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Single-mode fibers are advantageous over multi-mode fibers in many aspects, e.g., much smaller loss, much longer transmission distance, much greater bandwidth, and higher bit rates. We propose a kind of single-TM-mode Bragg fiber in which magnetic materials are introduced. The idea for designing this kind of Bragg fiber comes from the symmetry of TE modes and TM modes when permittivity and permeability are replaced by each other. Through the transfer matrix method, we demonstrated a special kind of single-TM-mode Bragg fiber in a wide frequency range. Guiding modes may be in the bandgaps, at the edges of bandgaps, and in some region in conduction bands, but much more strongly confined guiding TM modes are inside the bandgaps. In addition, the optimization of the structure is also discussed.
©2008 Optical Society of America
1. Introduction
Bragg fibers in which light is confined in the core by Bragg reflection were proposed by Yeh et al. in 1978 [1]. With the development of photonic crystals [2–3], researchers introduced the concept of photonic crystals into Bragg fibers, improving greatly their optical performances [4–7]. Meanwhile, besides the traditional Chew’s method, many new methods are developed for calculating the guiding modes in Bragg fibers [8–15]. Up to now, much attention has been paid to study the dispersion and the properties of guiding modes in Bragg fibers [16–28].
It is known that single-mode fibers are advantageous over multi-mode fibers in many aspects, e.g., much smaller loss, much longer transmission distance, much greater bandwidth, and higher bit rates. So, developing single-mode fibers is very important for optical communications.
In practice, Bragg fibers have a finite number of layers and support only leaky modes. But if one of these leaky modes has a sufficiently lower loss than any other modes, the fiber can be considered as a single-mode fiber. It has been demonstrated that TE modes in conventional Bragg fibers made of dielectric layers always undergo low loss than TM modes. Through suitable design, we can obtain a single-TE-mode Bragg fiber [19–20]. Until now, however, there is no good method to prepare a single-TM-mode Bragg fiber in a wide frequency range with low loss. In this paper we will realize it by introducing magnetic materials into traditional Bragg fibers.
In Ref. [18], single-TM-mode operation in the lower half of the bandgap was obtained by moving TM modes downwards in frequency through adding an additional coaxial core in the hollow of Bragg fibers. However, we can realize a single-TM mode in the whole bandgap. Furthermore, the loss of TM modes in the Bragg fibers proposed in this paper is several orders lower than that in Ref. [18] for the same index contrast and number of periods.
This paper is organized as follows. In section 2 we first give the basic principles and physical model. In section 3, we study the bandgaps of the corresponding 1-D PC, calculate the guiding modes in the proposed Bragg fiber, and analyze the results obtained. In section 4, we will discuss optimization of the structure. Finally, in section 5, we give a brief summary.
2. Basic Principles and Physical Model
Generally, when the core radius of a Bragg fiber is large enough, the multi-layers of periodic cladding can be viewed as a 1-D dielectric PC with a contrast of permittivity of the two media in a period. In such a photonic crystal made of dielectric materials, the bandgap for TE waves is always deeper and wider than that for TM waves, so that TE modes can be confined more effectively than TM modes in it.
According to Maxwell equations, we can get two equations respectively for TE and TM modes with the same wave vector in isotropic medium:
where the sub-script ⊥ indicates that the vector is perpendicular to the wave vector, and μ and ε are relative permittivity and permeability. If we exchange permittivity with permeability, the TE-mode electric vector E TE ⊥ with the TM-mode magnetic vector H ™ ⊥, the TE-mode magnetic vector H TE with the TM-mode electric vector(-E ™), these two equations change into each other. This means that, if dielectric materials are replaced by magnetic materials, the TM (TE) modes propagating in the magnetic media are of the same form as the TE (TM) modes in the dielectric media.
To go a step further, let us consider the propagation of waves from Medium 1 to 2. The reflectivities for TE and TM waves at the intersection of the two media can be written out according to Fresnel’s law:
where (ε 1, μ 1, θ 1) and (ε 2, μ 2, θ 2) are the relative permittivity, relative permeability, and the angle between the wave vector and the surface normal of Medium 1 and 2, respectively. If (ε 1,ε 2,μ 1,μ 2) is replaced by (μ 1,μ 2,ε 1,ε 2) respectively, from Eqs. (2) we will obtain
Equations (3–5) indicate that the photonic-bandgap properties for TE and TM waves will exchange with each other. They also clearly verify the symmetry phenomenon described in the above paragraph.
So we can deduce that a 1-D magnetic PC, which is a periodic arrangement of two magnetic materials with a contrast of permeability in a period, will achieve a deeper and wider bandgap for TM waves than for TE waves. Consequently, if we introduce magnetic materials into Bragg fibers, TM modes will be better confined than TE modes, and we will get single-TM-mode Bragg fibers.
The structure of this novel Bragg fiber is similar in shape to a conventional one. It may have various forms with different parameters. Here, we just discuss a structure which is created by replacing one of the two cladding media in a conventional Bragg fiber with a magnetic material, as shown in Fig. 1. The two media in Fig. 1 have the same permittivity, but different permeability with identical thickness. In the following numerical simulations, we suppose the relative permittivity of the two media to be ε 1=ε 2=1.5, the relative permeability of the two media to be μ 1=1.7 and μ 2=1, the thickness of two adjacent media to be d 1=d 2=0.5a, the number of periods to be 16, and the radius of the hollow core to be Rc=3.214a.
Fig. 1. The structure and the profiles of refractive index, relative permittivity, relative permeability of the Bragg fiber proposed
3. Numerical Simulations
Then we turn to the simulation of the proposed fiber. For the first step, we study the bandgaps of a 1-D PC composed of alternating layers of the two materials used in this Bragg fiber with the same width and number of layers. By the transmission matrix method [29] we can calculate the transmission spectrum of this PC for both TE-mode and TM-mode operations. Typical results for the incident angle θ at 0°, 40° and 80° are shown in Fig. 2, where θ is defined as the angle between the incident wave vector and the normal of the 1D-PC surface. From Fig. 2 we can see that when θ increases, the minimum transmissivity of each bandgap for TE waves will increase from 10-3 to 10-0.7, while that for TM- polarization will decrease from10-3 to 10-7, i.e., the TM bandgap is always deeper than the TE bandgap. We can also see that the bandgap may go deeper by several orders when the incident angle is getting large. Moreover, the bandgap goes deepest at the central frequency of each bandgap. So, TM modes near these frequencies in the Bragg fiber can be confined much better in the hollow core than TE modes, i.e., single-TM-mode transmission can be obtained in each bandgap. Furthermore, it can be seen that the first TM bandgap is deeper than the 1st and 2nd bandgaps, so only a single-TM mode in the wide frequency band will exist after some distance of transmission.
We now move to calculate the guiding modes in the proposed fiber. For simplicity, we just consider the frequency region ωa/2πc=0.3–1.35, which covers the first three bandgaps. Typical results are indicated in Figs. 3 and 4. The modes are calculated out using the matlab equation solver according to a straightforward transfer matrix method described in Ref. [13]. Each mode is characterized by a complex propagation constant β and an effective index neff=β/k, where k is the wave number in free space. In Fig. 3, the dispersion relations (the real part of the effective index Re(neff) vs. the normalized frequency ωa/2πc) and the transmission loss coefficients (proportional to the imaginary part of the effective index) are shown for TE and TM modes, respectively. Figure 4 shows the dispersion and loss for HE modes (m=1). The real part of the complex effective index is actually linked to α, the angle between the wave vector and the axis of the fiber, by Re(neff)=cosα which varies with frequency. The loss coefficient γ is defined as [23]:
Fig. 2. The TE (a, c, e) and TM (b, d, f) transmissivity spectrum of the 1-D PC corresponding to the Bragg fiber in Fig.1 for incident angle being 0° (a, b), 40° (c,d), and 80° (e, f)
For convenience of analysis, the bandgap regions are marked with background colors in Fig. 3. And the guiding modes in the bandgap regions are indicated in specified color lines, while the modes in other region are in blue lines.
Figure 3 tells us clearly that most modes are within or at the edge of the bandgaps. It is natural that bandgaps can avoid radial dissipation of waves, so that guiding modes can exist in the bandgaps. As for the guiding modes at the bandgap edges, it is also reasonable because reflections of outer layers still exist, and the waves can be confined to travel along the fiber axis with some acceptable losses.
