Abstract

Generalized Green’s Function Surface Integral Equation Method (G-GFSIEM) is used to study propagation of surface plasmon polariton waves at interface of two semi-infinite metal-dielectric waveguides. Reflection, transmission, and scattering coefficients for structures with different dielectric constants are calculated by using this method and by using energy conservation law. Conditions where scattering coefficient is maximized or minimized are studied. It is found that by using appropriate materials with specified dielectric constants, structures with required reflection, transmission, and scattering coefficients can be designed.

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Introduction

Surface Plasmon Polaritons, SPPs, are of great importance due to their interesting applications in subwavelength photonic circuits, spectroscopy, and different types of chemical and medical sensors. SPPs are localized p-polarized electromagnetic fields produced by coupling between electromagnetic wave and free electrons at the surface of a metal which propagate at the interface of the metal-dielectric media. They transfer energy in nanoscale range and propagate in distances comparable with their mean free path distance [1].

Having such properties for the SPP waves, one can envision two dimensional plasmonic circuits over which surface plasmon polaritons propagate [2,3].

Plasmonic circuits may consist of a variety of components, e.g. filters, couplers, modulators, splitters, and reflectors, which are connected to each other by plasmonic waveguides [4]. The simplest form of a plasmonic waveguide consists of a thick metal plate where SPP wave propagates over it. In order to design a nano circuit it is necessary to analyze interaction of incident SPP wave with the common interface of the waveguide with each component. In the simplest case such interface can nearly be considered as the interface of two metal-dielectric waveguides with common propagation axis. In addition of reflection and transmission of SPP wave a significant part of the energy at each interface, typically between 10 to 30 percent, converts to radiative waves and, thus, reduces efficiency [5].

Stegeman investigated scattering of SPP waves at the interface of two metal - dielectric waveguides by expanding the electric field based on normal modes of the structure. The resulting equations were solved for a discretized boundary to calculate reflection, transmission, and scattering, RTS, coefficients and, consequently, the electric field at any arbitrary point [6]. To increase precision of the results, number of points on the boundary should be increased. The main drawback of this method is that two metal plates at both ends of the waveguides should be assumed to ensure convergence of the expansion.

Mode matching technique, as a semi-analytical approach, is also used to study this structure [5]. In this method eigenmodes of each waveguide are calculated and the electric and the magnetic fields for each region are expanded by using those eigenmodes. By applying appropriate boundary conditions one can obtain coupled integral equations. By solving obtained equations the electric and the magnetic fields at any arbitrary point in space are calculated. In spite of its simplicity, the method is not appropriate for complicated structures because it is necessary to obtain a complete set of eigenmodes.

Finite Difference Time Domain Method, FDTD, and Finite Element Method, FEM, have also been used to study similar problem under condition where the scattering is eliminated [7,8]. Both methods can be applied when the scattering exists. But, one should calculate fields at all meshes. On the other hand, to increase precision of the result, mesh size should be reduced that is time consuming.

In this paper, interaction of the SPP waves with two semi-infinite metal-dielectric waveguides at their common interface is studied by using Generalized Green’s Function Surface Integral Equation Method (G-GFSIEM) [911]. The method is previously used to study scattering of the electromagnetic wave from a closed boundary, i.e. a metallic nanostrip located above a semi-infinite waveguide. We use this technique to study scattering of the SPP wave from interface of two metal-dielectric waveguide with open boundaries.

The method is similar to BIM [12] where the field at each arbitrary point inside a closed boundary is related to the field and its derivative on the boundary via second Green’s theorem. The advantage of both methods is that only the surface, and not the volume, of the scattering center is discretized. On the other hand, in BIM Green’s function of free space is used which may result to significant truncation error for boundaries extended to infinity, e.g. the boundary where the SPP waves propagate over it. But, in G-GFSIEM appropriate Green’s function which depends on the geometry of the structure and the type of the material is used. That takes into account effect of those boundaries in order to minimize such error. The set of the integral equations obtained by G-GFSIEM is then numerically solved by using Boundary Element Method, BEM [13]. The results illustrate that specified RTS coefficients can be obtained by choosing materials with appropriate dielectric constants. We have also found that for dielectric materials with equal skin depths the scattering coefficient becomes negligible.

The structure of the paper is as follows. First, we introduce theoretical analysis for the structure under study and obtain the field at the boundaries as well as the reflection and the transmission coefficients. Next, numerical results on calculating those coefficients are compared with the results presented in ref [5]. to verify correctness of our method. Then, structures consisting of two semi-infinite metal-dielectric waveguides with different dielectric constants are studied and their RTS coefficients are numerically calculated.

Review of theory

Assume a two semi-infinite metal-dielectric waveguides with conjunction plane located at x=0 (Fig. 1 ). All materials constructing the structure are assumed isotropic and non-magnetic (μ=1). The structure is assumed invariant in the z direction.

