Abstract

Transfer matrices for one-dimensional (1-D) multi-layered magneto-optical (MO) waveguides are formulated to analytically calculate the nonreciprocal phase shifts (NRPS). The Cauchy contour integration (CCI) method is introduced in detail to calculate the two complex roots of the transcendental equation corresponding to backward and forward waves. By virtue of perturbation theory and the variational principle, we also present the general upper limit of NRPSs in 1-D MO waveguides. These analytical results are applied to compare silicon-on-insulator (SOI) based MO waveguides. First, a three-layered waveguide system with MO medium is briefly examined and discussed to check the validity and efficiency of the above theory. Then we revisited the reported low-index-gap-enhanced NRPSs in MO waveguides and obtained substantially different results. Finally, the potential of common plasmonic waveguides to enhance the nonreciprocal effect is investigated by studying different waveguides composed of Metal, MO medium and dielectrics. Our study shows that the reasonable NRPSs can be optimized to some extent but not as much as claimed in previous publications.

© 2012 OSA

1. Introduction

Non-reciprocal photonic devices, including optical isolators and circulators, are always challenging but indispensable in building chip-scale opto-electronic integrated circuit (OEIC). The main obstacles are the large lattice and thermal mismatch between the magneto-optical (MO) garnets and the semiconductor substrates. Considerable efforts have been devoted to break the time-reversal symmetry and produce nonreciprocal effects on the silicon-on-insulator (SOI) platform. A possible solution to bypass the materials problem is to realize one-way mode transition through material index modulation [1, 2], which is appealing because no foreign material is involved beyond complementary-metal-oxide-semiconductor (CMOS) compatible materials. However, further study is required to reduce the device footprint and compensate for the accompanying loss. Hence, conventional structures containing magnetized MO materials are still dominant for producing reciprocal effect. The first successful experiment that brings MO materials to the SOI platform was conducted by direct wafer bonding, which enables the integration of a single-crystal MO film with thin a flat thin SOI waveguide [3]. Very recently, a polycrystalline garnet bilayer containing Ce-doped yttrium iron garnet (Ce:YIG) was directly deposited onto the silicon platform [4] and an optical isolator was monolithically integrated. This striking technology can be added into the back-end of CMOS process, which enables plenty of room to envision new structures on SOI platform.

The number of theoretical and numerical studies of SOI-based MO waveguides continue to increase [59]. It is well known that the Faraday rotation coefficients in widely-used low-loss MO materials (e.g. Ce:YIG) are so weak that the length of the constructed Mach-Zehnder-type isolator is usually at the scale of a millimeter [10, 11]. Therefore, in order to reduce the interaction length of light and MO medium, enhancing MO effect in waveguides has become a hot topic. Different resonators containing MO medium may be feasible since the required nonreciprocal phase shift (NRPS) only needs to shift the backward resonance spectrum away from the forward peaks with a scale of half maximum bandwidth (HMBW) [12,13]. But designs are limited by the operation bandwidth and only work as a nonreciprocal filter. In practice, most isolators are based on interferometers with NRPSs. There are mainly three approaches to enhance the NRPS in waveguides: (1) breaking the waveguide symmetry to cater to the NRPS of the concerned waveguide mode; (2) strongly confining light into a small volume and strengthening its interaction with the MO material and (3) creating additional magnetic domains. However, due to the difficulty in material synthesis, very few research groups have fabricated MO waveguides and most NRPS numerical studies have not been experimentally verified. Hence, analytical calculations are of great importance to study the fundamental physics in MO waveguides and some comparison analysis is useful to improve the reliability.

In this paper, the universal transfer matrix of a multilayered MO waveguide system was derived to accurately calculate the NRPSs, together with the detailed implementation of the uncommon Cauchy contour integration method to find solutions of transcendental equations. The analytical results are in good agreement with those predicted by perturbation theory. We found that there is an upper limit for the available NRPS using a fixed MO material. These theoretical predictions are applied to make comparisons between reported schemes to enhance nonreciprocal phase shifts, such as asymmetric three-layered waveguides, slot waveguides and plasmonic waveguides. By doing so, some ambiguity in previous publications about the calculation of NRPS are cleared up.

2. Transfer matrices for multi-layered magneto-optical waveguide system

Let us start from the time-dependent Maxwell equations to derive the transfer matrix of a homogenous MO layer. The configuration is shown in Fig.1. For the sake of convenience, we use the dimensionless notations in the whole text by normalizing a unit physical length (e.g. 1μm) to unity [14] and assume the time and z dependent phase factor to be exp [j2πν(tγz)], where ν is the normalized frequency and γ is the normalized propagation constant (also known as the effective refractive index). The latter can be negative for backward propagating waves. The Faraday’s law and Ampere’s law can be written as the following set of equations

