http://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&limit=500&action=history&feed=atomMaterials in Meep - Revision history2024-03-28T18:37:17ZRevision history for this page on the wikiMediaWiki 1.7.3http://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&diff=4863&oldid=prevArdavan: /* Loss and gain */2016-08-05T23:24:56Z<p><span class="autocomment">Loss and gain</span></p>
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<tr><td> </td><td style="background: #eee; font-size: smaller;">If &gamma; above is nonzero, then the dielectric function &epsilon;(&omega;) becomes ''complex'', where the imaginary part is associated with absorption loss in the material if it is positive, or gain if it is negative. Alternatively, a dissipation loss or gain may be added by a positive or negative conductivity, respectively&mdash;this is often convenient if you only care about the imaginary part of &epsilon; in a narrow bandwidth, and is described in detail below.</td><td> </td><td style="background: #eee; font-size: smaller;">If &gamma; above is nonzero, then the dielectric function &epsilon;(&omega;) becomes ''complex'', where the imaginary part is associated with absorption loss in the material if it is positive, or gain if it is negative. Alternatively, a dissipation loss or gain may be added by a positive or negative conductivity, respectively&mdash;this is often convenient if you only care about the imaginary part of &epsilon; in a narrow bandwidth, and is described in detail below.</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;">If you look at Maxwell's equations, then <math>d\mathbf{P}/dt</math> plays exactly the same role as a current <math>\mathbf{J}</math>. Just as <math>\mathbf{J} \cdot \mathbf{E}</math> is the rate of change of mechanical energy (the power expended by the electric field on moving the currents), therefore, the rate at which energy is lost to absorption <span style="color: red; font-weight: bold;">for </span>is given by:</td><td>+</td><td style="background: #cfc; font-size: smaller;">If you look at Maxwell's equations, then <math>d\mathbf{P}/dt</math> plays exactly the same role as a current <math>\mathbf{J}</math>. Just as <math>\mathbf{J} \cdot \mathbf{E}</math> is the rate of change of mechanical energy (the power expended by the electric field on moving the currents), therefore, the rate at which energy is lost to absorption is given by:</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">:absorption rate <math>\sim \frac{d\mathbf{P}}{dt} \cdot \mathbf{E}</math> </td><td> </td><td style="background: #eee; font-size: smaller;">:absorption rate <math>\sim \frac{d\mathbf{P}}{dt} \cdot \mathbf{E}</math> </td></tr>
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Ardavanhttp://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&diff=4862&oldid=prevArdavan at 19:41, 3 August 20162016-08-03T19:41:23Z<p></p>
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<tr><td> </td><td style="background: #eee; font-size: smaller;">If you specify a Lorentzian resonance <math>\omega_n</math> at too high a frequency relative to the time discretization <math>\Delta t</math>, the simulation becomes unstable. Essentially, the problem is that you are trying to model a <math>\mathbf{P}_n</math> that oscillates too fast compared with the time discretization for the discretization to work properly. If this happens, you have three options: increase the resolution (which increases the resolution in both space and time), decrease the Courant factor (which decreases <math>\Delta t</math> compared to <math>\Delta x</math>), or use a different model function for your dielectric response.</td><td> </td><td style="background: #eee; font-size: smaller;">If you specify a Lorentzian resonance <math>\omega_n</math> at too high a frequency relative to the time discretization <math>\Delta t</math>, the simulation becomes unstable. Essentially, the problem is that you are trying to model a <math>\mathbf{P}_n</math> that oscillates too fast compared with the time discretization for the discretization to work properly. If this happens, you have three options: increase the resolution (which increases the resolution in both space and time), decrease the Courant factor (which decreases <math>\Delta t</math> compared to <math>\Delta x</math>), or use a different model function for your dielectric response.