Vladislav Shteeman PhD student of professor Amos Hardy

Vladislav ShteemanPhD student of professor Amos Hardy

Analysis and design oftwo-dimensional

photonic crystal devices

OutlinePhotonic crystals:o the subject of studyo existing methods of analysis

Analysis of 2D photonic arrays with Coupled-mode Theory (CMT) :o finite photonic arrayso infinite photonic latticeso infinite photonic superlatticeso infinite photonic crystals with defects

a. point defectsb. linear defects

Photonic crystalsPhotonic crystals are materials that have a periodic variation in a

refractive index on a length scale of the light wavelength.

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.

2Dphotonic

arrays

Plane-wave expansion (PWE) (most popular and widespread)

Finite-difference time domain (FDTD)

Transfer matrix method (TMM)

Routine methods of photonic crystals analysis

Arrays of any complexity and dimensionality

could be analyzed.

High requirements for CPU power and memory size.

Long computation time.

Numerical instabilities for large and / or complex structures.

Convergence problems with the PWE

advantage disadvantageuniversality

3D 2D 1D

superlattice

Routine methods of photonic crystals analysis

arrays of parallelwaveguides

photonic crystal fibers (with index-guiding)

arrays of coupled lasers

arrays made by patter-ning the VCSEL mirror

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.

This class encompasses: light

propagation direction

z

y

x

Physical model: a 2D array of parallel waveguides


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.


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.

partial amplitudes, U0 .of the array mode full array mode (U0 X Solitary WG mode)

Array modes of a photonic crystal (partial amplitudes + propagation constants), should be found from the CMT eigenequation:

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

00ˆˆ UIUM

Analysis of propagation along z-axe

zUzMidzzdU ˆ


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 : o accurate analysis of 2D photonic devices of various degree of complexity o no divergence problemo time requirements are lower than in other methods


Assume that the photonic array is extended to infinity

(transit to the photonic crystal)

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)

An ability of a fast and an accurate analysis of a wide range of photonic arrays:

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.

Main advantages of the extended CMT

Coupled Mode Theory(extension to infinite arrays)

A finite 20 X 20 photonic crystalmade of identical multimode waveguides (5

guided modes in each waveguide) CMT equation for a finite array:

00ˆˆ UIUM

20002000ˆ

200052020

M

equationscoupled


Extended CMT equations:

00ˆ,,ˆ UIkkUkkM yxyx

55,ˆ5

yx kkM

equationscoupled

Time saving – 102 - 103

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).

Λ

u0


Assume now, that the same 20 X 20 photonic crystal is albeit-infinite and

enable periodic boundary conditions:

xikeu0xikeu0

yikeu0 yx ikikeu0

yx ikikeu0

yikeu0 yx ikikeu0

yx ikikeu0

Periodic boundary conditions

Translation symmetry

Bloch exponents due to the periodic boundary conditions

yx kkofpairsarrayfor

,40020202020

2D Band structure Band structure along high symmetry lines

ГГ Х М

1.468

1.470

1.472

1.486

1.488

1.490

1.500

1.502

Х

Мyk

xkГ

n eff =

σ/(2

π/λ 0)

n eff =

σ/(2

π/λ 0)

ky

○ CMT for finite arrays (20 X 20 photonic crystal)– CMT extended to infinite arrays of identical WGs


kx

Method Type of calculation

CMT for finite

arrays

CMT extended to

infinite arrays

OPW ( 32 X 32 plane waves per

unit cell)Full set of array modes (20X20 array of multimode WGs, each WG supports for 5 guided modes)

1 hour < 10-3 sec.

15 hoursBand structure (20X20 array of multimode WGs, each WG supports for 5 guided modes)

5 hours 1 min.

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.

V. Shteeman, I. Nusinsly, E. Kapon, A.A. Hardy. Extension of Coupled Mode analysis to infinite photonic superlattices. IEEE JQE, 44, No. 9, (2008).

