Waveguide

The uses of waveguides for transmitting signals were known even before the term was coined. The phenomenon of sound waves guided through a taut wire have been known for a long time, as well as sound through a hollow pipe such as a <a href="/facts/Cave/SNfJTJtM">cave</a> or medical <a href="/facts/Stethoscope/DrKjH4x1">stethoscope</a>. Other uses of waveguides are in transmitting power between the components of a system such as radio, radar or optical devices. Waveguides are the fundamental principle of <a href="/facts/Guided_wave_testing/PJdUyVLs">guided wave testing</a> (GWT), one of the many methods of <a href="/facts/Nondestructive_testing/Czz0Vt0B">non-destructive evaluation</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
Specific examples:

<ul><li>Optical fibers transmit light and signals for long distances with low attenuation and a wide usable range of wavelengths.</li>
<li>In a <a href="/facts/Microwave_oven/nS1B3wp2">microwave oven</a> a waveguide transfers power from the <a href="/facts/Magnetron/8qd9pfsj">magnetron</a>, where waves are formed, to the cooking chamber.</li>
<li>In a radar, a waveguide transfers radio frequency energy to and from the antenna, where the <a href="/facts/Electrical_impedance/I06NSXtN">impedance</a> needs to be matched for efficient power transmission (see below).</li>
<li>Rectangular and circular waveguides are commonly used to connect feeds of parabolic dishes to their electronics, either low-noise receivers or power amplifier/transmitters.</li>
<li>Waveguides are used in scientific instruments to measure optical, acoustic and elastic properties of materials and objects. The waveguide can be put in contact with the specimen (as in a <a href="/facts/Medical_ultrasonography/Ju5BFuuV">medical ultrasonography</a>), in which case the waveguide ensures that the power of the testing wave is conserved, or the specimen may be put inside the waveguide (as in a dielectric constant measurement, so that smaller objects can be tested and the accuracy is better.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li>
<li>A <a href="/facts/Transmission_line/cYlqaPo3">transmission line</a> is a commonly used specific type of waveguide.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li></ul>
<h2 id="history">History</h2>

The first structure for guiding waves was proposed by <a href="/facts/J._J._Thomson/Y4jyndJF">J. J. Thomson</a> in 1893, and was first experimentally tested by <a href="/facts/Oliver_Lodge/jYCILgvS">Oliver Lodge</a> in 1894. The first mathematical analysis of electromagnetic waves in a metal cylinder was performed by <a href="/facts/John_Strutt,_3rd_Baron_Rayleigh/HcJyYdk5">Lord Rayleigh</a> in 1897.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>: 8  For sound waves, Lord Rayleigh published a full mathematical analysis of <a href="/facts/Propagation_mode/80y1JV4M">propagation modes</a> in his seminal work, "The Theory of Sound".<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> <a href="/facts/Jagadish_Chandra_Bose/r2W5xbXS">Jagadish Chandra Bose</a> researched <a href="/facts/Extremely_high_frequency/l28Grdmp">millimeter</a> wavelengths using waveguides, and in 1897 described to the Royal Institution in London his research carried out in Kolkata.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>[9]
The study of dielectric waveguides (such as optical fibers, see below) began as early as the 1920s, by several people, most famous of which are Rayleigh, <a href="/facts/Arnold_Sommerfeld/cJtpVHvn">Sommerfeld</a> and <a href="/facts/Peter_Debye/C9rm7nx8">Debye</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> Optical fiber began to receive special attention in the 1960s due to its importance to the communications industry.
The development of radio communication initially occurred at the lower frequencies because these could be more easily propagated over large distances. The long wavelengths made these frequencies unsuitable for use in hollow metal waveguides because of the impractically large diameter tubes required. Consequently, research into hollow metal waveguides stalled and the work of Lord Rayleigh was forgotten for a time and had to be rediscovered by others. Practical investigations resumed in the 1930s by <a href="/facts/George_C._Southworth/uDwMTX5R">George C. Southworth</a> at <a href="/facts/Bell_Labs/fKP5Gk0H">Bell Labs</a> and <a href="/facts/Wilmer_L._Barrow/WISh2TN6">Wilmer L. Barrow</a> at <a href="/facts/MIT/Xk49vNA8">MIT</a>. Southworth at first took the theory from papers on waves in dielectric rods because the work of Lord Rayleigh was unknown to him. This misled him somewhat; some of his experiments failed because he was not aware of the phenomenon of <a href="/facts/Waveguide_cutoff_frequency/vc8veeXp">waveguide cutoff frequency</a> already found in Lord Rayleigh's work. Serious theoretical work was taken up by <a href="/facts/John_R._Carson/1GBlLwWl">John R. Carson</a> and <a href="/facts/Sallie_Pero_Mead/tXeUCbQP">Sallie P. Mead</a>. This work led to the discovery that for the TE01 mode in circular waveguide losses go down with frequency and at one time this was a serious contender for the format for long-distance telecommunications.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>: 544–548 
The importance of <a href="/facts/Radar/jk9BTbFA">radar</a> in <a href="/facts/World_War_II/2efV5L1U">World War II</a> gave a great impetus to waveguide research, at least on the <a href="/facts/Allies_of_World_War_II/YzCdtjDY">Allied</a> side. The <a href="/facts/Magnetron/8qd9pfsj">magnetron</a>, developed in 1940 by <a href="/facts/John_Randall_(physicist)/STtE751y">John Randall</a> and <a href="/facts/Harry_Boot/PIwERQ4Q">Harry Boot</a> at the University of Birmingham in the United Kingdom, provided a good power source and made microwave radar feasible. The most important centre of US research was at the <a href="/facts/Radiation_Laboratory_(MIT)/hYcQJtxR">Radiation Laboratory</a> (Rad Lab) at <a href="/facts/MIT/Xk49vNA8">MIT</a> but many others took part in the US, and in the UK such as the <a href="/facts/Telecommunications_Research_Establishment/K202wPDU">Telecommunications Research Establishment</a>. The head of the Fundamental Development Group at Rad Lab was <a href="/facts/Edward_Mills_Purcell/V9UC5ygc">Edward Mills Purcell</a>. His researchers included <a href="/facts/Julian_Schwinger/BdepoI1r">Julian Schwinger</a>, <a href="/facts/Nathan_Marcuvitz/S6WsjkQx">Nathan Marcuvitz</a>, Carol Gray Montgomery, and <a href="/facts/Robert_H._Dicke/6LM6oN79">Robert H. Dicke</a>. Much of the Rad Lab work concentrated on finding <a href="/facts/Lumped_element_model/Nqtt18v5">lumped element models</a> of waveguide structures so that components in waveguide could be analysed with standard circuit theory. <a href="/facts/Hans_Bethe/X6bkxRRI">Hans Bethe</a> was also briefly at Rad Lab, but while there he produced his small aperture theory which proved important for <a href="/facts/Waveguide_filter/ACjb5SjG">waveguide cavity filters</a>, first developed at Rad Lab. The German side, on the other hand, largely ignored the potential of waveguides in radar until very late in the war. So much so that when radar parts from a downed British plane were sent to <a href="/facts/Siemens_%26_Halske/lAsPJ9sk">Siemens & Halske</a> for analysis, even though they were recognised as microwave components, their purpose could not be identified.

