|
|
| Line 1: |
Line 1: |
| When an [[electromagnetic wave]] travels through a medium in which it gets absorbed (this is called an "[[opacity (optics)|opaque]]" or "[[attenuation constant|attenuating]]" medium), it undergoes [[exponential decay]] as described by the [[Beer–Lambert law]]. However, there are many possible ways to characterize the wave and how quickly it is absorbed. This article describes the mathematical relationships among:
| | By investing in a premium Word - Press theme, you're investing in the future of your website. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site. <br><br> |
| *[[Absorption coefficient]],
| |
| *[[Penetration depth]] and [[Skin depth]],
| |
| *[[Propagation constant]], [[attenuation constant]], [[phase constant]], and complex [[wavenumber]],
| |
| *[[Complex refractive index]] and extinction coefficient,
| |
| *[[Dielectric constant|Complex dielectric constant]],
| |
| *[[Alternating current|AC]] [[Electrical conductivity|conductivity]].
| |
| Note that in many of these cases there are multiple, conflicting definitions and conventions in common use. This article is not necessarily comprehensive or universal.
| |
|
| |
|
| == Background: Unattenuated wave ==
| | Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. Some of the Wordpress development services offered by us are:. In the event you beloved this information along with you would want to get more info regarding [http://d2.ae/wordpress_dropbox_backup_833049 wordpress dropbox backup] i implore you to visit the site. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. t need to use the back button or the URL to get to your home page. Word - Press makes it possible to successfully and manage your website. <br><br>It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory. <br><br>It has become a more prevalent cause of infertility and the fertility clinic are having more and more couples with infertility problems. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. To do this, you should link your posts to other relevant posts that you've created. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. Word - Press offers constant updated services and products, that too, absolutely free of cost. <br><br>There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. ) Remote Login: With the process of PSD to Wordpress conversion comes the advantage of flexibility. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
| {{main|Electromagnetic wave equation}}
| |
| | |
| A electromagnetic wave propagating in the +''z''-direction is conventionally described by the equation:
| |
| <math> \mathbf{E}(z,t) = \mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)})</math>
| |
| where
| |
| :'''E'''<sub>0</sub> is a vector in the ''x''-''y'' plane, with the units of an electric field (the vector is in general a [[complex vector]], to allow for all possible polarizations and phases),
| |
| :<math>\omega</math> is the [[angular frequency]] of the wave,
| |
| :''k'' is the [[angular wavenumber]] of the wave,
| |
| :Re indicates [[real part]]. | |
| :''e'' is [[e (mathematical constant)|Euler's number]]; see the article [[Complex exponential]] for information about how ''e'' is raised to complex exponents.
| |
| | |
| The [[wavelength]] is, by definition,
| |
| :<math> \lambda = \frac{2\pi}{k}</math> . | |
| For a given frequency, the wavelength of an electromagnetic wave is affected by the material in which it is propagating. The ''vacuum'' wavelength (the wavelength that a wave of this frequency would have if it were propagating in vacuum) is
| |
| :<math> \lambda_0 = \frac{2\pi c}{\omega}</math>
| |
| (''c'' is the [[speed of light|speed of light in vacuum]]). In the absence of attenuation, the [[index of refraction]] (also called [[refractive index]]) is the ratio of these two wavelengths, i.e.,
| |
| :<math>n = \frac{\lambda_0}{\lambda} = \frac{ck}{\omega}</math>.
| |
| | |
| The [[intensity (physics)|intensity]] of the wave is proportional to the square of the amplitude, time-averaged over many oscillations of the wave, which amounts to:
| |
| :<math> I(z) \propto |\mathbf{E}_0 e^{i(k z - \omega t)}|^2 = |\mathbf{E}_0|^2 </math>.
| |
| Note that this intensity is independent of the location ''z'', a sign that ''this'' wave is not attenuating with distance. We define ''I''<sub>0</sub> to equal this constant intensity:
| |
| :<math> I(z) = I_0 \propto |\mathbf{E}_0|^2</math>.
