Nanophotonics - UNESCO

NanophotonicsAtilla Ozgur Cakmak, PhD

Unit 3Lecture 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 engineeringRemember 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?

Light source

2 beam emergingSuperprism

Added corrugations at the exit

All thanks to dispersion engineering as we are going to discuss

Dispersion engineeringRemember 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

W. T. Lu et al. Opt. Express vol. 15, pp. 9166-9175 (2007)

Negative refraction?

Dispersion engineeringThe 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.

2 beam emerging

Negative refraction?

kz

kx

ω1

ω2

http://upload.wikimedia.org/wikipedia/commons/7/72/DoubleCone.png

E or ω

Ewald circles => 2 2

x zk k kv

Dispersion engineering

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.

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 engineeringNow, 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, 0.5205and 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)


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.

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, 0.5205and 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



Steps:6) Put your Poynting vector (pink thick

arrows) from the isofrequencycontours. The exact direction will be determined by the group velocity of the wave (vg). The group velocity is defined as:

7) Then, obviously Ponyting vector will be normal to the isofrequencycontour. Its direction should be towards right.

k gv r



Steps:8) We should check the sign of9) The isofrequency contours are

expanding outwards (getting larger) as frequency is increased for the isofrequency contours centered around M. It is the opposite for the isofrequency contours centered around Gamma point. Hence, vg

will be tangential but also outwards for M-centered contours.

k gv r



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.



focusing

collimation

R. Zengerle et al. Opt. Express vol. 13, pp. 5719-5730 (2005)

W. Y. Liang et al. Opt. Express vol. 15, pp. 1234-1239 (2007)

All of these scenarios are possible with dispersion engineering.

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)


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)


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 diffr

0

action orders in the form of

2

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.

m xk k mL

Topological photonicsTopological 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.htmlhttp://www.vi-ti.de/vi-ti/EN/Research/01-Introduction/01-Introduction_node.html

Topological photonicsChern 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. One way of breaking T symmetry is working with gyromagnetic photonic crystals.

Topological photonicsMagneto optical photonic crystals can be formed by assembling yttrium iron garnet photonic crystals at microwave frequencies. Unfortunately, such magneto optical effects are very difficult to attain at optical frequencies. An exemplary square crystal is shown below.


We can modulate the permeability of these materials. The permeability should now be written in tensor format. Based on the applied external magnetic field’s direction and magnitude these µ and k parameters change their sign and value. Hence, ∆C is created between 2nd and 3rd band. Notice that it is directional! (does not have a negative slope twin for –k).

missing!

Topological photonicsIf we make use of this property and change the polarization of the magnetic field applied in portions of the photonic crystal as shown below, we will have unidirectional propagation. Notice that these are topologically protected states (within bandgap) and cannot penetrate into the photonic crystal either when excited with a point source.

+ B

- B

Topological photonicsWe can change the polarization and this time we will have the propagation in the other direction.

+ B

- B

Topological photonicsNow, let us insert an obstacle (like a perfect electric conductor) in the middle of the path. As you can see, the protected edge modes are not scattered and stably continuing their journey.

+ B

- B

Slow lightAnother important phenomena in photonic crystals is the appearance of slow light. A conventional waveguide mode’s dispersion diagram is shown below (b). Remember that k gv

r

Under these conditions a conventional waveguide can slow down the light but not significantly. Check the slopes of the dispersion curves and compare with the light line’s slope. As for Photonic crystal waveguides we can attain dispersion curves where gets really small while the dispersion curve gets very flat.

k gv r

T. Baba, Nature Photonics vol. 2, pp. 465-473 (2008)

Photonic Crystal SlabsLight coupling for a dielectric waveguide will be trivial as we discussed in the previous sections. Modes will reside under the air line. Let us investigate the waveguide as segments.

Slab modes

Air line

Even mode

Odd mode

Scatterer modes

Continous modesCoupled to air

Photonic Crystal SlabsThen, the new appearing modes will have a different dispersion band because of the non-existing guidance in the new scenario. However, these modes would be very radiative and coupled to air modes outside of the air line. We have to consider the out of plane propagation.

