Realizations of higher-order dispersion

Figotin et al. [7] noted that dielectric stacks of anisotropic materials can generate third- or even fourth-order dispersion. In this case, anisotropy relates to the use of ferrite layers within the dielectric stacks. It has been demonstrated that spectral asymmetry, i.e.,

П w(-b), actually leads to third-order dispersion. As such, it can include an inflection point within its curve, referred to as the SIP in the ш-p diagram. The presence of SIP within the dispersion curve implies a number of properties:

• Wave group velocity slows down and even vanishes at the SIP


  • • Waves propagate only in a single direction due to the asymmetry of the dispersion curve. Specifically, one mode can propagate, but the other in the opposite direction is “frozen” at the SIP. That is, unidirectional propagation occurs in asymmetric media [10].
  • • Dispersion curves are of third order. Also, at the inflection

point, П 0, П 0, = 0 (see Figs. 3.9b and 3.10a).

др др2 dp3

The MPC dispersion and related wave slowdown have been used for RF antenna miniaturization, bandwidth enhancements, and for photonics applications [11, 21]. It was shown in Ref. [20] that at least two misaligned anisotropic layers (misalignment angle other then 0 or n/2) inside one period of the stack can lead to maximally flat dispersion behavior at the band edge. This is depicted in Figs. 3.9a and 3.11b. Four propagating waves are involved to create the subject ш-в diagram, implying a fourth-order dispersion. This special band edge is referred to as DBE. Its maximally flat dispersion curve is

... ... dw d2wd3wd4a>

associated with the conditions: П 0, П 0, П 0, = 0,

dp dp2 dp3 dp4 ,

as depicted in Fig. 3.9a. The implied field at amplification at the band edge was applied to improve the directivity of antennas.

In 2008, Yarga et al. [22] demonstrated an experimental realization of the DBE modes using volumetric stacks of dielectrics. To introduce anisotropy in the stack layers, thin metal stripes with different orientations were employed, as depicted in Fig. 3.2b. MPC “frozen modes” were also realized by Stephenson et al. [15] using CTLs on ferrite substrates. Mumcu et al. [23] were the first to realize DBE modes using coupled microstrip lines and used the concept in antenna miniaturization. Further, to achieve emulation of anisotropy, the dispersion relation of the MPC modes in Ref. [15] was calculated using the same approach as in Figotin's work [20]. However, the first experimental demonstration of the “frozen mode” using transmission lines was done by Apaydin et al. [5] and is depicted in Fig. 3.12. Two CTLs consisting of eight or nine periods were printed

(a) MPC with anisotropic layers (Aj and A) and a third ferrite

Figure 3.10 (a) MPC with anisotropic layers (Aj and A2) and a third ferrite

layer. Field enhancement and unidirectionality is also illustrated. (b) Realization of the MPC mode in printed form. Magnetic biasing is shown as circles and labeled as H0.

on calcium vanadium garnet (CVG) and an external magnet was used for biasing. This successful experiment demonstrated the existence of the MPC modes and also verified the theoretical predictions of the supported modes. The measured group velocity at the SIP was 286 times slower than the speed of light. Further, the contrast between forward and backward wave transmission was 75%, verifying the theoretical prediction of unidirectionality [5].

(a) Non-reciprocal dispersion diagram of the MPC, displaying the

Figure 3.11 (a) Non-reciprocal dispersion diagram of the MPC, displaying the

fSIP. (b) Fourth-order dispersion diagram showing the DBE resonance at the band edge [15].

Coupled transmission lines printed on a magnetic substrate used to generate MPC modes

Figure 3.12 Coupled transmission lines printed on a magnetic substrate used to generate MPC modes. (a) Unit cell of the periodic transmission line structure. (b) Finite printed MPC prototypes comprising nine and eight unit cells for calculating the dispersion behavior of the coupled printed transmission lines. Reprinted with permission from Ref. 5, Copyright 2012, IEEE.

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