Design of nonreciprocal waveguide devices based on two-dimensional magneto-optical photonic crystals

Design of nonreciprocal waveguide devices based on two-dimensional magneto-optical photonic crystals

Optics & Laser Technology 50 (2013) 195–201 Contents lists available at SciVerse ScienceDirect Optics & Laser Technology journal homepage: www.elsev...

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Optics & Laser Technology 50 (2013) 195–201

Contents lists available at SciVerse ScienceDirect

Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec

Design of nonreciprocal waveguide devices based on two-dimensional magneto-optical photonic crystals Le Zhang a, Dongxiao Yang a,b,n, Kan Chen a, Tao Li a, Song Xia a a b

Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China Research Center for Terahertz Technology, Zhejiang University, Hangzhou 310027, China

art ic l e i nf o

a b s t r a c t

Article history: Received 17 November 2012 Received in revised form 22 February 2013 Accepted 6 March 2013 Available online 10 April 2013

Abstract: Isolator, circulator and crossing waveguide devices based on two-dimensional magneto-optical photonic crystals were designed. The dispersion relation, mode distribution and transmission spectrum for these nonreciprocal devices were analysed using the finite element method. An isolator, a four-port circulator and a low-crosstalk crossing waveguide with a continual one-way transmission bandwidth of 10.6%, a circulation bandwidth of 4.7% and a low-crosstalk bandwidth of 16.6% were fabricated, respectively. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Photonic crystal Magneto-optical crystal Nonreciprocal waveguide

1. Introduction In recent years, a variety of photonic crystal (PC) structures that operate in the infrared and optical regimes have been vigorously researched, including filters, waveguides and cavities [1]. During the investigation of optical circuits, utilising the properties of magnetooptical (MO) crystals and PCs has been found to be a viable approach to the design of nonreciprocal devices [2]. An ultra-compact, nonreciprocal optical isolator based on a MO PC slab has been proposed with an operating bandwidth of 400 GHz at a telecommunication wavelength of 1.55 μm [3]. Using a combination of a nonreciprocal waveguide and a resonant cavity, multi-port circulators have been realised. A three-port optical circulator made from a MO cavity in a 2D PC with complete isolation and transmission has been thoroughly analysed [4]. An approach for the design of resonant cavities employed in magnetophotonic crystal circulators and isolators has been proposed [5]. Another new type of circulator, based on directional coupling between one-way photonic chiral edge states and conventional two-way waveguides, has been proposed, to which the operational principle and design procedures can be readily extended [6]. Moreover, a nonreciprocal optical divider based on a two-dimensional PC and a MO cavity has been reported. The divider accomplishes two functions: the division of the input signal and the isolation of the input port from the two output ports, which have a calculated bandwidth of 100 GHz at a wavelength of 1.5 μm [7]. Using the nonreciprocity and the unique characteristics of PCs, a type of nonreciprocal waveguide composed of Yttrium Iron Garnet

n

Corresponding author at: Zhejiang University, Department of Information Science and Electronic Engineering, No.38 Zheda road, Hangzhou, Zhejiang 310027, China Tel./fax: +86 571 879 52787. E-mail address: [email protected] (D. Yang). 0030-3992/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2013.03.003

(YIG) and Alumina (Al2O3) rod PCs is proposed in this paper. The theoretical foundation and the prototype gyromagnetic PC waveguide, based on the quantum Hall effect (QHE), were reported in 2008 [8–9]. The non-zero off-axis components of the permittivity and the permeability tensors lead to the non-zero Chern number of the MO PC band [10]. When two photonic band gap (PBG) regions with different Chern invariants of bands are adjacent to one another, unidirectionally propagating photon modes can occur along the separating edge [11]. The experimental demonstration was reported in Nature in 2009 [12]. Based on the nonreciprocal PC waveguide, a series of functional devices, including isolator, circulator and crossing waveguides, were designed. 2. Theory A line-defect waveguide in a PC structure with a rectangular lattice array of cylindrical rods, with a lattice constant a and a radius 0.2a, is proposed and shown in Fig. 1. The materials of the upper and lower parts of this structure are nonmagnetic Al2O3 (ε ¼8.9, μ¼ 1) and magnetic YIG rods (in red), respectively. It has been reported that the gyromagnetic response of the YIG used in our designs is dispersive [13], as described in the Landau–Lifschitz equation in the following form: 0 1 0 1 μxx μxy 0 0 μxx jβ Bμ C B C μ ¼ @ yx μyy 0 A ¼ @ −jβ μyy 0 A ð1Þ 0

