Progress In Electromagnetics Research B, Vol. 35, 241–261, 2011

IMPROVED FORMULATION OF SCATTERING MATRI-CES FOR SEMI-ANALYTICAL METHODS THAT IS CON-SISTENT WITH CONVENTION

R. C. Rumpf *

EM Lab, W. M. Keck Center for 3D Innovation, University of Texasat El Paso, El Paso, TX 79968, USA

Abstract—The literature describing scattering matrices for semi-analytical methods almost exclusively contains inefficient formulationsand formulations that deviate from long-standing convention in termsof how the scattering parameters are defined. This paper presentsa novel and highly improved formulation of scattering matrices thatis consistent with convention, more efficient to implement, and moreversatile than what has been otherwise presented in the literature.Semi-analytical methods represent a device as a stack of layers thatare uniform in the longitudinal direction. Scattering matrices arecalculated for each layer and are combined into a single overallscattering matrix that describes propagation through the entire device.Free space gaps with zero thickness are inserted between the layersand the scattering matrices are made to relate fields which existoutside of the layers, but directly on their boundaries. This frameworkproduces symmetric scattering matrices so only two parameters needto be calculated and stored instead of four. It also enables thescattering matrices to be arbitrarily interchanged and reused todescribe longitudinally periodic devices more efficiently. Numericalresults are presented that show speed and efficiency can be increasedby more than an order of magnitude using the improved formulation.

1. INTRODUCTION

Semi-analytical methods are highly efficient methods for solving partialdifferential equations where part of the problem is solved numericallyand the other part analytically. These methods are highly efficientwhen the analytical direction is very large or complicated. They have

Received 31 August 2011, Accepted 18 October 2011, Scheduled 31 October 2011* Corresponding author: Raymond C. Rumpf (rcrumpf@utep.edu).

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proven to be a very powerful tool in CEM and offer incredibly efficientanalysis of layered devices and structures with longitudinal periodicity.Semi-analytical methods represent devices as a stack of layers. Eachof these layers may have arbitrary complexity in the transverse plane,but must be uniform in the longitudinal direction. Typically the xand y axes are made to be the transverse directions while the z axis ismade to be the longitudinal direction. Devices with curved geometriesare represented by many thin layers using a staircase approximation.Analysis starts by analyzing the cross section of each unique layerto numerically calculate the set of electromagnetic eigen-modes thatexists in each layer. The eigen-vectors from this computation containthe amplitude functions of the eigen-modes. The eigen-values arethe propagation constants of the eigen-modes which describe howthey propagate in the longitudinal direction. After calculation, theeigen-modes can be analytically propagated forward and backwardthrough each layer. Because the longitudinal direction is analytical, thelayers can be of any length with no computational burden. Boundaryconditions at the interfaces between the layers are enforced by equatingtangential field components on either side. In this manner, propagationthrough the entire device is modeled rigorously.

Figure 1 shows four examples of devices that are efficientlymodeled by semi-analytical methods. Figure 1(a) is a microstriptransmission line divided into three layers that are each uniformin the vertical direction. Thin metal features combined with largephysical dimensions are ideally suited for efficient analysis by a semi-analytical method. Figure 1(b) is a rib waveguide grating. Oftenthese are composed of hundreds of grating periods, but are veryefficiently modeled using a semi-analytical method. Figure 1(c) showsa diffraction grating. These often contain small features in the grating,but with large dimensions in the thicknesses of the layers. Thisconfiguration is also ideally suited for analysis by a semi-analyticalmethod. Figures 1(d) and (e) show an electromagnetic band gap(EBG) material that is ten unit cells thick. Each unit cell is furtherdivided into 20 layers using the staircase approximation. The overalldevice is therefore composed of 200 layers, but only 10 are unique.Semi-analytical methods offer highly efficient analysis of structures likethis.

