1 Introduction

Circulators are widely used for protecting the light source from harmful reflections of load devices, extracting feedback signals for optical diagnosis or detecting, and designing optical bridges and optical add-drop multiplexers. In the past decade, great efforts have been devoted to optical circulators based on photonic crystals [1] due to their compactness and suitability for photonic integrated circuits. A kind of high-transmission three-port and four-port circulators based on a single magneto-optical (MO) cavity are first proposed by employing reversed external magnetic fields to enhance the resonance effect of cavity in photonic crystal [2, 3]. Then, highly compact Y-typed, W-typed and cross-typed circulators of different types are further designed with modulated magneto-optical cavities [4–7]. Recently, a side-coupled cavity has been adopted to solve the unsymmetrical problem for designing T-format photonic crystal circulators [8].

All of the above-mentioned circulators are based on single magneto-optical cavity, and their efficiencies and structural symmetries have attracted great attention. However, their bandwidth is generally narrow, which sets some limitations to their applications. To our knowledge, until now only one method has been proposed to realize broadband photonic crystal circulators [9]. It is based on directional coupling between one-way photonic crystal edge states and conventional two-way waveguides in square-lattice photonic crystals.

In our paper, another new approach is presented. As we know, for resonance-based optical filters, a single-defect structure always leads to Lorentzian response in which a sharp narrow peak exists in transmission spectra [10, 11]. It is known that a flat-top band can be obtained in high-order filters. Moreover, high-order filters can be created by cascading multiple resonators [12–14]. This idea can be transferred to enlarge the bandwidth of optical circulators. So here, we employ the coupling effect of cascaded magneto-optical defect rods to realize broadband circulators based on photonic crystals. Furthermore, with the rapid development of large-scale photonic integrated circuits, multi-port circulators play important roles in optimizing and enhancing the integration of systems and thus need to be studied. However, in this paper we mainly focus on the investigations of a six-port circulator as an example to show the design method of broadband circulators.

2 Theoretical considerations

Different from the circulators reported in the literatures, we propose to cascade six magneto-optical rods as a ring and couple them to six waveguides symmetrically, as shown in Fig. 1a. The MO rods and the waveguides are denoted by M i and W i (i = 1, 2, …, 6), respectively. And the six corresponding waveguide ports are denoted by P i . This circulator is designed to realize the nonreciprocal propagation among the six ports, denoted by P 1 → P 2, P 2 → P 3, P 3 → P 4, P 4 → P 5, P 5 → P 6 and P 6 → P 1.

Fig. 1
figure 1

a Schematic diagram of the six-port circulator; b the simplified model for the case of the input wave launched from port P 1

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In this structure, each MO rod can be regarded as a cavity, which can support degenerate odd mode |o〉 and even mode |e〉 without external magnetic field [8]. Their superposition forms a standing dipole mode at the MO rod. Thus, the wave at each MO rod will couple to its adjacent MO rods on both sides simultaneously owing to the structure symmetry. Furthermore, when an external magnetic field is applied, the modes |o〉 and |e〉 will couple with each other under the MO effect, and then, the rotation of the formed dipole mode at the MO rod takes place, so that the coupling orientation at the MO rod changes accordingly. For example, if an input wave is launched from port P 1, by choosing proper parameters of the magnetized MO rods, the dipole mode at MO rod M 1 can be rotated by 60°, just pointing to the MO rod M 2. It indicates that the coupling between the MO rods M 1 and M 2 becomes strong and the coupling between the MO rods M 1 and M 6 becomes weak. As a result, the input wave will transmit to port P 2 and be isolated from other ports, i.e., the case of P 1 → P 2. For this case, the transmission can be regarded as the coupling effect of two cascaded MO rods M 1 and M 2 in the two line-defect waveguides W 1 and W 2, as shown in Fig. 1b. Under the MO effect, the formed dipole modes are denoted by dashed circles. And the mode amplitudes at M 1 and M 2 denoted by a i (i = 1, 2) can be described by coupled mode theory in time domain [15] as follows:

