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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 6 — Mar. 22, 2004
  • pp: 1126–1135
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Full-vectorial analysis of complex refractive-index photonic crystal fibers

Ren Guobin, Wang Zhi, Lou Shuqin, Liu Yan, and Jian Shuisheng  »View Author Affiliations


Optics Express, Vol. 12, Issue 6, pp. 1126-1135 (2004)
http://dx.doi.org/10.1364/OPEX.12.001126


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Abstract

We investigated the modal properties of complex refractive-index core photonic crystal fibers (PCFs) with the supercell model. The validity of the approach is shown when we compare our results with those reported earlier on a step complex refractive-index profile. The imaginary part of the electric field results in wave-front distortion in the complex refractive-index profile PCFs, which means that the power flows out or into the doped region according to the sign of the imaginary part of the refractive index. A simple formula is proposed for calculating the gain or loss coefficients of these fibers. The numerical results obtained by the approximation formula agree well with the full-vectorial results.

© 2004 Optical Society of America

1. Introduction

The critical characteristics in fiber lasers or amplifiers are the gain and loss along the length of the fiber. Except for models based on rate and propagation equations, an alternative method is to analyze optical fibers whose refractive-index profile is described in terms of a complex function. The gain or loss is decided by the imaginary component of the complex propagation constant, which is critically dependent on the imaginary component of the complex refractive-index profile.

The propagation characteristics of optical fibers with a circular complex refractive-index profile have been reported in the literature [1

1. A. Reisinger, “Characteristics of optical guided modes in lossy waveguides,” Appl. Opt. 12, 1015–1025 (1973). [CrossRef] [PubMed]

5

5. R. Singh Sunanda and E. Khular Sharma, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,” IEEE J. Quantum Electron. 37, 635–640 (2001). [CrossRef]

]. These analyses are based on scalar wave equations and thus are applicable only for weakly guiding fibers. With the advent of photonic crystal fibers (PCFs) [6

6. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

] and Bragg fibers [7

7. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419(2000). [CrossRef] [PubMed]

], the numerical method must be applicable for fibers with complicated refractive-index or high-index contrast profiles. PCFs used as active fibers were first reported by Wadsworth et al. [8

8. W. J. Wadsworth, J.C. Knight, W. H. Reeves, P.S.J. Russell, and J. Arriaga, “Yb3+-doped photonic crystal fibre laser, ” Electronics Lett. 36, 1452–1454 (2000) [CrossRef]

], and they offer several advantages compared with standard fibers, such as mode control and high-dispersion tailoriability. Such PCFs have been used to design amplifiers with improved performance, such as low-threshold or high-power amplifiers and lasers [9

9. K. G. Hougaard, J. Broeng, and A. Bjarklev, “Low pump power photonic crystal fibre amplifiers,” Electron. Lett. , 39, 599–600 (2003) [CrossRef]

,10

10. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tunnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express 11, 818–823 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-818 [CrossRef] [PubMed]

].

2. Simulation method

For the case of compactness, two subscripts m and n are introduced to express the opposite parities of the mode electric field, which have the logical value 0 or 1, and are used to describe the symmetry of the x component ex (x,y) as ex (-x,y)=(-1) mex (x,y) and ex (x,-y)=(-1) nex (x,y). All the compositions of mn are [00, 01, 10, 11], which can completely express the symmetry of the mode electric field about both axes. To improve computational efficiency, the transverse electric field can be expanded with the localized orthonormal Hermite-Gaussian basis functions as follows:

ex(x,y)mn=a,b=0F1εabxψ2a+m(x)ψ2b+n(y),
(1)
ey(x,y)mn¯=a,b=0F1εabyψ2a+m¯(x)ψ2b+n¯(y),
(2)

where the bar over the subscript indicates the logical operator NOT and the subscript mn indicates that there are four sets of (ex , ey ) with different parity. F is the number of expansion terms, εabs (s=x,y) are the expansion coefficients, and ψi (s) is the ith-order orthonormal Hermite-Gaussian function [12

12. T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999). [CrossRef]

,13

13. W. Zhi, R.G. Bin, L.S. Qin, and J. S. Sheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract. cfm? URI=OPEX-11-9-980. [CrossRef] [PubMed]

,14

14. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310 [CrossRef] [PubMed]

].