0

μzz

0

0

μzz

where μxx ¼ μyy ¼ 1 þ ðω0 ωm =ω20 −ω2 Þ, β ¼ −ðωωm =ω20 −ω2 Þ, ω0 ¼2πγΗ0, ωm ¼2πγ(4πms), γ¼ 2.8  106 rad s−1 G−1, 4πms ¼ 1780 G and Η0 is the external magnetic field along the z-axis. For simplification, the YIG parameters shown in Eqs. (2), i.e., 4.28 GHz in an applied

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magnetic induction of 0.16 T in the direction of the z-axis, are used in our simulations [9]. Using this approximation, which ignores the material dispersion and loss, is reasonable, as discussed below. 0

14

B ε ¼ 15, μ ¼ @ −12:4j 0

12:4j

0

1

14

C 0A

0

1

ð2Þ

All of the calculations in this paper are conducted with the finite element method (FEM) using COMSOL Multiphysics. The bulk TM mode band structures of the rod square lattice before and after the application of an external magnetic field are compared in the insets of Fig. 1. Before the application of an external magnetic field, both the band structures of the upper cladding Al2O3 rod square lattice (in blue) and the lower cladding

YIG rod square lattice (in cyan) have relatively large band gaps between the first and second bands. The distance between the first band of upper PC and the second band of lower PC is the total band gap. After the application of a magnetic field, the band structure of the Al2O3 square lattice is unchanged, while that of the YIG square lattice changes significantly. The band gap between the first and second bands is pulled down and has no intersection with the band gap of the upper cladding lattice. Instead, the relatively small band gap between the second and third bands is expanded. Together with a small gap between the third and fourth bands, this leads to two intersection band gaps, as shown in orange. Fig. 2(a) shows the calculated band diagram of the line-defect waveguide. Because of the periodicity, only one period in the x-direction needs to be considered to solve the domain, as shown

Fig. 1. Two-dimensional PC structure of the waveguide. The upper half part consists of an Al2O3 rod array and the lower half part consists of a YIG rod array. The two insets show the bulk TM mode band structures of the rod square lattice before and after the application of an external magnetic field. In the following, the coordinate axis setting is the same as shown.

Fig. 2. (a) Band diagram of the PC waveguide. The bulk states of the upper and lower claddings are shown in blue and cyan, respectively, corresponding to the colours of the band structures in the insets of Fig. 1. The two band gaps shown in white are the overlapping parts of the upper and lower cladding band gaps. The red lines are waveguide modes in the band gaps. (b) Normalised electric field distribution at the frequency of 0.34c/a and the wave vectors k ¼ 0.268 (left) and 0.37c/a and k¼ -0.224 (right).

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in the middle frame of Fig. 1. The wave vector k uses 2π/a as its default unit; in the following, this unit is omitted. The curves clearly present the nonreciprocity of the MO crystal. Asymmetrical waveguide modes appear in both band gaps, as shown in red. Two frequencies, e.g., 0.34c/a and 0.37c/a, are required for a detailed analysis. At a frequency of 0.34c/a, there is only one real waveguide mode along the positive x-axis direction at k¼ 0.268, as demonstrated by the left-normalizised electric field distribution in Fig. 2(b). At 0.37c/a, only a real waveguide mode along the negative x-axis direction is found at k ¼-0.224, as demonstrated by the right field distribution in Fig. 2(b). At larger frequencies, such as 0.35c/a and 0.38c/a, there are two red waveguide modes with opposite group velocities; therefore, these modes are not one-way guiding. As a result, in some frequency regions, the proposed PC structure acts as a one-way waveguide.

