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1 Citation for published version: Chen, L & Bird, DM 2011, 'Guidance in Kagoe-like photonic crystal fibres II: perturbation theory for a realistic fibre structure' Optics Express, vol. 19, no. 7, pp DOI: /OE Publication date: 2011 Link to publication 2011 Optical Society of Aerica. This paper was published in Optics Express and is ade available as an electronic reprint with the perission of OSA. The paper can be found at the following URL on the OSA website: Systeatic or ultiple reproduction or distribution to ultiple locations via electronic or other eans is prohibited and is subject to penalties under law. University of Bath General rights Copyright and oral rights for the publications ade accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requireents associated with these rights. Take down policy If you believe that this docuent breaches copyright please contact us providing details, and we will reove access to the work iediately and investigate your clai. Download date: 23. Jan. 2019

2 Guidance in Kagoe-like photonic crystal fibres II: perturbation theory for a realistic fibre structure Lei Chen and David M. Bird Centre for Photonics and Photonic Materials, Departent of Physics, University of Bath, Bath, BA2 7AY, UK d.bird@bath.ac.uk Abstract: A perturbation theory is developed that treats a localised ode ebedded within a continuu of states. The ethod is applied to a odel rectangular hollow-core photonic crystal fibre structure, where the basic odes are derived fro an ideal, scalar odel and the perturbation ters include vector effects and structural difference between the ideal and realistic structures. An expression for the attenuation of the fundaental ode due to interactions with cladding odes is derived, and results are presented for a rectangular photonic crystal fibre structure. Attenuations calculated in this way are in good agreeent with nuerical siulations. The origin of the guidance in our odel structure is explained through this quantitative analysis. Further perspectives are obtained through investigating the influence of fibre paraeters on the attenuation Optical Society of Aerica OCIS codes: ( ) Fiber design and fabrication; ( ) Fiber properties. References and links 1. L. Chen, G. J. Pearce, T. A. Birks, and D. M. Bird, Guidance in Kagoe-like photonic crystal fibres I: analysis of an ideal fibre structure, Subitted to Opt. Express (2011). 2. P. W. Anderson, Localized agnetic states in etals, Phys. Rev. 124, (1961). 3. L. Chen, Modelling of photonic crystal fibres, Ph.D. thesis, University of Bath (2009). 4. S. Davison and M. Stesliska, Basic Theory of Surface States (Oxford U. Press, 1992). 5. E. Econoou, Green s Functions in Quantu Physics (Springer-Verlag, 1990). 6. F. Couny, F. Benabid, and P. S. Light, Large-pitch Kagoe-structured hollow-core photonic crystal fiber, Opt. Lett. 31, (2006). 7. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Rayer, Generation and Photonic Guidance of Multi-Octave Optical-Frequency Cobs, Science 318, (2007). 8. A. Argyros and J. Pla, Hollow-core polyer fibres with a kagoe lattice: potential for transission in the infrared, Opt. Express 15, (2007). 9. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, Square-lattice large-pitch hollow-core photonic crystal fiber, Opt. Express 16, (2008). 10. N. Guan, S. Habu, K. Takenaga, K. Hieno, and A. Wada, Boundary eleent ethod for analysis of holey optical fibers, J. Lightwave Technol. 21, (2003). 11. T.-L. Wu and C.-H. Chao, Photonic crystal fiber analysis through the vector boundary-eleent ethod: effect of elliptical air hole, IEEE Photon. Technol. Lett. 16, (2004). 12. X. Wang, J. Lou, C. Lu, C.-L. Zhao, and W. Ang, Modeling of pcf with ultiple reciprocity boundary eleent ethod, Opt. Express 12, (2004). 13. Y. Wang, F. Couny, P. Roberts, and F. Benabid, Low loss broadband transission in optiized core-shape kagoe hollow-core pcf, in Lasers and Electro-Optics (CLEO) and Quantu Electronics and Laser Science Conference (QELS), 2010 Conference on, (2010), pp (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6957