×E=j2πνH
and
×H=j2πνεrE,
respectively. By dropping the free-space impedance, the three-dimensional field vectors E and H are scaled to the same order of magnitude. Considering the MO effect, the general expression of the relative permittivity tensor εr reads
εr=(εxxjεxyjεxzjεxyεyyjεyzjεxzjεyzεzz).
The off-diagonal elements (i.e. gyrotropy) correspond to the first-order MO effects, where the subscripts denote the coupled field components. The εxy induced by a longitudinal magnetic induction Bz gives rise to the coupling between two transverse electric field components. This is the so-called Faraday rotation effect, which is not covered in this paper. The other two, εxz from the y-directed magnetization and εyz from x-directed produce an NRPS by coupling the longitudinal component Ez with one transcendental electric field component Ex and Ey, respectively. Since the element εyz can be treated similar to εxz, let us assume that εxy = 0 and εyz = 0 and only analyze the NRPS induced by the transverse magnetic (TM) mode. For such modes, the longitudinal magnetic field component is missing (Hz = 0) and there are only three non-trivial components Hy, Ex and Ez. Substituting Eq.(3) into Eq.(2) yields
zExxEz=j2πνHy
and
(zHyxHy)=j2πν(εxxjεxzjεxzεzz)(ExEz)
Clearly, the above equation can be inverted as
(ExEz)=1j2πν1εxxεzzεxz2(εzzjεxzjεxzεxx)(zHyxHy)
By substituting Eq.(6) into Eq.(4), we get the differential equation for Hy component as
x2Hy(2πν)2[εzzεxxγ2(εzzεxz2εxx)]Hy=0.
This equation was first reported in [15], indicating that in a homogenous MO medium without waveguiding, the properties of Hy are conserved and no NRPS exists except for a minor modification to the propagation constant. The NRPS must be attributed to the coupling of the two electric fields at the MO material interfaces. By defining εe=εzzεxz2εxx and recalling the momentum conservation, we can get the normalized transverse wave vector κ
κ=+εzzεxxγ2εe.
Now we focus on the fields in the n-th MO layer. According to Eq.(7), the solutions of the magnetic field Hy can be separated to a forward wave and a backward wave and take the form of
Hy(n)={Anexp[2πκn(xxn)]+Bnexp[2πκn(xxn)]}ei2πνγz
The positive sign in Eq.(8) assures that the An term is for the forward wave with phase retardation along x direction. Subsequently, by using z = − j2πνγ and substituting Eq.(9) into Eq.(6), the field Ez can be written as
Ez=1jεe(12πνxHyγεxzεxxHy)
At the interfaces between any two layers, both tangential field components Ez and Hy must be continuous, so
f(n)(dn)=f(n+1)(0),
where
f(n)(x)=(111εe(κεxzγεxx)1εe(κεxzγεxx))(Aexp(2πνκx)Bexp(+2πνκx))(n),
and the label (n) should be added to all the local variables on the right hand side except for γ. The origin of x is chosen as the left boundary of the n-th layer. From Eq.(11), we are able to extract the transfer matrix Sn of the n-th layer, which relates the weighted indices An and Bn with An+1 and Bn+1 by
(A(n+1)B(n+1))=Sn(A(n)B(n))
By simple algebraic manipulation, the transfer matrix can be written as
Sn=12((1+ab)exp(ϕn)(1ab)exp(ϕn)(1a+b)exp(ϕn)(1+a+b)exp(ϕn))
where the transverse phase delay in the n-th layer ϕn = 2πνκndn. The full expressions of a and b describe the jumping process from layer n to layer n + 1, which can be written as
an,n+1=εe(n+1)κnεe(n)κn+1
and
bn,n+1=(εxz(n+1)εxx(n+1)εe(n+1)εe(n)εxz(n)εxx(n))γκn+1,
respectively. If b = 0, the above equation returns to the case of non-magnetized medium. A nonzero b breaks the time-reversal symmetry and gives rise to the NRPS. Let us assume A0 = 1 to normalize all the weighted indices. The reflection coefficient R and transmission coefficient T can be obtained from the cascaded transfer matrix ST = SN SN−1 ···S1 by
(AN0)=ST(1B0).
In the two-port network indicated by the above equation, the physical meaning of B0=ST,21ST,22 is the reflection coefficient, whose poles correspond to the guided modes that exponentially decay away from the cladding boundaries. Hence, in essence, the calculation of NRPS involves solving the equation ST,22 = 0 and to obtain the two roots γ corresponding to backward and forward propagating waves. Subsequently, the NRPSs of unit length are obtained by the difference of the normalized propagation constants Δγ by NRPS = 2πνΔγ.

 

Fig. 1 Schematic of a multi-layered slab waveguide system composed of N + 1 isotropic layers, in which magnetooptical medium can be included. The propagation direction z points into the paper and the waveguide is infinite in y.

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3. Cauchy contour integration method

In later sections, we will study plasmonics-enhanced MO waveguides, in which material loss must be taken into account. Hence, the roots for propagation constants are complex. It is widely accepted that finding the roots of a transcendental equation in a complex plane is quite challenging. Researchers in plasmonics field usually utilize pure numerical root finding schemes, such as Nelder-Mead miniaturization method [16, 17] and reflection pole method [18]. Here we introduce a semi-analytical method to calculate the modes in complex plane by Cauchy contour integration (CCI) [19, 20]. We describe the method as semi-analytical because the root is obtained by an analytical formula of CCI but the integration itself would be done through numerical accumulations.

Originally, the CCI method was frequently used to calculate the residues of a complex function in complex analysis. The basic concept can be described as follow. In a complex plane, there is a domain D enclosed by a simple closed curve C. A single-valued function f(z) is analytical in D and has no null point on C. The count and the sum of roots of equation f(z) = 0 in domain D can be calculated by

N=12πiCf(z)f(z)dz
and
n=1Nznk=12πiCzkf(z)f(z)dz,
respectively. Naturally, the k = 1 case is adopted for the root searching. Special attention should be paid in the following issues. First, the function f(z) must be analytical in D, i.e. f(z) satisfies the Cauchy-Riemann condition. Second, one needs to choose a possible region where the roots may be located at. Third, in each possible region, there may be several roots so that we have to isolate each of them by dividing D into smaller and smaller domains. Instead of the reported Kuhn root-finding algorithm [19], we propose an intuitive and easy-to-implement numerical method to search for the roots. (1) In the complex plane we can plot the figure of 1f(z) numerically and define a trianglar C through P0P1P2P0 to enclose the domain D that contains the poles. (2) N is calculated by Eq.(18) to check the number of roots. Let us choose a reasonable numerical tolerance ε (in our case, ε = 10−4.) (3) If |N| < ε, stop here and recheck the predefined domain C since no root exists in it. (4) If |N − 1| < ε, return the root value evaluated by Eq.(19), else divide the triangle domain C into three child triangles as P0P1P3P0, P1P2P3P1 and P2P0P3P2. (5) For each child domain, go to step (2) for further calculation. The integration should be conducted clockwise otherwise the sign of the result would be reversed. Triangular domains are recommended because they approach the root faster than other polygons and maintain a similar shape between father and child domains.

As needed in Eq.(19), the transfer matrix of the derivative of ST,22 with respect to γ can also be easily derived by a simple linear algebraic manipulation from Eq.(14). All numerical integrations can be easily performed by built-in functions in Matlab. To obtain a continuous and smooth characteristic curve by parameter scanning, for example, the mode dispersion curves, a useful trick is that the triangular domain for the third root can be estimated from the two previous roots.

