Dielectric-fibre surface waveguides for optical frequencies K.C.Kao and G.A.Hockham Indexing terms: Optical fibres,Waveguides Abstract:A dielectric fibre with a refractive index higher than its surrounding region is a form of dielectric waveguide which represents a possible medium for the guided transmission of energy at optical frequencies.The particular type of dielectric-fibre waveguide discussed is one with a circular cross-section.The choice of the mode of propagation for a fibre waveguide used for communication purposes is governed by consideration of loss characteristics and information capacity.Dielectric loss,bending loss and radiation loss are discussed,and mode stability,dispersion and power handling are examined with respect to information capacity.Physical- realisation aspects are also discussed.Experimental investigations at both optical and microwave wavelengths are included. List of principle symbols 2 Dielectric-fibre waveguide nth-order Bessel function of the first kind The dielectric fibre with a circular cross-section can =nth-order modified Bessel function of the second support a family of Hom and Eom modes and a family of kind 2 hybrid HEm modes.Solving the Maxwell equations under phase cofficient of the waveguide 2元 the boundary conditions imposed by the physical struc- ture,the characteristic equations are as follows: J first derivative of J for HEm modes Ki=first derivative of Kn n2B2/1,12 hi radial wavenumber or decay coefficient 61 L()62 K(u2 relative permittivity lu J(u)u2 K(u2) Ko free-space propagation coefficient LJu+⊥K2 a radius of the fibre (1) longitudinal propagation coefficient lu1 J(u)u2 K (u2) Boltzman's constant for Eom modes 入 =absolute temperature,K isothermal compressibility E1 Jo(u1)82 Ko(u2) (2) wavelength ui Jo(u) uz Ko(u2) n refractive index Hi)=uth-order Hankel function of the ith type for Hom modes H derivation of H. 1J6(u1)_1Ko(u2) =azimuthal propagation coefficient =vi-jv2 u Jou) u2 Ko(u2) (3) modulation period Subscript n is an integer and subscript m refers to the mth The auxiliary equations defining the relationship between root of J =0 u and u,are ui+uz=(ko a)2(e1-E2) h好=y2+k好e1 1 Introduction -h好=y2+k6e2 A dielectric fibre with a refractive index higher than its surrounding region is a form of dielectric waveguide which ui=hia,i=1 and 2 represents a possible medium for the guided transmission where subscripts 1 and 2 refer to the fibre and the outer of energy at optical frequencies.This form of structure region,respectively. guides the electromagnetic waves along the definable boundary between the regions of different refractive All the modes exhibit cutoffs except the HE mode, which is the lowest-order hybrid mode.It can assume two indexes.The associated electromagnetic field is carried orthogonal polarisations,and it propagates with an partially inside the fibre and partially outside it.The exter- increasing percentage of energy outside the fibre as the nal field is evanescent in the direction normal to the direc- dimensions of the structure decrease.Thus,when operating tion of propagation,and it decays approximately exponentially to zero at infinity.Such structures are often the waveguide in the HE mode,it is possible to achieve a single-mode operation by reducing the diameter of the referred to as open waveguides,and the propagation is fibre sufficiently.Under this condition,a significant pro- known as the surface-wave mode.The particular type of portion of the energy is carried outside the fibre.If the dielectric-fibre waveguide to be discussed is one with a cir- outside medium is of a lower loss than the inside dielectric cular cross-section. medium,the attenuation of the waveguide is reduced.With these properties,HE1 mode operation is of particular interest. Paper 5033E was originally published in the Proceedings IEE,July 1966.It was first The physical and electromagnetic aspects of the received 24th November 1965 and in revised form 15th February 1966. The authors were formerly with Standard Telecommunication Laboratories Ltd dielectric-fibre waveguide carrying the HE mode for use Harlow,Essex.Prof.Kao is now with ITT and Dr.Hockham is with Plessey Com- at optical frequencies will now be studied in detail.Con- pany Ltd.,241 Station Road,Addlestone,Surrey,United Kingdom clusions are drawn as to the feasibility and the expected IEE PROCEEDINGS,Vol.133,Pt.J,No.3,JUNE 1986 191Dielectric-fibre surface