Analytic modeling of trench waveguides for channel plasmon-polariton using full-hybrid trial-field functions

A full-hybrid trial-field modeling, which was proposed earlier by the authors for plasmonic structures of rectangular cross-section, has been applied to trench waveguides for channel plasmon polariton. Most of the computed results of complex propagation constant agree well with those obtained by method of lines and effective index method, though the latter two results themselves show some discrepancy. The particular aspect in which there is varying degree of disagreement among the results obtained in the above three methods is the imaginary part of the effective index seen by the plasmon-polariton.


Introduction
In the field of nano-photonics the thrust area of research is to explore the possibilities of designing the optical circuitry beyond the diffraction limit (sub-k confinement) and to make the transmission possible over the length scales of Optical Integrated Circuit Chips (OICC) in the range of as many X as possible.Plasmonic waveguides have got remarkable prospect in achieving such goals in nano-optics and photonics technology.
Exhaustive reviews of SPP waveguides are available, for example, in [1,2,3] and the elaborate discussion on the immense scope of Plasmonics beyond diffraction limit in various nano technology applications can be found in [4].Several nano guiding structures have been proposed in the recent years including the metal strips embedded in homogeneous or inhomogeneous dielectrics.Among such guiding structures, Gap and Channel Plasmon waveguides offer several desirable features [5,6,7,8].Channel Plasmon Polariton or CPP waveguides are of two types depending upon the shape of the nano sized channel cut in the real metal -(i) when the channel is a V-shaped groove in a metal, the supported CPP is called V-groove CPP, (ii) when the channel is a narrow rectangular groove in a metal, the supported CPP is referred to as Trench-CPP.Photonic components based on both types of CPPs have been demonstrated in practice [9,10,11] and the most interesting feature, is to have subwavelength lateral confinement and relatively long propagation length, simultaneously in CPPs.The modeling of SP waveguide components and characterization of propagation modes in channel-SPP guides are generally pa-formed with different simulation techniques and also applying the effective index method (EIM), traditionally employed in dielectric waveguide problems [12,13].EIM is simple in the sense that 2D waveguides are characterized by combining results obtained from analysis of ID guides.The semi-analytic Method of Lines techniques has become a very useful tool for numerical modeling of plasmonic waveguides [14,15,16,17].Minh and others [18] recently have applied MoL to calculate the propagation characteristics of Trench-CPP waveguides and have compared their results with those obtained by EIM [13] and FDTD [19].In this chapter of the thesis the simple analytic modeling of Trench-CPP guides using full hybrid trial field functions is presented.This approximate yet quite accurate technique was earlier successfully applied 127 to metallic SPP guides of rectangular cross-section and to rectangular nano-holes in metals [20] and has been presented in chapter 5. Hie computed values of complex effective index and propagation length for various trench dimensions and dielectric filling are compared with those in [13,18].

Formulation of the problem
The geometry of the Trench SPP guide is shown in figure 6.1.

Figure 6.1 Geometry of Trench-CPP Waveguide
It is a rectangular groove of depth d and width 2w in a semi-infinite metal block.It can be bisected longitudinally by a symmetry plane at y=0.The groove is usually air-filled but can be filled with other dielectrics.All modal fields are assumed to have exp (-j(3z) dependence in the direction of propagation.The parameters are chosen as in [13] and [18]: the wavelengths of operation X= 1550 nm, 1033 tun, 775 nm; the metal is gold with dielectric constant of-131.95-jl2.65 at 1550 nm; trench width 2w= 300 nm, 500 nm; depth d is variable.
Figure 6.2One Half of Trench-CPP Guide segmented into six regions

Trial field modeling
In this approach, we would construct the trial transverse spatial functions for EyE and H™ fields corresponding to TEX (Ex = 0) and TMX (Hx = 0) modes, respectively, and derive all six field components of the hybrid Plasmon modes of the structure from superposition of the TEX and TMX mode families [14].While writing the trial field functions, first we consider the following points: (a) only one-half of the structure, shown in figure 6.2, that takes into account the symmetry plane at y=0 is used for formulation, (b) the reduced structure is segmented into six regions, R1 to R6, as shown figure 6.2, (c) i,n/\r ick.u trial fields for each region i (i = 1 to 6) are constructed separately such that each trial field yfi satisfies Helmholtz equation in respective region t, (d) continuity of the transverse wave-vector at any interface between two media is also assumed.Secondly, since the success of trial-field modeling depends on how well the trial functions approximate the true fields, in writing the trial field functions inside the trench (region Rl), the nature of field variation, particularly with depth, as reported in [13,18] was taken into consideration.Clearly, the symmetry plane at y=0 is electric for the dominant field configuration.Provision for magnetic symmetry, however, is retained in the formulation.
For x-dependence of the field inside the trench, with respect to the chosen coordinates, one has the option of writing the functional form as sink kxx or cosh kxx, both growing toward the top of the trench and being almost identical beyond the argument value of about two.Choice of sink function forces the main electric field Ey to be zero at the trench bottom irrespective of the trench depth.The reported variation [13,18], however, shows that the field is always very small (but not zero) at the bottom and this value depends on the depth d.For short depths, the field at the bottom is not negligibly small and penetration into the metal is also significant.For deep trenches, both are very small.
In fact, our computation with sink function in the trial field for different trench widths always yielded the propagation constant of the gap-SPP mode [13].Thus cosh dependence on x inside the trench appears to be the logical choice for the trial field function.On inspection and with the above considerations in mind, the trial field functions for different regions are written as follows.Although regions R5 and R6 are shown in the segmented structure, the trial fields in these two regions are not written.
Obviously, field matching at the R2-R6 and R5-R6 interfaces are not meaningful.Similar observations can also be made about field matching at R1-R3 and R3-R4 interfaces.
Furthermore, field matching at the trench bottom (R1-R5 interface) is not necessary because it does not yield any useful information.This is to be expected from the reported variation of field with depth [13], which shows that this interface is not plasmonic, that is, it does not support a surface electromagnetic wave.So finally we write the trial field functions only for three regions: Rl, R2 and R4.
The transverse wave numbers are related to f3 by the following equation The upper and lower functions correspond, respectively, to magnetic and electric walls at the symmetry plane y = 0.The expressions (1) -( 6) for the field components need some clarifications.The summation on n in expressions (1)-( 6) is for the purpose of writing the trial functions in generalized form and do not represent expansion in terms of orthogonal basis functions.Only the nature of a physical solution for a prospective plasmon mode is envisaged while constructing the trial functions.However, the eigenvalue equations, which are transcendental in nature and can be solved for a series of complex roots (knix, k"iy) which would give the same value of p for a particular mode and different P for different modes.Therefore it is sufficient to compute, say, knx and £iiy for computation of p of the lowest mode.However, the root set (knix, kniy) can be computed to obtain the field profiles in different regions.

