Magnetic materials have attracted considerable attention for their applications in high density magneto-optical storage devices, which is not only appealing scientifically, but also makes the magnetic materials promising for a wide range of applications [1-11]. Due to their unique optical properties, magnetic materials have been introduced into the photonic crystals, forming magnetophotonic crystals (MPCs), i.e., the photonic crystals with at least one magnetic material component [2,3]. Now the optical properties can be mediated and controlled by the external electric or magnetic field due to the existence of magnetic materials. The special importance of MPCs can be ascribed to the existence of magneto-optical effects, for example, Kerr effect and Faraday rotation [12-28]. They were discovered by Kerr and Faraday, respectively, and are now widely used in integrated optics and magneto-optical devices for magnetic domain imaging, mapping of hysteresis loops and high density recording .
Magnetic materials with large magneto-optical responses are always the attractive ones used in magnetophotonic crystals. In contrast to the corresponding three-dimensional magnetic structures, magnetic materials exhibit a larger magneto-optical effects due to the light's confinement in the MPCs, offering a genuine chance to put their optical responses into applications. The optimized MPCs have been shown to behave like mixed systems, with a coexistence of high transmittance and large magneto-optical effects [12-28]. All possible configurations are proposed to achieve strong magneto-opitcal effects, including the ordinary cavity-based, multilayered periodic and aperiodic structures. Furthermore, the diffracted magneto-optical enhancement is also demonstrated in the grating structures theoretically and experimentally, which greatly reduces the thickness of the device contrary to multilayered structures and miniaturizes magneto-optical devices in integrated optics [29,30].
The ability to tune the otpical properties by an external stimulus is a key issue of modern optoelectronics. Although most attention has so far been focused on the magneto-optical properties of the given structures, tunable magneto-optical devices have important applications and will be respected in optical switches and displays. There are few reports concerning with the tunable magnetophotonic crystals [31-33]. For example, it is possible to manipulate the magnetic order of magnetic conducting spheres using the magnetic field, thus forming the tunable magnetophotonic crystals . Semiconductor quantum well has been discussed as possible candidates for achieving artificial tunable MPCs . From the application-oriented perspective it would be very desirable if tunable magneto-optical effects could be achieved by controlling applied electric fields, thus making such effect a potential proposition. Therefore, to search the alternate scheme is of great importance in tunable magneto-optical electronics. It was known that nematics liquid crystals (NLCs) are good choices for tunable photonic crystals due to their unique sensitivity to temperature, the electric, magnetic field, or lights itself . However, seldom reports concerning tunable magneto-optical effects based on the NLCs have been presented in the literatures. Therefore, it would be interesting to investigate a possible way of creating electrically controlled magneto-optical effects in the MPCs with the NLCs. The application of liquid crystals in the MPCs offers new opportunities for the tunable optoelectronic devices.
Liquid crystals are materials that display a phase of matter whose properties lie between those of a conventional liquid and a solid crystal. They are a class of materials particularly attractive for liquid crystal displays and optical electronic applications due to their high sensitivity to the external stimulus and have been studied experimentally extensively [35-37]. There are three basis kinds of liquid crystals: NLCs, cholesteric liquid crystals and smectic liquid crystals [38-47]. Here we only focus attention on the NLCs, which is characterized by molecules that have no positional order but tend to align along the same direction. Due to thermal random motion, friction and collision between molecules, not all molecules align along a certain direction and their directions vary around the average direction randomly. This average direction is referred to as the orientation of the liquid crystal, which stands for the average direction of most molecules. This parameter, i.e., the director, is an important factor to denote the liquid crystal's properties. Normally, the light waves with electric fields perpendicular or parallel to the director of a NLC have ordinary
Moreover, the particular nonreciprocity of magnetic materials, i.e., the polarized state (right/left handed polarized light) will switch into the opposite one (left/right handed polarized light) when the direction of the incident light is reversed, makes the MPCs remarkable workbenches for the unidirectional electromagnetic transmission [55-61]. Such a character shown in the cavity-based, periodic and aperiodic multilayered structures has made them good candidates to be isolators, unidirectional transmitted MPCs and one way waveguides with MPCs [57-59]. It has been unraveled that both ternary and microcavity configurations combined with waveguides can generate the isolators and circulators . The asymmetric isolators are shown to be much efficient compared with symmetric ones . The performance of small antennas embedded within MPCs constructed from periodic arrangements of homogenous and anisotropic material layers was demonstrated that the extraordinary high gain and enhanced power reception can be achieved . Furthermore, it has been demonstrated that the optical Tamm states (OTS) can be observed in one dimensional MPCs in recent theoretical and experimental works [64-70]. There will be considerable interest to exploit and identify the OTS in the MPCs. The ability of creating and manipulating the surface states is central to the development of subwavelength microscopy. However, few reports concerning the tunable OTS in MPCs have been presented . Therefore, it is interesting to find whether the NLCs can be used to realize the OTS in MPCs, or more importantly, whether such a configuration will support the controllable OTS. Controlling the OTS by electrical means is particularly interesting, as that would allow the integration of magnetic materials and the NLC with conventional optical electronics. Due to these unique optical properties, the MPCs with the NLC will hold substantial promise for possible optoelectronic devices.