4. The upper limit of NRPS in 1-D waveguides

Perturbation theory is widely used to calculate the NRPS, especially for the irregular two dimensional (2-D) MO waveguides [21]. Usually the gyrotropy is two orders of magnitude smaller than the reciprocal permittivity, and thus the non-diagonal elements Δεr in the permittivity tensor can be treated as a first-order approximation. Δγ can be estimated by the expression [6, 22, 23]

Δγ=ν2E*ΔεrEdxdy,
where the fields are normalized to the time averaged poynting vector
S=14(E×H*+E*×H)zdxdy.
The dominant modes in common waveguides are nearly linearly polarized. The coupling between the two electric field components along the two vertical principal axis are so small that the semi-vectorial approximation can be made. In general, indicated in Sec. 2, only the modes whose principal electric field is normal to the magnetized direction has NRPS. Therefore, for the sake of clarity, it is recommended to disregard the distinction between quasi-TE or quasi-TM modes, and to always take the magnetized direction as y and the principal field as Ex. We can reduce Eq.(20) to cater to the numerical simulation as
Δγ2γ2ReεxzEx*1εexεeExdxdy[|Ex2|12πνγExx(1εexεeEx*)]dxdy
Because most of the waveguides we are concerned with have high refractive index contrast, the discontinuity of material index can not be neglected, thus we have to use ∇ · D = 0 instead of ∇ · E = 0 in [23]. Nevertheless, Eq.(20) can also be reduced to another simplified form with the principal magnetic field Hy as [22]
Δγ22πνεxzεxx2Hy*xHydxdy
where the field Hy is normalized to unity by 1εe|Hy2|dxdy=1. In order to avoid unwanted NRPS cancellation, the sign of εxzHy*xHy in the integrand should remain unchanged over the whole interval. Because Hy*xHy is always odd in a symmetric waveguide for any modes, an antisymmetric εxz is favorable to produce the largest NRPS. For a 90°-rotated system, i.e., a system with the non-diagonal permittivity elements of εyz, one just needs to interchange all the symbols x and y in Eq.(22) and Eq.(32).

Since the power carried by a waveguide must be finite, the field components and their derivatives should have maximum values in the waveguide cross section. We expect that there is an upper limit for the absolute value of Δγ. By removing the ∫ dy in Eq.(32), the formula reduces to the one-dimensional case and we have

|Δγ|1πνmax|εxzεxx2|MO|HyxHydx|

By neglecting the refractive index discontinuity and using a scalar approximation, the variational principle for eigenvalues show that there is stationary expression for γ2 with respect to small variations of the eigen-field [24]. For the transverse electric modes, we have the following relation

γ2=[Hy2ε(x)(12πνxHy)2]dx
With the aid of the Hölder inequality, one may get
HyxHydx2πνHy2ε(x)dxγ2
It is also well-known that the guided mode of any asymmetric waveguide has a cutoff condition for the case where γ = n2, where n2 is the higher refractive index of the cladding layers at the waveguide sides. Therefore, we have an upper limit of |Δγ| expressed as
|Δγ|2max(εxzεxx2)εmn22
where εm is the maximum dielectric constant in the waveguide system. This equation is very instructive to predict the greatest NRPS value one can obtain no matter how to optimize the waveguide structure. Also indicated is that larger NRPS may be expected in high-contrast MO waveguides.

5. Magneto-optical waveguides on SOI platform

In this section, we calculate the NRPS in many different types of MO waveguides on SOI platform to verify the above theory. Unless otherwise specified, the refractive index of Si, Si3N4, SiO2 and Ce:YIG are assumed to be 3.477, 2.0, 1.444 and 2.22, respectively. The values of Faraday rotation of Ce:YIG films was reported experimentally to be as high as −4500°cm−1 in the wavelength of interest, which corresponds to a non-diagonal element of 0.0086 in the permittivity tensor. For the comparison analysis with prior publications, we conservatively assume εxz = 0.005 in the whole text. Actually, all the calculated NRPSs can be scaled to εxz for any realistic MO material.

The non-uniform Yee-meshed full vectorial mode solver was used to calculate the field distributions of all the six mode components, which were substituted into Eq.(20) for NRPS calculation. A sufficiently large computation window surrounded by perfect electric conductor (PEC) boundary conditions were used to enable us to roughly distinguish between leaky wave and guided modes because the former may hit the PEC boundary. In order to reflect the influences of small waveguide dimensions, the minimum mesh size is chosen to be a fraction of the minimum linewidth. As we will see, the results are in good agreement with several individual publications.

5.1. Multi-layered MO waveguides

The three-layered structure is the earliest reported MO waveguide and is widely used in literatures and experiments. Here we calculated the NRPS of the dominant (lowest) modes in such waveguides that can be realized on the SOI platform. Basically, the MO medium can be placed either in the middle or top layers. In light of 1+bn,n+1an,n+1=an+1,nbn+1,n, the transcendental equation can be deduced from Eq.(14) and expressed as

exp(4πνκ1d1)=1(a01+b01)1+(a01+b01)1(a21b21)1+(a21b21)
where the subscripts 0, 1 and 2 are for the three layers, respectively. With the aid of arctan(x)=12iln1ix1+ix, one can get the arctan form of Eq. (28) as
2πνiκ1d1=atan[i(a01+b01)]+atan[i(a21b21)].
This equation covers the previously-reported transcendental equations for three-layer MO waveguide with MO material as core layer [15] or sided layer [3]. By taking b01 and b21 as small quantities, one can relate Δγ to γ0 by
Δγ2|c01b01+c21b21c01γa01+c21γa21+2πνγ0d1κ1|
where cm,1=(1am,12)1 and γ0 is the normalized propagation constant of the correspondent reciprocal waveguide with b01 = b21 = 0 in Eq. (29).

In Fig. 2 we plotted the dependence of NRPS on the waveguide core layer thickness. It can clearly be seen that there is an optimal thickness for the largest NRPS in each structure, whose value is influenced by the refractive index contrast and the extent of waveguide symmetry. Among these structures, those with symmetric refractive index and asymmetrical MO element distributions, i.e. MO(+)/Si/MO(−) and SiO2/MO(+)/MO(−)/SiO2, have the largest NRPSs due to the constructive contributions from both sides of the waveguide. However, such waveguides are more difficult to realize in practice due to the opposite magnetization required. As for the uni-magnetized structures, the MO/Si/Air structure has the highest NRPS due to the serious geometric asymmetry. The optimal thickness gradually decreases to zero in the process approaching the symmetric case. For each curve, the NRPS value decreases rapidly below the optimal thickness to zero since there exists a cutoff thickness for an asymmetric waveguide when γ is equal to the larger refractive index n2 of the two claddings, which can be written as

d1=12πνε1ε2atan(ε1ε2ε2ε0ε1ε2).