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the þx direction; four black lines represent the propagation along the -x direction. The curves of windows A, B, C and D are similar and represent almost equivalent spectra. According to the transmittance, at a frequency of 0.34c/a, most of the energy can propagate along the þx direction, but only a small amount of energy can propagate along the -x direction. At 0.37c/a, most of the energy can propagate along the -x direction, but a small amount of energy can propagate along the þx direction. The normalised electric field distribution of forward and backward transmissions, shown in Fig. 3(c), shows a one-way pass property, for example, at a frequency of 0.34c/a. These findings are in good agreement with the above analysis on the dispersion relation. The one-way pass band could be defined as the band where the transmittance is larger than 0.5 for the pass direction and smaller

3. Design and numerical demonstration 3.1. Isolator The whole PC waveguide structure of the proposed isolator is formed by 15 periods, as illustrated in Fig. 3(a). Fifteen periods were chosen for an appropriate design example. The following example shows that the period number should not be too small for performance consideration. In a harmonic propagation simulation, waves are input at the left red boundary when propagating along the þx direction and at the right red boundary when propagating along the -x direction. Four monitor windows, named A, B, C and D, obtain the power flow for calculating the transmittance at each window. Fig. 3(b) shows the calculated transmittance spectra in both the forward and backward directions. Four red lines represent the transmittance of the four windows with wave propagation along

Fig. 4. The transmittance spectra of the waveguide structure with only 3 periods (as the inset). The red curve represents forward propagation and the black one represents backward propagation.

Fig. 3. (a) The PC waveguide structure with 15 periods. (b) Calculated transmittance spectra of both forward and backward transmission on four monitor windows named A, B, C and D. The four red curves are forward transmittances with waves propagating from the left boundary to the right and four black lines are those with waves propagating from the right boundary to the left. (c) Normalised electric field distribution of forward and backward transmission at the frequency of 0.34c/a.

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than 0.1 for the stop direction. According to Fig. 3(b), the one-way pass band primarily lies within (0.328–0.344)c/a, with a relative bandwidth of 4.76% (the ratio of the bandwidth and the centre frequency), and within (0.364–0.372)c/a, with a relative bandwidth of 2.17%. The whole bandwidth and the relative bandwidth are 0.024c/a and 6.93%, respectively. If the waveguide structure is simplified to only three periods, the transmittance spectra change to those shown in Fig. 4. The isolation function is still working, but the transmittance of the pass direction drops to approximately 0.5 because the confinement becomes weaker using a small number of periods. As a result, the fewest number of periods should be used to ensure a good transmittance. This consideration requires a balanced choice in practical application. The isolator was optimised by changing the parameters of the structure to obtain a better performance, such as a wider band-

width and a higher isolation. An optimised isolator with a bandwidth of 0.041c/a between 0.369c/a and 0.410c/a has been designed using YIG radii of 0.16a and a position shift to the defect of 0.1a for the YIG row neighbouring the waveguide, as shown in Fig. 5(a). The dip in the band can be attributed to the wave resonance in the lower half of the YIG rods array at a specific frequency. The insertion loss is defined as L¼ -10 lgTout, where Tout is the transmittance along the pass direction. The isolation is defined as I ¼-10lgTiso, where Tiso is the transmittance along the stop direction. The performance of the optimised isolator is shown in Fig. 5(b). The isolation can reach 115 dB while the insertion loss is maintained at approximately 0.5 dB. This performance, for (0.369–0.410)c/a and a bandwidth of 10.6%, is better than that of the most recent MO PC isolators and is only poorer than (0.52– 0.58)c/a with a bandwidth of 10.9% [9]. However, the radii of the

Fig. 5. The transmittance (a) and the insertion loss and isolation (b) of the optimised isolator with YIG radii of 0.16a and position shift to the defect of 0.1a for the YIG row neighbouring the waveguide.

Fig. 6. (a) The structure of a four-port magneto-optical photonic crystal circulator. (b) The transmittance of three output ports. (c) Normalised electric field distribution at the frequency of 0.358c/a. (d) Normalised electric field distribution at 0.360c/a.

