Waveguide (optics)

<h2 id="total-internal-reflection">Total internal reflection</h2>

<p>The basic principles behind optical waveguides can be described using the concepts of <a href="/facts/Geometrical_optics/gIFu84Tm">geometrical or ray optics</a>, as illustrated in the diagram.
</p><p>Light passing into a medium with higher <a href="/facts/Refractive_index/yjkzvgeE">refractive index</a> bends toward the normal by the process of <a href="/facts/Refraction/S8kvSKzM">refraction</a> (Figure a.). Take, for example, light passing from air into glass. Similarly, light traveling in the opposite direction (from glass into air) takes the same path, bending away from the normal. This is a consequence of <a href="/facts/T-symmetry/Tgf2LJMc">time-reversal symmetry</a>. Each ray in air (black) can be mapped to a ray in the glass (blue), as shown in Figure b. There's a one-to-one correspondence. But because of refraction, some of the rays in the glass are left out (red). The remaining rays are trapped in the glass by a process called <i><a href="/facts/Total_internal_reflection/Tm0M62NQ">total internal reflection</a></i>. They are incident on the glass-air interface at an angle above the <i><a href="/facts/Critical_angle_(optics)/Tm0M62NQ">critical angle</a></i>. These extra rays correspond to a higher <i><a href="/facts/Density_of_states/bwVTyhTf">density of states</a></i> in more-advanced formulations based on the <a href="/facts/Green%27s_function_(many-body_theory)/1v3ethbc">Green's function</a>.
</p><p>Using total internal reflection, we can trap and guide the light in a dielectric waveguide (Figure c). The red rays bounce off both the top and bottom surface of the high index medium. They're guided even if the slab curves or bends, so long as it bends slowly. This is the basic principle behind <a href="/facts/Optical_fiber/5ruuIFbj">fiber optics</a> in which light is guided along a high index glass <i>core</i> in a lower index glass <i>cladding</i> (Figure d).
</p><p>Ray optics only gives a rough picture of how waveguides work. <a href="/facts/Maxwell%27s_equations/LfJ1drHS">Maxwell's equations</a> can be solved by analytical or numerical methods for a full-field description of a dielectric waveguide.
</p>
<h2 id="dielectric-slab-waveguide">Dielectric slab waveguide</h2>

<p>Perhaps the simplest optical waveguide is the dielectric slab waveguide,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> also called a planar waveguide.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> Owing to their simplicity, slab waveguides are often used as toy models but also find application in on-chip devices like <a href="/facts/Arrayed_waveguide_grating/6STcgm2v">arrayed waveguide gratings</a> and <a href="/facts/Acousto-optic_modulator/mjsK4cGj">acousto-optic filters and modulators</a>.
</p><p>The slab waveguide consists of three layers of materials with different dielectric constants, extending infinitely in the directions parallel to their interfaces. Light is confined in the middle layer by <a href="/facts/Total_internal_reflection/Tm0M62NQ">total internal reflection</a> if the <a href="/facts/Refractive_index/yjkzvgeE">refractive index</a> of the middle layer is larger than that of the surrounding layers.
</p><p>The slab waveguide is essentially a one-dimensional waveguide. It traps light only normal to the dielectric interfaces. For guided <a href="/facts/Eigenmode/XCsvQpuI">modes</a>, the field in domain II in the diagram is propagating and can be treated as a <a href="/facts/Plane_wave/0aDUbNhQ">plane wave</a>. The field in domains I and III <a href="/facts/Evanescent_field/svCyRKr7">evanescently</a> decay away from the slab. The plane wave in domain II bounces between the top and bottom interfaces at some angle typically specified by the 
  
    
      
        
          
            
              β
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    {\displaystyle {\vec {\beta }}}
  
, the <a href="/facts/Wave_vector/Jk8wUEft">wave vector</a> in the plane of the slab. Guided modes constructively interfere on one complete roundtrip in the slab. At each frequency, one or more modes can be found giving a set of eigenvalues 
  
    
      