3 1. Introduction In Ref. [1], an analytic ethod was developed for an ideal, scalar odel of a rectangular hollowcore photonic crystal fibre (PCF). It was found that the fundaental ode is perfectly confined over a wide range of frequencies, providing a baseline fro which we ai to explain the broadband guidance observed in a class of Kagoe-like PCFs. Our ai in this paper is to extend the analysis of Ref. [1] in order to calculate the attenuation of the fundaental ode in a realistic fibre structure. For the ideal, scalar odel it has been found that a perfectly localised fundaental ode exists both in photonic bands and in bandgaps. With a vector governing equation, we find that the bandgaps vanish and that there are cladding odes with the sae propagation constant as the fundaental ode for all frequencies. These cladding odes are glass-guided [1] and have a weak coupling with the fundaental ode owing to the isatch of the spatial frequencies of the odal fields. The key point of this work is the use of perturbation theory to include the effects of the vector ters in the governing equation and the structural difference (the high-index intersections) between the ideal and realistic odel structures. The ethod used will need to treat the case of a localised state (the fundaental ode) ebedded within a continuu of states (the cladding odes). This proble recalls a case in condensed atter physics. When a localised ipurity orbital is ebedded in an electron gas, the sharply defined ipurity level broadens into a resonance. The width of the resonance is related to the strength of coupling between the localised and continuu states. Anderson [2] analysed this case with the use of Green s function ethods and found that the effect of the interaction can be discussed in ters of an iaginary coponent of the energy of the localised state. This controls the width of the resonance and can be related to the lifetie of an electron initially localised on the ipurity. In our case the interaction of the localised and cladding odes has the effect of broadening the fundaental ode into a resonance. The resonance width can be related to an iaginary part of the propagation constant and therefore this leads directly to a easure of the attenuation of the fundaental ode which is caused by the perturbation. The content of this paper is organised as follows. In Section 2, the perturbation theory is developed to siulate guidance in a realistic odel PCF structure. An approxiate ethod to calculate the attenuation due to the interaction of the fundaental and cladding odes is presented. Section 3 gives the results of the perturbation calculations, based on the theory developed in Section 2 and the analysis presented in Ref. [1]. It gives the origin and quantitative explanation of the leakage in our odel PCF structure. The effects of the fibre paraeters are also investigated. Section 4 is the conclusion. 2. Perturbation theory for the realistic odel 2.1. Forulation of perturbation theory for the vector governing equation We begin our derivation by rewriting the governing equation in a copact for. The scalar governing equation of Eq. (2) in Ref. [1] can be expressed as L 0 h n = β 2 0nh n, (1) where L 0 denotes 2 t +n 2 i k2 0, and n2 i is the dielectric function for the ideal odel structure. Here, β 2 0n and h n are the eigenvalues and eigenstates of the operator L 0, corresponding to the square of the propagation constants and their associated fields, respectively. The subscript n labels the odes including their polarisation. The vector governing equation corresponding to Eq. (1) in Ref. [1] can be written as L 0 H + δl(h)=β 2 H, (2) (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6958

4 where the eigenvalue β 2 and eigenvector H now respectively denote the square of the propagation constant and field for the realistic syste. The perturbation ter, δl(h), is a linear vector function of H. In our particular odel structure, δl(h) takes the for δl(h)= ( t H ) t lnn 2 r + Δn 2 k 2 0H, (3) where n 2 r is the dielectric function for the realistic odel PCF and Δn 2 = n 2 r n 2 i. Note that this definition of Δn 2 is different fro that in Ref. [1]; in this paper Δn 2 is non-zero only at the intersections of the glass strips (where it takes the value n 2 a n 2 g, where n a and n g are refractive indices for air and glass respectively) and is zero elsewhere. The right-hand side of Eq. (3) consists of two parts: the first is the vector ter in the governing equation; the second coes fro the difference of the dielectric constant between the realistic and ideal odel structures. We refer to the as the vector ter and high-index ter, respectively. The full vector solution H can be expressed by using the scalar solutions as basis functions: H = n a n h n, where a n are expansion coefficients and H includes h n in both the x and y polarisations. Substituting this into Eq. (2) gives L 0( ) ( a n h n + δl n n ) a n h n = β 2 a n h n. (4) n By ultiplying h on both sides of Eq. (4) and using the ortho-noralisation forula h h n da = δ n within one supercell, we find where and (Ln 0 + δl n )a n = β 2 a, (5) n δl n = L 0 n = δ n β 2 0 (6) h δl(h n ) da. (7) Equation (5) is a perturbation atrix equation, where interactions between odes of the ideal, scalar structure are expressed via the atrix eleents. Substituting Eq. (3) into Eq. (6), the vector and high-index perturbation ters respectively becoe [ ( t δln a = h (h n ) ) ] t lnn 2 r da (8) and δln b = h (Δn 2 k 2 0)h n da. (9) The integrals are over one supercell, but in practice the integral in