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Al2O3 rods and YIG rods in our proposed isolator are 0.2a and 0.16a, respectively. These radii are larger than those rods with radii of 0.11a reported in [9]. Because of a small performance difference, the fabrication of the proposed isolator is expected to be less difficult, and the design is thus expected to be more applicable. Here and in the following designs, the material dispersion is not considered, primarily because it does not substantially influence the results. As discussed in a previous study [9], the dispersion narrows the operational bandwidth but has little impact on backscattering suppression or the confinement of the edge mode. Additionally, the material loss is not considered for the same reason. Under the assumption that YIG material with a dielectric loss tangent of 0.0002 is commercially available, the imaginary part of the waveguide complex propagation constant corresponds to a decay length of a large number of lattice constants, far exceeding the practical structural length.

3.2. Circulator Based on the above isolator, a type of magneto-optical photonic crystal circulator with four ports and the same lattice constant a is proposed, as shown in Fig. 6 (a). As shown in red, 5  5 YIG rods with a radius of 0.2a form the core of the circulator. An array of Al2O3 rods with a radius of 0.2a is surrounding the core to create four PC waveguides with four ports. Due to the rotational

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symmetry of the structure, any one of the four ports can be chosen as the wave source and will lead to the same result. Thus, the red boundary at the left port in Fig. 6 (a) is chosen as the input port in this example. At the other three ports, three monitors, named A, B and C, obtain the power flow for calculating the transmittance. The output is expected at ports A or C to acquire the circulation function. The simulation results are shown in Fig. 6(b). Except for three dips, the transmittance of port A is larger than 0.5 and that of the other two isolation ports is less than 0.1, in the frequency range (0.354–0.373)c/a. This is defined as the circulation bandwidth. The normalised electric field distribution at a frequency of 0.358c/a, illustrated in Fig. 6(c), shows that most of the energy cannot go straight at the first turning but can only turn into port A. In addition, the field distribution of one dip at 0.360c/a in Fig. 6 (d) shows that the energy concentrates in the YIG rod array and forms a resonant cavity that prevents one-way waveguides from running properly. As a result, sharp depressions appear in the transmittance curve of port A. In practical application, multiple signals may come into the circulator from different ports at the same time. If one of them turns into resonant state, its mode field intensity will greatly strengthen, which gives rise to nonlinear effect disturbing other signals. To eliminate this limitation, the core is simplified to only one YIG rod, thus breaking the condition of the resonance. To improve the performance, four Al2O3 rods closest to the centre YIG rod at

Fig. 7. (a) Improved structure of the simplified circulator. (b) The transmittance of the circulator when the centre YIG rod radius is kept at 0.19a and the radii of other four YIG rods is set to 0.21a. (c) The isolation of port B and C.

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the crossing are substituted for four YIG rods, as shown in Fig. 7(a). The radii of the YIG rods are changed to obtain a larger bandwidth and a more stable extinction ratio. An optimised circulator with a bandwidth of 4.5% between 0.367c/a and 0.384c/a has been designed with the centre YIG radius of 0.19a and the other four YIG radii of 0.21a, as shown in Fig. 7(b). Meanwhile, the transmittance of port A is as high as 0.8 and is stable without large oscillation. The isolation of the isolated port is defined as I ¼-10 lg (Tiso/Tout), where Tout and Tiso are the transmittances of the output port and the isolated port, respectively. Fig. 7(c) shows the isolation properties of the optimised circulator. The isolation of port C reaches 35 dB at 0.38 c/a. Furthermore, the PC circulator can be simplified to another type of waveguide cross-structure by removing the centre YIG rod and keeping the other four rods at the crossing, as illustrated in Fig. 8(a). The red boundary at the left port is taken as the wave source, and three monitors, named A, B and C, obtain the power flow at the other three ports to calculate the transmittance. The optimal circulation band (0.363–0.372)c/a with a bandwidth of 2.5% appears when four YIG radii are changed to 0.19a, but all other radii are unchanged, as shown in Fig. 8(b). The transmittance of port A is close to 0.8, and the extinction ratio is higher. The circulation effect can be observed from Fig. 8(c). 3.3. Crossing waveguide More functional devices can be derived from the above structure. It can be clearly observed from Fig. 8(b) that the transmittance of port B is greater than 0.75, while those of ports A and C are less than 0.05 in the frequency range (0.392–0.443)c/a.

