Optimal design of photonic crystals 5 3

optimization formulations to design the single-polarization single-mode fibers.Despite being governed by a different setup of the Maxwell’s equations, both the band width and band gap op

λhWG,1

λhWG,2 0.4

0.5

0.6

0.7

0.4 0.5 0.6 0.7

0.4 0.5 0.6 0.7

||H2(β−1)||2

||H2(β^1)||2

||H2(β−2)||2

||H1(β^1)||2

Figure 5.8: β2 = 1.5 The figures in the columns from left to right each represent the case: (left)initial configuration, with band width Jh = 0.0122 ( ˆ β1 = 1.45); (middle) optimal configuration when only cladding is optimized, with band width 0.048 ( ˆ β1 = 1.42); (right) optimal configuration when both cladding and core are optimized, with band width Jh = 0.15 ( ˆ β1 = 1.27) The first row illustrates the PCF cross-sections The second row shows the corresponding dispersion relations between [ ˆ β1, β2] From the third row to the last, the magnetic field intensities at both ˆ β1 and β2 are shown in 3D surf plots.

optimization formulations to design the single-polarization single-mode fibers.

Despite being governed by a different setup of the Maxwell’s equations, both the band width and band gap optimization problems relates to maximizing the difference between two consecutive eigenvalues over some parameter sets of wave vectors As a result, the algorithm based on approximate and reduced subspaces projection and SDP reformulation can be suitably applied here with a few moderate variations

The band width optimization problem was modeled by two formulations of three scenarios each, and some resulting band structures together with validation of the field variable inten-sity are presented The results clearly demonstrated the success of our convex optimization algorithm in these more complicated physical problems

Chapter 6

Conclusions

We will conclude the thesis in this final chapter by first summarizing all the work and contribu-tions to date; To put things in perspective, suggescontribu-tions and proposals on further improvements

as well as extensions to a wider range of applications will be provided in the end

The optimal design of photonic crystals has been the central theme of this work The aim is

to develop convex optimization formulations that are reliable and efficient to design various photonic crystal devices possessing important properties

The notion of photonic crystals was introduced in chapter 1 These are a new class of materials that are rapidly growing in popularity because of their unique features and superior properties Among these properties, the most important are the band gaps and the index guid-ing In the band gap phenomenon, a properly designed photonic crystal can exhibit a range

of prohibitive frequencies to the propagation of electromagnetic waves In the index guiding mechanism, the propagation of the electromagnetic waves of certain frequencies can be localized

in the core region of the photonic crystal device; while other frequencies are attenuated in the waveguide according to the design requirements The two features are similar in that their be-havior can be almost completely predicted by examining the corresponding dispersion relations, which provide us with the basic mathematical models for further analysis and optimization Having examined the previous work on the optimization of the band gaps, e.g., parameters study or gradient-based optimization approach, we proposed to reformulate the nonlinear, non-convex, large scale band gap optimization problem to tractable convex programs We have also identified another band width optimization problem based on index guiding in photonic crystal fiber, to which we plan to extend the convex formulations

Chapter 2 is a collection of reviews on some fundamental physical and mathematical con-cepts that are frequently used throughout this work The central theoretical framework for our modeling is based on the Maxwell’s equations – the governing equations for

electromag-netic waves propagating in dielectric material, as well as the symmetry theories and the Bloch theorem that are used to simplify and recast the Maxwell’s equations to Hermitian eigenvalue problems Next, functional analysis and the finite element method as the numerical techniques

to solve the Maxwell’s equations were also summarized Basic concepts on optimization, such

as, convex cone, generalized inequalities, and standard convex programs, for example, semidef-inite program and second-order cone program, were reviewed as well

As a first step to the optimal design, we strived to solve for the eigenvalues accurately and efficiently in both physical problems in Chapter 3 The band gap problem for the two-dimensional photonic crystal can fortunately be simplified to a scalar eigenvalue equation for either transverse magnetic or transverse electric polarization of the EM waves Standard fi-nite element method with linear nodal basis functions can be applied, together with carefully discretized wave vector and dielectric function spaces, to compute the eigenmodes with conver-gence rate up to twice the order of the interpolation basis functions used

We also introduced a simple adaptive mesh refinement procedure The strategy is to increase the discretization resolution to allocate more degrees of freedom, hence more com-putational nodes, in regions where the eigenfunctions have higher gradient, and to maintain

a coarser discretization in regions of smooth solutions Typically, the increased variation in the eigenfunctions can be expected along the interfaces of the dielectric materials An adaptive computation mesh was obtained by successively refining the elements on the material interfaces

to the desired resolution, while maintaining the conformability of the finite element method Solving the Hermitian eigenvalue equations on the adaptive meshes, satisfactory numerical so-lutions of the eigenmodes, and a convergence rate of as high as 9 has been observed in the few examples analyzed

