Perfect square lattice with a channel every three rows
Perturbed lattice with channels
Intensity of the scattered field at the frequency of the peak transmission above
Moving a transmission gap
If we reverse the roles of the periodic structure and the air, then we have a planar waveguide in a photonic (or acoustic) crystal. Guided modes may exist in the air at frequencies that are in a spectral gap for the perfect periodic structure. See Refs. 2 (Ammari/Santosa) and 10 (Kuchment/Ong) in [RaSh08] in my list of articles. These modes are understood to cause resonant scattering behavior when a crystal with a planar waveguide, truncated parallel to the guide, is illuminated by a plane wave. Some simulations of this resonant scattering behavior are shown in the part of Section 5 of this page called "Fabry-Perot mirror". Detailed numerical calculations of resonant transmission spikes are shown in Ref. 15 of [RaSh08].
In [RaSh08], we prove that a spectral gap emerges as an
interval of
zero transmission of energy through a thick slab of a periodic material as
the thickness tends to infinity. We also prove that transmission
anomalies occur at guided mode frequencies for one-dimensional
structures. We employ the Calderón boundary-integral projectors
based on the radiating Green function for the Helmholtz equation
in free space and the decaying Green function for the Helmholtz
equation in a periodic structure.
The Calderón boundary-integral projectors for elliptic differential operators are analogous to the Cauchy boundary-integral projections in the theory of complex variables. There, an arbitrary function defined on a curve that divides the complex plane into two domains is decomposed uniquely into the boundary value of a holomorphic function on one side of the curve plus the boundary value of a holomorphic function on the other side, with both functions decaying like 1/z as z→∞ (unless one of the domains is bounded). These projections are (I ± H), where H is the Hilbert transform on the boundary, and the identity I arises from the Sokhotskyi-Plemelj formula.
There is a similar decomposition involving, say, Helmholtz or Maxwell fields in two domains separated by an interface. The projections, called the Calderón projectors, involve single- and double-layer potentials of an appropriate Green function (analogous to the Cauchy kernel 1/(z-z0)). The choice of the Green function depends on the application. The reader is referred to Refs. 5 (Costabel/Stephan) and 11 (Nédélec) in [ShVe03] for the projectors for the Helmholtz and Maxwell equations associated with a bounded scatterer.
The Calderón projectors provide a natural and beautiful way to pose the problem of scattering of plane waves by an obstacle. This could be, say, a bounded obstacle in free space, a bounded obstacle in a wave-guide, or a periodic slab. In [ShVe03], we show how to formulate the problem in the case of a periodic slab and use the properties of the projectors to gain knowledge about guided modes. The idea of the formulation of the scattering problem is this: The total field on the boundary of the scatterer together with its normal derivative (its trace) have values coming from the interior of the scatterer and values coming from the exterior, and these value must be matched appropriately. Both interior and exterior traces are decomposed into traces of the source and scattered fields. In the interior, this decomposition coincides with the decomposition arising from the complementary Calderón projectors constructed from a Green function with the interior material properties, and in the exterior, it coincides with the decomposition arising from the projectors that use the radiating Green function with the exterior material properties. Projecting onto the traces of the (known) source fields in the exterior and interior gives an equation of the form (B + C)u = f, where B is bounded, C is compact, u is the trace of the total field, and f is essentially the trace of the sum of the source fields.
Seeing that B and C depend analytically on the frequency ω and the Bloch wave vector κ parallel to the slab, the equation (B + C)u = 0 gives rise to a complex dispersion relation between κ and ω corresponding to generalized guided modes of the slab. Real-valued pairs (κ,ω) correspond to true guided modes. Since the imaginary part of ω on the dispersion relation must be negative, typical pairs correspond to fields that are decaying in time but exponentially growing spatially with distance from the slab. These are analogous to the fields of the "scattering resonances" for the Helmholtz resonator; see Ref. 6 (J. T. Beale) of [ShVe03].
In [RaSh08], we extend the Calderón projectors to interfaces between two complementary subdomains of a periodic medium. We then use these to investigate the limiting behavior of transmission of harmonic fields at frequencies in a spectral gap of the medium.
For the simulations of fields in periodic slabs that you see on this page, we used a numerical code developed at Duke University by M. Haider, S. Shipman, and S. Venakides [HaShVe02]. It is based on the integral representations that are formulated most elegantly using the Calderón boundary-integral projectors mentioned above [ShVe03].
