Classification and properties of radiation and guided modes in Bragg fiber

Transcription

1 Optics Communications 250 (2005) Classification and properties of radiation and guided modes in Bragg fiber Intekhab Alam, Jun-ichi Sakai * Faculty of Science and Engineering, Ritsumeikan University, Noi-higashi, Kusatsu-city, Shiga , Japan Received 24 September 2004; received in revised form 7 February 2005; accepted 8 February 2005 Abstract This article classifies modes into radiation and guided modes in a Bragg fiber, which has a cylindrically symmetric and periodic structure. Classification of the above modes into TE and TM modes is possible by considering the envelope of cladding fields and by noting a particular parameter relating to Bloch wavenumber. In particular, discussions are made for electromagnetic fields of radiation mode, contour curves concerning to the distinction between guided and radiation modes, and optical confinement. It is shown that radiation modes in this fiber distribute for discrete spectrum of propagation constant unlike the index-guiding optical fiber. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Bragg fiber; Photonic crystal fiber; Hollow optical fiber; Radiation mode 1. Introduction * Corresponding author. Fax: address: sakai@se.ritsumei.ac.p (J.-i. Sakai). Silica-based optical fibers have attained their ultimate loss and have been used widely. Plastic fibers have also been put into practical use. Two inbuilt drawbacks of the conventional optical fibers are material absorption loss and dispersion. Introduction of the erbium-doped fiber reduced this loss problem. Loss and dispersion also depend on optical confinement. The conventional index-guiding fiber has a high refractive index core surrounded by a low index cladding. Its guiding principle is based on the total internal reflection and the light penetrates into the cladding region. It is desired to get an optical fiber where optical wave is strongly confined in the core. One trial of strong optical confinement is the use of photonic crystal fibers [1,2] having the periodic structure. Among them, optical wave in Bragg fiber is guided on the basis of the Bragg diffraction due to the periodic cladding [3,4]. The main feature of the Bragg fiber is its hollow core, /$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi: /.optcom

2 I. Alam, J.-i. Sakai / Optics Communications 250 (2005) which is surrounded by a high refractive index media with a specially designed cladding material. The Bragg fiber is supposed to have lower material absorption loss and overcome the problem of optical confinement. This hollow core property of Bragg fibers may have future prospects. Bragg fibers have been fabricated [4 6] and the present fiber makes use of high cladding index of 4.6 [5]. From the practical point of view, it is desirable to use the refractive index as low as possible. The Bragg fiber has a complicated cladding structure compared to conventional optical fibers. Hence, it is significant to determine an optimal fiber structure from the theoretical point of view. The Bragg fiber has been analyzed using the transfer matrix method [3]. An approximate analysis method using the asymptotic expansion has been presented to get solution with moderate amount of calculations [7 9]. These works were mainly devoted to deriving eigenvalue equations and electromagnetic fields. It is also important to investigate properties of radiation modes for determining the lower limit of the index difference between cladding bilayers. Although a leakage of optical power has been mentioned for Bragg fiber [3,10], a deeper understanding is needed. In this paper, electromagnetic fields for both radiation and guided modes are widely discussed by considering the cladding field. Calculations are made using an asymptotic expansion analysis [9]. It will be pointed out that a key parameter, Re(X S ), is useful for distinguishing radiation from guided modes. Re(X S ) is tightly related to the optical confinement. Contour curves of Re(X S ) is also shown to estimate the permissible deviation from the quarter wave stack (QWS) condition, namely the optimal confinement condition. 