Vladislav Shteeman PhD student of professor Amos Hardy

1. Vladislav Shteeman PhD student of professor Amos Hardy • Analysis and design of • two-dimensional photonic crystal devices

2. Photonic crystals: • the subject of study • existing methods of analysis Outline • Analysis of 2D photonic arrays with Coupled-mode Theory (CMT) : • finite photonic arrays • infinite photonic lattices • infinite photonic superlattices • infinite photonic crystals with defects • point defects • linear defects

3. Photonic crystals Photonic crystalsare materials that have a periodic variation in a refractive index on a length scale of the light wavelength. 2D photonic arrays The periodicity of the lattice prohibits light from propagating in certain directions or possessing certain frequencies and give rise a photonic band gap. Principle goal:light control & manipulation. Important applications:integrated optics (guides, couplers, filters, sharp bends etc.), low threshold photonic crystals lasers and photonic crystal fibers.

4. Routine methods of photonic crystals analysis Plane-wave expansion (PWE) (most popular and widespread) Finite-difference time domain (FDTD) Transfer matrix method (TMM)

5. Routine methods of photonic crystals analysis disadvantage advantage universality 3D 2D 1D • High requirements for CPU power and memory size. • Long computation time. • Numerical instabilities for large and / or complex structures. • Convergence problems with the PWE Arrays of any complexity and dimensionality could be analyzed. superlattice

6. Coupled Mode Theory (a brief review) There is a specific class of 2D photonic crystals, where guided light propagates in longitudal direction. This class is available for analysis with another method: Coupled Mode Theory. lightpropagation direction This class encompasses: arrays of coupled lasers arrays of parallel waveguides photonic crystal fibers (with index-guiding) arrays made by patter- ning the VCSEL mirror

7. Coupled Mode Theory (a brief review) • The vector formulation of CMT, presented here, is another representation of Maxwell equations, which is convenient for description of interaction and propagation of guided modes in arrays of coupled waveguides and lasers. z Physical model: a 2D array of parallel waveguides y x

8. Coupled Mode Theory (a brief review) • CMT assumes that guided modes of all solitary waveguides are known. • An array is considered as an assembly of solitary waveguides, interacting with each other. Therefore, the modes of the solitary waveguides experience perturbation due to the interaction with neighbor waveguides.

9. Coupled Mode Theory (a brief review) Array modes of a photonic crystal (partial amplitudes + propagation constants), should be found from the CMT eigenequation: partial amplitudes, U0 .of the array mode full array mode (U0 X Solitary WG mode) Analysis of propagation along z-axe To analyze a propagation of the optical signal along the z-axe of a photonic crystal one needs to solve a general CMT equation: Analysis of array modes

10. Coupled Mode Theory (a brief review) • BUT: • Large-sized photonic devices, accounting for a large number of waveguides or VCSELs (especially multimode), still unavailable for accurate analysis without a supercomputer resources. • The advantages of the CMT approach • over the other methods of photonic crystals analysis : • accurate analysis of 2D photonic devices of various degree of complexity • no divergence problem • time requirements are lower than in other methods

11. Principle ideas behind the solution for very large arrays • Employ Bloch theorem and translation symmetry for infinite structures • Develop CMT extension to the case of infinite photonic arrays (photonic crystals) • Assume that the photonic array is extended to infinity (transit to the photonic crystal)

12. Coupled Mode Theory (extension to infinite arrays) An ability of a fast and an accurate analysis of a wide range of photonic arrays: Main advantages of the extended CMT Stable convergence of computational process (both for arrays with real n2(x,y) and for arrays with gain and loss). Small requirements for computer resources.

13. Coupled Mode Theory (extension to infinite arrays) A finite 20 X 20 photonic crystal made of identical multimode waveguides (5 guided modes in each waveguide) CMT equation for a finite array:

14. Coupled Mode Theory (extension to infinite arrays) Assume now, that the same 20 X 20 photonic crystal is albeit-infinite and enable periodic boundary conditions: Bloch exponents due to the periodic boundary conditions u0 Λ Extended CMT equations: Time saving – 102 - 103 Periodic boundary conditions Translation symmetry V. Shteeman, D. Boiko, E. Kapon, A. A. Hardy. Extension of Coupled Mode analysis to periodic large arrays of identical waveguides for photonic crystals applications. IEEE JQE, 43 (4), pp. 215-224 (2007).

15. Coupled Mode Theory (extension to infinite arrays) • Band structure • along high symmetry lines • 2D Band structure 1.502 1.500 М neff= σ/(2π/λ0) ○ CMT for finite arrays (20 X 20 photonic crystal) – CMT extended to infinite arrays of identical WGs neff= σ/(2π/λ0) 1.490 Х Г 1.488 1.486 1.472 1.470 ky 1.468 kx М Г Х Г

16. Coupled Mode Theory (extension to infinite arrays) • Practical applications • of the extended CMT analysis: Analysis and design of optical properties (pass-bands, stop-bands, group velocity) of microstructured fibers (with index-guiding) and arrays of parallel waveguides (including arrays with gain and loss). Analysis and design of operation frequencies of arrays of coupled VCSELs.

