Bloch Wavenumber Research Articles

Tracking Rayleigh–Bloch waves swapping between Riemann sheets

Rayleigh–Bloch waves are modes localized to periodic arrays of scatterers with unbounded unit cells. Here, Rayleigh–Bloch waves are studied for line arrays of sound-hard circular scatterers embedded in a two-dimensional acoustic medium, for which it has recently been shown that Rayleigh–Bloch waves exist for higher frequencies than previously thought. Moreover, it was shown that Rayleigh–Bloch waves can cut off (disappear) and cut on (reappear), and additional Rayleigh–Bloch waves can cut on and interact with the existing ones. These complicated behaviours are reconsidered using a family of quasi-periodic Green’s functions that allow particular plane-wave components to become unbounded away from the array. The Green’s function formulation is combined with the block Sakurai–Sugiura method to trace the trajectories of the Rayleigh–Bloch wavenumbers as they swap between Riemann sheets that are categorized according to the unbounded plane wave(s). A detailed analysis is presented for three different scatterer radius values, and contrasting qualitative behaviours are identified. The findings are consistent with those published previously, extend to higher frequencies than allowed by the previous approach, and provide new understanding of Rayleigh–Bloch waves around the critical frequency intervals where they cut on/cut off/interact.

Interface state-based bound states in continuum and below-continuum-resonance modes with high-Q factors in the rotational periodic system

Abstract In thispaper, we have introduced a new approach to calculate the orbital angular momentum (OAM) of bound states in the continuum (BICs) and below-continuum-resonance (BCR) modes in the rotational periodic system nested inside and outside by transforming the Bloch wave number from the translational periodic system. We extensively classify and study these BICs and BCR modes, which exhibit high-quality (high-Q) factors, in different regions relative to the interface of system. These BICs and BCR modes with high-Q factor have been studied in detail based on the distinctive structural parameters and scattering theory. Outcomes of this research break the periodic limitation of interface state-based BICs, and realize more and higher symmetry interface state-based BICs and BCR modes. Moreover, we can control the region that light captured by adjusting the frequency, and show that Q factor of BICs is more closely related to the ordinal number of rings and rotational symmetry number of the system.

Isospectral open cavities and gratings

Open cavities are often an essential component in the design of ultra-thin subwavelength metasurfaces and a typical requirement is that cavities have precise, often low frequency, resonances while simultaneously being physically compact. To aid this design challenge, we develop a methodology to allow isospectral twinning of reference cavities with either smaller or larger ones, enforcing their spectra to coincide so that open resonators are identical in terms of their complex eigenfrequencies. For open systems, the spectrum is not purely discrete and real, and we pay special attention to the accurate twinning of leaky modes associated with complex-valued eigenfrequencies with an imaginary part orders of magnitude lower than the real part. We further consider twinning of two-dimensional gratings, and model these with Floquet–Bloch conditions along one direction and perfectly matched layers in the other one; complex eigenfrequencies of special interest are located in the vicinity of the positive real line and further depend upon the Bloch wavenumber. The isospectral behaviour is illustrated, and quantified, throughout by numerical simulation using finite-element analysis.

Towards the fast buckling load prediction of composite shells via Bloch wave based numerical vibration correlation technique

The buckling load revealing structural capacity bears significant value for the design of engineering composite shells, especially in the aerospace industry. Non-destructive methods are prevailing for critical buckling load prediction. However, the experimental costs and computational time for the analysis are unaffordable, especially for large-scale and complex shells. Despite the numerical vibration correlation technique (NVCT) and its modifications proposed for faster critical buckling load prediction, the efficiency requires further improvement. For composite shells with rotational periodicity, analyzing one substructure based on the Bloch wave method can be an alternate for the analysis of the entire shell. Since the NVCT comprises the linear buckling analysis and repeated frequency analyses, twofold acceleration can be performed. Thus, the Bloch wave based numerical vibration correlation technique (BW-NVCT) is proposed. Firstly, the formulations for the Bloch wave method and the NVCT are derived. As the curve of Bloch wave numbers and buckling loads is monotonous or principally comprise a single peak, the computations in the Bloch wave method to determine the minimum buckling load are further reduced. Then, the implementation of BW-NVCT is presented wherein the Bloch wave calculations are simplified compared with the Bloch wave buckling analysis method. Secondly, three types of composite shells including the Z33 composite cylindrical shell, the pressure vessel and the nozzle are analyzed by the proposed method. By applying Bloch boundary conditions to the substructure, the linear buckling loads, natural frequencies are in high accuracy with respect to results by finite element analyses directly. In addition, predicted critical buckling loads by the BW-NVCT are in excellent precision and most importantly, remarkable efficiency is achieved compared with both the NVCT and the dynamic explicit analysis. Results show that the proposed method can capture both global and local buckling modes, even when bend-twist coupling exists. It is concluded that the BW-NVCT is widely applicable for composite shells and is substantiated as an effective method for fast buckling load prediction.

