OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 22 — Nov. 3, 2003
  • pp: 2905–2910
« Show journal navigation

Reformulation of the plane wave method to model photonic crystals

Rossella Zoli, Marco Gnan, Davide Castaldini, Gaetano Bellanca, and Paolo Bassi  »View Author Affiliations


Optics Express, Vol. 11, Issue 22, pp. 2905-2910 (2003)
http://dx.doi.org/10.1364/OE.11.002905


View Full Text Article

Acrobat PDF (380 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A new formulation of the plane-wave method to study the characteristics (dispersion curves and field patterns) of photonic crystal structures is proposed. The expression of the dielectric constant is written using the superposition of two regular lattices, the former for the perfect structure and the latter for the defects. This turns out to be simpler to implement than the classical one, based on the supercell concept. Results on mode coupling effects in two-dimensional photonic crystal waveguides are studied and successfully compared with those provided by a Finite Difference Time Domain method. In particular the approach is shown able to determine the existence of “mini-stop bands” and the field patterns of the various interfering modes.

© 2003 Optical Society of America

1. Introduction

Photonic Crystals (PC’s) are periodic arrangements of dielectric or metallic materials. In these last years, dielectric PC’s have been intensively studied to realize optical devices, such as filters, resonators and waveguides, exploiting their ability to control light propagation [1

1. K. Sakoda, Optical properties of Photonic Crystals, Springer (2001).

2

2. S. G. Johnson and J. D. Joannopoulos, (Photonic crystals The Road from Theory to Practice, Kluwer Academic Publishers, 2001).

]. It is well known that the unperturbed PC’s can exhibit forbidden bands, i.e. frequency ranges where no electromagnetic (e.m.) field is allowed. A single lattice defect can support a localized e.m. field within the frequency band gap of the unperturbed PC and a line of single defects can guide an optical signal along its direction creating frequency-selective waveguides. The importance of fast and accurate software tools for PC’s analysis and design is then evident. Among the different techniques, the plane-wave method (PWM) [1

1. K. Sakoda, Optical properties of Photonic Crystals, Springer (2001).

5

5. S. Guo and S. Albin, “A simple plane wave implementation method for photonic crystal calculations,” Opt. Express 11, 167 (2003). [CrossRef] [PubMed]

] and the Finite Difference Time Domain (FDTD) approach [6

6. A. Taflove, Computational electrodynamics - The Finite Difference Time-Domain Method, (Artech House, Norwood, MA, 1995).

8

8. M. Kafesaki, M. Agio, and C. M. Soukoulis, “Waveguides in finite-height two-dimensional photonic crystals,” J. Opt. Soc. Am. B 19, 2232 (2002). [CrossRef]

] are probably the most used ones.

The PWM, based on the Fourier expansion of the periodic dielectric constant, allows straightforward analysis of perfect PC’s. When defects are present, the PW approach can still be applied, but after the introduction of the so called supercell [1

1. K. Sakoda, Optical properties of Photonic Crystals, Springer (2001).

4

4. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavity in photonic crystals: Mode symmetry, tunability and coupling efficiency,” Phy. Rev. B 54, 7837–7842 (1996). [CrossRef]

], a domain containing the defect and a part of the regular structure large enough to guarantee that the field vanishes at its boundary. It is then possible again to apply the PW expansion to a new structure made by periodically arranged supercells. In this case the complexity of the algorithm increases, in particular in the expression of the dielectric function, even if one can take advantage of the Fast Fourier Transform (FFT) [2

2. S. G. Johnson and J. D. Joannopoulos, (Photonic crystals The Road from Theory to Practice, Kluwer Academic Publishers, 2001).

] and of the shift property of the Fourier transform [5

5. S. Guo and S. Albin, “A simple plane wave implementation method for photonic crystal calculations,” Opt. Express 11, 167 (2003). [CrossRef] [PubMed]

] to reduce CPU times. Particular care must also be used to choose the size of the supercell and the finite number of plane waves approximating the infinite sum as a compromise guaranteeing both convergence to correct results (large size and number of terms desired) and CPU time reduction (small size and few terms desired).