        (
        ω
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              β
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    {\displaystyle (\omega ,{\vec {\beta }})}
  
 which can be used to construct a <a href="/facts/Band_diagram/3G59hpkj">band diagram</a> or <a href="/facts/Dispersion_relation/oo7RM5eg">dispersion relation</a>.
</p><p>Because guided modes are trapped in the slab, they cannot be excited by light incident on the top or bottom interfaces. Light can be <i>end-fire or butte coupled</i> by injecting it with a lens in the plane of the slab. Alternatively a coupling element may be used to couple light into the waveguide, such as a grating coupler or prism coupler.
</p><p>There are 2 technologies: diffractive waveguides and reflective waveguides.
</p>
<h2 id="two-dimensional-waveguide">Two-dimensional waveguide</h2>
<p class="note">See also: <a href="/facts/Planar_transmission_line/PtQSrIpB">Planar transmission line § Imageline</a></p>
<h3>Strip waveguide</h3>
<p>A strip waveguide is basically a strip of the layer confined between cladding layers. The simplest case is a rectangular waveguide, which is formed when the guiding layer of the slab waveguide is restricted in both transverse directions rather than just one. Rectangular waveguides are used in <a href="/facts/Integrated_optical_circuit/C7skzpMo">integrated optical circuits</a> and in <a href="/facts/Laser_diode/TjHzJteV">laser diodes</a>. They are commonly used as the basis of such optical components as <a href="/facts/Mach%E2%80%93Zehnder_interferometer/gP0B1j0w">Mach–Zehnder interferometers</a> and <a href="/facts/Wavelength-division_multiplexing/xSkNRxfu">wavelength division multiplexers</a>. The <a href="/facts/Optical_cavity/BITPG3ga">cavities</a> of <a href="/facts/Laser_diode/TjHzJteV">laser diodes</a> are frequently constructed as rectangular optical waveguides. Optical waveguides with rectangular geometry are produced by a variety of means, usually by a <a href="/facts/Planar_process/2YxvYRLX">planar process</a>.
</p><p>The field distribution in a rectangular waveguide cannot be solved analytically, however approximate solution methods, such as <a href="/facts/Marcatili%27s_method/Ub2lnjg0">Marcatili's method</a>,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> <a href="/facts/Marcatili%27s_method/Ub2lnjg0">Extended Marcatili's method</a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> and Kumar's method,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> are known.
</p>
<h3>Rib waveguide</h3>
<p>A rib waveguide is a waveguide in which the guiding layer basically consists of the slab with a strip (or several strips) superimposed onto it. Rib waveguides also provide confinement of the wave in two dimensions and near-unity confinement is possible in multi-layer rib structures.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>
<h3>Segmented waveguide and photonic crystal waveguide</h3>
<p>Optical waveguides typically maintain a constant cross-section along their direction of propagation. This is for example the case for strip and of rib waveguides. However, waveguides can also have periodic changes in their cross-section while still allowing lossless transmission of light via so-called Bloch modes. Such waveguides are referred to as segmented waveguides (with a 1D patterning along the direction of propagation<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>) or as photonic crystal waveguides (with a 2D or 3D patterning<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>).
</p>
<h3>Laser-inscribed waveguide</h3>
<p>Optical waveguides find their most important application in <a href="/facts/Photonics/6UHcPXbq">photonics</a>. Configuring the waveguides  in 3D space provides integration between electronic components on a chip and optical fibers. Such waveguides may be designed for a single mode propagation of infrared light at telecommunication wavelengths, and configured to deliver optical signal between input and output locations with very low loss.
</p>

<p>One of the methods for constructing such waveguides utilizes photorefractive effect in transparent materials. An increase in the refractive index of a material may be induced by nonlinear absorption of pulsed laser light. In order to maximize the increase of the refractive index, a very short (typically femtosecond) laser pulses are used, and focused with a high NA microscope objective. By translating the focal spot through a bulk transparent material the waveguides can be directly written.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> A variation of this method uses a low NA microscope objective and translates the focal spot along the beam axis. This improves the overlap between the focused laser beam and the photorefractive material, thus reducing power needed from the laser.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>
When transparent material is exposed to an unfocused laser beam of sufficient brightness to initiate photorefractive effect, the waveguides may start forming on their own as a result of an accumulated <a href="/facts/Self-focusing/UvB7UP1N">self-focusing</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> The formation of such waveguides leads to a breakup of the laser beam. Continued exposure results in a buildup of the refractive index towards the centerline of each waveguide, and collapse of the mode field diameter of the propagating light. Such waveguides remain permanently in the glass and can be photographed off-line (see the picture on the right).
</p>
<h2 id="light-pipe">Light pipe</h2>
<p class="note">Main article: <a href="/facts/Light_tube/e194Y4as">Light tube</a></p>
<p>Light pipes are tubes or cylinders of solid material used to guide light a short distance.  In electronics, plastic light pipes are used to guide light from <a href="/facts/LED/gE8x4MDb">LEDs</a> on a circuit board to the user interface surface.  In buildings, light pipes are used to transfer illumination from outside the building to where it is needed inside.
</p>
<h2 id="optical-fiber-waveguide">Optical fiber waveguide</h2>

<p class="note">Main article: <a href="/facts/Optical_fiber/5ruuIFbj">Optical fiber</a></p>
<p>Optical fiber is typically a circular cross-section <i>dielectric waveguide</i> consisting of a <a href="/facts/Dielectric/PnVt9P2G">dielectric</a> material surrounded by another dielectric material with a lower <a href="/facts/Refractive_index/yjkzvgeE">refractive index</a>. Optical fibers are most commonly made from <a href="/facts/Silica_glass/avtX1fy3">silica glass</a>, however other <a href="/facts/Glass/hezRsAGf">glass</a> materials are used for certain applications and <a href="/facts/Plastic_optical_fiber/9GtDGFS9">plastic optical fiber</a> can be used for short-distance applications.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/ARROW_waveguide/TuYshunc">ARROW waveguide</a></li>
<li><a href="/facts/Cutoff_wavelength/vc8veeXp">Cutoff wavelength</a></li>
<li><a href="/facts/Dielectric_constant/vnKbPTn1">Dielectric constant</a></li>
<li><a href="/facts/Digital_planar_holography/pJxu3bRJ">Digital planar holography</a></li>
<li><a href="/facts/Electromagnetic_radiation/Ynx7ujCW">Electromagnetic radiation</a></li>
<li><a href="/facts/Equilibrium_mode_distribution/Au6IK07o">Equilibrium mode distribution</a></li>
<li><a href="/facts/Erbium-doped_waveguide_amplifier/VmdC7jpg">Erbium-doped waveguide amplifier</a></li>
<li><a href="/facts/Leaky_mode/nGnbxoJU">Leaky mode</a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li>
<li><a href="/facts/Lightguide_display/s2P4TjnQ">Lightguide display</a></li>
<li><a href="/facts/Photonic_crystal/R763CtuW">Photonic crystal</a></li>
<li><a href="/facts/Photonic-crystal_fiber/Q9LFwXud">Photonic-crystal fiber</a></li>
<li><a href="/facts/Prism_coupler/TKERz2D6">Prism coupler</a></li>
<li><a href="/facts/Transmission_medium/yT1uXLmq">Transmission medium</a></li>
<li><a href="/facts/Waveguide_(radio_frequency)/M60l8qrm">Waveguide (radio frequency)</a></li>
<li><a href="/facts/Waveguide/tJS9Y0GH">Waveguide</a></li>
<li><a href="/facts/Zero-mode_waveguide/tQr4ToLr">Zero-mode waveguide</a></li></ul>

<p> 13. Yao Zhou, Jufan Zhang, Fengzhou Fang. Design of a dual-focal geometrical waveguide near-eye see-through display. Optics and Laser Technology, 2022, Volume 156, <a href="https://doi.org/10.1016/j.optlastec.2022.108546">https://doi.org/10.1016/j.optlastec.2022.108546</a>.
</p><p>14. Yao Zhou, Jufan Zhang, Fengzhou Fang. Design of a large field-of-view two-dimensional geometrical waveguide. Results in Optics, Volume 5, 2021, 100147, <a href="https://doi.org/10.1016/j.rio.2021.100147">https://doi.org/10.1016/j.rio.2021.100147</a>.
</p>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20190103002718/http://www.advr-inc.com/waveguide.html">AdvR nonlinear waveguides in rubidium-doped potassium titanyl phosphate</a> (<a href="/facts/Potassium_titanyl_phosphate/FHyrkdN0">KTP</a>)</li></ul>