$$\frac{{{\text{d}}a_{i} }}{{{\text{d}}t}} = \left( {j\omega_{i} - \beta_{i} } \right)a_{i} + \sqrt {2\beta_{i} } S_{ + i} - j\kappa a_{l}$$
(1)
$$S_{ - i} = - S_{ + i} + j\sqrt {2\beta_{i} } a_{i} \quad (i \ne l,\;i,l = 1,2)$$
(2)

where ω i (i = 1, 2) is the resonance frequency of the MO rod M i , β i is the coupling coefficient between the MO rod M i and its corresponding waveguide W i , S +i and S ‒i are the amplitudes of the incoming and the outgoing waves in the waveguide W i , respectively, and κ is the mutual coupling coefficient between the two MO rods M 1 and M 2. Considering the symmetry in the structure, we have ω 1 = ω 2 and β 1 = β 2. We can define symbols ω r  = ω 1 = ω 2 and β = β 1 = β 2. For the case of P 1 → P 2, it has S +2=0. Assuming that S ±i and a i (i = 1, 2) have exp(jωt) time dependence, we can obtain the following relations from Eqs. (1) and (2):

$$j\omega a_{1} = \left( {j\omega_{r} - \beta } \right)a_{1} + \sqrt {2\beta } S_{ + 1} - j\kappa a_{2}$$
(3)
$$j\omega a_{2} = \left( {j\omega_{r} - \beta } \right)a_{2} - j\kappa a_{1}$$
(4)
$$S_{ - 2} = j\sqrt {2\beta } a_{2}$$
(5)

Using Eq. (4), we can obtain:

$$\frac{{a_{2} }}{{a_{1} }} = \frac{ - j\kappa }{{(j\omega - j\omega_{r} + \beta )}}$$
(6)

From Eqs. (3) and (5), the transmission T = |S ‒2/S +1|2 can be calculated as:

$$T = \left| {\frac{{2j\beta a_{2} }}{{(j\omega - j\omega_{r} + \beta )a_{1} + j\kappa a_{2} }}} \right|^{2}$$
(7)

Substituting Eq. (6) in Eq. (7), we can obtain as:

$$T = \frac{{4\beta^{2} \kappa^{2} }}{{(\omega - \omega_{r} )^{4} + 2\left[ {\beta^{2} - \kappa^{2} } \right](\omega - \omega_{r} )^{2} + \left[ {\beta^{2} + \kappa^{2} } \right]^{2} }}$$
(8)

Note that the terms (ω‒ω r )4 and (ω‒ω r )2 appear in the denominator simultaneously. This is different from the single-cavity structure in which only (ω‒ω r )2 appears. The term (ω‒ω r )2 represents Lorentzian-shaped response with a sharp peak appears in transmission. If the term (ω‒ω r )2 does not exist, the higher-order term (ω‒ω r )4 indicates that the transmission T varies more slowly with ω than that in a Lorentzian response by noting that δω = ω‒ω r is small at the frequency near ω r . However, for the frequency far from ω r , δω is large, so that the transmission T varies more quickly with ω than that in a Lorentzian response. As a result, the proposed structure will give rise to a better flat-top response than that in single-cavity structure. Thus, circulators with a broadband transmission can be realized when the coefficient of the term (ω‒ω r )2 is equal to 0. We have:

$$\beta^{2} = \kappa^{2}$$
(9)

The above Eq. (9) suggests that the coupling effect between M 1 and M 2 (simply called as rod–rod coupling) must be equal to that between M 1 and its linking waveguide W 1, or M 2 and its linking waveguide W 2 (simply called as rod–waveguide coupling). This condition can be maximally satisfied by designing appropriate structure and optimizing the corresponding parameters. Another important reason for the better flat-top ability is that the quality factor of the rod cavity in the proposed structure is much less than that in circulators reported in literatures because the former is formed only by an MO rod and one reflecting rod on each of its side, while the latter is formed by an MO rod and several reflecting rods. Furthermore, we consider whether both the broadband condition and maximum transmission can be realized simultaneously. Substituting the relation β 2 = κ 2 in Eq. (8), we can see that T = 1 at the resonance frequency ω r . Therefore, we can obtain broadband and high transmission as well.

