Finite-Element Time Domain Method in Photonics

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167

4Finite-Element Time Domain Method

In this chapter we present finite-element time domain methods for photon-ics. We differentiate between methods that are used by the larger numericalelectromagnetics community and describe the approximations relevant forsimulation of photonic structures. We discuss the derivation of the timedomain wave equation and both implicit and explicit solution schemes for

this equation. We also present some illustrations of the method.

4.1 Time Domain Numerical Methods

Simulation of several electromagnetic phenomena and photonic devices requirenumerical methods that can model the evolution of the fields as a functionof time. Some generic cases include modeling of transient phenomena thatchange rapidly with time, characterization at several frequencies simultane-ously (broadband spectral response), and nonlinear effects. In general, wherevertime dependence plays a significant role, the temporal nature of the Maxwell’sequations cannot be ignored and approximated in a manner similar to con-tinuous wave operation. Tus methods that can handle the field variation inspace and time have been developed.

Tere are several time domain methods that can be categorized indifferent ways: those based on regular structured spatial grids such as the well-known finite-difference time domain (FDD) [1–9], finite-differencetime domain beam propagation method (FD-DBPM) [10–13], those basedon unstructured meshes such as the finite element methods [14–36], and the

finite volume time domain (FVD) methods [37 and references therein]

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168 Finite Element Modeling Methods for Photonics

based on unstructured grids that use Green’s formula to eliminate differential

operators. Another way to classify time domain methods is based on the equations

that are solved and the methodology employed. It is possible to solve directlyMaxwell’s equations (1.1 and 1.2) as is the case in the FDD method [1–9]and several FED methods [20–25, 27], where an explicit scheme is obtained(though implicit FDD schemes have been delivered [2–9]). Tese methodsusually have a leapfrog system [38] wherein the electric and magnetic fieldcomponents are located at different points on the spatial grid and are com-puted at different times. Te curl operator appearing in the Maxwell’s equa-

tions relates the circulation of H(E) with the time change in E(H) (B andD are eliminated by use of the constitutive relationships). Te curl operatordictates that the electric and magnetic fields surround each other. Tereforesuch methods cannot yield the components of both the E and H fields at thesame physical location and even at the same instant of time. Tis can makeit difficult to adapt the frequency domain methods discussed in Chapters 2and 3 to time domain approaches. Furthermore, for explicit formulations,the time step has to satisfy the Courant-Friedrich-Levy (CFL) [1] conditionand is usually small, leading to increased calculation. We refer to this class ofmethods as direct solvers for convenience in this chapter.

Te second class of methods (time dependent BPMs) solve a second-order time dependent wave equation (in terms of either the E or the H fields)[10–18]. Te solution of the wave equation gives the field at all points on thegrid at each time step and usually involves solution of a matrix equation ateach step. Tese methods can be implicit or explicit. For the implicit methods,combining the SVEA eliminates the CFL condition, allowing use of relativelylarge time steps and hence less computation. Tese time marching methodsbear some similarity to the frequency domain methods where the wave equa-tion is solved. In Section 1.4.2 we presented a brief derivation of the waveequations from the Maxwellian curl equations (assuming time dependenceto be absent), and solution of the vector wave equation was discussed in Sec-tion 3.3. Te time dependent wave equations (both scalar and vector) can beobtained in a similar manner.

In the photonics community, the time domain approaches that are most widely used are the FDD [1], FD-D BPM [10–13], FED BPM [14–18],multiresolution time domain method [39, 40], and the FVD [37]. Recently,a powerful approach based on the discrete Galerkin method has also beenproposed [41]. Te FDD method can be computationally very resourceintensive. As the complexity and size of the device/structure increase, the

resource requirement increases substantially for the FDD and other time

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Finite-Element Time Domain Method 169

domain methods. Tus, a big challenge in time domain methods has been to

improve efficiency without compromising accuracy.Te D-BPM methods often have lower resource requirement, as these

deal with either the E or H field components (rather than both for directsolvers). Where the slowly varying envelope approximation (SVEA) is used inimplicit formulations, these can be used with larger time steps as well. Amongsuch methods, it is possible to choose those that represent the boundaries andthe structure well, such as methods based on unstructured or irregular grids.Tere are several FE-based methods that fall into this category [29–35]. Wediscuss in the following section an FED-BPM method that solves the wave

equation.