Even mode 1 Even mode 2 Odd mode

y

zThis odd mode will alter its phase while propagating to right from left. Did not show that here.

Photonic Crystal Slabs

L=300nm, w= 150nm, d2=200nm, d1=20nm (left), d1=100nm (middle), d1=200nm (right)If we have a perfectly periodic grating excited with a plane wave, then we would only couple thelight till 500THz. The density of states gets increased for d1=200nm. For d1=20nm, the modesresemble the waveguide modes. They just get folded back at the Brillion corner.

folding Gets separated

Starts to getseparated

nSInSi (200nm)

air

L

wd1


Now the light line itself should be represented for projected bands to 3D Photonic Crystal Slabs (Quasi 2D). The modes can have parity, i.e., symmetry with respect to the y axis (along the rods). The modes with odd parity and even parity are given above. The odd parity resembles the TE and even parity resembles the TM case for the 2D structures. There is a bandgap only for the odd parity. The electric field, Ey has odd parity along the y direction in that case. The results are claimed to be converging for the guided modes that are highly localized inside the slab.

y

z

z

h

Rods in air with a=1um, r=0.2um, h=2a=2um, εrods=12

Photonic Crystal SlabsEa case needs to be considered specially for computational reasons.

Drilled holes into a slab that is sandwiched in between two symmetric claddings. a=1um, rholes=0.45um, h=0.6a=0.6um, εslab=12, εholes=1 (air), εcladdings=2

Periodic rods (slab layer) with periodic claddings. a=1um, r=0.2um, T=2a=2um, D=10a=10um, εrods/slab=12, εbackground=1 (air), εcladdings=2

The second configuration that we have analyzed is consisted of symmetric claddings sandwiching a slab that has periodically etched holes in a hexagonal lattice. The slope of the light line is now changed, since we have a solid for the cladding. We now have a gap for the even parity modes just as in the case of the TM modes with Hy along the holes in the 2D case.

The third structure is composed of

periodic cladding layers at the top and

bottom layers. For such a case the main

difference is the calculation of the light

line. The light line turns out to be the

lowest band of the 2D periodic

structure composed of the upper

cladding rods and air.


The fourth structure is a symmetry breaking configuration along the y dimension. Up till now all the structures had symmetry and the modes could be calculated quickly. In this case the substrate is not sandwiching the slab that is composed of etched holes in a hexagonal lattice. This is a more realistic case, since we have the substrate only at the bottom. We have air cladding at the top. Furthermore, this case solves for the situation when the etching is not properly controlled. The drilled holes reach relatively long distances inside the substrate. This will be another parameter in the calculations. We have to determine our substrate layer as well as we have to check our control over the etching distances by consulting the fabrication people. Then we have two different claddings, thereby we have to calculate two different light lines. One of the light lines is for the air cladding (solid gray line) and the other one is for the 2D periodic structure of hexagonal holes drilled inside the substrate (dashed gray line). The latter one is the determining light line, since it filters out more in comparison to the air light line.

a=1um, rholes=0.45um, h=0.6a=0.6um, hcladding=100a=100um, εslab=12, εbackground=1 (air), εcladding/substrate=2


Finally we compare the band diagrams of the 2D and 3D structures for the rods and holes. The slab thickness is an important parameter as we learned. The 2D/3D rods have a periodicity of a=1um with a rod radius of r=0.2a=0.2 um. The 3D rods have a thickness of 2a=2um. Apparently, 2a is a good choice for the band gap formation in 3D rod structures. The cladding is air (up and bottom). The 2D and 3D rods have a dielectric permittivity of 12. We have a symmetry and therefore we have parity. Finally, a slab of dielectric permittivity 12 with hexagonal lattice in air (up and down) is examined. The lattice constant is a=1um, radius of the holes r=0.4a, the slab thickness is 0.5a.

rods holes

Photonic Crystal FibersIndex guiding with conventional fiber optic cables offer only a very small region to work with for guiding the fundamental mode and has confinement problems compared to the photonic crystal fibers, which has been one of the big applications of photonic crystals directly to the industry.