 

Fig. 2 Nonreciprocal phase shifts in three-layered SOI-based waveguides, where the MO medium (Ce:YIG) is placed (a) at one side and (b) in the center, respectively. The calculated Δγ by Eq. (30) is also shown for symmetric MO(+)/Si/MO(−) waveguide in (a), where γ0 is numerically prepared by 1-D finite difference frequency domain (FDFD) method.

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Figure 2(b) provides us with similar information in the case of center-placed MO medium. The optimal thicknesses are greater than those in Fig. 2(a) due to weaker light confinement. The NRPS is only a small fraction of those in Fig. 2(a). This phenomena is due to the cancelation of MO effect at the two interfaces. Another more intuitive evidence for this explanation is that no NRPS exists in both the MO/Si/MO and SiO2/MO/SiO2 structures because of the cancelation. For a symmetric three-layered structure, we have a01=1a12 and b01 = −a01b12 to simplify Eq. (28) so that all the linear terms for γ are missing in the nominator and denominator of Eq. (28). As a consequence, NRPS vanishes.

According to Eq.(27), the largest NRPS one can obtain from any waveguide composed of MO medium, Si and a low-refractive-index material should be roughly smaller than 2π1.55μm2×0.0052.2223.47722.22222rad/mm. It should be noted that all the calculated NRPSs fall into this range. In Fig. 2(b), |Δγ| <13.7rad/mm for the SiO2/MO/Air structure. These results agree well with many reasonable results reported in previous publications [3, 10, 11, 15].

5.2. Gap-assisted MO waveguides

In recent years, low-refractive-index (LRI) slot have attracted much attention because of its ability to strongly confine light within a small volume. Therefore, the light interaction with materials can be greatly enhanced [7]. A nano-gap structure was claimed to be useful for enhancing the NRPSs to greater than 30rad/mm (around 52rad/mm for εxz = 0.0086) in a 2-D Ce:YIG/Gap/Si/SiO2 waveguide. If this effect is efficient, the air gap can be readily replaced by another LRI material that can be controlled very accurately in realistic fabrication, e.g. SiO2 film, to enhance the MO effect. However, our results show that NO enhancement can be made by such a sided slot.

Figure 3(a) shows the analytical results for the Ce:YIG/Air/Si/SiO2 structure, which is a four-layered problem. Our method avoids the integration of a discontinuous function in perturbation theory and excludes uncertainty in 2-D cross-section simulation arising from integrals being taken at infinitesimal distances from lines of discontinuity [7, 22]. The thickness of the high-index layer (Si layer) and the gap width were scanned from 0.1μm to 1μm, and from 1nm to 100nm, respectively. We can see that the NRPS decreases monotonically with increasing gap width, without any finite gap width specially good for enhancement.

 

Fig. 3 (a) The dependence of NRPS in a Ce:YIG/Air/Si/SiO2 waveguide on the structural parameters and (b) the field distributions of Ex (in the lower half plane) and Hy (upper) with increasing air gap width and a fixed silicon thickness of 200nm.

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In order to resolve this problem, the mode profiles for different gap widths d are shown in Fig. 3(b). For step-like dielectric waveguides, the double integral in Eq. (32) is converted into a sum of single integrals along material boundaries xh by [22]

Δγh[εxz(xh,y)εxx2(xh,y)|Hy(xh,y)|2εxz(xh+,y)εxx2(xh+,y)|Hy(xh+,y)|2]dy,
where ”+” and ”−” stand for right and left limits, respectively. By increasing the gap width, the energy spreading into the MO layer decreases very rapidly and |Hy|2 at the Ce:YIG/Gap boundary decreases as well. The NRPS value is optimal at d = 0 corresponding to no LRI gap and decreases monotonically as d decreases. Hence, the high Ex in the LRI slot is just an illusion to enhance NRPS and the presence of the LRI gap does not result in NRPS enhancement.

5.3. Slotted waveguide with a compensation wall

It was also reported that the NRPS can be greatly enhanced by introducing a compensation wall into a slotted waveguide, where the MO material with opposite magnetized direction are sandwiched by high-index dielectric (e.g. silicon). By doing so, 20 times NRPSs enhancement increment and up to 100rad/mm were reported, which seriously exceeds our prediction for the upper limit. After revisiting this problem, we present totally different results.

Firstly, for comparison, the NRPSs in a 1-D Air/Si/MO(+)/MO(−)/Si/Air waveguide resembling the 2-D case is analytical calculated, where ”+” and ”−” denote that the magnetization direction is along +y and −y direction, respectively. As shown in Fig. 4, for x = 0, the NRPS is zero with no MO material. Increasing the MO layer thickness will increase the the NRPS to a maximum value of 6rad/mm. The NRPSs will then decrease and stablize at a value of 5rad/mm for increasing the MO layer thickness. This tendency can be easily explained by the separation of the two silicon layers. If they are far away from each other, the slot effect is missing and the waveguide system is actually two isolated asymmetric waveguides, as shown in Fig. 5(a). The dotted line in Fig. 4 is for the realistic 2-D silicon/MO(+)/MO(−)/silicon waveguide with air cladding and oxide substrate. The NRPS monotonically increases with the increasing slot width instead of any enhancement due to slot effect and finally tend towards the same limit of the 1-D case due to separation of the two silicon rectangular waveguides, as shown in Fig 5(b–c). This result is totally opposite to that in [8]. The reason why there is no enhancement is because of the cancelation of NRPS at the central MO(+)/MO(−) wall (x = 0) and the two sided MO/Si walls ( x=±d2) of the slot. Thus, it is impossible to obtain a 20 times enhancement by introducing a compensation wall. Next, we also examined the MO rib waveguide with a compensation wall, which was investigated in [5] and borrowed as a benchmark in [8]. The varying of NRPS are also shown in Fig. 4 and Fig. 5(d), which are in very good agreement with the results in [5]. The relatively small NRPSs should be attributed to the low index contrast of the calculated model, where the guiding material is a doped YIG with a refractive index of 2.2 and the substrate GGG has an index of 1.9. The mode area in Fig.5(d) is much larger than those in Fig.5(b–c). So, we do not think it is reasonable to compare a large-index-contrast YIG/Si slotted waveguide with a low-index-contrast YIG/GGG rib waveguide.