3 Structure and numerical discussions

Now, we use a two-dimensional photonic crystal structure to demonstrate our design concept. It is composed of triangle-lattice background dielectric rods in air, as shown in Fig. 2. Note that the triangle-lattice structure is convenient for building circulators with large number of ports than square-lattice structures, especially for six-port circulators. The refractive index and the radius of the dielectric rods are n 0 = 3.4 and r 0 = 0.2a, respectively, where a is lattice constant. In this photonic crystal, six waveguides in 60° rotational symmetry are formed by removing some dielectric rods from the structure and the width of waveguide is adjusted to be d = 3a. The radius of the MO rods is denoted by r M . And a dielectric rod labeled D 0 is added at the center, and its cross section is set as hexagon by mainly considering the 60° rotational symmetry in this structure. Furthermore, in order to enhance the coupling effect of each MO rod with its linking waveguide, six triangle dielectric rods labeled D i (i = 1, 2, …, 6) are, respectively, placed at the rod–waveguide junctions. We choose ferrite material for the MO rods. The relative permittivity of the ferrite material is given by ε r  = 12.4, while its relative permeability takes a form of a second-rank tensor [μ] = [μ r , jμ k , 0, ‒jμ k , μ r , 0, 0, 0, 1] under an applied direct-current magnetic field in the z-direction [16], where μ r  = 1+ω m (ω 0 + iαω)/[(ω 0 + iαω)2‒ω 2] and μ k  = ω m ω/[(ω 0 + iαω)2‒ω 2] with ω 0 = μ 0 γH 0, ω m  = μ 0 γM s , γ = 1.759 × 105 C/kg. The parameters α = 3×10‒5 and M s  = 2.39 × 105 A/m are taken for the material. Here, the applied field is set to be H 0 = 3.4 × 105 A/m.

Fig. 2
figure 2

Structure of the six-port circulator in triangle-lattice photonic crystal

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Due to the structure symmetry, the properties of the circulator can be studied by choosing an arbitrary port as the source port. Here, we select port P 1 as the input port. When the electromagnetic wave is launched from port P 1, we collect the wave powers at the output port P 2 and the isolated ports P 3, P 4, P 5 and P 6, respectively. The insertion loss and the isolation of the device defined as 10log(1/T out) and 10log(1/T iso) are calculated according to the powers at the input, output and isolated ports, where T out and T iso denote the transmissions for the output port and the isolated port, respectively.

We first investigate the situation without introducing the dielectric rods D i (i = 1, 2, …, 6). The radius of the six uniform MO rods is optimized to achieve the high-efficient coupling of the two cascaded MO rods M 1 and M 2 in the photonic crystal circulator. For convenience, we define a parameter R = r M/r 0 denoting the ratio of the radius of MO rods to that of background dielectric rod. For different values of R, the transmissions are calculated through scanning the normalized frequency. The results are given in Fig. 3, in which the line with stars represents the insertion loss for the output port P 2, and the lines with squares, up triangles, down triangles and diamonds represent the isolations for the isolated ports P 3, P 4, P 5 and P 6, respectively. It shows that the optimal value is R = 1.45. We can see that the insertion loss for output port P 2 reaches to the lowest as 0.31 dB, and the isolations for isolated ports P 3, P 4, P 5 and P 6 reaches to the highest as 29.23, 32.02, 32.79 and 28.41 dB, respectively. Meanwhile, the side length of the cross section for dielectric rod D 0 is modulated to be 1.2a. The results indicate that the coupling between MO rods M 1 and M 2 is much stronger than that of MO rods M 1 and M 6. Then, the input wave launched from port P 1 can transmit to port P 2 through the coupling of the MO rods M 1 and M 2; i.e., each MO rod provides a 60° rotation of wave vector.

