Floquet’sUnitCellDesignforPeriodicStructuresat...

Hindawi Publishing CorporationInternational Journal of Microwave Science and TechnologyVolume 2009, Article ID 160321, 10 pagesdoi:10.1155/2009/160321

Research Article

Floquet’s Unit Cell Design for Periodic Structures atOptical Frequencies

Alessandro Massaro,1 Roberto Cingolani,1 Adriana Passaseo,1, 2 and Massimo De Vittorio1

1 National Nanotechnology Laboratory of CNR-INFM, Universita del Salento, Distretto Tecnologico-ISUFI,Via Arnesano, 73100 Lecce, Italy

2 IMM-CNR Sezione Lecce, University Campus, Lecce-Monteroni, 73100 Lecce, Italy

Correspondence should be addressed to Alessandro Massaro, [email protected]

Received 25 March 2009; Accepted 7 July 2009

Recommended by Kamya Yekeh Yazdandoost

We present a new theoretical approach regarding the design of 2D periodic structure at optical frequencies. The model is basedon Floquet’s theory and on the variational equivalent circuit. The distributed circuit model is developed through the use ofthe microwave network theory and the optical theory of the step discontinuities. This approach analyzes 2D dielectric periodicstructures with high dielectric contrast by the transmission line model including variational equivalent circuits. The 3D FiniteElement Method (FEM) model validates Floquet’s design theory of the grating resonance and provides the design optimization ofan optical GaAs periodic waveguide.

Copyright © 2009 Alessandro Massaro et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. Introduction

Various scientific and engineering applications [1–11] can berealized by utilizing dielectric periodic devices. Technologicalimprovements for high frequency periodic structures aredeveloped in order to fabricate photonic crystal structuresfor power dividers, filters [4–9], directional couplers, andwavelength division multiplexer [10, 11]. New classes ofdielectric materials which are often artificially fabricatedfor this purpose are referred as photonic crystals and oftenare characterized by high dielectric contrast. The effect ofthe high dielectric contrast, at high working frequencies, isrelevant in the frequency response especially for discontin-uous structures such as cascade of step discontinuities. Stepdiscontinuities in planar dielectric waveguides are commonlyused in integrated circuits ranging from submillimetre tooptical frequencies. The step discontinuities in dielectricslab waveguides are, in fact, tailored for several componentssuch as distributed feedback lasers, gratings, transformers,antenna feed, and others. It is thus important to haveaccurate and reliable theoretical prediction of the behaviorof this discontinuity. Step discontinuity model, presentedin this work, starts from the junction between two slabsof different heights as two lines with different impedances.

As a refinement of this model we introduce the variationalequivalent circuit [12, 13] which minimizes the effect ofthe discontinuity for a periodic structure. By modelinga periodic dielectric structure as a cascade of unit cellelements [12–17] (see Figures 1 and 2(a)) the error ofthe frequency response of a single cell increases drasticallyfor a long structure. Periodic structures with low dielectriccontrast can be analyzed by the simplified transmissionline models [16] reported in Figure 2(b), but, for highdielectric contrasts, an accurate modeling is requested. Inthis way it is necessary to introduce the transformationratio that takes into account the amplitude difference ofthe traveling signals along the discontinuous waveguide. Thetransformation ratio, obtained through the modal analysisof the optical waveguide, allows to analyze high dielectriccontrast waveguides such as air holes in GaAs material. Thisaspect is considered in the proposed theoretical approachwith the equivalent variational circuit of a step discontinuity[12, 13]. The frequency behavior of periodic structures atoptical frequencies was analyzed in previous works [18, 19]by impedance matching technique. In the presented workan alternative and rigorous technique is combined with theBrillouin theory. This approach provides a useful new toolfor photonic crystal design.

2 International Journal of Microwave Science and Technology

In particular in the presented work the following aspectsare investigated: (i) design of 2D periodic waveguides bydistributed transmission line model at optical frequencies,including Floquet’s theorem [15–17], Brillouin diagram[15], and equivalent variational circuit of the propagatingmodes [12]; (ii) analytical example of a periodic GaAs slabwaveguide at working frequency λ0 = 1.55μm; (iii) 3D finiteelement method (FEM) simulation of a 3D GaAs periodicstructure by analyzing the effect of the 3D cell shape onthe frequency response with respect to the 2D theoreticalapproach.

