Ministry of higher education, science and innovation of the republic of uzbekistan

MINISTRY OF HIGHER EDUCATION, SCIENCE AND INNOVATION OF THE REPUBLIC OF UZBEKISTAN
MINISTRY FOR THE DEVELOPMENT OF INFORMATION TECHNOLOGIES AND COMMUNICATIONS OF THE REPUBLIC OF UZBEKISTAN
SAMARKAND BRANCH OF TASHKENT UNIVERSITY OF INFORMATION TECHNOLOGIES NAMED AFTER MUHAMMAD AL-KHOREZMI
FACULTY OF TELECOMMUNICATION TECHNOLOGIES



INDEPENDENT WORK
On the topic:Investigation of the modes of operation of a coaxial waveguide
Fulfilled: ТТ-2103 groups
Jo’raboyev Shonazar
Accepted:LawrenxeViktoria CH.

Plan

1.An All-Dielectric CoaxialWaveguide
2.Power density in the electric field for guided modes

An All-Dielectric CoaxialWaveguide

An All-Dielectric CoaxialWaveguideM. Ibanescu,1Y. Fink,2S. Fan,1E. L. Thomas,2J. D. Joannopoulos1An all-dielectric coaxial waveguide that can overcome problems of polarizationrotation and pulse broadening in the transmission of optical light is presentedhere. It consists of a coaxial waveguiding region with a low index of refraction,bounded by two cylindrical, dielectric, multilayer, omnidirectional reflectingmirrors. The waveguide can be designed to support a single mode whoseproperties are very similar to the unique transverse electromagnetic mode ofa traditional metallic coaxial cable. The new mode has radial symmetry and apoint of zero dispersion. Moreover, because the light is not confined by totalinternal reflection, the waveguide can guide light around very sharp corners
Waveguides are the backbone of modern op-toelectronics and telecommunications sys-tems. There are currently two major, and verydistinct, types of waveguides (metallic anddielectric) that are used in two separate re-gimes of the electromagnetic spectrum. Forradio frequencies, the metallic coaxial cableis of greatest prominence (1). In this type ofcable, the entire electromagnetic field is con-fined between two coaxial metal cylinders.The important fundamental electromagneticmode of a coaxial cable is the transverseelectromagnetic (TEM) mode, which isunique in that it has radial symmetry in theelectric field distribution and a linear rela-tionship between frequency and wave vector.This gives the TEM mode two exceptionalproperties. First, the radial symmetry impliesthat one need not worry about possible rota-tions of the polarization of the field after itpasses through the waveguide. Second, thelinear relationship ensures that a pulse ofdifferent frequencies will retain its shape as itpropagates along the waveguide. The crucialdisadvantage of a coaxial metallic waveguideis that it is useless at optical wavelengthsbecause of heavy absorption losses in themetal. For this reason, optical waveguidingisrestricted to the use of dielectric materials.However, because of the differences inboundary conditions of the electromagneticfields at metal and dielectric surfaces, it hasnot previously been possible to recreate aTEM-like mode with all-dielectric materials.Consequently, optical waveguidingisdone with the traditional index-guiding (thatis, total internal reflection) mechanism, asexemplified by silica and chalcogenideopti-cal fibers. Such dielectric waveguides canachieve very low losses (2). Although thCenter for Materials Science and Engineering andDepartment of Physics, Massachusetts Institute ofTechnology, Cambridge, MA 02139, USA.2Depart-ment of Materials Science and Engineering, Massa-chusetts Institute of Technology, Cambridge, MA02139, USA

The dispersion relations for the firstfew modes supported by a metallic coaxialwaveguideare shown in Fig. 2A. For defi-niteness, the inner and outer radii of thewaveguiding region are taken to be ri⫽ 3.00a and ro⫽ 4.67 a, respectively, where a isanarbitrary unit of length to be defined later. Forany value of the wave vector, the lowestfrequency mode is the TEM mode for whichboth the electric and magnetic fields aretransverse to the direction of propagation.This mode has zero angular momentum,which means that the mode is invariant underrotations around the axial direction. Anotheruseful property of this mode is its constantgroup velocity, which makes it dispersionlessat any frequency. The other modes shown inthe plot are transverse electric (TEml) modesfor l ⫽ 1 and varying angular momenta m (8).The cutoff frequency of any of these modes isof the form␻cutoff⫽crof冉rori冊where f is the solution to a transcendentalequation for each value of the angular mo-mentum m and for each polarization (TE orTM) (1).

Designing an all-dielectric waveguidewith similar principles of operation as themetallic coaxial waveguide is not straightfor-ward, because the boundary conditions at adielectric-dielectric interface differ fromthose at an air (dielectric)–metal interface. Inparticular, specular reflections cannot be ob-tained on a dielectric-dielectric interfacewhen the ray of light comes from the regionwith a lower index of refraction. Thus, it hasgenerally been assumed that an all-dielectriccoaxial waveguide cannot be designed to sup-port a TEM-like mode, even in principle.However, recent research on the omnidirec-tional dielectric reflector (9) has opened newpossibilities for reflecting, confining, andguiding light with all-dielectric materials. In-deed, a dielectric hollow waveguide usingthis principle was recently fabricated andtested successfully at optical wavelengths(4). The omnidirectional dielectric reflector,or simply the dielectric mirror, is a periodic,multilayered planar structure consisting ofalternating layers of low and high indices ofrefraction. This structure can be designed sothat there is a range of frequencies at whichincoming light from any direction and of anypolarizationis reflected. Moreover, the elec-tric fields of the reflected light in this fre-quency range have corresponding phaseshifts that are quite close to those acquiredupon reflection from a metal. In fact, there isa frequency for each angle of incidence andeach polarization, for which the phase shift isidentical to that of a metal. This observation,together with the fact that high reflectivity ofthe omnidirectional dielectric mirror is main-tained for all angles of incidence, stronglysuggests exploration of the possibilities ofusing an omnidirectional mirror in lieu of ametal in coaxial cable designs. In effect, theomnidirectional dielectric mirror provides anew mechanism for guiding optical and in-frared light without incurring the inherentlosses of a metal

Fig. 2. Projected band structures along an axial direction. (A) Traditionalmetallic coaxial cable with inner and outer coaxial radii of ri⫽ 3.00 aandro⫽ 4.67 a, respectively. The red bands correspond to allowed guidedmodes. For any given wavevector, the lowest frequency mode is a TEMmode characterized by a perfectly linear dispersion relation. The six nexthighest bands correspond to transverse electric (TEml) modes with l ⫽ 1and increasing angular momentum m.(B) Omnidirectional, reflecting,all-dielectric multilayer film. Light-blue regions correspond to modes forwhich light is allowed to propagate within the dielectric mirror, anddark-blue regions correspond to modes for which light is forbidden topropagate within the dielectric mirror. The diagonal black line identifiesthe edge of the light cone. The horizontal gray lines mark the boundariesin frequency within which omnidirectional reflectivity is possible. (C)Coaxial omniguide A with inner and outer coaxial radii of ri⫽ 3.00 a andro⫽ 4.67 a, respectively, and bilayers consisting of indexes of refractionn1⫽ 4.6 and n2⫽ 1.6 and thickness d1⫽ 0.33 a and d2⫽ 0.67 a,respectively. The red and yellow bands indicate guided modes confinedto the coaxial region of the waveguide. The dashed lines indicate modeswith less than 20% localization within the coaxial region. There is closecorrespondence between the modes labeled m ⫽ 1tom ⫽ 6 and thoseof (A) labeled TE11to TE61. Also, the yellow m ⫽ 0 mode corresponds toa TEM-like mode, as discussed in the text.
Before we begin our investigation of themodes supported by the coaxial omniguide,it is instructive to first review the modes ofa planar omnidirectional dielectric mirror.The projected band structure of the omni-directional mirror is shown in Fig. 2B. Thelight blue regions represent allowed propa-gation modes of light within the dielectricmirror. The dark blue region representsmodes for which light is forbidden to prop-agate within the dielectric mirror. The thickblack line identifies the edge of the lightcone, and the horizontal gray lines demar-cate the frequency range of omnidirectionalreflectivity. It is precisely within this rangeof frequencies that one would expect thecoaxialomniguide to support modes thatare most reminiscent of those of the metal-lic coaxial cable. To calculate the frequen-cies and field patterns of the modes ofcoaxialomniguideA, we proceed as de-scribed below.As a result of the cylindrical symmetryof the system, there are two good “con-served quantities” that can be used to spec-ify and classify the various modes support-ed by this waveguide. These are kz, theaxial component of the wave vector, and m,the angular momentum (m ⫽ 0,1,2...).For a given mode, the radial and angularcomponents of the electric and magneticfieldscan be calculated from the corre-sponding z (axial) components (12). For agiven wave vector kzand angular momen-tum m, the axial field components in a layerof index n have the general form