In this paper we present (Section 2) a new approach to the PWM, which reformulates the expression of the dielectric constant through the superposition of two regular lattice, the former for the perfect structure and the latter for the defects. This method leads to a single expression of the dielectric constant, which can be inserted in the eigenvalue and eigenmodes calculation routine. In Section 3, the method will be used to study a waveguide (W1) formed by single line defects in a two-dimensional triangular lattice of air holes in a dielectric substrate and will be shown to be able to accurately determine dispersion curves and field patterns in presence of “mini-stop bands”, evidencing also coupling phenomena [9

9. M. Agio and C. M. Soukoulis, “Ministop bands in single defect photonic crystal waveguides,” Phys. Rev. E 64, 055603 (2001). [CrossRef]

13

13. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phy. Rev. B 64, 155113 (2001). [CrossRef]

] among fundamental and higher order modes (the so called curve “anticrossing”). These results will be compared also with those obtained by a 2D-FDTD method, to prove the correctness of our approach. Finally (Section 4), conclusions will be drawn.

2. Reformulation of the plane wave method

The method will be described, for simplicity, considering a 2D PC with square or triangular lattice, but it can easily be extended to any lattice shape and to 3D structures. We consider first the ideal 2D PC made by cylinders with square or triangular lattice, that we call the “perfect lattice”. The rods have their axis parallel to the z direction, lattice constant a, radius R and dielectric constant εa. The dielectric constant of the background material is εb. The PC is characterized by the lattice vectors {a1,a2} and the reciprocal lattice vectors {b1,b2} [1

1. K. Sakoda, Optical properties of Photonic Crystals, Springer (2001).

,2

2. S. G. Johnson and J. D. Joannopoulos, (Photonic crystals The Road from Theory to Practice, Kluwer Academic Publishers, 2001).

].

The relative dielectric constant of the structure without defects (denoted by subscript g), expanded in Fourier series in the reciprocal lattice, is

εg(r)=Gεg(G)ejG·r
(1)

being r is the spatial vector position and G=h1 b1+h2 b2, with h1,h2 arbitrary integer numbers. The coefficient εg(G) is defined as

εg(G)=εafg+εb(1fg)
(2)

for G=0, while, for G0, it holds:

εg(G)=(εaεb)fg2J1(G·R)G·R
(3)

where fg=πR2a1×a2 is the filling factor of the perfect structure and J 1 is the first order Bessel function.

Once the perfect lattice has been defined, we consider the presence of defects, corresponding, for example, to removal of rods. We then consider a new lattice formed by the defects repeated with proper periodicity along the two lattice directions, if single defects are present, or along one, if straight waveguides are considered. We call this new structure “defect perfect lattice”. The defect dielectric constant is εd=εb-εa while the background material has a dielectric constant equal to 0. The defect perfect lattice can be characterized by the vectors c1=n1 a1 and c2=n2 a2, with n1, n2 arbitrary integer numbers. The reciprocal lattice vectors are d1=b1/n1, d2=b2/n2. The generic vector in the reciprocal lattice is now defined as S=l1d1+l2d2, where l1,l2 are arbitrary integer numbers. The relative dielectric constant of the defect perfect lattice, denoted by subscript s, can then be expressed as

εs(r)=Sεs(S)ejS·r
(4)

where the coefficient εs (S) is then

εs(S)=εdfs
(5)

for S=0, while, for S0, it holds

εs(S)=εdfs2J1(S·R)S·R
(6)

being fs=πR2c1×c2=fgn1n2 the filling factor of the defects.

The PC with defects is then formed by the superimposed perfect lattice and defect perfect lattice. For example, considering a W1 waveguide, Fig. 1(a) shows the direct lattice, with defects painted in green, built with n1=1 and n2=3. Figure 1(b) reports the superposition of the two reciprocal lattices, the former (larger gray circles) referring to the perfect lattice, the latter (smaller green dots) referring to the defect perfect lattice. Note that the nodes of the perfect reciprocal lattice are a subset of the defect perfect reciprocal lattice.

The dielectric function formed by the perfect lattice and the defect perfect lattice, results adding the dielectric constant distributions (1) and (4) obtaining:

ε(r)=εg(r)+εs(r)=Sεc(S)ejS·r
(7)

where εc(S) is the coefficient that describes the superposition of two reciprocal lattice. For nodes belonging to both reciprocal lattices, εc(S) can be written

εc(S)=fs(εaεb)(n1n21)+εb
(8)

for S=0, while, for S0

εc(S)=fs(εaεb)(n1n2)·2J1(S·R)S·R
(9)

where l1=m n1 and l2=n n2, with m, n arbitrary integers different from 0.