waveguides for optical frequencies K.C. Kao and G.A. Hockham Indexing terms: Optical fibres, Waveguides Abstract: A dielectric fibre with a refractive index higher than its surrounding region is a form of dielectric waveguide which represents a possible medium for the guided transmission of energy at optical frequencies. The particular type of dielectric-fibre waveguide discussed is one with a circular cross-section. The choice of the mode of propagation for a fibre waveguide used for communication purposes is governed by consideration of loss characteristics and information capacity. Dielectric loss, bending loss and radiation loss are discussed, and mode stability, dispersion and power handling are examined with respect to information capacity. Physical￾realisation aspects are also discussed. Experimental investigations at both optical and microwave wavelengths are included. List of principle symbols Jn = nth-order Bessel function of the first kind Kn = nth-order modified Bessel function of the second kind 271 271 B — —, phase coefficient of the waveguide Xg }'n = first derivative of Jn K^, = first derivative of Kn hi = radial wavenumber or decay coefficient €,- = relative permittivity k0 = free-space propagation coefficient a = radius of the fibre y = longitudinal propagation coefficient k = Boltzman's constant T = absolute temperature, K j 5c = isothermal compressibility X = wavelength n = refractive index Hj,0 = uth-order Hankel function of the ith type H'v = derivation of Hu v = azimuthal propagation coefficient = i^ — jv2 L = modulation period Subscript n is an integer and subscript m refers to the mth root of L = 0 1 Introduction A dielectric fibre with a refractive index higher than its surrounding region is a form of dielectric waveguide which represents a possible medium for the guided transmission of energy at optical frequencies. This form of structure guides the electromagnetic waves along the definable boundary between the regions of different refractive indexes. The associated electromagnetic field is carried partially inside the fibre and partially outside it. The exter￾nal field is evanescent in the direction normal to the direc￾tion of propagation, and it decays approximately exponentially to zero at infinity. Such structures are often referred to as open waveguides, and the propagation is known as the surface-wave mode. The particular type of dielectric-fibre waveguide to be discussed is one with a cir￾cular cross-section. Paper 5O33E was originally published in the Proceedings IEE, July 1966. It was first received 24th November 1965 and in revised form 15th February 1966. The authors were formerly with Standard Telecommunication Laboratories Ltd., Harlow, Essex. Prof. Kao is now with ITT and Dr. Hockham is with Plessey Com￾pany Ltd., 241 Station Road, Addlestone, Surrey, United Kingdom 2 Dielectric-fibre waveguide The dielectric fibre with a circular cross-section can support a family of HOm and EOm modes and a family of hybrid HEnm modes. Solving the Maxwell equations under the boundary conditions imposed by the physical struc￾ture, the characteristic equations are as follows: for HEnm modes n2 /32 — + 1 ) =< — for EOm modes for HOm modes u2 K0 (u2 ) 1 K'p(u2) u2 K0(w2) (1) (2) (3) The auxiliary equations defining the relationship between u1 and u2 are ui + ul= (k0 a)2 (ei - e2) -h\ = y2 + k2 0s2 u( = h(a,i=\ and 2 where subscripts 1 and 2 refer to the fibre and the outer region, respectively. All the modes exhibit cutoffs except the HE M mode, which is the lowest-order hybrid mode. It can assume two orthogonal polarisations, and it propagates with an increasing percentage of energy outside the fibre as the dimensions of the structure decrease. Thus, when operating the waveguide in the HEX1 mode, it is possible to achieve a single-mode operation by reducing the diameter of the fibre sufficiently. Under this condition, a significant pro￾portion of the energy is carried outside the fibre. If the outside medium is of a lower loss than the inside dielectric medium, the attenuation of the waveguide is reduced. With these properties, HE n mode operation is of particular interest. The physical and electromagnetic aspects of the dielectric-fibre waveguide carrying the HE n mode for use at optical frequencies will now be studied in detail. Con￾clusions are drawn as to the feasibility and the expected IEE PROCEEDINGS, Vol. 133, Pt. J, No. 3, JUNE 1986 191