At that time, microwave techniques were badly neglected in Germany. It was generally believed that it was of no use for electronic warfare, and those who wanted to do research work in this field were not allowed to do so.— H. Mayer, wartime vice-president of Siemens & Halske
German academics were even allowed to continue publicly publishing their research in this field because it was not felt to be important.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>: 548–554 <a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>: 1055, 1057 
Immediately after World War II waveguide was the technology of choice in the microwave field. However, it has some problems; it is bulky, expensive to produce, and the cutoff frequency effect makes it difficult to produce wideband devices. Ridged waveguide can increase bandwidth beyond an octave, but a better solution is to use a technology working in <a href="/facts/Transverse_mode/nCIl5D6a">TEM mode</a> (that is, non-waveguide) such as <a href="/facts/Coaxial/Y4puzfrx">coaxial</a> conductors since TEM does not have a cutoff frequency. A shielded rectangular conductor can also be used and this has certain manufacturing advantages over coax and can be seen as the forerunner of the planar technologies (<a href="/facts/Stripline/RNdJWRQJ">stripline</a> and <a href="/facts/Microstrip/QkKwyAN7">microstrip</a>). However, planar technologies really started to take off when printed circuits were introduced. These methods are significantly cheaper than waveguide and have largely taken its place in most bands. However, waveguide is still favoured in the higher microwave bands from around <a href="/facts/Ku_band/czTuuqpT">Ku band</a> upwards.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>: 556–557 <a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>: 21–27, 21–50 

<h2 id="properties">Properties</h2>
<h3>Propagation modes and cutoff frequencies</h3>
A propagation <a href="/facts/Mode_(electromagnetism)/W8GuDxK1"> mode</a> in a waveguide is one solution of the wave equations, or, in other words, the form of the wave.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> Due to the constraints of the <a href="/facts/Boundary_conditions/QEfmUqmP">boundary conditions</a>, there are only limited frequencies and forms for the wave function which can propagate in the waveguide. The lowest frequency in which a certain mode can propagate is the <a href="/facts/Cutoff_frequency/vc8veeXp">cutoff frequency</a> of that mode. The mode with the lowest cutoff frequency is the fundamental mode of the waveguide, and its cutoff frequency is the waveguide cutoff frequency.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>: 38 
Propagation modes are computed by solving the <a href="/facts/Helmholtz_equation/SWfaXrTB">Helmholtz equation</a> alongside a set of boundary conditions depending on the geometrical shape and materials bounding the region. The usual assumption for infinitely long uniform waveguides allows us to assume a propagating form for the wave, i.e. stating that every field component has a known dependency on the propagation direction (i.e. 
 
 
 
 z
 
 
 {\displaystyle z}
 
). More specifically, the common approach is to first replace all unknown time-varying fields 
 
 
 
 u
 (
 x
 ,
 y
 ,
 z
 ,
 t
 )
 
 
 {\displaystyle u(x,y,z,t)}
 
 (assuming for simplicity to describe the fields in <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">cartesian</a> components) with their complex <a href="/facts/Phasor/LyjSSUDo">phasors</a> representation 
 
 
 
 U
 (
 x
 ,
 y
 ,
 z
 )
 
 
 {\displaystyle U(x,y,z)}
 
, sufficient to fully describe any infinitely long single-tone signal at frequency 
 
 
 
 f
 
 
 {\displaystyle f}
 
, (angular frequency 
 
 
 
 ω
 =
 2
 π
 f
 
 
 {\displaystyle \omega =2\pi f}
 
), and rewrite the Helmholtz equation and boundary conditions accordingly. Then, every unknown field is forced to have a form like 
 