| |
| | |
| === Complex conjugate ambiguity ===
| |
| | |
| Because
| |
| :<math> \mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)}) = \mathrm{Re} (\mathbf{E}_0^* e^{-i(k z - \omega t)}),</math>
| |
| either expression can be used interchangeably. Generally, physicists and chemists use the convention on the left (with <math>e^{-i\omega t}</math>), while electrical engineers use the convention on the right (with <math>e^{+i\omega t}</math>, for example see [[electrical impedance]]). The distinction is irrelevant for an unattenuated wave, but becomes relevant in some cases below. For example, there are two definitions of [[refractive index|complex refractive index]], one with a positive imaginary part and one with a negative imaginary part, derived from the two different conventions.<ref name=refractiveindexconjugate>For the definition of complex refractive index with a positive imaginary part, see [http://books.google.com/books?id=K9YJ950kBDsC&pg=PA6 ''Optical Properties of Solids'', by Mark Fox, p. 6]. For the definition of complex refractive index with a negative imaginary part, see [http://books.google.com/books?id=qFl1mSZTtIcC&pg=PA588 ''Handbook of infrared optical materials'', by Paul Klocek, p. 588].</ref> The two definitions are [[complex conjugate]]s of each other.
| |
| | |
| == Absorption coefficient ==
| |
| {{main|Absorption coefficient|Beer-Lambert law}}
| |
| | |
| One way to incorporate attenuation into the mathematical description of the wave is via an '''[[absorption coefficient]]''':<ref name="Griffiths9.4.3">Griffiths, section 9.4.3.</ref>
| |
| :<math> \mathbf{E}(z,t) = e^{-\alpha_{abs} z / 2} \mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)})</math>
| |
| where <math>\alpha_{abs}</math> is the absorption coefficient. The intensity in this case satisfies:
| |
| :<math>I(z) \propto |e^{-\alpha_{abs} z/2}\mathbf{E}_0 e^{i(k z - \omega t)}|^2 = |\mathbf{E}_0|^2 e^{-\alpha_{abs} z}</math>
| |
| i.e.,
| |
| :<math>I(z) = I_0 e^{-\alpha_{abs} z}</math>
| |
| | |
| The absorption coefficient, in turn, is simply related to several other quantities:
| |
| *'''Attenuation coefficient''' is essentially (but not quite always) synonymous with absorption coefficient; see [[attenuation coefficient]] for details.
| |
| *'''Molar absorption coefficient''' or '''Molar extinction coefficient''', also called '''molar absorptivity''', is the absorption coefficient divided by molarity (and usually multiplied by ln(10), i.e., decadic); see [[Beer-Lambert law]] and [[molar absorptivity]] for details.
| |
| *'''Mass attenuation coefficient''', also called '''mass extinction coefficient''', is the absorption coefficient divided by density; see [[mass attenuation coefficient]] for details.
| |
| *'''Absorption cross section''' and '''scattering cross section''' are both quantitatively related to the absorption coefficient (or attenuation coefficient); see [[absorption cross section]] and [[scattering cross section]] for details.
| |
| *The absorption coefficient is also sometimes called '''opacity'''; see [[opacity (optics)]].
| |
| | |
| == Penetration depth, skin depth ==
| |
| {{main|Penetration depth|Skin depth}}
| |
| | |
| A very similar approach uses the '''[[penetration depth]]''':<ref>[http://www.iupac.org/goldbook/D01605.pdf IUPAC Compendium of Chemical Terminology]</ref>
| |
| :<math> \mathbf{E}(z,t) = e^{-z / (2 \delta_{pen})} \mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)})</math>
| |
| :<math>I(z) = I_0 e^{-z/\delta_{pen}}</math>
| |
| where <math>\delta_{pen}</math> is the penetration depth.
| |
| | |
| The '''[[skin depth]]''' <math>\delta_{skin}</math> is defined so that the wave satisfies:<ref name="Griffiths9.4.1">Griffiths, section 9.4.1.</ref><ref name="Jackson5.18A">Jackson, Section 5.18A</ref>
| |
| :<math> \mathbf{E}(z,t) = e^{-z / \delta_{skin} } \mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)})</math>
| |
| :<math>I(z) = I_0 e^{-2z/\delta_{skin}}</math>
| |
| where <math>\delta_{skin}</math> is the skin depth.
