Group-velocity dispersion in a Bragg fiber

Transcription

1 414 J. Opt. Soc. Am. B/ Vol. 5, No. 3/ March 008 J.-I. Sakai and K. Kuramitsu Group-velocity dispersion in a Bragg fiber Jun-Ichi Sakai and Kazuki Kuramitsu Faculty of Science and Engineering, Ritsumeikan University, Noji-higashi, Kusatsu City, Shiga, Japan Received November 9, 007; accepted December 19, 007; posted January 3, 008 (Doc. ID 89457); published February 9, 008 A comprehensive study of the group-velocity dispersion of the TE, TM, and hybrid modes in a Bragg fiber is presented by showing many numerical examples. The dispersion can be represented in a simple form as the sum of the material and the waveguide dispersions under the quarter-wave stack condition. The performance of the dispersion can be well understood by relating it to the optical power confinement factor. The dispersion of the Bragg fiber is inversely proportional to the second power of its core radius. The simple dispersion formula can be conveniently used to evaluate the dispersion near the central region of the photonic band. Properties relating to the dispersion are equivalent to those of the conventional optical fibers in which all the power is confined to their core. The properties are formally the same as those in the circular metallic waveguide. 008 Optical Society of America OCIS codes: , , INTRODUCTION Much attention is paid to photonic crystal fibers because they may have properties superior to those of conventional optical fibers. A merit of photonic crystal fibers is the controllability of the group-velocity dispersion. The dispersion compensation has been presented in the microstructured optical fiber [1,] and the Bragg fiber [3 5]. As one of the photonic bandgap fibers, researchers have studied the Bragg fiber [6], or the omniguide fiber [7], that has a cylindrical symmetry and cladding with alternating high- and low-refractive indices. The groupvelocity dispersion of the Bragg fiber has been calculated for several modes using the transfer matrix [5,8], the beam propagation method (BPM) [3], and the asymptotic matrix methods [4,9]. Many works are accomplished on the group-velocity dispersion of the TE 01 mode [5,8] orthe dispersion and dispersion slope of the TE 01 mode in the c-band [9] as well as the dispersion compensation above. Large dispersion relates to soliton and femtosecond pulses [3]. Only a part of the various modes have been previously studied. Photonic crystal fibers usually resort to numerical calculations to study various characteristics owing to its structural complexity, as mentioned above. However, a physical understanding of the solution is also important. To understand the properties of a microstructured optical fiber, or a holey fiber, the concept of the effective V value has been developed in [10] and subsequent papers. When the Bragg fiber is operated under the quarter-wave stack (QWS) condition, it has an eigenvalue equation formally the same as the circular metallic waveguides in an asymptotic expansion method [11] and in geometrical optics [1]. The QWS condition is useful for a deep understanding of the Bragg fiber. The purpose of the present paper is to comprehensively investigate the group-velocity dispersion of the TE, TM, and hybrid modes and to elucidate properties of the dispersion under the QWS condition in the Bragg fiber. The propagation constant is calculated using the asymptotic expansion method [11,13], and the dispersion is discussed within its framework. This paper also makes a comparison with conventional optical fibers and the circular metallic waveguide.. BRIEF DESCRIPTION OF FUNDAMENTAL PROPERTIES IN THE BRAGG FIBER The Bragg fiber is assumed to have a cylindrically symmetric microstructure and a hollow core surrounded by an infinitely extending periodic cladding (see Fig. 1). The core radius is r c and its refractive index is n c. The cladding consists of alternating dielectric layers having high n a and low refractive indices n b n a n b n c. Their layer thicknesses are a and b, respectively, and the cladding period is set to be =a+b. A cylindrical coordinate system r,,z is used here, with z being the propagation direction of light. In the present paper, we use exact field expressions for the core and the asymptotic expansion for the cladding [11]. Note that the present method secures a high accuracy for r c / 0 1 because it uses the asymptotic expansion for the cladding field. Electromagnetic field components are assumed to have a spatiotemporal factor of expit z, with and being the angular frequency and propagation constant, respectively. Lateral propagation constants are defined by i k 0 n i /k 0 1/ i = a,b,c, 1 where k 0 =/ 0 denotes the wavenumber of vacuum and 0 denotes the wavelength of vacuum. The subscripts a, b, and c refer to the cladding layer a, cladding layer b, and the core, respectively. Let us consider the QWS condition where an optical wave is efficiently confined to the core owing to the Bragg diffraction in the periodic cladding. The condition can be written as /08/ /$ Optical Society of America

2 J.-I. Sakai and K. Kuramitsu Vol. 5, No. 3/March 008/J. Opt. Soc. Am. B 415 Equations (5a) and (5b) are derived by removing the propagation constant from an expression in Eq. (). Instead of Eq. (5b) one may use an alternative expression in which the cladding layer thickness a and the subscript a are replaced by the b in Eq. (5a). Equation (3) is formally identical with the propagation constant of the circular metallic waveguide that has only the TE and TM modes. Accordingly, we can obtain expressions of the guiding limit angular frequency, phase velocity v p, and group velocity v g similar to those of the circular metallic waveguide. From Eq. (3) of the Bragg fiber we can derive d n eff = n c N c = c, v p v g 6 where N i = n i + k 0 dn i = n i 0 dn i d 0 i = a,b,c 7 Fig. 1. Schematic of a Bragg fiber. r c, core radius; n c, refractive indexofcore;n a and n b, indices of layers with thickness a and b, respectively; and =a+b, period in the cladding. a a = b b = /. Then the eigenvalue