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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 19 — Sep. 15, 2008
  • pp: 14945–14953
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Full-vectorial coupled mode theory for the evaluation of macro-bending loss in multimode fibers. application to the hollow-core photonic bandgap fibers.

Maksim Skorobogatiy, Kunimasa Saitoh, and Masanori Koshiba  »View Author Affiliations


Optics Express, Vol. 16, Issue 19, pp. 14945-14953 (2008)
http://dx.doi.org/10.1364/OE.16.014945


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Abstract

In the hollow core photonic bandgap fibers, modal losses are strongly differentiated, potentially enabling effectively single mode guidance. However, in the presence of macro-bending, due to mode coupling, power in the low-loss mode launched into a bend is partially transferred into the modes with higher losses, thus resulting in increased propagation loss, and degradation of the beam quality. We show that coupled mode theory formulated in the curvilinear coordinates associated with a bend can describe correctly both the bending induced loss and beam degradation. Suggested approach works both in absorption dominated regime in which fiber modes are square integrable over the fiber crossection, as well as in radiation dominated regime in which leaky modes are not square integrable. It is important to stress that for multimode fibers, full-vectorial coupled mode theory developed in this work is not a simple approximation, but it is on par with such “exact” numerical approaches as finite element and finite difference methods for prediction of macro-bending induced losses.

© 2008 Optical Society of America

1. Introduction

Hollow photonic band gap (HPBG) fibers Refs. [1

1. P. Russell, “Photonic crystal fibers,” Science 299, 358 (2003). [CrossRef] [PubMed]

, 2

2. C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003). [CrossRef] [PubMed]

, 3

3. 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, 650 (2002). [CrossRef] [PubMed]

] guide light within hollow cores via reflection of the guided light from the walls of a surrounding dielectric multilayer mirror. Such fibers promise low transmission loss at almost any wavelength as confinement of the electromagnetic energy in the hollow core reduces considerably the effect of fiber material losses. Potential applications of such fibers include high power guiding in mid-IR, ultra-low nonlinearity fibers for telecommunications, high sensitivity detectors where sensing layers and analyte are placed inside of a hollow fiber core.

Fig. 1. (a) Schematic of a hollow core Bragg fiber. Sz is the transverse distribution of the longitudinal energy flux component for the Gaussian-like HE 11 core guided mode. (b) Spectrum of the modal propagation constants for the hollow core Bragg fiber shown in (a), evaluated at λ = 10.6µm.

2. Coupled Mode Theory for the leaky modes of a bent fiber

Interestingly, field distribution in the leaky modes of a bent multimode fiber can be well approximated by employing full-vectorial coupled mode theory (CMT) Ref. [9

9. M. Skorobogatiy, S. A. Jacobs, S.G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10, 1227 (2002). [PubMed]

]. This approximation works best inside, or in the near vicinity of a fiber core. CMT allows, for example, finding beam intensity distribution at the bend output. Advantage of the coupled mode theory is its simplicity as only eigen modes of a straight fiber have to be computed, for which many efficient numerical solvers exist. Moreover, as we will see in the following, coupling elements within CMT framework have to be computed only for a single value of a bending radius, while for any other value of a bending radius they have to be simply re-scaled. This allows efficient computation of the eigen spectra and total bending losses for multiple values of a bending radius. Finally, eigen value problem resulting from CMT involves full matrices of small order (typically less than 1000 × 1000).

To formulate CMT in the case of a bend we follow closely the method of perturbation matching detailed in Ref. [9

9. M. Skorobogatiy, S. A. Jacobs, S.G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10, 1227 (2002). [PubMed]

]. Within this method Maxwell equations are transformed into a curvilinear coordinate system where dielectric function becomes that of a straight reference fiber. One then uses the modes of a reference fiber as an expansion basis to solve for the scattering problem. Due to curvilinear transformation, Maxwell equations acquire additional terms responsible for coupling between the modes of a straight waveguide. In the particular case of a bend shown in Fig. 2(a) we use the following coordinate transformation:

x=Rb(Rbx)cos(sR),
z=(Rbx)sin(sR)
(1)

Ftβb(x,y,s)=exp(iβbs)βrCβrβbFtβr(x,y),
(2)

where βb and βr are the propagation constants of the modes of a bend and a reference straight fiber, respectively.