Two popular semi-analytical techniques are the method of lines(MOL) [1, 2] and rigorous coupled-wave analysis (RCWA) [3, 4]. TheMOL usually applies the finite-difference method to analyze thelayer cross sections, but other techniques like finite-elements canalso be used. It is well suited for analyzing metallic structures,multilayer waveguides and longitudinally inhomogeneous waveguide

Progress In Electromagnetics Research B, Vol. 35, 2011 243

(a) (b) (c)

(d) (e)

Figure 1. Examples of electromagnetic devices efficiently modeledby semi-analytical methods. (a) Microstrip transmission line.(b) Waveguide grating. (c) Diffraction grating. (d) Electromagneticband gap material. (e) Layers in the EBG unit cell.

devices. RCWA is the MOL performed in Fourier-space and hasbecome the dominant method for analyzing diffraction from periodicdielectric structures. Instead of using finite-differences, however, thetransverse coordinates are discretized using a Fourier transform leadingto a plane wave expansion inside each of the layers.

The boundary condition problem that results from semi-analyticalmethods can be numerically large, particularly when a device iscomposed of many layers. It is most efficient to use an algorithmthat solves the boundary conditions one layer at a time insteadof solving all layers simultaneously. To do this, CEM borrowedmatrix techniques from transmission line and network theory includingtransfer matrices [5–7], scattering matrices [8–13], hybrid matrices [14],and impedance matrices [15, 16]. Scattering matrices have emerged asthe dominate technique despite speed and efficiency benefits of theother techniques [4, 14, 16].

A scattering matrix enables a complex structure to be representedas a “black box.” It relates waves incident on the box to the waves

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Figure 2. Conventional definition of the scattering parameters.

scattered by the box in either direction. In fact, the electromagneticproperties of any linear structure can be summarized in a scatteringmatrix. The scattering matrix and physical interpretation of theparameters are illustrated in Figure 2. The S11 parameter quantifieshow much of the wave applied to Port 1 reflects from Port 1. S21

quantifies how much of that same applied wave transmits through thedevice to Port 2. S12 quantifies how much of a wave applied to Port 2will transmit through to Port 1. Finally, S22 quantifies how much ofthe second applied wave reflects from Port 2. This can be generalizedso that Smn represents how much of a wave applied to Port n exits fromPort m. In this manner it is commonly understood that S11 representsreflection and S21 represents transmission. For symmetric devices, itis expected that S11 = S22 and S21 = S12.

Scattering matrices in electromagnetics became popular in the1960s when Hewlett Packard introduced the first microwave networkanalyzers capable of swept amplitude and phase measurements [17].In addition, several key papers were published around this timepopularizing the concept [8, 10]. Today, scattering parameters are socommon that the terms “S11” and “S21” are often used synonymouslyfor “reflection” and “transmission.” Despite this strong and long-standing convention, the CEM community is adopting inefficient andunconventional formalisms for scattering matrices for use in semi-analytical methods [7, 16, 18–33]. The majority of the literature onthis topic appears in optics journals, where the benefits of semi-analytical methods are more pronounced and the use of scatteringparameters in experiments is less common. This, however, is nota universal convention used in optics; see Ref. [34]. Most of thesepapers define S11 and S22 as transmission quantities and S12 andS21 as reflection quantities. This practice is confusing, deviatessharply from convention, and it is no longer reliable to interpretthe indices of the scattering parameters to understand what is beingquantified. Further, the scattering matrices being adopted for semi-analytical methods are numerically inefficient and their order cannot bearbitrarily interchanged. This feature can be very useful to efficiently

Progress In Electromagnetics Research B, Vol. 35, 2011 245

model longitudinally inhomogeneous structures. The unconventionalscattering matrices also relate fields outside of the layer on one sideto fields inside the layer on the other side. This leads to scatteringmatrices that are inherently asymmetric regardless of what materialsare placed outside of the layer.

This paper formulates a highly improved version of scatteringmatrices for semi-analytical methods that also adopts the properconvention for the definition of the scattering parameters. The newformulation is stable and accurate for both 2D and 3D simulations.When the layers are composed of materials that obey reciprocity,implementation is more efficient because the scattering matrices aresymmetric. In this case, fewer calculations have to be performed andfewer parameters have to be stored in memory. Implementation is moreversatile because the order of the layers can be arbitrarily interchangedto describe complex and longitudinally periodic devices. Discussionstarts by establishing the numerical framework for scattering matricesin semi-analytical methods and unifying the MOL and RCWA. Basedon this framework, formulas to calculate scattering matrices for theindividual layers are derived along with the procedures to combinemultiple scattering matrices into a single scattering matrix. Equationsused to calculate transmitted and reflected fields are presented alongwith an equation for performing dispersion analysis of longitudinallyperiodic structures. Lastly, a numerical example is presented toillustrate the utility and efficiency of the improved scattering matrixformulation.

