref. [47] Modification of [48] the Rod Ref. Radius

77 68

Frequency

Frequency Bandwidth Bandwidth 0.444%

NDBP Values

NDBP Values 0.342

68

0.481% 0.444%

64

0.485%

0.311

60

0.481% 0.491%

0.295

Average Group 64 Index 22.85 60 26 40 Average Group Index 62

Frequency 0.485% Bandwidth 0.813% 0.491% 0.896% 0.65% Frequency 0.43% Bandwidth

0.327 0.342

0.327

NDBP0.311 Values 0.186 0.295 0.233 0.258 NDBP 0.268Values

Ref.Feature [31] 22.85 0.813% 0.186 3.3. Backscattering-Immune Ref. [46] 26 0.896% 0.233 As illustrated in Figure 4a, with an external 40 magnetic field applied along the +𝑧 direction, Ref. [47] 0.65% 0.258 the 0.268 tailored four PCWs still Ref. have[48] only positive group62 velocities (the slope 0.43% of the waveguide dispersions). It indicates that the waveguide supports only forward transmission and the opposite direction transmission is completely prohibited. The absence of backward propagation channel avoids the 3.3. Backscattering-Immune Feature possibility of backscattering loss and enables the electromagnetic wave to wrap around different As illustrated in Figure 4a, in with an external magnetic applied along disorders the +z direction, types of obstacles as verified Figure 7. We introduce fourfield types of interfacial into the the waveguide to investigate robustness the edge modes. For Figure 7a shows the 𝐸𝑧 tailored four PCWs still havethe only positiveofgroup velocities (thecomparison, slope of the waveguide dispersions). field distributions without introducing the only disorders. As shown in Figure 7b, light bypassesdirection the It indicates that the waveguide supports forward transmission andthe the opposite inserted metal reflector and maintain the forward transmission. This can never happen in a the transmission is completely prohibited. The absence of backward propagation channel avoids conventional PCW, as the metal reflector would cause strong scattering and even block the guide possibility of backscattering loss and enables the electromagnetic wave to wrap around different mode. However, in the proposed unidirectional PCW, a new interface between the metal reflector types of obstacles as verified in Figure 7. We introduce four types of interfacial disorders into the and the MOPC is formed, thus providing a way for the electromagnetic wave to flow around the waveguide to investigate the robustness of the edge modes. For comparison, Figure 7a shows the metal reflector and no backscattering takes place. The second type of defect is shown in Figure 7c, Ez field distributions introducing the disorders. Asby shown in Figure 7b, the when two YIG rodswithout at the interface of the MOPC are replaced the Copper rods with thelight same bypasses size, the inserted metal reflector and around maintain forward transmission. can neverThe happen the electromagnetic wave flow the the metal rods without introducingThis backscattering. third in a conventional PCW, the metal reflector cause strong scattering andalong eventhe block the guide type of defect is as a Z-shaped interface, as would shown in Figure 7d, the light travels Z-shaped

mode. However, in the proposed unidirectional PCW, a new interface between the metal reflector and the MOPC is formed, thus providing a way for the electromagnetic wave to flow around the metal reflector and no backscattering takes place. The second type of defect is shown in Figure 7c, when two YIG rods at the interface of the MOPC are replaced by the Copper rods with the same size, the electromagnetic wave flow around the metal rods without introducing backscattering. The third type of defect is a Z-shaped interface, as shown in Figure 7d, the light travels along the Z-shaped interface

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Sci. 2018, 8, x 8 of 12 and theAppl. forward propagation is not affected by the two sharp corners. In Figure 7e, two Alumina rods and two YIG rods at the interface of the waveguide (circled in the black rectangles) are replaced by interface and the forward propagation is not affected by the two sharp corners. In Figure 7e, two two Alumina rods with smaller rod radii r3 = 0.08a and two YIG rods with larger rod radii r4 = 0.11a, Alumina rods and two YIG rods at the interface of the waveguide (circled in the black rectangles) are respectively. It is that the unidirectional transmission is not by the interfacial replaced byfound two Alumina rods with smaller rod radii 𝑟3 = 0.08𝑎 andaffected two YIG rods with larger rod defects. This ideal robust unidirectional transmission feature is of great significance for slow radii 𝑟4 = 0.11𝑎, respectively. It is found that the unidirectional transmission is not affectedlight by thesystems. interfacial defects. This and idealrobustness robust unidirectional transmission featureallis four of great significance for The same unidirectionality of the edge modes from types of perturbation is slow lightthe systems. The magnetic same unidirectionality and robustness the edge modes from all four types 7b–e). observed when external field is applied in the −zofdirection (bottom plots in Figure of perturbation is observed when the external magnetic field is applied in the −z direction (bottom The propagation of the electromagnetic wave in the waveguide is switched to the opposite direction plots in Figure 7b–e). The propagation of the electromagnetic wave in the waveguide is switched to accordingly in all five cases. the opposite direction accordingly in all five cases.