equations, transcendental equations, for the TE, TM, and hybrid modes are markedly simplified [11] and give an expression of the propagation constant as with = k 0n c = j0,+1 j 0, U QWS =j1, j, 1 j, 1 U QWS 1/, 3 :TE 0 mode :TM 0 mode. 4 :HE mode :EH mode Here, j, and j, indicate the th zeros of the Bessel function J of order and its derivative J with respect to its argument, respectively. The corresponds to the azimuthal mode number. The EH 1 mode is always degenerate with the TE 0 mode under the QWS condition. It should be noted that Eq. (3) is apparently separated from the cladding parameters. Under the QWS condition fiber structural parameters are not arbitrarily selected but must simultaneously satisfy and 0 a = 4n a n c + U QWS 0 r c 1 1 a b 0 =16n a n b. 1/ 5a 5b is the group index, n eff /k 0 is the effective index, and c is the vacuum light velocity. By the way, for a dispersive step-index fiber with the core index n 1 and the cladding index n, we have a general relation [14] d n eff = c = n 1 N n N v p v g Here, N j is the group index (j=1, ), and 1 denotes the optical power confinement factor to the core. A comparison between Eqs. (6) and (8) shows that the QWS condition in the Bragg fiber formally corresponds to a case where all the optical power is confined to the core in the dispersive step-index fiber. This suggests that the QWS condition offers a condition in which the optical wave is well confined to the core. 3. GROUP-VELOCITY DISPERSION UNDER THE QUARTER-WAVE STACK CONDITION After the group-velocity dispersion is derived under the QWS condition, it will be discussed in conjunction with the optical confinement factor and the circular metallic waveguide in this section. A. Derivation of Group-Velocity Dispersion and Its Properties The group-velocity dispersion is defined by D 1 k d 0 c 0 dk. 0 When the Bragg fiber is operated under the QWS condition and its index dispersion is taken into account, the group-velocity dispersion is derived to be D QWS = 1 c 0 n c d n c N 0 n eff d c 3 0 U QWS 10 n eff by substituting Eq. (3) into Eq. (9). We can also get the same expression in Eq. (10) from Eq. (), as shown in Appendix A. The D QWS will be called a QWS dispersion for simplicity hereafter. 9

3 416 J. Opt. Soc. Am. B/ Vol. 5, No. 3/ March 008 J.-I. Sakai and K. Kuramitsu Results deduced from Eq. (10) are summarized as follows: (1) The first and second terms indicate the material D QWS,m and waveguide dispersions D QWS,w in the Bragg fiber with infinitely extending cladding. () It appears that the QWS dispersion does not depend on the cladding material but only depends on the core material. The cladding has an influence on the dispersion via the effective index n eff. (3) The material dispersion of the Bragg fiber is in formal agreement with that of the weakly guiding step-index fiber [15] except for the n eff. (4) If the index dispersion of the core is neglected, then the material dispersion disappears. (5) If the effective index n eff is regarded as constant at a fixed 0, then the waveguide dispersion D QWS,w is inversely proportional to r c. The n eff is expressed by n c U QWS / 1/ using Eq. (3). Hence, if r c / 0 U QWS /n c holds, we can make use of D QWS,w 1/r c with good accuracy. Here, the value of U QWS / corresponds to the guiding limit =0 of each mode [11]. Namely, if r c / 0 is about three times larger than the guiding limit, the proportionality D QWS,w 1/r c gives a good approximation. This proportionality has been shown for the TE 01 mode [8]. (6) The waveguide dispersion D QWS,w can be represented as a function of the normalized core radius r c / 0 except for a factor of c 0. If the normalized core radius r c / 0 is fixed, then the D QWS,w is inversely proportional to the operation wavelength. A small group-velocity dispersion is attainable by a long wavelength operation. (7) At the guiding limit 0, the U QWS / becomes n c, and hence the group-velocity dispersion D QWS goes toward infinity. This corresponds to the fact that the group velocity v g =0 at =0. When the core consists of air and its index dispersion is considered [16], the material dispersion D QWS,m of air is plotted in Fig. for several n eff. Above n eff =0.8 that is important for practical use, absolute values of the material dispersion are 0. ps/km nm in most of m. Fig.. n eff. Material dispersion of air in the Bragg fiber for several B. Relationship between the Waveguide Dispersion and the Optical Power Confinement Factor If the index dispersion of the core is considered negligible, then the N c reduces to the n c in the second term of Eq. (10). In that case, U QWS / and n c included in the waveguide dispersion D QWS,w are related with the optical power confinement factor QWS to the core [17]. For TE mode, we get U QWS n a n b r c 1 TE QWS = =n c n eff. 11 TE QWS When we consider a case of TE QWS 1.0 in Eq. (11), the lefthand side is regarded as proportional to the optical power flowing through the cladding. Equation (11) shows that the waveguide dispersion D QWS,w of the TE mode decreases as the QWS approaches unity and that D QWS,w reaches zero at QWS =1. It is possible to say that a large n eff leads to a large TE QWS. In the case of TM and EH modes, the n c can be written as n n a a n b b r c 1 TM QWS c = n a a 3 + n, 1 b b 3 TM QWS for TM mode, and n c = n a a n b b r c 1 EH QWS n a a 3 1+a/r c + n b b 3 1+b/r c EH QWS for EH mode. 13 As long as is not so large, a/r c =/U QWS n c /n a 1 and b/r c =/U QWS n c /n b 1 hold. Accordingly, the waveguide dispersion of the EH mode can be approximated by that of the TM mode. One obtains U QWS n a n b r c 1 /j, 1 HE QWS =, 14 HE C HE QWS for HE mode, where C HE a1+ a r c A HE + b1+ b r c B HE, 15a b a A HE 1+ n a a n b b j, n ar c bj, 1 4 n anaj c, n. 15b eff n c r c Here, the B HE in Eq. (15a) is obtained by an expression where only n a r c /bj, is replaced by n b r c /aj, in the A HE. Although Eq. (14) apparently includes the azimuthal mode number unlike the other modes, it has a QWS dependence of the waveguide dispersion similar to that of the TE mode.

