Electromagnetic Field Behavior in Isotropic Negative Phase Velocity/Negative Refractive Index Guided Wave Structures Compatible with Millimeter-Wave Monolithic Integrated Circuits

electromagnetic field distributions are found for frequencies slightly under one hundred giga Hertz, using a left handed substrate in a microstrip guided wave structure


I. Introduction
Tremendous interest in the last few years has occurred with the experimental realization of macroscopic demonstrations of left-handed media, predicted or at least suggested in the literature several decades ago [1]. Attention has followed on the focusing characteristics and related issues of left-handed media (LHM), with appropriate arrangements to accomplish such behavior, as shown by literature publications [2 -25].
But no attention has been directed toward what intrinsic left-handed media could do in propagating devices used in integrated circuit configurations. This is not to say that some work has not happened on applications using backward wave production or LHM properties in specialized microwave, devices which rely on reduced dimension negative phase velocity behavior [26] - [34]. (Also see references contained in [35] and [36] for other focusing and backward wave devices.) And much of that work has looked at macroscopic realizations, which may be amenable in the future with current efforts on metamaterials, to advancing microwave integrated circuit component technology utilizing left-handed media.
We are particularly interested here in what new physical properties are the result of using material which is intrinsically left-handed, or also variously referred to in the literature as negative phase velocity material (NPVM or NPV) or negative refractive index material (NRIM or NIM). There may be substantial interest in understanding the effects of left-handed media in guided wave structures since advances in integrated circuit technology, in passive components, control components, and active devices has increasingly been utilizing layers and arrangements of many differing materials. From heterostructures in active devices to complex materials like chiral, ferroelectric and ferromagnetic materials, in passive and control components, this trend has been rising.
Efforts on metamaterials is sure to further this trend.
A hint at the remarkably different field distributions has been disclosed recently using LHM substrates in guided wave devices [37]. Dispersion diagram description of the physics is provided in [35], and this diagram shows the effect of the RHM/LHM interface seen in the cross-sectional view on the propagation normal to the cross-section.
Bands of pure phase propagation and bands of evanescent propagation occur. Also, negative phase velocity behavior of the LHM interacts with the RHM to generate regions of both ordinary wave propagation as well as backward wave propagation in the negative phase velocity sense relative to the guided wave power flow. In [35] only the low end band is displayed in field distribution plots at 5 GHz. But the plots shown are instructive for the new physics they demonstrate: unusual field line or circulation characteristics for the electric or magnetic fields, startling intensity variation of the fields, counter intuitive charge arrangement on the guiding metal strip, and interesting visual display of opposed Poynting vectors in adjoining RHM and LHM regions for power flow down the device.
Attention to the new possibilities for electronic devices is given in [36], [38] when using LHM/NPV substrates. There, distributions up to 40 GHz are provided, somewhat over the beginning of the millimeter wave frequency band. However, nowhere have we made available the remarkable field distributions found at the higher millimeter wavelengths, and so in this article we would like to show for the first time what the fields look like at nearly 10 11 = 100 GHz (we will actually draw our attention to f = 80 GHz as a starting point). A new technique we have developed of lifting out the lower magnitude fields in order to visualize their directions in arrow distribution plots will be utilized here for the first time (Section V). This is particularly important in distribution plots where the field magnitudes may vary over many orders of magnitude. A number of field distribution plotting methods will be employed in this paper: arrow plots based upon linear representation of the field magnitude, arrow plots for both electric and magnetic fields based upon scaled representation of the field magnitude, line plots showing electric field behavior emanating from the strip and off of it, line plots showing magnetic field behavior circulating around the guiding strip and off of it, and magnitude plots of both the electric and magnetic fields.
Sections II and III provide short discussions of left-handed material properties (Section II) and the Green's function technique (Section III) used to solve the material physics/field problem. Once these preliminaries are out of the way, the eigenvalues (Section IV) and the field distributions (Section V) are determined.