 

Fig. 1 Schematic of the structure under study. Ωi, i = d1, d2, m1, and m2 present different regions. Solid lines present boundary between different regions. The boundaries B and B are, respectively, located at |x|= and |y|=. Reflection and transmission coefficients are calculated by using points A and B, respectively.

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Let a p-polarized SPP wave with time dependency of exp(iωt) and unit energy flow travels from left to right and impinges the interface between two waveguides at x=0. The wave is partially reflected, transmitted, and scattered due to its interaction with the interface. To analyze such interaction we start with Helmholtz wave equation for a p-polarized wave that is as:

[ε(r,ω).ε(r,ω)+2+k02ε(r,ω)]HZ(r,ω)=0.

The corresponding Green’s function is obtained by solving the following equation:

[ε(r,ω).ε(r,ω)+2+k02ε(r,ω)]g(r,r,ω)=δ(r,r).
where r=(x,y) and r=(x,y)are the position vectors of observation and source points. The Green’s function g(r,r,ω) that is introduced in ref [9] is the solution of Eq. (2) for a system consisting of a dielectric in the region y>0 and a metal in the region y<0 with a line source parallel to the z axis situated in one of these media. The Green’s function g(r,r,ω) can be obtained by solving Eq. (2) and applying appropriate boundary conditions on the Green’s function and its derivative at metal- dielectric interface, at infinity, and at r=r point.

By using Eqs. (1) and (2), boundary integral equation over boundaries of regions Ωd1 and Ωm1 are obtained as (see the Appendix A):

12HZ,1(r)=B1,2(g(r,r)xHZ,1(r)HZ,1(r)xg(r,r))dyB1(g(r,r)xHZ,1(r)HZ,1(r)xg(r,r))dy,
and corresponding integral equation over boundaries of regions Ωd2 and Ωm2 are obtained as:

12HZ,2(r)=B2,1(g(r,r)xHZ,2(r)HZ,2(r)xg(r,r))dy+B2(g(r,r)xHZ,2(r)HZ,2(r)xg(r,r))dy.

One should note that the left hand side of Eq. (3) and (4) are the Cauchy principal value integrals. After doing some mathematics (see the Appendix B), the Generalized Green Function Surface Integral Equation for the magnetic field over boundaries of regions Ωd1 and Ωm1 are obtained as:

12HZ,1(r)=B1,2[g(r,r)xHZ,1(r)HZ,1(r)xg(r,r)]dyB1[g(r,r)xHi(r)Hi(r)xg(r,r)]dyrB1[g(r,r)xHr(r)Hr(r)xg(r,r)]dy.
and corresponding integral equation over boundaries of regions Ωd2 and Ωm2 are obtained as:
12HZ,2(r)=B2,1[g(r,r)xHZ,2(r)HZ,2(r)xg(r,r)]dy+tB2[g(r,r)xHt(r)Ht(r)xg(r,r)]dy.
where r and t are amplitude reflection and transmission coefficients, respectively (see the Appendix B). Here, Hi, Hr, and Ht are solutions of Helmholtz equation which are, respectively, given by the following equations:
Hi=Niexp(iksppx)exp(ksppyεd1/εm1)Hr=Nrexp(iksppx)exp(ksppyεd1/εm1)Ht=Ntexp(iksppx)exp(ksppyεd2/εm2),
for positive y and by the following equations:
Hi=Niexp(iksppx)exp(ksppyεm1/εd1)Hr=Nrexp(iksppx)exp(ksppyεm1/εd1)Ht=Ntexp(iksppx)exp(ksppyεm2/εd2).
for negative y. In Eqs. (7) and (8), kspp=k0[εd1εm1/(εd1+εm1)]1/2 and kspp=k0[εd2εm2/(εd2+εm2)]1/2 are surface plasmon wave numbers of two waveguides at frequency ω and Ni, Nr, and Nt are normalization constants obtained from the relation ReSxdy=1. Then, the reflection and the transmission coefficients are obtained by using R=|r|2 and T=|t|2, respectively [5].

We numerically solve Eqs. (5) and (6) by using Boundary Element Method and by applying appropriate boundary conditions for each interface. Then, the boundaries of the structure under study are discretized and the integrals in Eqs. (5) and (6) are transformed as a summation of integrals over those elements. That results to 2N equations for N nodes located on the boundaries. Solving the resulting system of equations gives the field and its derivative at the nodal points. Next, by using Eqs. (24) and (27), reflection and transmission coefficients and, consequently, scattering coefficient are calculated. To calculate magnetic field inside Ωd1 and Ωm1 regions one should use the following equation:

HZ,1(r)=B1,2(g(r,r)xHZ,1(r)HZ,1(r)xg(r,r))dyB1(g(r,r)xHi(r)Hi(r)xg(r,r))dyrB1(g(r,r)xHr(r)Hr(r)xg(r,r))dy. (9).

