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ECE 1252 – INTRODUCTION TO COMPUTATIONAL ELECTRODYNAMICS 1

The Finite-Difference Frequency-Domain MethodHans-Dieter Lang, Student-Member, IEEE

Abstract—The finite-difference frequency-domain (FDFD)method is introduced and implemented in 1D, 2D and 2.5D forboth driven mode and eigenmode analysis. Simulation resultsand execution time performances are presented and comparedto those obtained using the well-known FDTD method.

Index Terms—finite-difference, frequency-domain, FDFD,eigenmode analysis, resonator modes

I. INTRODUCTION

FOR most applications, the finite-difference time-domain(FDTD) method is an effective and versatile technique

to solve Maxwell’s equations. However, for some cases, thefrequency-domain comes more natural and is more usefulthan the time-domain. Of course, results obtained using thelatter can be transformed into the frequency domain usingthe Fourier transform, but getting the right resolution orbeing able to pick some special frequency might require aprohibitively large amount of time steps or interpolation. Thus,it can be interesting to solve the models at the frequency ofinterest right from the beginning. Also, there are a few caseswhere the simulations are difficult to do or cannot be done(accurately enough) using time-domain methods. One exampleis simulating the resonance frequencies and quality factors ofhigh-𝑄 resonators [11].

The finite-difference frequency-domain (FDFD) method is asimple yet highly capable frequency-domain technique, basedon Yee’s cell [8], originally developed for the FDTD method.While often Lui and Cheng [3] are credited with introducingthe FDFD method, the idea has been around at least a fewyears longer and could have been introduced or at leastanalyzed first by Rappaport and McCartin in 1991 [1].

II. THE FDFD METHOD

A. Derivation in 1-D

To derive the FDFD method, one can start either fromMaxwell’s (curl) equations in the frequency domain [12]

∇×E = −𝑗𝜔𝜇H (1a)∇×H = 𝑗𝜔𝜀E+ J (1b)

or from the wave equations in the frequency domain, e.g.,(∇2 + 𝑘2

)E = 𝑗𝜔𝜇J , (2)

where, 𝑘2 = 𝜔2𝜀𝜇 is the wavenumber (squared). In thissection, Maxwell’s curl equations will be used to derive theFDFD in 1-D, whereas the wave equation will be used forthe derivation of the 2.5-D eigenmode algorithm in the nextsection.

Hans-Dieter (H. D.) Lang, ID: 1000182599, is a PhD candidate with theelectromagnetics group, at the University of Toronto, Ontario, Canada.Email: [email protected]. December 21, 2012, v1.1.

For a 𝑦-polarized wave propagating in 𝑥-direction, the curlequations result in

𝜕𝑥𝐸𝑦 = −𝑗𝜔𝜇𝐻𝑧 (3a)𝜕𝑥𝐻𝑧 = −𝑗𝜔𝜀𝐸𝑦 − 𝐽𝑦 (3b)

where the short notation 𝜕𝑥 = 𝜕/𝜕𝑥 is used. These equationscan then be discretized using finite differences, as known fromthe FDTD method:

𝐸𝑦𝑖+1 − 𝐸𝑦

𝑖

Δ𝑥= −𝑗𝜔𝜇𝐻𝑧

𝑖+1/2 (4a)

𝐻𝑧𝑖+1/2 −𝐻𝑧

𝑖−1/2

Δ𝑥= −𝑗𝜔𝜀𝐸𝑦

𝑖 − 𝐽𝑦𝑖 (4b)

where, similarly to the FDTD cases, the electrical field islocated at integer indices and the magnetic field 𝐻 in betweenthe 𝐸𝑖-field nodes, at half-integer indices. Since we will haveto assemble all the fields in a column vector, it makes senseto change the indices to pure integer index numbers, called𝑛𝑦 for the 𝑦-directed electric fields and 𝑛𝑧 for the 𝑧-directedmagnetic fields, as shown in Fig. 1. Note that 𝑛𝑦 = 1 ... 𝑁𝑥+1and 𝑛𝑧 = 1 ... 𝑁𝑥, where 𝑁𝑥 is the total amount of cells inthe 𝑥-direction.

𝑥𝑖 =

𝐸1=0

1

𝐸2

2

𝐸3

3

𝐸4

4

𝐸5

5

𝐻1

1.5

𝐻2

2.5

𝐻3

3.5

𝐻4

4.5

Δ𝑥

Fig. 1: Discretized space for electric and magnetic fields in 1-D.

As can be seen from Eqs. (4a) and (4b), unlike in theFDTD case1, these equations cannot be put into an updateequation for the electric (magnetic) field depending on theprevious values of the electric (magnetic) field and the updatedmagnetic (electric) fields. Instead, all equations are gatheredand put into matrix form2:

⎡⎢⎢⎢⎢⎢⎣1 0 0 0 . . .

1/Δ𝑥 𝑗𝜔𝜇 −1/Δ𝑥 0 . . .0 1/Δ𝑥 𝑗𝜔𝜀 −1/Δ𝑥

0 0 1/Δ𝑥 𝑗𝜔𝜇. . .

......

. . .. . .

⎤⎥⎥⎥⎥⎥⎦⏟ ⏞

A

⎡⎢⎢⎢⎢⎢⎣𝐸1

𝐻1

𝐸2

𝐻2

...

⎤⎥⎥⎥⎥⎥⎦⏟ ⏞

x

=

⎡⎢⎢⎢⎢⎢⎣0

0

−𝐽2

0

...

⎤⎥⎥⎥⎥⎥⎦⏟ ⏞

b

(5)

1This is only true for the first-order FDTD case, for higher-order FDTDschemes similar issues arise.