Fig. 1. Modeling of 2D triangular lattice PC waveguide W1: (a) direct lattice of the W1 and (b) reciprocal lattices of the perfect lattice (larger gray circles) and the defect perfect lattice (smaller green dots).

Otherwise, for nodes corresponding only to defects:

εc(S)=fs(εbεa)·2J1(S·R)S·R.
(10)

The structure usually obtained repeating the supercell is now built superposing two infinite periodic structures. This allows to calculate the dielectric constant distribution of the studied structure with a much simpler routine. Care must always be taken to choose the proper distance between the defects to avoid coupling phenomena, but this simply reduces to the best choice of n1 and n2, without the need to repeat the supercell construction procedure.

For the structure formed by the perfect lattice and the defect perfect lattice, the electromagnetic problem to be solved to determine eigenvalues and eigenmodes is the classical one [1

1. K. Sakoda, Optical properties of Photonic Crystals, Springer (2001).

,2

2. S. G. Johnson and J. D. Joannopoulos, (Photonic crystals The Road from Theory to Practice, Kluwer Academic Publishers, 2001).

] with the only formal difference that S is used instead of G:

Sε1(SS)(k+S)·(k+S)Hz(S)=ω2c2Hz(S)
(11)
Sε1(SS)k+S2Ez(S)=ω2c2Ez(S)
(12)

where ω is the eigen-angular frequency and k the wave vector while the eigenfunctions of the electric and magnetic fields can be expanded in Fourier series [1

1. K. Sakoda, Optical properties of Photonic Crystals, Springer (2001).

,2

2. S. G. Johnson and J. D. Joannopoulos, (Photonic crystals The Road from Theory to Practice, Kluwer Academic Publishers, 2001).

]:

E(r)=SE(S)ej(k+S)·r,H(r)=SH(S)ej(k+S)·r.
(13)

Note that the differences between this approach and the classical one, based on the supercell concept, refer only to the way to obtain the final Eqs. (1112

12. C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdrè, and U. Osterle “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001). [CrossRef]

), which is now simpler and faster to run, and not to the results, which are always the same. We have tested the algorithm for perfect structures, single defect resonant cavities and waveguides and it proved stable and rapidly converging.

3. Results and comparison between different methods

As an example of application of the method, we consider a two-dimensional single line photonic crystal waveguide (W1) of air circular holes in a dielectric substrate, with relative dielectric constant equal to 11.56. The crystal has a triangular lattice and a filling factor of 83%. This structure has a photonic band gap both for TE and TM polarized fields. In the following we will restrict our attention to TM polarized fields (E parallel to the rod axes) propagating along the k direction (in the x-y plane). The method is applied considering waveguides repeated every 6 lattice periods a and 1089 plane waves. In this case, few minutes of CPU time are enough to run the code on a 1800 MHz Pentium IV Computer.

Fig. 2. (a) Dispersion curves computed by the PW method and (b) transmission coefficient computed by FDTD of a W1 waveguide. (c) Local picture of the “mini-stop band” associated to the coupling among the fundamental mode and higher order modes. (d)-(e)-(f) Electric displacement pattern associated to the fundamental mode. (g)-(h) Electric displacement pattern associated to the high order modes.

The waveguide dispersion curves obtained with our method are shown in Fig. 2(a). The dispersion curves are calculated in the first Brillouin zone, in the y direction of Fig. 1, varying k between 0 and π/a. The gray areas are the projected band structure of the perfect crystal that limit the band gap between 0.46 a/λ and 0.54 a/λ. The solid lines are the dispersion curves of the waveguide modes. The red line, marked by index 1, refers to the fundamental mode.