The corresponding equation for magnetic field inside the regions Ωd2 and Ωm2 is as:

HZ,2(r)=Bd2,1(εd2εd1g(r,r)xHZ,1(r)HZ,1(r)xg(r,r))dyBm2,1(εm2εm1g(r,r)xHZ,1(r)HZ,1(r)xg(r,r))dy+tB2(g(r,r)xHt(r)Ht(r)xg(r,r))dy.

In Eqs. (5) and (6) there are integrals taken over boundaries extended to infinity (interface of two waveguide in Fig. 1) which should be truncated at appropriate points in order to discretize the boundary. Each truncation point should be chosen somewhere that is located much deeper than the skin depth inside the medium. The skin depths of the dielectric and the metal media are, respectively, given by:

dd=Re([(εd+εm)]1/2/k0εd),
and

dm=Re([(εd+εm)]1/2/k0εm).

Numerical results

We have numerically solved Eqs. (5) and (6) which are obtained by G-GFSIEM to calculate fields at the interface of two waveguides. First, variation of the z-component of the magnetic field for the structure presented in Fig. 1 are calculated by using G-GFSIEM and the result is compared with that obtained by mode matching technique [5]. The dielectric constants of the media are assumed to be the same as those mentioned in ref [5]. Figure 2 illustrates that there is a good agreement between our results with those presented in Ref [5]. Since the time variation of the magnetic field in our method is assumed as exp(iωt), the term Im(HZ)depicted in Fig. 2 has opposite sign as compared with corresponding result presented in ref [5].

 

Fig. 2 a) Magnetic field Hz, and b) Electric field Ey distributions at the interface for structure with εd1=2.25, εd2=1, εm1=εm2=18.30.5i. The wavelength of incident beam is λ=632.8nm.

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Furthermore, variation of RTS coefficients vs. dielectric constant of one dielectric medium when metal media are the same, i.e.εm1=εm2, and the other dielectric constant is kept fixed are also verified (Fig. 3 ). The results are in good agreement with those presented in Ref [5]. which confirms, again, correctness of the method.

 

Fig. 3 Variation of reflection, transmission, and scattering coefficients for a structure with a) εd2=1, εm1=εm2=18.30.5i vs. εd1 and b) εd1=1, εm1=εm2=18.30.5i vs. εd2. The wavelength of incident beam is λ=632.8nm.

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After verifying correctness of the method we study variation of the scattering coefficient as a function of dielectric constants of different media. One of the major mechanisms of energy dissipation in a plasmonic circuit is scattering from finite or infinite defects. The level of scattering depends on the structure and dielectric constants of the media which constitute the waveguide. Therefore it is worthwhile to investigate the effects of dielectric constants on RTS coefficients of the structure shown in Fig. 1.

Figure 4 illustrates variation of RTS coefficients vs. εd1 [Fig. 4(a), 4(b)] and εd2 [Fig. 4(c), 4(d)]. The wavelength of incident beam is λ=632.8nm. While in Figs. 4(a) and 4(d) scattering coefficient steadily increases by increasing dielectric coefficient, in Fig. 4(b), 4(c) it decreases by increasing εd1 (εd2), reaches to a minimum and gradually increases for larger values of dielectric constant. To interpret such behavior we first consider a simple case where both waveguides are the same, i.e. εd1=εd2 and εm1=εm2. Then, we expect transverse profile of the SPP wave for both waveguides to be the same and the scattering coefficient to be equal to zero. Deviation from such structure results to different transverse profiles for the SPP waves in two waveguides. Consequently, skin depths in the metallic and in the dielectric regions of both waveguides become different. Comparing variation of skin depth in two dielectric media, i.e. d1/d2, vs. dielectric constants of the constituting media in Fig. 4 illustrates when transverse profiles of the field in the dielectric media are the same [air and εd11.8 in Fig. 4(b) and air and εd21.8 in Fig. 4(c)], the scattering coefficient is nearly zero. That corresponds to condition where d1/d2=1 (filled circle in Fig. 5 ). On the other hand, when d1/d2 ratio is not equal to unity transverse profiles of the SPP field at two sides of the interface becomes different. Therefore, in order to satisfy boundary condition some part of the energy of the SPP wave should convert to radiative wave and, consequently, the scattering coefficient becomes non-zero. Figure 5 also confirms that for structures corresponding to Figs. 4(a) and 4(d), where scattering coefficient is always non-zero, d1/d2 ratio is never becomes equal to unity.

 

Fig. 4 Variation of RTS coefficients vs. εd1 (a, b) and εd2 (c, d). Dielectric constant of A and B are εA=202iand εB=6020i, respectively. The wavelength of the incident beam is λ=632.8nm.

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Fig. 5 Variation of d1/d2 vs. dielectric constant εd for structures investigated in Fig. 4.