Bifurcation of bound states in the continuum in periodic structures.

In lossless dielectric structures with a single periodic direction, a bound state in the continuum (BIC) is a special resonant mode with an infinite quality factor (Q factor). The Q factor of a resonant mode near a typical BIC satisfies Q∼1/(β-β ∗)2, where β and β ∗ are the Bloch wavenumbers of the resonant mode and the BIC, respectively. However, for some special BICs with β ∗=0 (referred to as super-BICs by some authors), the Q factor satisfies Q ∼ 1/β6. Although super-BICs are usually obtained by merging a few BICs through tuning a structural parameter, they can be precisely characterized by a mathematical condition. In this Letter, we consider arbitrary perturbations to structures supporting a super-BIC. The perturbation is given by δF(r), where δ is the amplitude and F(r) is the perturbation profile. We show that for a typical F(r), the BICs in the perturbed structure exhibit a pitchfork bifurcation around the super-BIC. The number of BICs changes from one to three as δ passes through zero. However, for some special profiles F(r), there is no bifurcation, i.e., there is only a single BIC for δ around zero. In that case, the super-BIC is not associated with a merging process for which δ is the parameter.

Zonostrophic instabilities in magnetohydrodynamic Kolmogorov flow

A classic stability problem relevant to many applications in geophysical and astrophysical fluid mechanics is that of Kolmogorov flow, a unidirectional purely sinusoidal velocity field written here as u = ( 0 , sin ⁡ x ) in the infinite ( x , y ) -plane. Near onset, instabilities take the form of large-scale transverse flows, in other words flows in the x-direction with a small wavenumber k in the y-direction. This is similar to the phenomenon known as zonostrophic instability, found in many examples of randomly forced fluid flows modelling geophysical and planetary systems. The present paper studies the effect of incorporating a magnetic field B 0 , in particular a y-directed “vertical” field or an x-directed “horizontal” field. The linear stability problem is truncated to determining the eigenvalues of finite matrices numerically, allowing exploration of the instability growth rate p as a function of the wavenumber k in the y-direction and a Bloch wavenumber ℓ in the x-direction, with − 1 / 2 < ℓ ≤ 1 / 2 . In parallel, asymptotic approximations are developed, valid in the limits k → 0 , ℓ → 0 , using matrix eigenvalue perturbation theory. Results are presented showing the robust suppression of the hydrodynamic Kolmogorov flow instability as the imposed magnetic field B 0 is increased from zero. However with increasing B 0 , further branches of instability become evident. For vertical field there is a strong-field branch of destabilised Alfvén waves present when the magnetic Prandtl number P m < 1 , as found recently by A.E. Fraser, I.G. Cresswell and P. Garaud (J. Fluid Mech. 949, A43, 2022), and a further branch for 1 $ ]]> P m > 1 in the presence of an additional imposed x-directed fluid flow U 0 . For horizontal magnetic field, a branch of field-driven, tearing mode instabilities emerges as B 0 increases. The above instabilities are present for Bloch wavenumber ℓ = 0 ; however allowing ℓ to be non-zero gives rise to a further branch of instabilities in the case of horizontal field. In some circumstances, even when the system is hydrodynamically stable arbitrarily weak magnetic fields can give growing modes, via the instability taking place on large scales in x and y. Detailed comparisons are given between theory for small k and ℓ, and numerical results.

Robust and non-robust bound states in the continuum in rotationally symmetric periodic waveguides.