2. SEMI-ANALYTICAL FRAMEWORK

Formulation of a rigorous semi-analytical method begins withMaxwell’s equations describing the fields inside a single uniform layerof the device. The magnitudes of the electric and magnetic fields arerelated through the material impedance η. For numerical robustness,it is good practice to normalize the field quantities so they are of thesame order of magnitude. To do this here, the magnetic field willbe normalized according to ~H = −jη0

~H where j =√−1 and η0 is

the impedance of free space. Maxwell’s equations with the normalizedmagnetic field are written as

∇× ~E = k0µr~H (1)

∇× ~H = k0εr~E (2)

The term k0 is the free space wave number and is equal to 2π/λ0, whereλ0 is the free space wavelength. The terms µr and εr are functions that

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describe the relative permeability and relative permittivity in the crosssection of the layer. Eqs. (1) and (2) are vector equations that can beexpanded into a set of six coupled partial differential equations. Aftereliminating the longitudinal field components Ez and Hz using backsubstitution, the remaining four equations are discretized in the x-yplane and cast into the following matrix form.

∂

∂z

[ex

ey

]= k0P

[hx

hy

](3)

∂

∂z

[hx

hy

]= k0Q

[ex

ey

](4)

For the MOL, ex, ey, hx and hy are column vectors containing the fieldcomponents at discrete points across the transverse plane at pointsdescribed by the Yee grid [35, 36]. For materials described by diagonaltensors, the matrices P and Q are computed according to

PMOL =[ −De

xε−1zz Dh

x

(µyy + De

xε−1zz Dh

x

)− (

µxx + Deyε−1zz Dh

y

)De

yε−1zz Dh

x

](5)

QMOL =[ −Dh

xµ−1zz De

y

(εyy + Dh

xµ−1zz De

x

)− (

εxx + Dhyµ−1

zz Dey

)Dh

yµ−1zz De

x

](6)

The matrices Dex and De

y are banded matrices that calculate spatialderivatives of the electric fields across the grid [37]. The matrices Dh

x

and Dhy are also banded matrices, but calculate spatial derivatives of

the magnetic fields across the grid. These matrices are related throughDe

x = −(Dhx)H and De

y = −(Dhy)H where the superscript H indicates

a complex transpose, or Hermitian operation. The terms εxx, εyy, εzz,µxx, µyy and µzz are diagonal matrices that perform point-by-pointmultiplications on the field terms. The dielectric function εxx (x, y),for example, is reshaped into to a linear array and placed along thediagonal of a sparse matrix to construct εxx.

For RCWA, ex, ey, hx and hy are column vectors containingthe amplitudes of the spatial harmonics of the field expansion. Formaterials described by diagonal tensors, the matrices P and Q arecomputed according to

PRCWA=

−Kx[εzz∗]−1Ky

([µyy∗]−Kx[εzz∗]−1Kx

)

−([µxx∗]−Ky [εzz∗]−1Ky

)Ky[εzz∗]−1Kx

(7)

QRCWA=

−Kx[µzz∗]−1Ky

([εyy∗]−Kx[µzz∗]−1Kx

)

−([εxx∗]−Ky [µzz∗]−1Ky

)Ky[µzz∗]−1Kx

(8)

Progress In Electromagnetics Research B, Vol. 35, 2011 247

The matrices Kx and Ky are diagonal matrices containing the wavevector components of the plane wave expansion. The matrices [εxx∗],[εyy∗], [εzz∗], [µxx∗], [µyy∗], and [µzz∗] are full matrices that performtwo-dimensional convolutions in Fourier-space. For one-dimensionalgratings the convolution matrices will have Toeplitz symmetry. Toimprove convergence of the algorithm, fast Fourier factorization rulesshould be incorporated into the P and Q matrices [38–41].