4.2 Finite-Element Time Domain (FETD) BPM Method

Tere are several FE-based time domain BPM that solve the wave equation.Many of these methods solve the vector wave equation, are often wide-bandin nature, and may employ vector/edge elements [32, 33, 42]. Te origin ofthese methods is in the field of numerical electromagnetics, which can differsomewhat from the photonics community in the applications the methods are

devised for. Te former deal extensively with radar, antennae, remote sensing,high-speed/high-frequency circuits, scattering and radiation, open boundaryproblems, and so forth. Te latter largely focus on guided wave structuressuch as waveguides, sources, and detectors, for which the constraints on themodeling methods can be different. Terefore, even though several timedomain methods have been put forward for electromagnetics problems, we will confine our focus to approaches suited to photonics.

In many photonic devices the modulation frequency of the propagating wave is often much lower than the frequency of the carrier wave. Tereforeit is sufficient to neglect the high-frequency carrier and consider the evolu-tion of the comparatively slowly varying envelope. We will primarily discussthe FED BPM that solves the scalar time dependent wave equation, whichdescribes the evolution of the envelope of the field.

Starting with Maxwell’s curl equations (1.1) and (1.2), it is possible toeliminate either the E or H fields in a procedure similar to that described inSection 1.4.2. Tis results in the time dependent vector wave equation (for E ,and a similar equation can be derived for the H ):

× ×

= −

∂

∂ −

∂

∂

1 2

2

t

J

t E

E —

m — e

(4.1)

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Solution of (4.1) involves both spatial and temporal variations for a vector

field with its three components, which can be computationally very heavy. Tus,in situations where the vector nature of the fields (coupling between componentsdue to boundary conditions, large magnitude of longitudinal components) canbe ignored, it is simpler to solve the scalar equation for a single component ofthe field. In many cases this is the situation, and only the reflections caused byboundaries/changes in refractive index or generation of new frequencies needsto be included with sufficient accuracy. Refer to the literature [29–35] for excel-lent work on the time dependent vector wave equation and direct solutions ofMaxwell’s equations [1–9]. We follow a treatment similar to [14, 18] here and

discuss the solution of the scalar wave equation in two dimensions.Equation (4.1) contains only the electric field, though it still containsthe curl operator which can be further simplified. We adopt the simplifica-tion of two-dimensional propagation in the yz plane with no variation in the

x direction. Proceeding in similar fashion to Section 3.3, we apply the curloperation with the nabla operator modified to include the perfectly matchedlayer (PML) boundary condition. We then obtain

∂∂

− ∂∂

− ∂

∂∂∂

− ∂

∂∂∂

− ∂

∂

+

− ∂∂

∂∂

− ∂

∂

= − ∂

∂

ˆ ˆ

ˆ

2

2

2

2

x s y

s

s

E

y s

z

s

s

E

z y s

z

s

s

E

y

s

s

E

z

z s

s y

s

s

E

y

s

s

E

z s

n

c t

y

y x z

z x z

y z z y

y y z z y E

(4.2)

the vector wave equation for the E field in the absence of current density, J . Asimilar procedure can be developed for theH field as well. We then make thescalar propagation assumption and consider the evaluation of only one field

component. We solve for the transverse modes and only need to consider thefirst term on the left-hand side of (4.2), which is dependent on a single fieldcomponent, to obtain

s y

ps

s y s

z p

s

s z s

q

c t y

y

z

z 0

2

2

2

∂∂

∂Φ∂

+ ∂

∂∂Φ∂

−

∂ Φ∂

= (4.3)

where

E p q n x , 1, for TE modes

2

Φ = = = (4.4)

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H p n q

x

, 1 1 for TM modes2Φ = = = (4.5)

where we have now written the more generic form for both the electric andmagnetic fields. Te parameter s is defined in accordance to (2.148). For thePML region perpendicular to the y axis, perpendicular to the z axis, or corners,s y = 1 and s z = s , s y = s , and s z = 1 or s y = 1 and s z = 1, respectively.

