The Problem of Quasiperiodic Photonic Structures Solved by Considering the Cut of 2D Periodic Structure

The physical objective of solving for eigen-modes of a 1D quasiperiodic structure in photonics has been achieved. This was achieved thru considering this structure as a 1D projection or cut of a 2D periodic structure. And the problem is solved in a manner similar to 2D periodic photonic structures. A mechanical analogy (quasiperiodic orbits) helps to bring conceptual clarity.


Introduction
The idea is to use the construction of a 1D quasi-periodic structure. The construction uses the diagonal sectioning of a (parent) 2D periodic tiling, the diagonal having an irrational slope (see for example [1]). This construction is used to come up with a (Fourier) series expansion. This series expansion may later be used to solve the EVP (Eigen-Value Problem) Master Equation of Photonics (see [2]). It is found that instead of solving for a single frequency/eigenvalue, we need to solve for two-one for each dimension in the associated 2D periodic tiling problem. It is found that to find the two frequencies, we need to solve the 2D periodic problem as well as the 1D quasi-periodic problem. The method of solution and the solution itself closely resembles that of periodic problems to the extent that concepts like reciprocal lattice, periodicity and a Bloch's Theorem still apply (in a modified form for the 1D quasi-periodic problem and exactly for the 2D periodic problem)! A mechanical analogy clarifies matters.

1) Mixing 2) Dense Quasiperiodic orbits 3) Sensitivity to Initial Conditions
Mixing means that if the motion passes through some region in phase space, then it will eventually pass through another region regardless of how far they are apart. A quantification of the idea of the orbit roaming throughout the entire phase space rather than being bounded or confined. A chaotic orbit that visits and revisits that is mixes with all regions of the available phase space is identified with what is called a strange attractor. Its association is not with a localized attractor such as a fixed point or a limit cycle, but rather with a very extended region of phase space, hence the designation strange.
Quasiperiodicorbit means that the orbiting mass repeatedly and irregularly pass through the whole range of the domain without ever closing on themselves, and without any particular time period associated with successive transits. They are dense because they pass through or arbitrarily close to every point of the domain, a property that conforms with the ergodic hypothesis. Like painting or covering a 2D square piece of canvas with very thin line-like brush strokes.
Sensitivity to Initial conditions: a small change in the initial conditions can result in a large change in position and velocity many transits or iterations later (having discretized the problem and the motion unfolds over discrete time steps). Moreover, this small change (perturbation say) could lead to switching from the linear bounded regime over to the chaotic non-linear one… as two initially close points grow exponentially apart. This exponential growth of the gap is quantified via the Liaponuv exponent, which is directly related to the fractal dimension of the strange attractor. This exponential growth of the gap is just the famous Butterfly Effect-a popular defining property of chaos. An example is a spaceship in an Earth orbit. A small rocket boost will move it to a nearby orbit whereas a strong boost could throw it out of orbit, heading for outer space. Another common example of how linear and chaotic motions differ when periodicity is not present is turbulence in water. While there is streamline flow, two nearby points in the water stay close together as they move along; after the onset of turbulence the same two points, on average, keep moving farther and farther apart. It is worth noting though that, they may now be related by Renormalization owing to fractal nature of turbulent flow. Moreover, renormalization rather than averaging pops up in QFT with its non-linear equations rather than the linear Schrodinger Equation of QM.

Sensitivity to Initial Conditions, Statistical
Mechanics and Quantization To drive the point home, observe that this means that Liouville's Theorem which is applied to ensembles in statistical mechanics (the conservation of density of states in phase space) does not hold for chaotic motion. For some initial region R, R may deform with time, but number of states within R is conserved like the constant density of a fluid. That is a system initially inside R never leaves it and a This may be thought of as a result of the linearity of Schrodinger's Equation even though the original classical system maybe non-linear (see [4] That is, quantization condition itself can be thought of as a bounding condition. Which physically makes sense. Quantization of the electron (say in Bohr Model via de Broglie hypothesis: viewing it as a wave rather than a particle) solved the instability problem of the classical atom-seen thru Maxwell's Equations-via removing the secular terms that lead to the instability of the orbit.
And quantization of light (viewing it as particles rather than waves) removes the UV divergences-helping the integral have a finite value or making the integral bounded... or a cutoff is introduced. And the whole idea of Renormalizationcentral to all quantum field theories-is again about handling this problem of unboundedness and infinities.
Further discussion and generalizations are available in [4]. The property of ergodicity, which involves covering all accessible regions of a domain, is shared by incommensurate non-chaotic orbits with respect to an ordinary attractor (for example, a torus), and by chaotic orbits with respect to a strange attractor. And it is interesting to note that ergodicity is invoked in statistical treatments too.