 

Fig. 4 The comparison of NRPSs. The soild line is for the dependence of analytical NRPSs to the MO layer width in a 1-D Air/180-nm Si/x-nm MO(+)/x-nm MO(−)/180-nm Si/Air waveguide,where the two silicon layer width is 180nm. The green line corresponds to the Fig. 4. in [8], where the waveguide height is 500nm and the width of the slotted MO layer is 2x nm. The red line is the same as Fig.5. in [5], where the rib height h is 200nm and the rib width is 2x nm.

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Fig. 5 The field distributions of the Ex and Hy mode components in different waveguide structures, including (a) 1-D Air/180-nm Si/MO(+)/MO(−)/180-nm Si/Air waveguides with slot widths of 30nm and 1200nm; 2-D Si/MO/Si slotted waveguides with x=15nm in (b) and x=600nm in (c); and (d) a 2-D Ce:YIG/GGG rib waveguide. The left and right columns in (b–d) are for the Ex and Hy components, respectively.

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5.4. NRPS in plasmonic-enhanced MO waveguides

As an important branch of nanophotonics, plasmonics have been developed rapidly for nearly one decade for their potentials in light routing and manipulation [25]. By breaking the diffraction limit to squeeze light into a subwavelength volume, weak physical effects in optical materials can be dramatically enhanced so that the light-matter interaction lengths or volumes can be greatly reduced. Not surprisingly, how the surface plasmon wave propagates in a plasmonic MO waveguide is an interesting topic. Researchers have investigated optical isolating action in surface plasmon polaritons [9], where large NRPSs can be obtained by eliminating the unwanted cancelation caused by waveguide symmetry. As an extension, here we study several types of mainstream 1-D plasmonic waveguides to discuss their nonreciprocal properties, including the cases of (1) Cu/Si/MO, (2) Cu/MO/Si, (3) Cu/Oxide/MO, (4) Cu/MO/Oxide, (5) Cu/MO(+)/MO(−)/Cu and (6–7) MO(+)/Cu/MO(−). The transverse electric field distributions of the modes of interest are all shown in Fig.6(c). For the MO(+)/Cu/MO(−) structure, there are two fundamental modes: (6) long-range plasmon polariton with higher γ and (7) short-range plasmon polariton with lower γ. For the reason of symmetry, a compensation wall is set for the case (6–7) to obtain the largest NRPS.

The CMOS-compatible metal for supporting plasmonic wave is assumed to be copper, whose material index is modeled according to experimental data [26]. In practice, an NRPS of π is usually necessary to be introduced into one Mach-Zehnder interferometric arm for complete isolation. The required physical length is defined as Lπ (for the case of push-pull scheme, Lπ2). Similar to [9], we also use 1-dB attenuation length L1dB of

20log10exp[2πνIm(γ)L1dB]=1
to define the length in which the propagation loss reaches 1dB, which is related to the well-defined propagation length Li after which the intensity decreases to 1e by Li=10log101e4.343L1dB. The calculated Lπ and L1dB for different waveguides are shown in Fig.6(a–b) by increasing the critical parameter of waveguide thickness. In general, more or less, there is finite accompanying non-reciprocal loss in plasmonic MO waveguides, but it is negligible compared with the reciprocal loss.

Let us read Fig. 6(a) first from the aspects of cutoff point, optimal thickness and limit. First, the cases (1–4) belong to asymmetric three-layer waveguides with metal cladding, so all of them have a cutoff thickness [27]. The cutoff thickness of the cases (2,3) are around 20nm due to quasi-slot effect of LRI guiding, whereas that of the cases (1,4) are greater than 100nm (not shown in figure). Moreover, because NRPS is seriously influenced by the extent of asymmetry, there are optimal thicknesses for NRPS in the cases (1,2). Second, when d = 0 for the cases (2,4), the MO layer vanishes; Lπ goes to infinity and nonreciprocal effect is missing. Conversely, when d = 0 for the cases (1,3) and d → ∞ for the cases (5–7), all the structures return to the simplest bilayer surface plasmon waveguide Cu/MO discussed in [9], where the transcendental problem of Eq. (17) reduces to

εzzεxxγ2εe+εxzεxxγ=εeεmγ2εm,
where εm is metal permittivity. Under the first-order approximation, f(γ) ≈ f(γspp) + f′(γspp)(γγspp) and γspp=εmεeεm+εe, its analytical solution can be expressed as
Δγεxzεeεm2εm2εe2γspp2εe.
At 1.55μm, εCu ≈ −68 + 10i (whereas εAg ≈ −87 + 8.7i); the Lπ is 618μm (705μm) and the L1dB is 2.15μm (4.07μm), corresponding to an insertion loss of 288dB (173dB). In theory the push-pull scheme can reduce the insertion loss by half. For the Cu/MO(+)/MO(−)/Cu structure, the NRPS is very stable when d > 20nm and rapidly increase when the MO layer thickness goes down. This is because the field intensity contrast at x = 0 and x=±d2 becomes more prominent and weakens the cancelation in the integral of Eq.(32). For the short-range surface plasmon polariton in the MO(+)/Cu/MO(−) waveguide, the NRPSs also rapidly increase with decreasing layer thickness due to the enhanced fields at x=d2. Third, more importantly, we need to compare the magnitudes of Lπ and L1dB, which is critical to the feasibility of combining plasmonics and MO medium. The data in Fig.6 (a) and (b) are plotted differently in magnitude by 3 order. As calculated for the bilayer Cu/MO case, an insertion loss at the scale of 100dB is almost meaningless for practical use in that even a dielectric waveguide can achieve the comparable Lπ without any metallic loss. Just considering enhancement, the case (7) may be appealing because an Lπ value of 100μm can be obtained when d ≈ 10nm, which is very challenging for dielectric MO waveguide. However, the loss problem is very serious due to the relatively fast decrease of L1dB. To conclude, the loss problem seriously constrains the use of plasmonics for enhancement of NRPS.

 

Fig. 6 The characteristic length for (a) π–NRPS and (b) 1-dB propagation loss and (c) the field distributions, of the dominant modes in several typical 1-D plasmonic waveguides. The field components are for Ex component that was normalized to their maximums. The modes are distinguished in color in all panels. The field distributions are calculated with a sandwiched layer thickness of 10nm.

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6. Conclusions

In conclusion, we analytically investigated the general properties of complex MO waveguides by establishing the transfer matrix of multilayered MO waveguide systems. The enhancement of NRPSs in asymmetric three-layered waveguides, nano-slot waveguides and plasmonic waveguides are revisited and comparisons between the schemes were made. These reported schemes are indeed useful for improving the MO effect, but from an optical analysis, it seems impossible to enhance the MO effect by an order of magnitude. How to bypass this criterion is left as an open issue.