17. Coupled Mode Theory (extension to infinite superlattices) Number of CMT equations = A total number of guided modes in the supercell Λ Bloch exponents due to the periodic boundary conditions Translation symmetry of SUPERLATTICE Time saving – ~103 Extended CMT equations: Periodic boundary conditions for SUPERLATTICE • V. Shteeman, I. Nusinsly, E. Kapon, A.A. Hardy. Extension of Coupled Mode analysis to infinite photonic superlattices. IEEE JQE, 44, No. 9, (2008).

18. Coupled Mode Theory (extension to infinite superlattices) • Band structure • along high symmetry lines • 2D Band structure 1.511 М 1.510 neff= σ/(2π/λ0) neff= σ/(2π/λ0) Х Г ○ CMT for finite arrays (45 X 45 photonic array) – CMT extended to infinite photonic superlattices 1.505 1.504 ky kx 1.500

19. Coupled Mode Theory (extension to infinite superlattices) • Practical applications • of the extended CMT analysis: Analysis and design of operation frequencies of arrays of coupled VCSELs and parallel waveguides, thresholdless lasers and patterned resonant cavities .

20. Coupled Mode Theory (analysis of infinite photonic arrays with point defects) • 2D band structure of bulk • Point defect states infinite array of identical WGs ( ( ) ) CMT extended to infinite arrays of identical WGs CMT extended to infinite photonic superlattices + supercell method ` ` infinite superlattice made of supercells with the same defect pattern ` Solve the extended CMT equations for only TWO pairs of {kx , ky} : min & max of the 1st Brillouin zone V. Shteeman, I. Nusinsly, E. Kapon, A.A. Hardy. Analysis of Photonic Crystals With Defects Using Coupled Mode Theory. Submitted to IEEE JQE.

21. Coupled Mode Theory (analysis of infinite photonic arrays with point defects) • Pass & stop bands • as a function of λ of incoming light + Helmholtz equation solution in finite differences ○ CMT for finite arrays (PCF containing 177 WGs ) – CMT extended to infinite arrays of identical WGs - - CMT extended to infinite photonic superlattices Gap states HE11 origin } 1st bandHE11 origin Gap states EH11 & HE31origin } Gap states TE01 & TM01 origin ` } 2nd – 3rd bandsTE01 &TM01origin } }

22. Coupled Mode Theory (analysis of infinite photonic arrays with point defects) • Pass & stop bands • as a function of λ of incoming light

23. Coupled Mode Theory (analysis of infinite photonic arrays with point defects) • Practical applications Defect modes engineering in micro- structured fibers with index-guiding (including fibers with gain and loss) and arrays of parallel waveguides. Analysis and design of arrays of coupled VCSELs with preprogrammed output frequencies, which are spatially separated in the desirable way.

24. Coupled Mode Theory (analysis of infinite photonic arrays with linear defects) • 2D band structure of bulk • 1D line defect curves ` infinite array of identical WGs ( ( ) ) CMT extended to infinite arrays of identical WGs CMT extended to infinite photonic superlattices + supercell method ` infinite 1D superlattice made of supercells with the same defect pattern ` ` V. Shteeman, I. Nusinsly, E. Kapon, A.A. Hardy. Analysis of Photonic Crystals With Defects Using Coupled Mode Theory. Submitted to IEEE Journal of Quantum Electronics.

25. Coupled Mode Theory (analysis of infinite photonic arrays with linear defects) Х' М Х Г • Band structure • along high symmetry lines (λ0= 0.9 μm) • strong light confinement inside the defect WGs (perfect 90º bent ) • group velocity ~ 4∙10-8c (slow light) 1D bands HE11origin ` 2D band HE11origin 1D bands HE11origin 1D bands TE01 and TM01origin ○ CMT for finite arrays (30 X 30 photonic crystal) – CMT extended to infinite arrays of identical WGs and infinite 1D photonic superlattices 1D bands TE01 and TM01origin 2D bands TE01 and TM01 origin

26. Coupled Mode Theory (analysis of infinite photonic arrays with linear defects) • Pass & stop bands structure • as a function of λ of incoming light ○ CMT for finite arrays (30 X 30 photonic crystal) – CMT extended to infinite arrays of identical WGs and infinite 1D photonic superlattices `

27. Coupled Mode Theory (continue of the research) z • Light propagation along z-direction Array mode – finite CMT For a finite photonic array holds a general form of CMT equation : y x 1 0 1 Array mode – Bloch function input light location To speed up the computational process

28. Thankyou for your attention.