We now define the elements of a normalization matrix B for the modes of a reference fiber are defined as:

Bβr,βr=ŝ·dxdy(Etβr×Htβr+Etβr×Htβr),
(3)

where t signifies transverse field components, and s signifies longitudinal field components. For the true guided modes, integration in Eq. (3) is over the whole 2D space. Moreover, for the two orthogonally polarized modes in the plane and perpendicular to the plane of a band, matrix B is diagonal even in the presence of material losses. In the case of radiating fibers (such as hollow core Bragg fibers) characterized by non-square-integrable leaky modes, integration in Eq. (3) is performed only in the finite region terminated by the interface between the last reflector layer and a cladding (a so called cut-off approximation). Finally, for the orthogonally polarized leaky modes, matrix B is dominantly diagonal, and, in practice, can be considered as strictly diagonal. These two approximations for the integrals involving leaky modes become exact in the limit of infinite number of bi-layers in which case the core mode becomes truly guided.

Elements of the coupling matrix ΔM for the modes of a reference fiber are defined as:

ΔMβr,βr=ωRbdxdy[(HsβrHsβr+Htβr·Htβr)+ε(x,y)(EsβrEsβr+Etβr·Etβr)]·x.
(4)

With these definitions, eigen modes of a bent fiber can be found by resolving the following eigen value problem Ref. [9

9. M. Skorobogatiy, S. A. Jacobs, S.G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10, 1227 (2002). [PubMed]

]:

βbBCβb=(BBr+ΔM)Cβb,
(5)

where 𝓑r is a diagonal matrix of the eigenvalues of a straight reference fiber Bβr,βrr=βr, while Cβb is a vector of unknown expansion coefficients in Eq. (2).

Fig. 2. (a) Schematic of a fiber bend. (b) Losses of the eigen modes of a bent hollow core Bragg fiber evaluated at λ = 10.6µm as a function of the bending radius. Material absorbtion dominated regime. (c) Losses of an HE 11-like mode of a bend as a function of the wavelength of operation for various values of the bending radius.

3. Total bending loss and beam degradation in a bent fiber

In practical applications, a more convenient measure of bending loss is given by the ratio of the total power at the bend output to the total power at the bend input. As HE 11 mode of a straight Bragg fiber is the one most compatible with the Gaussian-like mode of a laser source, we assume that all the power at the bend input is in the HE 11 mode. Traditionally, to calculate total bending loss under a given excitation condition, one would use a beam propagation method, which is, generally, more computationally intensive than a CMT. Using CMT to solve for the bending loss involves calculation of the total power at the bend output. Assuming that the vector In r defines excitation coefficients of the modes of a reference fiber at the bend input, then the vector of expansion coefficients (for the modes of a straight fiber) at the bend output is:

Outr=Cbexp(iBbRbθb)Cb1Inr,
(6)

Ftout(x,y)=βrOutβrrFtβr(x,y).
(7)

Finally, energy flux along the direction of a bend is Ss=ŝ·Re(E t×H*t)/2. Substituting Eq. (7) into the definition of the energy flux we finally get for the output power:

Pout=ŝ·dxdySsout(x,y)=βr,βrOutβrrOutβrr*ŝ·dxdyRe(Etβr(x,y)×Htβr*(x,y))2.
(8)

By substituting the output coefficients by the input coefficients InOut, expression Eq. (8) gives an input power Pin. Bend loss per unit of length is then defined as:

αbend[dBm]=10log10(PoutPin)(θbRb).
(9)

In Fig. 3(a) in solid curves we plot bend induced losses (αbendαHE11) as a function of the bending radius, assuming that only a HE 11 mode is launched at the bend input. As before, all the simulations are implemented for λ = 10.6µm. For comparison, in circles we present the same loss as calculated by the finite element beam propagation method Ref. [7

7. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to Photonic Crystal fibers,” IEEE J. Quantum Electron. 38, 297 (2002). [CrossRef]

], and observe an excellent match. Note that polarization in the plane of a bend is significantly lossier than polarization perpendicular to the plane of a bend. In Fig. 3(b) we present beam intensity distributions for the lossiest polarization at the bend output for three values of the bending radii. Note that mode mixing and beam quality degradation becomes substantial for bending radii smaller than 10cm.

Fig. 3. (a) Bending losses of a hollow core Bragg fiber under HE 11 launching condition for various values of bending radius (absorption dominated regime). (b) Plots of intensity distribution at the bend output, under HE 11 launching conditions for various values of bending radius. (c) Bending losses of a hollow core Bragg fiber under HE 11 launching condition for various values of bending radius (radiation dominated regime).

So far, we have considered Bragg fiber operating in the absorption dominated regime, and having square integrable modes. As was mentioned before, the CMT formalism developed in this paper is general, and it also works in the case of a fiber operating in the radiation dominated regime. To demonstrate this, in Fig. 3(c) we present bend induced losses for a Bragg fiber with the same structural parameters as before, however having only 19 layers in the reflector, and made of loss-less materials. Eigen modes of such a fiber are non square integrable leaky modes. In this case, normalization Eq. (3) and coupling elements Eq. (4) must be computed using the cut-off approximation by integration over a finite region of the fiber crossection confined by the boundary with the cladding. CMT results for the total bending loss presented as solid curves in Fig. 3(c) are compared with the predictions by the finite element beam propagation method Ref. [7

7. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to Photonic Crystal fibers,” IEEE J. Quantum Electron. 38, 297 (2002). [CrossRef]

], and a good match is observed.