Fig. 3
figure 3

Calculated insertion loss and isolations at different R = r M/r 0 when the input wave launched from port P 1

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After obtaining the efficient rod–rod coupling, the next procedure is to modulate the rod–waveguide coupling to match with the rod–rod coupling, according to the broadband condition predicted by Eq. (9). This can be realized by adding a triangle dielectric rod at the junction between each MO rod and its linking waveguide, i.e., the rods labeled D i (i = 1, 2, …, 6) shown in Fig. 2. Note that, we use triangle rods but not circular rods here. The reason is that the equivalent refractive index for the triangle rod is a monotonic change, while that for the circular rod is not. In order to modulate the rod–waveguide coupling, it is more suitable to use rods with a monotonic change in equivalent refractive index to match the rod and the waveguide. Through optimizing the rod parameters, detailed spectra of insertion loss and isolation with respect to normalized frequency are shown in Fig. 4a. For comparison, the results without adding triangle dielectric rods are given in Fig. 4b. As can be seen that, after introducing the triangle dielectric rods, the performance of the circulator becomes better, such as the insertion loss for port P 2 decreases to 0.017 dB from 0.31 dB, and the isolations for ports P 3, P 4, P 5 and P 6 increases from 29.84 to 29.23 dB, 33.10 to 32.02 dB, 32.96 to 32.79 dB and 29.05 to 28.41 dB, respectively. At the same time, the operating bandwidth is extended to a wider range; i.e., the normalized bandwidth frequency is from a/λ = 0.3080 to a/λ = 0.3086, corresponding to a relative bandwidth of 1.72 %. The reason for the improvements is that the rod–waveguide coupling and rod–rod coupling in structure are ideally matched through the careful parameter modulation. In the modulation, the side length of the cross section for the rods D i (i = 1, 2, …, 6) is adjusted to be 0.2a.

Fig. 4
figure 4

Insertion loss and the isolations calculated by scanning the normalized frequency: a with and b without adding the dielectric rods D i (i = 1, 2, …, 6)

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In order to check the feasibility of the circulator, the field patterns of E z for different cases are simulated, as shown in Fig. 5. The operating frequency of a/λ = 0.3083 is chosen in the operating band. For the example of P 1 → P 2 shown in Fig. 5a, the input wave launched from port P 1 is almost totally transmitted to port P 2 (the output port), isolating ports P 3, P 4, P 5 and P 6 (the isolated ports). According to the rotational symmetry in the structure, single direction circulation propagation can be accomplished among the six ports, i.e., P 1 → P 2, P 2 → P 3, P 3 → P 4, P 4 → P 5, P 5 → P 6, and P 6 → P 1. To save space, only two cases are shown, while the other four cases are omitted. Figure 5b shows the case for P 6 → P 1. In addition, the circulator has a characteristic of broad bandwidth, which provides more convenience in working with other devices in integrated photonic circuits. Note that, according to the requirements of users, broadband circulators with other number of ports (such as 3, 4, 5 or 7 ports) can also be realized, if the rod–waveguide coupling is equal to the rod–rod coupling in the designed structure, i.e., satisfying the broadband condition given in Eq. (9) in Sect. 2.

Fig. 5
figure 5

Distributions of electric field E z operating at the frequency of a/λ = 0.3083 for input wave launched a from port P 1 and b from port P 6

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4 Conclusions

In summary, we present a method for the design of broadband six-port circulator based on a magneto-optical-rod ring coupled to waveguides in photonic crystal and predict the broadband conditions by coupled mode theory. The results show that the insertion loss of the device can reach to 0.017 dB and isolations for the isolated ports are all above 29 dB. The proposed circulator with multiple ports can provide flexibility in realizing functions of directional control, optical routing or isolating, especially for modulating or improving the integration properties in large-scale optical circuit systems.