Optics Communications 285 (2012) 3763–3769

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Optics Communications

0030-40

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journal homepage: www.elsevier.com/locate/optcom

Magnetically tunable silicon-ferrite photonic crystals for terahertz circulator

Fei Fan, Sheng-Jiang Chang n, Chao Niu, Yu Hou, Xiang-Hui Wang

Institute of Modern Optics, Nankai University, Key Laboratory of Optoelectronic Information Science and Technology, Ministry of Education, Tianjin 300071, China

a r t i c l e i n f o

Article history:

Received 31 January 2012

Received in revised form

22 April 2012

Accepted 23 May 2012Available online 6 June 2012

Keywords:

Terahertz

Photonic crystals

Circulator

Magneto-optical material

18/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.optcom.2012.05.044

esponding author.

ail address: sjchang@nankai.edu.cn (S.-J. Chan

a b s t r a c t

The gyromagnetic properties of ferrite materials and the nonreciprocal property of a silicon-ferrite

photonic crystal cavity are investigated in the terahertz region. Through the structure optimization and

analysis of defect mode coupling, we design a magnetically tunable circulator, of which central

operating frequency can be tuned from 180 to 205 GHz and the maximum isolation is 65.2 dB.

Moreover, the further study shows that the gyrotropy, dispersion, and ferromagnetic loss of ferrite

materials under the different external magnetic fields greatly affect the transmission and isolation

property of this device. This circulator is flexible to realize functions of controllable splitting, routing,

filtering and isolation by changing the external magnetic field for the THz applications.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Terahertz (THz) communications have attracted a lot of inter-ests, especially in the sub-THz frequency range of 0.1–0.3 THz[1–3]. Photonic crystals (PCs) have been confirmed an effectiveway to transmit as well as control THz waves, and facilitateminiaturization of devices [4–6]. Non-reciprocity of gyrotropicPCs has been studied theoretically and experimentally in recentyears, known as unidirectionality or one-way waveguide [7–10].The key for realizing one-way waveguide is the breaking of time-reversal symmetry. These integrated nonreciprocal devices canplay important roles with several functions, such as splitting,directional control and isolation in the future THz communicationsystems.

Circulator is a kind of nonreciprocal devices. Circulators formedof magneto-photonic crystals (MPCs) in the optical wavelengthshave been studied [11–15], and by using the temporal coupled-mode theory, the geometric structure of the circulator has beenoptimized to achieve impedance matching and the maximumisolation [12,14] However, to our knowledge, any dispersion andlosses of magnetic materials have not been considered in theprevious reports, and there have not been any reports on the PCscirculator in the THz region so far. Several reports on the electro-magnetic parameters of ferrite materials have shown that somekinds of ferrites have low absorption loss in the THz region[16–18]. A small absorption loss and the existence of the ferro-magnetic resonance (FMR) [19,20] indicate that the ferrite is a kind

ll rights reserved.

g).

of promising materials for the THz wave. In our previous article[21], the transmission properties of ferrite PCs have been studied inthe THz region. Moreover, a tunable filter and a polarizationcontroller have been designed for the THz wave. In this paper wewill focus on the nonreciprocal properties of MPCs in the THzregion. Moreover, we will discuss the tuning property of theproposed circulator under the different external magnetic fields,and consider the effects of the ferromagnetic loss, dispersion, andgyrotropy of the ferrite in the discussions.

2. Theoretical analysis

2.1. Properties of the ferrite in the THz regime

Fully magnetized ferrite under an external static magneticfield is a gyromagnetic material, of which permeability tensor canbe derived from the Landau–Lifschitz equation in the followingform [22]:

2m ¼ ½m jk 0

�jk m 0

0 0 m0

� ð1Þ

The tensor element m and k are given by

m¼ m0ð1þoexom

o2ex�o2

Þ ð2aÞ

k¼ m0

oom

o2ex�o2

ð2bÞ

where oex¼gBex, om¼m0gMs, g is gyromagnitic ratio, m0 is thepermeability in the vacuum, o is the circular frequency of the

F. Fan et al. / Optics Communications 285 (2012) 3763–37693764

incident wave, Bex is the external magnetic flux density, and Ms isthe saturation magnetization of ferrite materials. Consideringferromagnetic loss, oex should be replaced by oex� jm0gDH/2 inEq. (2), where DH is the FMR linewidth of ferrite materials. Yanget al. [16] measured the absorption coefficient of LuBiIG singlecrystal with less than 0.3 cm�1 in the range of 0.2–1 THz byterahertz time-domain spectroscopy. LuBiIG has the dielectric con-stant of e¼4.85, the FMR linewidth of 2.8–5.1 Oe (1 Oe¼79.61 A/m)and the saturation magnetization of 1560 G (1 G¼10�4 T) [16,23].The electro-magnetic parameters of LuBiIG will be used in thefollowing calculations, and the ferromagnetic loss is determinedby the FMR linewidth of 5.1 Oe in our simulations.

