In this work a filter design acting on both TE and **TM** polarisation components is presented. The TE light is filtered using a cascade of ring resonator devices. Due to birefringence of the TE and **TM** **modes**, the cascaded rings present **TM** mode passbands within the TE stopbands. To attenuate the **TM** **modes** a metal layer is fabricated over the ring. In section 2 the design, technology and the linear response the filter is presented. The e ff ects of polarisation mode scattering on high extinction filters is shown in section 3. In section 4 the performance of the fabricated filter is demonstrated. Section 5 presents application of these filters in non-linear Four Wave Mixing generation and filtering fully on-chip. Conclusions are given in section 6.

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from 0 to 60 and very small in the case of (c) for all values of κa. These results give the influence of the internal interaction forces of the electron gas on the dispersion relation of **TM**-**modes**. On comparing this calculation with TE-**modes**, it can be seen that the internal interaction forces play an important role on the dispersion relation of **TM**-**modes**. If these forces are not included, the frequency will decrease rapidly as the wave number increases 35 . The above three

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Ω ) is large in the case of (a) for all values of κ a starting from 0 to 60 and very small in the case of (c) for all values of κ a. These results give the influence of the internal interaction forces of the electron gas on the dispersion relation of **TM**-**modes**. On comparing this calculation with TE-**modes**, it can be seen that the internal interaction forces play an important role on the dispersion relation of **TM**-**modes**. If these forces are not included, the frequency will decrease rapidly as the wave number increases 25 . The above three calculations were

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At present time, theory of bulk cyclotron waves is developed sufficiently well [see e.g., 1, 2], which is represented in a wide utilization of bulk electron cyclotron waves in nuclear fusion investigations [3–5] for additional plasma heating and plasma diagnostics. These waves are also applied to the development of new high frequency and high power electronic devices [6, 7]. A utilization of restricted plasma volumes for different practical purposes makes it possible to excite both bulk and surface types of waves [8]. In our previous articles [9, 10], we have studied the cases of surface electron cyclotron waves with extraordinary and ordinary polarization, correspondingly, and also their propagation under conditions, when an external magnetic field was assumed to be oriented parallel to a plasma-dielectric interface. Unlike these cases, here we study the case of perpendicular orientation of an external steady magnetic field and do not restrict our consideration by the wavelength range located near a limit of long wavelengths compared with Larmor radius of electron, as it has been done in [9, 10]. To derive a set of equations describing parametric excitation of surface electron cyclotron **TM**-**modes** (SECTM- **modes**) the non-linear boundary condition, which determines discontinuity of tangential magnetic field of studied **modes**, is formulated as done in papers [9, 10]. This discontinuity is determined by surface electric current, which is induced by external alternating electric field on the plasma interface.

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Hertzian potentials have been used to solve different electromagnetic problems: in the study of the properties of aperture array systems [1], non-linear waveguides [2], Green’s functions for multilayered media [3] and electromagnetic wave interaction with nanodevices [4]. They have also been applied for the determination of TE and **TM** **modes** of circular cylindrical cavities using magnetic-type and electric-type Hertzian potentials res- pectively [5]. Figure 1 shows the case of a cylindrical dielectric resonator enclosed by a metal shield, where b is the radius of the outer cylinder, a is the inner resonator radius and d is the height (length of the structure). The configuration can be regarded as a cylindrical waveguide enclosing a central sample of radius a, and terminated by perfectly conducting planes. The general solution for the axial E field in **TM** **modes** was discussed in [6], proposing a general solution for the axial E field for **TM** **modes**.

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Most of the applications proposed use at least partially fiberised set-ups, where the polarisation state of the light is random or modified as it travels through the fibres. This makes it very difficult both to set the polarisation to a known state and to measure how it is modified. But it is well known that SOAs are polarisation sensitive, with different gain and refractive indices in the TE and **TM** **modes**. Although considerable efforts have been made to reduce this sensitivity, using strain and different waveguide structures, and gain anisotropy of less than 2dB has been achieved, this does not imply preservation of the state of polarisation of the input beam.

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mode and the dominant TE mode cannot be equal to each other only by the division. So we introduce a ridge in the center of the sector guide. The eﬀect is the same with the ridge installed in the rectangular waveguide as shown in Fig. 3. That is, the cutoﬀ frequency of the dominant TE mode can be lowered by the ridge eﬀect. Figs. 4(a), (b), and (c) show the dispersion characteristics of the dominant TE mode in the 4-divided sector waveguide for dependence of the ridge height h , ridge width w , and wall thickness d , respectively. We can see from this ﬁgure that the dispersion characteristic of the dominant TE mode strongly depends on the ridge height h and wall thickness d , whereas it does not depend on the ridge width w so much. The trend of the dispersion characteristics for parameters h and d can be explained by the additional shunt capacitance due to the ridge and variation of the waveguide space due to the wall thickness, respectively. So we select ridge height h as a parameter controlling the cutoﬀ frequency of the dominant TE mode. As a result, it is possible to construct the waveguide section with negative permittivity which expresses a shunt inductor in the same frequency range with the cutoﬀ **TM** 01 -mode