2Evidently, (5) is not yet optimized and still contains unnecessary rows inorder to show the principle most clearly. Also, it can be more efficient to usea field vector x = [E,H]𝑇 instead of the shown form, with alternating 𝐸𝑖

and 𝐻𝑖+1/2.


where A is the operator matrix, x is the unknown field vectorand b is the source vector. This linear system is then solvedusing either

∙ Direct inversion

Ax = b ⇒ x = A−1b (6)

(for example using x=A\b in Matlab)∙ Iterative methods∙ Least-square (for badly or ill-conditioned matrixes)

Since the actual method to invert matrices is beyond the scopeof this project, we will not consider it any further. Neither willwe consider optimizing the operator matrix A accordingly.

B. PML for FDFD

The derivation of the PML for FDFD is straightforwardand similar to the PML for FDTD. The fields are absorbedin a lossy material; the electric fields in a lossy dielectricand the magnetic fields in a lossy magnetic material. Bothare achieved by adding a loss term (non-zero conductivity) tothe permittivity and permeability, respectively:

𝑗𝜔𝜀𝐸𝑦𝑖+1

𝑗𝜔𝜇𝐻𝑧𝑖+1/2

inside PML−−−−−−−→

(𝑗𝜔 +

𝜎𝑖+1

𝜀

)𝜀𝐸𝑦

𝑖+1(𝑗𝜔 +

𝜎𝑖+1/2

𝜀

)𝜇𝐻𝑦

𝑖+1/2

(7)

where 𝜎 is the electric conductivity and 𝜎/𝜀 = 𝜎*/𝜇 (𝜇* isthe magnetic conductivity) was used (similar to FDTD), tominimize reflections.

To minimize numerical artifacts, the conductivity 𝜎 insidethe PML region is gradually varied using a polynomial of order𝑝. Similar to the FDTD, the maximum conductivity 𝜎max alongwith the order of the polynomial 𝑝 are chosen according toempirical rules. The rules for optimizing PML for FDTD canbe applied and result in good overall performances, but otherrules [2], [4], [5] have been shown to be even better.

For more dimensional problems anisotropic PML are used,just as in FDTD.

C. Results

1) PML Verification: From the steady-state solution, thereflection coefficient can be easily calculated, for exampleusing

|Γ| = VSWR− 1

VSWR+ 1(8)

where VWSR is the voltage standing wave ratio. Here insteadof the voltages the electric fields are used directly. Thephase could be found by comparing the destinations of theminima (or zeros) and maxima compared to the distance tothe reflective interface. To see if the PML is working, severalone-dimensional simulations were made, an example is shownin Fig. 4. It shows the electric and magnetic fields alonga transmission line, once with shorted ends and once withthe PML absorber in place. Since in that case no reflectionoccurs, the result is the same as terminating the line withits characteristic impedance. Note that this termination has animpact all along the line, which can be seen by the envelopes(absolute values).

2) Dispersive media: As is well known, modeling dis-persive media in FDTD can become quite involved. Thefrequency-dependences are most commonly modeled usingtime derivatives of additionally introduced variables, resultingin auxiliary differential equations. The resulting computationalcomplexity is high and instability problems can arise, for somespecial values for the dispersion models, which might requirea finer spatial resolution and therefore even more increase incomputational demands.

In the frequency domain the situation is much simpler: sincethe frequency of interest 𝜔0 is chosen and put into Maxwell’sequations, the same can be done with media parameter modelwhich then results in a new single value for 𝜀(𝜔)|𝜔=𝜔0 and𝜇(𝜔)|𝜔=𝜔0 .

Reflection at an interface of air and a well-known Lorentzmedia with the frequency-dependent permittivity modeled by

𝜀(𝜔) = 𝜀∞ + (𝜀𝑠 − 𝜀∞)𝜔20

𝜔20 + 2𝑗𝜔𝛿 − 𝜔2

(9)

(using 𝜀∞ = 𝜀0, 𝜀𝑠 = 2.25 𝜀0, 𝛿 = 0.28 × 1016 s−1 and𝜔0 = 4.0×1016 rad/sec) was simulated both using FDTD andFDFD and the results are shown in Fig. 2. As can be seen,

0 0.5 1 1.5 2 2.5 3x 1016

0

0.2

0.4

0.6

0.8

1s1

1 (li

near

)

Frequency (Hz)

FDTD: s=0.9FDTD: s=1analyticFDFD

|Γ|(

linea

r)

Fig. 2: Reflection coefficient vs. frequency of FDFD (40 values,simulation time < 3 s), FDTD (8192 values, approx. 10 s) and

analytic methods.

the agreement is very good, even though for the FDFD lineonly 40 frequency steps were simulated (compared to 8192for the FDTD). Of course, with FDTD being a broadbandsimulation method, the reflection factor is obtained for a largefrequency span (depending on the time discretization and thetotal number of time steps), whereas with the FDFD onlya value for a single frequency results for each simulation.However, when it is looked for the reflection factor at a specialfrequency, a high number of time steps (or interpolation) mightbe required for FDTD, whereas for FDFD just one simulationat that frequency has to be carried out.

Fig. 3 shows snapshots from both simulation techniques:∙ In the time domain (FDTD, subfigure a) the wave (Gaus-

sian pulse) is launched at the blue dashed line (travelingboth to the left and right) and measured at the blackdashed line. First the incident wave is measured and thenthe reflection from the interface (red line) is measured


0 20 40 60 80 100 120 140 160 180 200

−0.1−0.05

00.05

0.1

Cell number

Fiel

d am

plitu

de (a

.u.)

𝜀 = 𝜀0 𝜀 = 2𝜀0

∙ No PML: shorted TL∙ VSWR → ∞∙ 𝐸 and 𝐻 (𝑉 and 𝐼) out of

phase → no real power flow

(a) Transmission line with shorted ends (PEC boundary)

0 20 40 60 80 100 120 140 160 180 200

−0.1−0.05

00.05

0.1

Cell number

Fiel

d am

plitu

de (a

.u.)

𝜀 = 𝜀0 𝜀 = 2𝜀0

∙ With PML: matched TL∙ VSWR → 1

∙ 𝐸 and 𝐻 (𝑉 and 𝐼) outin phase → real power flowtowards the ends (loads)

(b) Transmission line with matched ends (PML boundary)

0 20 40 60 80 100 120 140 160 180 200−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Re(E)Im(E)Abs(E)Re(H)Im(H)Abs(H)

Fig. 4: Evidence of a working PML in a 1-D FDFD transmission line simulation using the standing wave ratio of the envelopes.(Source at center, different permittivities/wave impedances: left 𝜀 = 𝜀0, 𝑍 = 𝜂; right 𝜀 = 2𝜀0, 𝑍 = 𝜂/

√2)

0 1000 2000 3000 4000 5000 6000

−0.1

0

0.1

0.2

0.3

0.4

Timestep n=9701

Source

Lorentz media

Transmittedpulse

Reflected pulse

(Pulse propagatingtowards left)

Ele

ctric

field

(µV

/m)

Cell number

(a) FDTD

0 100 200 300 400 500−5

0

5x 10−7

Lorentz media

VSWR ⇒ |Γ|Decay in the media

∝ e−𝛼𝑥

Source

Ele

ctric

field

(V/m

)

Cell number

(b) FDFD

Fig. 3: Lorentz media modeled in FDTD and FDFD.

and the two are compared after transforming them to thefrequency domain.

∙ Using FDFD (subfigure b) the wave is launched at theblue dashed line again and also travels to the left andright side. Since this is a steady-state simulation wecannot get away with moving the left boundary far awayenough, so that no reflection thereof is recorded. Instead,an absorbing boundary (PML) has to be used. The sameis true for the right end (inside the media), where thePML just uses the 𝜀Lorentz(𝜔) at the simulated frequency𝜔 = 𝜔0 and no problems as in dispersive PML for FDTDarise. The magnitude of the reflection coefficient can becalculated using the VSWR between the interface and thesource. Also, the exponential decay in the Lorentz media,due to the inherit loss is visual.

Evidently both methods lead to the same results here, butthe FDFD implementation is certainly much simpler andstraightforward than the one in FDTD. As discussed, absorbingboundaries (PML) can be considered more important for theFDTD method, since in the resulting steady-case we cannotget away with moving unwanted problems far enough awayfrom the probe and therefore the domain always has to beterminated by the respective boundary conditions.

III. FDFD EIGENFREQUENCY ANALYSIS

There are two kinds of eigenfrequency or eigenmode anal-yses that can be carried out:

∙ eigenfrequency analysis, to find the resonance frequencies𝜔𝑛 of some structure (e.g. a resonator)

∙ generalized eigenfrequency analysis (or eigenmode anal-ysis), to find some propagation constant 𝛽 according atsome frequency 𝜔


For both the corresponding eigenvectors (field distributions)can also be found3. The first is more general and was used tofind resonance modes in resonators.

The 2.5-D FDFD eigenmode analysis algorithm is based ona similar idea to the one of the 2.5-D FDTD method for guidedwave structures introduced by Xiao and Vahldieck 1992/93 [9],[10], based on Yee’s mesh [8].

A. FDFD Resonator Eigenfrequency Analysis

1) Dipole Resonances in 1-D: The eigenvalue problemin one dimension (e.g. eigenfrequency analysis of a simpletransmission line resonator) can be obtained by discretizingthe one-dimensional wave equation for the electric field ineigenvalue problem formulation:(

𝜕2𝑥 + 𝑘2

)𝐸𝑦 = 0

Eigenvalue−−−−−→ 𝜕2𝑥𝐸𝑦 = −𝜔2𝜀𝜇⏟ ⏞

𝜆

𝐸𝑦 (10)

(Note that in this section 𝜆 stands for the eigenvalue, notwavelength!). The second-order derivative is approximatedusing the second-order finite difference

𝜕2𝑥𝐸𝑦 ≈

𝐸𝑦𝑖+1 − 2𝐸𝑦

𝑖 + 𝐸𝑦𝑖−1

Δ𝑥2𝑖 = 1 ... 𝑁𝑥 , (11)

where 𝐸𝑦0 = 𝐸𝑦

𝑁𝑥+1 = 0, for now (implicit short or PECends). The full4 eigenvalue problem in matrix form becomes

1

Δ𝑥2

⎡⎢⎢⎢⎢⎢⎣2 −1 0 . . .

−1 2 −1. . .

0 −1 2. . .

.... . . . . . . . .

⎤⎥⎥⎥⎥⎥⎦⏟ ⏞

A

⎡⎢⎢⎢⎢⎢⎣𝐸1

𝐸2

𝐸3

...

⎤⎥⎥⎥⎥⎥⎦ = 𝜆

⎡⎢⎢⎢⎢⎢⎣𝐸1

𝐸2

𝐸3

...

⎤⎥⎥⎥⎥⎥⎦ . (12)

In this case the matrix A is not just sparse but a symmetrictridiagonal band matrix. Also, the eigenvalue can be replacedby 𝜆′ = 𝜆Δ𝑥2, which makes the whole problem independentof the actual discretization; the only remaining dependence ison the actual number of nodes 𝑁𝑥.

For the open ends, instead of the Dirichlet boundary con-ditions of implicit ends 𝐸𝑥

0 = 𝐸𝑥𝑁𝑥+1 = 0 we use Neumann

boundary conditions, i.e.,

𝜕𝑥𝐸𝑦 ≈ 1

Δ𝑥(𝐸𝑦

1 − 𝐸𝑦0 ) = 0 ⇒ 𝐸𝑦

0 = 𝐸𝑦1 (13)

and similarly for 𝐸𝑦𝑁𝑥+1. This changes the finite difference at

these positions from (11) to, e.g.,

𝜕2𝑥𝐸𝑦

𝑥=0

≈𝐸𝑦

𝑖+1 − 2𝐸𝑦𝑖 + 𝐸𝑦

𝑖−1

Δ𝑥2

𝑖=1

=𝐸𝑦

2 − 𝐸𝑦1

Δ𝑥2(14)

and, thus, in the first and last entries of the matrix A the twosare replaced by ones.

3In Matlab the function eig() is used to solve the eigenvalue problems.If the function is called using two variables (instead of just one for theeigenvalues), i.e. [eigenvectors eigenvalues]=eig(A);, both theeigenvalues and the corresponding eigenvectors (in the modal matrix) can befound using a single command.

4Note that, this example the matrix A is already optimized, meaningimplicitly reduced by zero rows and columns since these would be the entriesfor 𝐸𝑥

0 and 𝐸𝑦𝑁𝑥+1, which are not contained in the matrix.

Fig. 5 shows the eigenmodes of a 1-D transmission lineresonator of length 𝑙 = 150mm, with shorted (a) and open(b) ends. These results can be interpreted as the currentand electric field distributions along a dipole antenna (half-wavelength at around 1GHz) at various frequencies. Notethat, although the computation in this case runs very fast5,i.e. 𝑡sim < 0.01 s, because of the nice properties of the matrix,A is already of dimensions (𝑁𝑥 + 1)× (𝑁𝑥 + 1).

20 40 60 80 100−0.2

−0.1

0

0.1

0.2

Cell number

Fiel

d am

plitu

de (a

.u.)

𝑓GHz =0.97991.95562.93883.91734.8949

(a) Shorted ends, similar to current distributions of a dipole antenna.

20 40 60 80 100−0.2

−0.1

0

0.1

0.2

Cell number

Fiel

d am

plitu

de (a

.u.)

𝑓GHz =00.98961.9792.96793.956

(b) Open ends, similar to electric field amplitudes along a dipole antenna.

Fig. 5: Dipole resonance frequencies and field distributions.

2) 2-D Cavity Resonator Modes: The 2-D resonator eigen-frequency solver is obtained using the wave equation ofthe electric fields in two dimensions (for TE modes) andfollowing steps very similar to the ones presented for the one-dimensional case. For simplicity only PEC boundaries (zerotangential electric fields) are considered.

Figs. 6 and 7 show some results obtained using the FDFDeigenmode method in two dimensions. The total nodes ofthe electric field 𝐸𝑧 , thus total number of equations, are𝑁equations = (𝑁𝑥 + 1) × (𝑁𝑦 + 1), where 𝑁𝑥 and 𝑁𝑦 arethe number of cells in the respective dimension. The operatormatrix A is then of size 𝑁equations × 𝑁equations > (𝑁𝑥𝑁𝑦)

2.Thus, the matrix contains more entries than the square of thetotal number of cells. While the matrix can be optimized6

considerably (because of zero rows and columns, due to theboundary conditions), it becomes clear why the FDFD methodis known to require considerably more RAM than the FDTDmethod and why iterative matrix inversion algorithms are oftenmore practical than direct inversion. Since A is very sparse,especially for very large matrices the RAM consumption canbe greatly reduced by storing just the non-zero values, along

5Reference System: Intel Core i7 (Dual-Core) 2 GHz, 8 GB 1600 MHzDDR3 RAM, SSD

6In matlab this can be realized using:A(∼any(A,2),:)=[]; % delete zero rowsA(:,∼any(A,1))=[]; % delete zero columns


with their positions7.

10 20 30

10

20

30 Mode n=1

10 20 30

10

20

30 Mode n=2

10 20 30

10

20

30 Mode n=3

10 20 30

10

20

30 Mode n=4

10 20 30

10

20

30 Mode n=5

10 20 30

10

20

30 Mode n=6

0

0.02

0.04

0.06

−0.05

0

0.05

−0.05

0

0.05

−0.05

0

0.05

−0.05

0

0.05

−0.05

0

0.05

Fig. 6: The first six TE resonator modes of a rectangular cavity(35× 30 cells, Δ𝑥 = Δ𝑦 = 1mm).

At first glance, the diagonal modes in Fig. 7 might comesomewhat unexpected, but these field distributions are theresult of superposition of the second and third modes ofthe rectangular cavity when the widths in 𝑥-direction and 𝑦-direction become equal (and thus both modes are present at thesame (eigen-)frequency). The same is true for modes 5 and 6,where the fields cancel (mode 5) or add (mode 6) along bothdiagonals for the square cavity.

3) Comparison to FDTD: Compared to the eigenfrequencyanalysis using FDTD, the FDFD is again more straightfor-ward — and will find the modes with certainty. Using FDTD,the field in the cavity would be excited by a broadband pulseat some position inside, and be probed at some other point(s).To "ensure" that all modes are picked up, multiple field probescould be distributed randomly. An other way could be to locatethe probes at points of irrational factors of the aspect ratio ofthe resonator (e.g. 1/

√2), since the zeros always occur at

rational fractions of the width and length of the resonator.Since the grid is discretized, there are no real irrational pointsavailable and, thus, both methods cannot ensure to pick upall the modes reliably all the time (which should generallynot be a problem, however, if enough probes are used). Also,to reduce "noise" and make the modes separable enough,FDTD mode analysis can require a high number of time steps,especially for higher-order modes.

With the FDFD eigenmode analysis, depending on the ac-tual eigenvalue decomposition algorithm, all eigenfrequencies

7Matlab provides the command sparse() for this purpose (which isefficient for sparse matrices with densities of about 2/3 or lower, as givenin the help manual). However, some functions such as eig() cannot dealwith sparse matrices (if they do not have special properties, e.g. symmetry)and the command full() has to be used first, to again receive a commonfull matrix, thus reducing the effect for unoptimized applications.

10 20 30

10

20

30

Mode n=1

0

0.02

0.04

10 20 30

10

20

30

Mode n=2

−0.05

0

0.05

10 20 30

10

20

30

Mode n=3

−0.05

0

0.05

10 20 30

10

20

30

Mode n=4

−0.05

0

0.05

10 20 30

10

20

30

Mode n=5

−0.05

0

0.05

10 20 30

10

20

30

Mode n=6

−0.04−0.0200.020.040.06

Fig. 7: The first six TE resonator modes of a square cavity(35× 35 cells, Δ𝑥 = Δ𝑦 = 1mm).

(that can be resolved by the discretized spatial grid, accordingto the Nyquist-limit) or those close to a chosen value, arefound directly. With combined eigenvalue/eigenvector solversalso the field distributions can be found directly. Thus, also inthis case, the frequency-domain methods come more naturallyand straightforward than the time-domain counterparts.

B. 2.5-D FDFD Simulation of Propagation Constants of aGuided Wave Structure

1) Derivation: If we assume that the guided wave structureis uniform along the 𝑧-axis and the wave propagates in thepositive 𝑧-direction, the guided fields can be expressed as: [6]

E(𝑥, 𝑦, 𝑧)

=[x𝐸𝑥(𝑥, 𝑦) + y𝐸𝑦(𝑥, 𝑦) + z𝐸𝑧(𝑥, 𝑦)

]e−𝑗𝛽𝑧 , (15a)

H(𝑥, 𝑦, 𝑧)

=[x𝐻𝑥(𝑥, 𝑦) + y𝐻𝑦(𝑥, 𝑦) + z𝐻𝑧(𝑥, 𝑦)

]e−𝑗𝛽𝑧 , (15b)

where, as usual, the harmonic time variation e 𝑗𝜔𝑡 is sup-pressed. Further, derivatives with respect to 𝑧 can be takenanalytically and result in

𝜕𝑧𝐺𝑣 = −𝑗𝛽𝐺𝑣 (16)

for any field quantity 𝐺 in direction 𝑣. Thus follows forMaxwell’s curl equations

∇×E =

⎡⎣ 𝜕𝑦𝐸𝑧 + 𝑗𝛽𝐸𝑦

−𝑗𝛽𝐸𝑥 − 𝜕𝑥𝐸𝑧

𝜕𝑥𝐸𝑦 − 𝜕𝑦𝐸

𝑥

⎤⎦ = −𝑗𝜔𝜇

⎡⎣𝐻𝑥

𝐻𝑦

𝐻𝑧

⎤⎦ (17a)

⇓⎡⎣ 𝑗𝜔𝜇𝐻𝑦 − 𝜕𝑥𝐸𝑧

−𝜕𝑦𝐸𝑧 − 𝑗𝜔𝜇𝐻𝑥

𝜕𝑥𝐸𝑦 − 𝜕𝑦𝐸

𝑥 + 𝑗𝜔𝜇𝐻𝑧

⎤⎦ = 𝑗𝛽

⎡⎣𝐸𝑥

𝐸𝑦

0

⎤⎦ (17b)


for the electric field (Faraday’s law) and from the curl of themagnetic field (Ampère’s law)

∇×H =

⎡⎣ 𝜕𝑦𝐻𝑧 + 𝑗𝛽𝐻𝑦

−𝑗𝛽𝐻𝑥 − 𝜕𝑥𝐻𝑧

𝜕𝑥𝐻𝑦 − 𝜕𝑦𝐻

𝑥

⎤⎦ = 𝑗𝜔𝜇

⎡⎣𝐸𝑥

𝐸𝑦

𝐸𝑧

⎤⎦ (18a)

⇓⎡⎣ −𝜕𝑥𝐻𝑧 − 𝑗𝜔𝜀𝐸𝑦

𝑗𝜔𝜀𝐸𝑥 − 𝜕𝑦𝐻𝑧

𝜕𝑥𝐻𝑦 − 𝜕𝑦𝐻

𝑥 − 𝑗𝜔𝜀𝐸𝑧

⎤⎦ = 𝑗𝛽

⎡⎣𝐻𝑥

𝐻𝑦

0

⎤⎦ (18b)

These equations could already be implemented as⎡⎢⎢⎢⎢⎢⎢⎣0 0 −𝜕𝑥 0 𝑗𝜔𝜇 00 0 −𝜕𝑥 −𝑗𝜔𝜇 0 0

−𝜕𝑦 𝜕𝑥 0 0 0 𝑗𝜔𝜇0 −𝑗𝜔𝜀 0 0 0 −𝜕𝑥

𝑗𝜔𝜀 0 0 0 0 −𝜕𝑦

0 0 −𝑗𝜔𝜀 −𝜕𝑦 𝜕𝑥 0

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣E𝑥

E𝑦

E𝑧

H𝑥

H𝑦

H𝑧

⎤⎥⎥⎥⎥⎥⎥⎦ = 𝑗𝛽

⎡⎢⎢⎢⎢⎢⎢⎣E𝑥

E𝑦

0H𝑥

H𝑦

0

⎤⎥⎥⎥⎥⎥⎥⎦(19)

but since two (matrix) equations are independent of theeigenvalue 𝛽, the system can be optimized further. By solvingthe respective rows for the 𝑧-components and filling the resultsinto the respective spots, we get⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0𝜕𝑥𝜕𝑦𝑗𝜔𝜀

𝑗𝜔𝜇− 𝜕2𝑥

𝑗𝜔𝜀

0 0𝜕2𝑦

𝑗𝜔𝜀− 𝑗𝜔𝜇 −𝜕𝑦𝜕𝑥

𝑗𝜔𝜀

−𝜕𝑥𝜕𝑦𝑗𝜔𝜇

𝜕2𝑥

𝑗𝜔𝜇− 𝑗𝜔𝜀 0 0

𝑗𝜔𝜀−𝜕2𝑦

𝑗𝜔𝜇

𝜕𝑦𝜕𝑥𝑗𝜔𝜇

0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

·

⎡⎢⎢⎣E𝑥

E𝑦

H𝑥

H𝑦

⎤⎥⎥⎦ = 𝑗𝛽

⎡⎢⎢⎣E𝑥

E𝑦

H𝑥

H𝑦

⎤⎥⎥⎦ (20)

where we once again see the coupled nature of Maxwell’sequations: the electric field depends only on the magneticfield and vice versa. The equation can be further simplifiedby solving for the magnetic fields and filling them into theequations for the electric fields, resulting in the well-knownwave equations(

𝜔2𝜀𝜇+ 𝜕2𝑥 + 𝜕2

𝑦

) [E𝑥

E𝑦

]= 𝛽2

[E𝑥

E𝑦

]. (21)

which can be discretized straightforwardly using finite differ-ences for the second-order derivatives as discussed previously.

Note that this simplification assumed uniform 𝜇 and 𝜀,which is not true for most problems. In general, the respective𝜀𝑝,𝑞 and 𝜇𝑝,𝑞 (where 𝑥 = 𝑝 · Δ𝑥 and 𝑦 = 𝑞 · Δ𝑦 are thediscretized positions) have to be carried all the way throughthe derivation, leading to coupled formulations for the twoelectric field components of the form [7]

𝑎𝑥1𝐸𝑥𝑝+1,𝑞 + 𝑎𝑥2𝐸

𝑥𝑝−1,𝑞 + 𝑎𝑥3𝐸

𝑥𝑝,𝑞−1 + 𝑎𝑥4𝐸

𝑥𝑝,𝑞+1

+ 𝑎𝑥5𝐸𝑦𝑝+1,𝑞−1 + 𝑎𝑥6𝐸

𝑦𝑝+1,𝑞 + 𝑎𝑥7𝐸

𝑦𝑝,𝑞−1 + 𝑎𝑥8𝐸

𝑦𝑝,𝑞

+ 𝑎𝑥9𝐸𝑥𝑝,𝑞 = 𝛽2𝐸𝑥

𝑝,𝑞 (22)

where 𝑎𝑥𝑖 (𝑖 = 1 ... 9) are coefficients, related to an averagepermittivity around the node, e.g. [7]

𝑎𝑥1 = − 2

Δ𝑥2· 𝜀𝑝,𝑞 + 𝜀𝑝,𝑞+1

𝜀𝑝,𝑞 + 𝜀𝑝,𝑞−1 + 𝜀𝑝+1,𝑞 + 𝜀𝑝+1,𝑞−1. (23)

Eqs. (21) was implemented to simulate the propagationconstant 𝛽 of a common waveguide (PEC walls). Using finitedifferences for the derivatives follows

𝐸𝑡𝑝+1,𝑞 − 2𝐸𝑡

𝑝,𝑞 + 𝐸𝑡𝑝−1,𝑞

Δ𝑥2+

𝐸𝑡𝑝,𝑞+1 − 2𝐸𝑡

𝑝,𝑞 + 𝐸𝑡𝑝,𝑞−1

Δ𝑦2

+ 𝜔2𝜀𝜇𝐸𝑡𝑝,𝑞 = 𝛽2𝐸𝑡

𝑝,𝑞 (24)

To state this in a linear system consisting of a matrix Aacting on the electric fields in form of a vector x = [E𝑥,E𝑦]𝑇 ,the coordinate indices 𝑝 and 𝑞 are mapped for the 𝑥 and 𝑦components, respectively, according to

𝐸𝑥𝑝𝑥,𝑞𝑥 : (𝑝𝑥, 𝑞𝑥) ↦−→ 𝑛 = 𝑝𝑥 + (𝑞𝑥 − 1)𝑁𝑥 (25a)

𝐸𝑦𝑝𝑦,𝑞𝑦 : (𝑝𝑦, 𝑞𝑦) ↦−→ 𝑚 = 𝑝𝑦 + (𝑞𝑦 − 1)(𝑁𝑥 + 1) . (25b)

where

𝑝𝑥 = 1, ... , 𝑁𝑥 𝑞𝑥 = 1, ... , 𝑁𝑦 + 1

𝑝𝑦 = 1, ... , 𝑁𝑥 + 1 𝑞𝑦 = 1, ... , 𝑁𝑦

Note that both mappings contain the number of cells in 𝑥-direction 𝑁𝑥, since that is the dimension along which thevectors are numbered first. Thus, we get, for example for𝑁𝑥 = 2 and 𝑁𝑦 = 3 a situation as shown in Fig. 8.

𝐸𝑥1

𝐸𝑥2

𝐸𝑥3

𝐸𝑥4

𝐸𝑥5

𝐸𝑥6

𝐸𝑥7

𝐸𝑥8

𝐸𝑦1

𝐸𝑦2

𝐸𝑦3

𝐸𝑦4

𝐸𝑦5

𝐸𝑦6

𝐸𝑦7

𝐸𝑦8

𝐸𝑦9

𝐸𝑧1

𝐸𝑧2

𝐸𝑧3

𝐸𝑧4

𝐸𝑧5

𝐸𝑧6

𝐸𝑧7

𝐸𝑧8

𝐸𝑧9

𝐸𝑧10

𝐸𝑧11

𝐸𝑧12

𝑥, 𝑝

𝑦, 𝑞

Fig. 8: Simplified Yee’s cells in 2.5D using wave equations (𝐸fields only, 𝐸𝑧 is just shown for completeness, it is only implicitly

contained in the algorithm), for 𝑁𝑥 = 2 and 𝑁𝑦 = 3.

Using these mappings, the equations become

𝐸𝑥𝑛+1 − 2𝐸𝑥

𝑛 + 𝐸𝑥𝑛−1

Δ𝑥2+

𝐸𝑥𝑛+𝑁𝑥

− 2𝐸𝑥𝑛 + 𝐸𝑥

𝑛−𝑁𝑥

Δ𝑦2

+ 𝜔2𝜀𝜇𝐸𝑥𝑛 = 𝛽2𝐸𝑥

𝑛 𝑛 ∈ 1, 2, ..., 𝑁𝑥(𝑁𝑦 + 1) (26)

for the 𝑥-directed and

𝐸𝑦𝑚+1 − 2𝐸𝑦

𝑛 + 𝐸𝑦𝑚−1

Δ𝑥2+𝐸𝑦

𝑚+(𝑁𝑥+1) − 2𝐸𝑦𝑚 + 𝐸𝑦

𝑚−(𝑁𝑥+1)

Δ𝑦2

+ 𝜔2𝜀𝜇𝐸𝑦𝑚 = 𝛽2𝐸𝑦

𝑚 𝑚 ∈ 1, 2, ..., (𝑁𝑥 + 1)𝑁𝑦 (27)

for the 𝑦-directed fields.


2) Boundary conditions: The problem area (grid) has tobe of finite extent and at the boundaries we have to definesome physical relations for the fields. The simplest and mostcommon boundary is a perfect electric conductor (PEC), whichstates that tangential fields are zero at the boundary and normalcomponents can be dealt with using image theory. For 𝐸𝑥 theboundaries are

∙ 𝐸𝑥𝑝𝑥,𝑞𝑥 = 0 for 𝑞𝑥 = 1,∀ 𝑝𝑥

→ 𝐸𝑥𝑛 = 0 for 𝑛 = 1, ..., 𝑁𝑥

∙ 𝐸𝑥𝑝𝑥,𝑞𝑥 = 0 for 𝑞𝑥 = 𝑁𝑦 + 1,∀ 𝑝𝑥

→ 𝐸𝑥𝑛 = 0 for 𝑛 = 𝑁𝑥 𝑁𝑦 + 1, ... , 𝑁𝑥(𝑁𝑦 + 1)

∙ 𝐸𝑥𝑝𝑥−1,𝑞𝑥 = 𝐸𝑥

𝑝𝑥,𝑞𝑥 (image) for 𝑝𝑥 = 1,∀ 𝑞𝑥→ 𝐸𝑥

𝑛−1 = 𝐸𝑥𝑛 for 𝑛 = 1 + (𝑞𝑥 − 1)𝑁𝑥

∙ 𝐸𝑥𝑝𝑥+1,𝑞𝑥 = 𝐸𝑥

𝑝𝑥,𝑞𝑥 (image) for 𝑝𝑥 = 𝑁𝑥,∀ 𝑞𝑥→ 𝐸𝑥

𝑛+1 = 𝐸𝑥𝑛 for 𝑛 = 𝑞𝑥𝑁𝑥

and similarly for 𝐸𝑦 . The zeros dominate (are applied afterthe images) in case of boundaries on both 𝑝 and 𝑞.

3) Algorithm: The following steps are taken in the algo-rithm to assemble the matrix:

a) Initialization: preassign zero square matrices of di-mensions 𝑁𝑥(𝑁𝑦 + 1) by (𝑁𝑥 + 1)𝑁𝑦 for the 𝐸𝑥

and 𝐸𝑦 , respectively.b) Put 𝜔2𝜀𝜇−2/Δ𝑥2−2/Δ𝑦2 on the main diagonal of

the matrix and 1/Δ𝑥2 along the diagonals one to theleft and right of the main diagonal as well as 1/Δ𝑦2

to the diagonals at a ±𝑁𝑥 distance. Do the same forthe 𝐸𝑦 matrix, but use a distance of 𝑁𝑥 + 1 for thedistance to the diagonal in the later case.

c) Apply the correct boundaries: first apply the imagesby changing the respective positions on the maindiagonals and then set the right cells to zero to cancelthe tangential field parts.

d) At this point other boundary conditions (inside theproblem domain) could be applied.

4) Results: The propagation characteristics 𝛽 and 𝛼 of thefirst three TE𝑛0 modes were simulated using a finite-differenceimplementation of the wave equation of the electric field andthe results compared to analytical solutions, as shown in Fig. 9.The dimensions of the waveguide are 19.05mm × 9.525mm(to be able to compare the results to [7]), which was discretizedinto 24× 12 cells. The matrix dimension becomes 300× 300,but the simulation time is still shorter than one second. As canbe seen, the agreement is very good.

IV. CONCLUSIONS & OUTLOOK

A. SummaryThe FDFD method was introduced and applied to several

well-known problems, where the results and execution timeperformances were compared to those obtained with the well-known FDTD method. As shown, the FDFD method is astraightforward application of finite differences to the spatialderivatives (curls) of Maxwell’s equations in the frequencydomain (phasor form).

Solving Maxwell’s equations in the frequency domain canbe advantageous over the time domain for some applications,especially:

2 4 6 8 10 120

50

100

150

200

250

Frequency (GHz)

` (ra

d/m

), _

(Np/

m)

` analytic_ analytic` FDFD_ FDFDTEM limitcutoffs

TE10

TE20

TE01 TE30

Fig. 9: The propagation constants 𝛽 and attenuation constants 𝛼 forthe TE10, TE20, TE01 and TE30 modes of a rectangular waveguide.

Dimensions 19.05mm × 9.525mm, according to [7].

∙ Simulations using dispersive media∙ Eigenmode analysis∙ Simulations of high-𝑄 resonators and their properties

As shown, FDFD methods are quite fast when results have tobe obtained at a certain frequency, where with FDTD a largenumber of time steps (or interpolation or other enhancements)have to be simulated, to get the same resolution. Also, unlikethe FDTD method, FDFD is inherently stable, since it isessentially inverting a matrix to solve a linear system. On theother hand, this matrix, containing all the spatial derivativesof the fields in the model, grows very fast with increasingmodel sizes and complexities. That is why FDFD is knownto require (much) more RAM compared to FDTD, since thismatrix needs to be stored at once.

B. Personal Notes

On a personal note, doing simulations in FDFD (and ver-ifying and debugging an FDFD code) has proved to be verydifferent than FDTD. First of all, since FDFD simulates thesteady-state, we can no longer get away with certain issuesby "moving them far enough away" so that their effects donot compromise the simulation results. Thus, a good PMLboundary is much more important than in FDTD. Also, usingFDFD there is no way to start a simulation and then forexample observe what happens to the propagating wave tosee where a problem is located — a property of the FDTDmethod which the author grew to like quite a bit over manysimulations.

Although the FDFD method is commonly used in Photonics(since there simulations of dispersive media are very common)and other areas, literature on this topic is quite sparse. Mostreports focus on special cases and enhancements for specificapplications.

C. Outlook

For further investigations, for example simulation of dielec-tric resonators in multiple dimensions a working (anisotropic)PML suited for eigenmode analysis is required. Also, espe-cially for circular (disk) resonators, it could be interesting


to have a conformal mesh, e.g. like the polar coordinatesystem, where the FDFD could be implemented by discretizingMaxwell’s equations in cylindrical coordinates.

ACKNOWLEDGMENT

The author would like to thank Prof. Costas D. Sarris,Dr. G. Kevin Zhu and Neeraj Sood for their helpful hints andcomments.

REFERENCES

[1] C. M. Rappaport, B. J. McCartinFDFD Analysis of Electromagnetic Scattering in Anisotropic MediaUsing Unconstrained Triangular MeshesIEEE Transactions on Antennas and Propagation, Vol. 39, No. 3, March1991

[2] C. M. RappaportPerfectly Matched Absorbing Boundary Conditions Based on AnisotropicLossy Mapping of SpaceIEEE Microwave and Guided Wave Letters, Vol. 5, No. 3, March 1995

[3] M.-L. Lui, Z. ChengA direct computation of propagation constant using compact 2-Dfull-wave eigen-based finite-difference frequency-domain techniqueProceedings of the 1999 International Conference on ComputationalElectromagnetics and Its Applications (ICCEA ’99), p. 78-81, 1999

[4] E. A. Marengo, C. M. Rappaport, E. L. MillerOptimum PML ABC Conductivity Profile in FDFDIEEE Transactions on Magnetics, Vol. 35, No. 3, May 1999

[5] C. M. Rappaport, M. Kilmer, E. L. MillerAccuracy considerations in using the PML ABC with FDFD Helmholtzequation computationInternational Journal of Numerical Modelling: Electronic Networks,Devices and Fields, Vol. 13, No. 5, October 2000

[6] Y.-J. Zhao, K.-L. Wu, K.-K. M. ChengA Compact 2-D Full-Wave Finite-Difference Frequency-Domain Methodfor General Guided Wave StructuresIEEE Transactions on Microwave Theory and Techniques, Vol. 50, No.7, July 2002

[7] L.-Y. Li, J.-F. MaoAn Improved Compact 2-D Finite-Difference Frequency-DomainMethod for Guided Wave structuresIEEE Microwave and Wireless Components Letters, Vol. 13, No. 12,December 2003

[8] K. S. YeeNumerical solution of initial boundary value problems involvingMaxwells equations in isotropic mediaIEEE Transactions on Antennas and Propagation, Vol. 14, May 1966

[9] S. Xiao, R. Vahldieck, H. JinA fast 2-D FDTD full wave analyzer for arbitrary guided wavestructuresIEEE Microwave Guided Wave Letters, Vol. 2, No. 5, May 1992

[10] S. Xiao, R. VahldieckAn Efficient 2-D FDTD Algorithm Using Real VariablesIEEE Microwave and Guided Wave Letters, Vol. 3, No. 5, May 1993

[11] Aliaksandra IvinskayaFinite-Difference Frequency-Domain Method in NanophotonicsPhD Thesis, Technical University of Denmark, Lyngby, 2011

[12] Umran S. Inan, Robert A. MarshallNumerical Electromagnetics – The FDTD MethodCambridge University Press 2011

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