Finally, from Figs. 2,3 we note that bending loss scales as ~R -2 b for large bending radii when mode mixing is small, while bending loss scales as ~R -1 b for tight bending radii when mode mixing is significant. Scaling for large bending radii can be explained using perturbation theory (PT) Ref. [10

10. M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002). [CrossRef]

]. In this regime, from Eq. (4) it follows that ΔM ~R -1 b. Moreover, bending loss is mostly determined by the loss of a HE 11-like mode of a bend, whose complex propagation constant is given by the second order PT expression:

βbβr=βrβrΔMβr,βr2Bβr,βrBβr,βr1βrβr~Rb2.
(10)

4. Discussion

5. Conclusion

We demonstrated that full-vectorial coupled mode theory formulated in the curvilinear coordinate system associated with a bend can predict correctly bending induced radiation and absorption losses in photonic bandgap fibers. Results of the CMT for a bent hollow core Bragg fiber were compared with predictions of the finite element code and excellent agreement was found.

Acknowledgments

We would like to thank Prof. S. G. Johnson (MIT, Cambridge, USA) and Dr. S. A. Jacobs for their insights into the problem of Bragg fiber bending shared with M. Skorobogatiy during his tenure at OmniGuide Inc. (Cambridge, USA).

References and links

1.

P. Russell, “Photonic crystal fibers,” Science 299, 358 (2003). [CrossRef] [PubMed]

2.

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003). [CrossRef] [PubMed]

3.

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, 650 (2002). [CrossRef] [PubMed]

4.

S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weiseberg, 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, 748 (2001). [CrossRef] [PubMed]

5.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002). [CrossRef]

6.

E. Pone, A. Hassani, S. Lacroix, A. Kabashin, and M. Skorobogatiy, “Boundary integral method for the challenging problems in bandgap guiding, plasmonics and sensing,” Opt. Express 15, 10231 (2007). [CrossRef] [PubMed]

7.

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to Photonic Crystal fibers,” IEEE J. Quantum Electron. 38, 297 (2002). [CrossRef]

8.

D.M. Shyroki, J. Lgsgaard, and O. Bang, “Finite-difference modeling of Bragg Fibers with ultrathin cladding layers via adaptive coordinate transformation,” Proc. of SPIE 6728, 672830 (2007).

9.

M. Skorobogatiy, S. A. Jacobs, S.G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10, 1227 (2002). [PubMed]

10.

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(060.2400) Fiber optics and optical communications : Fiber properties
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: April 22, 2008
Revised Manuscript: July 31, 2008
Manuscript Accepted: August 14, 2008
Published: September 8, 2008

Citation
Maksim Skorobogatiy, Kunimasa Saitoh, and Masanori Koshiba, "Full-vectorial coupled mode theory for the evaluation of macro-bending loss in multimode fibers. application to the hollow-core photonic bandgap fibers," Opt. Express 16, 14945-14953 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14945


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References

  1. P. Russell, "Photonic crystal fibers," Science 299, 358 (2003). [CrossRef] [PubMed]
  2. C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, "Low-loss hollow-core silica/air photonic bandgap fibre," Nature 424, 657 (2003). [CrossRef] [PubMed]
  3. 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, 650 (2002). [CrossRef] [PubMed]
  4. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weiseberg, 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, 748 (2001). [CrossRef] [PubMed]
  5. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322 (2002). [CrossRef]
  6. E. Pone, A. Hassani, S. Lacroix, A. Kabashin, and M. Skorobogatiy, "Boundary integral method for the challenging problems in bandgap guiding, plasmonics and sensing," Opt. Express 15, 10231 (2007). [CrossRef] [PubMed]
  7. K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to Photonic Crystal fibers," IEEE J. Quantum Electron. 38, 297 (2002). [CrossRef]
  8. D. M. Shyroki, J. Lgsgaard and O. Bang, "Finite-difference modeling of Bragg Fibers with ultrathin cladding layers via adaptive coordinate transformation," Proc. SPIE 6728, 672830 (2007).
  9. M. Skorobogatiy, S. A. Jacobs, S. G. Johnson, and Y. Fink, "Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates," Opt. Express 10, 1227 (2002). [PubMed]
  10. M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, "Analysis of general geometric scaling perturbations in a transmitting waveguide," J. Opt. Soc. Am. B 19, 2867 (2002). [CrossRef]

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