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This paper investigates the numerical modelling of VLF Trimpis\ produced by a D region inhomogeneity on the great circle path[ Two di}erent codes are used to model Trimpis on the path NWC!Dunedin[ The _rst is a 1D Finite Element Method Code "FEM#\ whose solutions are rigorous and valid in the strong scattering or non!Born limit[ The second code is a 2D model that invokes the Born approximation[ The predicted Trimpis from these codes compare very closely\ thus con_rming the validity of both models[ The modal scattering matrices for both codes are analysed in some detail and are found to have a comparable structure[ They indicate strong scattering between the dominant **TM** **modes**[ Analysis of the scattering matrix from the FEM code shows that departure from linear Born behaviour occurs when the inhomogeneity has a horizontal scale size of about 099 km and a maximum electron density enhancement at 64 km altitude of about 5 electrons[ Þ 0887 Elsevier Science Ltd[ All rights reserved[

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Abstract—Two dimensional metallic photonic band gap (PBG) structures, which have higher power handling capability, have been analyzed for their dispersion characteristics. The analysis has been performed using finite difference time domain (FDTD) method based on the regular orthogonal Yee’s cell. A simplified unit cell of triangular lattice PBG structure has been considered for the TE and **TM** **modes** of propagation. The EM field equations in the standard central-difference form have been taken in FDTD method. Bloch’s periodic boundary conditions have been used by translating the boundary conditions along the direction of periodicity. For the source excitation, a wideband Gaussian pulse has been used to excite the possible **modes** in the computational domain. Fourier transform of the probed temporal fields has been calculated which provides the frequency spectrum for a set of wave vectors. The determination of eigenfrequencies from the peaks location in the frequency spectrum has been described. This yields the dispersion diagram which describes the stop and pass bands characteristics. Effort has been made to describe the estimation of defect bands introduced in the PBG structures. Further, the present orthogonal FDTD results obtained have been compared with those obtained by a more involved non-orthogonal FDTD method. The universal global band gap diagrams for the considered metal PBG structure have been obtained by varying the ratio of rod radius to lattice constant for both polarizations and are found identical with those obtained by other reported methods. Convergence of the analysis has been studied to establish the reliability of the method. Usefulness of these plots in designing the devices using 2-D metal PBG structure has also been illustrated.

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corrugated resonators, some researches on the eigen-mode have been carried out [18]. But, researches on the eigen-mode of the coaxial outer corrugated resonator have seldom been found. The main reason is that it is difficult to derive the mode coupling coefficients because the structure of the coaxial outer corrugated resonator is more complicated. On the other hand, the calculation of eigen-mode becomes complex by second order transmission line equations with mode coupling coefficients. To overcome these difficulties, the paper uses surface impendent theory to get eigen-equations of TE and **TM** **modes** and applies transmission and coupling wave theory to obtain the first order transmission line equations with mode coupling coefficients. The paper is organized as follows: In Section 2, dispersion equations of TE and **TM** **modes** are derived from surface impedance theory. In Section 3, the first-order transmission line equation with mode coupling coefficients is established by the transmission line theory. In Section 4, mode coupling coefficients are derived by the coupling wave theory. In Section 5, the resonant frequency, quality factor and field profiles geometry of the eigen-mode of coaxial outer corrugated resonators are calculated. Section 6 is the summary. 2. DISPERSION EQUATION

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back. The superlattice becomes a perfect mirror for TE **modes**, or a filter for **TM** **modes**, in this frequency range. An interesting result of Figure 5(b) is the existence of an absolute band gap of **TM** polarization, while such an omnidirectional gap cannot exist in ususal RHM superlattices. In these previous examples of Figures 3 and 5, we have shown that an appropriate choice of the material parameters and the LHM/RHM superlattice can display an omnidirectional gap for either TE or **TM** polarization. In the following, we propose another kind of structure to create complete gap for both wave polarizations in the one dimensional photonic crystal by the association of two superlattices composed by LHM-RHM. For the sake of briefness in this paper, we choose vacuum as the RHM and the other parameters are taken to be:

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From ahead experimental result, we can get the conclusion. Polarization state rotation always occurs through SOA, rotation degree is different follow different input power and input polarization state. Two different input polariza- tion states lead to output power and gain obvious differ- ence of x and y axis. Their phase difference, polarization Azimuth and extinction ratio are difference obviously. Due to x and y axis do not coincide TE and **TM** axis, obvious cosine variation curve is observed at Figure 8(a). It implies input polarization state of Figure 6(a) arise phase difference of TE and **TM** **modes** bigger change, and it arise rotation bigger degree and extinction ratio

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Abstract—The recurrence dispersion equation of coupled single-mode waveguides is modiﬁed by eliminating redundant singularities from the dispersion function. A recurrence zero-bracketing (RZB) technique is proposed in which the zeros of the dispersion function at one recurrence step bracket those of the next recurrence step. Numerical examples verify the utility of the RZB technique in computing the roots of the dispersion equation of the TE and **TM** **modes** of both uniform and non-uniform arrays.