2. Basic theory on Bragg fiber Schematic structure of a Bragg fiber is shown in Fig. 1. Here, n c is the refractive index of the core, and n a and n b denote the indices of cladding regions a and b, respectively. In addition, r c is the core radius, and a and b are the cladding material thicknesses. The period in the cladding is K = a + b. Fig. 1. Schematic diagram of Bragg fiber. r c, core radius; n c, refractive index of core; n a and n b, indices of layer with thickness a and b, respectively; K = a + b, period in the cladding; r a,m and r b,m, relative radial coordinates for cladding; a m and b m, amplitude coefficients for cladding layers a in TE mode; a 0 m and b0 m, amplitude coefficients for cladding layers b in TE mode. For TM mode, a m and b m should be replaced by c m and d m, respectively. We treat the TE and TM modes as our main study of this paper. Maxwell equations in the cylindrical form have been used to derive the solution for this fiber. Wave components to be connected at interfaces are given by H z A ¼ U tz D TE i i ðrþ ð1þ ie h B i for TE mode. Here, U tz = exp[i(xt bz)] denotes the spatio-temporal dependence with b being the propagation constant and z the propagation direction of wave. D TE i ðrþ is the 2 2 representation matrix whose elements are Hankel functions in the core region. The elements in the cladding region are expressed by the asymptotic expansion and correspond to forward and backward waves [9]. A i and B i are the amplitude coefficients and their subscripts are designated by c for the core. They are replaced by a m and b m in the cladding a region while the prime is added to them to express the amplitude coefficients for the cladding b region. For TM mode, field components, H z and ie h,in Eq. (1) must be replaced by E z and ih h. Amplitude coefficients, a m and b m, should be replaced by c m and d m, respectively, in the cladding a region. Eigenvalue equations in the Bragg fiber consist of two kinds of equation: The first eigenvalue equation holds for the matched value of the wave

3 86 I. Alam, J.-i. Sakai / Optics Communications 250 (2005) at the core-cladding interface [9], as is the case of the conventional optical fiber. J 0 0 ð cr c Þ J 0 ð c r c Þ þ cf c ½expð ik S KÞ X S Y S Š i a f a ½expð ik S KÞ X S þ Y S Š ¼ 0; ð2þ where 1 for TE mode; f i ¼ ð3þ 1=n 2 i for TM mode: Here, the suffix S applies to TE or TM mode, and the subscript i applies to a, b and c. J 0 is the Bessel function of order 0 and the order 0 corresponds to the azimuthal mode number m of TE ml or TM ml mode. The prime for J 0 indicates differentiation with respect to its argument. Bloch wavenumber K S in Eq. (2) is determined from the wave periodicity in a periodic cladding according to the Floquet theorem [11]. The phase factor expð ik S KÞ constitutes the second eigenvalue equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expð ik S KÞ¼ReðX SÞ ½ReðX S ÞŠ 2 1 ð ¼ 1; 2; S ¼ TE or TMÞ: ð4þ Here, = 1 applies to the upper sign and =2 to the lower sign in the double sign notation. Lateral propagation constants for each region are defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ðn i k 0 Þ 2 b 2 ði ¼ a; b; cþ ð5þ with k 0 =2p/k 0 being the vacuum wavenumber and k 0 the vacuum wavelength. In Eqs. (2) and (4), we set translational matrix elements as X S ¼ cosð b bþ i f b b þ f a a sinð b bþ 2 f a a f b b expð i a aþ; ð6þ Y S ¼ i f b b f a a sinð b bþ expði a aþ: 2 f a a f b b Eq. (4) tacitly includes the propagation constant b. By solving Eqs. (2) and (4) simultaneously, we can get b, electromagnetic fields and other characteristics. In particular, X S is the key parameter for featuring several properties, as can be found later. Amplitude coefficients a m and b m in the cladding a region, and a 0 m and b0 m in the cladding b region are shown in Appendix A. 3. Classification of radiation and guided modes based on the cladding fields 3.1. Relationship between cladding fields and Re(X S ) Using Eqs. (1) and (A.1) and after some mathematical manipulation, cladding fields for TE mode can be written in the following matrix form [9]: sffiffiffiffiffiffiffiffi H z 2 ¼ U tz exp½ iðm 1ÞK TE KŠ ie h p r¼r i r 1 1 ixl 0 = i ixl 0 = i expð i ir i;m Þ 0 a1 0 expði i r i;m Þ b 1 : ð7þ Here, l 0 is the magnetic permeability of vacuum. Relative