In this chapter, we present a theoretical investigation on the NLC-based MPCs by performing a 4×4 transfer matrix method [71-73]. The optical properties of the MPCs with the NLCs will be reminiscent of both components. The coupling between the incident waves and the magnetic materials creates the magneto-optical effects, while the component of the NLCs is now responsible for its tunability occurring in the present configurations. The chapter is organized as follows: In Sec. II, we give a general description of the 4×4 transfer matrix method, which is a consequence of the combination of Maxwell equations and constitutive relations. The relations between the reflected, transmitted field vectors and the incident field vectors are listed to elucidate the magnitude of magneto-optical effects. In Section III, the case of periodic structure is considered, i.e., one dimensional MPC whose unit cell is composed of alternating NLCs and magnetic materials. In this part, the NLC is treated as a simple isotropic dielectric material approximately . In Sec. IV, one-dimensional MPC infiltrated with the NLC is investigated, where the intrinsic anisotropic properties of the NLC are taken into account. Combined with transfer matrix method and a piecewise homogeneity approximation method for the NLC , the magneto-optical effects of MPCs with the NLC is definitely achieved. In Sec. V, the tunable OTS is reported theoretically in the MPCs with the NLC. Finally, we give a summary about this work.
Our investigations reveal a variety of interesting results which are the main characteristics of one-dimensional MPCs with the NLC. The magneto-optical effects appear the peaks due to the abnormal dispersion relations at the edge of the band gaps. It is shown the shift of the peaks and the enhanced values in the magneto-optical spectrum as the permittivity of NLCs increases, which demonstrates that the magneto-optical effects can be altered by changing the permittivity in the NLCs. When the NLC-based cavity structure is subject to the applied electric field, the magneto-optical effects are observed to exhibit the similar trends as those of the periodic one treating the NLC as an approximate isotropic dielectric material. A significant shift of the peaks in the magneto-optical spectrum is observed with the external voltage due to the directors' reorientation. In addition, the existence of intra-Brillouin-zone band gaps is demonstrated and they are predicted to be dependent on the applied voltages in the MPC with the NLC. If the present MPC is attached to the other photonic crystals with two dielectric materials, the OTS is also observed in the transmission spectrum and can be controlled by the applied voltages. Our results provide an approach to modify a interface state in a controllable manner, which is significant to its potential applications. Moreover, it is possible to design a larger tunable magneto-optical effects by simply selecting the appropriate liquid crystals in the MPCs with the NLC. The ability of creating the electrically-controlled magneto-optical effects can provide an extra space to be explored in the future. Such tunability of the magneto-optical effect may be useful for future application in electro-optical devices.
2. Theoretical treatment
A 4×4 transfer matrix method is an effective way to explore the optical properties of one dimensional multilayered configurations, which has been proved to be accurate enough to tackle the band structures of the uniaxial, bi-axially anisotropic dielectric materials and liquid crystals [71-72]. For the sake of convenience, we describe the theoretical method and detailed treatment in this section.
Fig. 1 shows a typical multilayered structure, which is sandwiched to the incident medium with the refractive index
For a monochrome light propagating along the
Eq. (2) has a solution of the form:
Within the frame of transfer matrix method, the field vector can be expressed as,
Due to the continuous boundary conditions of electronic and magnetic fields, the equation Ψ
where and are the Kerr/Faraday rotation angle and the ellipticity for the
3. Electrically-controlled magneto-optical effects in magnetophotonic crystals consisting of magnetic materials and NLC
In this part, we investigate the magneto-optical effects of a one dimensional MPC whose unit cell is composed of a magnetic material and NLC and give a detailed explanation to the obtained results.
Generally, the permittivity of NLC is provided byfor linear polarized light incident as an extraordinary wave onto it, where
Fig. 2 shows a schematic illustration of a one-dimensional finite MPC, i.e., (NLC/magnetic materials). In the calculations, the total periodic number
where the magnetization vector makes an angle Θ with the
The solid, dash and dotted lines in Fig. 3 (a) describe the behaviors of
Note that all the peaks in Fig. 3(a) shift towards longer wavelengths when the NLC's permittivity increases, which is particularly interesting since it reflects the good tunability of the Kerr effect in our model. A similar shift is also observed for the photonic band gaps as shown in Fig. 3(b). This result is reasonable since the positions of photonic band gaps depend on the wave impedance ratio of two components [82, 83], while the latter changes with the NLC's permittivity in the present case. For a rough estimation, the central wavelength of the NLC-based MPC structure can be evaluated by
Fig. 3(a) also shows the significant increase of peak height with the permittivity of NLC in the spectrums of the Kerr rotation angle, which can be explained according to the circular birefringence. Different refractive indices due to different localization conditions, i.e., the localization wavelengths for left- and right- circular polarized lights, leads to the rotation of the reflected light . In photonic crystals, the effective parameters of the unit cell can be expressed by the effective homogeneous anisotropic medium method . The effective permittivity of MPC consisting of magnetic materials and NLC is given by
where and Note that the diagonal components of the effective permittivity depend on the permittivity of NLC but the non-diagonal components do not. Therefore, with the increase of
Generally, refractive indices of liquid crystal are fundamentally interesting and practically useful parameters in the calculation of optical properties. By changing the permittivity of the liquid crystals, the magneto-optical effects varied in a controlled fashion, indicating that the engineering of magneto-optical devices relies on the control of the applied electric field. Here we focus attention on the change of the orientation of the molecules under the influence of the applied electric field at a fixed temperature. Actually, besides the external electric field, temperature is also an important factor affecting the liquid crystal refractive indices, which offers an alternative approach to control the magneto-optical effects, i.e., a temperature tunable MPC.