Eq. (9) is only over the intersections of the glass strips. In Ref. [1], the fields h and h n have been analytically expressed using trigonoetric functions. Moreover, the separation of the scalar fields along the x and y axes reduces the area integrals in Eqs. (8) and (9) to one diensional integrals. This akes the calculation of the atrix eleents very straightforward and efficient. The details of the derivation are provided in Ref. [3]. The perturbation ters δl n in Eq. (5) can be classified into three types. The first are diagonal eleents δl. These ters can be included in L 0 and cause a shift of the propagation constant for each ode. However, this effect does not lead to any interaction with other odes and therefore does not contribute to the loss. The diagonal ters of Eq. (5) can be rewritten as L 0 + δl = β 2 0, (10) (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6959

5 where β 0 is the shifted propagation constant of the ode, including both vector and highindex ters. As we shall see, these diagonal ters are substantially larger than the off-diagonal atrix eleents. When considering cladding odes that have propagation constants close to the fundaental ode, it is these shifted eigenvalues that will be analysed. It is also these shifts that reove the bandgaps that are present in a scalar solution of the ideal odel structure. The second type of atrix eleent, δl n, contains δl 0 n and δl n0, where 0 denotes 0x or 0y, representing the x or y polarised fundaental ode, and n labels a cladding ode. Coupling of this type leads to light leaking fro the fundaental ode into the cladding, giving rise to confineent loss. The third type, δl n, represents interactions between different cladding odes. In principle these is a fourth type of atrix eleent δl 0x 0y and δl 0y 0x which represent coupling between the two polarisations of the fundaental ode. In practice, however, these ters are zero due to syetry Attenuation due to ode interactions If we solve Eq. (5) directly this gives only a real propagation constant and therefore can not address attenuation. We therefore will use a Green s function ethod fro which attenuation can be derived. As entioned in the introduction, this is siilar to the proble for an ipurity ebedded in an electron gas; in our case the fundaental ode is the ipurity and the continuu of cladding odes is the electron gas. We follow Anderson s ethod [2], extending this to include vector effects. It is iportant to note that an analytic treatent is ipossible if all the perturbation ters are considered. If, however, we neglect the interaction ters δl between different cladding odes, an analytic analysis can be perfored. This approxiation should be applicable because the attenuation is ainly caused by the interaction between the fundaental ode and cladding odes. The cladding-cladding ters are likely to be less iportant in affecting the overall agnitude of the attenuation. In the following derivation, we focus on the x polarised fundaental ode; the expressions for the y polarisation are equivalent. Having dropped the δl ters fro Eq. (5), the Green s function is defined by (L 0 δ v + δl δ v + δl v)g vn = δ n. (11) v The equivalent diagonal Green s function is defined by and can be written as (L 0 + δl )G 0 = 1, (12) ( ) G 0 = β 2 β 0 2 1, + iε (13) where ε approaches zero fro below. The two Green s operators in Eqs. (11) and (12) are related by the Dyson equation [4] G ab = G 0 aaδ ab + G 0 apδl pqg qb, (14) p,q where we note that for δl pq to be non-zero, either p or q ust label a fundaental ode. When the subscripts {ab} of G ab are chosen as { 0x 0x }, { 0y 0x } and { 0x }, Eq. (14) becoes = G 0 0x 0x + G 0 0x 0x δl 0x G 0x, (15) G 0y 0x = G 0 0y 0y δl 0y G 0x, (16) (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6960

6 and G 0x = G 0 δl 0x + G 0 δl 0y G 0y 0x, (17) where we note that labels cladding odes only. Substituting Eq. (17) into Eqs. (15) and (16), we obtain = G 0 0x 0x + G 0 0x 0x V 0x 0x + G 0 0x 0x G 0y 0x V 0x 0y, (18) and G 0y 0x = G0 0y 0y V 0y 0x 1 G 0 0y 0y V 0y 0y, (19) where V ab = δl ag 0 δl b ; by using Eq. (13) it can be written as V ab = δl aδl b. (20) β 2 β iε Substituting Eq. (19) into Eq. (18), we obtain an expression for : = G 0 0x 0x 1 G 0 0x 0x V 0x 0X G0 0x 0x G 0 0y 0y V 0x 0y V 0y 0x 1 G 0 0y 0y V 0y 0y. (21) Because of the syetry of the two polarised fundaental odes, we have G 0 0x 0x = G 0 0y 0y ; we also note that V 0x 0x = V 0y 0y and V 0x 0y = V 0y 0x. Equation (21) then becoes = ( V 2 0x 0y (G 0 0x 0x ) 1 V 0x 0x (G 0 0x 0x ) 1 V 0x 0x ) 1. (22) In Eq. (22), the ter containing V 2 0x 0y can be neglected because, as discussed below, the agnitude of the atrix eleents is sall. Equations (13), (20), and (22) are then cobined to give ( ) = β 2 β 0 2 δl 0x + iε 0x δl 1 0x β 2 β iε, (23) where β 0 0x is deterined by Eq. (10) and denotes