For a dielectric constant structure with axially symmetric x and y, i.e., ε(-x,y)=ε(x,-y)=ε(x,y), the expression can be given as a sum of the cosine functions as

ε(r)=ε(x,y)=a,b=0P1Pabcos2πaxDxcos2πbyDy,
lnε(r)=lnε(x,y)=a,b=0P1Pablncos2πaxDxcos2πbyDy,
(3)

where P is the number of expansion items and Pab , Pabln are the expansion coefficients that can be analytically evaluated from the Fourier transform. Dx (Dy ) is the characteristic period in the x(y) direction. When fibers with a complex refractive-index profile are examined, the expansion coefficients Pab , Pabln are complex. This approach is convenient especially when one is investigating PCFs or Bragg fibers [14

14. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310 [CrossRef] [PubMed]

,15

15. W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, “Compact supercell method based on opposite parity for Bragg fibers,” Opt. Express 11, 3542–3549 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542 [CrossRef] [PubMed]

].

Lmn[εxεy][[M1+k2M2+M3x]mn[M4x]mn[M4y]mn¯[M1+k2M2+M3y]mn¯][εxεy]=β2[εxεy],
(4)

The complex refractive index of the fiber can generally be written as n(x, y)=nr (x, y)+i ni (x, y). The positive (negative) sign of the imaginary part indicates a gain (lossy) media. If we take the complex index as n *, where the asterisk represents a complex conjugate operation, it can be proved that the matrix Lmn will be Lmn* . According to Eq. (4), We have

Lmn*[εx*εy*]=β2*[εx*εy*].
(5)

3. Propagation properties of complex index PCFs

3.1 Validity of the method

Table 1. Comparison of complex mode index ne between Ref. [5] and our method for complex index step fibers

table-icon
View This Table

3.2 Propagation properties of complex index core PCF

Fig. 1. Cross section of PCF with complex refractive-index core.

After validating the full-vectorial method, we apply this method to analyze PCFs with a complex index core. One of the most important PCF configurations consists of a silica fiber with a solid core surrounded by a silica cladding pierced by rings of air holes, which are typically hexagonally packed. The parameter hole spacing Λ and relative hole size d/Λ are used to define the structure of the PCF (shown in Fig. 1). We assume that the fiber core is doped with core radius R and that its refractive index has an imaginary part. We take the fiber parameters as Λ=4.4 µm, R=2.2 µm and the relative hole size to be d/Λ=0.3. The refractive index of the fiber core is taken as ncore =1.475+ini , where ni =10-3, the index of the cladding matrix is nmatrix =1.458, and the index of air is nair =1. The complex refractive-index profile can be considered as a function of radial distance, pump and signal wavelength, and dopant profiles for active medium [4

4. Sunanda and E. K. Sharma, “Field variational analysis for modal gain in erbium-doped fiber amplifiers,” J. Opt. Soc. Am. B , 16, 1344–1347 (1999). [CrossRef]

]. For simplicity, we assume that the complex refractive index is a constant in fiber core throughout this paper.

Fig. 2. Real part and imaginary part of modal field HE11.
Fig. 3. Real part and imaginary part of modal field TM01.

Fig. 4. Phase distribution of mode HE11x.

We find that the phase distortion is principally dominated by ni —the imaginary part of the refractive index in the fiber core. Figure 5 shows the phase distribution along the x=0 axis with different ni for d/Λ=0.3, λ=1550nm. In Fig. 5, the phase is multiplied by 10 for ni =10-4 and 100 for ni =10-5. It is obvious that these curves are overlapped; therefore it can be concluded that the phase distribution P(x,y) satisfies P(x,y)∝ni . For different ni , the line types of these phase distributions are the same except for the amplitude.

Fig. 5. Phase distribution along x=0 axis with different ni for d/Λ=0.3, λ=1550 nm.