PCs structure is a kind of non-uniform medium. Therefore, fora non-uniform system including gyromagnetic materials, theMaxwell equation is in the form of

e�1Ur � ½

2m�1Ur � E

-� ¼o2=c2

UE-

ð3Þ

where e is a constant, while2m is a function of space position. We

will focus on transverse electric (TE) wave ðEx ¼ Ey ¼ 0, Eza0Þ inPCs. Because the magnetic field of transverse magnetic (TM) wavedoes not interact with the magnetic dipoles of the ferrite, thepermeability for TM wave is a scalar of m0. The magnetic field ofTE wave interacts with the magnetic dipoles of the ferrite, so thepermeability tensor for TE wave in PCs structures cannot besimply modeled by an effective isotropic medium with thepermeability of a scalar value. However, finite difference time-domain (FDTD) method or finite element method (FEM) can beused for anisotropic lossy mediums. By using these methods wecan consider off-diagonal element k and on-diagonal element m ofthe permeability tensor into our simulation.

Fig. 1. The permeability tensor of ferrite changes with the external magnetic field incre

(c) k/m as the function of magnetic fields and THz frequencies.

The tensor element m and k are calculated by Eq. (2) under thedifferent external magnetic fields. As shown in Fig. 1a, m and kdecrease with the increase of magnetic field. When the magneticfield is smaller than 7.15 T, the imaginary part of permeability islarger than zero, so the ferromagnetic loss cannot be neglectedwhen the frequency is close to the FMR. Furthermore, thegyrotropy of ferrite materials is characterized by k/m [24].Fig. 1b and Fig. 1c show that both m and k/m exhibit a largedispersion as a function of magnetic field and frequencies. Whenthe magnetic field becomes larger or THz frequency is lower,k/m is smaller, so the gyrotropy becomes relatively weak. Thecalculations above including the ferromagnetic loss and disper-sion of the ferrite will be considered in the following discussions.

2.2. Structure of the circulator

The structure of a three-port junction circulator is shown inFig. 2a, with three 1201 rotational symmetry branches of wave-guides coupled to a cavity at the center. The two-dimensionaltriangular PCs rods with the radius of r1¼0.25a, where a is thelattice period. The normalized frequency of a/l¼0.56 correspondsto the frequency of 0.2 THz with the lattice period of a¼840 mm.Three lines rods and the rod at the center are removed to form thewaveguides and the cavity, respectively. Because the loss of high-resistivity silicon is smaller than that of LuBiIG in the THz regime(absorption coefficient of 0.02 cm�1 vs. 0.3 cm�1), we use siliconrods as the PC waveguide (PCW), and LuBiIG rods as the PC cavity.When a uniform static magnetic field Hex ¼Hz perpendicular tothe x–y plane is applied, the function of circulator can be realizedas shown in Fig. 2a.

asing. (a) Real and imaginary part of m and k at 0.2 THz; the real part of (b) m and

Fig. 2. Schematic diagram of the circulator. The circles in red color indicate the silicon rods with the radius of r1, the circles in blue color indicate the silicon rods with r2

near the ferrite rods, and the circles in green color indicate the ferrite rods with r3. The arrows indicate the directions of energy flows excited at three ports. (a)–(c) is three

different cavity structures. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

F. Fan et al. / Optics Communications 285 (2012) 3763–3769 3765

Compared with three cavity structures shown in Fig. 2, thecavity shown in Fig. 2b has a low coupling coefficient betweenwaveguide and cavity due to the high dielectric constant e¼11.7of silicon, resulting in a low transmission from input to output[25]; the LuBiIG rod in the center shown in Fig. 2c has a verylow Q value due to its incomplete cavity structure, leading to aweak mode confinement [25]. The LuBiIG rods cavity shown inFig. 2a has a large coupling coefficient (e¼4.85 of LuBiIG) and acavity structure with a high Q. The following discussions willshow this cavity structure with a high transmittance and a veryhigh isolation.

A guided mode within the frequency range of 0.485–0.575 c/alocates in the band-gap of PCW, and the cavity without anexternal magnetic field has only one monopole mode with theresonance frequency of 0.495 c/a. The light wave at this resonancefrequency can be coupled between the cavity and waveguides.According to the previous reports [11–14], two orthogonal doublydegenerate dipole modes exist without an external field, whilethe external magnetic field makes them couple and forms twolinear combinations corresponding to left and right-rotatingmodes with frequencies oL and oR. Fig. 4b shows one of thesetwo rotating modes in our cavity structure. Only rotating modescan well support the function of the circulator, and the frequencyðoLþoRÞ=2 is just the central operating frequency of the circu-lator [12]. The largerDo between oL and oR is, the intensity ofmagneto-optical coupling is stronger. The magneto-optical cou-pling strength VLR between the two rotating modes can befollowed as:

VLR ¼i

2

ffiffiffiffiffiffiffiffiffiffiffiffiffioLoRp R

kzUð E!n

L � E!

RÞdVffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRm9 E!

L92dVRm9 E!

R92dV

q ð4Þ

where E!

L and E!

R are the electrical fields of left and right-rotating defect modes, respectively. We can see from Eq. (4)that the magneto-optical coupling strength is not only deter-mined by the gyrotropy k/m, but also closely related to thespatial overlap between the ferrite domain and the defectmodes field.

According to Eq. (4), the maximum magneto-optical couplingstrength can be obtained by structural optimization. The adjust-ment of the rods radii r2, r3 shown in Fig. 2 may change the spatialarrangement of the ferrite domain and the defect modes fields.The isolation directly reveals the magneto-optical couplingstrength, and it followsI¼�10lgðTiso=ToutÞ, where Tiso and Tout aretransmittances of the isolated port and output port respectively.

The optimization results of the cavity in Fig. 2a under themagnetic field of 7.17 T (m¼1.26 and k¼0.26 at 0.55 c/a) areshown in Fig. 3. When r3¼0.25a, there are two isolation peaks of65.2 dB at r2¼0.247a and 63.5 dB at r2¼0.254a as shown inFig. 3a. Other geometric parameters of r2 and r3 in Fig. 3a andFig. 3b lead to a significant decrease of the isolation. We alsooptimized the structures in Fig. 2b and c under the same externalfield of 7.17 T, which are shown here. For the cavity in Fig. 2b,when r2¼0.247a and r3¼0.29a, the best isolation of 28.2 dB isobtained, but transmission of output is only 50%; for the cavity inFig. 2c, when r2¼0.25a and r3¼0.3a, the best isolation of 15.8 dBis obtained, both far lower than the best isolation of 65.2 dB inFig. 3a. This confirms the previous analysis. Consequently,through the structure optimization, the isolation of the circulatoris greatly increased from 36.7 dB to 65.2 dB, higher than that ofany previous reports. Therefore, we select the geometric para-meters of r1¼0.25a, r2¼0.247a, r3¼0.25a in the followingdiscussions.

2.3. Defect mode analysis

The FEM method is used to find all of the eigenmodes in thecavity, and three typical ones at 7.17 T are shown in Fig. 4. Theroles of the rotating dipole modes shown in Fig. 4b have beendiscussed above. Unlike the previous reports, there are threedipoles rotating around each ferrite rods in this silicon-ferritecavity, and they together form the rotating modes E

!L and E

!R.

Because the field of each dipole mode has a good overlap with theferrite rod, the strong magneto-optical coupling is realized. Otherdefect modes cannot support the function of the circulator welldue to the weak magnet-optical coupling. For example, themonopole mode shown in Fig. 4a does not show any non-reciprocity, because its frequency is far from the FMR and there-fore the gyrotropy k/m is too small, according to the gyrotropydispersion shown in Fig. 1c. The hybrid mode in Fig. 4c onlyshows a weak non-reciprocity due to a small overlap between theferrite rods and the mode field in the cavity center. Therefore,although there are several defect modes in the silicon-ferritecavity, only the rotating dipole modes play a key role in thecirculator.

According to Fig. 1, the external magnetic field can vary theproperties of the silicon-ferrite PC cavity. We can see from Fig. 5that the operating frequency ðoLþoRÞ=2 can be tuned from 0.523to 0.563 c/a by changing external magnetic fields because of thedifferent permeability tensors. Because of the dispersion of k/m,

Fig. 3. Optimization results of the cavity. (a) Isolations of Cavity A with r2¼0.24–0.26a, r3¼0.25a, and their corresponding frequencies; (b) isolations of Cavity A with

r3¼0.245–0.255a, r2¼0.247a and 0.254a; (c) optimized isolation spectrum of Cavity B with r2¼0.247a and r3¼0.29a; (d) optimized isolation spectrum of Cavity C with

r2¼0.25a and r3¼0.3a.

Fig. 4. Electric field pattern of defect modes in the cavity at 7.17 T. (a) A monopole mode at the frequency of 0.492 c/a; (b) a rotating mode without any vibrations at 0.552

c/a; (c) a rotating mode with vibration 0.572 c/a. The arrows indicate the rotational component of the vector, and the symbol ‘z’ indicates the vibration component along

the z direction. LuBiIG rods are labeled by the symbol ‘A’, ‘B’ and ‘C’.