</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;">Roughly speaking, the <math>\mathbf{P}_n</math> equation becomes unstable for <math>\omega_n \Delta t / 2 > 1</math>. (Note that, in Meep frequency units, you specify <math>f_n = \omega_n/2\pi</math>, so this quantity should be less than <math>1/\pi \Delta t</math>.) <span style="color: red; font-weight: bold;">A future version of </span>Meep will check a necessary stability criterion automatically and halt with an error message if it is violated.</td><td>+</td><td style="background: #cfc; font-size: smaller;">Roughly speaking, the <math>\mathbf{P}_n</math> equation becomes unstable for <math>\omega_n \Delta t / 2 > 1</math>. (Note that, in Meep frequency units, you specify <math>f_n = \omega_n/2\pi</math>, so this quantity should be less than <math>1/\pi \Delta t</math>.) Meep will check a necessary stability criterion automatically and halt with an error message if it is violated.</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">== Loss and gain ==</td><td> </td><td style="background: #eee; font-size: smaller;">== Loss and gain ==</td></tr>
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Ardavanhttp://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&diff=4788&oldid=prevStevenj: /* Material dispersion */2015-02-12T16:31:19Z<p><span class="autocomment">Material dispersion</span></p>
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<tr><td> </td><td style="background: #eee; font-size: smaller;">:<math>\mathbf{D} = \varepsilon_\infty \mathbf{E} + \mathbf{P}</math></td><td> </td><td style="background: #eee; font-size: smaller;">:<math>\mathbf{D} = \varepsilon_\infty \mathbf{E} + \mathbf{P}</math></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;">where <math>\varepsilon_\infty</math> (which [[Meep FAQ|must be positive]]) is the ''instantaneous'' dielectric function (the infinite-frequency response) and '''P''' is the remaining frequency-dependent ''polarization'' density in the material. '''P''', in turn, has its ''own'' time-evolution equation, and the exact form of this equation determines the frequency-dependence &epsilon;(&omega;). [Note that Meep's definition of &omega; uses a sign convention <math>\exp(-i\omega t)</math> for the time <span style="color: red; font-weight: bold;">dependence</span>.] In particular, Meep supports any material dispersion of the form of a sum of harmonic resonances, plus a term from the frequency-independent electric conductivity:</td><td>+</td><td style="background: #cfc; font-size: smaller;">where <math>\varepsilon_\infty</math> (which [[Meep FAQ|must be positive]]) is the ''instantaneous'' dielectric function (the infinite-frequency response) and '''P''' is the remaining frequency-dependent ''polarization'' density in the material. '''P''', in turn, has its ''own'' time-evolution equation, and the exact form of this equation determines the frequency-dependence &epsilon;(&omega;). [<span style="color: red; font-weight: bold;">'''</span>Note<span style="color: red; font-weight: bold;">''' </span>that Meep's definition of &omega; uses a sign convention <math>\exp(-i\omega t)</math> for the time <span style="color: red; font-weight: bold;">dependence—&epsilon; formulas in engineering papers that use the opposite sign convention for &omega; will have a sign flip in all the imaginary terms below. If you are using parameters from the literature, you should use *positive* values of &gamma; and &sigma; as-is for loss; don't be confused by the difference in &omega; sign convention and flip the sign of the parameters</span>.] In particular, Meep supports any material dispersion of the form of a sum of harmonic resonances, plus a term from the frequency-independent electric conductivity:</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">:<math>\varepsilon(\omega,\mathbf{x}) = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{\omega} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{\sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega_n^2 - \omega^2 - i\omega\gamma_n} \right] ,</math></td><td> </td><td style="background: #eee; font-size: smaller;">:<math>\varepsilon(\omega,\mathbf{x}) = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{\omega} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{\sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega_n^2 - \omega^2 - i\omega\gamma_n} \right] ,</math></td></tr>
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Stevenjhttp://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&diff=4782&oldid=prevStevenj: /* Material dispersion */2014-07-09T16:18:17Z<p><span class="autocomment">Material dispersion</span></p>