 
 
 U
 (
 x
 ,
 y
 ,
 z
 )
 =
 
 
 
 U
 ^
 
 
 
 (
 x
 ,
 y
 )
 
 e
 
 −
 γ
 z
 
 
 
 
 {\displaystyle U(x,y,z)={\hat {U}}(x,y)e^{-\gamma z}}
 
, where the 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 term represents the propagation constant (still unknown) along the direction along which the waveguide extends to infinity. The Helmholtz equation can be rewritten to accommodate such form and the resulting equality needs to be solved for 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 and 
 
 
 
 
 
 
 U
 ^
 
 
 
 (
 x
 ,
 y
 )
 
 
 {\displaystyle {\hat {U}}(x,y)}
 
, yielding in the end an eigenvalue equation for 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 and a corresponding eigenfunction 
 
 
 
 
 
 
 U
 ^
 
 
 
 (
 x
 ,
 y
 
 )
 
 γ
 
 
 
 
 {\displaystyle {\hat {U}}(x,y)_{\gamma }}
 
for each solution of the former.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
The propagation constant 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 of the guided wave is complex, in general. For a lossless case, the propagation constant might be found to take on either real or imaginary values, depending on the chosen solution of the eigenvalue equation and on the angular frequency 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
. When 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 is purely real, the mode is said to be "below cutoff", since the amplitude of the field phasors tends to exponentially decrease with propagation; an imaginary 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
, instead, represents modes said to be "in propagation" or "above cutoff", as the complex amplitude of the phasors does not change with 
 
 
 
 z
 
 
 {\displaystyle z}
 
.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>

<h3>Impedance matching</h3>
In <a href="/facts/Circuit_theory/IjmVqbG8">circuit theory</a>, the <a href="/facts/Electrical_impedance/I06NSXtN">impedance</a> is a generalization of <a href="/facts/Electrical_resistance_and_conductance/hxSt0UAj">electrical resistance</a> in the case of <a href="/facts/Alternating_current/qxICS3xa">alternating current</a>, and is measured in <a href="/facts/Ohm/QNVrlGjw">ohms</a> (
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
).<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> A waveguide in circuit theory is described by a <a href="/facts/Transmission_line/cYlqaPo3">transmission line</a> having a length and <a href="/facts/Characteristic_impedance/oUg0mhah">characteristic impedance</a>.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>: 2–3, 6–12 [19]: 14 <a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> In other words, the impedance indicates the ratio of voltage to current of the circuit component (in this case a waveguide) during propagation of the wave. This description of the waveguide was originally intended for alternating current, but is also suitable for electromagnetic and sound waves, once the wave and material properties (such as <a href="/facts/Pressure/eyc37GRG">pressure</a>, <a href="/facts/Density/92p6hh9I">density</a>, <a href="/facts/Dielectric_constant/vnKbPTn1">dielectric constant</a>) are properly converted into electrical terms (<a href="/facts/Electric_current/Dc1xdnUX">current</a> and impedance for example).[21]: 14 
<a href="/facts/Impedance_matching/0mdy9TAp">Impedance matching</a> is important when components of an electric circuit are connected (waveguide to antenna for example): The impedance ratio determines how much of the wave is transmitted forward and how much is reflected. In connecting a waveguide to an antenna a complete transmission is usually required, so an effort is made to match their impedances.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a>
The <a href="/facts/Reflection_coefficient/se689p3C">reflection coefficient</a> can be calculated using: 
 
 
 
 Γ
 =
 
 
 
 
 Z
 
 2
 
 
 −
 
 Z
 
 1
 
 
 
 
 
 Z
 
 2
 
 
 +
 
 Z
 
 1
 
 
 
 
 
 
 
 {\displaystyle \Gamma ={\frac {Z_{2}-Z_{1}}{Z_{2}+Z_{1}}}}
 
, where 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 (Gamma) is the reflection coefficient (0 denotes full transmission, 1 full reflection, and 0.5 is a reflection of half the incoming voltage), 
 
 
 
 
 Z
 
 1
 
 
 
 
 {\displaystyle Z_{1}}
 
 and 
 
 
 
 
 Z
 
 2
 
 
 
 
 {\displaystyle Z_{2}}
 
 are the impedance of the first component (from which the wave enters) and the second component, respectively.[22]
An impedance mismatch creates a reflected wave, which added to the incoming waves creates a standing wave. An impedance mismatch can be also quantified with the <a href="/facts/Standing_wave_ratio/zYRofbkD">standing wave ratio</a> (SWR or VSWR for voltage), which is connected to the impedance ratio and reflection coefficient by: 
 
 
 
 
 V
 S
 W
 R
 
 =
 
 
 
 
 |
 
 V
 
 
 |
 
 
 
 m
 a
 x
 
 
 
 
 
 
 |
 
 V
 
 
 |
 
 
 
 m
 i
 n
 
 
 
 
 
 
 =
 
 
 
 1
 +
 
 |
 
 Γ
 
 |
 
 
 
 1
 −
 
 |
 
 Γ
 
 |
 
 
 
 
 
 
 {\displaystyle \mathrm {VSWR} ={\frac {|V|_{\rm {max}}}{|V|_{\rm {min}}}}={\frac {1+|\Gamma |}{1-|\Gamma |}}}
 
, where 
 
 
 