For practical implementation, single-mode fibers are generally required. It is shown that the PCF will be endlessly single mode when d/Λ is less than 0.406 [17

17. B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre,“Microstructured optical fibers: where’s the edge?” Opt. Express 10, 1285–1290 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285 [CrossRef] [PubMed]

] for standard PCF (the fiber core is undoped). The active PCF discussed in Figs. 2 and 3 (d/Λ=0.3) is multimode (including the fundamental mode and second-order modes) at wavelength λ=1550 nm. Because the fiber core is up-doped, it is reasonable that the critical value of d/Λ for endlessly single-mode operation will decrease and depend on the refractive index of the core. To extend the available parameter space, the refractive index of the fiber core should be depressed, for example, by codoping with fluorine, to compensate the refractive-index increase.

Fig. 6. Real part (ner ) and imaginary part (nei ) of the mode index of fundamental mode versus relative air hole size d/Λ at 980, 1310, and 1550 nm.

4. Approximate formula for calculation gain or loss coefficient

γ=2βi=2k(ε0μ0)12Anrnie2dARe{Ae×h*·ẑdA},
(6)

According to the integral expression for the propagation constant of nonabsorbing waveguides [16

16. A.W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

]

Anr2e×h*·ẑdA=βrk(ε0μ0)12Anr2e2dA,
(7)

the subscript r denote the real values. In the weakly guiding approximation, we have

Ae×h*·ẑdAner(ε0μ0)12Ae2dA,
(8)

where ner =βr/k. Substituting Eq. (8) into Eq. (6), we have

nei=βi/k1nerAnrnie2dAAe2dA,
(9)

Referring to the optical fiber with a circular step-index profile, we obtain the imaginary part of the mode index

nei1nerknkrnkiΓk,
(10)

where k denotes the kth layer from the fiber core to the out-cladding, nkr (nki ) is the real (imaginary) part of refractive index in the kth layer, and Γk is the optical power confinement factor in the kth layer. For the complex index core PCF discussed in this paper, we have

neiΓncorernerncorei,
(11)

The approximation expression of the imaginary part of the mode index in Eq. (10) is especially useful when a weakly absorbing or amplifying waveguide is discussed. Only the mode index and power confinement factor of the corresponding waveguide, in which the imaginary part of the refractive index is neglected, are needed for calculating the imaginary part of the propagation constant (the gain or loss). Hence the approximate formulas provide a simple approach for evaluating the gain or loss coefficients, and the complicated process for solving the complex propagation constants of bound modes of the eigenvalue equation can be avoided.

We first compare the mode index ne and the power confinement factor Γ between the complex refractive-index core PCF and the corresponding fibers, in which the imaginary part of the imaginary refractive index is neglected. For clarity, fiber A represents PCF with ncore =1.475+i10-3 and the refractive index of the cladding matrix is nmatrix =1.458. Fiber B represents PCF with ncore =1.475 and nmatrix =1.458. Figure 7 shows the mode indices (for fiber A, only the real part of mode index are demonstrated) and the confinement factors of fiber A and B as a function of d/Λ. The differences of mode index and confinement factor are also given in the figure. We note that as the relative air hole size d/Λ increases, the differences of two types of fiber decrease. The restraining assumption of the stand method of perturbation is that the gain or loss exhibited by the waveguide does not alter the field significantly [1

1. A. Reisinger, “Characteristics of optical guided modes in lossy waveguides,” Appl. Opt. 12, 1015–1025 (1973). [CrossRef] [PubMed]

]. The assumption is supported by Fig. 7, in which the difference of fiber A and B is of an extremely small magnitude: 10-5 for mode index, and 10-4 for power confinement factor. Hence for practical fibers, the perturbation method is also applicable.

Fig. 7. (a) Mode indices and difference of fibers A and B, (b) power confinement factors and difference of fibers A and B. For mode index ne and power confinement factor Γ, the solid curves indicate fiber A and the curves with circles indicate fiber B.
Fig. 8. (a) Imaginary part of the mode indices versus relative air hole size d/Λ at different wavelengths. (b) Imaginary part of the mode indices versus wavelength with different relative air hole size d/Λ. The solid curves show the numerical results from full-vectorial method, and dotted curves with circles indicate the results from Eq. (11).