4 J.-I. Sakai and K. Kuramitsu Vol. 5, No. 3/March 008/J. Opt. Soc. Am. B 417 We can deduce some results about the waveguide dispersion D QWS,w by substituting Eqs. (11) (14) into Eq. (10): (1) The waveguide dispersion D QWS,w is formally proportional to 1 QWS / QWS for the TE, TM, and hybrid modes under the QWS condition. This implies that when the optical confinement factor QWS approaches unity, the waveguide dispersion is reduced to zero. This reduction is because fields are entirely confined to the core, eventually vanishing the effect of the waveguide structure. () The waveguide dispersion is proportional to n a n b or n a a n b b depending on the mode in the Bragg fiber. These factors correspond to the relative index difference dependence of the waveguide dispersion in the conventional optical fibers. The waveguide dispersion of the step-index fiber is proportional to the relative index difference at a fixed normalized frequency [15]. This means that the waveguide dispersion of the Bragg fiber is by n a n b / larger than that of the step-index fiber. The n a n b is usually of the order of unity and values of are 0.% to 1.0%. It is expected that the waveguide dispersion of the Bragg fiber is at least 100 times larger than that of the step-index fiber. This is the reason why the dispersion of the Bragg fiber is markedly large, as reported herein [4,5]. (3) The waveguide dispersion is inversely proportional to the third power of the effective index n eff. A large value of n eff is effective for decreasing the dispersion because a large n eff results in a high QWS. C. Waveguide Dispersion at the Photonic Band Edge In the Bragg fiber the photonic band edge (PBE) corresponds to a situation where the decay rate of the envelope of electromagnetic fields is zero in the cladding [11]. The PBE takes place at n a =n b for the TE mode and at n a a =n b b for the TM and hybrid modes. Then the optical power confinement factor QWS reaches zero [17]. Since the expressions in Eqs. (11) (14) take on indeterminate forms at the PBE, further consideration is needed. In the case of the TE mode, the equality n a =n b leads to simply another equality a=b and has no influence on core parameters, as can be seen from Eqs. (5a) and (5b). Therefore, the propagation constant is determined by Eq. (3) regardless of the n a value. Namely, the waveguide dispersion is calculated by Eq. (10) even at the PBE and depends on the core parameters and the operation wavelength. At the PBE of the TM and hybrid modes, the propagation constant is represented by [11] 1 n + 1 1/ a n b. 16 k 0 = The substitution of Eq. (16) into Eq. (10) leads to an expression of the waveguide dispersion at the PBE as 3/ N c D QWS,w = 1 + n a 1n b c 0 1 n a + 1 n b n c In this case fiber parameters are given by a/ 0 =n a +n b 1/ /4n a, b/ 0 =n a +n b 1/ /4n b, and r c / 0 =U QWS / n c 1/n a +1/n b 1 1/ for a combination of n a, n b, and n c. An expression in Eq. (17) includes only the refractive indices except for a factor of c 0, showing evidence of the dispersion caused by the waveguide structure. The waveguide dispersion at the PBE is independent of the mode, and the discrepancy among the modes appears only in the core radius. The waveguide dispersion does not vanish at the PBE because fine changes in electromagnetic fields remain in the cladding unlike conventional optical fibers. Let us show a numerical example for the PBE. Since the PBE never arises from a condition n b nc,weset n a =1.4, n b =1., and n c =1.0. Then one obtains cut /k 0 = and D QWS,w =749 ps/km nm. The cladding thicknesses are a/ 0 =0.35 and b/ 0 =0.301, and the core radii are r c / 0 = for the HE 11 mode and r c / 0 =0.986 for the TM 01 mode. D. Numerical Examples of Waveguide Dispersion Figure 3 shows the waveguide dispersion D QWS,w of the TE, TM, and hybrid modes as a function of r c / 0. Assume that the index dispersion of the core is negligible and n c =1.0. All the modes are plotted that exist in the range of r c / For a certain r c / 0,Eq.(3) gives the effective index n eff that is included in Eq. (10). The ordinate is plotted for 0 =1.0 m. We can see from the figure that the D QWS,w is in rough proportion to 1/r c and that the D QWS,w goes toward zero with the increase of the normalized core radius. This is in accordance with the fact that the QWS approaches unity with the increase in the core radius. One might think that the HE 11 mode exhibits the lowest dispersion among all the modes for a fixed core radius. It should be noted that the normalized cladding thicknesses change according to Eqs. (5a) and (5b). Therefore, the cladding parameters must also be fixed to compare them exactly with values of the waveguide dispersion. Figure 3 is available for estimating the dispersion value at a wavelength other than 1.0 m. For a certain 0 in m unit, the dispersion is evaluated from Fig. 3 on the condition that only the ordinate is multiplied by 1/ 0. For example, for the TE 01 mode with 0 =10m and r c =0m we have D QWS,w =35.9 ps/km nm. Fig. 3. Waveguide dispersion of the TE, TM, and hybrid modes for the Bragg fiber under the QWS condition as a function of r c / 0. n c =1.0. The ordinate is for the dispersion measured at 0 =1.0m. Two digits designate the azimuthal and radial mode numbers.

