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Vol. 54, No. 33 / November 20 2015 / Applied Optics
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Compact photonic crystal circulator with flat-top transmission band created by cascading magneto-optical resonance cavities QIONG WANG, ZHENGBIAO OUYANG,* MI LIN,
AND
QIANG LIU
THz Technical Research Center and College of Electronic Science and Technology, Shenzhen University, Shenzhen Key Laboratory of Micro-Nano Photonic Information Technology, Shenzhen, Guangdong 518060, China *Corresponding author: [email protected] Received 29 July 2015; revised 19 September 2015; accepted 16 October 2015; posted 16 October 2015 (Doc. ID 247014); published 16 November 2015
A new type of compact three-port circulator with flat-top transmission band (FTTB) in a two-dimensional photonic crystal has been proposed, through coupling the cascaded magneto-optical resonance cavities to waveguides. The coupled-mode theory is applied to investigate the coupled structure and analyze the condition to achieve FTTB. According to the theoretical analysis, the structure is further optimized to ensure that the condition for achieving FTTB can be satisfied for both cavity–cavity coupling and cavity–waveguide coupling. Through the finite-element method, it is demonstrated that the design can realize a high quality, nonreciprocal circulating propagation of waves with an insertion loss of 0.023 dB and an isolation of 23.3 dB, covering a wide range of operation frequency. Such a wideband circulator has potential applications in large-scale integrated photonic circuits for guiding or isolating harmful optical reflections from load elements. © 2015 Optical Society of America OCIS codes: (230.5298) Photonic crystals; (130.5296) Photonic crystal waveguides; (230.3810) Magneto-optic systems. http://dx.doi.org/10.1364/AO.54.009741
1. INTRODUCTION Nonreciprocal optical devices, such as circulators or isolators, are important components in integrated photonic circuits as they have the special capability in guiding or isolating the reflections of electromagnetic waves [1–7]. In the past few decades, circulators based on photonic crystals [8] have been vigorously studied due to their ultracompact structures and excellent performances. In 2005, a three-port optical circulator with complete transmission and high isolation was first proposed by coupling a single magneto-optical (MO) cavity to three rotational symmetric waveguides [9]. And the same MO cavity can also be placed between two parallel waveguides, which can realize a high-quality circulator with four ports [10]. Then, a type of carefully modulated MO cavity was used to realize Y-format and W-format circulators which only requires a single-direction external magnetic field [11,12]. A cross-typed circulator has also been designed based on the MO effect of an array of ferrite rods [13]. Furthermore, in order to obtain a T-format circulator, a side-coupled cavity was employed to download and upload electromagnetic wave with the purpose to solve the asymmetric problem in the structure [14]. From above, it is evident that photonic crystals can provide great flexibilities in the designs of optical circulators. On the other hand, with the rapid development of optical systems, the integration of broadband circulators with multiple 1559-128X/15/339741-06$15/0$15.00 © 2015 Optical Society of America
components is in considerable need in order to get abundant functions. Until now, a type of broadband circulator has been proposed in magneto-optical photonic crystals. It is based on the directional coupling between one-way photonic chiral edge states and conventional two-way waveguides [15]. To the best of our knowledge, only the above-mentioned method has been reported to obtain broadband circulators in photonic crystal until now. The question of whether we have any other option is now still unanswered. In this paper, different from the reported literature that is based on a single MO cavity, we propose to use the coupling effect of multiple cascaded MO resonance cavities to investigate and realize a flat-top transmission band (FTTB) circulator. Both analytical coupled-mode theory and the finite-element method (FEM) are applied to study such a coupled structure and its performance as a three-port circulator. The design is further optimized according to the theoretical analysis; operation characteristics such as insert loss and isolation of the circulator are, respectively, discussed in detail. 2. THEORETICAL CONSIDERATIONS In optical circulators, at least three ports are needed, while a more-port-structure can be obtained by integration [16]. Therefore, we focus on the basic three-port circulators. For
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comparison, we first discuss the reported circulators by coupling a single MO cavity to three waveguides. The simplified model is shown in Fig. 1(a), where a solid circle labeled C 0 represents the MO cavity and three rectangles labeled W i i 1; 2; 3 represent the three waveguides. The three ports are denoted by P i i 1; 2; 3. According to the coupled-mode theory in the time domain, the amplitude of the cavity mode denoted by a0 can be described as follows [17–20], d a0 jω0 − γ 01 − γ 02 − γ 03 a0 dt pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi j 2γ 01 S 1 j 2γ 02 S 2 j 2γ 03 S 3 ; (1) pffiffiffiffiffiffiffiffi S −i −S i j 2γ 0i a0
i 1; 2; 3;
(2)
where ω0 is the cavity resonance frequency, S i S −i represents the amplitude of the incoming (outgoing) electromagnetic wave in the waveguide W i , and γ 0i is the coupling coefficient between the MO cavity C 0 and the waveguide W i . In this structure, a nonreciprocal, circulating propagation can be numerically realized, denoted as P 1 → P 2 , P 2 → P 3 and P 3 → P 1 . For the case of P 1 → P 2 , it means that the input wave from port P 1 will completely transmit to port P 2 while port P 3 is isolated from P 1 and P 2 , corresponding to γ 01 γ 02 ≫ γ 03 ≈ 0 and S 2 S 3 S −3 0. When there exists a very high isolation in the structure, the aforementioned case can be treated as a single-MO-cavity resonance structure coupled with the input waveguide (W 1 ) and the output waveguide (W 2 ), as shown in Fig. 1(b). Let γ 0 γ 01 γ 02 . The transmission (T ) can be calculated as S 2 4γ 20 : (3) T −2 S ω − ω 2 2γ 2 1
0
0
Note that the term ω − ω0 as shown in the denominator of Eq. (3) will lead to a Lorentzian response with a sharp peak in the transmission spectrum. In order to achieve FTTB for the circulator, we propose a resonance model by coupling three MO cavities to three 2
Fig. 1. (a) Schematic diagram of a single MO cavity coupled to three waveguides. (b) The simplified model for the input wave launched from port P 1 to P 2 .
waveguides symmetrically, as shown in Fig. 2. The three MO cavities are denoted by C i i 1; 2; 3, and the mode amplitudes (ai ) in cavity C i can be described by pffiffiffiffiffiffiffi d a1 jω1 − γ 1 a1 j 2γ 1 s 1 − jη12 a2 − jη13 a3 ; (4) dt pffiffiffiffiffiffiffi d a2 jω2 − γ 2 a2 j 2γ 2 s 2 − jη12 a1 − jη13 a3 ; dt
(5)
pffiffiffiffiffiffiffi d a3 jω3 − γ 3 a3 j 2γ 3 s 3 − jη13 a1 − jη23 a2 ; dt
(6)
pffiffiffiffiffiffi 2γ i ai
(7)
S −i −S i j
i 1; 2; 3;
where ωi i 1; 2; 3 is the resonator frequency of the MO cavity C i, γ i is the coupling coefficient between the MO cavity C i and its corresponding waveguide W i , and ηij is the mutual coupling coefficient between C i and C j . Without an external magnetic field applied in this structure, each MO cavity can support a dipole standing mode, denoted by the dashed circles in the figure. When an input wave is launched from port P 1 , C 2 and C 3 are equivalent with C 1 , so we have ω2 ω3 , γ 2 γ 3 , η12 η13 . As a result, the transmission from port P 1 to P 2 is equal to that from port P 1 to P 3 , i.e., T jS −2 ∕S 1 j2 jS −3 ∕S 1 j2 . It means that the structure functions as a power divider in which the energy is split equally into two parts. Here, the coupling direction of the dipole standing mode at cavity C 1 is along the symmetry axis of cavities C 2 and C 3 , and the coupling direction of the dipole standing mode at cavity C 2 C 3 is along the central axis of waveguide W 2 W 3 , as shown by the arrows in Fig. 2. When an external magnetic field is applied in the coupled structure, the dipole standing mode in a cavity produces a rotation due to the MO effect, so that the coupling direction will change accordingly. As a result, nonreciprocal propagations denoted as P 1 → P 2 , P 2 → P 3 and P 3 → P 1 can be realized in this structure. The case of P 1 → P 2 is also given as an example. By modulating the parameters of magnetized MO cavities, the coupling direction at MO cavity C 1 can be rotated by 30°, just pointing to MO cavity C 2 and deviating from MO cavity C 3,
Fig. 2. Schematic diagram of three MO cavities coupled to three waveguides.
Research Article which makes the coupling between C 1 and C 2 become strong while the coupling between C 1 and C 3 becomes weak. Thus, the input wave from port P 1 will transmit to port P 2 and be isolated from port P 3 . This can be described by a coupling model consisting of two cascaded cavities C 1 and C 2 that lie between waveguides W 1 and W 2 , as shown in Fig. 3(a). In Fig. 3(b), it shows the direction change of the wave vector: C 1 provides a 30° clockwise rotation and C 2 provides an additional 30° clockwise rotation. In this case, we have γ 1 ≫ γ 3 ≈ 0, γ 2 ≫ γ 3 ≈ 0, η12 ≫ η13 ≈ 0, η12 ≫ η23 ≈ 0, S 2 S 3 S −3 0. Due to the consideration of the symmetry in the structure, we have ω1 ω2 and γ 1 γ 2 . Thus we can define symbols ωc ω1 ω2 , γ c γ 1 γ 2 , and ηc η12 . The transmission T from port P 1 to P 2 can be calculated from Eqs. (4)–(7) as S −2 2 4γ 2c η2c : T S 1 ω − ωc 4 2γ 2c − η2c ω − ωc 2 γ 2c η2c 2 (8)
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Note that Eq. (8) contains the terms ω − ωc 4 and ω − ωc 2 in the denominator. For ω − ωc ≈ 0 and γ 2c − η2c ≠ 0, the term ω − ωc 4 can be neglected since it is much smaller than ω − ωc 2 , which corresponds to the Lorentzian response. In this case, from Eq. (8) we can calculate the changing rate of T with respect to ω out to be d T jρ1 j ≈ 16γ 2c η2c γ 2c η2c −4 jγ 2c − η2c ω − ωc j ∝ jω − ωc j: dω (9) For γ 2c − η2c 0, however, the term with ω − ωc 2 disappears and the term ω − ωc 4 will play a dominant role for ω − ωc ≈ 0. Here, from Eq. (8) we can calculate the changing rate of T with respect to ω out to be dT ≈ 16γ 2c η2c γ 2c η2c −4 jω − ωc 3 j ∝ jω − ωc j3 : jρ2 j d ω (10) Obviously, from Eqs. (9) and (10) we have jρ2 j ≪ jρ1 j when ω − ωc ≈ 0. This means that the transmission varies more slowly with ω for γ 2c − η2c 0 and ω − ωc ≈ 0, which gives rise to a better FTTB. In addition, the transmission can reach up to 100% at the flat region, as can be seen from Eq. (8) for γ 2c η2c . We can regard γ 2c η2c as the condition to obtain FTTB. Under this condition, it suggests that the coupling of MO cavity C 1 C 2 and its linked waveguide W 1 W 2 must be equivalent to that of the two cascaded MO cavities C 1 and C 2 . This can be simplified as that the cavity–waveguide coupling must be equivalent to the cavity–cavity coupling. Similarly, the simplified models for the cases of P 2 → P 3 and P 3 → P 1 are shown in Figs. 3(c) and 3(e), respectively, and their direction changes of wave vector are shown in Figs. 3(d) and 3(f ), respectively. Note that the models are suitable for application in the frequencies of very high isolation, i.e., for the transmission of the energy from one port to another with little leakage into the third port. According to the analytical result obtained from the coupled-mode theory, it may help pave the way for building a novel circulator that has FTTB. Note that the concrete parameters in the material are not considered in this section. In the following, the design of a photonic crystal circulator will be presented and investigation of frequency characteristics will be shown. 3. STRUCTURE AND NUMERICAL SIMULATIONS
Fig. 3. Simplified models for the input wave launched from port (a) P 1 , (c) P 2 , (e) P 3 , and the direction changes of the wave vector for the cases of (b) P 1 → P 2 , (d) P 2 → P 3 , (f) P 3 → P 1 . All MO cavities are under the same magnetic field.