In order to study the slowly varying envelope of the field and eliminatethe high-frequency carrier, we further substitute [14, 18]

y z t y z t j t o

( , , ) ( , , )exp( )f w Φ = (4.6)

into (4.3), where ω o is the frequency of the carrier, and obtain

s q

c t js

q

c t s

y p

s

s y s

z p

s

s z

q

c o

y

y

z z o

2 02

2

2 2 2

f w f f f w f−

∂∂

− ∂

∂ +

∂∂

∂∂

+ ∂

∂∂∂

+ =

(4.7)

Solution of (4.7) will give us the evolution of the envelope of the field, ϕ [11]. o obtain the solution, we first apply Galerkin’s procedure to convert theequation to a matrix equation containing mass and stiffness matrices

c M

d

dt j

c M

d

dt K

c M o o1

2 02

2

2 2 2

f w f w f

{ } { }{ }[ ] [ ] [ ] [ ] { }− − + +

= (4.8)

where {ϕ } is the global vector containing the field envelope values at thenodes and {0} is the global null vector. Te FE matrices in the equation can

be defined as

K ps

s

N

y

N

y p

s

s

N

z

N

z dydz

y T

z T

e e

2 2

∫∫ ∑[ ] { } { } { } { }

= − ∂

∂

∂

∂ −

∂

∂

∂

∂

(4.9)

M sq N N dydz T

e e ∫∫ ∑[ ] { }{ }= (4.10)

where the symbolse N

e e

, , , and { }

Σ ∫∫ have their usual meaning.

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4.2.1 Wide Band and Narrow Band Approximations

Equation (4.8) bears some similarity to the wide-angle BPM equation ofChapter 3. Te second-order derivative with respect to time is in some wayssimilar to the second-order derivative with respect to z (in the BPM equation),and it requires special consideration. Neglecting this term results in narrowband schemes, wherein the spectrum of the pulses that can be handled havea narrow frequency spectrum. Tis is analogous to the Fresnel approximationand leads to paraxial BPM-like schemes. It is possible to include/approximatethe second-order derivative term resulting in schemes with higher accuracythough accompanied by heavier computation.

A popular approach in handling the second-order derivative is to employthe Pade approximants, inspired by the wide-angle BPM (discussed in Section3.3.2). We use a similar approach (a step-by-step derivation is not given toavoid duplication) and obtain for the time domain equation, (4.8)

j c

M d

dz K

c M o o

2 02

2

2

w f w f

{ }{ }[ ] [ ]− + +

= (4.11)

where

M M c

K c

M

o

o

4

2

2

2

2w

w [ ] [ ] [ ] = − +

(4.12)

Tis class of wideband schemes allows accurate propagation of pulses with amuch broader frequency spectrum. Note that (4.11) becomes the narrow bandequation if M [ ] is replaced by [ M ].

Next, we discuss the methods for solution of the time domain equations.

4.2.2 Implementation of the FETD BPM Method: Implicit and ExplicitSchemes

A key part of the time domain algorithm is the solution of the matrix equa-tions that result from the discretization. Tere is more than one approach,and each has its corresponding advantages and disadvantages.

Te wide band equation, (4.11), is a linear differential equation and canbe solved accordingly. For example, by applying the Crank Nicholson methodto the time variable similar to the approach in [14], we obtain

A B i i i i 1f f{ } { }[ ] [ ]=

+ (4.13)

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where

A j c

M t K c

M i

o

i i o

i 2 0.5

2

2

2

w D

w [ ] [ ] [ ]= − + +

(4.14)

B j c

M t K c

M i

o

i i o

i 2 0.5

2

2

2

w D

w [ ] [ ] [ ]= − − +

(4.15)

Equation (4.13) can be solved by standard methods such as the conjugategradient method and others.Te use of the Crank Nicholson method to derive (4.13) results in a

scheme that is implicit and unconditionally stable. Terefore, we can expectthis method to be robust even with reasonably large time steps. However,the linear equation has to be solved at every time step, and for large matricesthis can be computationally demanding, especially where iterative methodsare used.

A proposed simplification [18, 28] uses the concept of matrix lumpingto simplify the full wide-band equation, (4.8). Lumping is a technique that

has been employed in numerical electromagnetics [28] as well as FE schemesfor other disciplines such as civil engineering [43] with some success, thoughthis can lead to instability and large errors as well [31]. Te computationaladvantage is that an explicit scheme results, requiring only matrix multiplica-tion for each propagation step, thus simplifying the method and making itfaster. We apply this technique directly to the full wide-band equation, (4.8),in a manner similar to the procedure used in Obayya [18].

Lumping involves summing up the matrix entries in each row and cre-ating a diagonal matrix where each diagonal element is the row-sum of thecorresponding row. Terefore, we have

M M i i i j

j , ,∑= (4.16)

Te equation containing the replacement diagonal matrix is then multipliedby the inverse of the diagonal matrix to obtain

c

d

dt

j

c

d

dt

M K

c

o o12 0

2

2

2 2

1

2

f w f w f

{ } { }{ }[ ] [ ] { }− − + +

=−

(4.17)

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Applying central differencing to approximate the time derivatives in (4.17)

gives

c t j

c t

M K c

I

n n n o n n

on

1 22

2

0

2

1 1

2 2

1 1

1

2

f f f

D

w f f

D

w f

{ } { } { } { } { }

{ }[ ] [ ] [ ] { }

− − +

− −

+

+

=

+ − + −

−

(4.18)

Collecting the like terms together we get

1

2 1

11

2

2

1

1

c t

j t t M K

c I

j t

j t no

n

o

on

f D

w D D

w f

w D

w D fo

2{ } { } { }[ ] [ ] [ ]=+

+ +

+

− +

++

−

−

(4.19)

for the full band case. For the narrow band case, the second-order derivativeor the first term in (4.17) is neglected and we obtain

c t

j

M K

c

I n

o

on n1

1

2 1f

D

w

w f f{ } { } { }[ ] [ ] [ ]= +

++

−

− (4.20)

o calculate the field ϕ n+1, after a single time step at n + 1, the fieldvalues at two previous time steps are required, {ϕ }n and {ϕ }n–1, and are multi-plied with the coefficients in (4.19) for the wide band case and (4.20) for thenarrow band case. After each completed propagation step, the field at n + 1is designated the field at n, and the field originally at n is designated the fieldat n − 1. Tus for any propagation step, only the field at two previous timesteps is required, while field values previous to that can be ignored. Te massand stiffness matrices in the equation are constant and do not change; hencethese only have to be computed once. Furthermore, these matrices are thestandard FE-related matrices that have been discussed in detail in Chapter 2.

Te lumping procedure leads to an explicit formulation that is easyto implement and quite efficient both in terms of storage as well as speed ofcomputation. However, it comes with restrictions on the size of the time stepand instability. Te lumping procedure can lead to numerical dispersion andgenerate zero and/or negative entries in the diagonal matrix, while it is mostlyapplicable only to linear finite elements [31]. Terefore, it is a procedure thatmust be used only when the errors introduced by it are sufficiently low. Te

time step in this scheme is dictated by the CFL condition [1]:

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t

h

c D < (4.21)

where h is the smallest length of an element side in the mesh.It is also possible to apply the differencing scheme directly to the full

wide-band equation, (4.8), without lumping and obtain an implicit schemein which the coefficients of {ϕ }n–1, {ϕ }n, {ϕ }n+1 contain the mass and stiffnessmatrices. Tis wide-band scheme is more accurate than the explicit scheme(with lumping) and the wide-band scheme with Pade approximants. It hasroughly the same computational cost as the latter while it is more computa-

tionally intensive than the former. We will consider next the application of the explicit scheme with lumping.

4.3 Practical Illustrations of FETD BPM Applied to PhotonicStructures/Devices

ime domain methods are particularly useful to characterize devices atseveral frequencies simultaneously by not having to run the simulation ateach individual frequency separately. Furthermore, it is possible to study the

propagation and evolution of pulses with time domain methods, and thusthe restriction to continuous wave operation is lifted. Another strength ofthese methods is the ability to model waves in both forward and backwarddirections. Tus reflections arising from abrupt boundaries or large changesin refractive index can be modeled accurately. We discuss two examples thatbring out these features mentioned.

4.3.1 Optical Grating

We apply the explicit FED BPM method to simulate pulse propagation inthe optical grating (see figure 4 [18]). Te grating structure consists of severalperiods of alternating low and high refractive indices, and a Gaussian pulseis made incident on this structure. Te distributed feedback structure of thegrating leads to multiple reflections, and their interference gives us the overallreflection/transmission characteristics.

Figure 4.1 shows the reflection coefficient calculated using the explicitFED method with lumping for the E mode with both wide-band andnarrow-band schemes. A Gaussian pulse of width W o = 2.0μ m with centralcarrier wavelength 1.5μ m is made incident on the structure from the left.

Te pulse is centred at Z o = 11.0μ m and the time step used is 0.01fs. Both

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the narrow-band and wide-band reflection spectra are very close to expected

characteristics, indicating that even in the narrow-band case for this example,the FED method handles the multiple reflections quite accurately.

Tis structure illustrates the power of the FED BPM method as itallows us to propagate a pulse, calculate the reflections from the grating, andcharacterize the spectral response through a single simulation run. Te indexcontrast between the low and high index periods is also fairly high.

4.3.2 90° Sharp Bends

Integrated optics (IO) circuits consist of various optical components/struc-tures on a planar surface. Materials such as AlGaAs, InGaP, Si, and siliconon insulator (SOI) polymers and now even chalcogenide glasses can be usedto fabricate components which have desired functionality. Amongst these, thedesire to use the well-developed, high-volume, relatively low cost semiconduc-tor technology to fabricate optical structures on a chip and integrate them with electronics has also been a driving force. Tus IO structures with highindex contrast in Si have received much attention [44].

Te large index contrast leads to strong confinement of the optical field.Tis effect is used to make bends and waveguide interconnects that have a very

small radius of curvature, to reduce the footprint and size of the componenton chip. With materials that offer smaller index contrast such tight bends would not be possible due to leakage of light. We study two such sharp bends

Figure 4.1 Normalized reflected power for wide-band and narrow-band FETD schemes.

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with bend angle 90°: the first is a plain bend (see inset figure 7 [18]) while

the second has a square resonant cavity (see inset figure 8 [18]). We use theexplicit technique with lumping for these calculations.

Figure 4.2 shows the power reflected and transmitted through the plain90° bend. We can see from Figure 4.2 that there is significant reflection due tothe abrupt directional change at the corner. Tis rotation in the direction of the wavefront causes perturbation and generation of radiation modes. Couplingbetween guided modes and radiation modes leads to loss of power, especiallyat the outer edge of the bend; thus, the mode loses power, and at the outputend the transmitted power is only about 35%.

Te second bend, though with the slightly modified structure, showsimproved performance. A square resonant cavity is added here, and thisstructure leads to coupling between the guided modes of the waveguide andthose of the resonator. Te transmission increases to above 60% (see Figure4.3) making it far more suitable as a practical waveguide interconnect.

In the two examples of sharp bends discussed here, the simulationmethod has had to meet several challenges. Tese include accurate calculationof modes and field evolution for a very high index contrast waveguide, correctfield propagation in the bend region where the field is strongly perturbed andradiation modes take away a significant portion of the power, and accurate

handling of the backward propagating waves that arise from reflections. In thecase of the bend with the resonant cavity, the resonant interaction between the

Figure 4.2 Transmitted and reflected power for the sharp 90° bend.

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waveguide modes and the cavity modes is crucial to the performance of thedevice. Te FED BPM as described in this chapter has been used to model

this interaction without any additional parameters/change in formulation toaccount for it. Tus we find that the FED BPM method is robust and generalenough to be applied to challenging structures. Some care has to be taken inrelating the time steps with the mesh discretization for the explicit scheme.Overall, the FED BPM schemes work well for time dependent phenomena.

Many photonic devices rely on or utilize physical effects such as theelectro-optic effect, acousto-optic effect, thermal effects, and others for desiredfunctionality and application. Incorporating these effects in the modelingdomain is extremely important, and therefore methods that can handle thechange in the optical field due to these physical effects are essential. Te mate-rial presented so far concerning frequency domain FE-based methods (FEMdescribed in Chapter 2 and BPM described in Chapter 3) and the FEDdoes not describe how to simulate devices where other physical effects may bepresent. Because each physical effect pertains to a different physical propertyof the material and its interaction with the optical properties and field can becomplex, we treat this topic separately. In the following chapter, we discusssome of the effects we have mentioned and show how FE-based methods canbe used to model optical devices where these phenomena are present.

Figure 4.3 Transmitted and reflected power for the sharp bend with square resonant

cavity.

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Finite-Element Time Domain Method in Photonics

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