 
 
 |
 V
 |
 
 
 
 m
 i
 n
 
 /
 
 m
 a
 x
 
 
 
 
 
 {\displaystyle \left|V\right|_{\rm {min/max}}}
 
 are the minimum and maximum values of the voltage <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a>, and the VSWR is the voltage standing wave ratio, which value of 1 denotes full transmission, without reflection and thus no standing wave, while very large values mean high reflection and standing wave pattern.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>

<h2 id="electromagnetic-waveguides">Electromagnetic waveguides</h2>
<h3>Radio-frequency waveguides</h3>
Main articles: <a href="/facts/Waveguide_(radio_frequency)/M60l8qrm">Waveguide (radio frequency)</a> and <a href="/facts/Transmission_line/cYlqaPo3">Transmission line</a>
Waveguides can be constructed to carry waves over a wide portion of the <a href="/facts/Electromagnetic_spectrum/24LARvFY">electromagnetic spectrum</a>, but are especially useful in the <a href="/facts/Microwave/HmVSN31Y">microwave</a> and <a href="/facts/Optics/p0vq4S1T">optical</a> frequency ranges. Depending on the frequency, they can be constructed from either <a href="/facts/Electrical_conduction/akyCKvMy">conductive</a> or <a href="/facts/Dielectric/PnVt9P2G">dielectric</a> materials. Waveguides are used for transferring both <a href="/facts/Power_(physics)/mkaEyRnn">power</a> and communication signals.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>: 1–3 <a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>: xiii–xiv

<h3>Optical waveguides</h3>
Main article: <a href="/facts/Waveguide_(optics)/CQ6AnRAl">Waveguide (optics)</a>
Waveguides used at optical frequencies are typically dielectric waveguides, structures in which a <a href="/facts/Dielectric/PnVt9P2G">dielectric</a> material with high <a href="/facts/Permittivity/Sd3WtgT1">permittivity</a>, and thus high <a href="/facts/Index_of_refraction/yjkzvgeE">index of refraction</a>, is surrounded by a material with lower permittivity. The structure guides optical waves by <a href="/facts/Total_internal_reflection/Tm0M62NQ">total internal reflection</a>. An example of an optical waveguide is <a href="/facts/Optical_fiber/5ruuIFbj">optical fiber</a>.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a>
Other types of optical waveguide are also used, including <a href="/facts/Photonic-crystal_fiber/Q9LFwXud">photonic-crystal fiber</a>, which guides waves by any of several distinct mechanisms. Guides in the form of a hollow tube with a highly reflective inner surface have also been used as <a href="/facts/Light_pipe/e194Y4as">light pipes</a> for illumination applications. The inner surfaces may be polished metal, or may be covered with a multilayer film that guides light by <a href="/facts/Bragg_reflection/qEYANzQg">Bragg reflection</a> (this is a special case of a photonic-crystal fiber). One can also use small <a href="/facts/Prism_(optics)/eodBFF70">prisms</a> around the pipe which reflect light via total internal reflection<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> —such confinement is necessarily imperfect, however, since total internal reflection can never truly guide light within a lower-index core (in the prism case, some light leaks out at the prism corners).<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a>

<h2 id="acoustic-waveguides">Acoustic waveguides</h2>
Main article: <a href="/facts/Waveguide_(acoustics)/eEz71ovu">Waveguide (acoustics)</a>
An acoustic waveguide is a physical structure for guiding sound waves. Sound in an acoustic waveguide behaves like electromagnetic waves on a <a href="/facts/Transmission_line/cYlqaPo3">transmission line</a>. Waves on a string, like the ones in a <a href="/facts/Tin_can_telephone/jbBpShYa">tin can telephone</a>, are a simple example of an acoustic waveguide. Another example are pressure waves in the pipes of an <a href="/facts/Pipe_organ/F3CIPMwS">organ</a>. The term acoustic waveguide is also used to describe elastic waves guided in micro-scale devices, like those employed in piezoelectric delay lines and in <a href="/facts/Stimulated_Brillouin_scattering/bnCg0jDW">stimulated Brillouin scattering</a>.

<h2 id="mathematical-waveguides">Mathematical waveguides</h2>
Main article: <a href="/facts/Wave_equation/J2ksD2Az">Wave equation</a>
Waveguides are interesting objects of study from a strictly mathematical perspective. A waveguide (or tube) is defined as type of boundary condition on the wave equation such that the wave function must be equal to zero on the boundary and that the allowed region is finite in all dimensions but one (an infinitely long cylinder is an example.) A large number of interesting results can be proven from these general conditions. It turns out that any tube with a bulge (where the width of the tube increases) admits at least one bound state that exist inside the mode gaps. The frequencies of all the bound states can be identified by using a pulse short in time. This can be shown using the variational principles. An interesting result by <a href="/facts/Jeffrey_Goldstone/1k9Vw5fd">Jeffrey Goldstone</a> and <a href="/facts/Robert_Jaffe_(physicist)/EtVSvK42">Robert Jaffe</a> is that any tube of constant width with a twist, admits a bound state.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a>