In this section, a two-dimensional photonic crystal is designed to build the FTTB circulator. As displayed in Fig. 4, it consists of a triangle-lattice array of dielectric rods with the refractive index n1 3.4 and the rod radius r 1 0.2a 2 × 10−3 m in air, where a is the lattice constant. As noted, some dielectric rods are removed to form three waveguides with a Y-shaped distribution. And the width of the waveguide is adjusted to be L 3a. Three uniform MO rods are symmetrically set at the cross region of the three waveguides. Each MO rod can be regarded as a resonance cavity that is labeled by C i i 1; 2; 3. The radius of the MO rods is denoted by r.
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Fig. 4. FTTB circulator designed in a two-dimensional trianglelattice photonic crystal with three uniform MO cavities C i i 1; 2; 3 coupled to three waveguides, and the triangle dielectric rods H i i 0; 1; 2; 3 added to optimize the coupled structure.
In order to enhance the coupling effect among the MO cavities, a dielectric rod labeled by H 0 is introduced and lies at the center. Furthermore, three dielectric rods denoted by H i i 1; 2; 3 are, respectively, placed between each MO cavity C i and its nearest contacting waveguide W i to modulate the cavity–waveguide coupling. Note that the added dielectric rods H i i 0; 1; 2; 3 have triangle cross sections due to the consideration of the 120° rotational symmetry in the structure. We use FEM to investigate the circulator shown in Fig. 4. The structure domain, surrounded by perfect matched layers, is computed by dividing about 8.5 × 105 grid cells. When an input wave is launched from one of the three ports, we collect the signal at the end of the transmitting (isolated) waveguide and compare it with the input signal. The insertion loss (isolation) is calculated. In order to achieve efficient coupling among MO cavities, we first consider the structure without the dielectric rods marked as H i i 1; 2; 3 in Fig. 4. Yttrium-iron-garnet (YIG) is used for the MO material. Under an applied magnetic field of H 3.4 × 105 A∕m, the relative permeability of YIG material can be expressed by a tensor [21] as # " μr jμk 0 (11) μ −jμk μr 0 ; 0 0 1 where μr 1 ωm ωr iαω∕ωr iαω2 − ω2 and μk ωm ω∕ωr iαω2 − ω2 with ωr μ0 γH , ωm μ0 γM s , γ 1.759 × 105 C∕kg, α 3 × 10−5 , and M s 2.39 × 105 A∕m. And the relative permittivity of YIG material is given by εr 12.4. As the material for MO cavities has been selected, the radius of rod C i i 1; 2; 3 becomes an important factor that affects the coupling efficiency among MO cavities. All MO cavities are under the same magnetic field. For convenience, a parameter K r∕r 1 is defined as the ratio between
Research Article the radius of MO cavities to that of background dielectric rods. Due to the structure symmetry, the properties of the circulator can be studied by choosing an arbitrary port as the input port. Here, for example, we select port P 1 as the input port. Through changing the values of K , the transmissions for the output port P 2 and the isolated port P 3 can be calculated out, as shown by the solid and dashed lines in Fig. 5, respectively. We can see that the transmission T 2 for the output port P 2 reaches the highest value (∼93%) at K 1.3 and the corresponding insertion loss has the lowest value of 10 log1∕T 2 0.31 dB. Meanwhile, the transmission T 3 for the isolated port P 3 reaches the lowest value (∼0.46%) and the corresponding isolation has the highest value of 10 log1∕T 3 21.3 dB. The frequency is 1.0449 · 1010 Hz. The sum of reflection and loss is calculated and shown by the dotted line. The results indicate that the input wave from port P 1 can be efficiently transmitted to port P 2 through strong coupling between the MO cavities C 1 and C 2 , while port P 3 is highly isolated in the system since the coupling between C 1 C 2 and C 3 is relatively weaker. Note that the transmission in Fig. 5 is very low when k r∕r 1 is below 1. This is because the MO rod with small size cannot support a resonance that is within the waveguide dispersion; then the reflection of the wave is very high. In addition, the side length of the cross-section triangle for dielectric rod H 0 is modulated to be 0.25a. Furthermore, as introduced in Section 2, the cavity– waveguide coupling is equivalent to the cavity–cavity coupling. As we know, the cavity–waveguide coupling is sensitive to the connection region of the MO cavity and its linked waveguide, so now we will investigate the structure with the triangle dielectric rods H i i 1; 2; 3 added at the connection regions, with the purpose to modulate the cavity–waveguide coupling to match with the MO cavity–cavity coupling, which has been optimized above. By tuning the size and position of dielectric rods H i i 1; 2; 3, the calculated isolation and insertion loss with respect to the frequency are shown by the solid lines in Fig. 6(a). For comparison, the results without triangle dielectric rods are also given. As can be seen, after introducing the triangle dielectric rods, the working bandwidth of the circulator can be extended to a
Fig. 5. Before adding dielectric rods H i i 1; 2; 3, the transmissions of the output port P 2 and the isolated port P 3 for different value of K r∕r 1 , and the sum of reflection and loss in the structure.