Λ xike xike

yike yx ikike yx ikike

yike yx ikike yx ikike

Number of CMT equations=

A total number of guided modes in the supercell

Coupled Mode Theory(extension to infinite superlattices)

Time saving – ~103

Periodic boundary conditions for SUPERLATTICE

Translation symmetry of SUPERLATTICE

Extended CMT equations:

00ˆ,,ˆ UIkkUkkM yxyx

Bloch exponents due to the periodic boundary conditions

2D Band structure Band structure along high symmetry lines

n eff =

σ/(2

π/λ 0)

Х

Мyk

xkГ

1.500

1.504

1.505

1.510

1.511

n eff =

σ/(2

π/λ 0)

○ CMT for finite arrays (45 X 45 photonic array)– CMT extended to infinite photonic superlattices


kykx

Method Type of calculation

CMT for finite arrays

CMT extended to

infinite arrays

OPW ( 81 X 81 plane waves per

supercell)Full set of eigenmodes (45 X 45 array of single mode waveguides)

35 min. < 10-3 sec.60 hoursBand structure (45X45

array of single mode waveguides )

2 hours 2 min.


Analysis and design of operation frequencies of arrays of coupled VCSELs and parallel waveguides, thresholdless lasers and patterned resonant cavities .

Practical applications of the extended CMT analysis:

`

V. Shteeman, I. Nusinsly, E. Kapon, A.A. Hardy. Analysis of Photonic Crystals With Defects Using Coupled Mode Theory. Submitted to IEEE JQE.

infinite array of identical WGs

`

`CMT extended to infinite arrays of identical WGs( )

infinite superlattice made of supercells with the same defect pattern

CMT extended to infinite photonic superlattices + supercell method( )

2D band structure of bulk

Point defect states

Coupled Mode Theory(analysis of infinite photonic arrays with point defects)

Solve the extended CMT equations for only TWO pairs of {kx , ky} : min & max of the 1st Brillouin zone

Pass & stop bandsas a function of λ of incoming light

Gap states HE11 origin

Gap states TE01 & TM01 origin

1st band HE11 origin

Gap states EH11

& HE31 origin

2nd – 3rd bands TE01

&TM01 origin

+ 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 state array mode

Mode index (Helmholtz) for

CMT infinite Relative error

CMT finite Relative error

Perturbation Theory

Relative error mode 1 1.51450 1E-4 1E-4 5E-2mode 2 1.51322 1E-4 1E-4 5E-2mode 3 1.51305 1E-4 1E-4 5E-2mode 4 1.51223 1E-4 1E-4 5E-2

`

Pass & stop bandsas a function of λ of incoming light


effn m 85.00

Gap state array mode

Mode index (Helmholtz) for

CMT infinite Relative error

CMT finite Relative error

Perturbation Theory

Relative error mode 1 1.51450 1E-4 1E-4 5E-2mode 2 1.51322 1E-4 1E-4 5E-2mode 3 1.51305 1E-4 1E-4 5E-2mode 4 1.51223 1E-4 1E-4 5E-2

Practical applications


Defect modes engineering in microstructured 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.

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.

`

`

`

infinite array of identical WGsCMT extended to infinite arrays of identical WGs( )

infinite 1D superlattice made of supercells with the same defect pattern

CMT extended to infinite photonic superlattices + supercell method( )

2D band structure of bulk

1D line defect curves

`

Coupled Mode Theory(analysis of infinite photonic arrays with linear defects)

`

1D bands HE11 origin

1D bands HE11 origin

2D band HE11 origin

1D bands TE01 and TM01 origin

1D bands TE01 and TM01 origin 2D bands TE01 and

TM01 origin

x

y

y

Х

М

Г

yk

xk

x

Х'

Band structure along high symmetry lines (λ0= 0.9 μm)

○ CMT for finite arrays (30 X 30 photonic crystal)– CMT extended to infinite arrays of identical WGs and infinite 1D photonic superlattices


group velocity ~ 4∙10-8c (slow light)

strong light confinement inside

the defect WGs (perfect 90º bent )

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


Light propagation along z-direction

z

y

xinput light

location

zUzMidzzdU ˆ

101

For a finite photonic array holds a general form of CMT equation :

Coupled Mode Theory(continue of the research)

Array mode – Bloch function

Array mode – finite CMT

To speed up the computational process

Thank you for your attention.

Vladislav Shteeman PhD student of professor Amos Hardy

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