## Transverse lightwave circuits in microstructured optical fibers: resonator arrays

Optics Express, Vol. 14, Issue 4, pp. 1439-1450 (2006)

http://dx.doi.org/10.1364/OE.14.001439

Acrobat PDF (307 KB)

### Abstract

Novel type of microstructured optical fiber couplers is introduced where energy transfer is enabled by transverse resonator arrays built into a fiber crossection. Such a design allows unlimited spatial separation between interacting fiber cores which, in turn, eliminates inter-core crosstalk via proximity coupling, thus enabling scalable integration of many fiber cores. Moreover, in the limit of weak inter-resonator coupling, resonator arrays exhibit moderate polarization dependence.

© 2006 Optical Society of America

## 1. Introduction

1. M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Transverse lightwave circuits in microstructured optical fibers: waveguides,” Opt. Express **13**, 7506–7515 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-19-7506 [CrossRef] [PubMed]

2. B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, and A.H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. **36**, 1358–1359 (2000). [CrossRef]

3. B.H. Lee, J.B. Eom, J. Kim, D.S. Moon, U.-C. Paek, and G.-H. Yang, “Photonic crystal fiber coupler,” Opt. Lett. **27**, 812–814 (2002). [CrossRef]

4. J. Canning, M. A. van Eijkelenborg, T. Ryan, M. Kristensen, and K. Lyytikainen, “Complex mode coupling within air-silica structured optical fibres and applications,” Opt. Commun. **185**, 321 (2000). [CrossRef]

5. W.E.P. Padden, M. A. van Eijkelenborg, A. Argyros, and N. A. Issa, “Coupling in a twin-core microstructured polymer optical fiber,” Appl. Phys. Lett. **84**, 1689–1691 (2004). [CrossRef]

6. H. Kim, J. Kim, U.-C. Paek, B.H. Lee, and K. T. Kim, “Tunable photonic crystal fiber coupler based on a side-polishing technique,” Opt. Lett. **29**, 1194–1196 (2004). [CrossRef] [PubMed]

7. J. Laegsgaard, O. Bang, and A. Bjarklev, “Photonic crystal fiber design for broadband directional coupling,” Opt. Lett. **29**, 2473–2475 (2004). [CrossRef] [PubMed]

1. M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Transverse lightwave circuits in microstructured optical fibers: waveguides,” Opt. Express **13**, 7506–7515 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-19-7506 [CrossRef] [PubMed]

## 2. Structure and definitions

*L*

_{c}, and is due to evanescent coupling of the modal fields. When two cores are spaced further apart coupling reduces dramatically resulting in an exponential increase of the coupling length making such a coupler unpractical. In what follows we demonstrate a resonant, rather than proximity, directional coupling which allows energy transfer between two fiber cores regardless of the separation between them. Moreover, suggested coupler geometry is only moderately polarization dependent. To demonstrate the robustness of our design and to investigate the importance of band gap confinement on coupler radiation losses we choose the most challenging case - design of long range coupling between two hollow core fibers guiding in the band gap of a surrounding 2D photonic crystal cladding. Structure and modal properties of individual hollow waveguides are detailed in [8

8. K. Saitoh and M. Koshiba, “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,” Opt. Express **11**, 3100 (2003). [CrossRef] [PubMed]

*n*= 1.45 by removing two rows of tubes and smoothing the resulting core edges. The pitch is ʌ = 2

*μ*m, hole diameter is

*d*/ʌ = 0.9, total number of hole layers is six. Fundamental bandgap where the core guided modes are found extends between 1.29

*μ*m <

*λ*< 140

*μ*m. To form a coupler we place two hollow cores

*N*periods apart from each other as in Fig. 1(a). Transverse waveguide is then introduced by slightly reducing the diameters

*d*

_{w}of some of the holes along the line joining the cores. In the rest of the paper we use

*d*

_{w}/ʌ = 0.7. As hollow core size is relatively small, in order to not disturb too much guided hollow core mode we keep sizes of the

*N*

_{wrr}≥ 1 holes closest to the cores unchanged. Periodic array of defects is then formed by reducing diameters of the appropriate holes so that the separation between them is the same and equal to

*N*

_{rr}≥ 1, Figs. 1(b)-(i-iii).

