## Full-vectorial coupled mode theory for the evaluation of macro-bending loss in multimode fibers. application to the hollow-core photonic bandgap fibers.

Optics Express, Vol. 16, Issue 19, pp. 14945-14953 (2008)

http://dx.doi.org/10.1364/OE.16.014945

Acrobat PDF (410 KB)

### 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

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]

*β*=

*β*+

_{Re}*iβ*. Due to

_{Im}**F**(

*x*,

*y*)

*exp*(

*iβ z*) dependence of the modal fields (

**F**stands for the electric or magnetic field vector), imaginary part of the propagation constant defines modal propagation loss, which is typically expressed in the units of dB/masα[dB/m] = 20

*βIm*/

*ln*(10).

*R*= 100

_{c}*µm*, layer refractive indices

*n*= 2.80+

_{h}*i*1.94·10

^{-6},

*nl*= 1.60+

*i*1.94·10

^{-3}, and layer thicknesses

*d*= 0.926

_{h}*µm*,

*d*= 1.939

_{l}*µm*, operating at the emission wavelength of a

*CO*

_{2}laser

*λ*= 10.6

*µm*. Optical parameters of the high refractive index layer are those of a chalcogenide glass with absorption loss of 10dB/m, while parameters of a low refractive index layer are those of a generic polymer with absorption loss of 10

^{4}

*dB*/

*m*. Such fibers are already used for guiding

*CO*

_{2}laser radiation Ref. [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]

*HE*

_{11}mode of thus defined Bragg fiber is guided by the periodic reflector bandgap centered at

*λ*= 10.6

*µm*. For circular symmetric fibers the eigen fields are most conveniently expressed in a cylindrical coordinate system as

**F**(

*ρ*)

*exp*(

*imθ*+

*iβz*), where

*m*is a modal angular momentum. Losses of the modes of a HPBG fiber are strongly differentiated. In Fig. 1(b) we present modal losses calculated by the standard transfer matrix theory as a function of the mode effective refractive index. As seen from Fig. 1(b), low angular momentum modes (

*m*= 0, 1) with effective refractive indices close to that of air have the tendency to exhibit the lowest losses. Moreover, in Fig. 1(b) one can also distinguish the low-loss (TE-like) and high-loss (TM-like) branches of modes. Among all the modes, the

*HE*

_{11}mode with

*m*= 1 is the easiest one to excite with a Gaussian laser source. It is important to note that thus defined fiber works in the absorption dominated regime, meaning that radiation losses of the modes are much smaller than their absorption losses. For example, absorption loss of the

*HE*

_{11}mode is

*α*= 3.43

_{abs}*dB*/

*m*, while the mode radiation loss is

*α*= 0.45

_{rad}*dB*/

*m*. We note that to differentiate between the modal radiation and absorption losses one has to perform two simulations with and without material absorption losses. Then, simulation without material absorption loss is going to give modal radiation loss, while simulation with material absorption loss is going to give a net of the radiation and absorption losses. For the reference, an effective way of decreasing fiber absorption and radiation losses is by increasing the size of a hollow core Ref. [4

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]

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

*R*. Schematic of a bend is shown in Fig. 2(a). It is well known that bent fibers support leaky eigen modes. Such modes have complex valued propagation constants even in the absence of material losses due to bending-induced radiation loss. Moreover, such modes are not square integrable as they have non-zero outgoing flux at infinity. When calculating eigen modes of a bend the choice of numerical methods is limited. One typically uses either finite difference Ref. [8] or finite element methods Ref. [7

_{b}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]

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]

*β*= 0) eigen modes of a reference fiber, it is not possible to predict bending losses. This is due to the fact that a purely real spectrum of the modal propagation constants of the modes of a reference fiber, results in a purely real spectrum of the eigen modes of a bent fiber when CMT is used. CMT, however, can correctly estimate bending losses when modes of a reference fiber are characterized by the complex propagation constants. This can happen either in the case of a fiber featuring absorbing materials, or in the case of a radiating fiber, such as hollow core fiber guiding by photonic bandgap effect. It is important to note that bending losses evaluated by CMT will somewhat underestimate the true bending losses as CMT only considers the effect of loss increase via mixing with higher-loss modes, and not due to bend induced radiation. Another important point is that for CMT to converge the expansion bases has to include enough modes; thus, the fiber in question has to be overall multimode. This, however, does not signify that bending losses of a single mode fiber can not be computed. One simply has to include the cladding or jacket modes into the consideration.

_{Im}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]

*x*′,

*y*′,

*s*) is a curvilinear coordinate system associated with a bend, and

*y*=

*y*′. In this coordinate system dielectric profile becomes that of a straight reference fiber with a crossection identical to that of a bent fiber. When bending radius is much larger than the core size of a fiber, transverse modal fields of a bend can be expended into the transverse fields of a reference fiber:

*β*and

_{b}*β*are the propagation constants of the modes of a bend and a reference straight fiber, respectively.