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Figure 6 illustrates variation of RTS coefficients vs. dielectric constants of metallic media. It shows that the reflection coefficient is nearly constant for different values of dielectric constant of the metallic media but, the transmission and the scattering coefficients behave differently. While the scattering coefficient increases by increasing |εm1| [Fig. 6(a)] it decreases by increasing |εm2|, reaches to a minimum, and gradually decreases for its larger values [Fig. 6(b)]. Such behavior can be used to adjust the level of the scattering without significant change on the reflection coefficient. Our calculation has also revealed that RTS coefficients are almost insensitive to the variation of the imaginary part of dielectric constants of the metallic media [Fig. 6(c), 6(d)]. Such behavior can also be explained by studying variation of d1/d2 vs. dielectric constants of the metallic media.

 

Fig. 6 Variation of RTS coefficients vs. a) |εm1|, b) |εm2|, c) |εm1|, and d) |εm2|. Dielectric constants of A and B are εA=202iand εB=6020i respectively. The wavelength of incident beam is λ=632.8nm.

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By using this method, it is also possible to study variation of RTS coefficients of different structures vs. the wavelength of the incident beam. Figure 7 illustrates such variation for a typical structure consisted of air, quartz, Ag, and Al [14]. If the right side and the left side waveguides are consisted of air, Ag and quartz, Al, respectively, the scattering coefficient is nearly negligible for all wavelengths [Fig. 7(a)]. By replacing metallic media of the right side and the left side of the waveguides with each other, the scattering coefficient becomes non-zero, but gradually decreases for longer wavelengths [Fig. 7(b)].

 

Fig. 7 Variation of RTS coefficient vs. wavelength of the incident light for two different structures depicted in the inset.

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Figure 8 shows axial component of the magnetic field profile, |Hz|, for wavelength of λ=563.6nm and for structures investigated in Fig. 7. There is negligible scattering when the left side and the right side metallic media are Ag and Al, respectively [Fig. 8(a)]. That is because skin depths of two dielectric media in that structure are nearly equal (i.e. d1/d2). By replacing the left side and the right side metallic media that ratio becomes larger than unity and it results to a non-negligible value for the scattering coefficient [Fig. 8(b)].

 

Fig. 8 Profile of the magnetic field along z-axis, |Hz| for wavelength of λ=563.6nm and for structures investigated in Fig. 7.

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On the other hand, the reflection coefficient of the structure investigated in Fig. 7 is very low. One practical method to increase reflectivity is using materials with larger dielectric constant. Si is a typical example with ε=12.04 [15] that is often used in different nano-plasmonic circuits. Figure 9 illustrates variation of RTS coefficient vs. wavelength for the structure investigated in Fig. 7 when quartz is replaced by silicon. It shows that the reflection coefficient increases up to 36% by using silicon in that structure. Therefore one can say that by using appropriate dielectric material it is possible to design structures with desired RTS coefficients.

 

Fig. 9 Variation of RTS coefficients vs. wavelength for structure consisted of air, Ag and silicon, Al.

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Conclusion

In this study we used G-GFSIEM method to investigate reflection, transmission, and scattering of SPP waves with interface between two metal-dielectric waveguides. Using this method, RTS coefficients and distribution of the magnetic field at different interfaces and at any arbitrary point in space are calculated. To validate the correctness of the method the results are compared with those obtained by using mode matching technique and good agreement between them is observed.

Effects of variation of dielectric constants of different media on RTS coefficients are studied and it is observed when the skin depths of the SPP wave in both dielectric media are equal the scattering coefficient becomes negligible. The results illustrate that by using materials with appropriate dielectric constants it is possible to design structures with desired values of RTS coefficients.

Appendix A

In this appendix, we drive Eqs. (3) and (4) by using Green’s second theorem. Let’s assume the magnetic field at each region as:

δ(rr)HZ(r)da=Bi[g(r,r)n.HZ(r)HZ(r)n.g(r,r)]dl,
where Bi is boundary of region Ωi and n is the outward unit normal vector. When the observation point is located inside (outside) region Ωi the left hand side of Eq. (13) becomes equal to Hz(r) (zero). On the other hand, if the observation point is located on the boundaries of the region Ωi Green’s function and its derivative have singularity at r=r. Considering the contribution of the singularity, the resulting term on the left hand side of Eq. (13) becomes as 12Hz(r) and the right rand side is written in the form of Cauchy principal values integral. The integration over the boundaries of the waveguides where the SPP waves propagate can be transformed to integration over boundaries located at infinity. Then, using appropriate truncation results to negligible error. To do so, we put observation point on the upper half plane. Consequently, for region Ωd1 Eq. (13) changes to the following:
HZ,d1(r)=Bd1,2[g(r,r)nd1.HZ,d1(r)HZ,d1(r)nd1.g(r,r)]dy+Bd1,m1[g(r,r)nd1.HZ,d1(r)HZ,d1(r)nd1.g(r,r)]dx+Bd1[g(r,r)nd1.HZ,d1(r)HZ,d1(r)nd1.g(r,r)]dy+Bd1[g(r,r)nd1.HZ,d1(r)HZ,d1(r)nd1.g(r,r)]dx,
and becomes as follows for region Ωm1:

