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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 3 — Feb. 10, 2014
  • pp: 2725–2734
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Modeling and optimization of photonic crystal devices based on transformation optics method

Yinghui Cao, Jun Xie, Yongmin Liu, and Zhenyu Liu  »View Author Affiliations


Optics Express, Vol. 22, Issue 3, pp. 2725-2734 (2014)
http://dx.doi.org/10.1364/OE.22.002725


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Abstract

In this paper, we propose a method for designing Photonic Crystal (PhC) devices that consist of dielectric rods with varying size. In the proposed design method, PhC devices are modeled with the Transformation Optics (TO) approach, and then they are optimized using the gradient method. By applying the TO technique, the original device model is transformed into an equivalent model that consists of uniform and fixed-sized rods, with parameterized permittivity and permeability distributions. Therefore, mesh refinement around small rods can be avoided, and PhC devices can be simulated more efficiently. In addition, gradient of the optimization object function is calculated with the Adjoint-Variable Method (AVM), which is very efficient for optimizing devices subject to multiple design variables. The proposed method opens up a new avenue to design and optimize a variety of photonic devices for optical computing and information processing.

© 2014 Optical Society of America

1. Introduction

Photonic Crystals (PhCs) are novel artificial materials that allow us to manipulate electromagnetic waves in an unprecedented manner [1

1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2 (Princeton University, 2008).

,2

2. I. A. Sukhoivanov and I. V. Guryev, Photonic Crystals: Physics and Practical Modeling (Springer-Verlag, Berlin, 2009). [CrossRef]

]. A fundamental property of PhCs is the Photonic Band Gap (PBG) [1

1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2 (Princeton University, 2008).

, 2

2. I. A. Sukhoivanov and I. V. Guryev, Photonic Crystals: Physics and Practical Modeling (Springer-Verlag, Berlin, 2009). [CrossRef]

], which prohibits light propagation in PhCs at certain frequency ranges and can be exploited to control light propagation. By introducing defects in PhCs, we can realize novel photonic components and devices, such as PhC waveguides, filters, demultiplexers, etc [1

1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2 (Princeton University, 2008).

, 2

2. I. A. Sukhoivanov and I. V. Guryev, Photonic Crystals: Physics and Practical Modeling (Springer-Verlag, Berlin, 2009). [CrossRef]

]. An important condition for the application of PhC waveguides is the efficient light coupling between PhC waveguides and other conventional optical waveguides. For some PhC line-defect waveguides that comprise air-holes in a dielectric medium, good light coupling can be achieved by directly connecting the PhC waveguide with a dielectric slab waveguide [3

3. A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal optical waveguides,” Phys. Rev. B 64, 033308 (2001). [CrossRef]

]. For PhC line-defect waveguides composed of dielectric rods, couplers with tapered structure, such as tapered PhC couplers [4

4. P. Bienstman, S. Assefa, S. G. Johnson, J. D. Joannopoulos, G. S. Petrich, and L. A. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguides,” J. Opt. Soc. Am. B 20, 1817–1821 (2003). [CrossRef]

, 5

5. K. Dossou, L. C. Botten, C. M. Sterke, R. C. McPhedran, A. A. Asatryan, S. Chen, and J. Brnovic, “Efficient couplers for photonic crystal waveguides,” Opt. Commun. 265, 207–219 (2006). [CrossRef]

] and tapered dielectric waveguides [6

6. A. Mekis and J. D. Joannopoulos, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,” J. Lightw. Technol. , 19(6), 861–865 (2001). [CrossRef]

] can be used for efficient optical mode coupling. In this work, we consider the possibility of a PhC coupler that consists of varying-sized dielectric rods. Comparing with tapered PhC coupler that normally have 7∼10 layers of rods [4

4. P. Bienstman, S. Assefa, S. G. Johnson, J. D. Joannopoulos, G. S. Petrich, and L. A. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguides,” J. Opt. Soc. Am. B 20, 1817–1821 (2003). [CrossRef]

, 5

5. K. Dossou, L. C. Botten, C. M. Sterke, R. C. McPhedran, A. A. Asatryan, S. Chen, and J. Brnovic, “Efficient couplers for photonic crystal waveguides,” Opt. Commun. 265, 207–219 (2006). [CrossRef]

], the proposed PhC coupler is very compact, consisting of only 3 layers of dielectric rods along the light propagation direction.