4. Voltage-controlled magneto-optical effects in cavity- based magnetophotonic crystal with the NLC
We have predicted that the opportunity of creating electrically controlled Kerr effect in magnetic multilayered structures . It was suggested that a one-dimensional MPC composed of alternating NLC and magnetic materials can create tunable Kerr effect by considering the properties of liquid crystals and thus provide for tunable MPCs. However, we just employed an approximate isotropic treatment of NLC to analyze magneto-optical effects, which is a rough theoretical evaluation. It is generally known that the directors of NLCs exhibit inhomogeneous distribution under the influence of the applied electrical field . Therefore, a rigorous anisotropic treatment of NLCs is employed to consider the NLC director’s spatially inhomogeneous property upon an applied external voltage, which is based on the Newton method and continuous elastic theories. Although we expect such the tunable magneto-optical effect to be seen with a variety of patterns containing NLCs, we investigated a multilayered structure infiltrated with NLC, as shown schematically in Fig. 4. The defect of the NLC ensures that the present structure is sensitive to the external electric field and therefore shows the tunability of magneto-optical effects for observations at normal incidence.
The gray and white regions are magnetic materials, yttrium-iron-garnet(YIG) and non-magnetic materials gadolinium-gallium garnet (GGG) with permittivity
For the one-dimensional multilayered structure (GGG|YIG)n(YIG|GGG)mNLC(YIG|GGG)n, at the near infrared wavelength
Fig. 5(a) corresponds to the distribution of
Thus, the controllable magneto-optical effects have been predicted in the MPCs with the NLC, which will be a critical problem in the optoelectronic applications.
5. Voltage-controlled tamm state in periodic magnetophotonic crystal with the NLC
The Tamm state proposed by Tamm is one of the most fundamental physical properties of the interface, which means electron states can occur in the energy band gap at a crystal surface . In analogy with the electronic case, the OTS can be realized at the interface between two different photonic crystals with all isotropic dielectric materials. Conventional surface waves with a wave vector exceeding that of light in an incident medium decay exponentially away from the surface. In comparison with the conventional surface waves, the OTS can be formed for both the
The parameters used in our calculations are taken as
Furthermore, two stacked photonic crystals are generally used to illuminate the existence of the OTS in experiments , which can be observed by studying the transmission/reflection spectra with a sharp narrow peak/dip in the finite photonic crystal. Thus, it is necessary for us to provide the transmission spectrum. We design the two stacked photonic crystals, shown in Fig. 6(b), one of which consists of two dielectric materials and the other is composed of the NLC and magnetic material. Now we use the realistic material parameters at the wavelength
The tunable magneto-optical effects in magnetophotonic crystals with the NLC have been reviewed in this chapter. Two types of MPCs are studied, i.e., MPCs consisting of alternate magnetic materials and the NLC; the cavity-based MPCs with the NLC. Both of them exhibit the tunable magneto-optical properties qualitatively different from those of the unvariable MPCs without the NLC. It is predicted theoretically that the magneto-optical effects can be manipulated by the applied voltages in the theoretical treatments, i.e., the 4×4 transfer matrix method in combination with a piecewise homogeneity approximation for liquid crystals. A significant shift of the peaks' positions in magneto-optical spectrum is observed toward high frequencies with the applied voltages due to the high sensitivity of the directors on the external electric fields. By applying a tunable electric field, the voltage-induced reorientation of the directors of the NLCs alters the values of the dielectric permittivity in the NLC, thus leading to alternating magneto-optical effects. In addition, the tunable intra-Brillouin-zone band gaps can be realized in the MPCs with the NLC, which implies the existence of the OTS at the interface between two photonic crystals. The methods described here may possible to be implemented on many other periodic multilayered structures, cavity-based periodic structure, quasi-periodic and aperiodic structures, thus, opening the possibility to systematically investigate the magneto-optical effects in a controlled way. The structures we propose here is not only interesting in itself but also allows easy access to fabricate it in the experiments due to the mature techniques for one-dimensional systems. In fact, apart from the external applied electric field, the variation of temperature can also change the permittivity in the NLC component, and then alter the magneto-optical effects. It is obvious that there are a great space for the potential application of the MPCs with NLC. These results highlight an intriguing avenue for future investigations in the development of tunable liquid crystal-based magnetophotonic crystals optoelectronic devices.
This project was supported by the National Natural Science Foundation of China under Grant No.10774107 and Jiangsu provincial innovation project under Grant No. ZY320607.