{{projectinfo|V&V|Modeling Dispersive Materials Using EM.Tempo|ART DISP title.png|In this article, several periodic structures involving different types of dispersive materials are simulated using EM.Tempo and EM.Picasso, and the results are validated by the published data.|
*[[EM.Tempo]]
*Dispersive Material
*Debye Pole
:<math> \varepsilon (\omega) = \varepsilon_\infty + \sum_{p=1}^N \dfrac{\Delta \varepsilon_p}{1 + j\omega \tau_p}, \quad \Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_\infty </math>
where <math>\varepsilon_{\infty}</math> is the value of the permittivity at infinite frequency, <math>\tau_p</math> is the relaxation time corresponding to the p''th'' pole having the unit of seconds, and <math>\varepsilon_{sp}</math> is the value of the static permittivity (at DC) corresponding to the p''th'' pole. <math>\Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty}</math> represents the change in permittivity due to the p''th'' pole. Water has a Debye pole with parameters τ = 9.4×10<sup>-12</sup>s, ε<sub>s</sub> = 81 and ε<sub>∞</sub> = 1.8. In this example, we consider a laterally infinite slab of water with a finite thickness of 6mm. A periodic unit cell with lateral periods of 3mm along both X and Y directions is are assumed. Figure 1 shows the geometry setup for the periodic unit cell of the water slab in [[EM.Tempo]]. The top and bottom domain walls are assumed to be convolutional perfectly matched layers (PML). The periodic structure is excited using a normally incident plane wave source. <table><tr><td>[[Image:ART DISP10.png|thumb|left|300px|Figure 1: Geometry of the periodic unit cell of the dispersive water slab in EM.Tempo.]]</td></tr></table>
The table below summarizes the simulation parameters:
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Figure 1 shows the geometry setup for the periodic unit cell of the water slab in [[EM.Tempo]]. The top and bottom domain walls are assumed to be convolutional perfectly matched layers (CPML). The periodic structure is excited using a normally incident plane wave source.
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[[Image:ART DISP10.png|thumb|left|300px|Figure 1: Geometry of the periodic unit cell of the dispersive water slab in EM.Tempo.]]
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Figure 2 shows the results for the reflection coefficient of the water slab as computed by [[EM.Tempo]] and compares them to the analytical data given by Ref. [2]. A very good agreement between the two data sets is observed.
== Drude Plasma Slab ==
Next, [[EM.Tempo]]'s periodic boundary condition will be used to simulate a plasma slab with infinite extents in the X and Y directions, but a finite Z-thickness of 1.5 cm. The Drude model often provides a good abstraction for an unmagnetized isotorpic non-magnetized plasma. The complex permittivity of a Drude material with N poles is given by:
:<math> \varepsilon(\omega) = \varepsilon_{\infty} - \sum_{p=1}^N \dfrac{{\omega_p}^2}{\omega^2 - j\omega \nu_p} </math>
where <math>\omega_p</math> and <math>\nu_p</math> are the angular plasma frequency and angular collision frequency corresponding to the p''th'' pole, respectively, and both are expressed in rad/s. For an unmagnetized isotropic non-magnetized plasma, <math>\varepsilon_{\infty} = 1</math>1. A Drude pole with ω<sub>p</sub> = 1.803×10<sup>11</sup> rad/s, and ν<sub>cp</sub> = 2×10<sup>10</sup>radcollisions/s is used as the dispersive model for this project. Figure 3 shows the geometry setup for the periodic unit cell of the Drude plasma slab in [[EM.Tempo]]. A box of dimensions 10mm × 10mm × 15mm is considered, with lateral periods of 10mm along both X and Y directions. The top and bottom domain walls are assumed to be convolutional perfectly matched layers (PML). The periodic structure is excited using a normally incident plane wave source. <table><tr><td>[[Image:ART DISP1.png|thumb|left|300px|Figure 3: Geometry of the periodic unit cell of a plasma slab modeled with a Drude pole in EM.Tempo.]]</td></tr></table>
The table below summarizes the simulation parameters:
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Figure 3 shows Note that, due to the geometry setup infinite nature of ε as frequency approaches zero in the Drude model, the excitation bandwidth is set up to avoid the very bottom of the band. Also, since energy in the low end of the band may take a very long time to dissipate (the magnitude of the conductivity is very high here), we set a relatively stringent termination criteria. Figures 4 and 5 show the results for the periodic unit cell magnitude and phase of the reflection coefficient of the Drude plasma slab in as computed by [[EM.Tempo]]and compare them to the exact data given by Ref. A box [1]. Figures 6 and 7 show the results for the magnitude and phase of dimensions 10mm × 10mm × 15mm is considered, with lateral periods the transmission coefficient of 10mm along both X and Y directionsthe plasma slab as computed by [[EM. The top Tempo]] and bottom domain walls are assumed compare them to be convolutional perfectly matched layers (CPML)the exact data given by Ref. The periodic structure [1]. A very good agreement between the two data sets is excited using a normally incident plane wave sourceobserved in all the figures.
