4.4 2D Photonic Crystals

• An ideal 2D PhC is periodic in two directions and homogeneous in the third.

• Example: square lattice of dielectric columns

• The crystal can haveband gaps in ��-direction

1

All pictures if not referenced

otherwise taken from J. D.

Joannopoulos et al., Photonic Crystals

– Molding the flow of light, Princeton

University Press (2008).

Taken from: A.R. Parker et al., Nature 409, 36 (2001)

20 cm

300 nm

The sea-mouse (a marine animal) has a spectacularly iridescent

spine, owing to a hexagonal PhC structure of voids in chitin

4.4.1 2D Photonic Crystals in nature

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• Let’s consider our example of the square lattice of dielectric rods

• As the system is homogeneous in z-direction we know that the modes must be oscillatory in that direction, with no restriction on the wave vector ��

• The system has discrete translational symmetry in ��.

• Indexing the modes of the crystal by ��, �‖ and the band

number �, they take the form of Bloch states:

�, �‖, � = �� �‖������ �, �‖, �where � is the projection of � on the �� plane and � �is a periodic function � � = � � + � for all (in-plan) lattice vectors �

• Modes that propagate parallel to the ��-plane must be invariant under reflections through the ��-plane

4.4.2 Two-dimensional Bloch states

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• Mirror symmetry allows to classify the modes by separating them into two distinct polarizations:

• Transverse electric (TE): � in the plane, H normal to the plane

• Transverse magnetic (TM): H in the plane, E normal to the plane

• The band structures for TE an TM modes can be completely different

• We restrict ourselves to in-plane (�� = 0) propagation and consider our square array of dielectric rods (� = 8.9)

in air with radius �

�= 0.2

• Band extrema and gaps almost always appear at high symmetry points of the Brillouin zone. Therefore it is usually enough to plot the band structure along the edges of the irreducible Brillouin zone.

• Example of a full band structure of a 2D photoic crystal for the two lowest bands and the corresponding is a projected band structure:

• All information on the band structure is represented by these projections when assuming monotonous dependence in between these borders = high symmetry of band structure. 5

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• Example: band structure of the square lattice of dielectric rods. The left inset shows the brillouin zone, the irreducible BZ shaded in light blue. The right inset shows a cross sectional view of the dielectric function.

No gap

Gap

(31%)

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• The origin of the band gap becomes obvious by observing the displacement field patterns

The second band has

more of its energy in the

air with a correspondingly

higher frequency

The first band has most of

its energy concentrated in

the dielectric regions

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• Concentration factor:

!. ". =# $%� �(�) �(�)(

)*+.,

# $%� �(�) �(�)(

• The dielectric-band TM mode has a concentration factore of 83%, the air-band TM mode of only 32%

• For the TE band we plot the magnetic field profiles and remember that the displacement field - is largest along nodal planes of

The field lines of - must be

continuous in the plane,

forcing them to penetrate

the air regions � both

dielectric and air-band

modes have significant (c.f.

= 23% and 9%, respectively)

amplitude in the air regions

Normalized quantities revisited

All parameters can be normalized to the lattice constant. Then the units of the physical quantities become:

• frequency " in units of [!// ]

• angular frequency 0 [2π!// ]

• wavevector � [2π// ]

• wavelength λ [/]

� A normalized result can be scaled to different particular problems

• in scaled unites: gap extends from frequency " = 0.2837 !//to 0.4183 !// with a mid-gap frequency of "7 = 0.3510 !//

• in real units: a crystal with a mid-gap wavelength λ7 of 1.55 µm has: " = !/(1.55 μ:) = 0.3510 !//� / = 0.3510 ∗ 1.55 μ: = 0.5440 μ:

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Scale invariance

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• A complementary 2D PhC structure to the square lattice of dielectric columns is a square lattice of dielectric veins (thickness 0.165/, � = 8.9), namely a connectedstructure.

• For this structure we find a 18.9% gap for the TE modes

4.4.3 A complete in-plane bandgap

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• We can again understand the occurrence of the gap by looking at the field profiles in the two lowest bands

• Displacement field of X-Point TM-modes. (c.f. 89% vs. 77%)

• Magnetic field of X-Point TE-modes. (c.f. 83% vs. 14%)

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• To design a PhC with a complete in-plane bandgap (for both polarizations) we use our understanding of what aspects of 2D PhCs lead to TE and TM bandgaps, respectively

• TM gaps are favoured in a lattice of high-� regions, TE gaps are favoured for a connected lattice

• Combine both features in a compromise:

• Photonic band structure of a triangular lattice of air columns in dielectric

The structure has photonic band gaps for both TE and TM

polarizations (18% for �

�= 0.48 and � = 13) 13

BZ of the

triangular

lattice

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• So far, we considered modes propagating in the plane of periodicity only (�� = 0)

• Band structure of the triangular lattice of air columns for �� > 0. This example illustrates many of the features of the out-of-plane band structure common to all 2D PhCs

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4.4.4 Defects in 2D PhCs• Within the band gap, no propagating modes can exist.

Thus, by perturbing the lattice, we can create localized modes that have frequencies in the gap

• We have many options to do that. For the example of dielectric columns in air we can e.g. remove or modify rods

• Mirror symmetry is still intact � TM and TE modes still decouple Cavity

surrounded by

reflecting walls.

Photonic

analogue to

impurities in

semiconductors

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• Example: TM-modes of a square lattice of alumina columns in air (= = 0.38/). Band structure:

• In analogy with atomic orbitals, bands can be characterized by the number of nodal planes in each column.