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

<ol>
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<li id="fn:2"><p>Ramo, Simon, John R. Whinnery, and Theodore van Duzer, Fields and Waves in Communications Electronics, 2 ed., John Wiley and Sons, New York, 1984. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Silicon Photonics", by Graham T. Reed, Andrew P. Knights <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Marcatili, E. A. J. (1969). "Dielectric rectangular waveguide and directional coupler for integrated optics". Bell Syst. Tech. J. 48 (7): 2071–2102. doi:10.1002/j.1538-7305.1969.tb01166.x. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Westerveld, W. J., Leinders, S. M., van Dongen, K. W. A., Urbach, H. P. and Yousefi, M (2012). "Extension of Marcatili's Analytical Approach for Rectangular Silicon Optical Waveguides". Journal of Lightwave Technology. 30 (14): 2388–2401. arXiv:1504.02963. Bibcode:2012JLwT...30.2388W. doi:10.1109/JLT.2012.2199464. S2CID 23182579.{{cite journal}}:  CS1 maint: multiple names: authors list (link) <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Kumar, A., K. Thyagarajan and A. K. Ghatak. (1983). "Analysis of rectangular-core dielectric waveguides—An accurate perturbation approach". Opt. Lett. 8 (1): 63–65. Bibcode:1983OptL....8...63K. doi:10.1364/ol.8.000063. PMID 19714136.{{cite journal}}:  CS1 maint: multiple names: authors list (link) <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Talukdar, Tahmid H.; Allen, Gabriel D.; Kravchenko, Ivan; Ryckman, Judson D. (2019-08-05). "Single-mode porous silicon waveguide interferometers with unity confinement factors for ultra-sensitive surface adlayer sensing". Optics Express. 27 (16): 22485–22498. Bibcode:2019OExpr..2722485T. doi:10.1364/OE.27.022485. ISSN 1094-4087. OSTI 1546510. PMID 31510540. <a href="https://opg.optica.org/oe/fulltext.cfm?uri=oe-27-16-22485" target="_blank">https://opg.optica.org/oe/fulltext.cfm?uri=oe-27-16-22485</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>M. Hochberg; T. Baehr-Jones; C. Walker; J. Witzens; C. Gunn; A. Scherer (2005). "Segmented Waveguides in Thin Silicon-on-Insulator" (PDF). Journal of the Optical Society of America B. 22 (7): 1493–1497. Bibcode:2005JOSAB..22.1493H. doi:10.1364/JOSAB.22.001493. <a href="https://authors.library.caltech.edu/7675/1/HOCjosab05.pdf" target="_blank">https://authors.library.caltech.edu/7675/1/HOCjosab05.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>S. Y. Lin; E. Chow; S. G. Johnson; J. D. Joannopoulos (2000). "Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-μm wavelength". Optics Letters. 25 (17): 1297–1299. Bibcode:2000OptL...25.1297L. doi:10.1364/ol.25.001297. OSTI 754329. PMID 18066198. <a href="https://zenodo.org/record/1235662" target="_blank">https://zenodo.org/record/1235662</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Meany, Thomas (2014). "Optical Manufacturing: Femtosecond-laser direct-written waveguides produce quantum circuits in glass". Laser Focus World. 50 (7). <a href="https://www.laserfocusworld.com/lasers-sources/article/16550196/optical-manufacturing-femtosecondlaser-directwritten-waveguides-produce-quantum-circuits-in-glass" target="_blank">https://www.laserfocusworld.com/lasers-sources/article/16550196/optical-manufacturing-femtosecondlaser-directwritten-waveguides-produce-quantum-circuits-in-glass</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Streltsov, AM; Borrelli, NF (1 January 2001). "Fabrication and analysis of a directional coupler written in glass by nanojoule femtosecond laser pulses". Optics Letters. 26 (1): 42–3. Bibcode:2001OptL...26...42S. doi:10.1364/OL.26.000042. PMID 18033501. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Khrapko, Rostislav; Lai, Changyi; Casey, Julie; Wood, William A.; Borrelli, Nicholas F. (15 December 2014). "Accumulated self-focusing of ultraviolet light in silica glass". Applied Physics Letters. 105 (24): 244110. Bibcode:2014ApPhL.105x4110K. doi:10.1063/1.4904098. <a href="https://doi.org/10.1063%2F1.4904098" target="_blank">https://doi.org/10.1063%2F1.4904098</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Liu, Hsuan-Hao; Chang, Hung-Chun (2013). "Leaky Surface Plasmon Polariton Modes at an Interface Between Metal and Uniaxially Anisotropic Materials". IEEE Photonics Journal. 5 (6): 4800806. Bibcode:2013IPhoJ...500806L. doi:10.1109/JPHOT.2013.2288298. <a href="https://doi.org/10.1109%2FJPHOT.2013.2288298" target="_blank">https://doi.org/10.1109%2FJPHOT.2013.2288298</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
</ol>

Waveguide (optics) open-in-new

Waveguide (optics)