ELECE 4810 Metamaterials and Nanophotonics D Exercise 2

1. Propose and draw a structure or structures you would like to use to

Dispersion diagrams What is the dispersion equation? Ordinary waves They define the eigenwaves, i.

Isofrequency surface Poynting Vector is normal to the isofrequency surface Proper wavevector and Poynting

4. i. If the material has a non-linear response: Answer: This can be done

5. Look at the permittivity of seawater in the figure below and answer the

5. ii. Which model can explain the frequency dispersion in region 2? Region 1

5. iii. In regions 3 and 5, there are several resonances. Which models can

5. v. Look at region 1 again, does the dispersion satisfy the Kramers-Kronig relation

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ELEC-E 4810 Metamaterials and Nanophotonics D Exercise 2 - Solutions

1. Propose and draw a structure or structures you would like to use to realize three materials with isotropic, uniaxial, and biaxial properties, respectively. Isotropic Uniaxial Biaxial Sphere Disk Ellipsoid Answer: One solution is the use of dielectric inclusions, which shapes exploits symmetries. A sphere inclusion has radial symmetry and its useful for isotropic materials. A disk has radial symmetry on the plane normal to its axis, granting uniaxial properties. On the other hand, an ellipsoid has different axes lengths, producing different effective permittivies along each axis. This particular solution remains its material properties (isotropy, uniaxial, biaxial) as long as the periodicity is comparably smaller than the wavelength. 2

Dispersion diagrams What is the dispersion equation? Ordinary waves They define the eigenwaves, i. e. they give information how the waves propagate in the medium. Extraordinary waves Relation between the wavenumbers and the frequency 4

Dispersion diagrams 5

Isofrequency surface Poynting Vector is normal to the isofrequency surface Proper wavevector and Poynting vector are chosen accordantly based on energy propagation of normal component, and following the isofrequency gradient by comparing the isolines with the ones at a slightly higher 9 frequency.

4. i. If the material has a non-linear response: Answer: This can be done by adding an additional non-linear component, for example 12

5. Look at the permittivity of seawater in the figure below and answer the following questions: Region 1 Region 3 Region 2 R. 4 R. 5 14

Region 1 Region 2 15

5. ii. Which model can explain the frequency dispersion in region 2? Region 1 Region 2 Answer: In region 2, the electric conductivity decays and the effect of polar molecules becomes dominant. In that scenario, sea water can be characterized based on the Debye model. 16

5. iii. In regions 3 and 5, there are several resonances. Which models can be used to explain these resonances? Region 3 R. 4 R. 5 Answer: In regions 3 and 5, Lorentz model can be used to characterize sea water permittivity as there are resonant electron oscillations in the atoms. 17

Region 3 R. 4 R. 5 18

5. v. Look at region 1 again, does the dispersion satisfy the Kramers-Kronig relation and why? Region 1 Region 2 19