Given Eqs. (3) and(4), a matrix wave equation can be derived forthe electric field quantities. This is accomplished by differentiatingEq. (3) with respect to z and then eliminating the magnetic field bysubstituting Eq. (4) into the new expression. After normalizing the zcoordinate according to z′ = k0z, the matrix wave equation is

∂2

∂z′2

[ex

ey

]−Ω2

[ex

ey

]=

[00

](9)

Ω2 = PQ (10)

After calculating the eigen-vectors W and the eigen-values λ2 of thematrix Ω2, the general solution to Eq. (9) can be written as

ψ(z′) =

ex(z′)ey(z′)hx(z′)hy(z′)

=

[W W−V V

] [e−λz′ 0

0 eλz′

] [c+

c−

](11)

V = QWλ−1 (12)

In this equation, W describes the eigen-modes of the electric fields andV describes the eigen-modes of the magnetic fields. The exponentialterms e−λz′ and eλz′ describe forward and backward propagationthrough the layer respectively. The column vectors c+ and c− areamplitude coefficients of the eigen-modes in the forward and backwarddirections respectively.

3. REFORMULATION OF SCATTERING MATRICES

Equation (11) can be used to calculate the field throughout a layerthat is uniform in the z-direction. All real devices are composed ofmultiple layers and eigen-modes must be calculated separately foreach layer. The eigen-vectors are the amplitude functions of themodes while the eigen-values are the propagation constants. With thisinformation, the field can be analytically propagated through each of

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the layers. To connect the layers, boundary conditions are enforcedby equating the tangential components of the fields on either side ofthe interfaces. In this manner, propagation through the entire devicecan be described rigorously. To solve the boundary condition problemefficiently, scattering matrices have arisen as the most popular methodand solve the boundary condition problem one layer at a time instead ofall layers simultaneously. To be consistent with convention, we definethe scattering matrix for the ith layer as

[c′−1c′+2

]=

[S(i)

11 S(i)12

S(i)21 S(i)

22

][c′+1c′−2

](13)

When the materials comprising the layer have no loss or gain, energymust be conserved. This makes the scattering matrix S unitary whereit obeys S∗S = SS∗ and S∗ = S−1.

Figure 3 illustrates the generalized mathematical framework ofthe improved scattering matrix for the ith layer. This construction isdifferent than what is found in the literature in that it is consistentwith convention and relates fields that exist outside of the layer onboth sides. In this framework, the materials outside of the layer canbe anything and can even be different on either side. The scatteringparameters S(i)

11 , S(i)12 , S(i)

21 , and S(i)22 are square matrices that quantify

how energy scatters among the different eigen-modes. The columnvectors c′+1 , c′−1 , c′+2 , and c′−2 contain the mode coefficients of the fieldimmediately outside of the ith layer in both forward and backwarddirections. Their subscripts indicate which side of the layer is beingdescribed while the signs in their superscripts indicate the directionof propagation. Finally, it should be noted how the mode coefficientsrelate to position along the z axis. From Eq. (11), it can be seenthat the mode coefficients completely describe the field when thelocal z coordinate is zero. Away from this point it is necessary toconsider propagation quantified by the exponential terms in Eq. (11).We are free to choose where z = 0 in each layer without loss ofgenerality because the values of the mode coefficients have not yetbeen calculated. In order to place the external fields exactly on theboundaries of the layer, we define the local z coordinates in the externalregions to be zero at the interfaces. The mode coefficients of theexternal fields contain apostrophes to highlight they are quantitieswhich reside outside of the layer. They are positioned in Figure 3consistent with where they completely define the field without needingthe complex exponential terms.

To derive expressions for the scattering parameters for the ithlayer, we start by writing two expressions that enforce the boundary

Progress In Electromagnetics Research B, Vol. 35, 2011 249

Figure 3. Mathematical framework for the scattering matrix of theith layer.

conditions at the first and second interfaces. Respectively, these are[W1 W1

V1 −V1

] [c′+1c′−1

]=

[Wi Wi

Vi −Vi

] [c+

i

c−i

](14)

[Wi Wi

Vi −Vi

] [e−λik0Li 0

0 eλik0Li

] [c+

i

c−i

]=

[W2 W2

V2 −V2

][c′+2c′−2

](15)

Next, Eq. (14) is combined with Eq. (15) and the terms are rearrangedto put the expression in the form of Eq. (13). When this done, thescattering matrix elements are found to be