<h2 id="sound-synthesis">Sound synthesis</h2>
Main article: <a href="/facts/Digital_waveguide_synthesis/yYsV8FdX">Digital waveguide synthesis</a>
Sound synthesis uses <a href="/facts/Digital_delay_line/oXsBfG2m">digital delay lines</a> as computational elements to simulate <a href="/facts/Wave_propagation/PXhWIt7k">wave propagation</a> in tubes of <a href="/facts/Wind_instruments/w1af9f1t">wind instruments</a> and the <a href="/facts/Vibrating_strings/IpRY862Y">vibrating strings</a> of <a href="/facts/String_instruments/srW2H8dV">string instruments</a>.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Circular_polarization/fYQvT0mI">Circular polarization</a></li>
<li><a href="/facts/Earth%E2%80%93ionosphere_waveguide/oVqCtUIK">Earth–ionosphere waveguide</a></li>
<li><a href="/facts/Linear_polarization/y4nJQKS1">Linear polarization</a></li>
<li><a href="/facts/Orthomode_transducer/5vTQX7JC">Orthomode transducer</a></li>
<li><a href="/facts/Polarization_(waves)/fP33m0eD">Polarization</a></li>
<li>Flap attenuator</li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/James_Baker-Jarvis/VA63Wo9k">Baker-Jarvis, James</a> (1990). <a href="https://nvlpubs.nist.gov/nistpubs/Legacy/TN/nbstechnicalnote1341.pdf">Transmission/Reflection and short-Circuit Line Permittivity Measurements</a> (PDF). Boulder, Colorado: National Institute of Standards and Technology.</li>
<li>Balanis, Constantine A. (1989). <a href="https://web.archive.org/web/20090514060247/http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471621943.html">Engineering Electromagnetics</a>. Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-62194-2. Archived from <a href="http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471621943.html">the original</a> on May 14, 2009.</li>
<li>Beranek, Leo Leroy; Mellow, Tim (2012). <a href="https://books.google.com/books?id=VYvS7MyaEE8C">Acoustics: Sound Fields and Transducers</a>. Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-391421-7.</li>
<li>Cronin, N. J. (1995). <a href="https://books.google.com/books?id=xq80UbJYqm8C">Microwave and Optical Waveguides</a>. CRC Press. p. 38. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7503-0216-6.</li>
<li>EETech Media, LLC. <a href="https://www.allaboutcircuits.com/textbook/alternating-current/chpt-14/waveguides/#:~:text=A%20waveguide%20is,capacitance%20">"Waveguides"</a>. All About Circuits. Retrieved December 31, 2023.</li>
<li>Emerson, D.T. (1997a). <a href="https://ieeexplore.ieee.org/abstract/document/602853/authors#authors">"The work of Jagadis Chandra Bose: 100 years of mm-wave research"</a>. 1997 IEEE MTT-S International Microwave Symposium Digest. Vol. 2. pp. 553–556. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2FMWSYM.1997.602853">10.1109/MWSYM.1997.602853</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7803-3814-6. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:9039614">9039614</a>.</li>
<li>Emerson, D.T. (1997b). <a href="https://books.google.com/books?id=09Zsv97IH1MC&pg=PA88">"The work of Jagadis Chandra Bose: 100 years of millimeter-wave research"</a>. IEEE Transactions on Microwave Theory and Techniques. 45 (12): 2267–2273. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1997ITMTT..45.2267E">1997ITMTT..45.2267E</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F22.643830">10.1109/22.643830</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780986488511.</li>
<li>Goldstone, J.; Jaffe, R. L. (1992). <a href="https://journals.aps.org/prb/abstract/10.1103/PhysRevB.45.14100">"Bound states in twisting tubes"</a>. Physical Review B. 45 (24): 14100–14107. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1992PhRvB..4514100G">1992PhRvB..4514100G</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FPhysRevB.45.14100">10.1103/PhysRevB.45.14100</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/10001530">10001530</a>.</li>
<li>Han, C.C.; Hwang, Y. (2012). "Satellite attenas". In Lo, Y. T.; Lee, S. W. (eds.). <a href="https://books.google.com/books?id=_0wOBwAAQBAJ">Antenna Handbook: Volume III Applications</a>. Springer Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4615-2638-4.</li>
<li>Herres, David. <a href="https://www.testandmeasurementtips.com/making-measurements-in-optical-fiber-faq/#:~:text=Optical%20fiber%20is%20actually%20a,over%20considerable%20distances%20without%20amplification.">"Optical Fiber: A waveguide for light and internal reflection"</a>. Test & Measurement Tips. Retrieved January 1, 2024.</li>
<li>Institute of Electrical and Electronics, Engineers; Radatz, Jane; Standards Coordinating Committee, Terms and Definitions; IEEE Computer Society, Standards Coordinating Committee (1997). <a href="https://books.google.com/books?id=6FbxAAAAMAAJ">The IEEE Standard Dictionary of Electrical and Electronics Terms</a> (6th ed.). New York, New York: Institute of Electrical and Electronics Engineers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-55937-833-8.</li>