The simulation and optimization of photonic components that contain both electrically large and small features is very challenging, because refined mesh grids around the small features may significantly increase the computational overhead [7

7. O. Ozgun and M. Kuzuoglu, “Software metamaterials: Transformation media based multiscale techniques for computational electromagnetics”, J. Comput. Phys. 236, 203–219 (2013). [CrossRef]

]. As shown in the design examples of section 3 and 4, PhC devices may contain dielectric rods that are much smaller than others, thus mesh refinements are needed to capture the local variations of the device geometry and electromagnetic field in conventional simulation and optimization process. This will lead to an increase of unknowns to be solved, and may cause problems of generating meshes of good quality around these small rods. The above difficulties, which are referred to as multi-scale problems [7

7. O. Ozgun and M. Kuzuoglu, “Software metamaterials: Transformation media based multiscale techniques for computational electromagnetics”, J. Comput. Phys. 236, 203–219 (2013). [CrossRef]

], may become severe when the PhC device contains several dozens to hundreds of small rods. To avoid the multi-scale problems, we can apply the recently developed TO technique [7

7. O. Ozgun and M. Kuzuoglu, “Software metamaterials: Transformation media based multiscale techniques for computational electromagnetics”, J. Comput. Phys. 236, 203–219 (2013). [CrossRef]

12

12. Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science & Applications 1, e38; (2012). [CrossRef]

] to transform the original device model into an equivalent model, which consists of uniform and fixed-sized rods, with parameterized permittivity and permeability distribution. Therefore, mesh refinement around small sized rods can be avoided, and PhC device can be simulated more efficiently.

Photonic devices with varying geometry can be optimized with heuristic methods, such as the genetic algorithm [13

13. P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Marti, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett. 34(18), 2760–2762 (2009). [CrossRef] [PubMed]

] and the particle swarm algorithm [14

14. Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express 21(1), 1310–1316 (2013). [CrossRef] [PubMed]

]. However, because of their high computational cost, heuristic methods are inefficient for devices with lots of design variables [15

15. C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design.” Opt. Express 21(18), 21693–21701 (2013). [CrossRef] [PubMed]

]. In this work, we use a gradient method to optimize PhC devices, where the gradient of optimization object is calculated with the Adjoint Variable Method (AVM) [16

16. N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible Adjoint Sensitivity Technique for EM Design Optimization,” IEEE Tans. Microwave Theory Tech. 50(12), 2751–2758 (2002). [CrossRef]

18

18. J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22(6), 1191–1198 (2005). [CrossRef]

]. By using the AVM method, for each optimization iteration only one forward problem and one adjoint problem need to be solved. Therefore, the optimization is very efficient for devices with complex structures [15

15. C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design.” Opt. Express 21(18), 21693–21701 (2013). [CrossRef] [PubMed]

, 16

16. N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible Adjoint Sensitivity Technique for EM Design Optimization,” IEEE Tans. Microwave Theory Tech. 50(12), 2751–2758 (2002). [CrossRef]

].

2. Transformed model of PhC device

Figure 1(a) shows the structure of a PhC device that consists of dielectric rods with varying size. To avoid the multi-scale problems, we use the TO technique [7

7. O. Ozgun and M. Kuzuoglu, “Software metamaterials: Transformation media based multiscale techniques for computational electromagnetics”, J. Comput. Phys. 236, 203–219 (2013). [CrossRef]

11

11. L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3:1217(2012). [PubMed]

] to transform the original device model into an equivalent model that consists of rods with a uniform and fixed size, as shown in Fig. 1(b). The transformed rods and surrounding medium have anisotropic permittivity and permeability distributions, parameterized by radii of the rods in real geometry. The parameterized permittivity and permeability distributions can be derived using the TO technique [7

7. O. Ozgun and M. Kuzuoglu, “Software metamaterials: Transformation media based multiscale techniques for computational electromagnetics”, J. Comput. Phys. 236, 203–219 (2013). [CrossRef]

11

11. L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3:1217(2012). [PubMed]

]. Therefore, optimization of the original device is replaced by optimization of the parameterized permittivity and permeability distribution of the transformed device model.

Fig. 1 (a) A PhC device consisting of varying-sized dielectric rods. (b) Transformed model of (a), in which the transformed rods have a uniform and fixed size. Gray color represents the permittivity distributions. (c) A single dielectric rod, surrounded by air. (d) Transformed model of (c).

A single dielectric rod and the transformed model is shown in Figs. 1(c) and 1(d), respectively. The dielectric rod in real geometry has a radius of rc, and is stretched to be rb in the transformed model. The surrounding air of the rod in Fig. 1(c) is transformed into the surrounding medium layer of the stretched rod in Fig. 1(d), while the outer boundary of the surrounding medium is kept as same as ra after the transformation. Region of the real dielectric rod and surrounding air are denoted by Ω1 and Ω2, respectively. In the transformed model, region of the stretched rod and the surrounding medium are denoted by Ω̃1 and Ω̃2, respectively. Position vector in the original (transformed) geometry is denoted by r(x, y) (r̃(, )). Therefore, the coordinate transformation from the original geometry of Fig. 1(c) to the transformed geometry of Fig. 1(d) can be defined as
r˜=rbrcr=rbrcr(rΩ1,r˜Ω˜1)
(1)
and
r˜=rarbrarc(rrc)+rb=rarbrarc(rrc)+rb(rΩ2,r˜Ω˜2)
(2)
With the above coordinate transformation, the permittivity and permeability distribution of the transformed model in Fig. 1(d) can be calculated as [7

7. O. Ozgun and M. Kuzuoglu, “Software metamaterials: Transformation media based multiscale techniques for computational electromagnetics”, J. Comput. Phys. 236, 203–219 (2013). [CrossRef]

, 11

11. L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3:1217(2012). [PubMed]

]
ε˜r=1detJεr,μ˜r=JTJdetJμr
(3)
where J is the Jacobian of the coordinate transformation
J=(x˜,y˜)(x,y)=(x˜/xx˜/yy˜/xy˜/y)
(4)

In this work, we use commercial FEM software package COMSOL [19

19. COMSOL Multiphysics, http://www.comsol.com/

] to simulate PhC devices, with the transformed device model. To verify the transformed model, we first simulate light scattering from the original dielectric rod in Fig. 1(c), and then compare it with simulation using the transformed model in Fig. 1(d). In the simulation we set ra = 0.45μm, rb = 0.3μm, and rc = 0.125μm. Refractive index of the dielectric rod is nrod = 3.4. We assume the incident light propagates along x-axis, at the wavelength of λ = 1μm. The incident light polarizes along z-axis (TM mode), with the electric field intensity of |E| = 1. The total electric field obtained from simulations using the original and transformed model is shown in Figs. 2(a) and 2(b), respectively. Comparing Fig. 2(b) with Fig. 2(a), we can see that the electric field distribution inside the transformed rod is stretched. However, in the region r > ra, the difference of the electric field between the two simulations is negligible (∼ 10−4), as shown in Fig. 2(c). Thus for an outside observer, the transformed rod in Fig. 2(b) “looks” identical to the original rod in Fig. 2(a).

Fig. 2 (a) Light scattering of a single dielectric rod. (b) Light scattering of the same dielectric rod, using the transformed model. (c) The difference of the field distribution between (a) and (b) in the region of r > ra, showing almost negligible difference (in the order of 10−4).

To evaluate the advantages of the TO based model, we compare the meshes of the original and transformed model of a single dielectric rod, in the region of rrb. rb is the radius of the transformed rod, which is set as rb = 0.3μm as above. Meshes in the region of rrb can be generated identically for the original and transformed model, so they are ignored in the comparison. In the region of rrb, triangular mesh grids are generated, and quadratic elements are used. Meshes of the transformed rod, which just occupy the region of rrb, are shown in Fig. 3(a). In the region of r ≤ rb, meshes of the original model are shown in Figs. 3(b)–3(e), with rod radius of rc = 0.2, 0.1, 0.05, 0.01μm, respectively. Comparing Fig. 3(a) with Figs. 3(b)–3(e), we can see that in order to accurately capture the small geometrical features, meshes around the rod in original model is much denser than the meshes of the transformed rod, especially for rod with smaller radius. Statistic data of meshes in region of rrb, including number of elements, mesh points, and degrees of freedom, are listed in Table 1. Data in Table 1 shows, for meshes of rod with radius of rc = 0.2 ∼ 0.001μm, the number of degrees of freedom is approximately 2 ∼ 7 times of that of the transformed model. From such a comparison, we can conclude that our TO based approach will be superior over conventional methods, especially for complex geometries with many variable and small features.

Fig. 3 (a) Meshes of the transformed rod. (b)–(e) Meshes of the original model, in region of rrb. Radius of the rod is rc = 0.2, 0.1, 0.05, 0.01μm, respectively.

Table 1. Statistic data of meshes of the original and transformed model, in the region of rrb. Unit for radius: μm.

table-icon
View This Table

3. Optimization of PhC coupler

Figure 4(a) is a schematic diagram of the proposed PhC coupler. An input coupler, denoted as coupler1, couples light between the input dielectric slab waveguide and the PhC line defect waveguide. Similarly, an output coupler, denoted as coupler2, couples light between the PhC waveguide and the output slab waveguide. The input and output coupler are symmetric, converting the slab waveguide mode into the PhC line defect waveguide mode or vice versa. Both the input and output slab waveguide are single mode waveguides, with core thickness of d = 1μm, and refractive index of nco = 1.5. The slab waveguide core is surrounded by air. The dispersion relation and guided mode of the slab waveguide can be obtained by solving the eigenvalue equation of the dielectric slab waveguide [20

20. D. Marcuse, Theory of Dielectric Optical Waveguides(Academic Press, 1974).

].

Fig. 4 (a) Schematic diagram for PhC coupler. PhC couplers, indicated by dashed lines, couple light between the PhC line defect waveguide and the dielectric slab waveguides. (b) Rods radii of the optimised coupler, which are optimized with sampling wavelengths obj1–obj3, as defined in Eq. (6).

We assume the period of bulk PhC is a = 0.6μm. and the radius of the dielectric rod in bulk PhC is r0 = 0.125a, with refractive index of nrod = 3.4. Band structure of the bulk PhC is calculated using MIT Photonic Bands (MPB) [21

21. S. G. Johnson and J. D. Joanopoulos, “Block-iterative frequency-domain methods for Maxwell’s equation in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]

], and Photonic Band Gap (PBG) is found in the range of normalized frequency 0.37335 ∼ 0.48741, which corresponds to wavelength 1.23μm ∼ 1.61μm. A PhC line defect waveguide is formed by removing one row of dielectric rods. Numerical simulations show that odd mode of the PhC line defect waveguide is supported in the wavelength range of 1.3μm ∼ 1.6μm.

To optimize PhC devices, an object function is defined as
J=i(1pi)2
(5)
where pi is the power transmission coefficient of the device at the i-th sampling wavelength. To evaluate the effectiveness of the proposed design method, we use three different sets of sampling wavelengths to define optimization objectives, namely
obj1:λi=1.5,1.525,1.55,1.575,1.6(μm)obj2:λi=1.545,1.55,1.555(μm)obj3:λi=1.55(μm)
(6)
For sampling wavelengths set obj1, which contains 5 different sampling wavelengths, the optimization procedure will check and optimize at all the 5 sampling wavelengths. Hence the optimized PhC device will have the broadest transmission band. For sampling wavelengths set obj3, which contains only one sampling wavelength, the optimized PhC device will have the highest transmission coefficient at the sampling wavelength, but with the narrowest transmission band.