The modes of the coaxial omniguidearecalculated with two different approaches. Thefirst is a semianalytic approach based on thetransfer matrix method (14 ). Starting fromMaxwell’s equations, the z components of theelectric and magnetic fields in each layer canbe written in the general form given by Eq. 1.For given kz, ␻, and m, the only variables thatdetermine the EzandHzfields are the fourcoefficients in front of the Bessel functions(two for Ezand two for Hz). The boundaryconditions at the interfaces between adjacentlayerscan be written in the form of a matrixequation

Fig. 3. Power density in the electric field for guided modes at kz⫽ 0.19 (2␲/a) in Fig. 2C. (A throughD) correspond to guided modes with angular momenta m ⫽ 0tom ⫽ 3, respectively. The colorbar indicates that power increases in going from black to dark red, to red, to orange, to yellow. Theblue circles identify the boundaries between the various dielectric shells and are included as a guideto the eye. Most of the power is confined to the coaxial region of the waveguide. The cylindricalsymmetry and radial dependence of the m ⫽ 0 mode are consistent with those of a TEM mode

In the projected band structure for coaxialomniguide A (Fig. 2C), the red and yellowbands represent guided modes localized with-in the region defined by the inner and outercoaxial radii of the waveguide. The dashedlines represent modes with less than 20%localization within the coaxial region. Thereis close correspondence between the modeswithin the omnidirectional reflectivity rangelabeled m ⫽ 1tom ⫽ 6 and those of thecoaxial cable labeled TE11to TE61(16). Them⫽ 0 mode appears to correspond to theTEM mode. Of course, for a coaxial omni-guide with a limited number of outer shells,these modes can only exist as resonances.Nevertheless, even with only 2.5 bilayers, wefind that they can be extremely well localizedresonances, and the leakage rate decreasesexponentially with the number of shells. Thestrong localization is shown in Fig. 3. Here,we plot the power density in the electric fieldfor the four lowest frequency modes at kz⫽0.19 (2␲/a). As the color bar indicates,powerincreases in going from black to dark red, tored, to orange, to yellow. The blue circlesidentify the boundaries between the variousdielectric shells and are included as a guide tothe eye. In all cases, the power is confinedprimarily within the coaxial region. This isparticularly true for the m ⫽ 0 mode, which isalso cylindrically symmetric, just like theTEM mode.
TM mode) possesses several of the character-istics of the TEM mode. First, as mentionedabove, it has zero angular momentum andhence a radially symmetric electric field distri-bution. Second, the electric and magnetic fieldswithin the coaxial waveguiding region (whereover 65% of the power is concentrated) arenearly identical to those of the metallic coaxialcable; for example, the predominant compo-nents are ErandHfand vary as 1/r. Finally, atthe point where the m ⫽ 0 dispersion curve(yellow line) crosses the light line, there is anexact correspondence between the electromag-netic fields of the coaxial omniguide and themetallic coaxial cable, inside the coaxial region.Moreover, the derivative of the group velocityis exactly zero near this point, leading to nearlydispersionless propagation throughout its vicin-ity (18).The characteristics described above are cer-tainly the attributes one would hope to achievein order to overcome problems with polariza-tion-rotation and pulse broadening.Butwhatabout single-mode behavior? The bands shownin Fig. 2C are clearly multimode; that is, for agiven frequency there are two or more guidedmodes that can be excited. To design a coaxialomniguide that can support single-mode behav-ior, we need only readjust our structural param-eters


