## Comparative study of air-core and coaxial Bragg fibers: single-mode transmission and dispersion characteristics

Optics Express, Vol. 9, Issue 13, pp. 733-747 (2001)

http://dx.doi.org/10.1364/OE.9.000733

Acrobat PDF (637 KB)

### Abstract

Using an asymptotic formalism we developed in an earlier paper, we compare the dispersion properties of the air-core Bragg fiber with those of the coaxial Bragg fiber. In particular we are interested in the way the inner core of the coaxial fiber influence the dispersion relation. It is shown that, given appropriate structural parameters, large single-mode frequency windows with a zero-dispersion point can be achieved for the TM mode in coaxial fibers. We provide an intuitive interpretation based on perturbation analysis and the results of our asymptotic calculations are confirmed by Finite Difference Time Domain (FDTD) simulations.

© Optical Society of America

## 1 Introduction

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Amer. **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]

6. 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]

6. 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]

6. 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]

*n*

_{1}=4.6 and a thickness

*d*

_{1}=0.333

*a*, whereas layer 2 has an index of refraction

*n*

_{2}=1.6 and a thickness

*d*

_{2}=0.667

*a*. Here,

*a*=

*d*

_{1}+

*d*

_{2}is the unit length of periodicity of the multilayered structure. The cores of the two fibers are different: whereas the Bragg fiber has only a uniform core with an index of refraction

*n*

_{o}=1.0 (i.e. an air-core Bragg fiber) and a radius

*r*

_{o}, the coaxial fiber core contains an additional inner core with an index of refraction

*n*

_{i}and a radius

*r*

_{i}. For simplicity we choose to vary only three parameters in this paper:

*n*

_{i},

*r*

_{i}, and

*r*

_{o}.

## 2 Theoretical analysis

### 2.1 Asymptotic calculations

*n*

_{i}=4.6,

*r*

_{o}=1.4a, and the cladding parameters given in Fig. 1, We plot the dispersion relations in Fig. 2 for different inner core radius

*r*

_{i}.

*r*

_{i}is increased from

*r*

_{i}=0 (air-core Bragg fiber) to

*r*

_{i}=0.267

*a*, the TM band undergoes a large downward shift in frequency whereas the

*m*=1 band moves very little, creating a large single-mode transmission window for the TM mode in the lower half of the bandgap.

*m*=1 modes and for later convenience we choose modes with the same propagation wavevector

*β*. Note the drastic reduction in field amplitude for the

*m*=1 mode inside the inner core of the coaxial fiber (Compare Fig. 3(B) with Fig. 3(D)).

### 2.2 Perturbation Analysis

*β*, let the unperturbed modes of an air-core Bragg fiber be:

*r, θ*)

*exp*[

*i*(

*βz*-

*ω*

_{m}

*t)*],

*m*=0, 1, 2, …,

8. Roy J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phy. Rev. A **43**, 467–491 (1991). [CrossRef]

*ε*(

*r*,

*θ*) induced by the high-index inner core. Let

*δ*

*δω*

_{m}be the changes in the mode functions and eigen frequencies, respectively. The eigenmode equation now takes the form:

*εδ*

*δ*

*ε*(

*r, θ*) may also be significant owing to the large index of refraction of the inner core (For

*n*

_{i}=4.6,Δ

*ε*=

*δω*

_{m}, provided the size of the inner core is small enough (See Fig. 2).

*δω*

_{m})

^{2}can be neglected, expand Eq. (3) and use Eq. (1), then Eq. (3) simplifies to:

*δ*

*a*

_{mn}’s are constants. Substituting Eq. (5) for

*δ*

*δω*

_{m}is a constant, it can be pulled outside the integral:

*m*=1 bands. First note that Δ

*ε*is a windowing function that is nonzero only inside the inner core, hence for the mode function (

*E*

_{z}component inside the inner core, which would make a significant contribution to the top integral in Eq. (8). This explains why the TM band makes a large downward shift in frequency even with the insertion of a relatively small inner core. In the case of the

*m*=1 band, we refer back to Fig. 3(D), which shows a very small

*δω*is to be expected, verified by Fig. 2 where the

*m*=1 band changes very little for the given range of

*r*

_{i}’s.