the shifted propagation constant of the x polarised fundaental ode. By coparing Eqs. (13) and (23), it can been seen that the perturbation ter in Eq. (23) acts as an additional shift in the square of the propagation constant of the fundaental ode, arising fro the interaction with the cladding odes. It is iportant to note that this perturbation ter is coplex. The real part is relatively uniportant (causing a sall shift in the propagation constant), but the existence of an iaginary part leads directly to an attenuation of the fundaental guided ode. By using the identity [5] 1 li y 0 x + iy = P1 iπδ(x), (24) x where P is the principal value and δ(x) is a Dirac delta function, the iaginary part of the perturbation ters in Eq. (23) can be expressed as ( ) Δβ 2 0x [Iag] = π δl 0x δl 0x δ β 2 β 0 2. (25) (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6961

7 Equation (25) is the key equation of this paper; it gives an analytic expression fro which the attenuation of the fundaental ode due to interaction with cladding odes can be derived. It can be seen that the attenuation depends on two( factors. One ) is the density of the cladding states which is expressed through the ter δ β 2 β 0 2 ; the second is the perturbation atrix eleents δl 0x δl 0x, where 0x and are labels of the x polarised fundaental ode and cladding odes, respectively. The values of both of these parts can be calculated using Eqs. (8) to (10). To coplete our analysis of the attenuation we write β 2 0x [Cplx] = β 0 2 0x [Real] + iδβ 2 0x [Iag]. (26) To first-order, the iaginary part of the propagation constant is then given by β 0x [Iag] = Δβ 2 0x [Iag] /(2β 0 0x [Real] ). (27) In a practical calculation of the attenuation, the Delta function in Eq. (25) needs to be broadened. We use a Gaussian soothing, and Eq. (25) then becoes Δβ 2 0x [Iag] = π 1 exp 2πσ { [( β 2 β 0 2 ) ] 2 /σ 2 } δl 0x δl 0x, (28) where 1/σ 2π is a noralisation factor. The width corresponding to each ode is controlled by σ; a sall value of σ represents slight soothing and a sharp peak in the plots. The value of σ will be deterined through convergence tests in the following. 3. Results for the odel PCF structure 3.1. Mode distribution and interaction As discussed in Ref. [1], the odel rectangular hollow-core PCF has a square cladding lattice, and the central defect is created by oving outward the four glass strips that enclose the central square air hole. We use a standard structure in which the thickness of the glass strips and the shift to create the central defect are set at 0.05Λ and 0.125Λ, respectively. In the reainder of this paper, all quantities will be ade diensionless by using the pitch Λ of the cladding lattice as the unit of length. We therefore use (βλ) 2 for the diensionless propagation constant and δl Λ 2 for the diensionless perturbation atrix eleent. Our investigation focuses on the cladding odes close to the air-line. In the previous paper, scalar odes have been calculated for an ideal odel structure which has a higher refractive index at the intersections of the glass strips [1]. Solutions for the vector governing equation and for the realistic structure (without the high-index intersections) have a shifted value of (βλ) 2, which is deterined by Eq. (10). These shifted odes are the basis states for our perturbation theory. We calculate the diagonal shift for all the scalar odes of the ideal structure and then consider only the shifted odes which are close to the air-line in the calculation of attenuation. We start by considering an 8 8 supercell with a noralised frequency k 0 Λ = 40; this was analysed in detail in Ref. [1]. The calculation of the diagonal shift gives 64 basis states which are air-guided odes (including the fundaental ode) and 48 basis states which are glassguided odes, that are located in the vicinity of the fundaental ode (within a distance of (βλ) 2 less than 20). After the diagonal shift, the (βλ) 2 value for the fundaental guided ode is , which is greater than the unperturbed value. The nearest air-guided and glass-guided odes are separated fro the fundaental ode by (βλ) 2 differences of and 0.199, respectively. (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6962