This leaves only those parameters that are com-mon to both the coaxial omniguide and themetallic coaxial cable: the inner and outer radiiof the coaxial waveguiding region. Single-mode operation for the TEM-like mode willonly be possible if all other modes are movedup in frequency so that the lowest nonzeroangular mode has its cutoff frequency inside thebandgap. To do this, we have to decrease theinner radius of the coaxial waveguidingregion.At the same time, the thickness of the bilayers,a, should remain constant, which means that wecan no longer accommodate three bilayers inthe inner part of the waveguide. Actually, theinner radius has to be decreased so much thatwe are forced to discard the periodic structurein the inner region and to replace it with a singledielectric rod. Loss of the inner-core mirrorstructure is not crucial, however. What is im-portant is to add a thin rod of dielectric in thecore in order to avoid the 1/r divergence of thefield at the origin and to use a dielectric of highenough contrast to localize the TEM-like modein the coaxial region. This approach, however,will not work if ri⬎ a, and one must then revertto a multilayer core. Testing different values forthe inner and outer radii of the waveguidingregion, we have found a configuration that hasthe desired properties. This new embodiment,coaxialomniguide B, is shown in Fig. 1C. Thecentral dielectric rod has an index of refractionn1⫽ 4.6 and a radius ri⫽ 0.40 a. The coaxialwaveguiding region has an outer radius ro⫽1.40 a, and the parameters of the outer bilayersare the same as those used for configuration A.In configuration B, there are two frequencyranges where the waveguide can operate in asingle-mode fashion. We plot the dispersioncurves for the modes supported by coaxial om-niguide B in Fig. 4. The yellow dots indicatemore than 50% confinement of the electric fieldpower, whereas the red dots represent confine-ment between 20 and 50%. (The dashed linesindicate confinement that is less than 20%.) Thetwo white boxes identify the frequency rangeswhere the m ⫽ 0 band is single-mode. A com-parison of Figs. 2C and 4 reveals that the cutofffrequency of the m ⫽ 1 band has shifted sig-nificantly upward, whereas the m ⫽ 0 bandremains relatively unchanged. The flatness ofthe m ⫽ 1 band (19) enables the TEM-like bandto be single-mode both above it and below it infrequency. The exact values of the parameterswere chosen so that, in the middle of the higherfrequency single-mode window [at ␻⫽0.205(2␲c/a)], the mode is also dispersionless (18).Figure 5 shows the distribution of theelectric field components for the m ⫽ 0 modeof coaxial omniguide B at kz⫽ 0.2 (2␲/a)and ␻⫽0.203 (2␲c/a). Because the (kz, ␻)point is very close to the light line, the elec-tric field in the waveguiding region is almostcompletely transverse to the direction ofpropagation. (The z component of the mag-netic field will always be zero because this isa pure TM mode.) The field distributionclearly reveals a high confinement of themode in the waveguiding region, as desired.Moreover, these values of ExandEylead to anet field distribution that is completely radi-ally symmetric, consistent with an angularcomponent that is exactly zero. All the fea-tures mentioned above attest to the closecorrespondence between the m ⫽0 modeanda pure TEM mode.There are several additional issues associat-ed with the coaxial omniguide. The first is apractical issue involving the coupling of lightinto the coaxial omniguide. One possible meth-od for coupling would be to begin with anomniguide with a very thin core that increasesgradually to match the core of the coaxial con-figuration. Because the electromagnetic field ofa laser source can have a TEM00mode, thisshould lead to efficient coupling into the TEM