<li>Khare, Rashmi; Nema, Rajesh (2012). <a href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d06f70454491af8aa887670d5fa75477f8ff1036">"Review of Impedance Matching Networks for Bandwidth Enhancement"</a>. International Journal of Emerging Technology and Advanced Engineering. 2 (1): 92–96.</li>
<li>Levy, R.; Cohn, S.B. (1984). <a href="https://ieeexplore.ieee.org/document/1132817">"A History of Microwave Filter Research, Design, and Development"</a>. IEEE Transactions on Microwave Theory and Techniques. 32 (9): 1055–1067. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1984ITMTT..32.1055L">1984ITMTT..32.1055L</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2FTMTT.1984.1132817">10.1109/TMTT.1984.1132817</a>.</li>
<li>Lo, Y. T.; Lee, S. W. (December 6, 2012). <a href="https://books.google.com/books?id=_0wOBwAAQBAJ">Antenna Handbook: Volume III Applications</a>. Springer Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4615-2638-4.</li>
<li>McLachlan, Norman William (1964). <a href="https://books.google.com/books?id=215knQEACAAJ">Theory and Application of Mathieu Functions, By N.W. Mclachlan</a>. New York, New York: Dover.</li>
<li>Marcuvitz, Nathan (1951). <a href="https://books.google.com/books?id=Ao34iFuNZgIC">Waveguide Handbook</a>. IET. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-86341-058-1.</li>
<li>Okamoto, Katsunari (2010). <a href="https://books.google.com/books?id=2grZ1AR9lMUC">Fundamentals of Optical Waveguides</a>. Elsevier. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-08-045506-8.</li>
<li>Oliner, Arthur A. (January 30, 2006). "The evolution of electromagnetic waveguides: from hollow metallic guides to microwave integrated circuits". In Sarkar, T. K.; Mailloux, Robert; Oliner, Arthur A.; <a href="/facts/Magdalena_Salazar_Palma/QbjALXhQ">Salazar-Palma, Magdalena</a>; Sengupta, Dipak L. (eds.). <a href="https://books.google.com/books?id=NBLEAA6QKYkC">History of Wireless</a>. John Wiley & Sons. pp. 543–566. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-78301-5.</li>
<li>Olisa, Samuel Chukwuemeka; Khan, Muhammad A.; Starr, Andrew (2021). <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7865912">"Review of Current Guided Wave Ultrasonic Testing (GWUT) Limitations and Future Directions"</a>. Sensors. 21 (3): 811. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2021Senso..21..811O">2021Senso..21..811O</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.3390%2Fs21030811">10.3390/s21030811</a>. <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7865912">7865912</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/33530407">33530407</a>.</li>
<li>Payne, R.; Webb, D. (1971). <a href="https://pubmed.ncbi.nlm.nih.gov/5288850/">"Orientation by means of long range acoustic signaling in baleen whales"</a>. Annals of the New York Academy of Sciences. 188 (1): 110–141. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1971NYASA.188..110P">1971NYASA.188..110P</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1111%2Fj.1749-6632.1971.tb13093.x">10.1111/j.1749-6632.1971.tb13093.x</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0077-8923">0077-8923</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/5288850">5288850</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:42324742">42324742</a>.</li>
<li>Pozar, David M. (2012). <a href="https://books.google.com/books?id=_YEbGAXCcAMC">Microwave Engineering</a>. John Wiley & Sons. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-470-63155-3.</li>
<li>Ramo, Simon; Whinnery, John R.; Van Duzer, Theodore (1994). <a href="https://books.google.com/books?id=lIFkQgAACAAJ">Fields and Waves in Communication Electronics</a>. New York: John Wiley and Sons. pp. 321–324. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-58551-0.</li>
<li>Rayleigh, John William Strutt Baron (1894). <a href="https://books.google.com/books?id=zTYIAAAAIAAJ">The Theory of Sound</a>. Macmillan.</li>
<li>Saxe, Steven G. (1989). <a href="https://www.sciencedirect.com/science/article/abs/pii/0165163389900269">"Prismatic film light guides: Performance and recent developments"</a>. Solar Energy Materials. 19 (1): 95–109. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0165-1633%2889%2990026-9">10.1016/0165-1633(89)90026-9</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0165-1633">0165-1633</a>.</li>
<li>Smith, Julius O. (1996). <a href="https://www.jstor.org/stable/3681331">"Physical Modeling Synthesis Update"</a>. Computer Music Journal. 20 (2): 44–56. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F3681331">10.2307/3681331</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0148-9267">0148-9267</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/3681331">3681331</a>.</li>
<li>Zhang, Hanqiao; Krooswyk, Steven; Ou, Jeff (2015). <a href="https://www.sciencedirect.com/science/article/abs/pii/B9780124186637000010">"Chapter 1 - Transmission line fundamentals"</a>. High Speed Digital Design. Morgan Kaufmann. pp. 1–26. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-418663-7.</li></ul>