A fiber grating and a one-dimensional (1D) periodic array of spheres are examples of rotationally symmetric periodic (RSP) waveguides. It is well known that bound states in the continuum (BICs) may exist in lossless dielectric RSP waveguides. Any guided mode in an RSP waveguide is characterized by an azimuthal index m, the frequency ω, and Bloch wavenumber β. A BIC is a guided mode, but for the same m, ω and β, cylindrical waves can propagate to or from infinity in the surrounding homogeneous medium. In this paper, we investigate the robustness of nondegenerate BICs in lossless dielectric RSP waveguides. The question is whether a BIC in an RSP waveguide with a reflection symmetry along its axis z, can continue its existence when the waveguide is perturbed by small but arbitrary structural perturbations that preserve the periodicity and the reflection symmetry in z. It is shown that for m = 0 and m ≠ 0, generic BICs with only a single propagating diffraction order are robust and non-robust, respectively, and a non-robust BIC with m ≠ 0 can continue to exist if the perturbation contains one tunable parameter. The theory is established by proving the existence of a BIC in the perturbed structure mathematically, where the perturbation is small but arbitrary, and contains an extra tunable parameter for the case of m ≠ 0. The theory is validated by numerical examples for propagating BICs with m ≠ 0 and β ≠ 0 in fiber gratings and 1D arrays of circular disks.

Negative refraction and exceptional point with Parity-Time symmetry in a piezoelectric mechanical metamaterial

In this work, negative refraction and exceptional point (EP) in piezoelectric mechanical metamaterials are considered, which corporates with the Parity–Time (PT) symmetry. The oblique incidence of anti–plane shear wave at the interface between a homogeneous medium and phononic crystal can excite multiple refraction modes. In periodic layers, propagation characteristics are illustrated by the dispersion relation and transfer matrix method. The normal mode decomposition is developed and time average Poynting vector is used to derive the refraction angle at the interface of piezoelectric metamaterials. The beam splitting is shown with different incident angles, as well as the pure negative refraction is achieved. Due to the mechanical and electric coupling, the complex conjugate of Bloch wave number appears in the dispersion relation and degenerates to the exceptional point. In addition, a defect layer is introduced to study the transmission coefficient of two phononic crystals with different arrangements. When the gain and loss are introduced into the metamaterials, the Parity–Time symmetry system is realized and a complete transmission with a unidirectional zero reflection at exceptional point can be achieved.

Frozen mode regime in an optical waveguide with a distributed Bragg reflector

We introduce a glide symmetric optical waveguide exhibiting a stationary inflection point (SIP) in the Bloch wavenumber dispersion relation. An SIP is a third-order exceptional point of degeneracy (EPD) where three Bloch eigenmodes coalesce to form a so-called frozen mode with vanishing group velocity and diverging amplitude. We show that the incorporation of chirped distributed Bragg reflectors and distributed coupling between waveguides in the periodic structure facilitates the SIP formation and greatly enhances the characteristics of the frozen mode regime. We confirm the existence of an SIP in two ways: by observing the flatness of the dispersion diagram and also by using a coalescence parameter describing the separation of the three eigenvectors collapsing on each other. We find that, in the absence of losses, both the quality factor and the group delay at the SIP grow with the cubic power of the cavity length. The frozen mode regime can be very attractive for light amplification and lasing in optical delay lines, sensors, and modulators.

Spectrum of exchange spin waves in a one-dimensional magnonic crystal with antiferromagnetic ordering

Purpose of the study is to show that the conditions for the propagation of exchanged spin waves (ESWs) in an asymmetric superlattice with antiferromagnetically ordered cells depend significantly on the chirality of the precession of the ESW magnetization (polarization, “magnon pseudospin”). Method. When constructing the EWS spectra, the Croning– Penny model (transfer-matrix method) and the Landau–Lifshitz equation are used to determine the nature of the waves in the cells. In the case of a uniaxial medium, there is only one type of ESW, therefore, when fields are joined at the boundary, the conservation of chirality is an essential factor due to which the ESW in one cell is always traveling, and in the other — evanescent. Thus, a superlattice for ESW is an effective periodic “potential” in which asymmetry can be realized either by applying an external field, or by a difference in the thickness and/or physical properties of the cell materials. Results. Based on the analysis of the spectrum, maps of the transmission zones for ESW of different chirality were constructed in three representations — “Bloch wave number – frequency”, “frequency – relative cell thickness”, as well as in the plane of cell wave numbers. It is shown that the presence of asymmetry leads to a difference in the width of the transmission zones for waves of different chirality. For a finite structure, the frequency dependences of the transmission and reflection coefficients of the ESW are plotted. An increase in the attenuation of the ESW near the boundaries of the transmission zones was also found. Conclusion. The results of the study can be used in the design of magnon valves and other devices based on ESW, in which their chirality can be controlled.