Acknowledgments

The author H. F. Zhou would like to thank Prof. Kazuaki Sakoda of National Institute for Materials Science (NIMS) in Japan for his enlightening discussion. This study was supported by Singapore SERC/A*STAR Grant 092-154-0098.

References and links

1. Z. F. Yu and S. H. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon . 3, 91–94 (2009). [CrossRef]  

2. L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011). [CrossRef]   [PubMed]  

3. R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood Jr., and H. Dotsch, “Magneto-optical noreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. 29, 941–943 (2004). [CrossRef]   [PubMed]  

4. L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011). [CrossRef]  

5. N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. 35, 250–253 (1999). [CrossRef]  

6. A. F. Popkov, “Nonreciprocal TE-mode phase shift by domain walls in magnetooptic rib waveguides,” Appl. Phys. Lett. 72, 2508–2510 (1998). [CrossRef]  

7. R. Y. Chen, G. M. Jiang, Y. L. Hao, J. Y. Yang, M. H. Wang, and X. Q. Jiang, “Enhancement of nonreciprocal phase shift by using nanoscale air gap,” Opt. Lett. 35, 1335–1337 (2010). [CrossRef]   [PubMed]  

8. W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. 98, 171109 (2011). [CrossRef]  

9. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006). [CrossRef]  

10. Y. Shoji and T. Mizumoto, “Ultra-wideband design of waveguide magnetooptical isolator operating in 1.31μm and 1.55μm band,” Opt. Express 15, 639–645 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-639. [CrossRef]   [PubMed]  

11. Y. Shoji and T. Mizumoto, “Wideband design of nonreciprocal phase shift magneto-optical isolators using phase adjustment in Mach-Zehnder interferometer,” Appl. Opt. 45, 7144–7150 (2006). [CrossRef]   [PubMed]  

12. M. C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. Bowers, “Silicon ring resonators with bonded nonreciprocal magneto-optic garnets,” Opt. Express 19, 11740–11745 (2011), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-639. [CrossRef]   [PubMed]  

13. N. Kono, K. Kakihara, K. Saitoh, and M. Koshiba, “Nonreciprocal microresonators for the miniaturization of optical waveguide isolators,” Opt. Express 15, 7737–7751 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7737. [CrossRef]   [PubMed]  

14. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2004).

15. H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992). [CrossRef]  

16. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005). [CrossRef]  

17. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

18. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999). [CrossRef]  

19. Z. K. Wang, “An implementation of Kuhn’s rootfinding algorithm for polynomials and related discussion,” J. Numer. Methods Comput. Appl. 3, 175–181 (1981).

20. Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl. 2, 88–92 (1994).

21. S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. 45, 882–888 (1974). [CrossRef]  

22. H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005). [CrossRef]  

23. M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999). [CrossRef]  

24. M. J. Adams, An Introduction to Optical Waveguides (John Wiley and Sons, 1980).

25. P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. 6, 16–24 (2012). [CrossRef]  

26. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

27. I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express 18, 348–363 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-1-348 [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. Z. F. Yu and S. H. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon.  3, 91–94 (2009).
    [CrossRef]
  2. L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
    [CrossRef] [PubMed]
  3. R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood, and H. Dotsch, “Magneto-optical noreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. 29, 941–943 (2004).
    [CrossRef] [PubMed]
  4. L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011).
    [CrossRef]
  5. N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. 35, 250–253 (1999).
    [CrossRef]
  6. A. F. Popkov, “Nonreciprocal TE-mode phase shift by domain walls in magnetooptic rib waveguides,” Appl. Phys. Lett. 72, 2508–2510 (1998).
    [CrossRef]
  7. R. Y. Chen, G. M. Jiang, Y. L. Hao, J. Y. Yang, M. H. Wang, and X. Q. Jiang, “Enhancement of nonreciprocal phase shift by using nanoscale air gap,” Opt. Lett. 35, 1335–1337 (2010).
    [CrossRef] [PubMed]
  8. W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. 98, 171109 (2011).
    [CrossRef]
  9. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006).
    [CrossRef]
  10. Y. Shoji and T. Mizumoto, “Ultra-wideband design of waveguide magnetooptical isolator operating in 1.31μm and 1.55μm band,” Opt. Express 15, 639–645 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-639 .
    [CrossRef] [PubMed]
  11. Y. Shoji and T. Mizumoto, “Wideband design of nonreciprocal phase shift magneto-optical isolators using phase adjustment in Mach-Zehnder interferometer,” Appl. Opt. 45, 7144–7150 (2006).
    [CrossRef] [PubMed]
  12. M. C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. Bowers, “Silicon ring resonators with bonded nonreciprocal magneto-optic garnets,” Opt. Express 19, 11740–11745 (2011), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-639 .
    [CrossRef] [PubMed]
  13. N. Kono, K. Kakihara, K. Saitoh, and M. Koshiba, “Nonreciprocal microresonators for the miniaturization of optical waveguide isolators,” Opt. Express 15, 7737–7751 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7737 .
    [CrossRef] [PubMed]
  14. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2004).
  15. H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
    [CrossRef]
  16. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
    [CrossRef]
  17. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
  18. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
    [CrossRef]
  19. Z. K. Wang, “An implementation of Kuhn’s rootfinding algorithm for polynomials and related discussion,” J. Numer. Methods Comput. Appl. 3, 175–181 (1981).
  20. Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl. 2, 88–92 (1994).
  21. S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. 45, 882–888 (1974).
    [CrossRef]
  22. H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
    [CrossRef]
  23. M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
    [CrossRef]
  24. M. J. Adams, An Introduction to Optical Waveguides (John Wiley and Sons, 1980).
  25. P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. 6, 16–24 (2012).
    [CrossRef]
  26. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).
  27. I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express 18, 348–363 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-1-348
    [CrossRef]

2012 (1)

P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. 6, 16–24 (2012).
[CrossRef]

2011 (4)

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[CrossRef] [PubMed]

W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. 98, 171109 (2011).
[CrossRef]

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011).
[CrossRef]

M. C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. Bowers, “Silicon ring resonators with bonded nonreciprocal magneto-optic garnets,” Opt. Express 19, 11740–11745 (2011), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-639 .
[CrossRef] [PubMed]

2010 (1)

2009 (2)

2007 (2)

2006 (2)

2005 (2)

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[CrossRef]

H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
[CrossRef]

2004 (1)

1999 (3)

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
[CrossRef]

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. 35, 250–253 (1999).
[CrossRef]

1998 (1)

A. F. Popkov, “Nonreciprocal TE-mode phase shift by domain walls in magnetooptic rib waveguides,” Appl. Phys. Lett. 72, 2508–2510 (1998).
[CrossRef]

1994 (1)

Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl. 2, 88–92 (1994).