0=Bm1,2[g(r,r)nm1.HZ,m1(r)HZ,m1(r)nm1.g(r,r)]dy+Bm1,d1[g(r,r)nm1.HZ,m1(r)HZ,m1(r)nm1.g(r,r)]dx+Bm1[g(r,r)nm1.HZ,m1(r)HZ,m1(r)nm1.g(r,r)]dy+Bm1[g(r,r)nm1.HZ,m1(r)HZ,m1(r)nm1.g(r,r)]dx.

By using Green’s function g(r,r) the following relations are obtained [9]:

g(x,y,x,y=0+)=εm1εd1g(x,y,x,y=0),
and:

ng(x,y,x,y=0+)=ng(x,y,x,y=0).

Then, the second integral on the right hand side of Eq. (15) is written as:

Bm1,d1[g(r,r)nm1.Hm1(r)Hm1(r)nm1.g(r,r)]dx=Bd1,m1[g(r,r)nm1.Hd1(r)Hd1(r)nm1.g(r,r)]dx=Bd1,m1[g(r,r)nd1.Hd1(r)Hd1(r)nd1.g(r,r)]dx,

By using Eqs. (15) and (18), Eq. (14) reduces to:

HZ,1(r)=Bd1,2[g(r,r)nd1.HZ,d1(r)HZ,d1(r)nd1.g(r,r)]dy+Bm1,2[g(r,r)nm1.HZ,m1(r)HZ,m1(r)nm1.g(r,r)]dy+Bm1[g(r,r)nm1.HZ,m1(r)HZ,m1(r)nm1.g(r,r)]dy+Bd1[g(r,r)nd1.HZ,d1(r)HZ,d1(r)nd1.g(r,r)]dy+Bm1[g(r,r)nm1.HZ,m1(r)HZ,m1(r)nm1.g(r,r)]dx+Bd1[g(r,r)nd1.HZ,d1(r)HZ,d1(r)nd1.g(r,r)]dx.

Because both radiative and SPP waves becomes zero at Bd1 and Bm1, the corresponding integrals could be neglected and, therefore, Eq. (19) is simplified to:

HZ,1(r)=B1,2[g(r,r)xHZ,1(r)HZ,1(r)xg(r,r)]dyB1[g(r,r)xHZ,1(r)HZ,1(r)xg(r,r)]dy.

Corresponding equation for the field in regions Ωd2, Ωm1, and Ωm2 can be derived in similar manner.

Appendix B

Equations (3) and 4 should be used to derive Eqs. (5) and (6). To solve the former two equations and calculate fields at different interfaces the part of integrals which are taken at infinity should be calculated. Then, an appropriate contour, that is a rectangle when considering an incident SPP wave, should be taken to calculate the integrals (Fig. 1.) A complete orthogonal set of normal modes of each waveguide consists of a single SPP mode and a continuous set of surface leaky modes which propagate in the forward and in the backward direction. SPP mode in metal and dielectric media has evanescent fields but surface leaky mode has radiative and evanescent fields for dielectric and metal region, respectively [5]. Then, the fields over boundaries Bd1 and Bm1 are written as:

H1=Hi+rHr+HS,
and the fields over boundaries Bd2 and Bm2 are as:
H2=tHt+HS.
where r and t are amplitude reflection and transmission coefficients, respectively. Here, Hi, Hr, and Ht are given by Eqs. (7) and (8). In Eqs. (21) and (22) Hs is linear combination of surface leaky modes. The leaky wave can be assumed equal to zero for boundaries Bi, i = d1, d2, m1, m2 which are located at infinity.

Therefore, to calculate fields over boundaries Bd1 and Bm1 (Bd2 and Bm2) only relations for amplitude reflection (transmission) coefficients at those boundaries are left to be derived. To calculate that coefficient we consider an arbitrary point inside Ωd2 or Ωm2region (e.g. point A in Fig. 1). Then, using Eqs. (3) and (21) gives:

0=B1,2[g(rA,r)xHZ,1(r)HZ,1(r)xg(rA,r)]dyB1[g(rA,r)x(Hi(r)+rHr(r))(Hi(r)+rHr(r))xg(rA,r)]dy.

One should note that the left hand side of Eq. (23) is zero because the observation point is located inside the right side waveguide. Rearranging Eq. (23) gives the amplitude reflection coefficient as:

r=fi(rA)+B12[g(rA,r)xHZ,1(r)HZ,1(r)xg(rA,r)]dyfr(rA),
where:
fi(rA)=B1[g(rA,r)xHi(r)Hi(r)xg(rA,r)]dy,
and:

fr(rA)=B1[g(rA,r)xHr(r)Hr(r)xg(rA,r)]dy.