Ordinary fiber

P. Russell Science, vol. 299 pp. 358, (2003)

Two main categories will be discussed: a) Solid core (A)b) Hollow core (G)

Solid core

Hollow core

Photonic Crystal FibersA) Propagation diagram for a conventional single-mode fiber with a Ge-doped silica core and a pure silica cladding. Guided modes form at points like R, where light is free to travel in the core but unable to penetrate the cladding (because total internal reflection operates there). The narrow red strip is where the whole of optical telecommunications operates. (B) Propagation diagram for a triangular lattice of air channels in silica glass with 45% air-filling fraction. In region (1), light is free to propagate in every region of the fiber [air, photonic crystal (PC), and silica]. In region (2), propagation is turned off in the air, and, in (3), it is turned off in the air and the PC. In (4), light is evanescent in every region. The black fingers represent the regions where full two-dimensional photonic band gaps exist. Guided modes of a solid-core PCF form at points such as Q, where light is free to travel in the core but unable to penetrate the PC. At point P, light is free to propagate in air but blocked from penetrating the cladding by the PBG; these are the conditions required for a hollow-core mode.


Photonic Crystal FibersFor solid core photonic crystal fibers, think of it as a decreased cladding effective index. Remember the effective index method that we had carried out earlier for ordinary waveguides. In fiber optic cables, the refractive index difference is small. Consequently, they are weakly guided. Now, we are boosting the contrast and refractive index contrast by drilling holes.

http://projects-web.engr.colostate.edu/lpdl/research/fiber/fiber.html


In a solid-core PCF, the pattern of air holes acts like a modal sieve. In (a), the fundamental mode is unable to escape because it cannot fit in the gaps between the air holes—its effective wavelength in the transverse plane is too large. In (b) and (c), the higher order modes are able to leak away because their transverse effective wavelength is smaller. If the diameter of the air holes is increased, the gaps between them shrink and more and more higher order modes become trapped in the “sieve.”

Endless single mode photonic crystal fiber

Photonic Crystal Fibers

The effective index contrast decreases at shorter wavelengths. Most of the higher order modes will start to become very leaky and we end up with the fundamental mode. We just see one dielectric (the core) at very short wavelength and makes the light cone to have a refractive index of 𝜀 = 1.45

Fundamental mode

Higher orders

Photonic Crystal Fibers

Can we get rid of the material related losses and dispersive effects with hallow core photonic crystals?Unfortunately, yes and no at the same time. Photonic band gap guiding is essential. However, the surface states cause significant losses.

Photonic Crystal FibersBragg fibers resemble our 1-D photonic crystal analysis. Intuitively, one checks for omnidirectional reflection. Guiding intense CO2 laser light at very large wavelengths is made possible with Bragg fibers. They also rely on photonic bandgap guiding.

knotted

straight

B. Temelkuran et al. Nature vol. 420, pp. 650 (2002)

Magneto-optic PCs (problem)

1) Please watch the associated video which will guide you through the basics of frequency domain solutions in Comsol environment. You will be obtaining the results that were given in the lecture notes, ie. directional guiding of the beams. The designs Xiaofei Zang, Chun Jiang, JOSA B vol. 28, pp. 554-557 (2011) in were duplicated to achieve topologically protected states.

Photonic Crystal Slabs (problem)

2) Now, we will investigate photonic crystal slabs depicted in Phys. Rev. B vol. 65, 235112 (2002). A square lattice of holes drilled in a slab is given as shown below. We will examine the transmission through such structures with Stanford’s S4 (RCWA) free software. The dispersion relations for the photonic crystal slabs can be found with scanning the incident angle range.

Documents

Nanophotonics - UNESCO