The governing Maxwell eigenvalue equations for the photonic crystal fiber problem dis-played some increased complexity One complication is that they lead to a system of equations involving both the transverse and longitudinal components of the field variables Moreover, the notorious spurious modes required special treatment of the transverse components of the field variables, i.e., H(curl, Ω) conforming bases had to be used in the finite element approximation This was done in addition to the H1(Ω) basis functions applied for the longitudinal component approximation We formed the so called “mixed formulation” When the lowest order of the H(curl, Ω) interpolation basis functions are used in addition to the linear H1(Ω) basis functions

in the mixed formulation, the computed eigenmodes on successively finer computation meshes are converging at a rate of 2 as well Fortunately, the additional use of H(curl, Ω) conforming basis functions did not require any special treatment to the mesh adaptivity procedure The convergence rate of the eigenvalues computed on the adaptive meshes was as high as 6 The formal optimization formulation for the band gap problem was developed in chap-ter 4 We started with a well-posed, but nonlinear, non-convex, and large-scale optimiza-tion statement, with low regularity and a non-differentiable objective Through restricoptimiza-tion to the appropriate eigenspaces, we reduced the large-scale non-convex optimization problem via

reparametrization to a sequence of small-scale convex semidefinite programs for which mod-ern optimization solvers can be efficiently applied Adaptive mesh refinement was naturally incorporated to the optimization procedure By initializing the optimizations with previously optimized structure on coarser meshes, we obtained the final well represented optimal struc-tures on adaptively refined meshes with improved computation cost Numerical results and extensive optimal designs of the two-dimensional photonic crystals are presented with optimal band gaps of various configurations, e.g., absolute band gaps, complete band gaps, and multiple band gaps Among all the results, we have obtained various photonic crystal structure with: single absolute band gap of gap-midgap ratio as high as 97.8%, as many as 4 multiple absolute band gaps, and up to two complete band gaps in both square and hexagonal lattices

In chapter 5, we studied the band width optimization problem arising in the photonic crys-tal fibers due to index guiding mechanism By proposing several formal convex optimization formulations for the design of the single-mode single polarization fibers, we demonstrated that the optimization recipes developed for the band gap optimization problem in chapter 4 could

be extended to solve this similar yet more complicated optimization problem We also pre-sented several optimal designs as a proof of principle, and verified the results by the intensity plots of the localized and attenuated fields corresponding to the guided and unguided modes respectively

While this thesis covers in great detail particular cases of the optimal design of photonic crystals, the following directions are of great interest for the further development of this field

Band gap optimization of three-dimensional photonic crystals

One of the most challenging problems in this field is the design of realistic photonic crystals, i.e., structures with full three-dimensional periodicity A 3D photonic crystal provides perfect dielectric confinement of light of any polarization in all three dimensions by exhibiting a com-plete band gap in its energy spectrum However, a comcom-plete 3D band gap is very rare, as it must smother the entire three-dimensional Brillouin zone An increased dielectric constant is very often not enough for all the directional gaps to be wide enough to create an overlap Hence, some lattice structures with nearly spherical Brillouin zone are preferred to construct the 3D photonic crystal, e.g., face-centered cubic and diamond lattices A number of 3D crystals have been discovered to yield sizable complete photonic crystal band gaps [61, 31, 29]

Besides the physical feasibility, numerical complications should be taken into account in the design of the photonic crystals To solve the three-dimensional Maxwell’s equations, not only will the degrees of freedom of the system increase, but also the number of decision vari-ables Moreover, a full vectorial, three-dimensional Maxwell’s equations also require the full H(curl, Ω) conforming basis functions Despite the 3D designs based on physical intuition

mentioned above, we have not come across any references on formal mathematical optimiza-tion formulaoptimiza-tions With the installment of subspace reducoptimiza-tion and mesh adaptivity, we are optimistic about the potential of our convex optimization algorithm in handling the 3D opti-mization problem

Phononic/photonic crystals

By analogy to the propagation of the electromagnetic waves in periodic dielectric material, the propagation of the elastic waves in materials with periodic mechanical properties has estab-lished another interesting field, phononic crystals Extension of the band gap phenomenon and optimization of the crystal structures of elastic materials have attracted considerable attention [58, 52, 34, 41] Another interesting design problem is the coupling of the two types of mate-rials, and the construction of structures possessing both the photonic and phononic material properties[40] Based on the successful story of our algorithm, the optimal design of this type

of crystal is certainly a new and promising area of research

Field localization

In all our optimization formulations, eigenvalues have been the solo actors of the objective func-tions They have been the natural and reasonable choices because the properties of eigenvalues dictate many fundamental behaviors of the system, e.g., the propagation of the field variables

In addition, the manipulation of eigenvalues can be easily redirected to the properties of the matrices in the corresponding discrete eigenvalue equations, which can be further handled by semidefinite cones and the associated generalized inequalities Nevertheless, all these seem like

an indirect and winding detour to achieve convex optimization One should be able to operate more directly on the vector spaces For example, the localization of the field variable can be easily formulated as a least square problem [23] which is a subset of convex optimization; More adventurously, one might even formulate it with first-order methods

Uncertainties and robustness

Another area of focus involves the more practical issue of manufacturablility During the stage

of fabrication, uncertainties or defects are prone to be introduced Moreover, we note that many

of the optimized crystal designs shown in chapter 4 involve intricate patterns of materials at the nano-level, and may be too expensive or even impossible to fabricate Simply incorporating fabrication constraints such as bounds on the curvature of boundaries or connectedness of materials easily yields combinatorially intractable optimization models Instead, we propose

to modify the basic optimization problem, so that a resulting solution is robust for fabrication

to account for both the uncertainties in the optimization formulation a priori, and to obtain optimal structures that retain the desirable properties amidst the manufacturing defects This

is somewhat in the spirit of robust convex optimization [8], and is the subject of future research

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