Similarly, to calculate amplitude transmission coefficient observation point is assumed to be located inside one of the regions Ωd1 or Ωm1 (e.g. point B in Fig. 1). Then, using Eqs. (4) and (22) gives:

t=B21[g(rB,r)xHZ,2(r)HZ,2(r)xg(rB,r)]dyft(rB),
where:

ft(rB)=B2[g(rB,r)xHt(r)Ht(r)xg(rB,r)]dy.

Inserting Eq. (21) in Eq. (3) and using Eq. (24), Generalized Green Function Surface Integral Equation for the magnetic field over boundaries of regions Ωd1 and Ωm1 are obtained as:

12HZ,1(r)=B12[g(r,r)xHZ,1(r)HZ,1(r)xg(r,r)]dyB1[g(r,r)xHi(r)Hi(r)xg(r,r)]dyrB1[g(r,r)xHr(r)Hr(r)xg(r,r)]dy.

Similarly, using Eqs. (4), (22), and (27) the corresponding equations over boundaries of regions Ωd2 and Ωm2 are obtained as:

12HZ,2(r)=B21[g(r,r)xHZ,2(r)HZ,2(r)xg(r,r)]dy+tB2[g(r,r)xHt(r)Ht(r)xg(r,r)]dy.

By using boundary conditions Eq. (30) reduces to:

12HZ,1(r)=Bd2,1[εd2εd1g(r,r)xHZ,1(r)HZ,1(r)xg(r,r)]dyBm2,1[εm2εm1g(r,r)xHZ,1(r)HZ,1(r)xg(r,r)]dy+tB2[g(r,r)xHt(r)Ht(r)xg(r,r)]dy.

Equations (29) and (31) can be solved by using BEM to calculate magnetic filed over interface x=0 in Fig. 1.

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11. J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009). [CrossRef]  

12. J. D. Jackson, Classical electrodynamics, (John Wiley & Sons, New York, 1999), p. 479.

13. F. Paris, and J. Canas, Boundary element method-fundamentals and applications (Oxford University Press, 1997).

14. E. D. Palik, ed., The Handbook of optical constants of solids (Academic Press, New York, NY 1997)

15. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005). [CrossRef]  

References

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  1. H. Raether, Surface plasmons on smooth and rough surfaces and on grating (Springer-Verlag, Berlin, 1988).
  2. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
    [Crossref]
  3. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
    [Crossref]
  4. S. I. Bozhevolnyi, ed., Plasmonic Nanoguides and Circuits (Pan Stanford, Singapore, 2009).
  5. R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
    [Crossref]
  6. G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
    [Crossref]
  7. M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
    [Crossref]
  8. J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008).
    [Crossref] [PubMed]
  9. J. Jung and T. Søndergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77(24), 245310 (2008).
    [Crossref]
  10. T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi 244(10), 3448–3462 (2007) (b).
    [Crossref]
  11. J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
    [Crossref]
  12. J. D. Jackson, Classical electrodynamics, (John Wiley & Sons, New York, 1999), p. 479.
  13. F. Paris, and J. Canas, Boundary element method-fundamentals and applications (Oxford University Press, 1997).
  14. E. D. Palik, ed., The Handbook of optical constants of solids (Academic Press, New York, NY 1997)
  15. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005).
    [Crossref]

2009 (1)

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
[Crossref]

2008 (3)

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008).
[Crossref] [PubMed]

J. Jung and T. Søndergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77(24), 245310 (2008).
[Crossref]

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

2007 (2)

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[Crossref]

T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi 244(10), 3448–3462 (2007) (b).
[Crossref]

2006 (2)

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[Crossref]

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[Crossref]

2005 (1)

1981 (1)

G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
[Crossref]

Boltasseva, A.

Bozhevolnyi, S. I.

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
[Crossref]

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005).
[Crossref]

Brongersma, M. L.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[Crossref]

Chandran, A.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[Crossref]

Ebbesen, T. W.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

Elser, J.

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008).
[Crossref] [PubMed]

Genet, C.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

Hai, M.

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[Crossref]

Hong-Gao, T.

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[Crossref]

Jung, J.

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
[Crossref]

J. Jung and T. Søndergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77(24), 245310 (2008).
[Crossref]

Kjaer, K.

Larsen, M. S.

Leosson, K.

Liu, Y.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[Crossref]

Maradudin, A. A.

G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
[Crossref]

Nikolajsen, T.

Oulton, R. F.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[Crossref]

Pei, W.

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[Crossref]

Pile, D. F.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[Crossref]

Podolskiy, V. A.

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008).
[Crossref] [PubMed]

Rahman, T. S.

G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
[Crossref]

Schuller, J. A.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[Crossref]

Søndergaard, T.

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
[Crossref]

J. Jung and T. Søndergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77(24), 245310 (2008).
[Crossref]

T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi 244(10), 3448–3462 (2007) (b).
[Crossref]

Stegeman, G. I.