8 Fro Eqs. (8) and (9), the vector and high-index perturbation atrix eleents can be calculated. For the vector ters, the average diagonal shifts of (βλ) 2 are and for the air-guided and glass-guided odes, respectively. By coparison, the off-diagonal eleents of the vector ter are considerably saller. Typical values are of order 10 1,10 0 and 10 3 respectively for air-air, glass-glass and air-glass interactions for the sae polarised odes. For the interaction between different polarisations, they are of order 10 5,10 0 and 10 2 for the three groups. The high-index ters exhibit a rather different pattern. The average diagonal shifts are and for the air-guided and glass-guided odes, which are significantly saller than for the vector ter. For the off-diagonal eleents of high-index ter, typical values are 10 5,10 0 and 10 2 for air-air, glass-glass and air-glass ode interactions with the sae polarisation; the interactions between differently polarised odes are identically zero. In general, we conclude that, apart fro the diagonal ter, the agnitude of the perturbation is relatively sall, which gives us confidence in the validity of our perturbation theory. This also justifies our neglect of the second order ters in Eq. (22). In general the vector ters tend to be larger than the high-index ters, but neither can be neglected in the perturbation calculation Attenuation calculations In order to use Eqs. (27) and (28) to calculate the attenuation, both the size of supercell and an appropriate σ value should first be deterined. For high precision, we want σ sufficiently sall so that only those states close to the fundaental ode are included. However, we also need an adequate nuber of these cladding states for coputational accuracy. In this case, the size of supercell should be as large as possible. However, a proble arises if the supercell becoes too large. As discussed in the previous paper, odes are found by searching for identical field values at the centres of neighbouring supercells [1]. When the size of supercell is enlarged, the transfer atrices ust pass through ore air and glass layers. For soe odes, the fields are confined only within a subset of layers and decay exponentially in others. To find these odes in a large supercell requires very high floating-point accuracy. For supercells exceeding in size, even quadruple precision is insufficient. In the deterination of σ values, we have therefore chosen a set of supercells no larger than Since our investigation concerns the properties of the fundaental ode, the (βλ) 2 value on the right-hand side of Eq. (28) should be chosen to be that of the fundaental ode. However, to investigate convergence of the calculations it is convenient to plot the iaginary part of Δ(βΛ) 2 for the fundaental ode as a function of (βλ) 2. In general, we find that the airguided odes have uch higher density of states than the glass-guided odes; we therefore consider the air-guided and glass-guided odes separately. A larger σ is required for the glassguided odes to give a sooth density of states. Plots of the iaginary part of Δ(βΛ) 2 for the fundaental ode are shown in Fig. 1 as a function of the soothing width σ and the size of the supercell. For the air-guided odes, we find that the sallest value of σ that provides well converged results is 0.3. In this case the difference between and supercells is less than 1% over the whole range shown in Fig. 1. The glass-guided odes show a broader distribution over a wide range of (βλ) 2, as shown in the right of Fig. 1. It is found that the sallest acceptable value of σ is 10 for the largest supercell we have used. In this case the difference between and supercells is less than 2% over the range shown in Fig. 1. Figure 1 shows an iportant result that the iaginary part of Δ(βΛ) 2 at the propagation constant of the fundaental guided ode is non-zero only for the glass-guided odes. The density of states of air-guided odes is zero at this propagation constant and so these odes do not contribute to the attenuation. We conclude that interaction with the glass-guided odes deterines attenuation in the high-transission region. This should also be valid for other ebers of the (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6963

9 Fig. 1. Variation of the iaginary part of Δ(β Λ)2 for the fundaental ode with soothing width σ for different sizes of supercells calculated using Eq. (28). The left and right panels show contributions fro air-guided and glass-guided cladding odes, respectively. The black arrows at (β Λ)2 = indicate the location of the shifted fundaental guided ode. class of PCFs that guide light due to weak coupling of odes, and thus provides quantitative support for the conclusions drawn for Kagoe [6 8] and square-lattice [9] hollow-core PCFs Frequency dependence of the attenuation Having developed a ethod to calculate the iaginary part of β, we now use it to analyse the dependence of the attenuation on frequency and fibre structure. At the saple frequency of k0 Λ = 40, the iaginary part of β Λ for the fundaental ode is calculated to be , indicating a low level of leakage. Figure 2(a) shows the attenuation over a range of noralised frequencies fro k0 Λ = 26 to 46 at a spacing of 0.2. Siilar to that observed in Kagoe and square-lattice hollow-core PCFs, the attenuation for rectangular hollow-core PCFs varies draatically as a function of frequency. In the selected range, it generally shows a decreasing trend with increasing frequency. This feature can also be seen via three peaks at k0 Λ = 28.0, 33.2 and 37.4, for which the iaginary β Λ declines fro to To test the validity of our perturbation calculation we have used the boundary eleent ethod [10 12] to calculate the attenuation. We use a odel rectangular hollow-core PCF, a scheatic of the fibre structure is shown in the inset of Fig. 2(a). All the structural paraeters are the sae as those used in the perturbation calculations, except for the details at the edge of the # $15.00 USD (C) 2011 OSA Received 27 Jan 2011; accepted 9 Mar 2011; published 25 Mar March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6964