The algorithm uses an acceleration technique that is similar to a
multipole algorithm, specialized to periodic structures. Most
efficiency is gained when the number of periods across the slab is very large.
References on multipole algorithms can be found on the website of
L. Greengard.
In [LiShVe03], we develop the variational calculus for the
transmission coefficient as a function of the material parameters
ε and μ for two-dimensional scattering of electromagnetic plane
waves by periodic slabs. The simulations below show how
transmission features can be optimized with the resulting
gradient-flow algorithm.
Perfect square lattice with no channel
Perfect square lattice with a channel every three rows
Perturbed lattice with channels
Intensity of the scattered field at the frequency of the peak
transmission above
Moving a transmission gap
I am presently working on implementing an algorithm for optimizing the "effective density of states" for a photonic crystal slab. This effective density is discussed in [Bendickson/Dowling/Scalora, Phys. Rev. E, 53 No. 4 p. 4107]. It is essentially the sensitivity of the effective wave number of a scattered field across the slab with respect to frequency and is thus directly related to the phase of the complex transmission coefficient. Anomalous behavior of this quantity coincides with resonant scattering in the structure and anomalous transmission of energy (the magnitude of the transmission coefficient).
Graph (a) below shows the transmission of energy through an infinite slab of a square-lattice photonic crystal of circular rods, five rods thick, as a function of reduced frequency (from 0 to 1) and angle of incidence (from 0° to 90°).
Notice the complete gap from frequencies of about 0.3 to 0.45 and the incomplete gap around 0.6. Graphs (b), (c), and (d) show realizations of how the transmission changes under random perturbations of the centers of the five rods in a period (the structure remains periodic---the perturbations are the same in each period). In (b), the perturbations are taken to uniformly distributed up to 10% of the distance between adjacent rods, in (c), it is 20% and in (d) it is 30%.
Observe that the complete gap is robust under these perturbations, whereas the incomplete gap is not.
Perfect 6x4 supercell
1. (1.54) 
|
2. (1.99)  |
1. (1.62) 
|
2. (137.1) 
|
3. (2.05) 
|
4. (2.77) 
|
1. (1.67) 
|
2. (108.2) 
|
3. (2.10) 
|
4. (3.70) 
|
1. (1.54) 
|
2. (1.76) 
|
3. (108.2) 
|
4. (61.2) 
|
5. (2.09) 
|
6. (2.67) 
|
7. (10.5) 
|
8. (2.26) 
|
9. (3.70) 
|
10. (2.24) 
|
11. (2.23) 
|
12. (2.22) 
|
The numbers in parentheses refer to the ratio of the maximal intensity of the field compared to the intensity of the incident field.
Perfect 1x10 supercell
1. (1.12)
2. (1.32)
3. (1.48)
4. (1.47)
5. (2.01)
6. (4.17)
7. (3.00)
8. (1.96)
9. (4.48)
10. (2.16)
11. (2.17)
12. (2.70)
1x10 supercell with 10% random perturbations of centers
1. (9.80)
2. (43.84)
3. (2.16)
1x10 supercell with 20% random perturbations of centers
1. (25.85)
2. (2.51)
3. (5.40)
1x10 supercell with 20% random perturbations of centers
1. (7.91)
2. (35.88)
3. (40.49)
1x10 supercell with 30% random perturbations of centers
1. (36.17)
2. (14.59)
3. (8.43)
1x10 supercell with 30% random perturbations of centers
1. (3.42)
2. (57.62)
3. (16.04)
1x10 supercell with 10% random perturbations of radii
1. (12.66)
2. (14.09)
3. (11.07)
1x10 supercell with 30% random perturbations of radii
1. (7.26)
2. (23.26)
3. (4.99)
1x10 supercell with 30% random perturbations of radii
1. (5.52)
2. (2.43)
3. (5.52)
1x10 supercell with 50% random perturbations of radii
1. (4.62)
2. (2.00)
3. (3.32)
4. (21.05)
5. (6.06)
6. (9.91)
1x10 supercell with 50% random perturbations of radii
1. (1.23)
2. (25.23)
3. (25.61)
4. (5.53)
5. (31.62)
6. (25.82)
5x5 supercell, 30% random perturbations of centers,
30° angle of incidence
(11.03)