*m*=1 mode can’t? The Appendix provides us with a clue. For the TM mode, we see from the Appendix that it has two

*E*

_{z}and

*E*

_{r}, which are proportional to the zeroth order Bessel’s function

*J*

_{0}(

*k*

_{co}

*r*) and first order Bessel’s function

*J*

_{1}(

*k*

_{co}

*r*) respectively, where

*k*

_{co}is the transverse wavevector inside the core. Assuming the core is small and knowing that

*J*

_{0}(0)=1 and

*J*

_{1}(0)=0, we see that

*E*

_{z}would necessarily be large and

*E*

_{r}would necessarily be small.

*m*=1 mode. From the Appendix, we observe that all three

*E*

_{z}

*, E*

_{r}, and

*E*

*θ*, are present in the inner core. The

*E*

_{z}component can be neglected because it is proportional to

*J*

_{1}(

*k*

_{co}

*r*), whereas

*E*

_{r}and

*Eθ*each contains a

*J*

_{0}(

*k*

_{co}

*r*) term and are equal in amplitude at

*r*=0. For a high-index core with

*n*

_{i}=4.6 and an air region outside, boundary matching imposes a large jump discontinuity at the boundary for

*E*

_{r}. In fact, at the boundary

*E*

_{r}is necessarily small in a high-index center core after the mode function has been properly normalized.

*Eθ*is also small in the core because it’s been tied down with

*E*

_{r}at

*r*=0, (i.e.,

*E*

_{θ}(0)=

*E*

_{r}(0) in amplitude). As a final remark, we note the same boundary conditions do not have the same “clamping” effect on the TM mode, where the predominant component is

*E*

_{z}, which is continuous across the core boundary.

*n*

_{i}=1.45 and

*r*

_{o}=1.4

*a*are plotted in Fig. 4, and the field profiles in Fig. 5. According to our above analysis, the clamping effect imposed by the boundary conditions on the

*E*

_{r}component of the

*m*=1 mode should not be nearly as prominent as before owing to the greatly reduced index of refraction in the core. As a result, the downward shifts of the TM band and the

*m*=1 band should be much smaller than before and of the same order of magnitude. This is exactly what we observe in Fig. 4, although the downward shift of the TM band is still larger because the inner core still provides some clamping even with the reduced index of refraction.

*r*

_{o}initially to push both the TM and the

*m*=1 bands higher up in the bandgap for the air-core Bragg fiber, then when we insert the inner core, we know the

*m*=1 band will pretty much stay high up in the bandgap while the TM band will come down all the way to the bottom. In this way we can extend the single-mode frequency window and hopefully it will also include a zero-dispersion point. This is exactly what we did and the resulting band diagrams for

*n*

_{i}=4.6 and

*r*

_{o}=0.867

*a*are plotted in Fig. 6. Note the arrow that points to the zero-dispersion point inside the single-mode window in Fig. 6(D).

### 2.3 FDTD simulations

9. F. Zepparelli, P. Mezzanotte, F. Alimenti, L. Roselli, R. Sorrentino, G. Tartarini, and P. Bassi, “Rigorous analysis of 3D optical and optoelectronic devices by the compact-2D-FDTD method.” Opt. and Quantum Electron. **31**, 827–841 (1999). [CrossRef]

*z*direction, the

*z*dependence of the fields is simply exp(

*iβz*). For example, we can write the electric field as

*x, y, t*)exp(

*iβz*) and the magnetic field as

*x, y, t*)exp(

*iβz*). As a result, the 3D Maxwell equations are reduced to 2D ones:

_{⊥}is defined as

11. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computat. Phys. **114**, 185–200 (1994). [CrossRef]

12. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. **44**, 1630–1639 (1996) [CrossRef]

9. F. Zepparelli, P. Mezzanotte, F. Alimenti, L. Roselli, R. Sorrentino, G. Tartarini, and P. Bassi, “Rigorous analysis of 3D optical and optoelectronic devices by the compact-2D-FDTD method.” Opt. and Quantum Electron. **31**, 827–841 (1999). [CrossRef]

*r*

_{o}whose band diagram is given in Fig. 6(D). Specifically the fiber parameters are given as follows:

*n*

_{i}=4.6,

*r*

_{i}=0.267

*a*,

*n*

_{o}=1,

*r*

_{o}=0.867

*a*,

*n*

_{1}=4.6,

*d*

_{1}=0.333

*a*,

*n*

_{2}=1.6 and

*d*

_{2}=0.667

*a*, where all the parameters are defined in Fig. 1. A total of 5 cladding pairs are used, which should provide good mode confinement based on our experiences.