Eigen Value Equations
Once all six field components in each region (Rl, R2, R4) are derived from the trial fields, the next task would be to apply the continuity conditions.The expressions for the L-XlrtT li^XY U field components necessary for tangential field matching are presented in Appendix A3.
It is not necessary to impose all the boundary conditions at all the interfaces which only over-defines the problem.We found that application of continuity of tangential field components at only the R1-R2 and R2-R4 interfeces (between two different media) yields the desired eigenvalue equations, given by ( 8) and ( 9), for all values of n..With the chosen parameters as mentioned in section 2, complex effective index and propagation length from its imaginary part were computed and are presented in figure 6.3 to figure 6.8.

Effect of Trench Depth
The variation of Re(rieff) with trench depth d, as shown in figure 6.3, is very much similar to the EIM results [13], though our computed values are always slightly higher.The computed propagation length L (figure 6.4) for 2w = 300 nm is almost same as that in [13].For 2w = 500 nm our values, however, are slightly lower, particularly at smaller depths.Though we have not shown comparison with MoL data in figures 6.3 and 6.4, an examination of figures 3(a) and (b) of [18] indicate that our data of Re(neg) agree better with the MoL data than with the EIM data.However, our computed values of propagation length agree much better with EIM data.The MoL value of propagation length hardly changes with trench depth and shows a small increase at small trench depth of about 1 pm.
The discrepancy in the nature of variation of the MoL values of L with respect to EIM data for small trench depths has been noted in [18].All the previously reported data and our results confirm that subwavelength lateral confinement as well as relatively long propagation length can be achieved with Trench-CPP waveguides with deep grooves.
Specifically, the fundamental Trench-CPP mode which show these features require large depth-to-width ratio.

Effect of Dielectric-filling of Trench
It has been shown [13,18] that in a trench waveguide with air-filled groove (which can be looked upon as two right-angled wedges hybridized by a finite-depth rectangular groove), the field is mainly confined at the two comers at the top of the trench.In [18] it was proposed that dielectric filling (ei>l) of the groove can be utilized for stronger  particularly for the lower trench depths (<1.5 pm), which is just the opposite of the variation obtained from MoL for air-filled trench.In [18], Im (n^) increases with decreasing trench depth which is just the opposite of the variation obtained by MoL for air-filled trench.It is difficult to understand why dielectric filling should cause this reversal in the nature of variation.The authors in [18] also have not offered any plausible explanation.The results obtained from trial-field modeling indicate the Im (n^) decreases with decreasing trench depth for both air-and dielectric-filled grooves.The decrease, however, is not exactly monotonic when ei>l.For small depths of both air-and dielectricfilled grooves, the confinement is poor, the loss is lower and hence the propagation length would be longer.

- 4 -Figure 6 . 3
Figure 6.3 Dispersion of the real part of the fundamental Trench-CPP mode effective index with depth of the air-filled trench.

Figure 6 . 4
Figure 6.4 Dispersion of the propagation length of the fundamental Trench-CPP mode with depth of the air-filled trench.
confinement of field inside the trench.Results of MoL computation have established the projected improvement in normalized phase constant.Unfortunately, this is accompanied by increase in normalized attenuation constant.However, there is still provision for trade off between the two characteristics obtained with dielectric filled trench.The trial-field modeling with segmented problem area is well-suited for computing the complex effective index of dielectric-filled trench configuration.The results of our computation are presented in figure6.5 and 6.6, where comparison with MoL data also is shown.The agreement of Re (n*e) is excellent.However, there is discrepancy in the imaginary part Propagation Length in pm.

Figure 6 . 5
Figure 6.5 Dispersion of the real part of the fundamental Trench-CPP mode effective index with trench depth for different dielectric inserts inside the trench (n = 3,2,1.44).

Figure 6 . 6 Figures 6 .Figure 6 . 7
Figure 6.6 Dispersion of the imaginary part of the fundamental Trench-CPP mode effective index with trench depth for different dielectric inserts inside the trench (n = 3,2, 1.44).