5 418 J. Opt. Soc. Am. B/ Vol. 5, No. 3/ March 008 J.-I. Sakai and K. Kuramitsu Figure 4 illustrates the wavelength dependence on the waveguide dispersion of the TE, TM, and hybrid modes at r c =.0 m. All the modes are shown that exist in the range of r c / For fixed r c, 0, and mode, one can evaluate the D QWS,w after the n eff is determined from Eq. (3). The values of the D QWS,w increase as the wavelength increases. The HE 11 mode exhibits the lowest dispersion among all the modes because it has a large n eff compared with other modes for a fixed core radius [11]. E. Correspondence to the Circular Metallic Waveguide For the Bragg fiber under the QWS condition, one obtains an eigenvalue equation and propagation properties that show a formal similarity to those of the circular metallic waveguide [11]. The propagation constant of the circular metallic waveguide is represented by with metal = k 01 U metal 0,+1 U metal =j j, 1 j, 0 :TE 0 mode 1/, 18 :TE mode. 19 :TM mode Although the dispersion does not arise from an ideal circular metallic waveguide, we can obtain a formal expression of its dispersion as D metal = c 0 U metal 0 n eff by applying Eq. (18) to Eq. (9). If we set n c =1.0 in Eq. (10) and the U QWS is corresponded to the U metal, then the QWS dispersion reduces to the dispersion of the circular metallic waveguide described in Eq. (0), that is, D QWS,w D metal holds for a corresponding mode. It has been pointed out by a numerical example that the group-velocity dispersion of the TE 01 mode resembles that of the circular metallic waveguide [8]. The resemblance in the dispersion between the Bragg Fig. 4. Waveguide dispersion of the TE, TM, and hybrid modes for the Bragg fiber under the QWS condition as a function of wavelength. 0 =1.55 m, r c =.0m, and n c =1.0. fiber and the circular metallic waveguide has a strong dependence on the optical power confinement factor, as stated in Subsection 3.B. 4. GROUP-VELOCITY DISPERSION UNDER THE QUASI QUARTER-WAVE STACK CONDITION Equation (10) shows the QWS dispersion in which fiber parameters always satisfy Eqs. (5a) and (5b) simultaneously. In actual circumstances, however, fiber structural parameters are fixed. This must also be kept in calculating the group-velocity dispersion. The propagation constant is evaluated by numerically solving the exact eigenvalue equations [11]. In calculating the dispersion by using Eq. (9), differentiation with respect to the wavenumber k 0 is performed using a numerical derivative formula within the least-squares approximation [18]. We used a five-points approximation. We impose the QWS condition on the central point of the calculation points. Then we keep this fiber structure for the calculation of the remaining four points. This is called a quasi-qws condition in this paper. Numerical results for the various modes will be shown below unlike previous works [4,5]. A. Core Radius Dependence of Dispersion Figures 5(a) and 5(b) show the core radius dependence of the group-velocity dispersion of the TE and TM modes, respectively, under the quasi-qws condition. Fiber parameters are prescribed at 0 =1.55 m and n b =1.5 such that the QWS condition is satisfied. For comparison, we also depict the waveguide dispersion D QWS,w under the QWS condition. The first five modes are shown here. We notice from Fig. 5(a) that the TE mode under the quasi-qws condition has a rough dependence, D1/r c, similar to that in the QWS condition. This dependence has been shown for the TE 01 mode [8]. In the TM mode we cannot always admit a good agreement between the two conditions. The agreement is high for higher-order modes in the TM mode. This is because higher-order modes exhibit a high optical confinement factor in the TM mode [19]. Increasing the cladding index difference n a n b brings a close agreement between the two conditions. These tendencies can be explained as follows: optical confinement increases as the decay length inside the Bragg stack decreases, namely when n a n b increases and is closer to the center of the photonic bandgap. The TE 0 mode should have better agreement because their fields are close to zero near the core radius, as is the case for the circular metallic waveguide. The group-velocity dispersion versus the core radius is shown in Figs. 6(a) and 6(b) for the HE and EH modes, respectively, with 0 =1.55 m and n a =.5. All the modes are plotted that exist in the range of r c / Characteristics of the HE mode under the quasi-qws condition are divided into two groups according to the radial mode number. The dispersion of the HE mode asymptotically goes toward zero with the increase of the core radius r c, as their QWS dispersion D QWS,w and the TE mode do as well. On the contrary, the dispersion of the HE 1 mode does not approach zero with the increase of r c, as the TM mode does not as well. These tendencies can