S(i)11 =

(Ai1 −XiBi2A−1

i2 XiBi1

)−1 (XiBi2A−1

i2 XiAi1 −Bi1

)

S(i)12 =

(Ai1 −XiBi2A−1

i2 XiBi1

)−1Xi

(Ai2 −Bi2A−1

i2 Bi2

)

S(i)21 =

(Ai2 −XiBi1A−1

i1 XiBi2

)−1Xi

(Ai1 −Bi1A−1

i1 Bi1

)

S(i)22 =

(Ai2 −XiBi1A−1

i1 XiBi2

)−1 (XiBi1A−1

i1 XiAi2 −Bi2

)(16)

Aij = W−1i Wj + V−1

i Vj

Bij = W−1i Wj −V−1

i Vj

(17)

Xi = e−λik0Li (18)

It is important to note that the scattering parameters are not onlya function of the materials inside the layer, but also the materialson either side of the layer. This makes intuitive sense since theexternal materials must be known in order to quantify reflection andtransmission. It does, however, prevent the scattering matrices fromhaving the ability to be arbitrarily interchanged. Conceptually, thisproblem is overcome by the novel approach of separating all of thelayers composing the device with gaps composed of free space [33]. As

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long as these gaps are given zero thickness, they will have no effect onthe performance of the device. In other words, two layers separatedby a zero thickness gap is physically the same thing as two adjacentlayers without any separation. Numerically, this is accomplishedby letting medium 1 and medium 2 in Figure 3 be free space. Inthis setting, the scattering matrices can be arbitrarily interchangedbecause each layer is surrounded by free space and all of the scatteringmatrices are relating mode coefficients of fields that exist in the samemedium. Given the eigen-modes W0 and V0 for free space, we haveW0 = W1 = W2 and V0 = V1 = V2 so Eqs. (16) and (17) reduce to

S(i)11 = S(i)

22 =(Ai −XiBiA−1

i XiBi

)−1 (XiBiA−1

i XiAi −Bi

)

S(i)12 = S(i)

21 =(Ai −XiBiA−1

i XiBi

)−1Xi

(Ai −BiA−1

i Bi

) (19)

Ai = W−1i W0 + V−1

i V0

Bi = W−1i W0 −V−1

i V0

(20)

When each layer is surrounded by free space and composed of materialsthat themselves obey reciprocity, the result scattering matrices becomesymmetric. For this reason, only two of the four scattering parametershave to be calculated for each layer. This reduces memory requirementsand makes computation faster and more efficient.

4. MULTILAYER DEVICES

4.1. Redheffer Star Product

To model a device composed of multiple layers, it is necessary tocombine multiple scattering matrices into a single scattering matrix.It is important to note that the combined scattering matrix typicallydoes not have any symmetry so it becomes necessary to store allfour scattering parameters in the combined matrices. Two scatteringmatrices can be combined as illustrated in Figure 4 using the Redhefferstar product [16, 26, 42]. It is derived by writing Eq. (13) for twoadjacent layers, combining the two equations to eliminate the commonmode coefficients, and then rearranging terms to relate the modecoefficients of the outermost ports.

Given a scattering matrix S(A) that is to be followed by ascattering matrix S(B), the combined scattering matrix can becalculated using the star product S(AB) = S(A) ⊗ S(B) as follows.

[S(AB)

11 S(AB)12

S(AB)21 S(AB)

22

]=

[S(A)

11 S(A)12

S(A)21 S(A)

22

]⊗

[S(B)

11 S(B)12

S(B)21 S(B)

22

](21)

Progress In Electromagnetics Research B, Vol. 35, 2011 251

Figure 4. Concept of combining scattering matrices with the starproduct.