radial coordinates, r a, m and r b, m, for cladding layers a and b are defined by r a, m = r [r c +(m 1)K] and r b, m = r [r c + a + (m 1)K]. The exponential terms in the matrix element represent the fine behavior in the cladding fields. Bloch wavenumber K TE in the first exponential term of Eq. (7) is common over the entire cladding region and dictates whether the wave is propagating or decaying along the r direction. This K TE also relates to the value of Re(X S ), which is inferable from Eq. (4). Regime where Re(X S ) < 1, corresponds to real K S. This condition makes the first exponential term an oscillatory one and results in the radiation mode [9]. Then the second eigenvalue Eq. (4) becomes a complex quantity and produces two Bloch wavenumbers, K S 1 and KS 2, out of phase by 180 with a real value. From this fact we can get two Bloch wave numbers of quasi radiation modes as

4 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 ReðK S 2 Þ¼1 1 ½ReðX S ÞŠ 2 K tan ReðX S Þ ¼ ReðK S 1 Þ: ð8þ This means that quasi radiation mode is constituted by superposition of forward and backward waves. Electromagnetic fields can be found by adding the field corresponding to two K S. On the other hand, Re(X S ) > 1 regime represents K S a complex quantity, which yields a decaying function in the cladding. The decaying property governs guided waves and growing function is not acceptable. In the guided mode either K S 1 or KS 2 is used. Re(X S) = 1 corresponds to ust the cutoff as well as the photonic bandgap edge. These three cases about Re(X S ) can be found by properly choosing the refractive indices and thicknesses of the cladding material, as is seen from ReðX S Þ¼cosð a aþ cosð b bþ 1 f b b þ f a a sinð a aþ sinð b bþ; 2 f a a f b b ð9þ where S = TE or TM. Eq. (9) includes only cladding parameters Parameters prescribed in calculation I. Alam, J.-i. Sakai / Optics Communications 250 (2005) whose n a Õs are 3.0 and 2.0. Using these parameters we have found a plot of Re(X S ) versus normalized propagation constant b/k 0 (see Fig. 2). The radial mode number l in TE 0l and TM 0l modes is labeled in order of decreasing b/k 0 value. From these data we get an idea that large n a value cannot produce any situation of radiation mode because Re(X S )< 1 is always satisfied for n a = 3.0. For n a = 3.0 both TE and TM modes have six guided modes. However, keeping the other parameters, such as n b, n c, n t, k 0 and r c, same and reducing n a value from 3.0 to 2.0, we can get the radiation modes. In case of n a = 2.0 and r c = 5.0 lm, the TE mode has one quasi radiation mode below the guided region whereas the TM mode has two and one radiation modes, respectively, above and below the guided region under the scale of b/k 0. Radiation modes are obtained for discrete b spectrum. This is a remarkable contrast to the usual index-guiding fiber whose radiation modes distribute for continuous b values [12]. So, the radiation mode in the Bragg fiber will be referred to as the quasi radiation mode. In the presently used theory, the cladding layers are assumed to extend infinitely. If the cladding It will be fixed in the present simulation that n c = 1.0, n b = 1.5, operating wavelength k 0 = 1.55 lm andr c =5.0lm. Here, we have used the quarter wave stack (QWS) condition from which we have calculated the cladding layer thickness a and b as p i ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ði ¼ a; bþ: ð10þ 2k 0 n 2 i n 2 t The tentative effective refractive index is defined by n t b/k 0. For n a = 2.0 and n b = 1.5, we can get a = lm, b = lm andk = lm at n t = 0.8. In addition, we have a = lm for n a = 3.0 and n t = Characteristics of Re(X S ) Before starting to discuss about Re(X S )wehave studied on two configurations of Bragg fibers, Fig. 2. Re(X S ) vs. normalized propagation constant b/k 0 for TE and TM modes. r c = 5.0 lm and n t = 0.8. Parameters n c = 1.0, n b = 1.5 and k 0 = 1.55 lm are fixed in all calculations. Solid and dashed lines indicate data for TE and TM modes, respectively, using Eq. (9). Symbols indicate the position of b for which eigenvalue equation (2) is satisfied for given parameters. Solid symbols indicate guided modes while open symbols indicate quasi radiation mode.