Figure 2(a) shows that the dispersion curve of the fundamental mode crosses the dispersion curves of other modes. In particular, around 0.425 a/λ and 0.49 a/λ, the so called “anticrossing” phenomenon occurs, involving higher order modes. This generates two “mini-stop bands” [9

9. M. Agio and C. M. Soukoulis, “Ministop bands in single defect photonic crystal waveguides,” Phys. Rev. E 64, 055603 (2001). [CrossRef]

13

13. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phy. Rev. B 64, 155113 (2001). [CrossRef]

], evidenced by the two circles, always in Fig. 2(a). Considering, in particular, the “mini-stop band” around 0.49 a/λ, Fig. 2(c) shows a close-up of the mini gap caused by the coupling between the fundamental mode and two higher order modes which are represented by blue and green lines, with index 2 and 3, respectively. Mode 3 is odd, while modes 1 and 2 are even. The field patterns of the waveguide modes corresponding to the five different points, marked in Fig. 2(c), are shown in the Figs. 2(d), 2(e), 2(f), 2(g), 2(h). In particular, Figs. 2(d), 2(e), 2(f) refer to the electric displacement of the fundamental mode. In the intermediate region (field of Fig. 2(e)), coupling of modes 1, 2 and 3 changes the field pattern, which results in a kind of zig-zag path caused by the interference of the odd and even modes. The even pattern of mode 2 is shown in Fig. 2(g), while the odd pattern of mode 3 is shown in Fig. 2(h).

Fig. 3. (1.7 MB) Movies of Electric field at (a) 0.47 a/λ and (b) 0.49 a/λ. [Media 2]

The presence of these two “mini-stop bands” has been confirmed using a 2D-FDTD code, based on the Yee’s algorithm with total field-scattered field approach [6

6. A. Taflove, Computational electrodynamics - The Finite Difference Time-Domain Method, (Artech House, Norwood, MA, 1995).

8

8. M. Kafesaki, M. Agio, and C. M. Soukoulis, “Waveguides in finite-height two-dimensional photonic crystals,” J. Opt. Soc. Am. B 19, 2232 (2002). [CrossRef]

]. Figure 2(b) reports the calculated transmission characteristic of the structure. The whole FDTD computational domain is divided into 372×623 cells with step of Δx=Δy=0.005µm and 8 GT-PML layers. Agreement between the two set of results is very good. Finally Fig. 3(a) and 3(b) show how the 2D-FDTD accounts for propagation of the electric field in the structure at 0.47 a/λ and 0.49 a/λ, which are respectively in a allowed and in a “mini-stop band”: the attenuation of the field in the latter case is evident.

4. Conclusions

In this paper we have presented a new approach of the plane-wave method for photonic crystal analysis. This procedure expresses the dielectric constant distribution as the superposition of two functions, one for the perfect lattice and one for the defect perfect lattice. This leads to a simpler expression of the dielectric constant to be inserted in the standard eigenvalue problem, thus reducing the code overall complexity. The approach is easy to implement and fast to run. As an example, a 2D triangular lattice photonic crystal waveguide was studied, determining its dispersion curves and the guided field distributions, allowing also a detailed study of mode coupling effects and “mini-stop bands” presence. These results have been successfully validated by comparison with those obtained using a FDTD approach.

Acknowledgments

Part of this work has been funded by MIUR (Italian Ministry of Education, University and Research).

References and links

1.

K. Sakoda, Optical properties of Photonic Crystals, Springer (2001).

2.

S. G. Johnson and J. D. Joannopoulos, (Photonic crystals The Road from Theory to Practice, Kluwer Academic Publishers, 2001).

3.

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band gap boundaries,” J. Appl. Phys. 79, 7483–7492 (1996). [CrossRef]

4.

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavity in photonic crystals: Mode symmetry, tunability and coupling efficiency,” Phy. Rev. B 54, 7837–7842 (1996). [CrossRef]

5.

S. Guo and S. Albin, “A simple plane wave implementation method for photonic crystal calculations,” Opt. Express 11, 167 (2003). [CrossRef] [PubMed]

6.

A. Taflove, Computational electrodynamics - The Finite Difference Time-Domain Method, (Artech House, Norwood, MA, 1995).

7.

F. Fogli, J. Pagazaurtundua Alberte, G. Bellanca, and P. Bassi, “Analysis of Finite 2-D Photonic Bandgap Lightwave Devices using the FD-TD Method,” Proc. of IEEE-WFOPC 2000, Pavia, June 8–9, 236–241 (2000).

8.