The first-order derivative of the object function J subject to the design variable rj is calculated as
Jrj=i2(1pi)pirj
(7)
where the design variable rj is the radius of the j-th rod of the PhC device. In the transformed device model rj represents a parameter of the permittivity and permeability distribution of the j-th transformed rod and its surrounding medium. The partial derivative term ∂pi/∂rj needs to be calculated for each design variable rj, and thus may results in intensive computational cost during the optimization procedure. In this work, the partial derivative term ∂pi/∂rj is calculated effectively by using the adjoint variable method [15

15. C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design.” Opt. Express 21(18), 21693–21701 (2013). [CrossRef] [PubMed]

18

18. J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22(6), 1191–1198 (2005). [CrossRef]

], where merely one forward problem and one additional adjoint problem need to be solved to obtain the partial derivative term ∂pi/∂rj for all the design variables. To ensure the symmetry of the optimized PhC devices, the partial derivative term ∂pi/∂rj of dielectric rods at symmetric positions are averaged in the optimization process.

At the beginning of the optimization process, we set all the 18 rods in each coupler with radius of r0 = 0.125a, which is the same as that of the bulk PhC. The transmission spectrum of the device with unoptimized PhC coupler is shown by the black dotted line in Fig. 5(a). Oscillations of the transmission spectrum of the unoptimized coupler indicates Fabry-Perot resonance caused by reflections at the unoptimized couplers, and the low power transmission coefficient (< 0.5) indicates an inefficient mode conversion by the unoptimized couplers. Figure 5(b) shows the profile of electric field Ez of the unoptimized PhC coupler, calculated at the wavelength of 1.55μm. We can see clearly that light is radiated from the two unoptimized PhC couplers.

Fig. 5 (a) Transmission spectra of the PhC coupler for a straight PhC waveguide. Spectrum of the unoptimized coupler is indicated by dotted line (black), spectrum of the coupler that optimized with obj1, obj2, and obj3 is indicated by dashed line (green), solid line (blue), and dash-dotted line (red) respectively. (b) Electric field Ez of the unoptimized PhC coupler, calculated at the wavelength of 1.55μm. (c) Electric field Ez of the coupler optimized with obj3.

For the optimization with obj1, obj2, obj3, the iteration step of optimization procedure is 71, 154, and 66, respectively. On a computer with Xeon E5645 CPU at 2.4GHz, the running time is 1016, 1302, 182 seconds, respectively. In this design example, at one sampling wavelength the computation time for the forward problem and the adjoint problem is approximately 2.3 and 0.5 seconds respectively. The optimized radii of the rods are shown in Fig. 4(b). Because of the symmetry of the couplers, only the radii of 9 rods of the coupler are listed in the table. The transmission spectra of the optimized coupler are shown in Fig. 5(a), where the spectra of couplers after optimization with obj1, obj2, and obj3 are indicated by the dashed, solid, and dash-dotted line respectively. Figure 5(a) shows, the PhC coupler optimized with obj1 has the broadest transmission band. For PhC coupler optimized with obj3, the power transmission coefficient at the central wavelength of 1.55μm (specifically 0.9877) is slightly higher than the couplers optimized with obj1 and obj2 (0.945 and 0.9876 respectively). As a verification, we choose the coupler that optimized with obj3, and simulate light propagation in the device. The obtained electric field Ez is depicted in Fig. 5(c), clearly showing the almost perfect light coupling.

4. Optimization of PhC coupler and PhC waveguide bend

Fig. 6 (a) Schematic diagram for PhC coupler and PhC waveguide bend. PhC couplers are indicated by dashed lines, and PhC waveguide bend is indicated by dotted line. (b) Rods radii of the optimised PhC coupler and PhC waveguide bend, which are optimized with sampling wavelengths obj1–obj3, as defined in Eq. (6).
Fig. 7 (a) Transmission spectra of the PhC coupler and PhC waveguide bend. Spectrum of the unoptimized device is indicated by dotted line, spectra of the device that optimized with obj1, obj2, and obj3 is indicated by dashed line, solid line, and dash-dotted line respectively. (b) Electric field Ez of PhC device optimized with obj3, calculated at the wavelength of 1.55μm. The rod of No. 12 and its counterpart at symmetric position are removed.

5. Perturbation analysis of the optimized device

Fig. 8 Magnitude of the normalized sensitivity of (a) the optimized PhC coupler in Section 3, (b) the optimized PhC coupler and waveguide bend in Section 4. Symbols “+”, “○”, and “×” represent sensitivity for PhC coupler and waveguide bend that are optimized with obj1, obj2, and obj3 respectively.