Frequency perturbation theory of bound states in the continuum in a periodic waveguide

In a lossless periodic structure, a bound state in the continuum (BIC) is characterized by a real frequency and a real Bloch wave vector for which there exist waves propagating to or from infinity in the surrounding media. For applications, it is important to analyze the high-$Q$ resonances that either exist naturally for wave vectors near that of the BIC or appear when the structure is perturbed. Existing theories provide quantitative results for the complex frequency (and the $Q$ factor) of resonant modes that appear or exist due to structural perturbations or wave vector variations. When a periodic structure is regarded as a periodic waveguide, eigenmodes are often analyzed for a given real frequency. In this paper, we consider periodic waveguides with a BIC and study the eigenmodes for a given real frequency near the frequency of the BIC. It turns out that such eigenmodes near the BIC always have a complex Bloch wave number, but they may or may not be leaky modes that radiate out power laterally to infinity. These eigenmodes can also be the so-called complex modes that decay exponentially in the lateral direction. Our study is relevant for applications of BICs in periodic optical waveguides, and it is also helpful for analyzing photonic devices operating near the frequency of a BIC.

A fast method to compute dispersion diagrams of three-dimensional photonic crystals with rectangular geometry

We propose a method and codes for fast computation of complex dispersion relations in three-dimensional photonic crystals (PCs) with rectangular geometry. The main idea of the method is to convert the eigenproblem to a nonlinear equation equivalent to the zero-determinant condition. This equation is then solved iteratively either by fixed-point iteration or by rational approximation method. Additional mathematical elements include fast-converging continued-fraction expansion to compute the interaction tensor (appearing in the above nonlinear equation) and efficient accounting for the rectangular geometry in matrix-vector multiplications, which are involved in computing the continued fraction coefficients. The method allows one to perform realistic three-dimensional computations on a typical laptop computer, including finding the Bloch wave vector in the band gaps and in evanescent mode bands. This paper is focused on the method and includes its detailed explanation and illustration with examples. The associated computational package contains a detailed user guide and a set of further demonstrations, which can be run with the help of provided scripts. Program summaryProgram title: RectDisp v.1-3CPC Library link to program files:https://doi.org/10.17632/k7nk86cfbs.1Code Ocean capsule:https://codeocean.com/capsule/7141738Licensing provisions: CC BY NC 3.0Programming language: Fortran 2003Nature of problem: The program computes the complex Bloch wave number q as a function of real frequency ω in photonic crystals (PCs) with rectangular geometry. The PC constituents are characterized by complex dispersive permittivities and are non-magnetic. Projection of q onto the XY-plane of the PC is fixed and can be specified by the user, and the projection onto the Z-axis is computed.Solution method: The method converts the eigenproblem of finding q to the nonlinear equation equivalent to the zero-determinant condition for, at most, a 3×3 matrix, and solves the latter by an iterative method (either fixed-point iteration or rational approximation or a combination of the two approaches). Additional features include continued-fraction expansion for computing the coefficients in the nonlinear equation and utilization of variable separability characteristic of the rectangular geometry for fast matrix-vector multiplication.Additional comments including restrictions and unusual features: The current version of the codes computes at most one value of q for each frequency. This excludes spurious modes arising due to the artificial band folding. Such modes are usually not coupled to incident radiation. However, some other modes can be missed. Under most circumstances, this can happen at higher frequencies, i.e., for ωh/c>π, where h is the PC period. Verification of mode coupling to external radiation and exhaustive search for all coupled modes are not currently implemented but we plan to add these features in future releases. If a solution is not found at some frequency, it can potentially be found by fine-tuning input parameters as described in the User Guide.