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[[Image:ART DISP1DISP2.png|thumb|left|300px550px|Figure 34: Geometry Magnitude of the periodic unit cell reflection coefficient of a plasma slab (as modeled with a Drude pole in ). Solid line: results computed by [[EM.Tempo]], symbols: data presented by Ref. [1].]]
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Note that, due to the infinite nature of ε as frequency approaches zero in the Drude model, the excitation bandwidth is set up to avoid the very bottom of the band<table><tr><td>[[Image:ART DISP4. Also, since energy in the low end png|thumb|left|550px|Figure 5: Phase of the band may take a very long time to dissipate (the magnitude of the conductivity is very high here), we set a relatively stringent termination criteria. Figures 4 and 5 show the results for the reflection coefficient of the a plasma slab (as modeled with a Drude pole). Solid line: results computed by [[EM.Tempo]] and compares them to the exact , symbols: data given presented by Ref. [1]. Figures 6 and 7 show the results for the transmission coefficient of the plasma slab as computed by [[EM.Tempo]] and compares them to the exact data given by Ref. [1]. A very good agreement between the two data sets is observed in all the figures.</td></tr></table>
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[[Image:ART DISP2DISP3.png|thumb|left|550px|Figure 46: Magnitude of the reflection transmission coefficient of a plasma slab (as modeled with a Drude pole). Solid line: results computed by [[EM.Tempo]], symbols: data presented by Ref. [1].]]
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[[Image:ART DISP4DISP5.png|thumb|left|550px|Figure 57: Phase of the reflection transmission coefficient of a plasma slab (as modeled with a Drude pole). Solid line: results computed by [[EM.Tempo]], symbols: data presented by Ref. [1].]]
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== Sandwiched Cross-Dipole Slot Frequency Selective Surface (FSS) ==
In the third example, we will model a sandwiched frequency selective surface (FSS) structure described in Ref. [2]. This periodic structure consists of a cross-dipole patterned metallic layer sandwiched between two Debye material layers. The Debye layers have an equal thickness of 2.5mm. The periodic unit cell has equal periods of 2mm along both X and Y directions as shown in Figure 8. The cross-dipole slot consists of two orthogonal rectangles with a length of 1.8mm and a width of 0.6mm as shown in Figure 9. The total thickness of the FSS structure is 5mm.
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[[Image:ART DISP3DISP25.png|thumb|left|550px300px|Figure 68: Magnitude Geometry of the transmission coefficient periodic unit cell of a plasma slab (as modeled with a Drude pole). Solid line: results computed by [[the sandwiched cross-dipole slot FSS in EM.Tempo]], symbols: data presented by Ref. [1].]]