10 Fig. 2. Iaginary part of the propagation constant of the fundaental ode over a range of frequencies in the first transission window. (a) Coparison between perturbation and boundary eleent ethods. The inset shows the scheatic odel PCF structure calculated by the boundary eleent ethod. (b) The effect of the perturbation ters, where the red line indicates the full perturbation result; the green and blue lines are results that include only the vector and high-index perturbation ters, respectively. cladding, where the boundary eleent ethod requires an enclosed glass jacket rather than a supercell geoetry. Details of the boundary eleent calculations are given in Ref. [3]. The coparison in Fig. 2(a) shows that results for the attenuation are in good agreeent in their order of agnitude and they exhibit a siilar variation with frequency. Although the details do not atch well, this is to be expected because the calculated structures are not identical. Leakages for Kagoe and square-lattice hollow-core PCFs were easured experientally in the sae high-transission window, although the resonance frequencies differ because of the fibre structures. In these experients, the confineent losses varied fro about 0.5 db/ to 1.5 db/ in Kagoe PCFs (with pitch Λ=10.9 μ) [6], and fro 1 db/ to 4 db/ in squarelattice hollow-core PCFs (with pitch Λ = 17μ) [9] over a continuous range of frequencies. In our odel structure, the typical level of the iaginary part of βλ is between 10 6 and The corresponding confineent loss varies fro 0.58 db/ to 5.8 db/ (if we choose the pitch Λ = 15μ), in good agreeent with the experientally easured values. Figure 2(b) shows the separate attenuations calculated using only the vector or high-index perturbation ters. This coparison shows that the vector ters are the doinant effect in the (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6965

11 attenuation, including the high-loss peaks. The influence of the high-index ters reains at a relatively low level, and only for the lowest losses is their effect coparable to that of the vector ters. This feature indicates that the coupling of the odes arising fro the air-glass interface in uch ore significant than that fro the intersections of the glass strips. For frequencies close to a resonance of the glass strips, the confineent loss of PCFs significantly increases [7, 8]. Assuing that the fundaental guided ode is located on the air-line, i.e. β = k 0, the resonance condition is given by k 0 Λ n 2 g n 2 a t/λ = jπ [7], where n g and n a are the refractive indices for the glass and air, respectively, t is the thickness of the glass strips, and j is an integer. In the vicinity of a resonance the attenuation is found to increase; in Kagoe hollow-core PCFs, a high loss region of width 200 n separates the high transission windows [7]. In our odel structure, the value of t/λ is 0.05; the lowest noralised resonance frequency is therefore k 0 Λ = The attenuations shown in Fig. 2 are located within the first high-transission window (i.e. between j = 0 and 1). In the following calculations, we focus on the guidance only within this window. Near to the resonance frequency, our calculated attenuation values becoe negative and therefore unphysical. In this case, the choice of considering only the glass-guided odes is not sufficient to calculate the attenuation. A nuber of odes are located very close to the fundaental guided ode. Their fields occupy both the air and the glass regions, showing that they are precursors to the set of higher-order glass-guided odes that are about to be trapped. These characteristics lead to a strong coupling with the fundaental guided ode, giving rise to contributions to attenuation that are not included in our odel Effect of the fibre structure The confineent loss can also be calculated as a function of the thickness of the glass strips. In this investigation, the centres of each air and glass region are unchanged, and the thicknesses of all the glass strips are kept the sae. Fig. 3. Iaginary part of the propagation constant as a function of the glass strip thickness in the first transission window when k 0 Λ = 40. The full, vector and high-index perturbation results are shown in red, green and blue, respectively. Figure 3 shows the attenuation of the fundaental guided ode versus the thickness of the glass strips at k 0 Λ = 40. Siilar peaks to those observed in Fig. 2 appear at t/λ = 0.030, and 0.044; a coparison between the two plots shows that reducing the thickness of the glass struts leads to a shift of the attenuation feature towards a lower wavelength. For exaple, the highest frequency peak for t/λ = 0.05 is at k 0 Λ = 37.4; for the structure with t/λ = 0.044, it oves to k 0 Λ = However, the overall appearance of the attenuation features reains (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6966