Bloch wavenumber identification of periodic structures using Prony’s method

Phononic crystals and metamaterials are spatially periodic waveguides that may exhibit wave attenuation bands and, therefore, can be applied as vibration attenuation devices. In this type of structure, the wave speed varies throughout the structural domain and, therefore, the local wavenumber also varies. Methods that allow the identification of representative wavenumbers are of key importance for the experimental characterization of these structures and their attenuation bands. In this context, Prony’s method can be a viable and efficient choice. In the present study, one- and two-dimensional Prony methods are, for the first time, addressed for estimating the Bloch wavenumbers of periodic structures. Bloch wavenumbers may be computed from a single unit cell by applying a periodic boundary condition. We show that the measurements used in Prony’s method must be taken at locations that are equally spaced by multiples of the spatial period of the structure. Otherwise, the spatially varying local wavenumber will cause the method to fail. The Total Least Squares Prony method is used to increase the number of observations per unit cell, as a strategy for noise reduction. This is a novel adaptation of Prony’s method for Bloch wavenumber estimation of periodic structures. A Bayesian approach is also proposed for highly noisy measurements. In the current study, we validate the numerical experiments using the theoretical wavenumber values of a periodic Euler–Bernoulli beam, a Levy-type plate periodic along one direction, and a thin square plate (Kirchhoff’s theory) with free boundary conditions periodic along two directions. The solutions for all proposed structures are obtained using the spectral element method and, for the square plate, with the finite element method. One experiment is performed for a periodic beam. Our simulation results show that the adapted Prony method, combined with the methodology to reduce the noise effect, performs very well for the considered phononic crystals.

Non-Hermitian waves in a continuous periodic model and application to photonic crystals

In some non-Hermitian systems, the eigenstates in the bulk are localized at the boundaries of the systems. This is called the non-Hermitian skin effect, and it has been studied mostly in discrete systems. In the present work, we study the non-Hermitian skin effect in a continuous periodic model. In a one-dimensional system, we show that the localization lengths are equal for all the eigenstates. Moreover, the localization length and the eigenspectra in a large system are independent of the types of open boundary conditions. These properties are also found in a non-Hermitian photonic crystal. Such remarkable behaviors in a continuous periodic model can be explained in terms of the non-Bloch band theory. By constructing the generalized Brillouin zone for a complex Bloch wave number, we derive the localization length and the eigenspectra under an open boundary condition. Furthermore, we show that the generalized Brillouin zone also has various physical properties, such as bulk-edge correspondence.

Rayleigh–Bloch waves above the cutoff

Extensions of Rayleigh–Bloch waves above the cutoff frequency are studied via the discrete spectrum of a transfer operator for a channel containing a single cylinder with quasi-periodic side-wall conditions. Above the cutoff, the Rayleigh–Bloch wavenumber becomes complex valued and an additional wavenumber appears. For small- to intermediate-radius values, the extended Rayleigh–Bloch waves are shown to connect the Neumann and Dirichlet trapped modes before embedding in the continuous spectrum. A homotopy method involving an artificial damping term is proposed to identify the discrete spectrum close to the embedding. Moreover, Rayleigh–Bloch waves vanish beyond some frequency but reappear at higher frequencies for small and large cylinders. The existence and properties of the Rayleigh–Bloch waves are connected with finite-array resonances.

A Non-Compact Effective Impedance Model for Can-to-Can Acoustic Communication: Analysis and Optimization of Damping Mechanisms

Abstract Can-annular combustors can feature azimuthal instabilities even if the acoustic coupling between the individual cans is weak. Recently, various studies have focused on modeling the acoustic communication between adjacent cans in can-annular systems. In this study, a coupling model is presented that, in contrast to previous models, includes the effect of density fluctuations, mean flow, and dissipative effects at the connection gaps. By assuming plane acoustic waves inside each can and exploiting the discrete rotational symmetry of the can-annular system, the acoustic can-to-can interaction can be represented by an effective Bloch-type impedance. A single can modeled with the effective impedance at the downstream end emulates the acoustic response of the entire can-annular arrangement. We then propose the idea of installing a liner just upstream of the first turbine stage to damp azimuthal instabilities. By using the proposed can-to-can coupling model, we discuss in detail the effect that the impedance of the liner has on the effective reflection coefficient for different Bloch wavenumbers. In the low-frequency limit, we derive an analytical condition for achieving maximum damping at a specific Bloch-number. We show that the damping of azimuthal modes depends on the porosity of the liner, mean flow parameters and the Bloch-structure of the mode. These results suggest the possibility of targeting the damping of modes of certain azimuthal order by geometric variations of the liner or of the connection gap. As an exemplary application of the theory, we setup a network model of a generic industrial 12-can combustor and investigate a cluster of acoustic and thermoacoustic eigenvalues for a varying liner porosity. The findings of this study provide a deeper understanding of the mechanisms that drive the can-to-can acoustic communication, and open the path for devising passive damping strategies aimed at stabilizing specific modes in can-annular combustors.