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[[Image:ART DISP5DISP26.png|thumb|left|550px300px|Figure 79: Phase The top view of the periodic unit cell of the sandwiched cross-dipole slot FSS.]]</td></tr></table> In this project, the Debye pole has parameters τ = 5.27×10<sup>-10</sup> s, ε<sub>s</sub> = 5.2 and ε<sub>∞</sub> = 3.7. The table below summarizes the simulation parameters: {| class="wikitable"|-! scope="col"| Center Frequency! scope="col"| Bandwidth! scope="col"| Grid Spacing! scope="col"| Termination Criterion|-| 5GHz| 10GHz| 0.1 mm | -100dB |} The transmission coefficient of a plasma slab the dispersive FSS computed by [[EM.Tempo]] is shown in Figure 10. The computed data matches the HFSS results presented by Ref. [2]. <table><tr><td>[[Image:ART DISP27.png|thumb|left|550px|Figure 10: Transmission coefficient of the sandwiched cross-dipole slot FSS. Solid line: results computed by [[EM.Tempo]], symbols: HFSS data presented by Ref. [2].]]</td></tr></table> In order to better understand the effect of the metal-slot FSS, it was removed from the structure of Figure 5 and the remaining thick Debye layer (as modeled with a Drude poleh = 5mm)was simulated using [[EM.Tempo]]. The computed results for the reflection coefficient of the combined Debye layers without the metal-slot FSS are shown in Figure 11, where the [[EM.Tempo]] data are compared again to the HFSS results presented by Ref. [2]. <table><tr><td>[[Image:ART DISP28.png|thumb|left|550px|Figure 11: Transmission coefficient of the combined Debye layers without the metal-slot FSS. Solid line: results computed by [[EM.Tempo]], symbols: HFSS data presented by Ref. [12].]]
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== Nano-Plasmonic Solar Cell ==
The third fourth structure to be considered in this article is a periodic nanoparticle-coated solar cell structure. A silicon-dioxide substrate is coated with an array of patterned silver-particle cuboids, which can be modeled with two distinct Debye poles. The nanoparticles are used to increase the optical absorption within semiconductor solar cells. The complex permittivity of a dispersive material with two Debye poles is given by:
:<math> \varepsilon(\omega) = \varepsilon_{\infty} + \dfrac{\varepsilon_{s1}-\varepsilon_{\infty}}{1 + j\omega \tau_1} + \dfrac{\varepsilon_{s2}-\varepsilon_{\infty}}{1 + j\omega \tau_2} </math>
The values of the parameters in the above expression corresponding to the silver nano-particles are given in the table below:
:<math> \varepsilon(\omega) = \varepsilon_{\infty}- \sum_{p| class=1}^N \dfrac{\Delta \varepsilon_p {\omega_p}^2}{\omega^2 "wikitable"|- 2j\omega \delta_p - {\omega_p}^2}, \quad \Delta \varepsilon_p ! scope= \varepsilon_{sp} - \varepsilon_{\infty} </math>"col"| Parameter! scope="col"| Valuewhere |-| ε<mathsub>\omega _p∞</mathsub> and | 4.391|-| τ<mathsub>\delta_p1</mathsub> are the angular resonant frequency and angular damping frequency corresponding to the p''th'' pole, respectively, and both are expressed in rad/s| 6. Similar to a Debye material, 4879×10<mathsup>\Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty}12</mathsup> represents the change in permittivity due to the p''th'' pole. In this project, the Lorentz material has the parameters s|-| &omegaepsilon;<sub>ps1</sub> = 2| 4.202548233×10<sup>127</sup> rad/s, |-| &deltatau;<sub>p2</sub> = 1| 3.45597×10<sup>13-14</sup> rad/s, and Δ|-| ε = 4.8e7. The silver<sub>s2</sub>| -particles increase the absorption of the solar cell [2]. 5072×10<sup>5</sup>|}
For this project, a silicon-dioxide substrate with a 30nm thickness and ε<sub>r</sub> = 3.9 is considered. The silver nano-particle cuboids have dimensions: 20nm × 20nm × 10nm as shown in Figure 12. The lateral periods are 30nm in both X and Y directions. The table below summarizes the simulation parameters:
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[[Image:ART DISP5DISP14.png|thumb|left|300px|Figure 312: Geometry of the periodic unit cell of the nano-particle coated solar cell in EM.Tempo.]]
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For this simulation, it is important to remember that ε(ω) will have a resonant peak near ω<sub>p</sub> = 2.20254×10<sup>12</sup> rad/s. The project bandwidth should be chosen to avoid this peak, since obtaining accurate results at these frequencies can be very difficult. The transmission coefficient for of this structure is shown in Figure 513, where the results computed by [[EM.Tempo]] are compared with the HFSS results of presented by Ref. [2], and a reasonable agreement is observed.
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[[Image:ART DISP6DISP15.png|thumb|left|550px|Figure 513: Transmission coefficient of the nano-particle-coated solar cell. Solid line: results computed by [[EM.Tempo]], symbols: data presented by Ref. [2].]]