12 unchanged. We now consider the effect of the size of the central defect; the full width of the central air hole is labelled D. The frequency k 0 Λ and the thickness of glass strips t/λ are fixed at 40 and 0.05, respectively. The central defect is ade by siultaneously oving the four glass strips nearest to the centre by a distance of between 0.03Λ and 0.185Λ. Fig. 4. (a) Dependence of attenuation on the size of the central defect. Full, vector and highindex perturbation results are shown in red, green and blue, respectively. (b) Attenuations for the two sizes of central defect as a function of frequency. The red and green lines respectively correspond to D = 1.20Λ and D = 1.11Λ. Figure 4(a) presents attenuations for different sizes of central defect. The values of the iaginary βλ vary ore soothly than those for the strut thickness, fro to The plots show a low-attenuation region, fro D = 1.1Λ to 1.15Λ, surrounded by two relatively high-loss areas. The iniu value of the iaginary βλ is about (i.e. a loss of 0.16 db/ for Λ = 15μ) at 1.11Λ. In this case, both the vector and the high-index perturbations are siultaneously suppressed. For a ore detailed view of this low-loss region, the attenuation for a central defect size of 1.11Λ has been plotted as a function of frequency. The results in Fig. 4(b) show a coparison of the attenuation with the forerly used central defect of D = 1.20Λ. They exhibit a very siilar variation with frequency, but for the saller core size, the attenuation characteristic is shifted towards a lower wavelength. This is siilar to what happens for the thinner glass strips. A coparison of the agnitude in Fig. 4(b) shows that, for a larger central defect, the average value of the leakage is relatively saller. (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6967

13 4. Conclusion The key technical developent of this study is the finding that a perturbation ethod is effective in the investigation of a class of PCFs which govern light due to the weak coupling of the fundaental ode and cladding odes. We have derived an expression for the attenuation which allows for rapid calculations and which is in good agreeent with large-scale nuerical calculations. Our quantitative analysis for rectangular hollow-core PCFs has shown that the fibre leakage arises fro the interaction between the fundaental ode and the cladding odes, and physically relates to the density of states of the cladding odes weighted by the agnitude of their coupling with the fundaental ode. In high-transission windows, the contribution of the air-guided odes to attenuation is found to be very sall. Although the glass-guided odes ay closely atch the fundaental ode in ters of their propagation constants, the large difference in the spatial frequencies of their fields greatly suppresses the leakage of light. In our odel structure, a saller central defect or thinner glass struts play a siilar role in pushing the attenuation to a shorter wavelength. Although these variations, for exaple the use of a larger central defect, can reduce the attenuation at a specific frequency, they are not effective for other frequencies and would not greatly affect the overall agnitude of the attenuation in a whole transission window. Our investigation shows that the ost of the confineent loss coes fro the coupling between the fundaental ode and the cladding odes situated in the glass layers closest to the central air defect. To effectively reduce the attenuation of the fundaental ode in Kagoelike PCFs, an optiised design of the core-surround is iportant. A suppressed ode coupling ay be achieved by odulating the field profile of the fundaental ode across the glass strips nearest to the central defect. This understanding also provides theoretical support for the recent experient in a Kagoe PCF with a hypocycloid shaped air core [13], for which the attenuation is significantly lower than that in a standard Kagoe PCF with a circular shaped central air defect. Acknowledgents We thank John Roberts and Ti Birks for helpful discussions in the course of this work and John Roberts for providing the boundary eleent code used in the calculations for Fig. 2(a). (C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6968