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<tr><td> </td><td style="background: #eee; font-size: smaller;">:<math>\mathbf{D} = \varepsilon_\infty \mathbf{E} + \mathbf{P}</math></td><td> </td><td style="background: #eee; font-size: smaller;">:<math>\mathbf{D} = \varepsilon_\infty \mathbf{E} + \mathbf{P}</math></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;">where <math>\varepsilon_\infty</math> is the ''instantaneous'' dielectric function (the infinite-frequency response) and '''P''' is the remaining frequency-dependent ''polarization'' density in the material. '''P''', in turn, has its ''own'' time-evolution equation, and the exact form of this equation determines the frequency-dependence &epsilon;(&omega;). [Note that Meep's definition of &omega; uses a sign convention <math>\exp(-i\omega t)</math> for the time dependence.] In particular, Meep supports any material dispersion of the form of a sum of harmonic resonances, plus a term from the frequency-independent electric conductivity:</td><td>+</td><td style="background: #cfc; font-size: smaller;">where <math>\varepsilon_\infty</math> <span style="color: red; font-weight: bold;">(which [[Meep FAQ|must be positive]]) </span>is the ''instantaneous'' dielectric function (the infinite-frequency response) and '''P''' is the remaining frequency-dependent ''polarization'' density in the material. '''P''', in turn, has its ''own'' time-evolution equation, and the exact form of this equation determines the frequency-dependence &epsilon;(&omega;). [Note that Meep's definition of &omega; uses a sign convention <math>\exp(-i\omega t)</math> for the time dependence.] In particular, Meep supports any material dispersion of the form of a sum of harmonic resonances, plus a term from the frequency-independent electric conductivity:</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">:<math>\varepsilon(\omega,\mathbf{x}) = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{\omega} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{\sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega_n^2 - \omega^2 - i\omega\gamma_n} \right] ,</math></td><td> </td><td style="background: #eee; font-size: smaller;">:<math>\varepsilon(\omega,\mathbf{x}) = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{\omega} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{\sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega_n^2 - \omega^2 - i\omega\gamma_n} \right] ,</math></td></tr>
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Stevenjhttp://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&diff=4459&oldid=prevStevenj: /* Material dispersion */2012-07-20T21:27:40Z<p><span class="autocomment">Material dispersion</span></p>
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<tr><td> </td><td style="background: #eee; font-size: smaller;">:<math>\frac{i \sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega (\gamma_n- i\omega)}</math>,</td><td> </td><td style="background: #eee; font-size: smaller;">:<math>\frac{i \sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega (\gamma_n- i\omega)}</math>,</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;">which is equivalent to the Lorentzian model except that the <math>\omega_n^2</math> term has been omitted from the denominator, and asymptotes to a conductivity <<span style="color: red; font-weight: bold;">code</span>>\sigma_n \omega_n^2 / \gamma_n</<span style="color: red; font-weight: bold;">code</span>> as <math>\omega\to 0</math>. In this case, <math>\omega_n^2</math> is just a dimensional scale factor and has no interpretation as a resonance frequency.</td><td>+</td><td style="background: #cfc; font-size: smaller;">which is equivalent to the Lorentzian model except that the <math>\omega_n^2</math> term has been omitted from the denominator, and asymptotes to a conductivity <<span style="color: red; font-weight: bold;">math</span>>\sigma_n \omega_n^2 / \gamma_n</<span style="color: red; font-weight: bold;">math</span>> as <math>\omega\to 0</math>. In this case, <math>\omega_n^2</math> is just a dimensional scale factor and has no interpretation as a resonance frequency.</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">==Numerical stability==</td><td> </td><td style="background: #eee; font-size: smaller;">==Numerical stability==</td></tr>