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== JerusalemReflection from a Two-Cross Frequency Selective Surface (FSS) Pole Lorentz Medium ==
In the second example, we will model a sandwiched frequency selective surface (FSS) The last structure described to be considered in Ref. [2]. A Jerusalem-cross patterned metallic layer this article is sandwiched between a two Debye material layers-pole Lorentz medium occupying the half-space Z < 0. The complex permittivity of a Debye dispersive material with N two Lorentz poles is given by:
:<math> \varepsilon (\omega) = \varepsilon_{\infty + } - G_1 \sum_dfrac{p=1(\varepsilon_{s1}^N - \dfracvarepsilon_{\Delta infty}){\varepsilon_pomega_1}^2}{1 + j\omega ^2 - 2j\tau_p}, omega \quad delta_1 - {\Delta omega_1}^2} - G_2 \varepsilon_p = dfrac{(\varepsilon_{sps2} - \varepsilon_{\infty }){\omega_2}^2}{\omega^2 - 2j\omega \delta_2 - {\omega_2}^2} </math>
where <math>\varepsilon_{\infty}omega _p</math> is the value of the permittivity at infinite frequency, and <math>\tau_pdelta_p</math> is are the relaxation time angular resonant frequency and angular damping frequency corresponding to the p''th'' pole having the unit of seconds, respectively, and <math>\varepsilon_{sp}<both are expressed in rad/math> is the value of the static permittivity (at DC) corresponding s. Similar to the p''th'' pole. a Debye material, <math>\Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty}</math> represents the change in permittivity due to the p''th'' pole. The coefficients G<sub>1</sub> and G<sub>2</sub> are the weights used for the two pole terms. In this projectorder to model the half-space, a periodic unit cell of dimensions 2mm × 2mm × 50mm is considered as shown in Figure 14. The lateral periods are 2mm in both X and Y directions. The values of the Debye pole has parameters in the above expression are given in the table below: {| class="wikitable"|-! scope="col"| Parameter! scope="col"| Value|-| &tauepsilon; = <sub>∞</sub>| 1.5|-| ω<sub>1</sub>| 1.272566×10<sup>11</sup> rad/s|-| δ<sub>1</sub>| 1.2566×10<sup>10</sup> and rad/s|-| &Deltaepsilon;<sub>s1</sub>| 3|-| G<sub>1</sub>| 0.4|-| ω<sub>2</sub>| 3.1416×10<sup>11</sup> rad/s|-| δ<sub>2</sub>| 3.1416×10<sup>10</sup> rad/s|-| ε = 1<sub>s2</sub>| 3|-| G<sub>2</sub>| 0.56|} <table><tr><td>[[Image:ART DISP19. png|thumb|left|300px|Figure 14: Geometry of the periodic unit cell of the two-pole Lorentz half-space medium in EM.Tempo.]]</td></tr></table>
The table below summarizes the simulation parameters:
! scope="col"| Termination Criterion
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| 5GHz500 THz| 10GHz500 THz| 0.1 mm .5nm | -100dB 50dB
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Figures 15 and 16 show the results for the magnitude and phase of the reflection coefficient of the two-pole Lorentz medium, respectively. Both figures compare the simulation data computed by [[EM.Tempo]] and the exact data given by Ref. [1].
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[[Image:ART DISP3DISP20.png|thumb|left|300px550px|Figure 315: Geometry Magnitude of the periodic unit cell reflection coefficient of the Jerusalemtwo-cross FSS on a dispersive substrate in pole Lorentz medium. Solid line: results computed by [[EM.Tempo]], symbols: data presented by Ref. [1].]]
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[[Image:ART DISP4DISP21.png|thumb|left|550px|Figure 316: Transmission Phase of the reflection coefficient of the Jerusalemtwo-cross FSSpole Lorentz medium. Solid line: results computed by [[EM.Tempo]], symbols: data presented by Ref. [21].]]
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The transmission coefficient of the Jerusalem-cross FSS computed by [[EM.Tempo]] is shown in Figure 3. The computed data matches the result from Ref. [2]. These data are quite different from the transmission coefficient of a structure where the sandwhich layers have a frequency-independent permittivity.
Figure 3:
== References ==
[2] K. El-Mahgoub, F. Yang, and A. Elsherbeni, Scattering Analysis of Periodic Structures Using Finite-Difference Time-Domain Method. Morgan & Claypool Publishers, 2012.
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