• More nodal planes � larger amplitude in low-� region� larger frequency

10.1% gap

"?-like”

"@-like”

• We now introduce a defect by decreasing the dielectric constant of a single column (∆� = �BCDEFGB − �IJKJLM)

• Defect states are pulled out of the @-band

• Similarly, when increasing the dielectric constant of a single column (∆� < 0), defect states are pulled out of the ?-band

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• Numerically calculated displacement fields of point defects:

Reduction of the dielectric constant of

a single rod (air defect). The mode

preserves the character of the @-band

where it originates from, apart from

the central column with no nodal plane

� monopole mode

Increase of the dielectric constant

of a single rod (dielectric defect)

The mode preserves the character

of the ?-band where it originates

from � quadrupole mode

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• The so far discussed defect modes are only weakly localized because of the rather small gap.

• To maximize localization one works with the largest possible gap. Example: the fundamental gap (31.4%) of the same structure but with = = 0.2/

• We vary the defect radius. Evolution of the localized modes:

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• Electric-field (O�) patterns of the defect states

• The modes are localized much more strongly than in the previous example

• Despite the breach of discrete translational symmetry, rotational symmetries can be preserved upon introduction of the defect.

• Modes with dipole symmetry are degenerate pairs

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• By modifying a sequence of unit cells we can create linear defects.

• While point defects can trap light, linear defects can guide light from one location to another � waveguide

• Light propagating in the waveguide with a frequency in the band gap is confined to the defect and can be directed along the defect

• A system with a linear defect still has one direction (here �) in which discrete translational symmetry is preserved.

• Thus, the corresponding wave vector is still a preserved quantity

• Maintaing also contiuous translational symmetry in z-direction, �� is conserved

• We restrict ourselves to in-plane propagation and consider only TM polarization

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• Band diagram of the 2D photonic crystal shown in the inset

• Introduction of the line defect results in a discrete guided band lying in the TM-bandgap

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• Electric field pattern associated with a linear defect formed by removing a column of rods from an otherwise perfect

square lattice of dielectric columns in air for �P = 0.3((Q

�)

• Key difference between point and line defects: For a point defect a mode can be localized when its frequency is inside the gap. A guided mode only needs to have a combination

�P , 0R that is disallowed in the crystal.

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• To test for the existence of a guided mode we pick a specific

�P, 0R and ask if there is any �S that will put that mode on a

band.

• If we can, there is at least one extended state in the crystal

with that combination �P, 0R . If we tried to establish a

guided mode with these parameters, it would leak into the crystal.

• The process of selecting a value of �P (which is conserved) and

examining all possible �S (which is no longer conserved) is called projecting the band structure (blue regions)

• Modes propagate with group velocity $0/$�P

• Remarkable property: PhC waveguides can guide light primarily in air! Not possible for index guiding.

• Our choice of removing a single line of columns results in a single-mode waveguide. Removing multiple modes results in a multi-mode waveguide.

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4.4.5 Photonic Crystal Slabs

• Structures with 2D periodicity and a finite thickness

• The finite thichness introduces qualitatively new behaviour as compared to ideal 2D PhC

• PhC slabs can confine light by a combination of index guiding and band gaps.

• Examples: A rod slab (= = 0.2/, $ = 2/) and a hole slab (= =0.3/, $ = 0.6/)

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• The in-plane wave vector �‖ = (�S, �P) is conserved due to

periodicity, but the vertical wave vector �� is not conserved.

• The system is invariant under reflections through the U = 0plane � Modes will be either TE-like or TM-like

• In the projected band structure, the extended modes form a

light cone for 0 > ! �‖ . Below the light cone appear discrete

guided bands. The respective and gaps are incomplete.

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• The size of the gap depends on the slab thickness

• It is ideal, when the fundamental index-guided mode is well confined, but no higher order modes are yet supported � the

ideal thickness is approximately V

( WXX

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• We can form a waveguide by reducing the radius of all rods in a particular row.

• Fabrication example (typically EBL with proximity correction):

• Removing a row of rods completely, as we did for 2D PhCs, the mode would not be guided, as it would not be vertically confined by index guiding

• Generally, PhC slabs cannot confine light to the low-index medium

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Band diagram of the wave-guide in the rod slab for a variation of the decreased radius

O�-field cross sections of the reduced-radius line defect for = = 0.14/ and Y�

(Q= 0.42

� = 0U = 0

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• The rod slab always requires a substrate.

• The influence of the substrate:It breaks the U = 0 reflection symmetry � TE and TM modes can couple and destroy any TM/TE-only bandgap.

• Modes that were confined in such a gap become leaky and radiate within the plane of periodicity.

• The polarization mixing can be reduced by etching the substrate together with the rods.

• Alternatively, the symmetry can be restored by adding a superstrate on top of the slab with similar optical properties as the substrate.

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• In the hole slab, on the other hand, we can form a waveguide by removing a row of holes.

• This waveguide has a series of guided modes

• Symmetry in the � = 0 plane means we can classify these modes as even or odd under reflection in this plane

Index-guided vertically, laterally confined by index-guiding

confined by index-guiding only

• Fabrication example:

• A free-standing membrane is achieved by underetching

32Optics Express Vol. 21, 7258-7275 (2013)

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• Z� field cross sections for the missing-hole waveguide at

�S =Q

�, only (c) is shown at �S = 0.32

Q

�

• If two defect modes couple to each other at an intersection point, this leads to an anti-crossing with drastic changes in the field pattern with �S