| |
| | |
| Physically, the penetration depth is the distance which the wave can travel before its ''intensity'' reduces by a factor of <math>1/e \approx 0.37</math>. The skin depth is the distance which the wave can travel before its ''amplitude'' reduces by that same factor.
| |
| | |
| The absorption coefficient is related to the penetration depth and skin depth by
| |
| | |
| :<math>\alpha_{abs} = 1/\delta_{pen} = 2/\delta_{skin}</math>
| |
| | |
| == Complex wavenumber, propagation constant ==
| |
| {{main|Propagation constant}}
| |
| | |
| Another way to incorporate attenuation is to use essentially the original expression:
| |
| :<math> \mathbf{E}(z,t) = \mathrm{Re} (\mathbf{E}_0 e^{i(\tilde{k} z - \omega t)})</math>
| |
| but with a '''complex [[wavenumber]]''' (as indicated by writing it as <math>\tilde{k}</math> instead of ''k'').<ref name="Griffiths9.4.1"/><ref name="Jackson7.5B">Jackson, Section 7.5.B</ref> Then the intensity of the wave satisfies:
| |
| :<math> I(z) \propto |\mathbf{E}_0 e^{i(\tilde{k} z - \omega t)}|^2</math>
| |
| i.e.,
| |
| :<math>I(z) = I_0 e^{-2z \mathrm{Im}(\tilde{k})}</math>
| |
| Therefore, comparing this to the absorption coefficient approach,<ref name="Griffiths9.4.3"/>
| |
| :<math> \mathrm{Im}(\tilde{k}) = \alpha_{abs}/2 </math>, <math>\mathrm{Re}(\tilde{k}) = k</math>
| |
| (''k'' is the standard (real) [[angular wavenumber]], as used in any of the previous formulations.) In accordance with the [[#Complex conjugate ambiguity|ambiguity noted above]], some authors use the complex conjugate definition, <math> \mathrm{Im}(\tilde{k}) = -\alpha_{abs}/2.</math><ref name=Lifante35>[http://books.google.com/books?id=Uq924mcshMkC&pg=PA35''Integrated Photonics: Fundamentals'', by Ginés Lifante, p.35]</ref>
| |
| | |
| A closely related approach, especially common in the theory of [[transmission line]]s, uses the '''[[propagation constant]]''':<ref>[http://www.atis.org/glossary/definition.aspx?id=2371 "Propagation constant", in ATIS Telecom Glossary 2007]</ref><ref>[http://books.google.com/books?id=AzLYk1qaaz8C&pg=PA93 ''Advances in imaging and electron physics, Volume 92'', by P. W. Hawkes and B. Kazan, p.93]</ref>
| |
| :<math> \mathbf{E}(z,t) = \mathrm{Re} (\mathbf{E}_0 e^{-\gamma z + i \omega t})</math>
| |
| :<math>I(z) = I_0 e^{-2z \mathrm{Re}(\gamma)}</math>
| |
| where <math>\gamma</math> is the propagation constant.
| |
| | |
| Comparing the two equations, the propagation constant and complex wavenumber are related by:
| |
| :<math>\gamma^* = -i\tilde{k}</math>
| |
| (where the * denotes [[complex conjugation]]), or more specifically:
| |
| :<math>\mathrm{Re}(\gamma) = \mathrm{Im}(\tilde{k}) = \alpha_{abs}/2</math>
| |
| (This quantity is also called the '''[[attenuation constant]]''',<ref name=Lifante35/><ref name=Sivanagaraju132/> sometimes denoted <math>\alpha</math>.)