Complex modes in an open lossless periodic waveguide.

Guided modes of an open periodic waveguide, with a periodicity in the main propagation direction, are Bloch modes confined around the waveguide core with no radiation loss in the transverse directions. Some guided modes can have a complex propagation constant, i.e., a complex Bloch wavenumber, even when the periodic waveguide is lossless (no absorption loss). These so-called complex modes are physical solutions that can be excited by incident waves whenever the waveguide has discontinuities or defects. We show that the complex modes in an open dielectric periodic waveguide form bands, and the endpoints of the bands can be classified to a small number of cases, including extrema on dispersion curves of the regular guided modes, bound states in the continuum, degenerate complex modes, and special diffraction solutions with blazing properties. Our study provides an improved theoretical understanding of periodic waveguides and a useful guidance to their practical applications.

Perturbation theories for symmetry-protected bound states in the continuum on two-dimensional periodic structures

On dielectric periodic structures with a reflection symmetry in a periodic direction, there can be antisymmetric standing waves (ASWs) that are symmetry-protected bound states in the continuum (BICs). The BICs have found many applications, mainly because they give rise to resonant modes of extremely large quality-factors ($Q$-factors). The ASWs are robust to symmetric perturbations of the structure, but they become resonant modes if the perturbation is non-symmetric. The $Q$-factor of a resonant mode on a perturbed structure is typically $O(1/\delta^2)$ where $\delta$ is the amplitude of the perturbation, but special perturbations can produce resonant modes with larger $Q$-factors. For two-dimensional (2D) periodic structures with a 1D periodicity, we derive conditions on the perturbation profile such that the $Q$-factors are $O(1/\delta^4)$ or $O(1/\delta^6)$. For the unperturbed structure, an ASW is surrounded by resonant modes with a nonzero Bloch wave vector. For 2D periodic structures, the $Q$-factors of nearby resonant modes are typically $O(1/\beta^2)$, where $\beta$ is the Bloch wavenumber. We show that the $Q$-factors can be $O(1/\beta^6)$ if the ASW satisfies a simple condition.

Indirect link between resonant and guided modes on uniform and periodic slabs

A uniform or periodic dielectric slab can serve as an optical waveguide for which guided modes are important, and it can also be used as a diffraction structure for which resonant modes with complex frequencies are relevant. Guided modes are normally studied below the lightline where they exist continuously and emerge from points on the lightline, but isolated guided modes may exist above the lightline and they are the so-called bound states in the continuum. Resonant modes are usually studied above the lightline (defined using the real part of the complex frequency), but they are not connected to the guided modes on the lightline. In this work, through analytic and numerical calculations for uniform and periodic slabs, we establish an indirect link between the resonant and guided modes. It is shown that as the (Bloch) wavenumber is increased, a resonant mode continues its existence below the lightline, until it reaches its end at an Exceptional Point (EP) where a pair of improper modes emerge, and one branch of improper modes eventually approaches the lightline at the starting point of a guided mode. Leaky modes with a real frequency and a complex (Bloch) wavenumber (propagation constant) are also related to the improper modes. They emerge at EPs in eigenvalue formulations where the frequency is regarded as a parameter. Our study is based on a non-Hermitian eigenvalue formulation that includes resonant, improper and leaky modes, and provides a complete picture for different kinds of eigenmodes on uniform and periodic slabs.

Dispersion behavior of electromagnetic wave near the resonance in 1D magnetized ferrite photonic crystals

In the present article, we have analyzed the dispersion of electromagnetic wave in the one dimensional magnetized ferrite photonic crystals near the resonance in the permeability of the constituent materials for transverse magnetization in the transverse electric mode. The dispersion relation is obtained by transfer matrix method. It is observed that in the vicinity of resonant frequency, large numbers of oscillations occur in the normalized Bloch wave number. These oscillations in the Bloch wave number are strongly dependent on external magnetic fields, filling factor, and damping constant. The frequency regime of these oscillations is found to be shifted in higher frequency range with increase in the magnitude of the magnetic fields. With increase in the filling factor keeping length of periods fixed, number of oscillations is found to be increased. Near the resonance, effect of incident angle is negligible. It is demonstrated that these nearly equidistant oscillations occurring in the vicinity of resonance may be used for making filter in micro wave frequency range.