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Stevenjhttp://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&diff=4458&oldid=prevStevenj: /* Material dispersion */2012-07-20T21:27:11Z<p><span class="autocomment">Material dispersion</span></p>
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<tr><td> </td><td style="background: #eee; font-size: smaller;">:<math>\frac{d^2\mathbf{P}_n}{dt^2} + \gamma_n \frac{d\mathbf{P}_n}{dt} = \sigma_n(\mathbf{x}) \omega_n^2 \mathbf{E}</math></td><td> </td><td style="background: #eee; font-size: smaller;">:<math>\frac{d^2\mathbf{P}_n}{dt^2} + \gamma_n \frac{d\mathbf{P}_n}{dt} = \sigma_n(\mathbf{x}) \omega_n^2 \mathbf{E}</math></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;">which corresponds to a term of the following form in &epsilon's &sum;<sub>''n''</sub>:</td><td>+</td><td style="background: #cfc; font-size: smaller;">which corresponds to a term of the following form in &epsilon<span style="color: red; font-weight: bold;">;</span>'s &sum;<sub>''n''</sub>:</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">:<math>\frac{i \sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega (\gamma_n- i\omega)}</math>,</td><td> </td><td style="background: #eee; font-size: smaller;">:<math>\frac{i \sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega (\gamma_n- i\omega)}</math>,</td></tr>
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Stevenjhttp://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&diff=4457&oldid=prevStevenj: /* Material dispersion */2012-07-20T21:26:57Z<p><span class="autocomment">Material dispersion</span></p>
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<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">Note that the conductivity <math>\sigma_D</math> corresponds to an imaginary part of &epsilon; given by (not including the harmonic-resonance terms) <math>i \varepsilon_\infty \sigma_D / \omega</math>. When you specify frequency in Meep units, however, you are specifying ''f'' without the 2&pi;, so the imaginary part of &epsilon; is <math>i \varepsilon_\infty \sigma_D / 2\pi f</math>.</td><td> </td><td style="background: #eee; font-size: smaller;">Note that the conductivity <math>\sigma_D</math> corresponds to an imaginary part of &epsilon; given by (not including the harmonic-resonance terms) <math>i \varepsilon_\infty \sigma_D / \omega</math>. When you specify frequency in Meep units, however, you are specifying ''f'' without the 2&pi;, so the imaginary part of &epsilon; is <math>i \varepsilon_\infty \sigma_D / 2\pi f</math>.</td></tr>
<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;">Meep also supports polarizations of the [[w:Drude model|Drude]] form, typically used for metals:</td></tr>
<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;">:<math>\frac{d^2\mathbf{P}_n}{dt^2} + \gamma_n \frac{d\mathbf{P}_n}{dt} = \sigma_n(\mathbf{x}) \omega_n^2 \mathbf{E}</math></td></tr>
<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;">which corresponds to a term of the following form in &epsilon's &sum;<sub>''n''</sub>:</td></tr>
<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;">:<math>\frac{i \sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega (\gamma_n- i\omega)}</math>,</td></tr>
<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;">which is equivalent to the Lorentzian model except that the <math>\omega_n^2</math> term has been omitted from the denominator, and asymptotes to a conductivity <code>\sigma_n \omega_n^2 / \gamma_n</code> as <math>\omega\to 0</math>. In this case, <math>\omega_n^2</math> is just a dimensional scale factor and has no interpretation as a resonance frequency.</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">==Numerical stability==</td><td> </td><td style="background: #eee; font-size: smaller;">==Numerical stability==</td></tr>
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Stevenjhttp://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&diff=4453&oldid=prevStevenj: /* Material dispersion */2012-07-20T21:05:47Z<p><span class="autocomment">Material dispersion</span></p>