2. Floquet’s Theory

The periodic structure of Figure 1(a) is analyzed by con-sidering the study of the resonance in the propagatingdirections. The theoretical model considers the propagationcharacteristic of the infinite loaded line as cascade of unitcells shown in Figure 1(b). Each unit cell of this line consistsof transmission lines with length L1 and L2 which representthe planar dielectric waveguide with high and low refractiveindex, respectively, and of two ideal transformers whichmodels the step discontinuities at optical frequencies. Thewave-amplitude transmission matrix (ABCD) is used for theunit cell of Figure 1, where

⎛⎝V1

V2

⎞⎠ =

⎛⎝A B

C D

⎞⎠u

·⎛⎝I1I2

⎞⎠, (1)

where u indicates the unit cell transmission matrix whichincludes the step discontinuity, and is expressed as

⎛⎝A B

C D

⎞⎠u

=⎛⎜⎝

cos(θ′) iZ′0 sin(θ′)

i sin(θ′)Z′0

cos(θ′)

⎞⎟⎠ ·

⎛⎜⎝r 0

01r

⎞⎟⎠

·⎛⎜⎝

cos(θ) iZ0 sin(θ)

i sin(θ)Z0

cos(θ)

⎞⎟⎠ ·

⎛⎜⎝

1r

0

0 r

⎞⎟⎠

·⎛⎜⎝

cos(θ′) iZ′0 sin(θ′)

i sin(θ′)Z′0

cos(θ′)

⎞⎟⎠,

(2)

where r is the transformation ratio. If the periodic structureis capable of supporting a propagating wave, it is necessaryfor the voltage and current at the (m+ 1)th cell terminal [12–15] to be equal to the voltage and current at themth terminal,apart from a phase delay due to a finite propagation time.Hence, let us assume that

Vm+1 = Vme−γl, Im+1 = Ime

−γl, (3)

where γ = α + iβz is the propagation constant along the z-propagating direction of the periodic structure, and l = Λ =2L1 + L2 is the unit-cell length. For a Bloch wave [12], theeigenvalue equation is given by

AD + e2γl − (A +D)eγl − BC = 0. (4)

zn1

L1 Λ = 2L1 + L2

n2

L2

I2I1 r : 1 1 : r

z

Air holes Unit cell

(a)

(b)

V1 V2θ′, Z′0 θ′, Z′0θ, Z0

Figure 1: (a) Periodic structure with air holes in dielectric material;(b) equivalent 1D transmission line of the unit cell: Z0 and Z′0 arethe characteristic impedances and θ′ = β′zL1 and θ = βzL2 are theelectrical lengths.

Being the determinant of the unit cell transmission matrixequal to 1, then (4) becomes

cosh(γl) = A +D

2. (5)

In the hypothesis of propagation along one direction (βz /= 0)without losses (α = 0), then (5) represents the followingFloquet’s condition:

A +D

2≤ 1, (6)

which indicates the passband region of the periodic struc-ture. For a cascade of unit cells the total ABCD matrix refersto the product of all the unit cell transmission matrices.

3. Brillouin Diagrams and 2D Model

When studying the passband and the stopband characteristicof a periodic structure, it is useful to plot the propagationconstant βz versus the propagation constant n2k0 of theunloaded line (uniform planar waveguide), where k0 is thefree space wavelength. Such a graph is called Brillouin [15]diagram. The diagram plots the dispersion relation for theplanar waveguide mode

βz =√

(n2k0)2 − k2c (7)

or

n2k0 =√β2z + k2

c , (8)

where n2 is the refractive index of the core planar waveguide,and kc is the cutoff wavenumber of the propagating mode.The Brillouin diagram is also useful in order to interpret thevarious wave velocities associated with a dispersive structure.In particular it is possible to evaluate the phase velocity

vp = ω

βz(9)

International Journal of Microwave Science and Technology 3

zd

D

Air hole

x

Region I Region II

n1

n2

n3

(a)

Z1 Z2

β2zβ1z

(b)

Y2 = 1/ Z2Y1 = 1/ Z1Y0

Yin

1 : r

(c)

Double stepdiscontinuity

r : 1

r : 1

1 : r

(d)

Figure 2: (a) Step discontinuity: junction between two dielectric slabs with core thicknesses d and D (longitudinal section of the unit cellshown in Figure 1(a)). (b) Simplified equivalent circuit for small step discontinuity and low dielectric contrast Δn = n2 − n1. (c) Variationalequivalent circuit for a step discontinuity. (d) Example of variational equivalent circuit of a double step discontinuity with one guided modein region I and two guided modes in region II.

Propagation

Operating points

Cutoff

n 2k 0

n2k0 = βz

kc1

kc2

β2z

βz (m−1)

β1z

14

12

10

8

6

6

4

4

2

20

0

× 106

14 12108× 106

Slope = vg/c

Slope = vp/c

Figure 3: Brillouin diagram for a GaAs (n2 = 3.408) junction ona substrate with n3 = 3.1. The operating points refer to a workingwavelength λ0 = 1.55μm; β1z and β2z are the effective propagationconstants along the z-propagating direction for the fundamental TEmodes in regions I and II of Figure 2(a), respectively; kc1 and kc2 arethe cutoff wavenumbers.

and the group velocity

vg = dω

dβz. (10)

For a generic 3D case the Brillouin diagram takes intoaccount all the propagation constants in the x, y, z directionsby the wavenumber conservation law [12]

n2k0 =√β2z + β2

x + β2y. (11)

The conservation law (11) applied to a bidimensionalwaveguide, and Floquet’s condition (5) allow to define theunit cell dimensions for a 2D periodic pattern [16].

4. Step Discontinuity:Variational Equivalent Circuit

Periodic air holes in GaAs material can be modeled by theanalogy with the cascade step discontinuities. A simple andeffective equivalent circuit for the small step (dielectric junc-tions) between two slabs waveguide is shown in Figure 2. Thesingle step discontinuity of a planar waveguide is reported


D

d

Air holes x, βx

y, βy

z, βz

(a)

Cutoff

Propagation

Plot A

Plot B

20

20

20

15

15

10

10

10

5

5

× 106

× 106

× 106

0

0

βz (m−1)

βy (m

−1 )

n 2k 0

(b)

Figure 4: Brillouin diagram for 3D GaAs periodic structures withair holes and different periodicity along the y- and z-axis. (a) 3Dperiodic structure; (b) Brillouin diagram: the dashed line representsthe set of operating points at working wavelength λ0 = 1.55μm; plotA refers to the waveguide with core thickness d; plot B refers to thewaveguide with core thickness D.

in Figure 2(a) where d and D are the core thicknesses of thetwo waveguides. In Figures 2(b) and 2(c) are reported thesimple equivalent circuit for a small step discontinuity (lowdielectric contrast or small difference between D and d) andfor the variational equivalent circuit, respectively. If moremodes are involved, the generic variational equivalent circuitconsiders each line for each propagating mode. In particularin Figure 2(d) is shown the case of two guided modes in theregion with D core thickness, for a double step discontinuity.For The transverse electric (TE) mode excitation defines thefollowing three field components:

Hx(x, z) = 1iωμ

∂

∂zEy(x, z), Hz(x, z) = − 1

iωμ

∂

∂xEy(x, z),

(12)

20

20

20

15

15

10

10

10

5

5

× 106

× 106

× 106

0

0

βz (m−1)

β v (m

−1 )

n 2k 0

Cutoff

Propagation

Plot B

Plot A

Figure 5: Brillouin diagram for 3D GaAs periodic structures withair holes and same periodicity along the y- and z-axis. The dashedline represents the set of operating points at working wavelengthλ0 = 1.55μm; plot A refers to the waveguide with core thickness d;plot B refers to the waveguide with core thickness D.

2 4 6 8

2

4

6

8

10

12

14

r = 10

r = 5

r = 1

|(A

+ D

)/2|

θ

Figure 6: Single unit cell Floquet’s resonance condition (passbandcondition) for different ratio transformation r. The passbandcondition is below the dashed line: in this case, n2 = 3.408, L1 =1μm, λ0 = 1.55μm, and L2 can be fixed in order to select theresonance condition.

where Ey(x, z) is represented by the modal expansion [12]

Ey(x, z) =⎧⎨⎩∑

k

akψk(x) +∫∞

0b(kx)φ(x; kx)dkx

⎫⎬⎭e

−iβzz,

(13)

where ak and b(kx) are the amplitudes of the guided and con-tinuum modes, respectively. For pronounced discontinuities,a considerable mode mixing characterizes the junction. Aneffective way to consider such mode mixing is to derive anequivalent circuit by using a variational approach [12, 13].Let us indicate by ψk the guided TE wave and by φ the modefunctions pertaining to the continuous spectrum in regions


0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

1

1.2

m

m = 1

m = 2

m = 3

m = 4m = 5

m

Stop-band

|(A + D)/2|

T

Stop-band

θ

Figure 7: Floquet’s resonance condition (passband condition) fordifferent unit cell numbers m. The passband condition is below thedashed line: in this case, r = 2, Z′0 = 2, Z0 = 1, and θ′ = θ.

I and II of Figure 2(a), respectively. By using (13) the field atthe left and at the right of the discontinuity is the following:

Eleft = (1 + R)∑

k

akψk(x) +∫∞

0A(kx)φ(x; kx)dkx,

Eright = T∑

k′a′k′ψ

′k′(x) +

∫∞0B(kx)φ′

(x; k′x

)dk′x,

(14)

where R indicates the reflection coefficient, and T is thetransmission coefficient. The orthonormality property of themodes provides the following relationships:

1 + R =∫ +∞

−∞E(x)

∑

k

ψk(x; kx)dx,

A(kx) =∫ +∞

−∞E(x)φk(x; kx)dx,

T =∫ +∞

−∞E(x)

∑

k′ψ′k′(x; k′x

)dx,

B(k′x) =

∫ +∞

−∞E(x)φk′

(x; k′x

)dx.

(15)

However, the transverse magnetic field at each side of thediscontinuity must be continuous, thus requiring

(1− R)∑

k

Y1kψk(x)−∫∞

0Y1k(kx)A(kx)φk(x; kx)dkx

= T∑

k′Y2k′ψ

′k′(x) +

∫∞0B(k′x)φ′k′(x; k′x

)dk′x,

(16)

where Y1k and Y2k′ are the characteristic modal admittancesof the transmission lines related to regions I and II,respectively, (see Figure 2(c)). Let us assume for simplicitythat only the guided mode Ψ1 exists in region I, and Ψ2 ispresent in region II. Let us also indicate by φ1, φ2 the modefunctions pertaining to the continuous spectrum of regions

I and II of Figure 2(a). By multiplying (16) by E (electricfield at junction), by integrating over x, and by dividing by(1 + R)2 it is possible to evaluate the input admittance ofFigure 2(c) as

Yin = 1− R1 + R

= Y2

Y1

[∫ +∞−∞E(x)ψ2(x; k′x)dx∫ +∞−∞E(x)ψ1(x; kx)dx

]2

+1Y1

∫∞0 Y1(kx)

[∫ +∞−∞E(x)φ1(x; kx)dx

]2dkx

[∫ +∞−∞E(x)ψ1(x; kx)dx

]2

+1Y1

∫∞0 Y2

(k′x)[∫ +∞

−∞E(x)φ2(x; k′x)dx]2dk′x[∫ +∞

−∞E(x)ψ1(x; kx)dx]2 ,

(17)

which can be rewritten as

Yin = r2Y2

Y1+Y0

Y1(18)

with

r2 =[∫ +∞

−∞E(x)ψ2(x; k′x)dx∫ +∞−∞E(x)ψ1(x; kx)dx

]2

, (19)

Y0 =∫∞

0 Y1(kx)[∫ +∞−∞E(x)φ1(x; kx)dx

]2dkx

[∫ +∞−∞E(x)ψ1(x; kx)dx

]2

+

∫∞0 Y2(kx)

[∫ +∞−∞E(x)φ2(x; k′x)dx

]2dk′x[∫ +∞

−∞E(x)ψ1(x; kx)dx]2 ,

(20)

suggesting the equivalent variation circuit of Figure 2(c). It isnoted that in this circuit the admittanceY0 includes radiationand reactive effects of the modes which cannot propagate. Byconsidering more propagating modes (19) becomes

r =∫ +∞−∞E(x)

∑k′ a

′k′ψ

′k′(x)dx∫ +∞

−∞E(x)∑

k akψk(x)dx. (21)

Equation (21) shows that the generic ratio transformationis a function of the propagating modes. Moreover, theimpedances characteristics of the transmission lines relatedto the regions I and II of Figure 2(a) are functions of theeffective refractive indices [12], in particular

Z1,2 = 1neI,eII

√μ0

ε0, (22)

where neI and neII are the effective refractive index of theregion I and region II, respectively.


1.3 1.4 1.5 1.6 1.7 1.80

0.2

0.4

0.6

0.8

1

T

θ

(a)

1.3 1.4 1.5 1.6 1.7 1.80

0.2

0.4

0.6

0.8

1

T

θ

(b)

1.3 1.4 1.5 1.6 1.7 1.80

0.2

0.4

0.6

0.8

1

T

θ

(c)

1.3 1.4 1.5 1.6 1.7 1.80

0.2

0.4

0.6

0.8

1

T

θ

(d)

1.3 1.4 1.5 1.6 1.7 1.80

0.2

0.4

0.6

0.8

1

T

θ

(e)

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

T

|(A + D)/2|

θ

(f)

Figure 8: 1D practical case of GaAs (n2 = 3.408) slab waveguide with L1 = 0.1163μm, L2 = 0.1246μm at central wavelength (resonancewavelength) and λ0 = 1.55μm. (a) Transmittivity of 1 unit cell, (b) transmittivity of 2 unit cells, (c) transmittivity of 3 unit cells, (d)transmittivity of 4 unit cells, (e) transmittivity of 5 unit cells, and (f) Floquet’s condition for 1 unit cell.

The TE modal profile reported by (19) and (21) is givenby [12]

ψ(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ake−p(x−d), x ≥ d,

akcos(kxx − ϕ

)

cos(kxd − ϕ

) , 0 ≤ x ≤ d,

akcos(ϕ)

cos(kxd − ϕ

) eqx, x ≤ 0,

ϕ = tan−1(q

kx

),

ak =√

21/p + d + 1/q

cos(kxd − ϕ

),

(23)

where p, q, and kx are the x-transverse propagation (seeFigure 2(a)) constant in the cladding, in the core and in


1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.750

0.2

0.4

0.6

0.8

1

1.2

1.4

T Passband

|(A + D)/2|

θ

(a)

1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.640

0.2

0.4

0.6

0.8

1

1.2

1.4

T

A B

|(A + D)/2|

θ

(b)

Figure 9: (a) Figures 8(a), 8(b), 8(c), 8(d), 8(e), and 8(f) on the same graph. (b) Floquet’s conditions for different unit cells number.

1.4

1.6

1.8

1

2

31.4

1.6

1.8

2

3

1

2

4

6

0.5

1

1.5

|(A + D)|/2 |(A + D)|/2

θθ

θ′θ′

Floquet’s design

(a)

1.4

1.6

1.8

1

2

3

0.5

1

T

θθ′

(b)

1.8

1.8

1.6

1.4

1.4

1.6

3

32

21

1

0.5

12

4

6

Floquet’s design

T

|(A + D)|/2

|(A + D)|/2

θ

θ’

θθ’

(c)

Figure 10: (a) Floquet’s region |A +D|/2 < 1 for a single unit cell. (b) Floquet’s region for five unit cells.


the substrate, respectively. The TE modal profiles Ψ′ areobtained from (22) by substituting k

′x to kx, p′ to p, q′ to

q, and a′k to ak and d to D. The same model can be appliedalso to transverse magnetic (TM) fields.

5. Analytical Results: Floquet’s Design

As example an asymmetrical slab waveguide (Figure 2(a))with n1 = 1, n2 = 3.408, n3 = 3.1, d = 0.24μm, D = 0.8μmis analyzed. By supposing a wavelength grating resonance atλ0 = 1.55μm, the Brillouin diagram of Figure 3 is obtained.In this diagram are considered the fundamental TE modes inregions I and II of Figure 2(a). The evaluated propagationconstant along the z-direction is βz1 = 1.2625 ∗ 107 m−1

in region I and βz2 = 1.3492 ∗ 107 m−1 in region II.The propagation constants are evaluated by considering thedispersion equation for a multilayer dielectric structure andthrough the effective dielectric constant method [12, 17].In this case the phase velocity and the group velocity arenear the asymptotic condition n2k0 = βz (see Figure 3).The Brillouin diagram of the 3D GaAs periodic structure isreported in Figure 4 in which two different periodicities areconsidered along the y- and z-axis. Moreover Figure 5 showsthe Brillouin diagram of the same GaAs structure in the caseof same periodicity along the y- and z-axis. Before to analyzethe transmittivity and Floquet’s resonance condition someaspects of the theoretical model are illustrated: in particularFigure 6 reports how Floquet’s condition (6) changes withthe ratio transformation r by fixing L1 and the resonantwavelength λ0 (more choices of L2 are possible with lowratio transformation); moreover the effect of the cell numberon the transition bands is shown in Figure 7 where thetransmittivity responses versus θ for different unit cellnumber are considered. In order to validate our model for amultimode analysis a multimode 2D periodic slab waveguidewith bandpass centred in λ0 = 1.55μm is considered.Regarding this example the same periodicity along the y-z-propagation directions and the wavenumber conservationlaw of (11) are considered. By taking into account theeffective refractive indices and (22), the 2D case becomesa simplified case of the 3D case. The Brillouin diagram ofFigure 5 shows the propagation conditions related to thisexample. For a resonant central wavelength of λ0 = 1.55μm,a possible geometrical configuration is the unit cell (along theperiodic yz plane) with L1 = 0.1163μm, L2 = 0.1246μm.By (21) a ratio transformation of r = 5 is calculated byconsidering n2 = 3.408, n1 = 1, n3 = 3.1, d = 0.24μm,D = 0.8μm and all the TE propagating modes in the GaAsslab with core thickness d and D (in particular one TEpropagating mode in the region with thickness d and twoTE propagating modes in the region with thickness D areconsidered). Figures 8 and 9 show the transmittivity andFloquet’s conditions for this example. Moreover Figure 9(b)indicates that the limit points A and B of Floquet’s conditionremain fixed by increasing the unit cell number; therefore,the single unit cell gives information about the passbandcentred at the resonance wavelength λ0. In Figures 10(a),10(b), and 10(c) is reported how it is possible to localize