II. Left -Handed Material Characteristics
It is expected that the left-handed medium's characteristic to alter the electromagnetic field based upon its new properties contained in its tensors describing permittivity and permeability will not only lead to new structures enlisting just this new material, but eventually allow the creation of multi-layered devices containing various substances including left-handed media. Here we report on the new physics associated with left-handed media in guided wave propagating structures which are applicable to microwave and millimeter wave integrated circuits, although the focus here is primarily on the millimeter wave region. Here we address the use of the left-handed media with its general bianisotropic crystalline properties reduced to scalars, that is with anisotropic permittivity ε tensor set equal to the isotropic permittivity value ε 1 , and anisotropic permeability µ tensor set equal to the isotropic permeability value µ 1 . Consideration of the anisotropic or bianisotropic crystalline case is examined elsewhere [39]. Suffice it to say here that just as in the case of optical or lower frequency focusing, isotropy is what allows proper organization of all the wave fronts (or rays in the geometric optics limit).
But in a guided wave structure, what may be the most critical issue, is the assumption of isotropy to allow arbitrary field contouring or sculpting [40]. Individual unit cell construction and repetitive cells in all directions can lead to isotropy, as well as materials with intrinsic isotropic crystalline properties. The scalar relative permittivity and permeability ε and µ seen in the literature have frequency dependence ε(ω) and µ(ω).
Whether this is a narrow or wide band phenomenon will not be addressed here, other than to note that there may be both metaobject construction as well as intrinsic material methods to adjust the bandwidth. There is every indication today that these two implementation categories may provide enough design possibilities to make such bandwidth adjustment realistic. So whether one uses nonresonant objects or resonant objects, or microscopic properties of crystals or nanoscale materials, there is no reason to doubt that the frequency region ∆ω over which the desired behavior occurs may be viewed as being subject to the choice of the physicist or engineer for some intended use.
So in order to study what the field distributions for a LHM substrate would do in a certain configuration at a particular frequency, we need to set Re[ε(ω )] = -ε r and Re[µ(ω )] = -µ r where ε r = real positive constant and µ r = real positive constant.
Fundamental mode is sought, which has an eigenvalue even as ω → . This is the simplest problem one can solve for in our inhomogeneous boundary condition problem. Hope for obtaining wideband behavior is now supported by recent results showing that negative refraction can be obtained by using heterostuctures of intrinsic crystals with negligible dispersion [41] - [44] . Narrowest behavior occurs with resonant structures like split ring resonator-rod combinations. Even for these structures, which are for example, the permeability being derivable from [45]), because ω p , ω 0 and F are subject to the designer's control, one can always, for a desired setting for ε and µ at a particular frequency ω, solve the two equations implicit in these depictions for the three unknowns, with a rich multiplicity of solutions. Finally, photonic crystals which provide the negative refraction using ordinary RHM crystals with RHM inclusions, may have bandwidths somewhere between non-dispersive and highly dispersive (as in the resonant structures).

III. Green's Function for Left -Handed Guided Wave Structure
Green's function for the problem is a self-consistent one for a driving surface current vector Dirac delta function applied at the guiding microstrip metal, with x 0 = point on the strip. Figure 1 (a) shows a crosssection of the structure with a right-handed material (RHM/PPV -ordinary material; PPV = positive phase velocity) used for the substrate and Fig. 1 (b) shows a cross-section of the structure with a left-handed material (LHM/NPV). The Green's function is a dyadic, constructed as a 2 2 × array relating tangential x-and z-components of surface current density to tangential electric field components. This Green's function is used to solve for the propagation constant (see [46] for a recent use of this type of Green's function).
Determination of the field components is done in a second stage of processing, which in effect creates a large rectangular Green's function array, of size 6 2 × , in order to generate all electromagnetic field components, including those in the y -direction normal to the structure layers. The governing equation of the problem can be stated as