Mechanics, Symmetry and Quasiperiodic Functions
As is well-known, there exists a connection between the existence of additional algebraic constants of the motion, or higher-symmetry groups, and degeneracy in the motions of the system.
In the case of the Kepler and isotropic harmonic oscillator problems, the additional constants of the motion are related to parameters of the orbit. Unless the orbit is closed, that is, the motion is confined to a single curve, we can hardly talk of such orbital parameters. Only when the various components of the motion have commensurate periods will the orbit be closed. The classic example is With the two frequency incommensurate …. That is,

Algebraic Formulation of Cut-Projection Method
Now, the periodicity of the dielectric 2D square periodic structure ( ) , x y ε permits us to expand it in a Fourier Series: With 1 2 , G G taking values from a discrete set: Later on, we shall suppress the labels periodic and quasiperiodic when it is clear the function in two variables is 2D parent periodic and the function is single variable is the 1D quasi-periodic.
Comparing the harmonics in (2.4) to the one in (1.4), we find that (and making the switch Moreover observe that: A. El Houshy And as , n m are integers and τ is irrational, then Or, . And Cahn says, see [5], that However the two together could only have a common period if there is a Lowest Common Multiple of the two periods. But this requires that they be commensurate; this is not made possible due to the irrationality of τ . This will be made clearer when considering the example of two different planets orbiting the same star and trying to find when they will be realigned again with the central star.

Similarity of My Equation to Electron Density for Quasi-Crystals
Also note that if substitute (2.2) in (2.4) we get: But now τ is any general irrational number not necessarily the Golden Ratio as in our case. Thus, the expansion we use is in a single variable but in TWO indices stressing the projection of a TWO-dimensional periodic lattice. Unlike the expansion used in Kittle [6], this is just a quasi-periodic expansion in a single index AND a single variable. In essence, the advantage of our expansion is that it allows us to draw analogies with two dimensional periodic problem.
To get a understanding of what the condition (2.3) means, consider again our 2D parent periodic structure. The quasi-periodic structure of direct physical interest lies along the line y x τ = .
And so, what we are interested in is the dielec- x y ε at the following points (Table 1).
That is, we are sampling the parent 2D periodic structure along the cut ( Figure 3).

A. El Houshy Journal of Applied Mathematics and Physics
And removing the subscript off the general point, shows that indeed the geometric operation of projecting along the cut y x τ = amounts to an algebraic substitution affecting the transformation: ; since with Dilation Operator D we have: , 3) was used to reach the second equality.

Master Equation in 1D
NOTE: traditionally in a 1D periodic multi-layer thin film problem, the propagation direction is taken along the z-axis, which also the direction along which the dielectric varies. That is, we are really solving for And the Master Equation should really read: However, we choose to work with the form in (3.1) as it emphasizes the cut-projection correspondence between the 1D quasi-periodic problem and the 2D parent periodic problem which is central to this work. Moreover, that neither form affects the final result in reciprocal space, where we have a relation between the wavenumbers, eigenvalues/frequencies and Fourier Coefficients. This is because in Fourier Space, all knowledge of spatial variables of configuration is forgotten thanks to linear independence of the Harmonics.
That is 1 2 , G G are both reciprocal lattices of the square lattice.
And so the 1D quasi-periodic problem is turned into a 2D Periodic Problem.
It is natural to expect a quasi-periodic response: Now the coefficient of ( on the LHS is (via the transformation While the coefficient of (