*x*or

*y*dominant component. We call such fields as

*x*or

*y*polarized. When second identical core is introduced, interaction between the core modes of same polarization leads to appearance of even and odd (with respect to a reflection in OY axis) supermodes with effective refractive indexes defined respectively as

*Re*

*λ*/(2Δ

*P*(0) = exp(-

*πmax*(

*Im*(

*N*= 4 coupling length is ~ 10cm, for

*N*= 7 it is ~ 1m, while for

*N*= 11 it is ~ 20/

*m*. Moreover, at a given frequency coupling lengths for different modal polarizations can differ significantly resulting in a strong polarization dependence of a coupler. In the following we demonstrate coupling via weakly polarization dependent resonator arrays exhibiting coupling lengths on a cm scale for arbitrary inter-core separations.

## 3. Weakly coupled fiber arrays

9. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic vrystal fibers,” J. Quantum Electron. **38**, 927–933 (2002). [CrossRef]

10. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method with perfectly matched layers for anysotropic optical waveguides,” J. Lightwave Technol. **19**, 405–413 (2001). [CrossRef]

11. H.A. Haus, W.P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. **5**, 16 (1987). [CrossRef]

12. 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. The fundamental connection between polarization mode dispersion and group-velocity dispersion,” J. Opt. Soc. Am. B **19**, 2867–2875 (2002). [CrossRef]

*ε*(

**r**

_{t}) of a multicore waveguide is comprised of two hollow core waveguides coupled via

*N*

_{r}weakly interacting resonator waveguides. Dielectric profiles of stand alone waveguides are defined as

*ε*

_{i}(

**r**

_{t}),

*i*= [1,

*N*

_{r}+ 2]. Efficient coupling between several cores requires phase matching of the corresponding core modes of stand alone fibers. At a particular wavelength

*λ*

_{0}, phase matching condition implies that the propagation constants

*λ*

_{0}) of all the core modes of stand alone fibers are equal to each other. For wavelengths close to a phase matching point

*λ*~

*λ*

_{0}, electromagnetic fields

**F**

_{t}of a supermode with a propagation constant

*β*(

*λ*) can be represented as linear combinations of the modal fields

*P*

_{i}(

*λ*):

*λ*

_{0}. By substituting this expansion into Maxwell Hamiltonian [12

12. 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. The fundamental connection between polarization mode dispersion and group-velocity dispersion,” J. Opt. Soc. Am. B **19**, 2867–2875 (2002). [CrossRef]

*β*(

*λ*) and a vector of expansion coefficients

*P*¯:

*N*and coupling Δ

*H*matrixes, and diagonal matrix

*D*(

*λ*) are defined as:

*N*and Δ

*H*assume a tridiagonal form. Thus, denoting left and right hollow waveguides by indexes “Lw” and “Rw”, denoting the identical equally spaced resonators by index “r”, and assuming only the nearest neighbor interactions we arrive at the following general form of the normalization and coupling matrices:

*H*

_{Lw,r},

*H*

_{r,Lw},

*H*

_{Rw,r},

*H*

_{r, Rw}, while inter-resonator coupling is described by the parameters

*H*,

_{ri,ri-1}*H*; mode normalization is assumed to be

_{ri,ri+1}*N*

_{i,i}= 1. Two dominant coupling mechanisms in this system are due to the hollow core - resonator and inter-resonator interactions. Thus, without the loss of generality to make qualitative analysis simpler we further assume that six complex coupling parameters can be reduced to just two real parameters

*C*

_{wr}=

*C*

_{rr}=

*x*any

*y*polarized modes. Therefore, for a complete coupler, in the absence of interaction between coupler sub-components, the point of phase matching between the hollow waveguide and resonator modes will be the same for both polarizations. From perturbation theory it follows that in the case of a weak coupling between coupler sub-waveguides, wavelengths of the coupling resonances for both polarizations will remain close to the same phase matching frequency in the absence of coupling. Coupler polarization dependence will then arise through the polarization dependence of the normalization and coupling matrices (4) and will manifest itself in the somewhat different resonance frequencies and coupling lengths at resonance.

*x*and

*y*polarizations, only one polarization of a hollow waveguide mode will be able to couple to a resonator singlet due to symmetry considerations, thus resonant coupling will only be observed for one of the polarizations. In the rest of the paper we will concentrate on the case of moderate polarization dependence of a coupler in which interaction is between the degenerate doublets of the hollow core and resonator waveguides.

*N*

_{i,j};

*j*≠

*j*due to overlap of the modes situated on the different cores. We further assume that hollow core - resonator coupling is much weaker than inter-resonator coupling

*C*

_{wr}≪

*C*

_{rr}so not to disturb considerably the fields of the hollow core modes. For the brevity of presentation, all the derivations in this paper are done for the case of an even resonator mode. With these assumptions, coupling between hollow cores mediated by a periodic resonator array can be readily understood. In Fig. 2 we present schematics of dispersion relations of the coupler supermodes modes as well as their propagation constants and expansion coefficients at a phase mathcing point for the case of one, two and three resonators.

### 3.1. Coupling via one resonator

*β*(

*λ*) of the supermodes relative to a dispersion relation of a core mode of a stand alone hollow core fiber

*λ*) is presented. Fine black dotted line represents dispersion relation of a guided mode of a stand alone resonator which at

*λ*

_{0}is phase matched with core guided modes of the hollow waveguides. By solving eigen problem (2) near

*λ*

_{0}we find that at the point of phase matching three degenerate modes split. Corresponding supermodes are formed by proper linear combinations of the three modes of the corresponding stand alone waveguides. Assuming that a resonator mode is even, then there will be one odd and two even supermodes. Field distribution in the odd supermode (dotted blue curve) presents an antisymmetric combination of the fields of the two hollow core modes with no contribution from a resonator mode. Not surprisingly, the propagation constant of such a supermode (point 2) will be equal to the propagation constant of a mode of a stand alone hollow core fiber. The two other supermodes (red curves) will be even, exhibiting avoiding crossing in the region of phase matching. Field distributions in such supermodes show strong mixing (interaction) of the hollow core modes with the resonator modes.

*λ*

_{0}at which distance between the points 1 and 2 is the same as the one between the points 2 and 3. At

*λ*

_{0}, which we also call a resonant wavelength, we define

*s*

_{1},

*s*

_{2}, and

*s*

_{3}to be the fields of the supermodes with propagation constants

*β*

_{1}=

*C*

_{wr},

*β*

_{2}=

*β*

_{3}=

*C*

_{wr}(Fig. 2(a)). Then, linear combination (

*s*

_{1}

*s*

^{3})/2 +

*s*

_{2}corresponds to a field distribution of a core guided mode in the left hollow waveguide only. After the propagation distance

*L*

_{c}=

*π*/(√2

*C*

_{wr}) =

*π*/(

*β*

_{3}-

*β*

_{2}) =

*π*/(

*β*

_{2}-

*β*

_{1}), modal composition becomes (- (

*s*

_{1}+

*s*

_{3})/2 +

*s*

_{2})

*exp*(

*jβ*

_{2}

*L*

_{c}), which corresponds to the field distribution of a core guided mode in the right hollow waveguide only. Note that coupling length is inversely proportional to the waveguide - resonator coupling strength.

*λ*≠

*λ*

_{0}we adopt the following definition of a coupling length:

*L*

_{d}and power loss per one coupling length as:

*m*. After introduction of a resonator, coupling length is reduced by almost two decades to 7

*cm*for

*y*polarization and 12

*cm*for

*x*polarization. As expected, resonant wavelengths for the

*x*and

*y*polarizations are almost the same. In the lower part of Fig. 3(a) intensities of the modal electric fields are presented for a mode 1 of Fig. 2(a) showing agreement with the predictions of a coupled mode theory (2). The widths of the resonances are proportional to the hollow waveguide - resonator coupling strength

*C*

_{wr}. Total power loss over one coupling length are predicted to be 25% for

*y*and 20% for

*x*polarizations. Out of the resonance both coupling length and loss rapidly increase.