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Fig. 7. Distributions of E z electric field in the three-port circulator operating at the frequency of 1.0449 · 1010 Hz for the input light launched (a) from port P 1 , (b) from port P 2 , and (c) from port P 3 .
Fig. 6. (a) The insertion loss and the isolation. (b) The sum of reflection and loss calculated by scanning frequency. The solid and dashed lines represent cases with and without the dielectric rods H i i 1; 2; 3 added at the junction of each magneto-optical cavity and its corresponding waveguide, respectively.
wider range, i.e., a FTTB with frequency from 1.0374 · 1010 Hz to 1.0524 · 1010 Hz appears in the calculated spectra. In addition, compared to the case without dielectric rods, the isolation increases from 21.3 to 23.3 dB and the insertion loss decreases from 0.31 to 0.023 dB. We also calculated the sum of reflection and loss in Fig. 6(b), in which the solid and dashed lines denote cases with and without dielectric rods H i i 1; 2; 3, respectively. It is evident that the cavity– waveguide coupling and cavity–cavity coupling in the structure are greatly matched, resulting in an increment of transmission. Here, the side length of the cross-section triangle for dielectric rods H i i 1; 2; 3 is adjusted to be 0.2a, and the distance between dielectric rod H i i 1; 2; 3 and the center rod H 0 is optimized as 1.3a. In order to examine the performance of the circulator with more details, the field distributions are calculated, as shown in Fig. 7. We can choose an arbitrary operation frequency in the FTTB; here the frequency of 1.0449 · 1010 Hz is selected to give an example. As can be seen in Fig. 7(a), the input wave launched from port P 1 is almost totally transmitted to port P 2 through the coupling of the MO cavities C 1 and C 2 .
Port P 3 is isolated. Similarly, the wave from port P 2 P 3 will transmit to the output port P 3 P 1 through the coupling of the MO cavities C 2 C 3 and C 3 C 1 , isolating port P 1 P 2 , as shown in Figs. 7(b) and 7(c), respectively. These results demonstrate that the device functions as an optical circulator in which the transmission with a single-direction circulation of P 1 → P 2 , P 2 → P 3 and P 3 → P 1 is realized. Wave propagation is in good agreement shown in Section 2. Note that each MO cavity can provide a 30° rotation of the wave vector; thus, two cascaded MO cavities can realize a 60° rotation of the wave vector between the input and the output waveguides. In addition, this type of circulator has a compact structure and can provide a broad operation bandwidth, such that it may provide a good platform for constructing integrated photonic circuits. 4. CONCLUSIONS In conclusion, we propose a new type of FTTB circulator by coupling the cascaded MO cavities to input and output waveguides, and analyze the condition for achieving FTTB in such a structure theoretically and numerically. A three-port circulator as an example has been designed in a twodimensional triangle-lattice photonic crystal. In order to achieve excellent FTTB, the structure is optimized to make the cavity–waveguide coupling perfectly match with the cavity–cavity coupling. Results indicate that a low insertion loss and a high isolation can be obtained. The circulator with a large operation bandwidth presented here can bring great convenience in integrating photonic devices in large-scale optical circuit systems, such as optical routing or isolating uses. Funding. National Natural Science Foundation of China (NSFC) (61275043, 61307048); Natural Science Foundation of SZU (201456); Open Fund of Shenzhen
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Key Lab of Micro-Nano Photonic Information Technology (201406); Shenzhen Kexin Ju funds (CXB201105050064A, JCYJ20140828163633988). REFERENCES 1. O. Zhuromskyy, H. Dötsch, M. Lohmeyer, L. Wilkens, and P. Hertel, “Magnetooptical waveguides with polarization-independent nonreciprocal phase shift,” J. Lightwave Technol. 19, 214–221 (2001). 2. Y. H. Lu, M. H. Cho, J. B. Kim, G. J. Lee, Y. P. Lee, and J. Y. Rhee, “Magneto-optical enhancement through gyrotropic gratings,” Opt. Express 16, 5378–5384 (2008). 3. H. B. Zhu and C. Jiang, “Nonreciprocal extraordinary optical transmission through subwavelength slits in metallic film,” Opt. Lett. 36, 1308–1310 (2011). 4. Y. Poo, R. X. Wu, Z. F. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011). 5. Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21, 15373–15381 (2013). 6. P. Pintus, F. D. Pasquale, and J. E. Bowers, “Integrated TE and TM optical circulators on ultra-low-loss silicon nitride platform,” Opt. Express 21, 5041–5052 (2013). 7. S. Chen, F. Fan, X. H. Wang, P. F. Wu, H. Zhang, and S. J. Chang, “Terahertz isolator based on nonreciprocal magneto-metasurface,” Opt. Express 23, 1015–1024 (2015). 8. S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice (Kluwer Academic, 2002). 9. Z. Wang and S. Fan, “Optical circulators in two-dimensional magnetooptical photonic crystals,” Opt. Lett. 30, 1989–1991 (2005). 10. Z. Wang and S. Fan, “Suppressing the effect of disorders using timereversal symmetry breaking in magneto-optical photonic crystals: an
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Compact photonic crystal circulator with flat-top transmission band created by cascading magneto-optical resonance cavities QIONG WANG, ZHENGBIAO OUYANG,* MI LIN,
AND
QIANG LIU
THz Technical Research Center and College of Electronic Science and Technology, Shenzhen University, Shenzhen Key Laboratory of Micro-Nano Photonic Information Technology, Shenzhen, Guangdong 518060, China *Corresponding author: [email protected] Received 29 July 2015; revised 19 September 2015; accepted 16 October 2015; posted 16 October 2015 (Doc. ID 247014); published 16 November 2015