6 J.-I. Sakai and K. Kuramitsu Vol. 5, No. 3/March 008/J. Opt. Soc. Am. B 419 Fig. 5. Core radius dependence of group-velocity dispersion of the TE and TM modes under the quasi-qws condition as a function of several n a. (a) TE mode and (b) TM mode. 0 =1.55 m and n b =1.5. also be understood by the optical confinement factor [19]. The group-velocity dispersion of the EH modes does not approach zero with the increase of r c, as the TM mode does not as well. For a small core radius of any of the modes, we find an excellent agreement between the quasi- QWS and QWS conditions. A fact that is also confirmed in the HE and EH modes is that an increase in the n a n b improves an agreement in characteristics between the quasi-qws and QWS conditions, although they are not shown here. B. Wavelength Dependence of Dispersion Since the Bragg fiber has an air core, it can be used in a wavelength other than the low loss region in conventional silica fibers. In this subsection, the core radius and the wavelength are set to be r c =.0 m and QWS =1.0 m where the QWS condition is satisfied. All the modes are plotted below that appear in the above condition. If r c / 0 is kept constant, then the dispersion is roughly in inverse proportion to the operation wavelength, as shown in Subsection 3.A. Hence, the dispersion at 1.0 m can provide much information. Wavelength dependence of the group-velocity dispersion is illustrated in Figs. 7(a) and 7(b) for the TE and TM modes, respectively, as a function of three n a values. The values of the dispersion only exist in the photonic band for each mode, and they rapidly change in the vicinity of the edges of the photonic band. A nearly flat dispersion region against the wavelength can be seen near the central part of the photonic band. The TE 01 mode has the widest flat region among all the modes shown. This is caused by the fact that the TE 01 mode exhibits the highest optical confinement factor among all the modes [19]. The photonic band is wide for the lower-order modes of the TE mode or for the higher-order modes of the TM mode. The larger the cladding index difference n a n b is, the wider the photonic band and the flat region are. The waveguide dispersion D QWS,w of the TE mode is also plotted in Fig. 7(a) for reference. A resemblance between the QWS and quasi-qws conditions is high for the lower-order mode in the TE mode except near the edges of the photonic band. For example, for n a =.5 and 0 =1.0 m we have D=517 ps/km nm and D QWS,w =359 ps/km nm for the TE 01 mode, and D Fig. 6. Core radius dependence of the dispersion of the HE and EH modes under the quasi-qws condition. (a) HE mode and (b) EH mode. 0 =1.55 m, n a =.5, and n b =1.5. Two digits designate the azimuthal and radial mode numbers.

7 40 J. Opt. Soc. Am. B/ Vol. 5, No. 3/ March 008 J.-I. Sakai and K. Kuramitsu Fig. 7. Wavelength dependence of the group-velocity dispersion of the TE and TM modes as a function of several n a. (a) TE mode and (b) TM mode. QWS =1.0m, r c =.0m, and n b =1.5. Fig. 8. Wavelength dependence of the dispersion of the HE and EH modes. (a) HE mode and (b) EH mode. QWS =1.0m, r c =.0m, n a =.5, and n b =1.5. Fig. 9. Comparison of the dispersion between the quasi-qws and QWS condition for the TE, TM, and hybrid modes as a function of cladding high-index n a. (a) TE and TM modes and (b) HE and EH modes. 0 =1.55 m, n t =0.8, n b =1.5, and n c =1.0. =497 ps/km nm and D QWS,w =181 ps/km nm for the TE 0 mode. The Bragg fiber has an excellent resemblance in the dispersion of the TE 01 mode to the circular metallic waveguide near the center of the photonic band, and it is due to a good optical confinement. Figures 8(a) and 8(b) show the wavelength dependence of the group-velocity dispersion of the HE and EH modes, respectively. Only the case of n a =.5 is shown here to avoid the complexity because many modes appear in the hybrid modes. Tendencies about the photonic bandwidth and the flatness in the dispersion can also be found in the hybrid modes. The HE mode is inclined to have

8 J.-I. Sakai and K. Kuramitsu Vol. 5, No. 3/March 008/J. Opt. Soc. Am. B 41 flat characteristics rather than the HE 1 mode near the central region of the photonic band. This performance can be understood by the optical power confinement factor in the same way as in Fig. 6(a). One can see from Figs. 7 and 8 that the TE 01 mode has the widest flat characteristics and shows relatively small dispersion values among the TE, TM, and hybrid modes. C. Cladding High-Index Dependence of Dispersion In the following we fix parameters at 0 =1.55 m, n b =1.5, and n c =1.0. The core radius r c is determined in the quasi-qws condition such that it satisfies Eq. (3) for a tentative index n t =/k 0 =0.8, where n t is used to determine the cladding structure. Figure 9 shows the cladding high-index dependence of the dispersion for the TE, TM, and hybrid modes. The cladding layer thickness a changes every n a and every mode so as to satisfy Eq. (5a). The index dispersion of the core