Therefore, a low-crosstalk crossing waveguide is proposed, as illustrated in Fig. 9(a). At each turning, three YIG rods are used rather than one YIG rod, as shown in red. The transmittance of three output ports is shown in Fig. 9(b). Not including the dip caused by resonance in the YIG rods at 0.428c/a, the usable band is (0.375–0.443)c/a, with a large bandwidth of 16.6%. Furthermore, the transmittance of ports A and C is very close to zero in the band, and a higher extinction ratio is thus obtained. When multiple signals come into this crossing, the crosstalk should be very low between them. The normalised electric field distribution at a frequency of 0.4c/a, shown in Fig. 9 (c), shows that most of the input energy goes straight and outputs at port B. For comparison, Fig. 9(d) illustrates the significant crosstalk of the waveguide cross structure without using YIG rods. Input energies were output at these three ports in an average manner.

4. Conclusion In summary, isolator, circulator and crossing waveguides based on two-dimensional magneto-optical photonic crystals were designed. The isolator uses several YIG rods to gain a continual one-way pass bandwidth as large as 10.6% between 0.369c/a and 0.410c/a, with a maximum isolation of 115 dB. This bandwidth performance is slightly poorer than that reported in [9], but may be more applicable because larger-sized rods are usually easier to fabricate. Based on the isolator, the first type of circulators use five YIG rods to obtain a usable circulation bandwidth of 4.7% between 0.367c/a and 0.384c/a, with an isolation reaching 35 dB.

Fig. 8. (a) The waveguide cross structure of another circulator. (b) The transmittance of three output ports. (c) Normalised electric field distribution at the frequency of 0.365c/a.

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Fig. 9. (a) The structure of the low-crosstalk crossing waveguide. (b) The transmittance of three output ports when using twelve YIG rods. (c) Normalised electric field distribution at the frequency of 0.4c/a. (d) The transmittance of three output ports without using YIG rods.

Through simplification, another type of waveguide cross structure circulator uses four YIG rods to gain a circulation bandwidth of 2.5% between 0.363c/a and 0.372c/a. Finally, a derivative lowcrosstalk crossing waveguide is proposed. It uses twelve YIG rods to achieve a large usable bandwidth of 16.6% between 0.375c/a and 0.443c/a and has low crosstalk. These designs are expected to provide the basis for a series of potentially useful functional photonic devices. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant Nos. 60671006 and 60971059. D. X. Yang’s e-mail address is [email protected]. Reference [1] Rao L, Yang DX, Hong Z. Guiding terahertz wave within a line defect of photonic crystal slab. Microwave and Optical Technology Letters 2012;54: 2856–2858. [2] Khanikaev AB, Steel MJ. Low-symmetry magnetic photonic crystals for nonreciprocal and unidirectional devices. Optics Express 2009;17:5265–72.

[3] Fang KJ, Yu ZF, Liu V, et al. Ultracompact nonreciprocal optical isolator based on guided resonance in a magneto-optical photonic crystal slab. Optics letters 2011;36:4254–6. [4] Wang Z, Fan SH. Optical circulators in two-dimensional magneto-optical photonic crystals. Optics Letters 2005;30:1989–91. [5] Smigaj W, Romero-Vivas J, Gralak B, et al. Magneto-optical circulator designed for operation in a uniform external magnetic field. Optics Letters 2010;35: 568–570. [6] Qiu WJ, Wang Z, Soljacic M. Broadband circulators based on directional coupling of one-way waveguides. Optics Express 2011;19:22248. [7] Dmitriev V, Kawakatsu MN. Nonreciprocal optical divider based on twodimensional photonic crystal and magneto-optical cavity. Applied Optics 2012;51:5917–20. [8] Haldane FDM, Raghu S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Physical Review Letters 2008;100:013904. [9] Wang Z, Chong YD, Joannopoulos JD, et al. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Physical Review Letters 2008;100: 013905. [10] Hatsugai Y. Chern Number and Edge States in the Integer Quantum Hall Effect. Physical Review Letters 1993;71:3697–700. [11] Raghu S, Haldane F. Analogs of quantum-Hall-effect edge states in photonic crystals. Physical Review A 2008;78:033834. [12] Wang Z, Chong YD, Joannopoulos JD, et al. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 2009;461: 772–775. [13] Fu JX, Liu RJ, Li ZY. Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces. Applied Physics Letters 2010;97:041112.