<h2 id="external-links">External links</h2>

Look up waveguide in Wiktionary, the free dictionary.

Wikimedia Commons has media related to Waveguides.

<ul><li><a href="http://www.ece.rutgers.edu/~orfanidi/ewa/ch09.pdf">Electromagnetic Waves and Antennas: Waveguides</a> Sophocles J. Orfanidis, Department of Electrical and Computer Engineering, Rutgers University</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Institute of Electrical and Electronics et al. 1997. - Institute of Electrical and Electronics, Engineers; Radatz, Jane; Standards Coordinating Committee, Terms and Definitions; IEEE Computer Society, Standards Coordinating Committee (1997). The IEEE Standard Dictionary of Electrical and Electronics Terms (6th ed.). New York, New York: Institute of Electrical and Electronics Engineers. ISBN 978-1-55937-833-8. <a href="https://books.google.com/books?id=6FbxAAAAMAAJ" target="_blank">https://books.google.com/books?id=6FbxAAAAMAAJ</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Payne & Webb 1971. - Payne, R.; Webb, D. (1971). "Orientation by means of long range acoustic signaling in baleen whales". Annals of the New York Academy of Sciences. 188 (1): 110–141. Bibcode:1971NYASA.188..110P. doi:10.1111/j.1749-6632.1971.tb13093.x. ISSN 0077-8923. PMID 5288850. S2CID 42324742. <a href="https://pubmed.ncbi.nlm.nih.gov/5288850/" target="_blank">https://pubmed.ncbi.nlm.nih.gov/5288850/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Olisa, Khan & Starr 2021. - Olisa, Samuel Chukwuemeka; Khan, Muhammad A.; Starr, Andrew (2021). "Review of Current Guided Wave Ultrasonic Testing (GWUT) Limitations and Future Directions". Sensors. 21 (3): 811. Bibcode:2021Senso..21..811O. doi:10.3390/s21030811. PMC 7865912. PMID 33530407. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7865912" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7865912</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Baker-Jarvis 1990. - Baker-Jarvis, James (1990). Transmission/Reflection and short-Circuit Line Permittivity Measurements (PDF). Boulder, Colorado: National Institute of Standards and Technology. <a href="https://nvlpubs.nist.gov/nistpubs/Legacy/TN/nbstechnicalnote1341.pdf" target="_blank">https://nvlpubs.nist.gov/nistpubs/Legacy/TN/nbstechnicalnote1341.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">EETech Media. - EETech Media, LLC. "Waveguides". All About Circuits. Retrieved December 31, 2023. <a href="https://www.allaboutcircuits.com/textbook/alternating-current/chpt-14/waveguides/#:~:text=A%20waveguide%20is,capacitance%20" target="_blank">https://www.allaboutcircuits.com/textbook/alternating-current/chpt-14/waveguides/#:~:text=A%20waveguide%20is,capacitance%20</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">McLachlan 1964. - McLachlan, Norman William (1964). Theory and Application of Mathieu Functions, By N.W. Mclachlan. New York, New York: Dover. <a href="https://books.google.com/books?id=215knQEACAAJ" target="_blank">https://books.google.com/books?id=215knQEACAAJ</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Rayleigh 1894. - Rayleigh, John William Strutt Baron (1894). The Theory of Sound. Macmillan. <a href="https://books.google.com/books?id=zTYIAAAAIAAJ" target="_blank">https://books.google.com/books?id=zTYIAAAAIAAJ</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Emerson 1997a. - Emerson, D.T. (1997a). "The work of Jagadis Chandra Bose: 100 years of mm-wave research". 1997 IEEE MTT-S International Microwave Symposium Digest. Vol. 2. pp. 553–556. doi:10.1109/MWSYM.1997.602853. ISBN 0-7803-3814-6. S2CID 9039614. <a href="https://ieeexplore.ieee.org/abstract/document/602853/authors#authors" target="_blank">https://ieeexplore.ieee.org/abstract/document/602853/authors#authors</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Balanis 1989. - Balanis, Constantine A. (1989). Engineering Electromagnetics. Wiley. ISBN 978-0-471-62194-2. Archived from the original on May 14, 2009. <a href="https://web.archive.org/web/20090514060247/http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471621943.html" target="_blank">https://web.archive.org/web/20090514060247/http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471621943.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Oliner 2006, Reprint. - Oliner, Arthur A. (January 30, 2006). "The evolution of electromagnetic waveguides: from hollow metallic guides to microwave integrated circuits". In Sarkar, T. K.; Mailloux, Robert; Oliner, Arthur A.; Salazar-Palma, Magdalena; Sengupta, Dipak L. (eds.). History of Wireless. John Wiley & Sons. pp. 543–566. ISBN 978-0-471-78301-5. <a href="https://books.google.com/books?id=NBLEAA6QKYkC" target="_blank">https://books.google.com/books?id=NBLEAA6QKYkC</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Oliner 2006. - Oliner, Arthur A. (January 30, 2006). "The evolution of electromagnetic waveguides: from hollow metallic guides to microwave integrated circuits". In Sarkar, T. K.; Mailloux, Robert; Oliner, Arthur A.; Salazar-Palma, Magdalena; Sengupta, Dipak L. (eds.). History of Wireless. John Wiley & Sons. pp. 543–566. ISBN 978-0-471-78301-5. <a href="https://books.google.com/books?id=NBLEAA6QKYkC" target="_blank">https://books.google.com/books?id=NBLEAA6QKYkC</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Levy & Cohn 1984. - Levy, R.; Cohn, S.B. (1984). "A History of Microwave Filter Research, Design, and Development". IEEE Transactions on Microwave Theory and Techniques. 