Higher-Order Dispersion Engineering
Graphical analysis
As noted earlier, second-order dispersion is a consequence of simple periodic stacks of isotropic dielectrics. Further, these stacks can be modeled as CTLs. A very special property of the CTLs is that each of the lines is identical to each other and, therefore, associated with the same phase and group velocity. However, if the transmission lines forming the CTLs are not electrically identical, we would then expect different behavior in the dispersion curve. In this section, we consider the analysis of such coupled non-identical transmission lines. The goal is to generate higher-order dispersion curves. As a first step, we proceed to formulate the fourth-order dispersion (DBE mode) since it is symmetric in nature, i.e., ш(в) = ш(-р) and comparatively easy to derive. Unlike the second-order dispersion, the calculation of fourth-order dispersion is complex and cumbersome. However, the second-order dispersion provides a building block for the derivation of the fourth-order dispersion.
To begin with, we first consider a pair of uncoupled transmission lines, each characterized by the parameters (L_{C} C_{1}) and (L_{C}, C_{2}), as shown in Fig. 3.7a (blue and red transmission lines). Here we choose L_{2} = 0.4L_{C} and C_{2} = 0.4C_{1} as shown. This choice implies that the wave velocity in each of the lines is unequal leading to four uncoupled waves in the dispersion diagram (Fig. 3.7b). Among these, two are forward traveling waves and two are backward traveling waves. The four uncoupled waves can be written as:
Now let us proceed to assume that the transmission lines are coupled. The zones where possible coupling can occur are marked by green circles and denoted by '1' and '2'. As shown in Fig. 3.7b, a coupling is considered between a forward wave mode b_{a} = wj L_{1}C_{1}
2p
and a backward wave mode b_{d} = - w L_{2}C_{2} in a similar manner
P
as derived in Eqs. (3.16) and (3.17). The coupling coefficient chosen for zone '1' is K_{C}. Similarly, another coupling is considered between
a forward wave mode b_{c} = O)^JL_{2}C_{2} and a backward wave mode
2p
b_{b} = - O L,C, . The coupling coefficient chosen for zone ‘2’ is K_{2}.
P
The coupling zones exhibit that uncoupled waves are not expected to couple at the bp = p point (marked as pink vertical line in Fig. 3.7). Rather, they couple at two separate locations equally distanced from the p point. This is the consequence of unequal phase velocities of two non-identical lines. Unlike the case of second-order dispersion, a forward wave and a backward wave with different phase velocities are expected to couple. The coupling coefficients (K_{1}, K_{2}) can be attributed to the natural coupling between a forward and a backward wave mode associated with any periodic structures, and (K_{1}, K_{2}) can be modeled by coupled inductance L_{M} or capacitance ^{C}M between two transmission lines similar to the circuit model shown in Fig. 3.8a. However, if the coupled modes experience anisotropic medium while propagating along the transmission lines, they proceed to couple further and generate a new fourth-order dispersion characteristic. To model and enumerate the coupling due to anisotropy, we employ a third kind of coupling coefficient, K_{3}, that couples two second-order modes together. The final dispersion relations of the resultant coupled modes are given in Eq. (3.19). It turns out that setting all K’s equal to each other, i.e., K_{1} = K_{2} = K_{3}, leads to a special type of fourth-order dispersion curve associated with the double band edge (DbBE) mode (Fig. 3.8b). The name double band edge is attributed to the two different band edges observed at the same frequency.
Figure 3.7 Two non-identical uncoupled transmission lines with four solutions of the propagation constants. (a) Two non-identical uncoupled transmission lines are marked by blue and red colors with circuit elements per unit length characterized by (L_{1(} CJ and (L_{2}, C_{2}). (b) Uncoupled forward and backward waves corresponding to blue and red transmission lines. Possible coupling is marked by green circles as '1' and '2' with coupling coefficients K_{1} and K_{2}, respectively.
Figure 3.8 Two non-identical coupled transmission lines with coupling introduced by anisotropy and the formation of DbBE modes. (a) Two nonidentical coupled transmission lines are marked by blue and red colors with circuit elements per unit length characterized by (L_{1(} CJ and (L_{2}, C_{2}). The coupling inductance and capacitance are (L_{M}, C_{M}) and can be attributed to the coupling coefficients (K_{1t} K_{2}). (b) Formation of DbBE mode is shown with coupling due to anisotropy depicted by the shadowed box. The coupling coefficient is marked as K_{3}, which relates to the coupling between two second-order modes. For DbBE formation, all three Ks are chosen as equal, i.e., K_{2} = K_{2} = K_{3}.
Although the coupling parameters (K_{v} K_{2}) are found from the coupled elements (L_{M}, C_{M}), the third coupling coefficient K_{3} is attributed only to the existence of anisotropy, as mentioned by Figotin et al. [7]. In fact, it is related to the polarization of two oppositely directed waves propagating in an appropriate anisotropic medium. Usually, anisotropy is realized when using magnetic materials to form the propagating medium, e.g., ferrites with DC bias. Actually, the dispersion relations written above provide for three of the known dispersion behaviors, depending on the choice of (K_{x}, K_{2}, K_{3}), but more possibilities exist. Specifically, when all there coupling coefficients, i.e., K_{1} = K_{2} = K_{3}, are the same, a DbBE is observed (Fig. 3.8b). But if we choose the case of K_{1} = K_{2}, K_{1} n K_{3}, a fourth-order maximally flat dispersion profile results. The associated mode is referred to as the DBE mode with the corresponding dispersion curve shown in Fig. 3.9a. This flat profile of the DBE modes is useful for antenna miniaturization and increased directivity, but leads to narrow bandwidths.
It is noted that spectral symmetry, i.e., w(b) = w(-b) is inherent in the DbBE and DBE modes. However, when the coupling parameters are chosen such that K_{1} n K_{2}, K_{1} = K_{3}, an unusual dispersion behavior is found, as shown in Fig. 3.9b. The resulting modes from the solution to Eqs. (3.19) are referred to as MPC modes. This is because in volumetric media, magnetic materials are required to realize these modes. As shown in Fig. 3.9b, the MPC mode dispersion is asymmetric, i.e., w(b) n w(-b), in nature and of third order with an inflection point. This inflection point is evocative of the stationary inflection point (SIP) in a typical steepest decent path and contributes to bandwidth improvements for antennas. Table 3.1 relates DbBE, DBE, and MPC modes to the three aforementioned parameters.
Table 3.1 Classification of dispersion engineering based on K parameters
Case 1 |
Double band edge (DbBE) mode (Fig. 3.8a) |
(«1 = K = K3) |
Case 2 |
Degenerate band edge (DBE) mode (Fig. 3.8b) |
(K1 = K_{2}, Ki П K3) |
Case 3 |
Magnetic photonic crystals (MPC) mode (Fig. 3.9) |
(«1 П «2, Ki = K3) |
Figure 3.9 Higher-order dispersion attributed to the coupled, but nonidentical, transmission lines. (a) Degenerate band edge (DBE) mode dispersion diagram, leading to a fourth-order curve. (b) Third-order dispersion diagram of the MPC mode showing the presence of the stationary inflection point.