School of Electrical Engineering, Sharif University of Technology, Tehran, Iran
Received Date: April 15, 2015 Accepted: June 23, 2015 Published: June 26, 2015
Citation: Khorasani S (2015) On The Bloch Theorem and Orthogonality Relations. J Laser Opt Photonics 2:118. doi:10.4172/2469-410X.1000118
Copyright: © 2015 Khorasani S. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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Bloch theorem for a periodic operator is being revisited here, and we notice extra orthogonality relationships. It is shown that solutions are bi-periodic, in the sense that eigenfunctions are periodic with respect to one argument, and pseudo-periodic with respect to the other. An additional kind of symmetry between r-space and k-space exists between the envelope and eignfunctions not apparently noticed before, which allows to define new invertible modified Wannier functions. As opposed to the Wannier functions in r-space, these modified Wannier functions are defined in the k-spaces, but satisfy similar basic properties. These could result in new algorithms and other novel applications in the computational tools of periodic structures.
Bloch-Floquet theorem; Periodic media; Photonic crystals; Plasmonic crystals; Electronic crystals
Consider a self-adjoint linear operator problem in the real physical space, denoted by the -space, as
where , λ are eigenvalues and are eigenfunctions. We define a translation operator  as
Now, we take the basis vectors , with the non-zero triple product . We may also define the discrete vectors where, n1, n2, n3 are integers. The set of all such discrete vectors are referred to as the lattice sites [2-4]. Now, suppose that these two operators commute
Then these two operators share the same set of eigenfunctions. Let satisfy the eigenvalue equation
The relationship between the sets of eigenvalues μ and λ is generally complicated and highly nontrivial .
Defining a reciprocal lattice vector we may call the set of all such discrete reciprocal lattice vectors as the reciprocal lattice points. For any choice of the discrete vectors R and K we now get the fundamental relationship
For any given discrete lattice vector R, this relationship allows infinite discrete solutions for K at the reciprocal lattice vectors, and vice versa. In a three-dimensional space, this will dictate a three-fold degeneracy on the eigenvalues and eigenfunctions, albeit in the form of discrete translational symmetry to be discussed later below.
Where K is defined as the Bloch wave vector, and correspondingly, the set of all such vectors is referred to as the K-space. In other words, is pseudo-periodic in the r-space. Accordingly, the eigenfunctions satisfy
Here, the so-called envelope functions are periodic in -space, that is
We respectively refer and to as the wave and envelope functions.
Referring to the above, we may see that both eigenvalues ì and ë are actually functions of K as and This leads us to the fact that
Extensions to the bloch theorem
A unit-cell of the k-space which constitutes the periodicity of λ (k) is known as the Brillouin Zone. Evidently, this unit-cell is extended in the K-space across the basis vectors, b1,b2 and b3.
For every given vector K, there are infinitely many of eigenvalue functions λ (k) in general. So to be precise, one would need to consider with n being a natural number referred to as the band index. This is exactly what truly happens for the case of every periodic media, such as photonic and plasmonic crystals [6-8] electronic crystals [2-4,9-12] and even photonic crystals .
If we take advantage of the periodicity of these eigenvalues, we may express λ (k) always has , in such a way that k-K would belong to the first Brillouin Zone. This notation with the aid of band-index n makes λn (k) a single-valued function of its argument. Similarly, as long as k were restricted to a single Brillouin Zone, then the band index integer n would be needed to resolve the eigenfunctions corresponding to the eigenvalues λn (k) at equivalent k points, where any two k1 and k2 points are said to be equivalent if with k being a discrete reciprocal lattice vector. From this point on, we drop the explicit dependence of eigenfunctions on the band index n for the sake of brevity, unless needed.
Further properties of the bloch theorem
The above relationships when put together with the basic Bloch theorem, we obtain unexpected result for the first time
That is, must be pseudo-periodic in the k-space, too. This is to be compared to the pseudo-periodicity of in the r-space.
When these two properties are combined, we arrive at the following conjugate results
These state that both of the wave and envelope functions are biperiodic, however, the wave is pseudo periodic in r-space and periodic in k-space, while conversely the envelope is periodic in r-space and pseudoperiodic in k-space.
These properties furthermore allow a trivial translationalinvariance in phase, such as the replacements
Would provide another equivalent set of eigenfunctions, as long as are real-valued and double-periodic as
Choice of is non-trivial for the optimum generation of Wannier functions, to be discussed later in Section4.
Now, both of the wave and envelope functions should satisfy orthogonality relationships in r-space and k-space respectively as
With ak and bk being some normalization constants. The inner products are here defined as
After normalization and expansion of the inner products, we get
While the first relation is simply known as the orthogonality of wave functions, the second one actually is an expression of the completeness relationship.
In the reduced-zone scheme, we may rewrite and renormalize the above equations as
where the integration of inner products are r estricted to unit cells as
Here, r- cell and k-cell refer actually to the lattice’s Unit Cell and Brillouin Zone, respectively.
Modified Wannier functions
As a result, we obtain the shift-properties and This allows us to ignore the discrete vector indices, and simplify the definitions as
where appropriate. Wannier functions now turn out to be readily orthogonal as
Finally, the original wave and envelope functions may be reconstructed from the respective Wannier functions through discrete summations over the lattice points and reciprocal lattice points as
Alternatively, we may rewrite the above definitions as
This allows us to obtain the direct transformation pairs between these Wannier functions as
Here, we note the use of identical band indices on both sides.
It should be mentioned that the translationally-invariant phase takes significant part in the integration over eigenfunctions, which may be illustrated explicitly as
Since there are infinitely many choices for Wannier functions would not be unique, and normally need to be constructed in such a way to be maximally-localized. This is normally done through global minimization of a spread functional; the reader is referred to literature for in-depth discussion of this issue [14,9,16-22] for plasmonic, photonic, phononic, and electronic crystals, respectively.
Two-dimensional photonic crystals
Consider a two-dimensional (2D) photonic crystal (PC) with in-plane propagation, which allows separation of E-polarized and H-polarized modes. Here, the 2D relative permittivity and permeability function makes the whole structure periodic as and with The operator for the normal component of the electric field would become 
The scalar operator and eigenvalue is in the extended zone now given as
With Furthermore, is the E-polarization band structure, and is easy to be shown to be self-adjoint operator under these criteria, hence the eigenfunctions obeying the orthogonality relationships after corresponding displaying the band indices
Similarly, the relationships of the H-polarized components could be found. As for the scalar operator of the normal component of the magnetic field , one would
is the H-polarization band structure, and is selfadjoint. Hence the eigenfunctions obey the orthogonality relationships after corresponding displaying the band indices
Where V actually denotes the area of the unit-cell; for this to happen, one may simply set Similarly, we have for the 2D PC. The relevant pair of H-polarization Wannier functions would be
Three-dimensional photonic crystals
If the permittivity and permeability functions are three-dimensional (3D) periodic tensor of space, then within the frame of reference which becomes diagonal (under no loss and no optical activity [9-11], one would have the following relation for the vector operator
Where is properly defined because of its diagonal form. Similarly, we obtain
Which leads to the orthogonality relations
Here,represents the tensor outer product, and is the 3× 3 unit tensor. From these equations the additional forms are inferred by taking the trace
Finally, when the dielectric is isotropic, one would reach the scalar equations
The relevant pair of vector E-polarization Wannier functions would be obtainable from generalization of the vector Wannier functions 
Identical deductions may be made for the magnetic field eigenfunctions once is diagonalized, which may be found by interchanging permittivity and permeability, as well as Electric and Magnetic fields everywhere.
In this article, we have presented an in-depth discussion on the basic properties of Bloch waves in periodic media, and obtained a novel biperiodicity property in the eigenfunctions for the first time. We showed that the envelope and wave functions both satisfy mutual periodic and pseudoperiodic properties in the physical and reciprocal spaces, which shows there exist much more similarity between these two spaces than thought before. Extensions of these properties were found to be applicable to defining a new set of Wannier functions in reciprocal space, rather than the conventional definition in physical space, while these two Wannier functions are directly connected through unorthodox transformations. It is envisioned that the future investigation of these pair of Wannier functions would result in more efficient analytical tools of periodic media, and simplify the construction of maximally-localized band functions.
The author is indebted to Dr. Ali Naqavi at the École polytechnique fédérale de Lausanne for reading the initial manuscript and providing insightful comments.
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