6. Conclusions

We use the transformation optics modeling technique and gradient-based optimization method to simulate and optimize PhC devices with varying-sized dielectric rods. By using the transformation optics approach, the original device model is transformed into an equivalent model that consists of uniform and fixed-sized rods, with parameterized permittivity and permeability distributions. Therefore, mesh refinement around small rods can be avoided, and PhC devices can be simulated more efficiently. In addition, the adjoint variable method is used to calculated the gradient of the optimization object, which is very efficient for optimizing devices with multiple design variables. The proposed method is expected to find important applications in designing and optimizing other photonic and optical devices with complex structures.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grand No. 51275504), and the Startup Fund for Postdoctoral Scientific Research Program of Jilin Province.

References and links

1.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2 (Princeton University, 2008).

2.

I. A. Sukhoivanov and I. V. Guryev, Photonic Crystals: Physics and Practical Modeling (Springer-Verlag, Berlin, 2009). [CrossRef]

3.

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal optical waveguides,” Phys. Rev. B 64, 033308 (2001). [CrossRef]

4.

P. Bienstman, S. Assefa, S. G. Johnson, J. D. Joannopoulos, G. S. Petrich, and L. A. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguides,” J. Opt. Soc. Am. B 20, 1817–1821 (2003). [CrossRef]

5.

K. Dossou, L. C. Botten, C. M. Sterke, R. C. McPhedran, A. A. Asatryan, S. Chen, and J. Brnovic, “Efficient couplers for photonic crystal waveguides,” Opt. Commun. 265, 207–219 (2006). [CrossRef]

6.

A. Mekis and J. D. Joannopoulos, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,” J. Lightw. Technol. , 19(6), 861–865 (2001). [CrossRef]

7.

O. Ozgun and M. Kuzuoglu, “Software metamaterials: Transformation media based multiscale techniques for computational electromagnetics”, J. Comput. Phys. 236, 203–219 (2013). [CrossRef]

8.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equation,” J. Mod. Opt. 43(4), 773–793 (1996). [CrossRef]

9.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

10.

Y. M. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012). [CrossRef] [PubMed]

11.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3:1217(2012). [PubMed]

12.

Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science & Applications 1, e38; (2012). [CrossRef]

13.

P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Marti, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett. 34(18), 2760–2762 (2009). [CrossRef] [PubMed]

14.

Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express 21(1), 1310–1316 (2013). [CrossRef] [PubMed]

15.

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design.” Opt. Express 21(18), 21693–21701 (2013). [CrossRef] [PubMed]

16.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible Adjoint Sensitivity Technique for EM Design Optimization,” IEEE Tans. Microwave Theory Tech. 50(12), 2751–2758 (2002). [CrossRef]

17.

G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. 29(19), 2288–2290 (2004). [CrossRef] [PubMed]

18.

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22(6), 1191–1198 (2005). [CrossRef]

19.

COMSOL Multiphysics, http://www.comsol.com/

20.

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic Press, 1974).

21.

S. G. Johnson and J. D. Joanopoulos, “Block-iterative frequency-domain methods for Maxwell’s equation in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]

22.

M. Schmidt, http://www.di.ens.fr/~Emschmidt/Software/minConf.html

23.

M. Minkov and V. Savona, “Effect of hole-shape irregularities on photonic crystal waveguides,” Opt. Lett. 37(15), 3108–3110 (2012). [CrossRef] [PubMed]

24.

V. Savona, “Electromagnetic modes of a disordered photonic crystal,” Phys. Rev. B 83, 085301 (2011). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(130.5296) Integrated optics : Photonic crystal waveguides

ToC Category:
Photonic Crystals

History
Original Manuscript: November 27, 2013
Revised Manuscript: January 6, 2014
Manuscript Accepted: January 9, 2014
Published: January 30, 2014

Citation
Yinghui Cao, Jun Xie, Yongmin Liu, and Zhenyu Liu, "Modeling and optimization of photonic crystal devices based on transformation optics method," Opt. Express 22, 2725-2734 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-2725


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References

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