School information

A course of the European School on Metamaterials will be held right after the Conference (31 August - 1 September 2018).The lectures will be held in hall D (the same venue as the conference).

The school topic is:

Nonreciprocal and Time-Modulated Metamaterials and Metasurfaces

The course will be focused on emerging engineered materials with broken time-reversal symmetry. Strong current interest in this topic is motivated by new possibilities to engineer and optimize nonreciprocal response of materials and open up possibilities to create new types of nonreciprocal devices. We will discuss the use of nonlinear, active, moving, and time-modulated structures, extending the metamaterials paradigm into the four-dimensional space.

After the course, the participants will

• Understand the concepts of time-reversal symmetry and reciprocity, the consequences of time-reversal symmetry and the limitations imposed by time-reversal symmetry;
• Understand fundamental differences in response of reciprocal and nonreciprocal metamaterials and metasurfaces;
• Understand different mechanisms for breaking time-reversal symmetry and enabling nonreciprocal response: magnetic bias, nonlinearity, time modulation, mechanical movements,…
• Know examples of designs of new devices based on time-modulated and nonlinear metamaterials and metasurfaces;
• Get acquainted with the latest ideas and developments in the field of nonreciprocal photonics and acoustics.


Andrea Alù (University of Texas at Austin, USA)  

Nader Engheta (University of Pennsylvania, USA)

Romain Fleury (École polytechnique fédérale de LausanneSwitzerland)

Dimitrios Sounas (University of Texas at Austin and Wayne State University, USA)

Sergei Tretyakov (Aalto University, Finland)

Richard Ziolkowski (University of Arizona, USA)

Preliminary lecture plan:

Lecture 1. Sergei Tretyakov. General introduction and fundamentals. Time-reversal symmetry of Maxwell’s equations. Definition and origin of reciprocity. Lorentz reciprocity theorem for general linear (bianisotropic) media. Onsager-Casimir symmetry relations. Time- and space-reversal symmetries working together.

Lecture 2. Andrea Alù. Fundamentals of space-time modulation, nonlinearity, and topological phenomena for time-reversal symmetry breaking and nonreciprocity. Nonreciprocity through time-space modulation.

Lecture 3. Nader Engheta. Magnetized epsilon-near-zero (ENZ) material and its nonreciprocal response, including the topological one-way solution at the boundary of magnetized ENZ.  Nonreciprocity in the near field in magnetized plasmonic scenarios, that creates energy flow vortex in the near field.  Nonreciprocal nonlinear metasurfaces (with varactors and chiral elements).

Lecture 4. Richard Ziolkowski. Nonreciprocity via magnetic bias and active devices. Cross-flux theorem in electromagnetics. How magneto-optic materials break Rayleigh-Carson reciprocity. Reciprocal and nonreciprocal behavior associated time domain antennas. Antenna and array practical applications.

Lecture 5. Dimitrios Sounas. Nonreciprocity and topological insulators through nonlinearity. Nonreciprocal isolators, circulators and antennas through space-time modulation.

Lecture 6. Romain Fleury. Numerical methods for analyzing time-modulated systems. (i) Review of static nonreciprocal systems and examples in acoustics. (ii) Theory of time-modulated systems. Floquet-Bloch theorem and coupled mode theory. Time-Floquet lattices. Numerical methods, analytical models versus finite-element simulations. (iii) Application to current topics. Floquet topological insulators. Zero-index time-varying metamaterials. Nonreciprocal mirrors. Nonreciprocal parametric amplification. Link with parity-time symmetry.

There will be also time for discussions with the lecturers, school dinner, and a small test.

School time schedule




08:00 - 08:50



08:50 - 09:00



09:00 - 10:30

Lecture 1

Lecture 4

10:30 - 11:00

Coffee Break

11:00 - 12:30

Lecture 2

Lecture 5

12:30 - 14:00


14:00 - 15:30

Lecture 3

Lecture 6

15:30 - 17:00

Coffee and discussions with the lecturers

Coffee and exam

18:00 – 21:00

School dinner


Click here to download the flyer of the school.

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