To verify Eq. (11), we illustrate both the full-vectorial results and the approximate results obtained with Eq. (11) in Fig. 8. In Fig. 8(a) the imaginary parts of the effective mode indexes are shown as a function of relative air hole size d/Λ at wavelengths of 980, 1310, and 1550 nm; in Fig. 8(b), the imaginary parts of the effective mode indices are shown as a function of wavelength for different relative air hole size d/Λ (0.1, 0.3 0.5 0.7). As shown in Fig. 8, Eq. (11) agrees well with the full-vectorial method. We find that the difference of the two groups of curves is less than 10-6. It is observed that the difference between the approximate formula and the full-vectorial method is small at short wavelengths but increases for long wavelengths. Further, the difference increases as the air hole size d/Λ increases. This is understood to occur because the weakly guiding approximation used in Eq. (11) brings more error for the large air hole and long wavelength.

5. Conclusion

We investigated the propagation properties of complex refractive-index core PCF with the supercell method. The validity of the approach is shown by comparison of the numerical results with those reported earlier on a step complex refractive-index profile. The imaginary part of the electric field results in wave-front distortion in the complex refractive-index profile PCF, which means that there is a power flow out or into the doped region according to the sign of the imaginary part of the refractive index. A simple formula is proposed for calculating gain or loss coefficient; with this formula the complicated process for solving the complex eigenvalue equation can be avoided. It is shown that the gain or loss coefficients are critically dependent on the power confinement factor and the mode index when the refractive-index profile is fixed. The fact that the numerical results obtained by the approximation formula are in good agreement with the full-vectorial results shows the validity of the formula.

Acknowledgments

This research has been supported by the Foundation of Beijing Jiaotong University (grant NJTU PD238, PD241), China.

References and links

1.

A. Reisinger, “Characteristics of optical guided modes in lossy waveguides,” Appl. Opt. 12, 1015–1025 (1973). [CrossRef] [PubMed]

2.

J. E. Sader, “Method for analysis of complex refractive index profile fibers,” Opt. Lett. 15, 107–109 (1990). [CrossRef]

3.

E. K. Sharma, Mukesh. P. Singh, and A. Sharma, “Variational analysis of optical fibers with loss or gain,” Opt. Lett. 18, 2096–2098 (1993). [CrossRef] [PubMed]

4.

Sunanda and E. K. Sharma, “Field variational analysis for modal gain in erbium-doped fiber amplifiers,” J. Opt. Soc. Am. B , 16, 1344–1347 (1999). [CrossRef]

5.

R. Singh Sunanda and E. Khular Sharma, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,” IEEE J. Quantum Electron. 37, 635–640 (2001). [CrossRef]

6.

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

7.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419(2000). [CrossRef] [PubMed]

8.

W. J. Wadsworth, J.C. Knight, W. H. Reeves, P.S.J. Russell, and J. Arriaga, “Yb3+-doped photonic crystal fibre laser, ” Electronics Lett. 36, 1452–1454 (2000) [CrossRef]

9.

K. G. Hougaard, J. Broeng, and A. Bjarklev, “Low pump power photonic crystal fibre amplifiers,” Electron. Lett. , 39, 599–600 (2003) [CrossRef]

10.

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tunnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express 11, 818–823 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-818 [CrossRef] [PubMed]

11.

Yu-Li Hsueh, E. S.T. Hu, M. E. Marhic, and G. Kazovsky, “Opposite-parity orthonormal function expansion for efficient full-vectorial modeling of holey optical fibers,” Opt. Lett. 28, 1188–1190 (2003). [CrossRef] [PubMed]

12.

T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999). [CrossRef]

13.

W. Zhi, R.G. Bin, L.S. Qin, and J. S. Sheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract. cfm? URI=OPEX-11-9-980. [CrossRef] [PubMed]

14.

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310 [CrossRef] [PubMed]

15.

