• Which Products are you interested in ?

    CST offers a wide range of EM simulation software to address design challenges across the electromagnetic spectrum, from static and low frequency to microwave and RF, for a range of applications, including EDA & electronics, EMC & EMI and charged particle dynamics.

  • Antenna Magus
  • IdEM
  • FEST3D
  • Optenni Lab
  • Looking for a Training, Workshop or eSeminar ?

    CST STUDIO SUITE® is being demonstrated at trade shows and workshops all over the world. Take a look at the list of conferences and exhibitions CST will be attending and get further information regarding CST workshops, eSeminars and training days.

  • TrainingsRegular training courses are held in CST's offices in Asia, Europe, and North America. Please check our trainings section for detail of trainings in all over the globe. Advance registration is normally required.

  • WorkshopsCST hosts workshops in multiple languages and in countries around the world. Workshops provide an opportunity to learn about specific applications and refresh your skills with experienced CST support staff. Make sure you visit our workshop section.

  • eSeminarsThroughout the year, CST simulation experts present eSeminars on the applications, features and usage of our software. You can also view past eSeminars by searching our archive and filtering for the markets or industries that interest you most.

  • Check our latest Events
  • Why create a MyCST Account ?

    A MyCST account may facilitate your access to many of the offerings on the CST website, for example the registration for eSeminars and the watching of eSeminars recordings, setting email preferences, and there is more functionality to come. It is required to participate in workshops and trainings.

  • Personal PreferencesAllows you to update your email preferences and areas of interest. It helps us to personalize your experience.

  • EventsSearch for events by location, industry and application. Once you are registered, you will be able to manage your registrations and check important details about your events. This section also provides you with a repository for Workshop & Training material.

  • LibraryYou can collect articles you find on the CST website to reference or read later by clicking on the “Add this article” button at the bottom of the article page.

  • Create Your Own Account
  • Need technical Support ?

    Customers can customize their accounts once they have completed the account creation process. This platform acts as vivid interface between CST and our customers.

    We therefore offer access to the latest Service Packs (including an automatic notification that a new Service Pack is available), a steadily growing database of Frequently Asked Questions (FAQs), Application Notes and Training Videos, as well as an individual FTP section for easy exchange of large files with our support team.

  • Do I need an Account?To access the Support Site a valid maintenance contract and a one-time registration is required.

    Please note that your Support login does not work for the MyCST account.

  • Get Support
  • How to request a Trial License ?

    Get your license in only three steps:

    1. Fill in the required fields in the contact form on the right and click "Send Us Your Request".

    2. Lean back and wait until one of our CST Experts contacts you.

    3. Enjoy a our trial license.

  • Student Edition

    Student Edition The CST STUDIO SUITE® Student Edition has been developed with the aim of introducing you to the world of electromagnetic simulation, making Maxwell’s equations easier to understand than ever. With this edition you have, bar some restrictions, access to our powerful visualization engine and some of the most advanced solvers of CST STUDIO SUITE.

    Student Edition

CST – Computer Simulation Technology

Investigation of Backward Wave Propagation on LHM Split Ring Resonators (SRRs)

Recent works [1,2,3] have predicted that a dense array of Split Ring Resonators (SRRs) can be considered as a gyrotropic bianisotropic media with magneto-electric coupling effects. This application article describes work carried out at the Laboratoire d'Electronique et Electromagnetisme (L2E), where simulations of such SRRs with CST MICROWAVE STUDIO® (CST MWS) have demonstrated the propagation of a backward wave component in a certain frequency range where the structure behaves like a medium with a negative refractive index.

The periodic structure under consideration is shown in Figure 1. A plane wave is introduced along the z-axis whereas the electric and magnetic fields have been polarized along the x- and y-axes respectively.

Figure 1: Dense array of copper SRRs backed by an epoxy substrate...

The frequency dependent effective refractive index of the equivalent metamaterial structure can be extracted by considering a single element rather than performing a time-consuming simulation of the entire array.

The relations for the refractive index n and wave impedance Z are given by the expressions in Figure 2, where S21 and S11 represent the transmission and reflection coefficients, and j and k are the complex number and the wavenumber respectively of the incident wave in free space.