32 (9): 1055–1067. Bibcode:1984ITMTT..32.1055L. doi:10.1109/TMTT.1984.1132817. <a href="https://ieeexplore.ieee.org/document/1132817" target="_blank">https://ieeexplore.ieee.org/document/1132817</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Oliner 2006. - Oliner, Arthur A. (January 30, 2006). "The evolution of electromagnetic waveguides: from hollow metallic guides to microwave integrated circuits". In Sarkar, T. K.; Mailloux, Robert; Oliner, Arthur A.; Salazar-Palma, Magdalena; Sengupta, Dipak L. (eds.). History of Wireless. John Wiley & Sons. pp. 543–566. ISBN 978-0-471-78301-5. <a href="https://books.google.com/books?id=NBLEAA6QKYkC" target="_blank">https://books.google.com/books?id=NBLEAA6QKYkC</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Han & Hwang 2012. - Han, C.C.; Hwang, Y. (2012). "Satellite attenas". In Lo, Y. T.; Lee, S. W. (eds.). Antenna Handbook: Volume III Applications. Springer Science & Business Media. ISBN 978-1-4615-2638-4. <a href="https://books.google.com/books?id=_0wOBwAAQBAJ" target="_blank">https://books.google.com/books?id=_0wOBwAAQBAJ</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Balanis 1989. - Balanis, Constantine A. (1989). Engineering Electromagnetics. Wiley. ISBN 978-0-471-62194-2. Archived from the original on May 14, 2009. <a href="https://web.archive.org/web/20090514060247/http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471621943.html" target="_blank">https://web.archive.org/web/20090514060247/http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471621943.html</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Cronin 1995. - Cronin, N. J. (1995). Microwave and Optical Waveguides. CRC Press. p. 38. ISBN 978-0-7503-0216-6. <a href="https://books.google.com/books?id=xq80UbJYqm8C" target="_blank">https://books.google.com/books?id=xq80UbJYqm8C</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Pozar 2012. - Pozar, David M. (2012). Microwave Engineering. John Wiley & Sons. ISBN 978-0-470-63155-3. <a href="https://books.google.com/books?id=_YEbGAXCcAMC" target="_blank">https://books.google.com/books?id=_YEbGAXCcAMC</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Ramo, Whinnery & Van Duzer 1994. - Ramo, Simon; Whinnery, John R.; Van Duzer, Theodore (1994). Fields and Waves in Communication Electronics. New York: John Wiley and Sons. pp. 321–324. ISBN 978-0-471-58551-0. <a href="https://books.google.com/books?id=lIFkQgAACAAJ" target="_blank">https://books.google.com/books?id=lIFkQgAACAAJ</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Balanis 1989. - Balanis, Constantine A. (1989). Engineering Electromagnetics. Wiley. ISBN 978-0-471-62194-2. Archived from the original on May 14, 2009. <a href="https://web.archive.org/web/20090514060247/http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471621943.html" target="_blank">https://web.archive.org/web/20090514060247/http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471621943.html</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Marcuvitz 1951. - Marcuvitz, Nathan (1951). Waveguide Handbook. IET. ISBN 978-0-86341-058-1. <a href="https://books.google.com/books?id=Ao34iFuNZgIC" target="_blank">https://books.google.com/books?id=Ao34iFuNZgIC</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Khare & Nema 2012. - Khare, Rashmi; Nema, Rajesh (2012). "Review of Impedance Matching Networks for Bandwidth Enhancement". International Journal of Emerging Technology and Advanced Engineering. 2 (1): 92–96. <a href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d06f70454491af8aa887670d5fa75477f8ff1036" target="_blank">https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d06f70454491af8aa887670d5fa75477f8ff1036</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Khare & Nema 2012. - Khare, Rashmi; Nema, Rajesh (2012). "Review of Impedance Matching Networks for Bandwidth Enhancement". International Journal of Emerging Technology and Advanced Engineering. 2 (1): 92–96. <a href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d06f70454491af8aa887670d5fa75477f8ff1036" target="_blank">https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d06f70454491af8aa887670d5fa75477f8ff1036</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Khare & Nema 2012. - Khare, Rashmi; Nema, Rajesh (2012). "Review of Impedance Matching Networks for Bandwidth Enhancement". International Journal of Emerging Technology and Advanced Engineering. 2 (1): 92–96. <a href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d06f70454491af8aa887670d5fa75477f8ff1036" target="_blank">https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d06f70454491af8aa887670d5fa75477f8ff1036</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Cronin 1995. - Cronin, N. J. (1995). Microwave and Optical Waveguides. CRC Press. p. 38. ISBN 978-0-7503-0216-6. <a href="https://books.google.com/books?id=xq80UbJYqm8C" target="_blank">https://books.google.com/books?id=xq80UbJYqm8C</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Okamoto 2010. - Okamoto, Katsunari (2010). Fundamentals of Optical Waveguides. Elsevier. ISBN 978-0-08-045506-8. <a href="https://books.google.com/books?id=2grZ1AR9lMUC" target="_blank">https://books.google.com/books?id=2grZ1AR9lMUC</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Herres. - Herres, David. "Optical Fiber: A waveguide for light and internal reflection". Test & Measurement Tips. Retrieved January 1, 2024. <a href="https://www.testandmeasurementtips.com/making-measurements-in-optical-fiber-faq/#:~:text=Optical%20fiber%20is%20actually%20a,over%20considerable%20distances%20without%20amplification." target="_blank">https://www.testandmeasurementtips.com/making-measurements-in-optical-fiber-faq/#:~:text=Optical%20fiber%20is%20actually%20a,over%20considerable%20distances%20without%20amplification.</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">"Light Pipe Technologies". Archived from the original on March 14, 2007. <a href="https://web.archive.org/web/20070314173338/http://www.physics.ubc.ca/ssp/research/lightpipe.htm" target="_blank">https://web.archive.org/web/20070314173338/http://www.physics.ubc.ca/ssp/research/lightpipe.htm</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Saxe 1989. - Saxe, Steven G. (1989). "Prismatic film light guides: Performance and recent developments". Solar Energy Materials. 19 (1): 95–109. doi:10.1016/0165-1633(89)90026-9. ISSN 0165-1633. <a href="https://www.sciencedirect.com/science/article/abs/pii/0165163389900269" target="_blank">https://www.sciencedirect.com/science/article/abs/pii/0165163389900269</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
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</ol>

Waveguide open-in-new

Waveguide