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On The Bloch Theorem and Orthogonality Relations | OMICS International
ISSN: 2469-410X
Journal of Lasers, Optics & Photonics
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On The Bloch Theorem and Orthogonality Relations

Sina Khorasani*

School of Electrical Engineering, Sharif University of Technology, Tehran, Iran

*Corresponding Author:
Sina Khorasani
School of Electrical Engineering
Sharif University of Technology
Tehran, P.O. Box 11365-9363, Iran
Tel: +98-912-304- 3142
Fax: +98 (21) 66 02 27 15
E-mail: [email protected]

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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Abstract

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.

Keywords

Bloch-Floquet theorem; Periodic media; Photonic crystals; Plasmonic crystals; Electronic crystals

Introduction

Consider a self-adjoint linear operator problem in the real physical space, denoted by the Equation -space, as

Equation

where Equation , λ are eigenvalues and Equation are eigenfunctions. We define a translation operator Equation [1] as

Equation

Now, we take the basis vectors Equation , with the non-zero triple product .Equation We may also define the discrete vectors Equation 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

Equation

Then these two operators share the same set of eigenfunctions. Let Equation satisfy the eigenvalue equation

Equation

The relationship between the sets of eigenvalues μ and λ is generally complicated and highly nontrivial [5].

We may now define a set of reciprocal lattice vectors Equation, Equation , and Equation with the triple product Equation [2-4].

Defining a reciprocal lattice vector Equation 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

Equation

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.

Bloch Theorem

The basis of the well-known Bloch-Floquet theorem [2-7] is that by which the eigenfunctions of Equation take on the property

Equation

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, Equation is pseudo-periodic in the r-space. Accordingly, the eigenfunctions satisfy

Equation

Here, the so-called envelope functions Equation are periodic in Equation -space, that is

Equation

We respectively refer Equation and Equation 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 Equation and EquationThis leads us to the fact that

Equation

Extensions to the bloch theorem

The eigenvalues λ (k) turnout to be multi-valued periodic functions in the reciprocal space, so that [2-4]

Equation

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 Equation 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 [13].

If we take advantage of the periodicity of these eigenvalues, we may express λ (k) always has Equation , 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 Equation corresponding to the eigenvalues λn (k) at equivalent k points, where any two k1 and k2 points are said to be equivalent if Equation 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

Equation

That is, Equation must be pseudo-periodic in the k-space, too. This is to be compared to the pseudo-periodicity of Equation in the r-space.

When these two properties are combined, we arrive at the following conjugate results

Equation

Equation

These state that both of the wave Equation and envelope Equation functions are biperiodic, however, the wave Equation is pseudo periodic in r-space and periodic in k-space, while conversely the envelope Equation is periodic in r-space and pseudoperiodic in k-space.

These properties furthermore allow a trivial translationalinvariance in phase, such as the replacements

Equation

Would provide another equivalent set of eigenfunctions, as long as Equation are real-valued and double-periodic as

Equation

Choice of Equation is non-trivial for the optimum generation of Wannier functions, to be discussed later in Section4.

Orthogonality relations

Now, both of the wave and envelope functions should satisfy orthogonality relationships in r-space and k-space respectively as

Equation

Equation

With ak and bk being some normalization constants. The inner products are here defined as

Equation

Equation

After normalization and expansion of the inner products, we get

Equation

Equation

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

Equation

Equation

where the integration of inner products are r estricted to unit cells as

Equation

Equation

Here, r- cell and k-cell refer actually to the lattice’s Unit Cell and Brillouin Zone, respectively.

Modified Wannier functions

The orthogonality relationships provide us with two sets of related, yet different, Wannier functions [9,13,14-17] in in r-space and k -space, respectively defined in the reduced zone-schemes as

Equation

Equation

As a result, we obtain the shift-properties Equation and Equation This allows us to ignore the discrete vector indices, and simplify the definitions as

Equation

Equation

where appropriate. Wannier functions now turn out to be readily orthogonal as

Equation

Equation

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

Equation

Equation

Alternatively, we may rewrite the above definitions as

Equation

Equation

This allows us to obtain the direct transformation pairs between these Wannier functions as

Equation

Equation

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

Equation

Equation

Since there are infinitely many choices for Equation 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.

Example

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 Equation and Equation with Equation The operator for the normal component of the electric field Equation would become [6]

Equation

The scalar operator and eigenvalue is in the extended zone now given as

Equation

Equation

With Equation Furthermore, Equation is the E-polarization band structure, and Equation 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

Equation

Equation

Where V actually denotes the area of the unit-cell; for this to happen, one may simply set Equation Note that, we have Equation for the 2D PC. The relevant pair of E-polarization Wannier functions [19,20] would be

Equation

Equation

Similarly, the relationships of the H-polarized components could be found. As for the scalar operator of the normal component of the magnetic field Equation , one would

Equation

Equation

Equation is the H-polarization band structure, and Equation is selfadjoint. Hence the eigenfunctions obey the orthogonality relationships after corresponding displaying the band indices

Equation

Equation

Where V actually denotes the area of the unit-cell; for this to happen, one may simply set Equation Similarly, we have Equation for the 2D PC. The relevant pair of H-polarization Wannier functions would be

Equation

Equation

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 Equation becomes diagonal (under no loss and no optical activity [9-11], one would have the following relation for the vector operator

Equation

Where Equation is properly defined because of its diagonal form. Similarly, we obtain

Equation

Which leads to the orthogonality relations

Equation

Equation

Here,Equationrepresents the tensor outer product, and Equation is the 3× 3 unit tensor. From these equations the additional forms are inferred by taking the trace

Equation

Equation

Finally, when the dielectric is isotropic, one would reach the scalar equations

Equation

Equation

The relevant pair of vector E-polarization Wannier functions would be obtainable from generalization of the vector Wannier functions [19]

Equation

Equation

Identical deductions may be made for the magnetic field eigenfunctions Equation once Equation is diagonalized, which may be found by interchanging permittivity and permeability, as well as Electric and Magnetic fields everywhere.

Conclusion

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.

Acknowledgement

The author is indebted to Dr. Ali Naqavi at the École polytechnique fédérale de Lausanne for reading the initial manuscript and providing insightful comments.

References

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