M. Kafesaki, M. Agio, and C. M. Soukoulis, “Waveguides in finite-height two-dimensional photonic crystals,” J. Opt. Soc. Am. B 19, 2232 (2002). [CrossRef]

9.

M. Agio and C. M. Soukoulis, “Ministop bands in single defect photonic crystal waveguides,” Phys. Rev. E 64, 055603 (2001). [CrossRef]

10.

S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, and R. Houdrè, “Coupled-mode theory and propagation losses in photonic crystal waveguides,” Opt. Express 11, 1490–1496 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1490. [CrossRef] [PubMed]

11.

S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Osterle, and R. Houdrè, “Mini-stopbands of one-dimensional system: The channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B. 63, 11311 (2001). [CrossRef]

12.

C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdrè, and U. Osterle “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001). [CrossRef]

13.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phy. Rev. B 64, 155113 (2001). [CrossRef]

OCIS Codes
(130.0130) Integrated optics : Integrated optics
(230.7380) Optical devices : Waveguides, channeled

ToC Category:
Research Papers

History
Original Manuscript: September 15, 2003
Revised Manuscript: October 24, 2003
Published: November 3, 2003

Citation
Rossella Zoli, Marco Gnan, Davide Castaldini, Gaetano Bellanca, and Paolo Bassi, "Reformulation of the plane wave method to model photonic crystals," Opt. Express 11, 2905-2910 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-22-2905


Sort:  Journal  |  Reset  

References

  1. K. Sakoda, Optical properties of Photonic Crystals, Springer (2001).
  2. S. G. Johnson, J. D. Joannopoulos, Photonic crystals The Road from Theory to Practice, Kluwer Academic Publishers, (2001).
  3. H. Benisty, �??Modal analysis of optical guides with two-dimensional photonic band gap boundaries,�?? J. Appl. Phys. 79, 7483-7492 (1996). [CrossRef]
  4. P. R. Villeneuve, S. Fan, J. D. Joannopoulos, �??Microcavity in photonic crystals: Mode symmetry, tenability and coupling efficiency,�?? Phy. Rev. B 54, 7837-7842 (1996). [CrossRef]
  5. S. Guo, S. Albin, �??A simple plane wave implementation method for photonic crystal calculations,�?? Opt. Express 11, 167 (2003). [CrossRef] [PubMed]
  6. A.Taflove, Computational electrodynamics �?? The Finite Difference Time-Domain Method, (Artech House, Norwood, MA, 1995).
  7. F. Fogli, J. Pagazaurtundua Alberte, G. Bellanca, P. Bassi, �??Analysis of Finite 2-D Photonic Bandgap Lightwave Devices using the FD-TD Method,�?? Proc. of IEEE-WFOPC 2000, Pavia, June 8-9, 236-241 (2000).
  8. M. Kafesaki, M. Agio, C. M. Soukoulis, �??Waveguides in finite-height two-dimensional photonic crystals,�?? J. Opt. Soc. Am. B 19, 2232 (2002). [CrossRef]
  9. M. Agio, C. M. Soukoulis, �??Ministop bands in single defect photonic crystal waveguides,�?? Phys. Rev. E 64, 055603 (2001). [CrossRef]
  10. S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F . Krauss, R. Houdrè, �??Coupled-mode theory and propagation losses in photonic crystal waveguides,�?? Opt. Express 11, 1490-1496 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1490.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1490.</a> [CrossRef] [PubMed]
  11. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Osterle, R. Houdrè, �??Mini-stopbands of one-dimensional system: The channel waveguide in a two-dimensional photonic crystal,�?? Phys. Rev. B. 63, 11311 (2001). [CrossRef]
  12. C. J. M. Smith, R. M. De La Rue, M. Rattier ,S. Olivier, H. Benisty, C. Weisbuch, T. F . Krauss, R. Houdrè, U. Osterle �??Coupled guide and cavity in a two-dimensional photonic crystal,�?? Appl. Phys. Lett. 78, 1487-1489 (2001). [CrossRef]
  13. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, B. Jaskorzynska, �??Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,�?? Phy. Rev. B 64, 155113 (2001). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 

Supplementary Material


» Media 1: AVI (1743 KB)     
» Media 2: AVI (1718 KB)     

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited