Nanophotonics Atilla Ozgur Cakmak, PhD Unit 3 Lecture 33: Photonic Crystals-Part4 Outline • Dispersion Engineering • Topological Photonics • Slow Light • Photonic Crystal Slabs • Photonic Crystal Fibers A couple of words… We continue and complete the discussion of Photonic Crystals by studying different research fields/communities that are born as a result of rich physics in Photonic Crystals. Suggested readings/skimming: “Photonic Crystals: Molding the Flow of Light” by John D. Joannopoulos et al. Chs. 7-9 2nd edition (this one is not a reading recommendation but more of a skimming and scanning suggestion) Dispersion engineering Remember the complicated optical networks that are highly employed in current technology. Can we intelligently guide the light with nanophotonic means, which were not available in conventional technology? Superprism 2 beam emerging Added corrugations at the exit Light source All thanks to dispersion engineering as we are going to discuss Dispersion engineering Remember the complicated optical networks that are highly employed in current technology. Can we intelligently guide the light with nanophotonic means, which were not available in conventional technology? 2 beam emerging Negative refraction? W. T. Lu et al. Opt. Express vol. 15, pp. 9166-9175 (2007) Dispersion engineering The bands that we have been examining have actually a 3-D shape. The dispersion curve for air line (even) has this property. Let us take two slices at two different frequencies, we will end up with Ewald circles ,again. E or ω 22 Ewald circles => k kxz k 2 beam emergingv ω2 kx ω1 Negative refraction? http://upload.wikimedia.org/wikipedia/commons/7/72/DoubleCone.png kz Dispersion engineering Normalized Frequency As for this case, for the triangular lattice we will have a more complicated band surfaces (right). Instead of scanning only the irreducible Brillouin zone, let us say we are scanning the complete Brillouin zone of k vectors. The first 4 surface bands are shown for the triangular lattice. Now, like the Ewald circles, take slices from different frequencies. Dispersion engineering Normalized Frequency For every different slice, we will end up with a different shape of isofrequency contours. The first 3 bands have been sliced. Try to imagine these slices from the 3D shape. Ignore X-Y it is on kx-kz plane. Dispersion engineering Normalized Frequency For every different slice, we will end up with a different shape of isofrequency contours. The first 3 bands have been sliced. Try to imagine these slices from the 3D shape. Ignore X-Y it is on kx-kz plane. Dispersion engineering Normalized Frequency For every different slice, we will end up with a different shape of isofrequency contours. The first 3 bands have been sliced. Try to imagine these slices from the 3D shape. Ignore X-Y it is on kx-kz plane. Dispersion engineering Now, we are approaching to the surface with an angle, the photonic crystal is made up of square lattice of rods. At a certain frequency, the isofrequency contours look like the following for three normalized frequencies (0.5078, 0.5205 and 0.5321): Steps: 1) Put the ewald circle of the medium that you are approaching from. In this case it is air. Gamma point will be at the center. This is represented with green circle. 2) Draw your incident wavevector k0 based on your incident angle (ψ). This is the wave coming from the transmitter antenna. (The interface is along Gamma-X) Dispersion engineering Now, we are approaching to the surface with an angle, the photonic crystal is made up of square lattice of rods. At a certain frequency, the isofrequency contours look like the following for three normalized frequencies (0.5078, 0.5205 and 0.5321): Steps: 3) Now, estimate the tangential wavevector based on k0sinψ. This tangential component has to be reserved at the surface (phase matching). Put a construction line that will cross the ewald circle together with photonic isofrequency contours. Dispersion engineering Now, we are approaching to the surface with an angle, the photonic crystal is made up of square lattice of rods. At a certain frequency, the isofrequency contours look like the following for three normalized frequencies (0.5078, 0.5205 and 0.5321): Steps: 4) The point(s) where the isofrequency contours are crossed are the coupling points. The incident light will couple to the photonic crystal modes at this frequency from this particle k- point. 