F. Fan et al. / Optics Communications 285 (2012) 3763–37693766

the magneto-optical coupling becomes weak when the magneticfield is larger and the THz frequency is lower, thus the maximummagneto-optical coupling occurs in the middle of the frequencyrange. The mode confinement becomes weak in the edge of thebandgap, so the minimum magneto-optical coupling will belocated in the upper edge of the range.

3. Numerical demonstrations of circulator

The steady-state electric field patterns of the device are givenby the FEM method. Without an external magnetic field applied,the device realizes a Y-type splitter at the frequency of 0.492 c/a,shown in Fig. 6a. Under the external magnetic field of 7.17 T,

F. Fan et al. / Optics Communications 285 (2012) 3763–3769 3767

the patterns of the input excited at Port A, B and C are shown inFig. 6b, c and d, respectively. The reverse magnetic field case isalso shown in Fig. 6e. The results show a good transmission to theoutput port and zero transmission to the isolated port at thefrequency of 0.554 c/a, so this device can realize the function ofcirculator at the operating frequencyðoLþoRÞ=2. It agrees wellwith the results shown in Figs. 5 and 7.

4. Discussion

To analyze the tuning property under the different externalmagnetic fields, the transmission and isolation spectra excited at

Fig. 5. The position of defect modes changes with the external magnetic field.

Fig. 6. Electric field patterns of the circulator. (a) The pattern at the frequency of 0.495 c

0.5545 c/a under the magnetic field of 7.17 T are excited at Port A, B and C, respective

Port A are shown in Fig. 7 by the FDTD method. Firstly, we focuson the transmission properties of the device. All of the maximumtransmittances at output port are higher than 85%, and thepassband of the output port moves to higher frequencies withthe external field increasing. The results show that the device canrealize the function of a tunable filter, of which tuning range isfrom 180–205 GHz. The results coincide with the movement ofdefect modes shown in Fig. 5. In addition to the role of thedispersion and cavity, the ferromagnetic loss at higher THzfrequencies near the FMR also has effects on the transmittanceand the line-shape of the spectra.

Secondly, the isolation properties of the device can be revealedin Fig. 7b. Because we optimize the structure under 7.17 T inSection 2.2, the maximum isolation is obtained at this magneticfield. The isolation initially increases with the increasing magneticfield, and declines dramatically after reaching the maximum at7.17 T. The main reason for the low isolation of the 7.24 T is dueto a weak gyrotropy k/m far from the FMR and a weak modeconfinement at the edge of the photonic bandgap. Fig. 8a showsthe rotating mode under the magnetic field of 7.24 T. Differentfrom the dipole shown in Fig. 4b, it also has left or right-handrotating components at x–y plane, while each dipole is vibratingalong the z direction. The gyrotropy is not strong enough to breakthe degeneracy of the two modes. The field pattern of thecirculator shown in Fig. 8b also agrees well with the resultsshown in Fig. 7b. The THz waves partly transmit through theisolated Port C.

Therefore, on the one hand, the PC cavity affects the modeconfinement and the spatial overlap of the magnetic materialdomain and the defect modes. This point can be optimizedthrough adjusting the cavity structure. On the other hand, thegyrotropy and ferromagnetic loss of the ferrite also greatly affectthe transmission and isolation property of this device as afunction of magnetic field and THz frequency. To advance theperformance of the device, novel magnetic materials shouldprovide a higher gyrotropy and lower loss in the THz regime,and should reduce the magnetic field applied.

/a without external magnetic field. (b), (c) and (d) The patterns at the frequency of

ly. (e) The patterns applied a reverse magnetic field of 7.17 T excited at Port B.

Fig. 7. Transmission and isolation spectra under the different external magnetic fields. (a) Transmission spectra from Port A to Port B; (b) isolation spectra between Port C

and Port B.

Fig. 8. (a) The defect mode pattern and (b) the field pattern of the circulator under the external field of 7.24 T.

F. Fan et al. / Optics Communications 285 (2012) 3763–37693768

5. Conclusions

In summary, we investgated the gyromagnetic properties of asilicon-ferrite PCs defect in the sub-THz region. Through thestructure optimization and analysis of defect mode coupling, wedesigned a magnetically tunable circulator, of which centraloperating frequency can be tuned from 180 to 205 GHz and themaximum isolation is 65.2 dB. The further study shows that thegyrotropy, dispersion, and ferromagnetic loss of ferrite materialsunder the different external magnetic fields greatly affect thetransmission and isolation property of this device. This circulatorcan realize the controllable splitting, filtering and isolating forTHz applications.

Acknowledgment

This work is supported by the National High Technology Researchand Development Program of China (Grant No. 2011AA010205), theNational Natural Science Foundation of China (Grant No. 61171027),the Natural Science Foundation of Tianjin of China (Grant No.10JCZDJC15200) and the Doctoral Fund of Ministry of Education ofChina (Grant No. 20090031110033).

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