A new type of compact three-port circulator with flat-top transmission band (FTTB) in a two-dimensional photonic crystal has been proposed, through coupling the cascaded magneto-optical resonance cavities to waveguides. The coupled-mode theory is applied to investigate the coupled structure and analyze the condition to achieve FTTB. According to the theoretical analysis, the structure is further optimized to ensure that the condition for achieving FTTB can be satisfied for both cavity–cavity coupling and cavity–waveguide coupling. Through the finite-element method, it is demonstrated that the design can realize a high quality, nonreciprocal circulating propagation of waves with an insertion loss of 0.023 dB and an isolation of 23.3 dB, covering a wide range of operation frequency. Such a wideband circulator has potential applications in large-scale integrated photonic circuits for guiding or isolating harmful optical reflections from load elements. © 2015 Optical Society of America OCIS codes: (230.5298) Photonic crystals; (130.5296) Photonic crystal waveguides; (230.3810) Magneto-optic systems. http://dx.doi.org/10.1364/AO.54.009741
1. INTRODUCTION Nonreciprocal optical devices, such as circulators or isolators, are important components in integrated photonic circuits as they have the special capability in guiding or isolating the reflections of electromagnetic waves [1–7]. In the past few decades, circulators based on photonic crystals [8] have been vigorously studied due to their ultracompact structures and excellent performances. In 2005, a three-port optical circulator with complete transmission and high isolation was first proposed by coupling a single magneto-optical (MO) cavity to three rotational symmetric waveguides [9]. And the same MO cavity can also be placed between two parallel waveguides, which can realize a high-quality circulator with four ports [10]. Then, a type of carefully modulated MO cavity was used to realize Y-format and W-format circulators which only requires a single-direction external magnetic field [11,12]. A cross-typed circulator has also been designed based on the MO effect of an array of ferrite rods [13]. Furthermore, in order to obtain a T-format circulator, a side-coupled cavity was employed to download and upload electromagnetic wave with the purpose to solve the asymmetric problem in the structure [14]. From above, it is evident that photonic crystals can provide great flexibilities in the designs of optical circulators. On the other hand, with the rapid development of optical systems, the integration of broadband circulators with multiple 1559-128X/15/339741-06$15/0$15.00 © 2015 Optical Society of America
components is in considerable need in order to get abundant functions. Until now, a type of broadband circulator has been proposed in magneto-optical photonic crystals. It is based on the directional coupling between one-way photonic chiral edge states and conventional two-way waveguides [15]. To the best of our knowledge, only the above-mentioned method has been reported to obtain broadband circulators in photonic crystal until now. The question of whether we have any other option is now still unanswered. In this paper, different from the reported literature that is based on a single MO cavity, we propose to use the coupling effect of multiple cascaded MO resonance cavities to investigate and realize a flat-top transmission band (FTTB) circulator. Both analytical coupled-mode theory and the finite-element method (FEM) are applied to study such a coupled structure and its performance as a three-port circulator. The design is further optimized according to the theoretical analysis; operation characteristics such as insert loss and isolation of the circulator are, respectively, discussed in detail. 2. THEORETICAL CONSIDERATIONS In optical circulators, at least three ports are needed, while a more-port-structure can be obtained by integration [16]. Therefore, we focus on the basic three-port circulators. For
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comparison, we first discuss the reported circulators by coupling a single MO cavity to three waveguides. The simplified model is shown in Fig. 1(a), where a solid circle labeled C 0 represents the MO cavity and three rectangles labeled W i i 1; 2; 3 represent the three waveguides. The three ports are denoted by P i i 1; 2; 3. According to the coupled-mode theory in the time domain, the amplitude of the cavity mode denoted by a0 can be described as follows [17–20], d a0 jω0 − γ 01 − γ 02 − γ 03 a0 dt pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi j 2γ 01 S 1 j 2γ 02 S 2 j 2γ 03 S 3 ; (1) pffiffiffiffiffiffiffiffi S −i −S i j 2γ 0i a0
i 1; 2; 3;
(2)
where ω0 is the cavity resonance frequency, S i S −i represents the amplitude of the incoming (outgoing) electromagnetic wave in the waveguide W i , and γ 0i is the coupling coefficient between the MO cavity C 0 and the waveguide W i . In this structure, a nonreciprocal, circulating propagation can be numerically realized, denoted as P 1 → P 2 , P 2 → P 3 and P 3 → P 1 . For the case of P 1 → P 2 , it means that the input wave from port P 1 will completely transmit to port P 2 while port P 3 is isolated from P 1 and P 2 , corresponding to γ 01 γ 02 ≫ γ 03 ≈ 0 and S 2 S 3 S −3 0. When there exists a very high isolation in the structure, the aforementioned case can be treated as a single-MO-cavity resonance structure coupled with the input waveguide (W 1 ) and the output waveguide (W 2 ), as shown in Fig. 1(b). Let γ 0 γ 01 γ 02 . The transmission (T ) can be calculated as S 2 4γ 20 : (3) T −2 S ω − ω 2 2γ 2 1
0
0
Note that the term ω − ω0 as shown in the denominator of Eq. (3) will lead to a Lorentzian response with a sharp peak in the transmission spectrum. In order to achieve FTTB for the circulator, we propose a resonance model by coupling three MO cavities to three 2
Fig. 1. (a) Schematic diagram of a single MO cavity coupled to three waveguides. (b) The simplified model for the input wave launched from port P 1 to P 2 .