G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
[Crossref]

Yong, C.

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[Crossref]

Zhang, X.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[Crossref]

Zhong-Tuan, M.

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[Crossref]

Zia, R.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[Crossref]

Chin. Phys. Lett. (1)

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[Crossref]

J. Lightwave Technol. (1)

Mater. Today (1)

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[Crossref]

Phys. Rev. B (4)

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[Crossref]

G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
[Crossref]

J. Jung and T. Søndergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77(24), 245310 (2008).
[Crossref]

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
[Crossref]

Phys. Rev. Lett. (1)

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008).
[Crossref] [PubMed]

Phys. Status Solidi (1)

T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi 244(10), 3448–3462 (2007) (b).
[Crossref]

Phys. Today (1)

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

Other (5)

S. I. Bozhevolnyi, ed., Plasmonic Nanoguides and Circuits (Pan Stanford, Singapore, 2009).

J. D. Jackson, Classical electrodynamics, (John Wiley & Sons, New York, 1999), p. 479.

F. Paris, and J. Canas, Boundary element method-fundamentals and applications (Oxford University Press, 1997).

E. D. Palik, ed., The Handbook of optical constants of solids (Academic Press, New York, NY 1997)

H. Raether, Surface plasmons on smooth and rough surfaces and on grating (Springer-Verlag, Berlin, 1988).

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Figures (9)

Fig. 1
Fig. 1

Schematic of the structure under study. Ω i , i = d1, d2, m1, and m2 present different regions. Solid lines present boundary between different regions. The boundaries B and B are, respectively, located at | x | = and | y | = . Reflection and transmission coefficients are calculated by using points A and B, respectively.

Fig. 2
Fig. 2

a) Magnetic field H z , and b) Electric field E y distributions at the interface for structure with ε d 1 = 2.25 , ε d 2 = 1 , ε m 1 = ε m 2 = 18.3 0.5 i . The wavelength of incident beam is λ = 632.8 n m .

Fig. 3
Fig. 3

Variation of reflection, transmission, and scattering coefficients for a structure with a) ε d 2 = 1 , ε m 1 = ε m 2 = 18.3 0.5 i vs. ε d 1 and b) ε d 1 = 1 , ε m 1 = ε m 2 = 18.3 0.5 i vs. ε d 2 . The wavelength of incident beam is λ = 632.8 n m .

Fig. 4
Fig. 4

Variation of RTS coefficients vs. ε d 1 (a, b) and ε d 2 (c, d). Dielectric constant of A and B are ε A = 20 2 i and ε B = 60 20 i , respectively. The wavelength of the incident beam is λ = 632.8 n m .

Fig. 5
Fig. 5

Variation of d 1 / d 2 vs. dielectric constant ε d for structures investigated in Fig. 4.

Fig. 6
Fig. 6

Variation of RTS coefficients vs. a) | ε m 1 | , b) | ε m 2 | , c) | ε m 1 | , and d) | ε m 2 | . Dielectric constants of A and B are ε A = 20 2 i and ε B = 60 20 i respectively. The wavelength of incident beam is λ = 632.8 n m .

Fig. 7
Fig. 7

Variation of RTS coefficient vs. wavelength of the incident light for two different structures depicted in the inset.

Fig. 8
Fig. 8

Profile of the magnetic field along z-axis, | H z | for wavelength of λ = 563.6 n m and for structures investigated in Fig. 7.

Fig. 9
Fig. 9

Variation of RTS coefficients vs. wavelength for structure consisted of air, Ag and silicon, Al.

Equations (31)