| |
| :<math>\mathrm{Im}(\gamma) = \mathrm{Re}(\tilde{k}) = k</math>
| |
| (This quantity is also called the '''[[phase constant]]''', sometimes denoted <math>\beta</math>.)<ref name=Sivanagaraju132>[http://books.google.com/books?id=KpY1hpKKwdQC&pg=PA132 ''Electric Power Transmission and Distribution'', by S. Sivanagaraju, p.132]</ref>
| |
| | |
| Unfortunately, the notation is not always consistent. For example, <math>\tilde{k}</math> is sometimes called "propagation constant" instead of <math>\gamma</math>, which swaps the real and imaginary parts.<ref>See, for example, [http://www.rp-photonics.com/propagation_constant.html Encyclopedia of laser physics and technology]</ref>
| |
| | |
| == Complex refractive index, extinction coefficient ==
| |
| {{main|Refractive index}}
| |
| | |
| Recall that in nonattenuating media, the [[refractive index]] and wavenumber are related by:
| |
| :<math>n = \frac{ck}{\omega}</math>
| |
| A '''complex refractive index''' can therefore be defined in terms of the complex wavenumber defined above:
| |
| :<math>\tilde{n} = \frac{c\tilde{k}}{\omega}</math>.
| |
| In other words, the wave is required to satisfy
| |
| :<math> \mathbf{E}(z,t) = \mathrm{Re} (\mathbf{E}_0 e^{i\omega((\tilde{n} z/c) - t)})</math>.
| |
| Comparing to the preceding section, we have
| |
| :<math>\mathrm{Re}(\tilde{n}) = \frac{ck}{\omega}</math>, and <math>\mathrm{Im}(\tilde{n}) = \frac{c \alpha_{abs}}{2\omega}=\frac{\lambda_0 \alpha_{abs}}{4\pi}</math>.
| |
| The real part of <math>\tilde{n}</math> is often (ambiguously) called simply the ''refractive index''. The imaginary part is called the '''[[Optical extinction coefficient|extinction coefficient]]'''.
| |
| | |
| In accordance with the [[#Complex conjugate ambiguity|ambiguity noted above]], some authors use the complex conjugate definition, where the (still positive) extinction coefficient is ''minus'' the imaginary part of <math>\tilde{n}</math>.<ref name=refractiveindexconjugate/><ref>Pankove, pp. 87-89</ref>
| |
| | |
| == Complex permittivity ==
| |
| {{main|Complex permittivity}}
| |
| | |
| In nonattenuating media, the [[permittivity]] and [[refractive index]] are related by:
| |
| :<math>n = c \sqrt{\mu \epsilon}</math> ([[SI]]), <math>n = \sqrt{\mu \epsilon}</math> ([[Gaussian units|cgs]])
| |
| where <math>\mu</math> is the [[magnetic permeability|permeability]] and <math>\epsilon</math> is the [[permittivity]]. In attenuating media, the same relation is used, but the permittivity is allowed to be a complex number, called '''[[complex permittivity]]''':<ref name="Griffiths9.4.3"/>
| |
| :<math>\tilde{n} = c \sqrt{\mu \tilde{\epsilon}}</math> ([[SI]]), <math>\tilde{n} = \sqrt{\mu \tilde{\epsilon}}</math> ([[Gaussian units|cgs]]). | |
| Squaring both sides and using the results of the previous section gives:<ref name="Jackson7.5B"/>
| |
| :<math>\mathrm{Re}(\tilde{\epsilon}/\epsilon_0) = \frac{c^2}{(\omega^2)(\mu/\mu_0)}(k^2-\frac{\alpha_{abs}^2}{4})</math>
| |
| :<math>\mathrm{Im}(\tilde{\epsilon}/\epsilon_0) = \frac{c^2}{(\omega^2)(\mu/\mu_0)}(k\alpha_{abs})</math>
| |
| (this is in SI; in cgs, drop the <math>\epsilon_0</math> and <math>\mu_0</math>).
| |
| | |
| This approach is also called the '''complex dielectric constant'''; the [[dielectric constant]] is synonymous with <math>\epsilon/\epsilon_0</math> in SI, or simply <math>\epsilon</math> in cgs.