W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, “Compact supercell method based on opposite parity for Bragg fibers,” Opt. Express 11, 3542–3549 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542 [CrossRef] [PubMed]

16.

A.W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

17.

B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre,“Microstructured optical fibers: where’s the edge?” Opt. Express 10, 1285–1290 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285 [CrossRef] [PubMed]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2310) Fiber optics and optical communications : Fiber optics
(230.7370) Optical devices : Waveguides

ToC Category:
Research Papers

History
Original Manuscript: January 16, 2004
Revised Manuscript: February 25, 2004
Published: March 22, 2004

Citation
Ren Guobin, Wang Zhi, Lou Shuqin, Liu Yan, and Jian Shuisheng, "Full-vectorial analysis of complex refractive index photonic crystal fibers," Opt. Express 12, 1126-1135 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-6-1126


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References

  1. A. Reisinger, �??Characteristics of optical guided modes in lossy waveguides,�?? Appl. Opt. 12, 1015-1025 (1973). [CrossRef] [PubMed]
  2. J.E. Sader, �??�??Method for analysis of complex refractive index profile fibers,�??�?? Opt. Lett. 15, 107-109 (1990). [CrossRef]
  3. E. K. Sharma, Mukesh. P. Singh, and A. Sharma, �??Variational analysis of optical fibers with loss or gain,�?? Opt. Lett. 18, 2096-2098 (1993). [CrossRef] [PubMed]
  4. Sunanda and E. K. Sharma, �??Field variational analysis for modal gain in erbium-doped fiber amplifiers,�?? J. Opt. Soc. Am. B, 16, 1344�??1347 (1999). [CrossRef]
  5. R. Singh Sunanda, and E. Khular Sharma, �??Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,�?? IEEE J. Quantum Electron. 37, 635-640 (2001). [CrossRef]
  6. T. A. Birks, J. C. Knight, and P. St. J. Russell, �??Endlessly single-mode photonic crystal fiber,�?? Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  7. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, �??An all-dielectric coaxial waveguide,�?? Science 289, 415-419(2000). [CrossRef] [PubMed]
  8. W. J. Wadsworth, J.C. Knight, W. H. Reeves, P.S.J. Russell, and J Arriaga, "Yb3+-doped photonic crystal fibre laser," Electronics Lett. 36, 1452 �?? 1454 (2000) [CrossRef]
  9. K. G. Hougaard, J. Broeng, and A. Bjarklev, �??Low pump power photonic crystal fibre amplifiers,�?? Electron. Lett., 39, 599-600 (2003) [CrossRef]
  10. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tunnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, "High-power air-clad large-mode-area photonic crystal fiber laser," Opt. Express 11, 818-823 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-818">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-818</a> [CrossRef] [PubMed]
  11. Yu-Li Hsueh, E. S.T. Hu, M. E. Marhic, and G. Kazovsky, �??Opposite- arity orthonormal function expansion for efficient full-vectorial modeling of holey optical fibers,�?? Opt. Lett. 28, 1188-1190 (2003). [CrossRef] [PubMed]
  12. T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P. J. Bennett, �??Holey optical fibers: an efficient modal model,�?? J. Lightwave Technol. 17, 1093-1102 (1999). [CrossRef]
  13. W. Zhi, R.G. Bin, L.S. Qin, and J. S. Sheng, �??Supercell lattice method for photonic crystal fibers,�?? Opt. Express 11, 980-991 (2003), <a href=" http://www.opticsexpress.org/ abstract. cfm? URI=OPEX-11-9-980.">http://www.opticsexpress.org/ abstract. cfm? URI=OPEX-11-9-980.</a> [CrossRef] [PubMed]
  14. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, "Mode classification and degeneracy in photonic crystal fibers," Opt. Express 11, 1310-1321 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310</a> [CrossRef] [PubMed]
  15. W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, "Compact supercell method based on opposite parity for Bragg fibers," Opt. Express 11, 3542-3549 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542</a> [CrossRef] [PubMed]
  16. A.W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).
  17. B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, "Microstructured optical fibers: where�??s the edge?" Opt. Express 10, 1285-1290 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285</a> [CrossRef] [PubMed]

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