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[ ε ( r , ω ) . ε ( r , ω ) + 2 + k 0 2 ε ( r , ω ) ] H Z ( r , ω ) = 0.
[ ε ( r , ω ) . ε ( r , ω ) + 2 + k 0 2 ε ( r , ω ) ] g ( r , r , ω ) = δ ( r , r ) .
1 2 H Z , 1 ( r ) = B 1 , 2 ( g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ) d y B 1 ( g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ) d y ,
1 2 H Z , 2 ( r ) = B 2 , 1 ( g ( r , r ) x H Z , 2 ( r ) H Z , 2 ( r ) x g ( r , r ) ) d y + B 2 ( g ( r , r ) x H Z , 2 ( r ) H Z , 2 ( r ) x g ( r , r ) ) d y .
1 2 H Z , 1 ( r ) = B 1 , 2 [ g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y B 1 [ g ( r , r ) x H i ( r ) H i ( r ) x g ( r , r ) ] d y r B 1 [ g ( r , r ) x H r ( r ) H r ( r ) x g ( r , r ) ] d y .
1 2 H Z , 2 ( r ) = B 2 , 1 [ g ( r , r ) x H Z , 2 ( r ) H Z , 2 ( r ) x g ( r , r ) ] d y + t B 2 [ g ( r , r ) x H t ( r ) H t ( r ) x g ( r , r ) ] d y .
H i = N i exp ( i k s p p x ) exp ( k s p p y ε d 1 / ε m 1 ) H r = N r exp ( i k s p p x ) exp ( k s p p y ε d 1 / ε m 1 ) H t = N t exp ( i k s p p x ) exp ( k s p p y ε d 2 / ε m 2 ) ,
H i = N i exp ( i k s p p x ) exp ( k s p p y ε m 1 / ε d 1 ) H r = N r exp ( i k s p p x ) exp ( k s p p y ε m 1 / ε d 1 ) H t = N t exp ( i k s p p x ) exp ( k s p p y ε m 2 / ε d 2 ) .
H Z , 1 ( r ) = B 1 , 2 ( g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ) d y B 1 ( g ( r , r ) x H i ( r ) H i ( r ) x g ( r , r ) ) d y r B 1 ( g ( r , r ) x H r ( r ) H r ( r ) x g ( r , r ) ) d y .
H Z , 2 ( r ) = B d 2 , 1 ( ε d 2 ε d 1 g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ) d y B m 2 , 1 ( ε m 2 ε m 1 g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ) d y + t B 2 ( g ( r , r ) x H t ( r ) H t ( r ) x g ( r , r ) ) d y .
d d = Re ( [ ( ε d + ε m ) ] 1 / 2 / k 0 ε d ) ,
d m = Re ( [ ( ε d + ε m ) ] 1 / 2 / k 0 ε m ) .
δ ( r r ) H Z ( r ) d a = B i [ g ( r , r ) n . H Z ( r ) H Z ( r ) n . g ( r , r ) ] d l ,
H Z , d 1 ( r ) = B d 1 , 2 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d y + B d 1 , m 1 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d x + B d 1 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d y + B d 1 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d x ,
0 = B m 1 , 2 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d y + B m 1 , d 1 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d x + B m 1 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d y + B m 1 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d x .
g ( x , y , x , y = 0 + ) = ε m 1 ε d 1 g ( x , y , x , y = 0 ) ,
n g ( x , y , x , y = 0 + ) = n g ( x , y , x , y = 0 ) .
B m 1 , d 1 [ g ( r , r ) n m 1 . H m 1 ( r ) H m 1 ( r ) n m 1 . g ( r , r ) ] d x = B d 1 , m 1 [ g ( r , r ) n m 1 . H d 1 ( r ) H d 1 ( r ) n m 1 . g ( r , r ) ] d x = B d 1 , m 1 [ g ( r , r ) n d 1 . H d 1 ( r ) H d 1 ( r ) n d 1 . g ( r , r ) ] d x ,
H Z , 1 ( r ) = B d 1 , 2 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d y + B m 1 , 2 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d y + B m 1 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d y + B d 1 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d y + B m 1 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d x + B d 1 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d x .
H Z , 1 ( r ) = B 1 , 2 [ g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y B 1 [ g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y .
H 1 = H i + r H r + H S ,
H 2 = t H t + H S .
0 = B 1 , 2 [ g ( r A , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r A , r ) ] d y B 1 [ g ( r A , r ) x ( H i ( r ) + r H r ( r ) ) ( H i ( r ) + r H r ( r ) ) x g ( r A , r ) ] d y .
r = f i ( r A ) + B 12 [ g ( r A , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r A , r ) ] d y f r ( r A ) ,
f i ( r A ) = B 1 [ g ( r A , r ) x H i ( r ) H i ( r ) x g ( r A , r ) ] d y ,
f r ( r A ) = B 1 [ g ( r A , r ) x H r ( r ) H r ( r ) x g ( r A , r ) ] d y .
t = B 21 [ g ( r B , r ) x H Z , 2 ( r ) H Z , 2 ( r ) x g ( r B , r ) ] d y f t ( r B ) ,
f t ( r B ) = B 2 [ g ( r B , r ) x H t ( r ) H t ( r ) x g ( r B , r ) ] d y .
1 2 H Z , 1 ( r ) = B 12 [ g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y B 1 [ g ( r , r ) x H i ( r ) H i ( r ) x g ( r , r ) ] d y r B 1 [ g ( r , r ) x H r ( r ) H r ( r ) x g ( r , r ) ] d y .
1 2 H Z , 2 ( r ) = B 21 [ g ( r , r ) x H Z , 2 ( r ) H Z , 2 ( r ) x g ( r , r ) ] d y + t B 2 [ g ( r , r ) x H t ( r ) H t ( r ) x g ( r , r ) ] d y .
1 2 H Z , 1 ( r ) = B d 2 , 1 [ ε d 2 ε d 1 g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y B m 2 , 1 [ ε m 2 ε m 1 g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y + t B 2 [ g ( r , r ) x H t ( r ) H t ( r ) x g ( r , r ) ] d y .

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