5 88 I. Alam, J.-i. Sakai / Optics Communications 250 (2005) layer is limited to a finite thickness, all modes tend to tunnel through the confinement region in the Bragg fiber [3,10] and the holey fiber [13]. Then modes become leaky and can have complex values of b, imaginary part of which relates to the power attenuation. However, radiation loss decreases exponentially as the number of cladding layers increases [10]. The present theory shows that the Bragg fiber can produce radiation modes even in the infinitely extending cladding. We see from Fig. 2 that the TM mode is inclined to become cutoff more than TE mode because Re(X TM ) of TM mode is less than that of TE mode. In addition, it should be noted that Re(X S ) for b = 0 shows an identical value for TE and TM modes and it depends only on n a value under the prescribed values. This is because the value inside parentheses in Eq. (9) reduces to (n b /n a + n a /n b ) for the above case of both modes. Fig. 3 shows the presence of radiation modes from the perspective of refractive index contrast and reveals that radiation mode can be found for n a < 2.11 and n b = 1.5 in case of TE 06 mode. We see that six TE 0l modes for n a values of 2.0 and 3.0 correspond to symbol positions in Fig. 2. Re(X TE ) vs. n a curves are crossing with decreasing n a.ifn a value is reduced to 1.6, then we get four radiation modes and two guided modes: these two are very close to the cutoff. The line of Re(X S )= 1 can be told as the line of cutoff [14], since for this value of Re(X S ) the electromagnetic field ust goes to quasi radiation mode, as shown later. Another interesting point is that the number of quasi radiation modes increases as n a value decreases (see Fig. 3). Re(X TM ) for TM mode is shown in Fig. 4 as a function of n a. The lower-order mode in TM mode exhibits a small Re(X TM ) for relatively large n a value unlike the TE mode, as can be seen from Figs. 3 and 4. This opposite characteristics can be also explained from the slope of each curve in Fig. 2. Re(X TM ) value decreases with decreasing the n a value. Radiation mode appears for n a < 2.23 in case of TM 01 mode. The lower limits of n a for TE 05 and TE 06 are 1.62 and 1.89, respectively. This is due to the fact that the TM mode tends to become cutoff Electromagnetic fields for TE and TM radiation modes Electromagnetic fields for several modes shown in Fig. 2 will be illustrated here. Fig. 5 shows the radiated electromagnetic fields of TE 06 mode for n a = 2.0 and r c = 5.0 lm up to 40 cladding pairs Fig. 3. Re(X TE ) vs. refractive index contrast for TE mode. n a = 1.6 to 3, r c = 5.0 lm and n t = 0.8. Data for TE 01 to TE 06 are arranged according to the order of radial mode number l in the right hand side. Fig. 4. Re(X TM ) vs. refractive index contrast for TM mode. Parameters are same as those in Fig. 3. Data for TM 01 to TE 06 are arranged according to the order of radial mode number l in the right hand side.

6 I. Alam, J.-i. Sakai / Optics Communications 250 (2005) Fig. 6. Electromagnetic fields, ie h and H z, for guided TE 05 mode. Parameters are same as those in Fig. 5 except for Re(X TE )= , b = lm 1 and b/k 0 = Fig. 7. Electromagnetic fields, ie h and H z, at nearly cutoff for quasi radiation TE 01 mode. n a = 1.78, r c = 5.0 lm, K = lm, k 0 = 1.55 lm and n t = 0.8. Re(X TE )= , b = lm 1 and b/k 0 = Fig. 5. Electromagnetic Fields, ie h and H z, for quasi radiation TE 06 mode. n a = 2.0, r c = 5.0 lm, K = lm, k 0 = 1.55 lm and n t = 0.8. Re(X TE )= , b = lm 1 and b/ k 0 = to elucidate the oscillatory waves in the envelope of the cladding field. The period of the cladding field envelope is about 7.56 lm which is much