### 3.2. Coupling via two resonators

*β*

_{4}-

*β*

_{1}) ~

*C*

_{rr}being proportional to the inter-resonator coupling strength. Field distribution in these supermodes will be dominated by the fields of the excited resonator modes. Modes 2 and 3 will be split the least with a difference in their propagation constants (

*β*

_{3}-

*β*

_{2}) ~

*C*

_{rr}≪

*C*

_{wr}. Field distribution in these supermodes will be dominated by the fields of the hollow waveguide core modes.

*λ*

_{±1}corresponding to the points of avoiding crossing of the resonator supermodes 1 and 4 with the hollow core waveguide supermodes 3 and 2. In the vicinity of the resonances

*λ*

_{±1}band diagram (Fig. 2(b)) is similar to that of a single resonator coupler (Fig. 2(a)), suggesting that coupling between two hollow cores is mediated by the excitation of one of the collective resonances of a two resonator structure. As before, we define resonant wavelengths

*λ*

_{±1}as the ones at which propagation constants of the three supermodes closest to a propagation constant of a mode of a stand alone hollow core fiber are equally spaced from each other. Spacing between such resonances will be proportional to the inter-resonator coupling strength |

*λ*

_{±1}-

*λ*

_{0}| /

*λ*

_{0}~

*λ*

_{0}

*v*

_{g}

*C*

_{rr}, where

*v*

_{g}is a group velocity of the resonator modes in the units of

*c*. Same definition of coupling length as in (5) can be used to analyze power transfer between the hollow cores in the vicinity of such resonances. Complete power transfer between the hollow cores is again possible at the resonances

*λ*

_{±1}, with the coupling length being inversely proportional to the hollow waveguide - resonator coupling strength

*L*

_{c}~

*cm*for

*y*polarization and 5

*cm*for

*x*polarization. By comparison with a single resonator array we observe that coupling length at resonance for a two resonator array is further reduced due to an increase in the hollow waveguide - resonator coupling as waveguide - resonator separation

*N*

_{wr}becomes smaller (from 5 to 3). Resonant wavelengths for the

*x*and

*y*polarizations are somewhat different. In the lower part of Fig. 3(b) intensities of the

*x*and

*y*polarized modal electric fields are presented for the mode 1 of Fig. 2(b) at the resonance

*λ*

_{-1}showing an agreement with coupled mode theory predictions. Note that resonator fields in the

*x*polarized mode (point 1

_{x}) are extended more toward each other than corresponding fields in the y polarized modes (point 1

_{y}), which explains why

*x*polarization. The widths of the resonances are proportional to the hollow waveguide - resonator coupling strength

*C*

_{wr}which are also somewhat different for both polarizations. Total power loss over one coupling length at resonances are ~ 6% for

*y*and 10% for x polarizations. Out of resonance, coupling length and loss of the power per coupling length rapidly increase. In between the two resonances coupling length and radiation losses remain relatively small for y polarization, while they become considerable for x polarization, making such a coupler polarization dependent.

### 3.3. Coupling via three resonators

*λ*

_{-1,0,1}corresponding to the points of avoiding crossing of the resonator supermodes with the hollow core waveguide supermodes. In the vicinity of the resonances band diagram is similar to that of a single resonator coupler, suggesting that coupling between two hollow cores is mediated by the excitation of one of the collective resonances of a three resonator structure. Spacing between coupler resonances will be proportional to the inter-resonator coupling strength. Complete power transfer between the hollow cores is possible at the resonances, with the coupling length inversely proportional to the hollow waveguide - resonator coupling strength, and independent of the inter-resonator coupling strength.

*cm*and 3

*cm*for y polarization, and 5

*cm*and 4

*cm*for

*x*polarization. Total power loss over one coupling length at resonances are 25% and 7% for

*y*polarization, and 10% and 18% for

*x*polarization. Out of resonance, coupling length and loss of the power per coupling length rapidly increase. In the lower part of Fig. 3(c) intensities of the

*x*and

*y*polarized modal electric fields are presented for the supermodes 1 and 2 of Fig. 2(c) at resonances

*λ*

_{-1}(points 1x,y on Fig. 3(c)) and

*λ*

_{0}(points 2

_{x,y}on Fig. 3(c)) showing an agreement with coupled mode theory predictions.

