The Propagation Characteristics Of The Optical Waveguides Including Left-handed Materials

Now,the research for the left-handed material(LHM) is one of hot points for the electromagnetic fields.As electromagnetic wave propagates in the material,its electric field,magnetic field and wave vector construct left system.However,for convensional materials,its electric field,magnetic field and wave vector construct right system.The paper investigates the propagation characteristics for the LHM optical waveguides through Maxwell equations.The contents include that:guided modes in the three-layer and five-layer slab waveguide;guided modes and surface modes in a single clad fiber;guided modes in a doubly clad fiber and the coupling between optical waveguides.1).Guided modes in a slab waveguide including LHMA slab waveguide is the simplest in all waveguides.The paper discusses the guided modes(TE and TM) in the three-layer slab waveguide including LHM and their dispersive equations are obtained.According the dispersive equations,we plot their dispersive curves.Compared these curves,we can find that:the three-layer slab waveguide with LHM cannot exist TEo mode and TMo mode;the dispersive curves for TE₁₁ mode and TM₁₁ mode are abnormal;the effective thickness H for the guided modes can be less zero;for the same normalized frequency,some guided modes have two different values near cutoff.Secondly,the TE modes for the five-layer slab waveguide including LHM are investigated and their dispersive equations are obtained.According to these dispersive equations,we plot some dispersive curves.The curves are studied and found that:for TE₂₁ mode,the core has no energy near cutoff,however,as the normalized fi'equency adds,the energy shifts from claddings to the core; for TE₁₁ mode,the energy in the core is the same as that in the clad,but,as the normalized frequency adds,energy can shift from the core to the clad,in the end,all energy is all in the clad.That benefits us to build new couplers in the future.2).The guided modes and surface modes in the single clad fiber with left-handed materialThe fiber is the most important for the optical transmission.This paper investigates the single fiber with LHM and obtains some characteristic equations for the oscillating modes(TE modes,TM modes,HE modes and EH modes).These equations are discussed near cutoff and far from cutoff.Besides,their dispersive curves are plotted and some new characteristics are found.Such as:in the LHM fiber, there are TE₀ mode and TMo mode and that is different with that of LHM slab waveguide;HE₁₁ is no longer the lowest mode.Secondly,the surface modes for the TE,TM,HE and EH modes are discussed and their dispersive equations are obtained.By using these equations,we plot their dispersive curves.Then,the curves are compared with that of relative guided modes. They show that the curves for the surface mode TEo and the guided mode TE₀₁ are continuous;the curves for the surface modes HE₁,HE₂,HE₃ and the guided modes EH₁₁,EH₂₁,EH₃₁ are continuous,respectively.3).Guided modes in the doubly clad fiber with LHMThe doubly clad fiber with LHM is studied carefully.First,by an accurate way, we obtain the characteristic equations for hybrid modes,TE modes and TM modes in that fiber.According to the equation of TE mode,we plot some dispersive curves for the different refractive index and different thickness of inner cladding.Compare these curves,we can find that:as the refractive index of the inner cladding adds,the cutoff frequency becomes big;as the thickness of the inner cladding adds,the cutoff frequency becomes small.Second,by an approximate way,we obtain the characteristic equations for the guided modes in that fiber.These equations are simplified far from cutoff.Their dispersive curves are plotted and the simple conditions are obtained.Besides, compare the curves,we find that:the dispersive characteristics for fundamental modes are abnormal;the cutoff frequencies of all guided modes can exist,etc.4).The coupling between the LHM waveguidesThe research of the coupling between LHM waveguides has more significance for its applications.First,we obtain the extended coupling wave equations and extended coupling coefficient between LHM waveguides.The expression for the extended coupling coefficient includes electric field and magnetic field,so,it is more complicated than that of common waveguides.We use the extended coupling coefficient to deal with the coupling between the three-layer slab waveguides with LHM.The coupling coefficient for TE modes is found and its coupling curves are plotted,too.Compare these curves,we find that:the nearer distance two waveguides, the stronger the coupling;the smaller the mode order,the stronger the coupling.Second,we investigate the coupling between five-layer slab waveguide including LHM through the extended coupling coefficient.The extended coupling coefficient for the even TE mode is obtained.Clearly,the coefficient expression is more complicated than that of common slab waveguide.In the end,using the extended coupling coefficient,we study the coupling between optical fibers i.e.the core is lefc-handed and the cladding fight-handed material.The extended coupling coefficient for TE₀₁ mode is obtained and some coupling curves are plotted.Researching these coupling curves,we find that:the bigger the absolute of refractive index in the core,the stronger the coupling.