waveguides symmetrically, as shown in Fig. 2. The three MO cavities are denoted by C i i 1; 2; 3, and the mode amplitudes (ai ) in cavity C i can be described by pffiffiffiffiffiffiffi d a1 jω1 − γ 1 a1 j 2γ 1 s 1 − jη12 a2 − jη13 a3 ; (4) dt pffiffiffiffiffiffiffi d a2 jω2 − γ 2 a2 j 2γ 2 s 2 − jη12 a1 − jη13 a3 ; dt
(5)
pffiffiffiffiffiffiffi d a3 jω3 − γ 3 a3 j 2γ 3 s 3 − jη13 a1 − jη23 a2 ; dt
(6)
pffiffiffiffiffiffi 2γ i ai
(7)
S −i −S i j
i 1; 2; 3;
where ωi i 1; 2; 3 is the resonator frequency of the MO cavity C i, γ i is the coupling coefficient between the MO cavity C i and its corresponding waveguide W i , and ηij is the mutual coupling coefficient between C i and C j . Without an external magnetic field applied in this structure, each MO cavity can support a dipole standing mode, denoted by the dashed circles in the figure. When an input wave is launched from port P 1 , C 2 and C 3 are equivalent with C 1 , so we have ω2 ω3 , γ 2 γ 3 , η12 η13 . As a result, the transmission from port P 1 to P 2 is equal to that from port P 1 to P 3 , i.e., T jS −2 ∕S 1 j2 jS −3 ∕S 1 j2 . It means that the structure functions as a power divider in which the energy is split equally into two parts. Here, the coupling direction of the dipole standing mode at cavity C 1 is along the symmetry axis of cavities C 2 and C 3 , and the coupling direction of the dipole standing mode at cavity C 2 C 3 is along the central axis of waveguide W 2 W 3 , as shown by the arrows in Fig. 2. When an external magnetic field is applied in the coupled structure, the dipole standing mode in a cavity produces a rotation due to the MO effect, so that the coupling direction will change accordingly. As a result, nonreciprocal propagations denoted as P 1 → P 2 , P 2 → P 3 and P 3 → P 1 can be realized in this structure. The case of P 1 → P 2 is also given as an example. By modulating the parameters of magnetized MO cavities, the coupling direction at MO cavity C 1 can be rotated by 30°, just pointing to MO cavity C 2 and deviating from MO cavity C 3,
Fig. 2. Schematic diagram of three MO cavities coupled to three waveguides.