1992 (1)

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

1981 (1)

Z. K. Wang, “An implementation of Kuhn’s rootfinding algorithm for polynomials and related discussion,” J. Numer. Methods Comput. Appl. 3, 175–181 (1981).

1974 (1)

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. 45, 882–888 (1974).
[CrossRef]

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (John Wiley and Sons, 1980).

Anemogiannis, E.

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Avrutsky, I.

Ayache, M.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[CrossRef] [PubMed]

Bahlmann, N.

H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
[CrossRef]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
[CrossRef]

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. 35, 250–253 (1999).
[CrossRef]

Berini, P.

P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. 6, 16–24 (2012).
[CrossRef]

Bi, L.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011).
[CrossRef]

Bowers, J.

Buchwald, W.

Chen, R. Y.

Chen, Y. F.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[CrossRef] [PubMed]

Dionne, G. F.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011).
[CrossRef]

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Dotsch, H.

H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
[CrossRef]

R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood, and H. Dotsch, “Magneto-optical noreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. 29, 941–943 (2004).
[CrossRef] [PubMed]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
[CrossRef]

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. 35, 250–253 (1999).
[CrossRef]

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

Espinola, R. L.

Fainman, Y.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[CrossRef] [PubMed]

Fan, S. H.

Z. F. Yu and S. H. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon.  3, 91–94 (2009).
[CrossRef]

Fehndrich, M.

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
[CrossRef]

Feng, L.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[CrossRef] [PubMed]

Gaylord, T. K.

Gerhardt, R.

Glytsis, E. N.

Hammer, M.

Hao, Y. L.

Hertel, P.

H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
[CrossRef]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
[CrossRef]

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. 35, 250–253 (1999).
[CrossRef]

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

Hu, J.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011).
[CrossRef]

Huang, J. Q.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[CrossRef] [PubMed]

Huang, W. P.

W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. 98, 171109 (2011).
[CrossRef]

Izuhara, T.

Jiang, G. M.

Jiang, P.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011).
[CrossRef]

Jiang, X. Q.

Josef, A.

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
[CrossRef]

Kakihara, K.

Khurgin, J. B.

J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006).
[CrossRef]

Kim, D

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011).
[CrossRef]

Kimerling, L. C.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011).
[CrossRef]

Kleine-Borger, J.

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
[CrossRef]

Kono, N.

Koshiba, M.

Kromer, H.

Leon, I. D.

P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. 6, 16–24 (2012).
[CrossRef]

Lohmeyer, M.

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
[CrossRef]

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. 35, 250–253 (1999).
[CrossRef]

Long, Y. L.

Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl. 2, 88–92 (1994).

Lu, M. H.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[CrossRef] [PubMed]

Luhrmann, B.

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

Makimoto, T.

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. 45, 882–888 (1974).
[CrossRef]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Mizumoto, T.

Mu, J. W.

W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. 98, 171109 (2011).
[CrossRef]

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Osgood, R. M.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

Pintus, P.

Polman, A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Popkov, A. F.

Ross, C. A.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011).
[CrossRef]

Saitoh, K.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2004).

Scherer, A.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[CrossRef] [PubMed]

Shoji, Y.

Soref, R.

Sure, S.

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Tien, M. C.

Tsai, M. C.

Wang, M. H.

Wang, Z. K.

Z. K. Wang, “An implementation of Kuhn’s rootfinding algorithm for polynomials and related discussion,” J. Numer. Methods Comput. Appl. 3, 175–181 (1981).

Wen, X. L.

Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl. 2, 88–92 (1994).

Wilkens, L.

H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
[CrossRef]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
[CrossRef]

Winkler, H. P.

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

Xie, C. F.

Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl. 2, 88–92 (1994).

Xu, Y. L.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[CrossRef] [PubMed]

Yamamoto, S.

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. 45, 882–888 (1974).
[CrossRef]

Yang, J. Y.

Ye, M.

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

Yu, Z. F.

Z. F. Yu and S. H. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon.  3, 91–94 (2009).
[CrossRef]

Zhang, W. F.

W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. 98, 171109 (2011).
[CrossRef]

Zhao, W.

W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. 98, 171109 (2011).
[CrossRef]

Zhuromskyy, O.

Appl. Opt. (1)

Appl. Phys. Lett. (4)

A. F. Popkov, “Nonreciprocal TE-mode phase shift by domain walls in magnetooptic rib waveguides,” Appl. Phys. Lett. 72, 2508–2510 (1998).
[CrossRef]

W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. 98, 171109 (2011).
[CrossRef]

J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006).
[CrossRef]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. 74, 2918–2920 (1999).
[CrossRef]

Comput. J. (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

IEEE J. Quantum Electron. (1)

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. 35, 250–253 (1999).
[CrossRef]

IEEE Trans. Magn. (1)

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. 28, 2979–2984 (1992).
[CrossRef]

J. Appl. Phys. (1)

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. 45, 882–888 (1974).
[CrossRef]

J. Lightwave Technol. (1)

J. Numer. Methods Comput. Appl. (2)

Z. K. Wang, “An implementation of Kuhn’s rootfinding algorithm for polynomials and related discussion,” J. Numer. Methods Comput. Appl. 3, 175–181 (1981).

Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl. 2, 88–92 (1994).

J. Opt. Soc. Am. B (1)

Nature Photon (1)

Z. F. Yu and S. H. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon.  3, 91–94 (2009).
[CrossRef]

Nature Photon. (2)

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. 5, 758–762 (2011).
[CrossRef]

P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. 6, 16–24 (2012).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. B (1)

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Science (1)

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[CrossRef] [PubMed]

Other (3)

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2004).

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

M. J. Adams, An Introduction to Optical Waveguides (John Wiley and Sons, 1980).

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Figures (6)

Fig. 1
Fig. 1

Schematic of a multi-layered slab waveguide system composed of N + 1 isotropic layers, in which magnetooptical medium can be included. The propagation direction z points into the paper and the waveguide is infinite in y.

