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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 9 — May. 5, 2003
  • pp: 1039–1049
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Asymptotic analysis of silicon based Bragg fibers

Yong Xu, Amnon Yariv, James G. Fleming, and Shawn-Yu Lin  »View Author Affiliations


Optics Express, Vol. 11, Issue 9, pp. 1039-1049 (2003)
http://dx.doi.org/10.1364/OE.11.001039


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Abstract

We developed an asymptotic formalism that fully characterizes the propagation and loss properties of a Bragg fiber with finite cladding layers. The formalism is subsequently applied to miniature air-core Bragg fibers with Silicon-based cladding mirrors. The fiber performance is analyzed as a function of the Bragg cladding geometries, the core radius and the material absorption. The problems of fiber core deformation and other defects in Bragg fibers are also addressed using a finite-difference time-domain analysis and a Gaussian beam approximation, respectively.

© 2003 Optical Society of America

1. Introduction

Bragg fibers, which are composed of a low index core (possibly air) surrounded by alternating annular layers with different dielectric constants, were first proposed by Yeh et al. [1

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

] in 1978. A schematic of a Bragg fiber is shown in Fig. 1. The possibility of guiding light in air by Bragg fibers [2

2. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. 17, 2039–2041 (1999). [CrossRef]

, 3

3. S. G. Johnson et al., “Low loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express 9, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

, 4

4. Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. 25, 1756–1758 (2000). [CrossRef]

, 5

5. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. 20, 428–440 (2002). [CrossRef]

, 6

6. J. Marcou, F. Brechet, and P. Roy, “Design of weakly guiding Bragg fibres for chromatic dispersion shifting towards short wavelengths,” J. Opt. A , 3, S144–S153 (2001). [CrossRef]

, 7

7. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express, 10, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

, 8

8. J. G. Fleming, S. Y. Lin, and R. Hadley, in Proceeding of the solid-state sensor actuator and microsystems workshop, p.p. 173, (Hilton Head, S.C., 2002).

] or photonic crystal fibers [9

9. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

] has recently attracted a lot of attention. Omni-fibers [2

2. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. 17, 2039–2041 (1999). [CrossRef]

, 3

3. S. G. Johnson et al., “Low loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express 9, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

], which are Bragg fibers with very large cladding indices contrast, have been experimentally demonstrated. Miniature Bragg fibers with silicon based cladding materials such as Si and Si3N4, were recently fabricated by J. Fleming et al. [8

8. J. G. Fleming, S. Y. Lin, and R. Hadley, in Proceeding of the solid-state sensor actuator and microsystems workshop, p.p. 173, (Hilton Head, S.C., 2002).

] using combinations of etching and CVD (chemical vapor deposition). The silicon based miniature Bragg fibers are developed for integrated optics applications such as thermo-optical switches and BioMEMS devices, which require properties quite different from other types of Bragg fibers intended for telecommunication applications. For example, Bragg fibers for integrated optics applications can tolerate propagation loss of the order of dB/cm, rather than <dB/km demanded by telecommunication fibers.

Fig. 1. Schematics of a Bragg fiber. In this paper, we assume the low index core is air (nco =1) and has radius of rco . The refractive index and the thickness of the cladding layers are respectively n 1, L 1, and n 2, L 2. The dashed line represents the interface between the “exact solution” region and the “asymptotic solution” region.

Due to the fabrication processes, it is difficult to realize the miniature Bragg fibers with perfect cylindrical symmetry. Using the finite difference time domain (FDTD) simulations, we investigate a deformed Bragg fiber and demonstrate that the Bragg fiber dispersion is insensitive to the air core deformation. However, the guided modes in the deformed fiber are no longer cylindrically symmetric, which may lead to a higher propagation loss. Due to the CVD processes, the Bragg fibers also have gas inlet ports along the fibers at intervals about hundreds of microns. Drawing analogy with free space diffraction of a Gaussian beam, we estimate the excessive loss associated with the presence of the gas inlet ports.

2. Asymptotic analysis

It can be shown that the guided Bragg fiber mode can be determined from the four electromagnetic field components Ez , H θ, Hz , and E θ [1

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

]. From symmetry consideration alone, we can label each guided mode according to its angular frequency ω, propagations constant β, and the “azimuthal quantum number” m, with the functional dependence of exp(iωt-iβz)exp(imθ). In the nth dielectric layer, the radial dependence of the guided mode can be written as [1

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

]:

[EzHθHzEθ]=Mn(r)[AnBnCnDn],
(1)

where Mn (r) is a 4×4 matrix, and the field amplitude constants An, Bn, Cn , and Dn are constant within the nth dielectric layer.

In the asymptotic formalism, we separate the dielectric layers of the Bragg fiber into two groups, where the exact solutions are used for the layers of the inner region, while the asymptotic approximation of the exact solutions is used for the outer region. (see Fig. 1). In the “exact solution” region, the 4×4 matrix Mn (r) is defined as [1

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

]:

Mn(r)=[Jm(knr)Ym(knr)00iωε0εnknJm(knr)iωε0εnknYm(knr)mβkn2rJm(knr)mβkn2rYm(knr)00Jm(knr)Ym(knr)mβkn2rJm(knr)mβkn2rYm(knr)iωμ0knJm(knr)iωμ0knYm(knr)],
(2)

where ε n is the dielectric constant of the nth layer, and kn is

kn=εnω2c2β2.
(3)

It should be emphasized that εn can be a complex number, with its imaginary part accounting for the material absorption in the nth dielectric layer. If the nth dielectric layer belongs to the “asymptotic solution” region, the 4×4 matrix Mn (r) take the form of [4

4. Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. 25, 1756–1758 (2000). [CrossRef]

, 5

5. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. 20, 428–440 (2002). [CrossRef]

]:

Mn(r)=1r[eiknreiknr00ωε0εnkneiknrωε0εnkneiknr0000eiknreiknr00ωμ0kneiknrωμ0kneiknr].
(4)

We notice that Eq. (4) is block-diagonal. As a result, the TM modes with field components Ez and H θ, are decoupled from the TE modes which consist of Hz and E θ.

The four field amplitude constants (A n+1, B n+1, C n+1, and D n+1) in the (n+1)th layer can be derived from the corresponding quantities in the adjacent nth layer by requiring the continuity of Ez, H θ, Hz , and H θ across the interface between the two dielectric layers [5

5. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. 20, 428–440 (2002). [CrossRef]

]:

[An+1Bn+1Cn+1Dn+1]=[Mn+1(ρn)]1Mn(ρn)[AnBnCnDn],
(5)

where ρn is the radius of the interface between the nth dielectric layer and the (n+1)th dielectric layer, Mn and M n+1 are defined according to Eq. (2) or Eq. (4), depending on whether the dielectric layer belongs to the inner “exact solution” region or to the outer “asymptotic solution” region. We can apply Eq. (5) iteratively and relate the field amplitude constants outside of the Bragg fiber (Aout, Bout, Cout , and Dout ) to the corresponding quantities within the low index core (Acore, Bcore, Ccore , and Dcore ) via a 4×4 transfer matrix:

[AoutBoutCoutDout]=[t11t12t13t14t21t22t23t24t31t32t33t34t41t42t43t44][AcoreBcoreCcoreDcore].
(6)

We require that the electromagnetic field must be finite within the low index core and that the electromagnetic field outside of the fiber consists of only the outgoing radiation field. Such boundary conditions lead to Bcore =Dcore =0 and Bout =Dout =0, respectively. Substituting these two requirements into Eq. (6), we obtain:

T[AcoreCcore]=0,T=[t21t23t41t43].
(7)

The complex propagation constant β of any guided mode is given by the condition of det(T)=0, and the field distribution can be obtained from the eigenvector [Acore Bcore ] that corresponds to the zero eigenvalue of the 2×2 matrix T.

3. Guiding in silicon-based miniature Bragg fibers

3.1 Dispersion and loss in a specific Bragg fiber geometry

We first apply the asymptotic formalism to analyze the dispersion and propagation loss of the guided modes in a Bragg fiber as reported by J. Fleming et al. in Ref. [8

8. J. G. Fleming, S. Y. Lin, and R. Hadley, in Proceeding of the solid-state sensor actuator and microsystems workshop, p.p. 173, (Hilton Head, S.C., 2002).

]. The fiber consists of four pairs of Si/Si3N4 mirror stacks, with Si being the innermost cladding layer. For the polysilicon layers, we use a refractive index n 1=3.5 and thickness L1=0.11µm. The refractive index of the Si3N4 layer is n 2=2.0 and its thickness is L2=0.21µm. The air core radius is rco =7.5µm. The bulk absorption in the polysilicon layer is estimated to be about 10dB/cm. In our analysis, we assume that the “exact solution” region consists of the five innermost dielectric layers, whereas the rest of the structure is described by the asymptotic solutions. With a large air core radius, the Bragg fiber supports multiple guided modes, which are labeled according to their “azimuthal quantum number” m followed by the number of zeros of the electromagnetic field in the air core. We limit ourselves to the first two TE modes (TE01 and TE02), the fundamental TM mode (TM01), and the fundamental mixed polarization mode (HE11).

In Fig. 2(a), we plot the E θ field of the TE01 mode at the wavelength of 1.65µm. We notice that the E θ component is zero at the interface between the air core and the Si layer, which has been explained by drawing an analogy to perfect metal [3

3. S. G. Johnson et al., “Low loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express 9, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

, 7

7. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express, 10, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

]. From a more fundamental point of view, the condition of E θ=0 at the air/Si interface leads to maximum reduction of the E θ field amplitude within the Si cladding layer [1

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

], which in turn results in minimum propagation loss. Combining the condition of E θ=0 at r=rco with the fact that TE modes take the form of J1(ω2c2β2r) within the air core, we find

neffTE0i=1(x1iλ2πrco)2,
(8)

where x 1i is the ith zero of the first order Bessel functions, i.e. J 1(x 1i)=0. This relation has been given in Ref. [7

7. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express, 10, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

]. Using the asymptotic approach, we calculate the effective indices (defined as βr c/ω) of the TE01, TE02, TM01, and HE11 modes and plot the results in Fig. 2(b). The effective indices of the TE01 and TE02 modes, predicted by Eq. (8), are also shown in Fig. 2(b) as solid lines. It is clear that Eq. (8) is in excellent agreement with the calculated dispersion of the TE modes.

Fig. 2. The asymptotic results of an air-core Bragg fiber with rco =7.5µm, n 1=3.5 and L1=0.11µm, n 2=2.0 and L2=0.21µm. (a) The E θ component of the TE01 mode at λ=1.65µm. (b) The effective indices of the TE01, TE02, TM01, and HE11 modes. (c) The loss of the TE01, TE02, TM01, and HE11 modes. In (a), (b) and (c), we set the absorption loss in the Si layer to be 10dB/cm. (d) The loss of the TE01 mode at λ=1.55µm with Si layer loss varying from 0dB/cm to 10dB/cm. The dashed line is a linear fit of the asymptotic results.

Fig. 3. The loss of the TE01, TE02, TM01, and HE11 modes at λ=1.55µm. The refractive index and thickness of the cladding layers are the same as in Fig. 2. In (a) we choose the air core radius rco =8µm and vary the Bragg cladding pair number. The dashed lines are the fitting of the asymptotic results using Eq. (9). In (b), we use 4 pairs of Bragg cladding layers and vary the air core radius. The dashed lines are the fitting of the asymptotic results for the TM01 and HE11 modes using Eq. (11). The solid lines are the estimation given by Eq. (12).

3.2 Loss dependence on Bragg cladding pair number and air core radius

ModalLossΔN,
(9)

where Δ is a constant. From fitting the asymptotic results using Eq. (9), we find ΔTE01TE02=0.29, and ΔTM01HE11=0.43. The fitting results are shown in Fig. 3(a) as dashed lines, which are in excellent agreement with the asymptotic results. We notice that the TE modes have similar Δ parameters, whereas the mixed polarization mode and the TM mode share a different Δ parameter. This interesting phenomenon can be understood as a direct result of the decoupling of the TE and the TM components in the Bragg cladding layers. Mathematically speaking, when m≠0 the presence of the off-diagonal terms such as (mβ/kn2r)Jm (knr) and (mβ/kn2r)Ym (knr) in Eq. (2) mixes the TE component and the TM component in the “exact solution” region. On the other hand, in the “asymptotic solution” region, the 4×4 transfer matrix becomes block-diagonal, according to Eq. (4). As a consequence, the TE component and the TM component decay independently in the asymptotic region even for the mixed polarization modes. Since in general the TE component decays much faster than the TM component (see the following discussion), the propagation loss of the mixed polarization modes should share the characteristics of the TM modes.

ΔTE=εl1εh1,ΔnonTE=εl2(εh1)εh2(εl1),
(10)

where we have substituted the propagation constant β by ω/c. Due to the large core area, this is an excellent approximation, as can also be seen from Fig. 2(b). Using ε l =4.0 and εh=12.25 for the Si/Si3N4 cladding, Eq. (10) gives ΔTE=0.27 and Δnon-TE=0.40, quite close to the values of ΔTE=0.29 and Δnon-TE=0.43 obtained from fitting Fig. 3(a) using Eq. (9). The difference might be due to the slight deviation of the Bragg cladding from a quarter-wave stack at λ=1.55µm.

Modal Loss(1rco)α.
(11)

We find αTE01=3.61, αTE02=3.49, αTM01=2.20, and αHE11=2.64.

It has been demonstrated in Ref. [7

7. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express, 10, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

] that the loss behavior of the TE modes can be modeled by a 1D Bragg stack. In this picture, we can think of the ith TE mode as formed by photons zigzagging within the air core with incident angle π-θ, with θ=x 1iλ/2πrco according to Eq. (8). As shown in Fig. 4, the distance between two consecutive “bounces” should be 2rco /θ, assuming a small θ. Denoting the amplitude reflection coefficient of the Bragg cladding is ℛ, we have exp(-4β irco /θ)=|ℛ|2. For |ℛ|2 approaching unity, this relation can be rewritten in the dB unit as

Modal Loss=3.46×103×x1iλrco2(12)(dBcm),
(12)

Fig. 4. Estimation of the modal loss from the picture of photons zigzagging within the Bragg fiber.

3.3 Loss dependence on cladding layer thickness

In designing Bragg fibers, it is generally assumed that the quarter-wave stack provides the greatest confinement as cladding layers. For a cladding layer with dielectric constant ε and thickness L, we define a parameter Φ=4ε1Lλ. . For a Bragg fiber with a large air core and consequently β≈2π/λ, the parameter Φ equals to unity if the cladding layer has exactly quarter wave thickness. For the fiber studied in Fig. 2(c), we find Φ Si =0.98 for the Si layer and ΦSi3N4=0.97 for the Si3N4 layer at λ=1.5µm. At λ=1.7µm, the corresponding quantities are, respectively, Φ Si =0.87 and ΦSi3N4=0.86 . Yet at 1.5µm the Bragg fiber is about four times as lossy as at 1.7µm. This suggests that the quarter-wave stack condition may not lead to lowest propagation loss.

To further investigate this phenomenon, we changed the Si3N4 layer thickness from 0.21µm to 0.19µm, while keeping the rest of the fiber parameters the same as in Fig. 2(c). At λ=1.5µm, the new Si3N4 layer thickness corresponds to ΦSi3N4=0.88 . We again calculate the Bragg fiber loss with λ between 1.5µm and 1.7µm and plot the results in Fig. 5(b). For the purpose of comparison, we copy Fig. 2(c) as Fig. 5(a). From Fig. 5, it is clear that the reduction of the Si3N4 layer thickness leads to both lower loss for the TE01 mode and larger loss penalty for the higher order modes. The solid lines are the loss estimate for the TE modes given by Eq. (12), which again are in excellent agreement with the asymptotic result.

4. Influence of the fiber deformation

In the asymptotic analysis, we assume that the Bragg fibers possess perfect cylindrical symmetry and are uniform along the propagation direction. The experimentally realized Bragg fibers as reported in Ref. [8

8. J. G. Fleming, S. Y. Lin, and R. Hadley, in Proceeding of the solid-state sensor actuator and microsystems workshop, p.p. 173, (Hilton Head, S.C., 2002).

], however, retain neither of these two properties: The fiber cross-section deviates considerably from the circular shape, and there are gas inlet ports along the fiber propagation direction.

We first use the finite difference time domain (FDTD) algorithm to simulate a Bragg fiber without the cylindrical symmetry but retains uniformity along the propagation direction. Taking advantage of the translational symmetry, we can assume the z dependence of the guided mode as exp(-iβz), where β is the propagation constant. This allows us to take out the z dimension in our FDTD calculations and consider in the simulation only the cross-section of the Bragg fibers. For more details on the FDTD algorithm, the readers can consult Ref. [10

10. A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, (Artech house, Boston, 2000).

].

Fig. 5. Asymptotic results for the TE01, TE02, TM01, and HE11 modal loss. The solid lines are the estimation of the TE mode loss given by Eq. (12). In (a), the fiber parameters are the same as used in Fig. 2(c). In (b), we change the Si3N4 layer thickness to 0.19µm, while the rest of the parameters remain the same as in (a).

The CVD process in the fabrication of the miniature Bragg fiber requires the placement of the gas inlet ports about every 1mm along the fiber propagation direction. The gas inlet port can be regarded as a special case of the Bragg fiber inter-connect as shown in Fig. 7. We can think of channel 1 as the Bragg fiber and channel 2 as the gas inlet port that feeds the gas to be deposited on the surface of the Bragg fiber. In this language, the excessive propagation loss caused by the gas inlet port is the throughput loss of the Bragg fiber inter-connect.

As in Fig. 7, we assume the Bragg fiber has an air core radius of r 1 and the gas inlet port has an air core radius of r 2. One method of calculating the throughput loss is to find the azimuthal component of the electric field at the beginning of the throughput port (denoted by E θ,t in Fig. (7)). Since the electric field of the TE01 mode has only the azimuthal component E θ,TE01, the throughput loss is simply given by 1-|∫ E θ,t E θ,TE01 rdrdθ|2. The process of finding E θ,t, however, can be very complicated. In the following discussion, we present a much simpler approach by drawing analogy with the free space diffraction of the Gaussian beam. Since the E θ,t field results from the diffraction of the TE01 mode, we expect the following discussion should at least give us an order of magnitude estimate.

Fig. 6. (a) Dispersion of a Bragg fiber, with air core radius 5.3µm, Si layer thickness 0.11µm and Si3N4 layer thickness 0.21µm. The diamonds and the circles respectively represent the dispersion of the cylindrically symmetric fibers and the deformed fibers, calculated from FDTD simulations. The solid line is given by Eq. (8). In (b) and (c), we show the Bz field of the TE01 mode at λ=1.55µm in the circularly symmetric Bragg fiber and the deformed Bragg fiber.

For a Gaussian beam with minimal spot size w 0, the evolution of the beam spot size w(z) follows [11

11. See for example, A. Yariv, Optical electronics in modern communications, Chapter 2, (Oxford university press, New York, 1997).

]:

[w(z)]2=w02(1+z2z02),z0=πw02λ.
(13)

In order to approximate the diffraction of the TE01 mode by a Gaussian beam, we need to find its equivalent minimal spot size w 0. According to Eq. (8), the far field divergence angle θ of the TE01 mode is θ=x 11λ/(2πr 1), whereas the far field divergence angle of a Gaussian beam is given by θ=λ/πw 0 [11

11. See for example, A. Yariv, Optical electronics in modern communications, Chapter 2, (Oxford university press, New York, 1997).

]. Equalizing the two divergence angles, we find w 0/r 1=2/x 11≈0.52, which has the right order of magnitude since the minimal spot size w 0 should be less than the Bragg fiber radius r 1. Substituting this result into Eq. (13), we find

[w0w(z)]2=11+z2λ2x11416π2r14.
(14)

Since w(z)2 is proportional to the cross-section area of the Gaussian beam, the amount of light that is coupled back into the throughput port can be approximated by [w 0/w(z)]2. Set z=2r 2 and use dB unit, we find:

ThroughtputLoss=23.7r22λ2r14(dB).
(15)
Fig. 7. Schematics of a Bragg fiber inter-connect.

Use the value of r 1=7.5µm, r 2=r 1/2, and λ=1.55µm, we find the throughput loss is 0.25dB. We notice that the the loss has a 1/r14 dependence. If we keep the rest of the parameters and increase air core radius by 50%, the throughput loss is decreased by a factor 5.

5. Conclusion

References and links

1.

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

2.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. 17, 2039–2041 (1999). [CrossRef]

3.

S. G. Johnson et al., “Low loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express 9, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

4.

Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. 25, 1756–1758 (2000). [CrossRef]

5.

Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. 20, 428–440 (2002). [CrossRef]

6.

J. Marcou, F. Brechet, and P. Roy, “Design of weakly guiding Bragg fibres for chromatic dispersion shifting towards short wavelengths,” J. Opt. A , 3, S144–S153 (2001). [CrossRef]

7.

A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express, 10, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

8.

J. G. Fleming, S. Y. Lin, and R. Hadley, in Proceeding of the solid-state sensor actuator and microsystems workshop, p.p. 173, (Hilton Head, S.C., 2002).

9.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

10.

A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, (Artech house, Boston, 2000).

11.

See for example, A. Yariv, Optical electronics in modern communications, Chapter 2, (Oxford university press, New York, 1997).

OCIS Codes
(130.2790) Integrated optics : Guided waves
(220.0220) Optical design and fabrication : Optical design and fabrication

ToC Category:
Research Papers

History
Original Manuscript: March 18, 2003
Revised Manuscript: April 25, 2003
Published: May 5, 2003

Citation
Yong Xu, Amnon Yariv, James Fleming, and Shawn-Yu Lin, "Asymptotic analysis of silicon based Bragg fibers," Opt. Express 11, 1039-1049 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-9-1039


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References

  1. P. Yeh, A. Yariv, and E. Marom, �??Theory of Bragg fiber,�?? J. Opt. Soc. Am. 68, 1196-1201 (1978). [CrossRef]
  2. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, �??Guiding optical light in air using an all-dielectric structure,�?? J. Lightwave Technol. 17, 2039-2041 (1999). [CrossRef]
  3. S. G. Johnson et al., �??Low loss asymptotically single-mode propagation in large-core Omniguide fibers,�?? Opt. Express 9, 748-779 (2001). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748</a> [CrossRef] [PubMed]
  4. Y. Xu, R. K. Lee, and A. Yariv, �??Asymptotic analysis of Bragg fibers,�?? Opt. Lett. 25, 1756-1758 (2000). [CrossRef]
  5. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, �??Asymptotic matrix theory of Bragg fibers,�?? J. Lightwave Technol. 20, 428-440 (2002). [CrossRef]
  6. J. Marcou, F. Brechet, and P. Roy, �??Design of weakly guiding Bragg fibres for chromatic dispersion shifting towards short wavelengths,�?? J. Opt. A, 3, S144-S153 (2001). [CrossRef]
  7. A. Argyros, �??Guided modes and loss in Bragg fibres,�?? Opt. Express, 10, 1411-1417, (2002). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411</a> [CrossRef]
  8. J. G. Fleming, S. Y. Lin, and R. Hadley, in Proceeding of the solid-state sensor actuator and microsystems workshop, p.p. 173, (Hilton Head, S.C., 2002).
  9. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St J. Russell, P. J. Roberts, and D. C. Allan, �??Single-mode photonic band gap guidance of light in air,�?? Science 285, 1537-1539 (1999). [CrossRef] [PubMed]
  10. A. Taflove, S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, (Artech house, Boston, 2000).
  11. See for example, A. Yariv, Optical electronics in modern communications, (Oxford university press, New York, 1997) Chapter 2.

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