Research Article which makes the coupling between C 1 and C 2 become strong while the coupling between C 1 and C 3 becomes weak. Thus, the input wave from port P 1 will transmit to port P 2 and be isolated from port P 3 . This can be described by a coupling model consisting of two cascaded cavities C 1 and C 2 that lie between waveguides W 1 and W 2 , as shown in Fig. 3(a). In Fig. 3(b), it shows the direction change of the wave vector: C 1 provides a 30° clockwise rotation and C 2 provides an additional 30° clockwise rotation. In this case, we have γ 1 ≫ γ 3 ≈ 0, γ 2 ≫ γ 3 ≈ 0, η12 ≫ η13 ≈ 0, η12 ≫ η23 ≈ 0, S 2 S 3 S −3 0. Due to the consideration of the symmetry in the structure, we have ω1 ω2 and γ 1 γ 2 . Thus we can define symbols ωc ω1 ω2 , γ c γ 1 γ 2 , and ηc η12 . The transmission T from port P 1 to P 2 can be calculated from Eqs. (4)–(7) as S −2 2 4γ 2c η2c : T S 1 ω − ωc 4 2γ 2c − η2c ω − ωc 2 γ 2c η2c 2 (8)
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Note that Eq. (8) contains the terms ω − ωc 4 and ω − ωc 2 in the denominator. For ω − ωc ≈ 0 and γ 2c − η2c ≠ 0, the term ω − ωc 4 can be neglected since it is much smaller than ω − ωc 2 , which corresponds to the Lorentzian response. In this case, from Eq. (8) we can calculate the changing rate of T with respect to ω out to be d T jρ1 j ≈ 16γ 2c η2c γ 2c η2c −4 jγ 2c − η2c ω − ωc j ∝ jω − ωc j: dω (9) For γ 2c − η2c 0, however, the term with ω − ωc 2 disappears and the term ω − ωc 4 will play a dominant role for ω − ωc ≈ 0. Here, from Eq. (8) we can calculate the changing rate of T with respect to ω out to be dT ≈ 16γ 2c η2c γ 2c η2c −4 jω − ωc 3 j ∝ jω − ωc j3 : jρ2 j d ω (10) Obviously, from Eqs. (9) and (10) we have jρ2 j ≪ jρ1 j when ω − ωc ≈ 0. This means that the transmission varies more slowly with ω for γ 2c − η2c 0 and ω − ωc ≈ 0, which gives rise to a better FTTB. In addition, the transmission can reach up to 100% at the flat region, as can be seen from Eq. (8) for γ 2c η2c . We can regard γ 2c η2c as the condition to obtain FTTB. Under this condition, it suggests that the coupling of MO cavity C 1 C 2 and its linked waveguide W 1 W 2 must be equivalent to that of the two cascaded MO cavities C 1 and C 2 . This can be simplified as that the cavity–waveguide coupling must be equivalent to the cavity–cavity coupling. Similarly, the simplified models for the cases of P 2 → P 3 and P 3 → P 1 are shown in Figs. 3(c) and 3(e), respectively, and their direction changes of wave vector are shown in Figs. 3(d) and 3(f ), respectively. Note that the models are suitable for application in the frequencies of very high isolation, i.e., for the transmission of the energy from one port to another with little leakage into the third port. According to the analytical result obtained from the coupled-mode theory, it may help pave the way for building a novel circulator that has FTTB. Note that the concrete parameters in the material are not considered in this section. In the following, the design of a photonic crystal circulator will be presented and investigation of frequency characteristics will be shown. 3. STRUCTURE AND NUMERICAL SIMULATIONS
Fig. 3. Simplified models for the input wave launched from port (a) P 1 , (c) P 2 , (e) P 3 , and the direction changes of the wave vector for the cases of (b) P 1 → P 2 , (d) P 2 → P 3 , (f) P 3 → P 1 . All MO cavities are under the same magnetic field.
In this section, a two-dimensional photonic crystal is designed to build the FTTB circulator. As displayed in Fig. 4, it consists of a triangle-lattice array of dielectric rods with the refractive index n1 3.4 and the rod radius r 1 0.2a 2 × 10−3 m in air, where a is the lattice constant. As noted, some dielectric rods are removed to form three waveguides with a Y-shaped distribution. And the width of the waveguide is adjusted to be L 3a. Three uniform MO rods are symmetrically set at the cross region of the three waveguides. Each MO rod can be regarded as a resonance cavity that is labeled by C i i 1; 2; 3. The radius of the MO rods is denoted by r.
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Fig. 4. FTTB circulator designed in a two-dimensional trianglelattice photonic crystal with three uniform MO cavities C i i 1; 2; 3 coupled to three waveguides, and the triangle dielectric rods H i i 0; 1; 2; 3 added to optimize the coupled structure.
In order to enhance the coupling effect among the MO cavities, a dielectric rod labeled by H 0 is introduced and lies at the center. Furthermore, three dielectric rods denoted by H i i 1; 2; 3 are, respectively, placed between each MO cavity C i and its nearest contacting waveguide W i to modulate the cavity–waveguide coupling. Note that the added dielectric rods H i i 0; 1; 2; 3 have triangle cross sections due to the consideration of the 120° rotational symmetry in the structure. We use FEM to investigate the circulator shown in Fig. 4. The structure domain, surrounded by perfect matched layers, is computed by dividing about 8.5 × 105 grid cells. When an input wave is launched from one of the three ports, we collect the signal at the end of the transmitting (isolated) waveguide and compare it with the input signal. The insertion loss (isolation) is calculated. In order to achieve efficient coupling among MO cavities, we first consider the structure without the dielectric rods marked as H i i 1; 2; 3 in Fig. 4. Yttrium-iron-garnet (YIG) is used for the MO material. Under an applied magnetic field of H 3.4 × 105 A∕m, the relative permeability of YIG material can be expressed by a tensor [21] as # " μr jμk 0 (11) μ −jμk μr 0 ; 0 0 1 where μr 1 ωm ωr iαω∕ωr iαω2 − ω2 and μk ωm ω∕ωr iαω2 − ω2 with ωr μ0 γH , ωm μ0 γM s , γ 1.759 × 105 C∕kg, α 3 × 10−5 , and M s 2.39 × 105 A∕m. And the relative permittivity of YIG material is given by εr 12.4. As the material for MO cavities has been selected, the radius of rod C i i 1; 2; 3 becomes an important factor that affects the coupling efficiency among MO cavities. All MO cavities are under the same magnetic field. For convenience, a parameter K r∕r 1 is defined as the ratio between