is neglected here. We employ such a large cladding high index as physically unrealizable for comparison as described below. Five modes are plotted in order from the lowest-order mode of each mode group. For comparison, we plot the waveguide dispersion D QWS,w shown in Eq. (10). In this case the D QWS,w has no n a dependence of the mode groups, although the core radius has a different value for individual modes. For n t =0.8 we have D QWS,w =1513 ps/km nm for any of the modes. One finds from the figures that the dispersion values under the quasi-qws condition approach a certain QWS dispersion with the increase of n a. This tendency can be explained as follows: we can say that the QWS condition is a case where the optical wave is well confined to the core. If the cladding index difference n a n b increases under the quasi-qws condition, then this leads to a good optical confinement. Fig. 10. Wavelength dependence of the dispersion of the TE 01 mode as a function of several n a. QWS =1.0m and n b =1.5. Values of thickness b are 0.3 and 0.33 m for r c =5.0 and 10.0 m, respectively. Values of thickness a at r c =5.0m are , , and m for n a =.5, 3.5, and 4.5, respectively. 5. GROUP-VELOCITY DISPERSION OF THE TE 01 MODE A. Numerical Examples on the 1.0 m Band The TE 01 mode is expected as a candidate for the single mode transmission in the Bragg fiber [9], although further study is needed. The group-velocity dispersion of the TE 01 mode has been analyzed by several researchers [5,8]. In this subsection, the group-velocity dispersion of the TE 01 mode will be related to the QWS condition and the optical confinement because the TE 01 mode shows the highest optical power confinement factor among all the modes [19]. Figure 10 depicts the wavelength dependence of the dispersion of the TE 01 mode as a function of n a and r c.we use QWS =1.0 m and n b =1.5. Note that the ordinate scale in this case is two orders smaller than that in Fig. 7(a). One finds from Figs. 7(a) and 10 that the dispersion of the TE 01 mode is markedly reduced as the core radius increases at a fixed n a. The nearly flat region of the photonic band against the wavelength increases with the increase of r c. An increase in n a n b has semiquantitatively the same influence on the dispersion as an increase in r c. Table 1 shows the numerical results of the dispersion near the flat wavelength region. These values were evaluated from a straight line region in Fig. 10. For reference, we show the waveguide dispersion D QWS,w at 0 =1.0 m. The values of the dispersion near the flat region range roughly from 45 to 70 ps/km nm at r c =5.0 m, and they range from 10 to 0 ps/km nm at r c =10.0 m. These values are relatively close to values of D QWS,w. This means that the dispersion under the QWS condition can be conveniently used to estimate a dispersion for a general case without solving transcendental equations. Let us consider a possible single mode transmission of the TE 01 mode. The TM and hybrid modes are inclined to be cut off more readily than the TE mode [11]. In addition, radiation losses of the TM and hybrid modes are about five orders larger than that of the TE 01 mode [0]. If only the TE mode is retained, then the first higher-order mode is the TE 0 mode. A single mode transmission is attained for 0.610=j 1,1 /n c r c / =j 1, /n c under the QWS condition. The cladding layer thicknesses are calculated by a= 0 /4n a and b= 0 /4n b in this case. Figure 11 shows the dispersion versus the wavelength in such TE 01 single mode transmission for several n a. The parameters are set to be r c =1.117 m and n b =1.5 such that the single mode operation is possible above 0 =1.0 m. As the n a increases, the dispersion and the dispersion slope become small near the central region of the photonic band. The value of the dispersion amounts to ps/km nm at 0 =1.0 m, and it is much larger than that of the conventional optical fibers. Values of D at 0 =1.0 m are 4873 and 3086 ps/km nm for n a =.5 and 3.0, respectively. In the single mode transmission, the n b dependence of the dispersion is shown in Fig. 1 with QWS =1.0 m, r c =1.117 m, and n a =3.5. The width of the flat region increases with the increase of the cladding index difference n a n b even when the n b is changed. Values of D at 0 =1.0 m are 3085 and 4753 ps/km nm for n b =.0 and.5, respectively. A comparison between Figs. 11 and 1 reveals that even if the n a n b are identical with each other, say 1.0 or 1.5, the photonic band is wider for a low n b. This is because low cladding indices lead to wide cladding layer thicknesses at the same core radius and wavelength under the QWS condition, resulting in little field