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<tr><td> </td><td style="background: #eee; font-size: smaller;">:::<math> = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{2\pi f} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{\sigma_n(\mathbf{x}) \cdot f_n^2 }{f_n^2 - f^2 - if\gamma_n/2\pi} \right] ,</math></td><td> </td><td style="background: #eee; font-size: smaller;">:::<math> = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{2\pi f} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{\sigma_n(\mathbf{x}) \cdot f_n^2 }{f_n^2 - f^2 - if\gamma_n/2\pi} \right] ,</math></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;">where <math>\sigma_D</math> is the electric conductivity, <math>\omega_n</math> and <math>\gamma_n</math> are user-specified constants (or actually, the numbers that one specifies are <math>f_n = \omega_n / 2\pi</math> and <math>\gamma_n / 2\pi</math>), and the <math>\sigma_n(\mathbf{x})</math> is a user-specified function of position giving the strength of the ''n''-th resonance. The &sigma; parameters can be anisotropic <span style="color: red; font-weight: bold;">tensors </span>(<span style="color: red; font-weight: bold;">although currently only diagonal </span>tensors <span style="color: red; font-weight: bold;">are supported)</span>, while the frequency-independent term <math>\varepsilon_\infty</math> can be an arbitrary tensor <span style="color: red; font-weight: bold;">(not necessarily diagonal)</span>. This corresponds to evolving '''P''' via the equations:</td><td>+</td><td style="background: #cfc; font-size: smaller;">where <math>\sigma_D</math> is the electric conductivity, <math>\omega_n</math> and <math>\gamma_n</math> are user-specified constants (or actually, the numbers that one specifies are <math>f_n = \omega_n / 2\pi</math> and <math>\gamma_n / 2\pi</math>), and the <math>\sigma_n(\mathbf{x})</math> is a user-specified function of position giving the strength of the ''n''-th resonance. The &sigma; parameters can be anisotropic (<span style="color: red; font-weight: bold;">real-symmetric) </span>tensors, while the frequency-independent term <math>\varepsilon_\infty</math> can be an arbitrary <span style="color: red; font-weight: bold;">real-symmetric </span>tensor <span style="color: red; font-weight: bold;">as well</span>. This corresponds to evolving '''P''' via the equations:</td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">:<math>\mathbf{P} = \sum_n \mathbf{P}_n</math></td><td> </td><td style="background: #eee; font-size: smaller;">:<math>\mathbf{P} = \sum_n \mathbf{P}_n</math></td></tr>
</table>
Stevenjhttp://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&diff=3759&oldid=prevStevenj: /* Material dispersion */2009-09-10T21:41:19Z<p><span class="autocomment">Material dispersion</span></p>
<table border='0' width='98%' cellpadding='0' cellspacing='4' style="background-color: white;">
<tr>
<td colspan='2' width='50%' align='center' style="background-color: white;">←Older revision</td>
<td colspan='2' width='50%' align='center' style="background-color: white;">Revision as of 21:41, 10 September 2009</td>
</tr>
<tr><td colspan="2" align="left"><strong>Line 18:</strong></td>
<td colspan="2" align="left"><strong>Line 18:</strong></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">:<math>\varepsilon(\omega,\mathbf{x}) = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{\omega} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{\sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega_n^2 - \omega^2 - i\omega\gamma_n} \right] ,</math></td><td> </td><td style="background: #eee; font-size: smaller;">:<math>\varepsilon(\omega,\mathbf{x}) = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{\omega} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{\sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega_n^2 - \omega^2 - i\omega\gamma_n} \right] ,</math></td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;">:::<math> = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{2\pi f} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{<span style="color: red; font-weight: bold;">f_n</span>(\mathbf{x}) \cdot f_n^2 }{f_n^2 - f^2 - if\gamma_n/2\pi} \right] ,</math></td><td>+</td><td style="background: #cfc; font-size: smaller;">:::<math> = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{2\pi f} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{<span style="color: red; font-weight: bold;">\sigma_n</span>(\mathbf{x}) \cdot f_n^2 }{f_n^2 - f^2 - if\gamma_n/2\pi} \right] ,</math></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">where <math>\sigma_D</math> is the electric conductivity, <math>\omega_n</math> and <math>\gamma_n</math> are user-specified constants (or actually, the numbers that one specifies are <math>f_n = \omega_n / 2\pi</math> and <math>\gamma_n / 2\pi</math>), and the <math>\sigma_n(\mathbf{x})</math> is a user-specified function of position giving the strength of the ''n''-th resonance. The &sigma; parameters