Does Bloch Theorem Still Apply?
The short answer is no. But the periodicity of the parent structure does still have some interesting implications for the quasiperiodic structure (And we still have periodicity in k-space).
Following [6], We may first solve (3.9) for the Coefficients of the components of the Field. We may then expand k H in the reciprocal lattice vectors 1 2 , G G of the parent 2D periodic square lattice ( k H the photonic analogue of k ψ in Solid State Physics): where C − k G is a shorthand for  where r the set of non-zero integers, by definition of a reciprocal lattice vector.
That is , 2, 1, 0,1, 2, r = − −   . However n is fixed (as well as m). Whence, there exists a subset of r which will coincide with the multiples of n. Namely, the sub- Where in the last step we used the definition of ( ) k u x for a slope p; and in the step before last we used the first of (3.15) and (3.22). Which makes sense, as geometrically: a rational cut with slope p m n = of a parent periodic structure with period a, produces a 1D structure with period na.
And the physics (eigenfunctions) reflects the geometry. For the irrational case… well we can't write p m n = … but we can make successive approximations. For example for the case of 1 5 2 τ + = , we can make the following decimal approximations with the corresponding fractions ( Table  2). And here the periodicity of the parent lattice kicks in! Explaining why we That is, the field may be multiplicatively decomposed into a periodic part and a phase factor(responsible for Periodic Boundary Condition … removed after translating N steps from first unit cell till the last cell… as length of lattice is L Na = ), which is the content of Bloch Theorem for Periodic Systems.

H x
To make more sense of the Bloch result above, and understand how 2D periodic systems relate to 1D quasi-periodic systems… let's take another look at the parent 2D square lattice:  Which the form we used above in (3.5). And now it is clearer why is it that translating the field by a lattice constant a means the field picks up a phase factor without changing form (Bloch's Theorem). This is something inherited from the parent periodic-lattice. Memory of periodicity is retained. That the field and the lattice have the same functional form simplifies the Algebra as it allows "recombination" of the Fourier terms, allowing us to invoke linear dependence to have a system of linear equations in the coefficients, see the steps leading to (3.9).

A Clearer Conceptual Explanation of the Failure of Bloch Theorem
Now we know from the periodic case for the 2D parent lattice that Bloch's Theorem Applies: i k x k y k k k k H x y u x y u x y u x Na y Na Applying the same Geometric Transformation y x τ = (3.36) that took a diagonal section along the 2D periodic lattice to go over to the 1D quasi-periodic lattice: And it was verified above that We may see now how the form of the physical field relates to the geometry of the photonic structure. And how the relation of the parent periodic lattice to the quasi-periodic structure affects the field in the latter. That is, the same relation  And the last step requires that slope = integer or rational (with adjusted periodicity) otherwise multivaluedness of roots of unity kicks in and in our case slope = τ = irrational But we still have a relation between Geometry and Physics(and in fact quasiperiodic periodic problem needs to be solved in conjunction with periodic problem two unknown frequencies require two equations see below):

A. El Houshy
Projective Periodicity of Geometry => Projective Periodicity of Eigenfunction We note that thankfully, the two equations in (3.40) are scalar … and for the case of the TM mode, we have the same scalar unknown as the 1D case which was also really solving for z H , see note at the beginning of the Physics section.
And so it is this TM mode that is fit for comparison.
We now use (3.40) to solve the 2D parent periodic problem. We make the following Fourier Expansions: Now the coefficient of ( ) Comparing this to the analogous result (3.9) of 1D quasiperiodic structure: And so we may use the solution of periodic problem (where we already calculate the Fourier Coefficients F and solve for the coefficients C rather than just stop at finding the eigenvalues, that is we solve for the eigenvalues and the eigenvectors) to find the solution of the quasi-periodic problem (same k 1 , k 2 and G 1 , G 2 and identical F's and C's)

Meaning of having a Parent Periodic Structure
Symmetries (Translational and otherwise) in Reciprocal Space and implied symmetries in the Brillouin Zone ( Figure 4).
Condition for Reciprocal lattice vectors: (3.49) We can't immediately apply the standard equation for 3D lattices as that involves a cross-product that may lead to having a reciprocal lattice vector along z-axis.
Instead we find by inspection that:

Propagation
First, it is instructive to observe how a wave in 2D would propagate in Cartesian  (3.51) Changing without form means: (3.52) And so, However, note that then we run into a contradiction due to the second of

Physical Interpretation of Two Different Frequencies
The question now that arises is: What is the physical meaning of the AHO tem-  The basic idea is that we shall derive the governing Master Equation for incom-  (3.74) And now the two curl equations become: A. El Houshy