_{r}*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.

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]

*𝓑*is a diagonal matrix of the eigenvalues of a straight reference fiber

^{r}*m*= 0 – 10 - in total 362 modes. All the calculations are performed at

*λ*= 10.6

*µm*. Two orthogonally polarized eigen modes of a reference Bragg fiber are found using transfer matrix technique Ref. [4

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]

**Fm**(

*ρ*)

*exp*(

*imθ*)±

**F**-

**(**

^{m}*ρ*)

*exp*(-

*imθ*)]

*exp*(

*iβz*). In what follows such defined polarizations are said to be polarized either in the plane of a bend (XZ plane), or perpendicularly to the plane of a bend. In Fig. 2(b) in dashed lines we present losses of the perpendicularly polarized eigen modes of a bent fiber as a function of the bending radius. In fact, the figure presents losses relative to the loss of an

*HE*

_{11}mode of a straight fiber. When bending radius increases, losses of one of the modes of a bend approaches that of a

*HE*

_{11}mode of a reference fiber, which allows us to identify such a mode as

*HE*

_{11}-like. However, as bending radius decreases below 3cm such identification becomes challenging as the

*HE*

_{11}-like mode experiences a large number of anticrossings with other modes, thus becoming strongly hybridized. For comparison, in solid curve we present losses of a

*HE*

_{11}-like eigen mode of a bend computed with finite element 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]

*HE*

_{11}-like mode of a bend as a function of the wavelength of operation for various values of the bending radius. From this plot we note that as long as

*HE*

_{11}-like mode of a bend can be clearly identified (bending radii larger than 3cm), such a mode is bandgap guided with the center of a bandgap practically unchanged. Therefore, loss increase of an

*HE*

_{11}-like mode of a bend can be rationalized as being primarily due to mode mixing of the

*HE*

_{11}mode of a straight fiber with much lossier higher order modes, and not due to band induced shift in the bandgap position. A word of caution is that the Bragg fiber considered in this work exhibits a very high refractive index contrast, and is highly multimode. Therefore, we do not expect that the last conclusion holds for a general photonic bandgap fiber.

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

*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:

*C*is a matrix of the bend eigenvectors of Eq. (5), while

_{b}*𝓑*is a diagonal matrix of the corresponding eigenvalues of the bend eigen modes

^{b}*C*

^{-1}

_{b}

**In**

^{r}defines modal expansion coefficients in terms of the modes of a bend that matches the excitation profile

**In**

*defined in terms of the modes of an input straight waveguide. Then,*

^{r}*exp*(

*i𝓑*) propagates thus excited modes of a bend to the bend output end. Finally multiplication by

^{b}R_{b}θ_{b}*C*at the end of a bend, converts the expansion in terms of the eigen modes of a bend into the expansion in terms of the eigen modes of an output straight waveguide. Note that expression Eq. (5) assumes that there is no back reflection at the bend input and output ends due to the modal field mismatch. Although this assumption is true for the moderate and large bending radii (

_{b}*R*> 1

_{b}*cm*in our case), for very tight bends this approximation might not be valid, and the region of applicability of Eq. (5), generally, deserves further study.

*S*=ŝ·

_{s}*Re*(

**E**

*×*

_{t}**H***

*)/2. Substituting Eq. (7) into the definition of the energy flux we finally get for the output power:*

_{t}*In*→

*Out*, expression Eq. (8) gives an input power

*P*. Bend loss per unit of length is then defined as:

^{in}*HE*

_{11}mode is launched at the bend input. As before, all the simulations are implemented for

*λ*= 10.6

*µm*. For comparison, in circles we present the same loss as calculated by the finite element beam propagation method Ref. [7

**38**, 297 (2002). [CrossRef]

**38**, 297 (2002). [CrossRef]

*R*

^{-2}

*for large bending radii when mode mixing is small, while bending loss scales as ~*

_{b}*R*

^{-1}

*for tight bending radii when mode mixing is significant. Scaling for large bending radii can be explained using perturbation theory (PT) Ref. [10*

_{b}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]

*M*~

*R*

^{-1}

*. Moreover, bending loss is mostly determined by the loss of a*

_{b}*HE*

_{11}-like mode of a bend, whose complex propagation constant is given by the second order PT expression:

## 4. Discussion

## 5. Conclusion

## Acknowledgments

## References and links

1. | P. Russell, “Photonic crystal fibers,” Science |

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 |

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 |

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 |

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 |

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 |

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. |

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 |

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. | 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 |

**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

Sort: Year | Journal | Reset

### References

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

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.