5) Poynting vector dictates that energy flow direction should not change. The flow: left->right Dispersion engineering Now, we are approaching to the surface with an angle, the photonic crystal is made up of square lattice of rods. At a certain frequency, the isofrequency contours look like the following for three normalized frequencies (0.5078, 0.5205 and 0.5321): Steps: 6) Put your Poynting vector (pink thick arrows) from the isofrequency contours. The exact direction will be determined by the group velocity of the wave (vgr). The group velocity is defined as: kg v 7) Then, obviously Ponyting vector will be normal to the isofrequency contour. Its direction should be towards right. Dispersion engineering Now, we are approaching to the surface with an angle, the photonic crystal is made up of square lattice of rods. At a certain frequency, the isofrequency contours look like the following for three normalized frequencies (0.5078, 0.5205 and 0.5321): Steps: 8) We should check the sign of 9) The isofrequency contours are expanding outwards (getting larger) as frequency is increased for the isofrequency contoursr centered around M. It is the oppositekg vfor the isofrequency contours centered around Gamma point. Hence, vg will be tangential but also outwards for M-centered contours. Dispersion engineering Now, we are approaching to the surface with an angle, the photonic crystal is made up of square lattice of rods. At a certain frequency, the isofrequency contours look like the following for three normalized frequencies (0.5078, 0.5205 and 0.5321): Steps: 10) It is the exact opposite for Gamma centered isofrequency contours. vg will be directed towards the center of the circles (inwards). 11) Once you connect the coupling point with the Gamma point with another arrow (purple), it gives you your phase velocity vp. People can imitate negative refraction, get spatial coherence, guide the light. Dispersion engineering collimation W. Y. Liang et al. Opt. Express vol. 15, pp. 1234-1239 (2007) focusing All of these scenarios are possible with dispersion engineering. R. Zengerle et al. Opt. Express vol. 13, pp. 5719-5730 (2005) Dispersion engineering (problem) 1) Based on what we have learned about dispersion engineering and also what you already know about surface gratings, comment on what is happening in the schematic shown below. 0 means the 0th order incoming light and +-1 are the diffraction orders. The light is incident on both directions (from top to down and from bottom to top). Dispersion engineering (solution) 1) Based on what we have learned about dispersion engineering and also what you already know about surface gratings, comment on what is happening in the schematic shown below. 0 means the 0th order incoming light and +-1 are the diffraction orders. The light is incident on both directions (from top to down and from bottom to top). This photonic crystal is an electromagnetic diode, it only conducts the light coming from one direction (top to down for the dispersion figure). Dispersion engineering (solution) 1) Based on what we have learned about dispersion engineering and also what you already know about surface gratings, comment on what is happening in the schematic shown below. 0 means the 0th order incoming light and +-1 are the diffraction orders. The light is incident on both directions (from top to down and from bottom to top). Approach from top to down, the Ewald circle is crossed by 0th order incoming light. However, it cannot couple to any Photonic crystal isofrequency bands. Remember that this incoming light will have diffraction orders in the form of 2 k k m mx0 L For a surface grating of period L We have coupling for -1st order, so a photonic crystal with gratings on one single end would act as a diode. Topological photonics Topological insulators are a big subject in solid state (below). There is a surface state between the valance and conduction bands. Thanks to this cone shaped dispersion for the electrons, the electrons cannot penetrate into the bulk material. In other words, the material at hand losses one dimension. A 3D bulk material will have only a 2D metallic (conducting) surface. Can we create a similar picture for nanophotonics for light? http://web.stanford.edu/group/fisher/research/TI.html http://www.vi-ti.de/vi-ti/EN/Research/01-Introduction/01-Introduction_node.html Topological photonics Chern number (C) is a very important number determining the topology. Spoon, cup and tea pot have different topologies. Their equivalent topology is represented with a sphere, toroid and connected toroids. For an ordinary waveguide, we have C=0, same topology in the configuration. This also includes the photonic crystal based waveguiding we have discussed so far. We end up seeing bandgaps and a waveguide dispersion (as shown in d left picture). If we can change the topology still within the same structure, there will be a change in C number (∆C). This change will enable topologically protected states within the bandgap (as shown in d right picture). Creating such topologically different systems will involve breaking P(airty) or T(ime) symmetry. L. Lu et al. Nature Photonics vol. 8, pp. 821-829 (2014) Topological photonics We are going to introduce the case of breaking T symmetry. Due to reciprocity, many of the electromagnetic systems should be T symmetric.