*E*

_{z}field distribution of a TM mode, filtered out by FDTD simulation. The frequency and propagation constant of the mode are respectively

*ω*=0.2238(2

*πc/a*) and

*β*=0.2(2

*π/a*). Fig. 8 clearly shows that the guided mode has no θ dependence, just as what we would expect from a TM mode function.

## 3 Conclusion

## A Appendix

*z*axis as the direction of propagation, then every field component has the form

*ψ*can be

*E*

_{z}

*, E*

_{r},

*E*

_{θ}

*, H*

_{z}

*, H*

_{r}

*, H*

_{θ}

*. ω*is the angular frequency and

*β*is the propagation constant.

*E*

_{z}and

*H*

_{z}. In particular,

*E*

_{r}and

*E*

_{θ}can be written as [1

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

*E*

_{r}is the only transverse

*k*

_{co}is the transverse wavevector inside the core,

*l*=1 mode:

## References and links

1. | P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Amer. |

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. |

3. | M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, “Fabrication of germanium-coated nickel hollow waveguides for infrared transmission,” Appl. Phys. Lett. |

4. | N. Croitoru, J. Dror, and I. Gannot, “Characterization of hollow fibers for the transmission of infrared radiation,” Appl. Opt. |

5. | 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 |

6. | M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science |

7. | Y. Xu, G. Ouyang, R. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” (submitted to J. Lightwave Technol.). |

8. | Roy J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phy. Rev. A |

9. | F. Zepparelli, P. Mezzanotte, F. Alimenti, L. Roselli, R. Sorrentino, G. Tartarini, and P. Bassi, “Rigorous analysis of 3D optical and optoelectronic devices by the compact-2D-FDTD method.” Opt. and Quantum Electron. |

10. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

11. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computat. Phys. |

12. | S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. |

**OCIS Codes**

(060.2430) Fiber optics and optical communications : Fibers, single-mode

(230.1480) Optical devices : Bragg reflectors

**ToC Category:**

Focus Issue: Photonic crystal fiber

**History**

Original Manuscript: November 7, 2001

Published: December 17, 2001

**Citation**

George Ouyang, Yong Xu, and Amnon Yariv, "Comparative study of air-core and coaxial Bragg fibers:
single-mode transmission and dispersion characteristics," Opt. Express **9**, 733-747 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-13-733

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### References

- P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978). [CrossRef]
- 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]
- M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, "Fabrication of germanium-coated nickel hollow waveguides for infrared transmission," Appl. Phys. Lett. 43, 430-432 (1983). [CrossRef]
- N. Croitoru, J. Dror, and I. Gannot, "Characterization of hollow fibers for the transmission of infrared radiation," Appl. Opt. 29, 1805-1809 (1990). [CrossRef] [PubMed]
- 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]
- M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000). [CrossRef] [PubMed]
- Y. Xu, G. Ouyang, R. Lee, and A. Yariv, "Asymptotic matrix theory of Bragg fibers," (submitted to J. Lightwave Technol.).
- Roy J. Glauber and M. Lewenstein, "Quantum optics of dielectric media," Phy. Rev. A 43, 467-491 (1991). [CrossRef]
- F. Zepparelli, P. Mezzanotte, F. Alimenti, L. Roselli, R. Sorrentino, G. Tartarini, and P. Bassi, "Rigorous analysis of 3D optical and optoelectronic devices by the compact-2D-FDTD method." Opt. and Quantum Electron. 31, 827-841 (1999). [CrossRef]
- K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
- J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Computat. Phys. 114, 185-200 (1994). [CrossRef]
- S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996). [CrossRef]

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