| |
| | |
| == AC conductivity ==
| |
| {{main|Electrical conductivity}}
| |
| | |
| Another way to incorporate attenuation is through the conductivity, as follows.<ref name="Jackson7.5C">Jackson, section 7.5C</ref>
| |
| | |
| One of the equations governing electromagnetic wave propagation is the [[Ampere's law|Maxwell-Ampere law]]:
| |
| :<math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{d\mathbf{D}}{dt}</math> (SI) <math>\nabla \times \mathbf{H} = \frac{4\pi}{c} \mathbf{J} + \frac{1}{c}\frac{d\mathbf{D}}{dt}</math> (cgs)
| |
| where '''D''' is the [[Electric displacement field|displacement field]]. Plugging in [[Ohm's law]] and the definition of (real) [[permittivity]]
| |
| :<math>\nabla \times \mathbf{H} = \sigma \mathbf{E} + \epsilon \frac{d\mathbf{E}}{dt}</math> (SI) <math>\nabla \times \mathbf{H} = \frac{4\pi \sigma}{c} \mathbf{E} + \frac{\epsilon}{c}\frac{d\mathbf{E}}{dt}</math> (cgs)
| |
| where <math>\sigma</math> is the (real, but frequency-dependent) conductivity, called '''[[alternating current|AC]] conductivity'''. With sinusoidal time dependence on all quantities, i.e. <math>\mathbf{H} = \mathrm{Re}(\mathbf{H}_0 e^{-i\omega t})</math> and <math>\mathbf{E} = \mathrm{Re}(\mathbf{E}_0 e^{-i\omega t})</math>, the result is
| |
| :<math>\nabla \times \mathbf{H}_0 = -i\omega\mathbf{E}_0(\epsilon + i\frac{\sigma}{\omega})</math> (SI) <math>\nabla \times \mathbf{H}_0 = \frac{-i\omega}{c} \mathbf{E}_0(\epsilon + i\frac{4\pi \sigma}{\omega})</math> (cgs)
| |
| If the current '''J''' was not included explicitly (through Ohm's law), but only implicitly (through a complex permittivity), the quantity in parentheses would be simply the complex permittivity. Therefore,
| |
| :<math>\tilde{\epsilon} = \epsilon + i \frac{\sigma}{\omega}</math> (SI) <math>\tilde{\epsilon} = \epsilon + i\frac{4\pi \sigma}{\omega}</math> (cgs).
| |
| Comparing to the previous section, the AC conductivity satisfies
| |
| :<math>\sigma = \frac{k\alpha_{abs}}{\omega\mu}</math> (SI) <math>\sigma = \frac{k\alpha_{abs}c^2}{4\pi\omega\mu}</math> (cgs).
| |
| | |
| == References and footnotes ==
| |
| *{{cite book | author=Jackson, John David | authorlink = J. D. Jackson | title=Classical Electrodynamics | edition=3rd ed. | location=New York | publisher=Wiley | year=1999 | isbn=0-471-30932-X}}
| |
| *{{cite book | author=Griffiths, David J. | authorlink=David Griffiths (physicist) | title=Introduction to Electrodynamics (3rd ed.) | publisher=Prentice Hall | year=1998 | isbn=0-13-805326-X}}
| |
| * J. I. Pankove, ''Optical Processes in Semiconductors'', Dover Publications Inc. New York (1971).
| |
| {{reflist}}
| |
| | |
| [[Category:Electromagnetic radiation]]
| |
| [[Category:Scattering, absorption and radiative transfer (optics)]]
| |
| [[Category:Optics]]
| |
By investing in a premium Word - Press theme, you're investing in the future of your website. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site.
Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. Some of the Wordpress development services offered by us are:. In the event you beloved this information along with you would want to get more info regarding wordpress dropbox backup i implore you to visit the site. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. t need to use the back button or the URL to get to your home page. Word - Press makes it possible to successfully and manage your website.
It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory.
It has become a more prevalent cause of infertility and the fertility clinic are having more and more couples with infertility problems. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. To do this, you should link your posts to other relevant posts that you've created. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. Word - Press offers constant updated services and products, that too, absolutely free of cost.
There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. ) Remote Login: With the process of PSD to Wordpress conversion comes the advantage of flexibility. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins.