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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 18 — Aug. 30, 2010
  • pp: 18671–18684
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Broadband transmission in hollow-core Bragg fibers with geometrically distributed multilayered cladding

Dora Juan Juan Hu, Gandhi Alagappan, Yong-Kee Yeo, Perry Ping Shum, and Ping Wu  »View Author Affiliations


Optics Express, Vol. 18, Issue 18, pp. 18671-18684 (2010)
http://dx.doi.org/10.1364/OE.18.018671


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Abstract

For the first time, the quasiperiodic Bragg fibers with geometrically distributed multilayered cladding are proposed and analyzed. We demonstrate that hollow-core Bragg fibers with quasiperiodic dielectric multilayer cladding can achieve low loss transmission over a broadband wavelength range of more than an octave (from 0.81 μ m to 1.7 μ m ). The periods of the Bragg blocks follows a geometrical progression with a common ratio r<rc , where rc is defined as the critical ratio of the periods of two adjacent Bragg blocks. The arrangement of the quasiperiodic cladding can significantly modify the characteristics of the fiber, leading to a broadening of the guiding range compared to a hollow Bragg fiber with uniform periodic multilayer cladding structure. In general, a larger r value results in a broader guiding range. More Bragg blocks in the cladding and more unit cells in each Bragg block lead to a lower fiber modal loss.

© 2010 OSA

1. Introduction

The hollow-core Bragg fiber which was firstly introduced in 1978 [1

1. P. Yeh, A. Yariv, and E. Maron, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68(9), 1196–1201 (1978). [CrossRef]

], has attracted much interest in recent years. These fibers are potentially useful for high-power laser systems as the nonlinear effects are significantly suppressed [2

2. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420(6916), 650–653 (2002). [CrossRef] [PubMed]

]. They are also promising in applications such as sensing, telecommunication, medicine and surgery [3

3. D. Passaro, M. Foroni, F. Poli, A. Cucinotta, S. Selleri, J. Laegsgaard, and A. O. Bjarklev, “All-Silica Hollow-Core Microstructured Bragg Fibers for Biosensor Application,” IEEE Sens. J. 8(7), 1280–1286 (2008). [CrossRef]

5

5. G. Vienne, Y. Xu, C. Jakobsen, H.-J. Deyerl, J. Jensen, T. Sorensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express 12(15), 3500–3508 (2004). [CrossRef] [PubMed]

]. The hollow-core guidance is due to the bandgap effect resulting from a periodic arrangement of two alternating materials stacked along the radial direction in the cladding. There were attempts to enlarge the spectral range of the hollow core guidance, for instance, a chirped multilayer Bragg fiber has been proposed with the help of genetic algorithm calculation [6

6. A. Husakov and J. Herrmann, “Chirped multilayer hollow waveguides with broadband transmission,” Opt. Express 17(5), 3025–3024 (2009). [CrossRef]

]. It was shown that the hetero-structure of the Bragg layers leads to a stretching of the transmission range at the expense of higher propagation loss. Recently, Yan et al. reported a low loss hollow fiber incorporating metal-wire based metamaterial at infrared frequencies [7

7. M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express 17(17), 14851–14864 (2009). [CrossRef] [PubMed]

].

2. Photonic bandgap of quasiperiodic multilayers

A schematic diagram of a Bragg fiber cross-section is shown in Fig. 1
Fig. 1 (a) Schematic diagram of the hollow-core Bragg fiber cross-section. (b) The quasiperiodic cladding is formed by m numbers of Bragg blocks; there are N unit cells in each block, i.e. N pairs of two dielectric materials alternating in the block. The period for k-th Bragg block is pk. The ratio of the periods in two adjacent Bragg block is r = pk/ pk-1. (c) An example of a quasiperiodic Bragg fiber with m = 2, N = 4, r = 1.5, and (d) its refractive index profile in radial direction.
. The cladding of the fiber is formed by alternating two different dielectric materials in a quasiperiodic arrangement. We assume that the refractive indices of the materials in the cladding are nH, nL (nH>nL) and the refractive index of the hollow core is n0 = 1. There are m numbers of Bragg blocks and the period for k-th block is pk (pk-1< pk< pk + 1). N unit cells exist in each block, i.e. N pairs of two dielectric materials are alternating in the block. The periods of the Bragg blocks are arranged in a geometrical distribution, i.e. the common ratio of the periods of two adjacent blocks r = pk + 1/ pk. For instance, the fiber structure with m = 2, N = 4 and its refractive index profile in radial direction are shown in Figs. 1(c) and 1(d).

The thickness of each dielectric layer is governed by the quarter-wave condition along the light line of air [9

9. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9(13), 748–779 (2001). [CrossRef] [PubMed]

]. First of all, the effective transverse wavelength λt,i in i-th layer can be estimated by
kt,i2=ki2kz2
(1)
where kt,i=2π/λt,i, ki=2π/λni, and kz=2π/λneff. Here λ is the targeted operating wavelength in middle of the bandgap, ni is the refractive index of the material in i-th layer, and neff is the effective mode index which is less than but very close to n0. For Bragg fibers with a large core size compared to the wavelength which, most of the mode energy resides in air and it is reasonable approximation to use neff=n0. The optimum layer thickness can be computed as
li=λt,i4+sλt,i2s=0,1,2...
(2)
When s = 0, the fundamental Bragg condition is satisfied. Other s values correspond to higher-order Bragg conditions. We chose s = 0 in this work, so that we have an explicit expression to estimate lH/L, i.e. Equation (3) shows the optimal layer thickness of the two materials with high/low refractive indices nH/L in the bi-layer PC cladding.

lH/L=λ4nH/L2n02=λ4nH/L21
(3)

The bandgap generated by the multilayer cladding in the Bragg fiber, can be approximated to that of a planar multilayer structure. The bandgap of the 1D PC with an infinite number of unit cells can be calculated by the plane wave expansion method [10

10. J. D. Joannapolous, R. D. Meade, and J. N. Winn, Photonic Crystals—Molding the Flow of Light (Princeton University Press, 1995).

] or transfer matrix method [11

11. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience 2003)

]. The 1D PC can be formed by finite number of unit cells, or a series of PC blocks comprising a finite number of unit cells. In order to treat the structure in a more generic manner, we rely on TMM technique. The transfer matrix for the bilayer of two materials with high/low refractive index can be written as [12

12. S. J. Orfanidis, Electromagnetic Waves and Antennas (Rutgers University, 2008)

]:
MT=11ρT2[ej(δH+δL)ρT2ej(δHδL)2jρTejδHsinδL2jρTejδHsinδLej(δH+δL)ρT2ej(δHδL)]
(4)
where ρT=nHTnLTnHT+nLT, δH=kHlHcosθH, δL=kLlLcosθL, kH=2π/λnH, kL=2π/λnL, and
nHT={nHcosθHnHcosθHnLT={nLcosθLnLcosθL TM polarizationTE polarization
(5)
θH and θL are angle of refraction in the layer of high/low refractive index, and they can be obtained by using Snell’s law,
sinθH/L=n0nH/Lsinθinc=sinθincnH/L
(6)
θinc is the angle of incidence, which can vary over 0°θinc90°. It is to be noted that cosθH/L can be expressed as 1sin2θH/L or 1sin2θincnH/L2. The eigenvalues of MT, i.e. λ±=ejKl, can be either both real-valued or both unit-magnitude complex-valued. K is the Bloch wavenumber, where K=cos1(a)/l, l=lH+lL, a is the trace of MT:

a=cos(δH+δL)ρT2cos(δHδL)1ρT2
(7)

The multilayer structure is primarily reflecting if K is imaginary and the eigenvalues λ± are real-valued, and it is primarily transmitting if K is real and the eigenvalues λ± are pure phases. Therefore, the condition a=1 determines the bandedge wavelengths of the high reflection bands, which is equivalent to:
cos2(δH+δL2)=ρT2cos2(δHδL2)
(8)
Notice that we can define the parameters L±=nHlHcosθH±nLlLcosθL, so that
δH±δL2=(kHlHcosθH±kLlLcosθL)2=kL±2
(9)
where k=2π/λ, is free space wavenumber. Taking the square root of Eq. (8) on both sides, we have
cos(πλ1L+)=|ρT|cos(πλ1L)cos(πλ2L+)=|ρT|cos(πλ2L)
(10)
Here λ1 and λ2 refer to the lower and upper bandgap edge wavelength respectively. The coefficients L± and ρT are dependent on polarization state (TE or TM) and the incidence angleθinc. Equation (10) is the general expression to obtain the solution of bandgap edge wavelengths for all incidence angles, i.e. 0°θinc90°, and for both TM and TE polarizations. They can be solved numerically.

Note that at grazing incidence or incidence angle near 90°, which is the case for the propagating light incident on the core/cladding interface inside the Bragg fiber, if we employ the fiber design of quarter-wave condition along the light line of air [9

9. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9(13), 748–779 (2001). [CrossRef] [PubMed]

], it is fairly accurate to set L=0. Therefore, we can have an approximate analytical solution for Eq. (10). Note that the lower and upper bandgap edges of the k-th PC block with period of pkin the Bragg fiber with hetero-structured cladding shown in Fig. 1 can be denoted as λk, and λk,+. The approximate analytical solution is given in Eq. (11). On the other hand, L0 for other θinc values, no analytical solution is available and Eq. (10) has to be solved numerically.
λk,=λ1=πL+cos1(|ρT|)=πL+π/2+sin1(|ρT|)λk,+=λ2=πL+cos1(|ρT|)=πL+π/2sin1(|ρT|)
(11)
In order to verify the accuracy of the Eq. (10), we also consider the special case with the same 1D PC structure discussed in [8

8. G. Alagappan and P. Wu, “Geometrically distributed one-dimensional photonic crystals for light-reflection in all angles,” Opt. Express 17(14), 11550–11557 (2009). [CrossRef] [PubMed]

]. When θinc=0°, and the 1D PC is designed with quarter wavelength thickness at normal incidence on the planar multilayer structure, i.e.
lH=pknLnH+nLlL=pknHnH+nL
(12)
pk is the period of the PC. We can deriveL+=nHlH+nLlL, L=0, ρT=nHnLnH+nL. We observe that Eq. (11) is still the analytical solution in this special case. It is also equivalent to the expression derived in [8

8. G. Alagappan and P. Wu, “Geometrically distributed one-dimensional photonic crystals for light-reflection in all angles,” Opt. Express 17(14), 11550–11557 (2009). [CrossRef] [PubMed]

] at θinc=0°. Having obtained the expressions for bandgap edge wavelengths of the 1D PC, we are ready to employ the idea of creating large bandgap by hetero-structured PC blocks with periods in geometrical distribution in the Bragg fiber. The critical common ratio, rc, would lead to the threshold condition of bandgap of adjacent blocks overlapping, i.e. λk1,+=λk,. From Eq. (11), rc is defined as
rc=π/2+sin1(ρT)π/2sin1(ρT)
(13)
Equation (13) is the general expression of the critical common ratio for all incidence angles, and for both TE and TM polarizations. The stretched bandgap of the hetero-structured cladding, is determined by the lower band edge wavelength of the first Bragg block λ1,, and the upper band edge wavelength of the last Bragg block λm,+. The common ratio r, is r=(pm/p1)1/m1. The condition, r < rc, has to be satisfied in the hetero-structured cladding design [8

8. G. Alagappan and P. Wu, “Geometrically distributed one-dimensional photonic crystals for light-reflection in all angles,” Opt. Express 17(14), 11550–11557 (2009). [CrossRef] [PubMed]

].

Figure 2
Fig. 2 rc as a function of incidence angel θinc and polarization. The refractive index nL = 1.625, nH = 2.896, 3.6, 4.6 respectively.
shows the critical common ratio rc, for the quasiperiodic cladding with materials of refractive index nL = 1.625, nH = 2.896, 3.6, and 4.6 respectively. There are a few observations from Fig. 2. First of all, the rc values for TM polarization are always smaller than that for TE polarization. Second, the minimum rc value occur at θinc=0°, 90° for TE and TM polarization respectively. This is because for TE polarization, the bandgap is getting broader with increasing θinc, whereas for TM polarization, the light line lies above the point where the TM bandgap becomes zero [8

8. G. Alagappan and P. Wu, “Geometrically distributed one-dimensional photonic crystals for light-reflection in all angles,” Opt. Express 17(14), 11550–11557 (2009). [CrossRef] [PubMed]

]. Third, the rc curves are shifted to higher values when the refractive index difference between the two materials are larger.

Next, we set nH=2.896, nL=1.625, and assume the Bragg cladding has infinite number of unit cells of the two materials. The projected cladding bandgap map, which is approximated by the planar Bragg stack in the limit of large core radius [9

9. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9(13), 748–779 (2001). [CrossRef] [PubMed]

], is calculated by TMM technique and shown in Fig. 3
Fig. 3 Projected band structure of the planar periodic multilayer structure. The refractive index of the two materials in the Bragg cladding nL = 1.625, nH = 2.896. The solid line is the light line. The gray and black regions correspond to TE allowed band, and TM allowed band respectively. The white regions correspond to the total bandgaps.
. β is the propagation constant. The TE allowed band and TM allowed band are plotted in gray and black regions, and the total bandgap is plotted in the white regions. It is clear that TM bandgap is narrower than TE bandgap. The omni-directional reflection at all angles and for both polarizations can occur within the range plotted by dash-dot lines.

The bandgap map of the quasiperiodic cladding cannot be obtained by assuming the structure has the infinite unit cells of uniform periodicity. Instead, by using transfer matrix method (TMM), we can treat the realistic hetero-structured cladding with finite layers, consisting of several Bragg blocks with different periods. Figure 4
Fig. 4 The reflection R versus wavelength λ and incidence angle θ for both TE and TM polarization. The Bragg cladding has the following parameters: nL = 1.625, nH = 2.896, m = 5, N = 6, (a) periodic 1D PC, p1=p2==pm=287nm. (b) quasiperiodic Bragg cladding, p1=287 nm, pm/p1=1.2.
plots the reflection factor distribution as a function of wavelength and incidence angle for both TE and TM polarizations. The quasiperiodic Bragg fiber has parameters m = 5, N = 6, nH=2.896, nL=1.625. Figure 4(a) is plotted with p1=p2==pm=287nm, corresponding to the periodic 1D PC, with 30 unit cells. Figure 4(b) is plotted with p1=287 nm, pm/p1=1.2, corresponding to the quasiperiodic Bragg cladding. Notice that in the fundamental bandgap for both polarizations, the wavelength range enclosed by the dash-dot lines shown in (a) for illustration, can be used for omni-directional reflection for all incidence angles and for both polarizations. A broadened bandgap is observed in the quasiperiodic Bragg cladding.

Figure 5(a)
Fig. 5 (a) The bandedge wavelengths of the fundamental bandgap for the periodic 1D PC (upper), and the quasiperiodic 1D PC (lower). There is obvious broadening in omni-reflection range marked by the solid-dash lines. (b) The transmission at incidence angle θ=90° for TE (dashed) and TM (solid) polarizations.
plots the bandedge wavelengths of the fundamental bandgap for periodic 1D PC (upper), quasiperiodic 1D PC (lower). The omni-reflection range is marked by the solid-dash lines. The transmission spectrum at incidence angle θ=90° for TE (dashed) and TM (solid) polarizations are plotted in Fig. 5(b). Notice that the lower bandgap edge wavelength is determined by the first block which has the same period in both periodic PC and quasiperiodic PC, so that it is the same in both cases. The upper bandgap edge wavelength is determined by the last block, so that we can observe an apparent red shift in quasiperiodic PC for both polarizations. Therefore, for each polarization states, the bandgap range is broadened in quasiperiodic 1D PC.

3. Guided modes in the quasiperiodic Bragg fibers

It should be noted that the hollow-core Bragg fiber consisting of PES and As2Se3 in the cladding can be fabricated by preform preparation and fiber drawing technique [1

1. P. Yeh, A. Yariv, and E. Maron, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68(9), 1196–1201 (1978). [CrossRef]

]. Recently, a hollow-core Bragg fiber with all-solid multilayer composite meso-structure having as many as 35 periods, i.e. 70 layers is reported [15

15. K. Kuriki, O. Shapira, S. D. Hart, G. Benoit, Y. Kuriki, J. F. Viens, M. Bayindir, J. D. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express 12(8), 1510–1517 (2004). [CrossRef] [PubMed]

]. Therefore, we believe that the proposed hollow-core quasiperiodic Bragg fiber with 60 layers in the cladding is within the capability of current fabrication technology.

4. Conclusions

Acknowledgement

References and links

1.

P. Yeh, A. Yariv, and E. Maron, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68(9), 1196–1201 (1978). [CrossRef]

2.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420(6916), 650–653 (2002). [CrossRef] [PubMed]

3.

D. Passaro, M. Foroni, F. Poli, A. Cucinotta, S. Selleri, J. Laegsgaard, and A. O. Bjarklev, “All-Silica Hollow-Core Microstructured Bragg Fibers for Biosensor Application,” IEEE Sens. J. 8(7), 1280–1286 (2008). [CrossRef]

4.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998). [CrossRef] [PubMed]

5.

G. Vienne, Y. Xu, C. Jakobsen, H.-J. Deyerl, J. Jensen, T. Sorensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express 12(15), 3500–3508 (2004). [CrossRef] [PubMed]

6.

A. Husakov and J. Herrmann, “Chirped multilayer hollow waveguides with broadband transmission,” Opt. Express 17(5), 3025–3024 (2009). [CrossRef]

7.

M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express 17(17), 14851–14864 (2009). [CrossRef] [PubMed]

8.

G. Alagappan and P. Wu, “Geometrically distributed one-dimensional photonic crystals for light-reflection in all angles,” Opt. Express 17(14), 11550–11557 (2009). [CrossRef] [PubMed]

9.

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9(13), 748–779 (2001). [CrossRef] [PubMed]

10.

J. D. Joannapolous, R. D. Meade, and J. N. Winn, Photonic Crystals—Molding the Flow of Light (Princeton University Press, 1995).

11.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience 2003)

12.

S. J. Orfanidis, Electromagnetic Waves and Antennas (Rutgers University, 2008)

13.

Published Database of optical constants by Photonic Bandgap Fibers & Devices Group [Online]. Available: http://mit-pbg.mit.edu/pages/database.html

14.

G. Xu, W. Zhang, Y. Huang, and J. Peng, “Loss characteristics of single-HE11-mode Bragg fiber,” J. Lightwave Technol. 25(1), 359–366 (2007). [CrossRef]

15.

K. Kuriki, O. Shapira, S. D. Hart, G. Benoit, Y. Kuriki, J. F. Viens, M. Bayindir, J. D. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express 12(8), 1510–1517 (2004). [CrossRef] [PubMed]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(230.7370) Optical devices : Waveguides
(230.5298) Optical devices : Photonic crystals

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: January 13, 2010
Revised Manuscript: January 24, 2010
Manuscript Accepted: January 30, 2010
Published: August 18, 2010

Citation
Dora Juan Juan Hu, Gandhi Alagappan, Yong-Kee Yeo, Perry Ping Shum, and Ping Wu, "Broadband transmission in hollow-core Bragg fibers with geometrically distributed multilayered cladding," Opt. Express 18, 18671-18684 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-18671


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References

  1. P. Yeh, A. Yariv, and E. Maron, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68(9), 1196–1201 (1978). [CrossRef]
  2. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420(6916), 650–653 (2002). [CrossRef] [PubMed]
  3. D. Passaro, M. Foroni, F. Poli, A. Cucinotta, S. Selleri, J. Laegsgaard, and A. O. Bjarklev, “All-Silica Hollow-Core Microstructured Bragg Fibers for Biosensor Application,” IEEE Sens. J. 8(7), 1280–1286 (2008). [CrossRef]
  4. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998). [CrossRef] [PubMed]
  5. G. Vienne, Y. Xu, C. Jakobsen, H.-J. Deyerl, J. Jensen, T. Sorensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express 12(15), 3500–3508 (2004). [CrossRef] [PubMed]
  6. A. Husakov and J. Herrmann, “Chirped multilayer hollow waveguides with broadband transmission,” Opt. Express 17(5), 3025–3024 (2009). [CrossRef]
  7. M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express 17(17), 14851–14864 (2009). [CrossRef] [PubMed]
  8. G. Alagappan and P. Wu, “Geometrically distributed one-dimensional photonic crystals for light-reflection in all angles,” Opt. Express 17(14), 11550–11557 (2009). [CrossRef] [PubMed]
  9. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9(13), 748–779 (2001). [CrossRef] [PubMed]
  10. J. D. Joannapolous, R. D. Meade, and J. N. Winn, Photonic Crystals—Molding the Flow of Light (Princeton University Press, 1995).
  11. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience 2003)
  12. S. J. Orfanidis, Electromagnetic Waves and Antennas (Rutgers University, 2008)
  13. Published Database of optical constants by Photonic Bandgap Fibers & Devices Group [Online]. Available: http://mit-pbg.mit.edu/pages/database.html
  14. G. Xu, W. Zhang, Y. Huang, and J. Peng, “Loss characteristics of single-HE11-mode Bragg fiber,” J. Lightwave Technol. 25(1), 359–366 (2007). [CrossRef]
  15. K. Kuriki, O. Shapira, S. D. Hart, G. Benoit, Y. Kuriki, J. F. Viens, M. Bayindir, J. D. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express 12(8), 1510–1517 (2004). [CrossRef] [PubMed]

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