IV. Eigenvalues of LHM and RHM Devices at Millimeter Waves
To gain some idea of the general value of the propagation constant, an eigenvalue is sought for an ordinary medium substrate at the millimeter-wave frequency for a device

V. Electromagnetic Field Distributions for Left -Handed Devices
In order to correctly perceive the intensity of the fields,

A. RHM/PPV Comparison Structure
Once we have identified the β value and know the point in the diagram about which we wish to operate, we can proceed on to making a field distribution plot.  Firstly, the E and H intensity distributions differ in appearance significantly for the LHM/NPV substrate structure compared to the relatively simple field pattern of the RHM/NPV structure. Secondly, in Fig. 3(a), the electric field arrows E t do not point into (or out of) the metal strip but point roughly (this interpretation is modified by examining the field line distribution patterns to be discussed below in reference to Fig. 5 for the LHM/NPV structure) in one direction above and below the strip, indicating that this branch still has a single charge (as was found for the RHM/PPV case in Fig. 3) due to the reversed effect of the displacement electric field continuity condition normal to the interface. Thirdly, in Fig. 3(a), electric field E t arrows, away from the metal strip near the interface, point in opposite directions as they cross the interface in terms of their normal components. Fourthly, in Fig. 3(b), magnetic field H t arrows circulate around the strip in one direction above the interface and in the opposite direction below it, and the magnetic field arrows H t when crossing the interface point in opposite directions in terms of their y -components. Fifthly, in Fig. 4(a), the electric field arrows E t point roughly (again this interpretation is modified later by the field line distribution patterns to be presented below in Fig. 5) into (or out of) the metal guiding strip indicating a situation only possible if an infinitesimal dipolar charge arrangement exists in the vertical sense [35].

B. LHM/NPV Structure
Of course, in addition to these noted differences between RHM/PPV and LHM/NPV substrates on guided wave behavior, the two solutions have significantly different field line distributions patterns as seen in Fig. 5 for the LHM/NPV structure. This is clear from the nearly circular circulating magnetic lines (first three) around the strip for the higher β value forward wave [ Fig. 5(b)] compared to the broadly extended magnetic field lines for the lower β value backward wave [ Fig. 5(a)] (This is evident by looking at the region above the substrate.). The electric field lines exhibit a dense pattern near the strip for the higher β value somewhat contained in a "shell," compared to that of the lower β value mini -"shell" which barely shows this pattern emerging (examine the region just above the metal strip in the air zone). For the lower β value [ Fig. 5(a)], there is positive charge on the bottom half of the strip in the LHM/NPV (field lines enter into the strip) as well as on the ends (field lines exit from the last quarter length of the total strip length on either side of the strip) of the top part of the strip. But the mini -"shell" has its E t field lines emanating from the inner edges of the positive top charge at the boundary between this positive charge and an inside region of the top strip which is negatively charged. Upper β solution has its "shell" extending about 20 % beyond the strip width, with the E t field lines emanating from the "shell" surface with the lines terminating nearest the strip center, originating near the intersection of the "shell" surface and the interface. Outside of the "shell" in both Figs. 5(a) and (b), the electric field lines E t revert back to the simpler case of them seemingly to arise from a single uniform charge on both sides of the strip. Because the "shell" is not seen at considerably lower frequencies, this seems to imply that at higher frequencies the structure is trying to become more like an ordinary media layered device. Finally, we note that both the electric E t and magnetic H t field line patterns are much more intricate for the higher β Lastly, in Fig. 7 is provided the electric field E magnitude distribution for the LHM/NPV structure, overlaid with electric field lines E t for the upper eigenvalue at 80 GHz. This figure combines some of the information from Fig. 4 (a) and Fig. 5 (b), in such a way to assist visualization and understanding of the field behavior, allowing one to see the directional information at a glance while being able to assess the strength of the field.

VI. Conclusion
In