9 4 J. Opt. Soc. Am. B/ Vol. 5, No. 3/ March 008 J.-I. Sakai and K. Kuramitsu Table 1. Numerical Examples of Dispersion of the TE 01 Mode r c m n a Quasi QWS Condition Flat Wavelength Region m Dispersion ps/km nm D at 1.0 m ps/km nm D QWS,w at 1.0 m ps/km nm Fig. 11. Cladding high-index dependence of the dispersion for the single-mode transmission of the TE 01 mode. r c =1.117 m, n b =1.5, and b= m. The single-mode operation is possible above QWS =1.0m. Values of thickness a are 0.1, , and m for n a =.5, 3.5, and 4.5, respectively. Fig. 1. Cladding low-index dependence of the dispersion for the single-mode transmission of the TE 01 mode. QWS =1.0 m, r c =1.117 m, n a =3.5, and a= m. Values of thickness b are , 0.15, 0.1, and m for n b =1.5,.0,.5, and 3.0, respectively. penetration into the cladding, hence, the wide photonic band. However, the dispersion value at QWS is nearly the same for an identical n a n b even if the n a value is different from each other. The index dispersion of the cladding material was neglected in calculating the above numerical examples. The cladding material is not specified here to show more general results. If we consider the index dispersion of cladding nearly identical with that of silica, the dispersion changes near the edges of the photonic band owing to a change in the photonic band. However, the index dispersion has little influence on the group-velocity dispersion near the QWS wavelength. Relative errors were 1% in most cases. It has been pointed out that an effect of the material dispersion in the cladding is 0.1 ps/km nm over most of the bandwidth in the TE 01 mode [8]. B. Numerical Example on the 10.6 m Band The group-velocity dispersion will be estimated here for realistic cladding material. The Bragg fiber was fabricated for use at 10.6 m [1]. Let us consider the TE 01 mode transmission using this fiber. The cladding materials are arsenic triselenide As Se 3 and poly (ether sulphone) (PES) and their refractive indices are.8 and 1.55, respectively. Although their thicknesses are 70 and 900 nm for a fundamental photonic bandgap at 3.55 m, they are not stated for the 10.6 m use. As a first example, the cladding layer thicknesses are multiplied by 3 because 10.6/ The core radius is 350 m. As a second example, the cladding layer thicknesses are selected such that the QWS condition is satisfied for n a =.8, n b =1.55, and 0 =10.6 m. In this quasi-qws we obtain r c / 0 =33.0 and /k 0 = at 0 =10.6 m. The dispersion for the two examples is demonstrated in Fig. 13. The dispersion values markedly decrease due to their large core radius ratio and long wavelength compared to the 1.0 m band. We can find no appreciable difference in the dispersion between the two examples near the central region of the photonic band. The photonic band is wider for the quasi-qws condition. 6. COMPARISON WITH TRANSFER MATRIX METHOD Figures 14 and 15 show the group-velocity dispersion of the TE 01 and HE 11 modes, respectively, to compare the present with the transfer matrix methods [8,]. The parameters used are the same as those in their citations. The index dispersion of the core and the cladding is neglected. As for the number of cladding layers, the present method assumes infinity, while the transfer matrix

10 J.-I. Sakai and K. Kuramitsu Vol. 5, No. 3/March 008/ J. Opt. Soc. Am. B 43 method assumes 17. For the TE 01 mode we have r c / 0 =8.46 at 0 =1.55 m. Note the unit of ps/m nm in the HE 11 mode. In the HE 11 mode, the core radius is relatively small, namely, r c / 0 =0.317 at 0 =1.55 m. We can see an excellent agreement between the present and the transfer matrix methods except for the PBEs in spite of both the difference in the number of cladding layers and the small core radius. Fig. 13. Group-velocity dispersion of the TE 01 mode in the 10.6 m band. Parameters r c =350 m, n a =.8, and n b =1.55 are common for both curves. Dashed curve (first example), a =810 nm and b=.7m; and solid curve (second example), a =1.013 m and b=.374 m. Fig. 14. Comparison of the dispersion between the present and the transfer matrix methods for the TE 01 mode. n a =4.6, n b =1.6, =0.434 m, r c =30, a=0., and b=0.78. The present method assumes infinitely extending cladding, while the transfer matrix method assumes 17 cladding layers. 7. SUMMARY Many numerical results were shown to comprehensively investigate the group-velocity dispersion of the TE, TM, and hybrid modes and to compare with dispersion D QWS under the QWS condition in the Bragg fiber. We have explicitly shown the dispersion D QWS in terms of the material and the waveguide dispersions, as shown in Eq. (10). Roughly speaking, the waveguide dispersion D QWS,w is inversely proportional to the second power of the core radius r c. It was shown that the waveguide dispersion D QWS,w closely relates to the optical power confinement factor QWS under the QWS condition in all the modes. Dispersion values approach D QWS as the cladding index difference n a n b or the core radius r c increases. The QWS dispersion is useful for evaluating the dispersion of actual circumstances when a mode is operated in the neighborhood of the central region of the photonic band, and it has a good