Research Article the radius of MO cavities to that of background dielectric rods. Due to the structure symmetry, the properties of the circulator can be studied by choosing an arbitrary port as the input port. Here, for example, we select port P 1 as the input port. Through changing the values of K , the transmissions for the output port P 2 and the isolated port P 3 can be calculated out, as shown by the solid and dashed lines in Fig. 5, respectively. We can see that the transmission T 2 for the output port P 2 reaches the highest value (∼93%) at K 1.3 and the corresponding insertion loss has the lowest value of 10 log1∕T 2 0.31 dB. Meanwhile, the transmission T 3 for the isolated port P 3 reaches the lowest value (∼0.46%) and the corresponding isolation has the highest value of 10 log1∕T 3 21.3 dB. The frequency is 1.0449 · 1010 Hz. The sum of reflection and loss is calculated and shown by the dotted line. The results indicate that the input wave from port P 1 can be efficiently transmitted to port P 2 through strong coupling between the MO cavities C 1 and C 2 , while port P 3 is highly isolated in the system since the coupling between C 1 C 2 and C 3 is relatively weaker. Note that the transmission in Fig. 5 is very low when k r∕r 1 is below 1. This is because the MO rod with small size cannot support a resonance that is within the waveguide dispersion; then the reflection of the wave is very high. In addition, the side length of the cross-section triangle for dielectric rod H 0 is modulated to be 0.25a. Furthermore, as introduced in Section 2, the cavity– waveguide coupling is equivalent to the cavity–cavity coupling. As we know, the cavity–waveguide coupling is sensitive to the connection region of the MO cavity and its linked waveguide, so now we will investigate the structure with the triangle dielectric rods H i i 1; 2; 3 added at the connection regions, with the purpose to modulate the cavity–waveguide coupling to match with the MO cavity–cavity coupling, which has been optimized above. By tuning the size and position of dielectric rods H i i 1; 2; 3, the calculated isolation and insertion loss with respect to the frequency are shown by the solid lines in Fig. 6(a). For comparison, the results without triangle dielectric rods are also given. As can be seen, after introducing the triangle dielectric rods, the working bandwidth of the circulator can be extended to a
Fig. 5. Before adding dielectric rods H i i 1; 2; 3, the transmissions of the output port P 2 and the isolated port P 3 for different value of K r∕r 1 , and the sum of reflection and loss in the structure.
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Fig. 7. Distributions of E z electric field in the three-port circulator operating at the frequency of 1.0449 · 1010 Hz for the input light launched (a) from port P 1 , (b) from port P 2 , and (c) from port P 3 .
Fig. 6. (a) The insertion loss and the isolation. (b) The sum of reflection and loss calculated by scanning frequency. The solid and dashed lines represent cases with and without the dielectric rods H i i 1; 2; 3 added at the junction of each magneto-optical cavity and its corresponding waveguide, respectively.
wider range, i.e., a FTTB with frequency from 1.0374 · 1010 Hz to 1.0524 · 1010 Hz appears in the calculated spectra. In addition, compared to the case without dielectric rods, the isolation increases from 21.3 to 23.3 dB and the insertion loss decreases from 0.31 to 0.023 dB. We also calculated the sum of reflection and loss in Fig. 6(b), in which the solid and dashed lines denote cases with and without dielectric rods H i i 1; 2; 3, respectively. It is evident that the cavity– waveguide coupling and cavity–cavity coupling in the structure are greatly matched, resulting in an increment of transmission. Here, the side length of the cross-section triangle for dielectric rods H i i 1; 2; 3 is adjusted to be 0.2a, and the distance between dielectric rod H i i 1; 2; 3 and the center rod H 0 is optimized as 1.3a. In order to examine the performance of the circulator with more details, the field distributions are calculated, as shown in Fig. 7. We can choose an arbitrary operation frequency in the FTTB; here the frequency of 1.0449 · 1010 Hz is selected to give an example. As can be seen in Fig. 7(a), the input wave launched from port P 1 is almost totally transmitted to port P 2 through the coupling of the MO cavities C 1 and C 2 .
Port P 3 is isolated. Similarly, the wave from port P 2 P 3 will transmit to the output port P 3 P 1 through the coupling of the MO cavities C 2 C 3 and C 3 C 1 , isolating port P 1 P 2 , as shown in Figs. 7(b) and 7(c), respectively. These results demonstrate that the device functions as an optical circulator in which the transmission with a single-direction circulation of P 1 → P 2 , P 2 → P 3 and P 3 → P 1 is realized. Wave propagation is in good agreement shown in Section 2. Note that each MO cavity can provide a 30° rotation of the wave vector; thus, two cascaded MO cavities can realize a 60° rotation of the wave vector between the input and the output waveguides. In addition, this type of circulator has a compact structure and can provide a broad operation bandwidth, such that it may provide a good platform for constructing integrated photonic circuits. 4. CONCLUSIONS In conclusion, we propose a new type of FTTB circulator by coupling the cascaded MO cavities to input and output waveguides, and analyze the condition for achieving FTTB in such a structure theoretically and numerically. A three-port circulator as an example has been designed in a twodimensional triangle-lattice photonic crystal. In order to achieve excellent FTTB, the structure is optimized to make the cavity–waveguide coupling perfectly match with the cavity–cavity coupling. Results indicate that a low insertion loss and a high isolation can be obtained. The circulator with a large operation bandwidth presented here can bring great convenience in integrating photonic devices in large-scale optical circuit systems, such as optical routing or isolating uses. Funding. National Natural Science Foundation of China (NSFC) (61275043, 61307048); Natural Science Foundation of SZU (201456); Open Fund of Shenzhen
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