Figure 2: Calculation of refractive index n and wave impedance Z

The S-parameters are calculated by simulating a single element of the array using the frequency domain solver in CST MWS. Unit cell boundary conditions with a full Floquet port mode implementation allow the consideration of the infinite SRR structure as shown in Figure 3. The resulting S-parameters are shown in Figure 4, and these then lead to the effective refractive index in Figure 5.

Figure 3: Single element of the periodic structure with the Unit Cell boundary conditions taking into account the EM-field periodicity given by the Floquet modes. The structure is described as an infinite periodic SRR array

Figure 4: S-parameters of the structure depicted in Figure 3. Red: S11; Green: S21

Figure 5: Effective refractive index versus frequency, as calculated from the S-parameters given in Figure 4

The effective permittivity and permeability are generally calculated from the relations εeff = n/Z and µeff = n.Z, but for bianisotropic media the extraction is more complicated since a magneto-electric parameter must be integrated [3].

From Figures 4 and 5 one can observe that the first and second resonances reveal all frequency bands where the refractive index is negative. Nevertheless, the backward wave propagation occurs only in the second resonance since the phase velocity behaves as a negative eigenmode [4]. Figure 6 confirms the evidence of this backward wave propagation: the phase of the right-to-left propagating wave appears to move from left to right within the structure.

Figure 6: Evidence of backward wave propagation inside the SRR array showing the negative refractive index of the equivalent medium

The backward wave propagation is due to the so-called Faraday Effect, since an induced quasistatic magnetic Hz field oscillates around a mean value of about 1.1 mA/m, as shown in Figure 7. This result has been extracted automatically from CST MWS simulation results using magnetic probes placed along the structure. This effect is reciprocal since the same Hz field is spontaneously induced along the z-axis.

Figure 7: Faraday effect induced along the structure

Looking carefully into the structure, one can see that the distribution of E-fields is the sum of two elliptically polarized electric fields (right and left handed) along the z-axis as shown in Figure 8. In fact each SRR cell behaves as a pseudo-chiral ferrite.

Figure 8: The linearly polarized E-field introduced along the x-axis leads to elliptically polarized electric fields inside the SRR structure indicating a gyrotropic effect

As an example, the lens of negative refractive index in Figure 9 is illuminated by a plane wave at 14.85 GHz and shows a focal point. A positive index would lead to the well known divergent lens.

Figure 9: Negative lens illuminated by a plane wave showing a focalization similar to a convergent lens

A gyrotropic bianisotropic medium has composed of a SRR array been proposed. An analysis using an extraction method calculating the phase velocity parameter has predicted the frequency range where the refractive index can be negative, thus leading to a backward wave propagation phenomenon. The results given by CST MWS have demonstrated the strong contribution of the Faraday Effect in the equivalent bianisotropic medium.


[1] R. Marques, F. Medina, R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials”, Phys. Rev. B 65 (2002) 144440-1-6

[2] C.W. Qiu, H.Y Yao, L.W Li; S. Zouhdi, T.S. Yeo, “Routes to left-handed materials by magnetoelectric couplings”, Phys. Review B75, 245214, 2007.

[3] S. A. Tretyakov, C. R. Simovski and M. Hudlicka, “Bianisotropic route to the realization and matching of backward-wave metamaterial slabs”, Phys. Review B75, 14504, 2007.

[4] H. Talleb, Z. E. Djeffal, D. Lautru, V. F. Hanna , “ Investigation of backward-wave propagation on LHM Split Ring Resonators”, Conference Meta’10, February 2010, Cairo, Egypt.

Rate this Article

0 of 5 Stars
5 Stars
4 Stars
3 Stars
2 Stars
1 Stars
contact support

Your session has expired. Redirecting you to the login page...

We use cookie to operate this website, improve its usability, personalize your experience, and track visits. By continuing to use this site, you are consenting to use of cookies. You have the possibility to manage the parameters and choose whether to accept certain cookies while on the site. For more information, please read our updated privacy policy

Cookie Management

When you browse our website, cookies are enabled by default and data may be read or stored locally on your device. You can set your preferences below:

Functional cookies

These cookies enable additional functionality like saving preferences, allowing social interactions and analyzing usage for site optimization.

Advertising cookies

These cookies enable us and third parties to serve ads that are relevant to your interests.