longer than the cladding period K = lm. For comparison, both electric and magnetic fields for guided mode are shown in Fig. 6 with same number of cladding pairs. Fields strongly decay in the cladding region and penetrate into only 10 cladding layer pairs. Envelope of the cladding field decays exponentially along the r direction. An n a value was sought for which b satisfies not only eigenvalue Eq. (2) but also Re(X TE 1 for given parameters. Two candidates, TE 01 and TE 05 modes, can be found near n a = 1.78 from Fig. 3. Electromagnetic fields of TE 01 quasi radiation mode at nearly cutoff are shown in Fig. 7 up to 40 cladding pairs. Envelope of the cladding field monotonically p and slowly decays in proportion to roughly 1= ffiffi r. This type of quasi radiation mode can not be found in the usual index-guiding fiber. In order to compare with electromagnetic fields for quasi radiation mode, Fig. 8 illustrates those of TE 01 mode at nearly b. 0 where eigenvalue Eq. (2) is satisfied for n a = 2.5 and r c = lm. Although fields in this case penetrate into the cladding a little bit, they decay in the cladding. This fact is very contrast to the usual index-guiding fiber where field spreads up to the infinity at the lower limit of b [12]. This is because an approximate QWS condition is still maintained for b. 0 in the present Bragg fiber. In fact, Re(X TE ) is equal to in this case. Whether a certain mode is guided or not depends on the Re(X TE ) value rather than b/k 0 in the Bragg fiber. Electromagnetic fields for TM 06 quasi radiation mode are shown in Fig. 9 up to 40 cladding pairs. Fiber parameters are the same as those in Fig. 5.

7 90 I. Alam, J.-i. Sakai / Optics Communications 250 (2005) TM modes near the QWS condition [9]. This means that guided modes are always obtained near the QWS condition. There is a question; How much shift can be permitted from the QWS condition to get guided modes? 4.1. Contour curve of Re(X TE ) for TE mode Fig. 8. Electromagnetic fields, ie h and H z, at nearly b. 0 for TE 01 mode. n a = 2.5, r c = lm, K = lm, k 0 = 1.55 lm and n t = 0.8. Re(X TE )= , b = lm 1 and b/k 0 = The contour line of Re(X TE ) will be figured out using Eq. (9), which can provide result without the mode designation. To get a clear concept about the contour line, two types of contour diagram will be generated. Fig. 10 shows the first contour diagram where a and b are kept constant while cladding layer thicknesses, a and b, are varied. This diagram is symmetrical with respect to b b = a a. This can be explained as follows; value in the parentheses of Eq. (9) is constant in the present case. So, we can easily see the symmetrical property mentioned above. If both a a and b b are shifted by p simultaneously in the a a b b plane, one obtains the same Re(X TE ) value as the previous one. The curve of 1 is the line of cutoff, and inside this region we can get the solution for guided modes. Moreover, a closer view to the contour diagram in Fig. 10 shows that optical confinement is Fig. 9. Electromagnetic fields, E z and ih h, for quasi radiation TM 06 mode. Parameters are same as those in Fig. 5 except for Re(X TM )= , b = lm 1 and b/k 0 = Although envelope of the cladding fields decays slowly, the envelope repeats the oscillation. The field behavior of TM mode is similar to that of TE mode. 4. Contour curves of Re(X S ) So far we have concentrated on electromagnetic fields with the variation of Re(X S ). By the way, it is found that Re(X S ) 6 1 holds for both TE and Fig. 10. Contour diagrams of Re(X TE ) for TE mode in the a a b b plane, where a and b are fixed. n a = 3.0 and n t = Dashed line satisfies a relation, a a + b b = p.