can be anisotropic tensors (although currently only diagonal tensors are supported), while the frequency-independent term <math>\varepsilon_\infty</math> can be an arbitrary tensor (not necessarily diagonal). This corresponds to evolving '''P''' via the equations:</td><td> </td><td style="background: #eee; font-size: smaller;">where <math>\sigma_D</math> is the electric conductivity, <math>\omega_n</math> and <math>\gamma_n</math> are user-specified constants (or actually, the numbers that one specifies are <math>f_n = \omega_n / 2\pi</math> and <math>\gamma_n / 2\pi</math>), and the <math>\sigma_n(\mathbf{x})</math> is a user-specified function of position giving the strength of the ''n''-th resonance. The &sigma; parameters can be anisotropic tensors (although currently only diagonal tensors are supported), while the frequency-independent term <math>\varepsilon_\infty</math> can be an arbitrary tensor (not necessarily diagonal). This corresponds to evolving '''P''' via the equations:</td></tr>
</table>
Stevenjhttp://ab-initio.mit.edu/wiki/index.php?title=Materials_in_Meep&diff=3758&oldid=prevStevenj: /* Material dispersion */2009-09-10T21:40:42Z<p><span class="autocomment">Material dispersion</span></p>
<table border='0' width='98%' cellpadding='0' cellspacing='4' style="background-color: white;">
<tr>
<td colspan='2' width='50%' align='center' style="background-color: white;">←Older revision</td>
<td colspan='2' width='50%' align='center' style="background-color: white;">Revision as of 21:40, 10 September 2009</td>
</tr>
<tr><td colspan="2" align="left"><strong>Line 18:</strong></td>
<td colspan="2" align="left"><strong>Line 18:</strong></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">:<math>\varepsilon(\omega,\mathbf{x}) = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{\omega} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{\sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega_n^2 - \omega^2 - i\omega\gamma_n} \right] ,</math></td><td> </td><td style="background: #eee; font-size: smaller;">:<math>\varepsilon(\omega,\mathbf{x}) = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{\omega} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{\sigma_n(\mathbf{x}) \cdot \omega_n^2 }{\omega_n^2 - \omega^2 - i\omega\gamma_n} \right] ,</math></td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;">::<math> = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{2\pi f} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{f_n(\mathbf{x}) \cdot f_n^2 }{f_n^2 - f^2 - if\gamma_n/2\pi} \right] ,</math></td><td>+</td><td style="background: #cfc; font-size: smaller;"><span style="color: red; font-weight: bold;">:</span>::<math> = \left( 1 + \frac{i \cdot \sigma_D(\mathbf{x})}{2\pi f} \right) \left[ \varepsilon_\infty(\mathbf{x}) + \sum_n \frac{f_n(\mathbf{x}) \cdot f_n^2 }{f_n^2 - f^2 - if\gamma_n/2\pi} \right] ,</math></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></td><td> </td><td style="background: #eee; font-size: smaller;"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;">where <math>\sigma_D</math> is the electric conductivity, <math>\omega_n</math> and <math>\gamma_n</math> are user-specified constants (or actually, the numbers that one specifies are <math>f_n = \omega_n / 2\pi</math> and <math>\gamma_n / 2\pi</math>), and the <math>\sigma_n(\mathbf{x})</math> is a user-specified function of position giving the strength of the ''n''-th resonance. The &sigma; parameters can be anisotropic tensors (although currently only diagonal tensors are supported), while the frequency-independent term <math>\varepsilon_\infty</math> can be an arbitrary tensor (not necessarily diagonal). This corresponds to evolving '''P''' via the equations:</td><td> </td><td style="background: #eee; font-size: smaller;">where <math>\sigma_D</math> is the electric conductivity, <math>\omega_n</math> and <math>\gamma_n</math> are user-specified constants (or actually, the numbers that one specifies are <math>f_n = \omega_n / 2\pi</math> and <math>\gamma_n / 2\pi</math>), and the <math>\sigma_n(\mathbf{x})</math> is a user-specified function of position giving the strength of the ''n''-th resonance. The &sigma; parameters can be anisotropic tensors (although currently only diagonal tensors are supported), while the frequency-independent term <math>\varepsilon_\infty</math> can be an arbitrary tensor (not necessarily diagonal). This corresponds to evolving '''P''' via the equations:</td></tr>
</table>
Stevenj