S(AB)11 = S(A)

11 + S(A)12

[I− S(B)

11 S(A)22

]−1S(B)

11 S(A)21

S(AB)12 = S(A)

12

[I− S(B)

11 S(A)22

]−1S(B)

12

S(AB)21 = S(B)

21

[I− S(A)

22 S(B)11

]−1S(A)

21

S(AB)22 = S(B)

22 + S(B)21

[I− S(A)

22 S(B)11

]−1S(A)

22 S(B)12

(22)

4.2. Doubling Algorithm for Scattering Matrices

For devices that are periodic in the longitudinal direction, the improvedscattering matrix formulation enables a very simple, fast and efficientalgorithm to model devices composed of many hundreds or thousandsof periods [2]. First, the scattering matrix for one unit cell isconstructed, which may be the combination of any number of scatteringmatrices. Second, it is combined with itself using the star product toobtain a scattering matrix that describes two unit cells in cascade. Thenew scattering matrix is combined with itself to obtain a scatteringmatrix that describes four unit cells. This process is continued untilan overall scattering matrix is calculated that describes the desirednumber of unit cells in powers of two. Figure 5 illustrates the doublingprocedure.

It is possible to generalize this procedure to calculate scatteringmatrices for any integer number of cascaded unit cells. This isaccomplished by converting the desired number of unit cells into binaryand constructing the overall scattering matrix from only the doubledscattering matrices that correspond to the required binary digits. Forexample, suppose a scattering matrix is needed to describe a devicecomposed of 166 unit cells. Converting this number to binary weget 10100110. Each binary digit of ‘1’ indicates a doubled scatteringmatrix that is needed to model exactly 166 unit cells. Altogether, the

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×1( ) (1) (N)S = S ⊗... ⊗S

×(2 )S = S ⊗ S

×1( ) ×1( )

×(4 )S = S ⊗ S

×(2 ) ×(2 )

×(4 ) ×(4 )S⊗= S×(8 )S

×(8 )S×(8 )S×(16 )S = ⊗

Figure 5. Doubling procedure.

combined scattering matrix is

S(×166)=S(×128)⊗∖/S(×64)⊗S(×32)⊗

∖/S(×16)⊗

∖/S(×8)⊗S(×4)⊗S(×2)⊗

∖/S(×1)

= S(×128) ⊗ S(×32) ⊗ S(×4) ⊗ S(×2) (23)

4.3. External Regions

After the entire device has been reduced to a single scattering matrixusing the procedures described above, the device resides in free space bydefault. Sometimes it is desired to have a material other than free spaceoutside of the device as an infinite half-space. This is common practicein photonics where a device resides on a very thick substrate. Toaccount for this, additional scattering matrices must be incorporatedthat “connect” the device to the external regions. Given a scatteringmatrix S(ref) that connects the device to the reflection region and ascattering matrix S(trn) that connects the device to the transmissionregion, the final scattering matrix is calculated according to Eq. (24)using the star product.

S(global) = S(ref) ⊗[S(1) ⊗ S(2) ⊗ · · · ⊗ S(N)

]

︸ ︷︷ ︸Device in free space

⊗S(trn) (24)

It is now necessary to derive equations to calculate the scatteringmatrices which connect the device to the external regions. In order topreserve the correct phase information in the global scattering matrix,the additional scattering matrices are made to describe layers with zerothickness. This is in addition to including free space gaps with zerothickness. The concept is illustrated in Figure 6. By definition, S(ref)

and S(trn) are not symmetric so all four scattering parameters must becalculated and so Eqs. (16)–(18) are used to derive them.

The equations to calculate the four scattering parameters for S(ref)

are derived from Eqs. (16)–(18) by letting L = 0, W1 = Wi = Wref ,

Progress In Electromagnetics Research B, Vol. 35, 2011 253

S S

S(1)S(2) S(3) S(N)

Figure 6. Construction of the global scattering matrix and connectionto external regions.

V1 = Vi = Vref , W2 = W0 and V2 = V0. This scattering matrix isonly needed when the reflection region is composed of a material otherthan free space.

S(ref)11 = −A−1

refBref

S(ref)12 = 2A−1

ref

S(ref)21 = 0.5

(Aref −BrefA−1

refBref

)

S(ref)22 = BrefA−1

ref

(25)

Aref = W−10 Wref + V−1

0 Vref

Bref = W−10 Wref −V−1

0 Vref

(26)

The equations to calculate the four scattering parameters for S(trn) arederived from Eqs. (16)–(18) by letting L = 0, W1 = W0, V1 = V0,Wi = W2 = Wtrn and Vi = V2 = Vtrn This scattering matrix is onlyneeded when the transmission region is composed of a material otherthan free space.