8 I. Alam, J.-i. Sakai / Optics Communications 250 (2005) optimum at a point a a = b b = p/2; this point corresponds to the QWS condition. This criterion can be seen from Fig. 11, which plots Re(X TE ) for several n a values as a function of a a subect to a a + b b = p. We see that optical confinement is improved as the cladding index difference (n a n b ) increases. From these figures we can estimate the permissible deviation from the QWS condition. In the second contour diagram (see Fig. 12), the cladding layer thicknesses, a and b, are fixed at and lm, respectively, for n a = 3.0, n b = 1.5, n t = 0.8 and k 0 = 1.55 lm according to Eq. (10). Afterward a and b are varied. This diagram is complicated unlike Fig. 10, as can be understood from Eq. (9). For reference, the solutions shown in Fig. 2 are inserted in Fig. 12. We see that these solutions are situated near the QWS condition but donõt always satisfy the QWS condition exactly. Figs. 10 and 12 help us to design the Bragg fiber under the actual situation. In the vicinity of the QWS condition of Fig. 12 we may have solutions for the radiation modes which are outside the contour line 1. Tentative effective refractive index n t has little influence on these plots and core radius r c has no effect. However, refractive index of the cladding has a prominent impact on the contour plot, which is fairly inferable. If the refractive index n a is lowered, then the contour area marked with 1 will also change. Fig. 12. Contour diagram of Re(X TE ) in the a a b b plane, where a and b are fixed. n a = 3.0 and n t = Six solid squares near the QWS condition ( a a = b b = p/2) indicate guided modes which correspond to the solutions shown in Fig. 2. This is shown in Figs. 13 and 14. It is found from Figs. 10 and 13 that decrease in n a leads to decrease in the guided mode region, namely Re(X TE ) > 1. A comparison of Fig. 14 with Fig. Fig. 11. Re(X TE ) vs. a a or b b curve subect to a a + b b = p, where r c = 5.0 lm, k 0 = 1.55 lm and n t = 0.8. Fig. 13. Contour diagrams of Re(X TE ) in the a a b b plane, where a and b are fixed. Parameters are same as those in Fig. 10 but for n a = 2.0.

9 92 I. Alam, J.-i. Sakai / Optics Communications 250 (2005) Fig. 14. Contour diagrams of Re(X TE ) in the a a b b plane, where a and b are fixed. Parameters are same as those in Fig. 12 but for n a = 2.0. Five solid and one open circles near the QWS condition indicate guided and radiation modes, respectively, which correspond to the solutions shown in Fig. 2. Fig. 15. Isolines of Re(X TE ) in the a a b b plane from the solution of Eq. (2), which is the exact one. n a = 2.0, r c = 5.0 lm, k 0 = 1.55 lm and n t = yields a similar conclusion to the above. For reference, the solutions obtained in Fig. 2 are added in Fig. 14. These are also located in the neighborhood of the QWS condition Approximation of contour plots The contour curves so far plotted are drawn by using Eq. (6), which bears the information about the cladding region alone but no information about the core. Fig. 15 is a contour curve of Re(X TE ), wich is evaluated from b through solving exact eigenvalue Eq. (2) for fiber parameters. Five curves correspond to the solutions for respective TE guided modes. Interpolation of these curves is shown in Fig. 16 to compare with contour curves obtained from approximate Eq. (6). Persistence of our vision in a small extent (for example, a a = and b b = ) reveals that there is small difference between the exact isolines and approximate isolines. Agreement between them is excellent. Hence, we can conveniently use the approximate contour solution given in Figs with respective set of parameters. Fig. 16. Interpolation of exact Re(X TE ) from solutions of Eq. (2) with approximate Re(X TE ) using Eq. (6). n a = Contour curve of Re(X TM ) for TM mode For a single set of parameters, both TE and TM modes persist with two different set of Re(X S ) values. So, the confinement is also changed between TE and TM modes. Cladding thicknesses, a and b, were fixed for n a = 2.0 and n t = 0.8 in the similar way to that in Fig. 12. The contour plot is figured out in Fig. 17 for TM mode by varying a and b. From the plot we can see that the contour line criterion is opposite from the TE mode. However, guided and radiation modes can be also found for TM mode. Confinement of the light for TM

10 I. Alam, J.-i. Sakai / Optics Communications 250 (2005) Decrease in n a leads to decrease in the guided mode region. Contour diagram can be accurately calculated by Eq. (6) including only cladding parameters and wavelength. It was confirmed on the contour diagram that the solutions derived in Section 3 are situated near the QWS condition. High refractive index material is difficult to fabricate although optical confinement using this material is better. On the other hand, low refractive index material has radiation modes or loss. This study will help us to fabricate a perfect low-loss Bragg fiber in the future. Acknowledgments Fig. 17. Contour diagram of Re(X TM ) for TM mode in the a a b b plane, where a and b are fixed. n a = 2.0 and n t = 0.8. Three solid and three open triangles indicate guided and radiation modes, respectively, which correspond to the solutions shown in Fig. 2. mode is also dependent on Re(X TM ). An approximation of these plots is also acceptable. The solutions obtained in Fig. 2 are inserted in Fig. 17 for reference. 5. Conclusions Various properties of radiation and guided modes have been clarified for TE and TM modes in the Bragg fiber. Discussions have been mainly made on quasi radiation modes and cutoff by showing electromagnetic fields and Re(X S ) peculiar to Bloch wavenumber. Classification between radiation and guided modes is possible by considering the value of Re(X S ) and its value governs the envelope of cladding fields. The envelope for radiation modes is oscillatory as well as their fine structure. It should be noted that quasi radiation modes in the Bragg fiber distribute against discrete spectrum of propagation constant unlike the conventional index-guiding optical fiber. Whether a certain mode is guided or not depends on the Re(X S ) value rather than b/k 0 in the Bragg fiber. Contour curves for Re(X S ) have been shown in the a a b b plane to elucidate the permissible deviation from the quarter wave stack condition. We express our special thanks to Mr. Jyumpei Sasaki for his valuable helps. Appendix A. Amplitude coefficients in the cladding region Amplitude coefficients in mth cladding layer for TE mode can be represented by! a m b m ¼n TE m Y TE expð ik TE ¼ exp½ iðm 1ÞK TE KŠ KÞ X TE a 1 b 1 for the cladding a region and a 0 m b 0 ¼ 1 rffiffiffiffiffi! b h S 11 h S 12 am m 2 a h S 21 h S b 22 m for the cladding b region. Here, h S 11 ¼ 1 þ f b b expð i a aþ; f a a h S 12 ¼ 1 f b b expði a aþ; f a a h S 21 ¼ 1 f b b expð i a aþ; f a a h S 22 ¼ 1 þ f b b expði a aþ: f a a ða:1þ ða:2þ ða:3þ Coefficient n TE 1 for the most inner cladding layer appearing in Eq. (A.1) is given by

11 94 I. Alam, J.-i. Sakai / Optics Communications 250 (2005) n TE 1 ¼ A c 2J 0 ð c r c Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=ðp a r c Þf½expð ik TE KÞ X TE ŠþY TE Šg ; ða:4þ where A c denotes the amplitude coefficient in the core. For TM mode, amplitude coefficients, a m, b m, a 0 m ; b0 m and nte m, should be replaced by c m, d m, c 0 m ; d0 m, respectively, in the above equations. References and ntm m [1] P. Russell, Science 299 (2003) 358. [2] J.C. Knight, Nature 424 (2003) 847. [3] P. Yeh, A. Yariv, E. Marom, J. Opt. Soc. Am. 68 (1978) [4] M. Ibanescu, Y. Fink, S. Fan, E.L. Thomas, J.D. Joannopoulos, Science 289 (2000) 415. [5] Y. Fink, D.J. Ripin, S. Fan, C. Chen, J.D. Joannopoulos, E.L. Thomas, J. Lightwave Technol. 17 (1999) [6] B. Temelkuran, S.D. Hart, G. Benoit, J.D. Joannopoulos, Y. Fink, Nature 420 (2002) 650. [7] Y. Xu, R.K. Lee, A. Yariv, Opt. Lett. 25 (2000) [8] Y. Xu, G.X. Ouyang, R.K. Lee, A. Yariv, J. Lightwave Technol. 20 (2002) 428. [9] J. Sakai, P. Nouchi, Opt. Commun., to be published; doi: /.optcom [10] S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Solac ić, S.A. Jacobs, J.D. Joannopoulos, Y. Fink, Opt. Exp. 9 (2001) 748. [11] P. Yeh, Optical Waves in Layered Media, Wiley, New York, 1988 (Chapter 6). [12] D. Marcuse, Theory of Dielectric Optical Waveguides, Academic Press, New York, [13] B.T. Kuhlmey, R.C. McPhedran, C.M. Sterke, P.A. Robinson, G. Renversez, D. Maystre, Opt. Exp. 10 (2002) [14] J. Sakai, J. Sasaki, K. Kawai, to be submitted.