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Abstract—In this paper, a general multilayer circular cavity with N slabs is analyzed analytically, obtaining characteristic equations for TE and **TM** **modes** to compute the complex resonant frequency eﬃciently using an algorithm based on Chebyshev’s root ﬁnder. The accuracy of the solutions is compared with full-wave circuit method, and the computational speed to achieve the roots of the characteristic equations is also compared with Cauchy Integral Method, which is commonly used to obtain complex roots. Furthermore, the relationship between the amplitudes of the diﬀerent regions is obtained, whereby the whole structure can be analyzed as a single one from now on.

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In the previous section (Section 3.1), the vertical and horizontal slot guides can only guide the quasi-TE and **TM** **modes**, respectively i.e. they are highly polarization dependent. This difficulties could be resolved with a cross- slot waveguide. It is also useful for biochemical sensing which supports the much stronger field enhancement in the slot region for both quasi-TE and **TM** **modes**. The cross-slot design contains both vertical and horizontal slots simultaneously shown in Fig. 3(a). Figures 3(b) and (c) show the FV-FEM simulated E x field profiles of

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In this paper, an analytical model for SPPs **modes** supported by metal nanowire is studied. The dispersion relation for SPPs **modes** is obtained by solving the classical Helmholtz equations, and the eigen **modes** are classiﬁed by periodic azimuthal ﬁeld distribution. To conﬁrm the accuracy of our method, we calculate the eﬀective indexes of **TM** **modes** and HE **modes** at 633 nm using our proposed analytical model and COMSOL Multiphysics. COMSOL Multiphysics is a popular numerical simulation tool for analyzing waveguides and gives precise results upon extremely ﬁne mesh. For a speciﬁc waveguide structure, the fundamental mode changes over time according to a simple set of rules, and it is possible to anticipate future behavior of the ﬁeld distribution. These simpliﬁcations of complex ﬁeld distributions ease the signal processing requirements for the communication systems [19]. Therefore, waveguide structures for long-distance communication normally work on single-mode condition (all other **modes** are cutoﬀ). Here, we analyze the cutoﬀ radius a c along with the relative permittivity

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Abstract—Corrugated elliptic waveguide in actually extensive application is analyzed by using the mode matching method and Mathieu function. Considering space harmonics in the interior and higher order **modes** in the slot region of the corrugated elliptic waveguide, the dispersion equation of even **TM** **modes** is derived. The dispersion and attenuation characteristics as well as the inﬂuence of passband and stopband properties with the changes of structural parameter are investigated in detail. The calculated results in good agreement with ones in the relevant references are of very important values in theoretical studies and actual applications of corrugated elliptic waveguide for microwave engineering.

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Microwave Waveguides: Introduction, Rectangular waveguides : solution of wave equation, rectangular coordinates, TE modes in rectangular waveguides, TM modes in rectangular wav[r]

with Grounded Co-Planar Waveguide (GCPW) feeding has been presented in [3], which resonates at a frequency of 6.78 GHz. Here, a size reduction of 50% is achieved compared with full mode SIW. A spoon-shaped slot antenna [4] is designed for circular polarization using SIW and HMSIW in which two hybrid **modes** are utilized and modulated to get improvement in impedance bandwidth. A compact frequency reconﬁgurable HMSIW antenna [5] has been designed whose resonant frequency can be varied by a varactor-loaded inter-digital capacitor. It resonates from 2.99 to 3.59 GHz by changing the bias voltage from 0 to 30 V. Transverse slots have been placed on both top and bottom surfaces of an HMSIW cavity [6] in which the wave radiates along a permanent magnetic wall. A compact S-shaped slot has been placed on the top of the HMSIW cavity in [7] to radiate at Ku-band. Two HMSIW antennas have been proposed in [8] to operate in X-band and Ka-band. Here eight slots are placed in perpendicular to the magnetic wall line, and a simple line feeding is used for each antenna. Three antennas have been proposed in [9] using FMSIW circular cavity to operate at X-band in which grounded coplanar waveguide feeding is employed. A dual-band antenna [10] has been proposed to operate at X-band and Ku-band in which an L-shaped slot is placed on the top of the cavity. One of the two arms of an L-shaped slot radiates at X-band, and the other radiates at Ku-band.

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A beam propagation method (BPM), which must be fully vectorial [21], can be used to calculate the power conversion between the two polarization states. On the other hand, a junction analysis approach can also be used, as the proposed polarization rotator structure is composed of only two butt-coupled uniform waveguide sections with only two discrete interfaces between them. A powerful numerical approach, the Least Squares Boundary Residual (LSBR) method [22] can also be used, which rigorously satisfies the continuity of the tangential electric and magnetic fields at the junction interface in a least squares sense, and obtains the modal coefficients of the transmitted and reflected fully hybrid **modes** at the discontinuity interface. The LSBR method looks for a stationary solution to satisfy the continuity conditions by minimizing the error energy functional, J, as given by [22]

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