Research

Photonic crystals in the terahertz spectral range

Photonic crystals are materials with a spatial periodicity of their dielectric function. For wavelengths in the range of the lattice constant, such structures show a special optical behavior. For example, photonic crystals may exhibit frequency or polarization selectivity or operate as waveguides.

one-, two-, and three-dimensional photonic crystal — <b>Fig. 1:</b> One-, two-, and three-dimensional photonic crystal

The optical properties of photonic crystals are determined by their photonic band structure, which, in analogy to the electronic bands in solids, is caused by the spatial periodicity of the dielectric function. The band structure defines the modes of photon propagation inside the material. If photonic band gaps exist, propagation in certain directions and frequency ranges is prohibited.

The following investigations have been conducted with two-dimensional photonic crystals consisting of steel cylinders with diameters between 1 mm and 2 mm. The average distance between the cylinders is less than 10 µm.

<b>Fig. 2:</b> Photonic crystal operating in the THz range, assembled from metallic cylinders

To investigate the optical properties of the structures, a THz time domain spectroscopy setup is used. Short, broadband THz pulses are generated by focusing ultrashort laser pulses from a Ti:sapphire laser on a photoconductive antenna. The progression of the pulses after transmission through a sample is measured using a second photoconductive antenna. The transmission spectrum is calculated from the time signal with a fast Fourier transform. Spectra in the frequency range between 0.1 THz and 1.6 THz (wavelengths between 0.2 mm and 3 mm) are obtained.

<b>Fig. 3:</b> Principle of time domain spectroscopy

The spectra show that transmission occurs only in certain narrow frequency ranges. The position and shape of these bands changes with the geometry of the samples. To understand this behavior, one has to examine the electromagnetic eigenmodes of the structures. These are determined by the equation

which is directly derived from Maxwell's equations. The following figure shows numerical calculations of the first transverse electric (TE) and transverse magnetic (TM) modes of a square lattice of cylinders. The fields are confined to the cavities between the cylinders.

<b>Fig. 4:</b> Eigenmodes of a photonic crystal

Transmission of electromagnetic radiation through a photonic crystal can only take place if an eigenmode with the frequency of the incident wave exists inside the structure. If the frequency lies in a band gap, the fields inside the crystal decay rapidly. To illustrate this behavior, the following figure shows numerical calculations of the fields inside the crystal for these two cases.

<b>Fig. 5:</b> Field pattern inside a photonic crystal with a plane wave incident from the left side. (<a class="download" href="t3://file?uid=5932">animation, 390 kB</a>)

Comparison of the transmission spectrum of a photonic crystal with its photonic band structure shows that the frequencies at which transmission occurs indeed coincide with the frequencies of the eigenmodes.

<b>Fig. 6:</b> Measured transmission spectrum (left) and calculated band structure (right) of a photonic crystal

Keywords:

terahertz (THz), photonic crystals, time domain spectroscopy (TDS), eigenmodes, transmission spectrum

references:

[1] "Band structure of terahertz metallic photonic crystals with high metal filling factor", B. Reinhard, G. Torosyan, R. Beigang, Applied Physics Letters 92, 201107 (2008)

[2] "Optical Properties of Photonic Crystals", K. Sakoda, Springer, (2001)

[3] "Photonic Crystals: Molding the Flow of Light", J. D. Joannopoulos, R. D. Meade, J. N. Winn, Princeton University Press, (1995)

[4] "Metallic photonic band-gap materials", M. M. Sigalas, C. T. Chan, K. M. Ho,and C. M. Soukoulis, Physical Review B 52, 11744?11751 (1995)