1.54 1.59

0.20.1

0.30.40.50.60.70.80.9

1

Tran

smit

tivi

ty

TheoreticalFEM 3D 5 × 5 circular holesFEM 3D 5 × 5 square holes

1.39 1.44 1.49 1.64 1.69

λ (μm)

Figure 11: Transmittivity response of the 2D theoretical and 3DFEM modeling.

Floquet’s region for five unit cells: Floquet’s region of thesingle cell is superimposed to the transmission characteristicof the periodic structure by defining the passband condition.

6. Bandpass Optimization: FEM Modeling

After the design of the 2D periodic structure we optimizethe periodic structure by analyzing the diffraction effect onthe frequency response. The different air hole geometries andthe air hole depths may change the bandwidth and the cutofffrequencies position. The full wave FEM method considersall these aspects also for complex 3D periodic structures withhigh unit cells number. The numerical modelling optimizesthe bandpass position around the theoretical prediction.

In Figure 11 is presented the comparison of the passbandbetween the theoretical model and the 3D FEM simulation:the theoretical model takes into account the same periodicityalong the y- and z-axis and the Brillouin diagram of Figure 5.Also in the 3D FEM simulation a passband around λ0 =1.55μm is checked. The difference of the bandwidth and thedifferent cutoff frequencies with respect to the theoreticalmodel is in the particular geometry of the periodic pattern(scattering effects of circular and square air holes with adefined depth). Figure 11 shows that the 2D theoreticalpassband is always centred, and that the 3D model is usefulin order to increase the bandwidth or to shift the cutofffrequencies (optimization process). In the FEM simulator theexcitations are wave ports placed at the boundary interfacein order to provide a window (cross sections) that couplesthe model device to the external space. The electromagneticsimulator assumes that the structure is excited by the naturalfield patterns (TE mode profiles according to the theoreticalhypothesis) associated with these cross-sections. The 2D fieldsolution generated for each wave port is the exact modalboundary condition at those ports for the 3D problem.Figure 12 proves the 3D resonance at λ0 = 1.55μm byconsidering the unit cell dimensions of the 2D theoreticalmodel (same unit cells along the y- and z-direction).


TEexcitation

(a)

TEexcitation

(b)

TEexcitation

(c)

TEexcitation

(d)

Figure 12: Volumetric electric field distribution of 5 × 5 unit cells with L1 = 0.1163μm, L2 = 0.1246μm: (a) top view of 5 × 5 square unitcells, (b) longitudinal section of 5× 5 square unit cells, (c) top view of 5× 5 circular unit cells, and (d) longitudinal section of 5× 5 circularunit cells.

7. Conclusion

To summarize we presented in this work a direct and accuratetheoretical approach regarding the resonance frequenciesdesign of dielectric periodic structures with high dielectriccontrast. The rigorous method is based on Floquet’s trans-mission line theory and on the equivalent variational circuitwhich models the discontinuity effect on the frequencyresponse at optical frequencies. The design is performed bythe Brillouin diagram which defines, through the wavenum-ber conservation law, the unit cell dimension of the resonat-ing structure. The 3D FEM simulation verifies the resonanceof a GaAs photonic crystal with air holes obtained by thetheoretical model around a working wavelength of λ0 =1.55μm. The FEM numerical tool optimizes the bandwidthand the cutoff frequencies position by considering differentair hole geometries. The model can be also applied to designphotonic band gap devices with microcavities.

References

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