S(trn)11 = BtrnA−1

trn

S(trn)12 = 0.5(Atrn −BtrnA−1

trnBtrn)

S(trn)21 = 2A−1

trn

S(trn)22 = −A−1

trnBtrn

(27)

Atrn = W−10 Wtrn + V−1

0 Vtrn

Btrn = W−10 Wtrn −V−1

0 Vtrn

(28)

4.4. Scattering Matrix Algorithm

Equation (24) and the discussion above lead to an efficient algorithmfor implementing semi-analytical methods using scattering matrices.

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While many variations to the algorithm exist, the most basic goes asfollows. Prior to the main loop, the eigen-modes of free space W0 andV0 are calculated and the global scattering matrix S(global) is initializedby setting S(global)

11 = S(global)22 = 0 and S(global)

12 = S(global)21 = I

where I is the identity matrix. The main loop of the algorithmstarts at the first layer by calculating its scattering matrix S(1). Itthen updates the global scattering matrix using the star product asS(global) = S(global)⊗S(1). The main loop then iterates a second time bycalculating the scattering matrix for the second layer S(2) and updatingthe global scattering matrix as S(global) = S(global) ⊗ S(2). After themain loop iterates through all of the layers, the global scatteringmatrix describes propagation through the entire device. If the deviceis periodic in the longitudinal direction, the main loop should iteratethrough all of the layers of one unit cell. The doubling algorithm canthen be applied to efficiently cascade any number of periods into asingle scattering matrix. When all of this is finished, the entire deviceresides in free space by default. If the materials outside the device aresomething other than free space, the global scattering matrix must beconnected to the external regions as S(global) = S(ref)⊗S(global)⊗S(trn).Now the global scattering matrix is complete and a solution to theproblem can be found.

Some variations exist that can make the algorithm even moreefficient. For example, it is not necessary to surround each layerby free space. In the improved framework, it is only necessary tosurround all the layers with the same external medium. The externalmedium can be conveniently chosen to be the transmission or reflectionregion. When this is done, it is no longer necessary to incorporatethe “connection” scattering matrix for that region because the globalscattering matrix will calculate fields in that medium by default. Asecond variation is to calculate the scattering matrices of the firstand last layers separately using Eqs. (16)–(18). By doing this, the“connection” scattering matrices are not needed so they do not haveto be calculated.

5. CALCULATING THE SOLUTION

5.1. Reflected Fields and Transmitted Fields

The global scattering matrix S(global) relates the mode coefficients ofan incident wave cinc to the mode coefficients of the reflected cref andtransmitted waves ctrn. Assuming there is no wave incident from the

Progress In Electromagnetics Research B, Vol. 35, 2011 255

transmission region, this relation is[cref

ctrn

]=

[S(global)

11 S(global)12

S(global)21 S(global)

22

][cinc

0

](29)

From Eq. (29), we can write two formulas to calculate the modecoefficients of the transmitted and reflected waves given the modecoefficients of the incident wave and the scattering parameters.

cref = S(global)11 cinc (30)

ctrn = S(global)21 cinc (31)

The mode coefficients are related to the fields through Eq. (11) asfollows. [

eincx

eincy

]= Wrefcinc (32)

[eref

x

erefy

]= Wrefcref (33)

[etrn

x

etrny

]= Wtrnctrn (34)

From the above equations, it is straightforward to calculate thereflected and transmitted fields given the incident field [einc

x eincy ]T .

[eref

x

erefy

]= WrefS

(global)11 W−1

ref

[einc

x

eincy

](35)

[etrn

x

etrny

]= WtrnS

(global)21 W−1

ref

[einc

x

eincy

](36)

If needed, the longitudinal field components can be calculated fromthese results using Maxwell’s divergence equations.

5.2. Dispersion Analysis

It is possible to calculate the dispersion through a longitudinallyperiodic device using scattering matrices. This information can beused to construct electromagnetic band diagrams [7], to homogenizeperiodic structures [43], or to calculate effective material properties ofmetamaterials [44–46]. For dispersion analysis, no source is needed.