optical confinement. The TE 01 mode has the widest flat characteristics against the wavelength and shows relatively small dispersion values among the TE, TM, and hybrid modes. It was revealed that properties relating to the dispersion of the Bragg fiber under the QWS condition are equivalent to those of the conventional optical fibers where all the power is confined to their core. The properties are formally equivalent to those in the circular metallic waveguide. APPENDIX A: DERIVATION OF THE GROUP-VELOCITY DISPERSION SHOWN IN EQ. (10) One obtains an expression for the propagation constant as = k 0 n a /ak 0 1/ + n b /bk 0 1/ A1 from Eq. () including only the cladding parameters. Differentiating Eq. (A1) with respect to k 0 we have d = 1 n eff n a N a + n b N b, A with the group index N i. On the other hand, the derivative of Eq. (3) yields Fig. 15. Comparison of the dispersion between the present and the transfer matrix methods for the HE 11 mode. n a =.8, n b =1.55, =0.41 m, r c =1., a=0.33, and b=0.67. The number of cladding layers is the same as that in Fig. 14. d = n cn c n eff. A3 A comparison between Eqs. (A) and (A3) leads to an equality

11 44 J. Opt. Soc. Am. B/ Vol. 5, No. 3/ March 008 J.-I. Sakai and K. Kuramitsu n c N c = 1 n an a + n b N b A4 under the QWS condition. The differentiation of Eq. (A), moreover, by k 0 and the use of Eq. (7) give k 0 d = 1 n effn a k 0 dn a + N a + n b k 0 dn b 1 n n an a + n b N b eff. A5 + N b From the derivative of Eq. (A4) with respect to k 0 one obtains n c k 0 dn c + N c = 1 n a k 0 dn a + N a + n b k 0 dn b + N b. A6 The substitution of Eqs. (A4) and (A6) into Eq. (A5) produces d k 0 dk = 1 0 n eff n dn c ck 0 + N c 1 dk n cn c 0. A7 n eff Equation (A7) is also derivable by directly differentiating Eq. (A3) by k 0. This means that the expression given in Eq. () is equivalent to the expression in Eq. (3). The application of Eq. (3) to the second and third terms of Eq. (A7) yields the waveguide dispersion in Eq. (10). A further modification of the first term of Eq. (A7) produces the material dispersion. REFERENCES 1. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, Groupvelocity dispersion in photonic crystal fibers, Opt. Lett. 3, (1998).. A. Ferrando, E. Silvestre, P. Andés, J. J. Miret, and M. V. Andés, Designing the properties of dispersion-flattened photonic crystal fibers, Opt. Express 9, (001). 3. J. Marcou, F. Brechet, and P. Roy, Design of weakly guiding Bragg fibres for chromatic dispersion shifting towards short wavelengths, J. Opt. A, Pure Appl. Opt. 3, S114 S153 (001). 4. G. Ouyang, Y. Xu, and A. Yariv, Theoretical study on dispersion compensation in air-core Bragg fibers, Opt. Express 10, (00). 5. T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, Dispersion tailoring and compensation by modal interactions in OmniGuide fibers, Opt. Express 11, (003). 6. P. Yeh, A. Yariv, and E. Marom, Theory of Bragg fiber, J. Opt. Soc. Am. 68, (1978). 7. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, An all-dielectric coaxial waveguide, Science 89, (000). 8. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, Low-loss asymptotically singlemode propagation in large-core omniguide fibers, Opt. Express 9, (001). 9. B. P. Pal, S. Dasgupta, and M. R. Shenoy, Bragg fiber design for transparent metro networks, Opt. Express 13, (005). 10. J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, Properties of photonic crystal fiber and the effective index model, J. Opt. Soc. Am. A 15, (1998). 11. J. Sakai, Hybrid modes in a Bragg fiber: general properties and formulas under the quarter-wave stack condition, J. Opt. Soc. Am. B, (005). 1. J. Sakai, Electromagnetic interpretation of the quarterwave stack condition by means of the phase calculation in Bragg fibers, J. Opt. Soc. Am. B 3, (006). 13. Y. Xu, R. K. Lee, and A. Yariv, Asymptotic analysis of Bragg fibers, Opt. Lett. 5, (000). 14. Y. Kokubun, Optical Wave Engineering (Kyoritsu, 1999), p. 49 (in Japanese). 15. D. Gloge, Dispersion in weakly guiding fibers, Appl. Opt. 10, (1971). 16. D. R. Lide, CRC Handbook of Chemistry and Physics, 81st ed. (CRC Press, 000), p J. Sakai, Optical power confinement factor in a Bragg fiber: 1. formulation and general properties, J. Opt. Soc. Am. B 4, 9 19 (007). 18. A. Savitzky and M. J. K. Goley, Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem. 36, (1964). 19. J. Sakai, J. Sasaki, and K. Kawai, Optical power confinement factor in a Bragg fiber:. Numerical results, J. Opt. Soc. Am. B 4, 0 7 (007). 0. M. Yan and P. Shum, Analysis of perturbed Bragg fibers with an extended transfer matrix method, Opt. Express 14, (006). 1. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, Wavelength-scalable hollow optical fibers with large photonic bandgaps for CO laser transmission, Nature 40, (00).. C. Lin, W. Zhang, Y. Huang, and J. Peng, Zero dispersion slow light with low leakage loss in defect Bragg fiber, Appl. Phys. Lett. 90, (007).