(a)

(b)

(c)

(d)

(e) Figure The electric distributionsfor for E𝐸z𝑧 at at 4.2926 4.2926 GHz in in thethe unidirectional slow light Figure 7. The7.electric fieldfield distributions GHz unidirectional slowPCW light PCW for parameters 𝑟 = 0.078𝑎, 𝑟 = 0.115𝑎 (a) in the absence of the obstacles; (b) with an inserted metal metal 1 2 for parameters r1 = 0.078a, r2 = 0.115a (a) in the absence of the obstacles; (b) with an inserted reflector (indicated by the yellow rectangle); (c) with two YIG rods replaced by Copper rods (circled reflector (indicated by the yellow rectangle); (c) with two YIG rods replaced by Copper rods (circled in in the black rectangles) of the same size; (d) with a Z-shaped interface; (e) with rod defects (circled in the black rectangles) of the same size; (d) with a Z-shaped interface; (e) with rod defects (circled in the the black rectangles and the corresponding rod radii are 𝑟3 = 0.08𝑎 and 𝑟4 = 0.11𝑎 respectively). black rectangles and the corresponding rod radii are r3 = 0.08a and r4 = 0.11a respectively). The top and the bottom plots in (a–e) correspond to the results of the externally applied magnetic field along +z and −z direction, respectively. The point source is indicated by the black star. Blue and red colors represent negative and positive field strength values.

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Appl. Sci. 2018, 1858and the bottom plots in (a–e) correspond to the results of the externally applied magnetic The8,top field along +𝑧 and −𝑧 direction, respectively. The point source is indicated by the black star. Blue and red colors represent negative and positive field strength values.

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Notably, as illustrated in the top plots of Figure 8, in all five cases, the power flow is cycling anticlockwise on the edge of the PC of and clockwise onfive thecases, edgethe of the MOPC. The opposite Notably, as illustrated in Alumina the top plots Figure 8, in all power flow is cycling anticlockwise onthe the power edge of flow the Alumina and and clockwise on theboundaries edge of the MOPC. opposite core cycling direction of on the PC upper the lower of the The waveguide cycling direction the power flow of onthe theunidirectional upper and the lower boundaries of the core may indicate the slowoflight properties waveguide. Under the waveguide reversed externally maymagnetic indicate the slowinlight of the unidirectional waveguide. Underto thethe reversed externally applied field theproperties −z direction, the power flow is switched opposite direction applied magnetic field in the −z direction, the power flow is switched to the opposite direction accordingly as demonstrated in the bottom plots in Figure 8. The results in Figures 7 and 8 are obtained accordingly as demonstrated in the bottom plots in Figure 8. The results in Figures 7 and 8 are by using a finite-element frequency domain analysis. The point source is located in the interface obtained by using a finite-element frequency domain analysis. The point source is located in the between two types of PC structures and perfectly matched absorbing boundary layers are applied to interface between two types of PC structures and perfectly matched absorbing boundary layers are the surroundings the structure in the FEM simulation. applied to theof surroundings of the structure in the FEM simulation.

(a)

(b)

(c)

(d)

(e) Figure 8. Poynting The Poynting vectors distributionsat at4.2926 4.2926 GHz slow lightlight PCWPCW for for Figure 8. The vectors distributions GHzin inthe theunidirectional unidirectional slow parameters = 0.078𝑎, 𝑟2 = 0.115𝑎(a)(a)ininthe theabsence absence of of the with an an inserted metal parameters r1 =𝑟10.078a, r2 = 0.115a the obstacles; obstacles;(b)(b) with inserted metal reflector (indicated by the yellow rectangle); (c) with two YIG rods replaced by Copper rods (circled in the black rectangles) of the same size; (d) with a Z-shaped interface; (e) with rod defects (circled in the black rectangles and the corresponding rod radii are r3 = 0.08a and r4 = 0.11a respectively). The top and the bottom plots in (a–e) correspond to the results of the externally applied magnetic field along +z and −z direction, respectively.

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4. Conclusions In this paper, we propose a slow light PCW with unique unidirectionality and robustness characteristics. By choosing appropriate rod radii of the first two rods of the MOPC adjacent to the waveguide core, a high NDBP of 0.342 with a group index of 77 and frequency bandwidth of 0.444% is obtained. This value is much higher than previously reported works based on radius adjustment. The corresponding GVD parameters β 2 is zero in the range of 4.2786 GHz to 4.2925 GHz. By further adjusting the magnitude of the externally applied magnetic field, much larger NDBP values of 0.394 is obtained. Furthermore, due to the absence of the backward-propagation channel, the backscattering is completely suppressed. Our simulation results show that the edge modes are robust against the introduced interfacial defects. It is of great significance to solving the scattering loss limitations in slow light systems. Author Contributions: Supervision, X.L.; Writing—Original Draft Preparation, Q.Z.; Writing—Review and Editing, Q.Z. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

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