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First, the scattering matrix for a single unit cell S(cell) is constructedusing procedures described above. This scattering matrix is related toits external mode coefficients through

[c′−1c′+2

]=

[S(cell)

11 S(cell)12

S(cell)21 S(cell)

22

][c′+1c′−2

](37)

The terms in this equation are rearranged almost to the form of atransfer matrix. That is, the new form relates the fields on either sideof the unit cell instead of relating input and output waves.

[0 −S(cell)

12

I −S(cell)22

][c′+2c′−2

]=

[S(cell)

11 −I

S(cell)21 0

][c′+1c′−1

](38)

Assuming the device is infinitely periodic in the z direction with periodΛ, the following periodic boundary condition must hold where β is thepropagation constant of the Bloch wave.

[c′+2c′−2

]= e−jβΛ

[c′+1c′−1

](39)

Substituting this periodic boundary condition into Eq. (38) leads to ageneralized eigen-value problem Ax = λBx that can be solved to findsolutions of β as well as the Bloch modes if they are desired.

[0 −S(cell)

12

I −S(cell)22

][c′+1c′−1

]= ejβΛ

[S(cell)

11 −I

S(cell)21 0

][c′+1c′−1

](40)

6. NUMERICAL EXAMPLE

To demonstrate the utility and performance enhancement of theapproach described in this paper, a three dimensional electromagneticband gap (EBG) material was modeled using RCWA. A structurewith face-centered-cubic (FCC) symmetry was chosen for this examplebecause this has the highest symmetry of all the Bravais lattices andis of great interest to the research community. The slab and one of itsten unit cells are depicted in Figures 1(d) and (e) respectively. Theunit cell was modeled as being a dielectric with εr = 6.0 with airfilled spheres cut away from the dielectric body so as to form FCCsymmetry. The unit cell was divided into 20 uniform layers usinga staircase approximation. Periodic boundary conditions were usedto approximate a slab that was infinitely periodic in the transversedirections.

Progress In Electromagnetics Research B, Vol. 35, 2011 257

Figure 7. Simulation times for three different scattering matrixformulations.

The EBG material was modeled using three different versionsof RCWA. The first version was based on the scattering matrixformulation found throughout most of the literature describing semi-analytical methods. Best effort was made to use as representativeof an implementation as possible. This was chosen to be that ofRef. [21]. The second model was based on the improved scatteringmatrix formulation presented in this paper. The only differencebetween these algorithms was the manner in which the layer scatteringmatrices were calculated. This second version was chosen to illustratethe efficiency improvement due solely to the symmetry afforded by theimproved scattering matrices. The third version was also based on theimproved scattering matrix formulation, but it exploited the ability tointerchange layers and perform doubling to more efficiently analyze thestructure. Due to symmetry, there existed only 10 unique layers in theentire structure so even fewer calculations were necessary in this case.Other than the scattering matrix manipulations, the three versions ofRCWA were kept identical. The codes were run on a single core of acomputer with an Intel i5 CPU running at 2.53 GHz.

The time required to calculate the transmittance and reflectanceat a single frequency was recorded for each version of the modelas the number of spatial harmonics was increased from 1 to 441.This data is provided in Figure 7. The improved scattering matrixformulation performed 20% faster than the standard formulation. Thiswas due solely to there being fewer computations when calculating thelayer scattering matrices. When the other benefits of the improvedscattering matrix formulation were exploited, the speed increased bymore than 23× over the standard formulation.

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7. CONCLUSION

The computational electromagnetics community has been adoptinginefficient and unconventional scattering matrix formulations for semi-analytical methods. The scattering parameters are being defineddifferent than long-standing convention. In many formulations, S11

is defined as a transmission quantity instead of reflection and S21 isdefined as a reflection quantity instead of transmission. To addressthis situation, scattering matrices in this paper were reformulatedconsistent with convention. In addition, each layer was separated byfree space gaps with zero thickness enabling scattering matrices foreach layer to be arbitrarily interchanged. The symmetry imposed bythis procedure leads to symmetric scattering matrices where S11 = S22

and S12 = S21 for all of the layers composing the device. This makesthe simulation faster, more efficient, and require less memory. Bysimulating transmittance and reflectance for an electromagnetic bandgap material, speed was increased by 20% due solely to the improvedsymmetry of the scattering matrices. A 23 fold increase in simulationspeed was observed when the ability to interchange layers was exploitedin the algorithm.

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