Figure 3 shows that the frequencies of the guiding modes go up when the real part of the effective index Re(neff) increases. To understand this phenomenon, it should be noted that larger Re(neff) implies smaller values of the angle between the wave vector and the axial direction of the fiber. This means that larger Re(neff) corresponds to greater incident angle of the wave on the multilayer surfaces. It is well known from the theory of optical multilayer that the bandgap moves toward the higher frequency region as the wave incident angle increases.
Figure 3 demonstrates that the guiding modes within the bandgaps have smaller loss coefficients than the modes at the bandgap edges both for TE modes and for TM modes. This is natural because the bandgap provide much stronger confinement for the guiding wave, so that dissipation in the radial directions within the bandgap is much less than that at the bandgap edges.
Comparing the results shown in Figs. 3 and 4, we can see that the loss coefficients of the guiding mode in the bandgaps for TM modes are several orders in magnitude less than that for TE and HE modes. Thus, TM modes are the preferred guiding modes in the proposed Bragg fiber. Since the loss coefficient takes a minimum in each bandgap region, the proposed fiber can operate in a single-TM mode in each of the regions that includes a bandgap and the conduction band around it.
Fig. 3. The dispersion relation (a) and loss coefficient (c) of TE modes and the dispersion relation (b) and loss coefficient (d) of TM modes. The transmissivities for the region marked with green, olive, orange, and chocolate are 0.1~0.3, 0.1~0.01, 0.01~0.001, and <0.001, respectively. The modes in the 1st, 2nd, and 3rd bandgap are in purple, cyan, and brown respectively, while all other modes are in blue.
From Fig. 3(d) we see that there exist some modes with very small loss coefficient within the second bandgap. It, however, corresponds to very small angles between the wave vector and the fiber axis. Such modes are usually very difficulty to excite up in the fiber. Regardless of these useless modes, we see from Fig. 3 that the guiding mode in the first bandgap region shows a smaller loss in transmission, so that for a certain distance of transmission, the proposed fiber can operate in a single-TM mode in a very wide region crossing a few bandgap regions.
To see the effect of confinement in the proposed Bragg fiber, the field distribution of the TM mode with smallest loss in the first bandgap is indicated in Fig. 5. The normalized angular frequency for this mode is ωa/2πc=0.4575, the effective index is neff=0.848178. In Fig. 5, the fields are normalized to their maximum.
4. Discussion
We note that there are many modes in the conduction bands in Fig. 3 when Re(neff) (equal to cosα) is large enough. When α is close to 0, the component of Poynting vector on the axis of the fiber dominates, most energy of the waves flows along the axis, so that some kind of guiding modes can exist when α is close to 0. But the loss of these modes is still much higher than those in the bandgaps due to their weak confinement in the radial directions. Moreover, due to their small transmission angle along the fiber axis, these modes are difficulty to be excited up from outside sources, i.e., these modes may not exist in practice.
For the optimization of this fiber structure, we can make the following considerations. First, high permittivity-ratio of the two materials inside a period leads to deeper and wider bandgaps, so it can broaden the single-mode-transmission band and decrease the loss of transmission in the fiber. Second, we can increase the number of layers in the fiber to reduce the loss of all modes since more layers mean deeper bandgap and stronger confinement of waves. Third, the width-ratio of the two layers in a period should have impact on the modes. Generally, the case that the two layers in a period have the same optical length corresponds to a maximum bandgap which is favorable for optimum operation of the fiber. Fourth, we can also increase the radius of the hollow core of the fiber to reduce the loss of the guiding modes [21]. Fifth, we can even add coaxial cylinder in the hollow core to adjust the wavelength of the single mode to a value desired [18].
5. Conclusion
In conclusion, we have proposed and demonstrated a special kind of Bragg fibers that can achieve single-TM-mode operation in a wide frequency band by adding magnetic materials into conventional Bragg fibers. An example of these fibers has been simulated to demonstrate this property, and the optimization is discussed. The symmetry in Maxwell equations is the theoretical basis for our work.
Acknowledgments
This project is supported by the National Natural Science Foundation of China (Grant No. 60471047), the Natural Science Foundation of Guangdong Province (Grant No. 04011308), and the Shenzhen Bureau of Science and Technology.
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