Fig. 2
Fig. 2

Nonreciprocal phase shifts in three-layered SOI-based waveguides, where the MO medium (Ce:YIG) is placed (a) at one side and (b) in the center, respectively. The calculated Δγ by Eq. (30) is also shown for symmetric MO(+)/Si/MO(−) waveguide in (a), where γ0 is numerically prepared by 1-D finite difference frequency domain (FDFD) method.

Fig. 3
Fig. 3

(a) The dependence of NRPS in a Ce:YIG/Air/Si/SiO2 waveguide on the structural parameters and (b) the field distributions of Ex (in the lower half plane) and Hy (upper) with increasing air gap width and a fixed silicon thickness of 200nm.

Fig. 4
Fig. 4

The comparison of NRPSs. The soild line is for the dependence of analytical NRPSs to the MO layer width in a 1-D Air/180-nm Si/x-nm MO(+)/x-nm MO(−)/180-nm Si/Air waveguide,where the two silicon layer width is 180nm. The green line corresponds to the Fig. 4. in [8], where the waveguide height is 500nm and the width of the slotted MO layer is 2x nm. The red line is the same as Fig.5. in [5], where the rib height h is 200nm and the rib width is 2x nm.

Fig. 5
Fig. 5

The field distributions of the Ex and Hy mode components in different waveguide structures, including (a) 1-D Air/180-nm Si/MO(+)/MO(−)/180-nm Si/Air waveguides with slot widths of 30nm and 1200nm; 2-D Si/MO/Si slotted waveguides with x=15nm in (b) and x=600nm in (c); and (d) a 2-D Ce:YIG/GGG rib waveguide. The left and right columns in (b–d) are for the Ex and Hy components, respectively.

Fig. 6
Fig. 6

The characteristic length for (a) π–NRPS and (b) 1-dB propagation loss and (c) the field distributions, of the dominant modes in several typical 1-D plasmonic waveguides. The field components are for Ex component that was normalized to their maximums. The modes are distinguished in color in all panels. The field distributions are calculated with a sandwiched layer thickness of 10nm.

Equations (35)

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× E = j 2 π ν H
× H = j 2 π ν ε r E ,
ε r = ( ε x x j ε x y j ε x z j ε x y ε y y j ε y z j ε x z j ε y z ε z z ) .
z E x x E z = j 2 π ν H y
( z H y x H y ) = j 2 π ν ( ε x x j ε x z j ε x z ε z z ) ( E x E z )
( E x E z ) = 1 j 2 π ν 1 ε x x ε z z ε x z 2 ( ε z z j ε x z j ε x z ε x x ) ( z H y x H y )
x 2 H y ( 2 π ν ) 2 [ ε z z ε x x γ 2 ( ε z z ε x z 2 ε x x ) ] H y = 0.
κ = + ε z z ε x x γ 2 ε e .
H y ( n ) = { A n exp [ 2 π κ n ( x x n ) ] + B n exp [ 2 π κ n ( x x n ) ] } e i 2 π ν γ z
E z = 1 j ε e ( 1 2 π ν x H y γ ε x z ε x x H y )
f ( n ) ( d n ) = f ( n + 1 ) ( 0 ) ,
f ( n ) ( x ) = ( 1 1 1 ε e ( κ ε x z γ ε x x ) 1 ε e ( κ ε x z γ ε x x ) ) ( A exp ( 2 π ν κ x ) B exp ( + 2 π ν κ x ) ) ( n ) ,
( A ( n + 1 ) B ( n + 1 ) ) = S n ( A ( n ) B ( n ) )
S n = 1 2 ( ( 1 + a b ) exp ( ϕ n ) ( 1 a b ) exp ( ϕ n ) ( 1 a + b ) exp ( ϕ n ) ( 1 + a + b ) exp ( ϕ n ) )
a n , n + 1 = ε e ( n + 1 ) κ n ε e ( n ) κ n + 1
b n , n + 1 = ( ε x z ( n + 1 ) ε x x ( n + 1 ) ε e ( n + 1 ) ε e ( n ) ε x z ( n ) ε x x ( n ) ) γ κ n + 1 ,
( A N 0 ) = S T ( 1 B 0 ) .
N = 1 2 π i C f ( z ) f ( z ) d z
n = 1 N z n k = 1 2 π i C z k f ( z ) f ( z ) d z ,
Δ γ = ν 2 E * Δ ε r E d x d y ,
S = 1 4 ( E × H * + E * × H ) z d x d y .
Δ γ 2 γ 2 Re ε x z E x * 1 ε e x ε e E x d x d y [ | E x 2 | 1 2 π ν γ E x x ( 1 ε e x ε e E x * ) ] d x d y
Δ γ 2 2 π ν ε x z ε x x 2 H y * x H y d x d y
| Δ γ | 1 π ν max | ε x z ε x x 2 | M O | H y x H y d x |
γ 2 = [ H y 2 ε ( x ) ( 1 2 π ν x H y ) 2 ] d x
H y x H y d x 2 π ν H y 2 ε ( x ) d x γ 2
| Δ γ | 2 max ( ε x z ε x x 2 ) ε m n 2 2
exp ( 4 π ν κ 1 d 1 ) = 1 ( a 01 + b 01 ) 1 + ( a 01 + b 01 ) 1 ( a 21 b 21 ) 1 + ( a 21 b 21 )
2 π ν i κ 1 d 1 = atan [ i ( a 01 + b 01 ) ] + atan [ i ( a 21 b 21 ) ] .
Δ γ 2 | c 01 b 01 + c 21 b 21 c 01 γ a 01 + c 21 γ a 21 + 2 π ν γ 0 d 1 κ 1 |
d 1 = 1 2 π ν ε 1 ε 2 atan ( ε 1 ε 2 ε 2 ε 0 ε 1 ε 2 ) .
Δ γ h [ ε x z ( x h , y ) ε x x 2 ( x h , y ) | H y ( x h , y ) | 2 ε x z ( x h + , y ) ε x x 2 ( x h + , y ) | H y ( x h + , y ) | 2 ] d y ,
20 log 10 exp [ 2 π ν Im ( γ ) L 1 d B ] = 1
ε z z ε x x γ 2 ε e + ε x z ε x x γ = ε e ε m γ 2 ε m ,
Δ γ ε x z ε e ε m 2 ε m 2 ε e 2 γ spp 2 ε e .

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