### 3.4. Coupling via more than three resonators

*N*

_{r}> 3 analytical expressions for the propagation constants at a phase matching point become cumbersome. However, in the limit when

*C*

_{wr}≪

*C*

_{rr}/

*N*

_{r}coupling via an array of

*N*

_{r}resonators can still be readily understood (Fig. 4). First of all, at a phase matching point

*λ*

_{0}degenerate modes will split into

*N*

_{r}+ 2 supermodes. Field distribution in most of them will be dominated by various mixing of resonator modes with only a very small content of the modes of hollow waveguides; corresponding propagation constants of such modes will be close to

*β*-

*C*

_{rr}

*sin*(

*πi*/(

*N*

_{r}+1)), where

*i*= -[

*N*

_{r}/2], [

*N*

_{r}/2]. Field distributions in the two or three modes with the propagation constants closest to the propagation constant

*C*

_{rr}≪

*C*

_{wr}. In the case of odd number of resonators (Fig. 4), the split in the propagation constants of the three supermodes closest to

*C*

_{wr}.

*Nr*resonances. Spacing between such resonances will be proportional to the inter-resonator coupling strength divided by the total number of resonators, while |

*λ*

_{i}, -

*λ*

_{0}|/

*λ*

_{0}~

*λ*

_{0}

*v*

_{g}2

*C*

_{rr}

*sin*(

*πi*/(

*N*

_{r}+ 1)),

*i*= - [

*N*

_{r}/2], [

*N*

_{r}/2]. In the vicinity of the resonances band diagram is similar to that of a single resonator coupler, suggesting that coupling between two hollow cores is mediated by the excitation of one of the collective resonances of an

*N*

_{r}resonator structure. Thus, at any resonance supermode analysis can be performed taking into account only three supermodes with propagation constants closest to the propagation constant of a mode of a stand alone hollow core waveguide. Such an analysis will be valid assuming that higher order modes are sufficiently far away

*C*

_{wr}≪

*C*

_{rr}/

*N*

_{r}. Split in the propagation constants of interacting supermodes will be proportional to the hollow waveguide - resonator coupling strength

*C*

_{wr}. At such resonant wavelengths complete power transfer from one hollow core into the other is possible after propagation over a coupling length

*L*

_{c}~

## 4. Comparison of a supermode analysis and beam propagation method

10. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method with perfectly matched layers for anysotropic optical waveguides,” J. Lightwave Technol. **19**, 405–413 (2001). [CrossRef]

*L*

_{c}there is no power left in the left core with all the remaining power transferred into the right core. Out of resonance, only partial power transfer is expected. In Fig. 5(c) in blue dotted lines for

*x*polarization and in red solid lines for

*y*polarization we plot the power remaining in the left core and the power transferred into the right core after propagation over one coupling length (note that at each wavelength coupling length is different Fig. 5(a)). In black lines we plot total power in the two cores remaining after propagation over one coupling length. We observe that even out of resonances, in the whole mode interaction region 1.311

*μm*<

*λ*< 1314

*μm*power transfer between the cores after one coupling length is in the range of 70 – 90%, the power remaining in the first core is in the range of 0 – 20%, while the total power loss is almost constant and equal 10%.

*x*nor for

*y*polarizations. Thus, an interesting question is to characterize partial power transfer from the left core into the right one close to the wavelengths of polarization independent operation. We demonstrate our analysis for

*λ*≃ 1.3118

*μm*. In Fig. 5(d) in blue dotted lines with circles for

*x*polarization and in red solid lines with circles for

*y*polarization we plot the power remaining in the left core and the power transferred into the right core after propagation over a fixed propagation length

*L*

_{c}= 10.5

*cm*corresponding to the coupling length for the

*x*and

*y*polarized modes at the point of polarization independent operation. At resonance, for both polarizations, the transmitted power in the right core is 80% with only 5% remaining in the left core. Out of the resonance, power transfer efficiency diminishes. We call a bandwidth of such a coupler a wavelength region where the power transferred into the right core is larger than the power remaining in the left core. With this definition we find the coupler bandwidths of 0.25

*nm*for

*x*and 0.5

*nm*for

*y*polarizations. In principle, to increase coupler bandwidth one has to augment hollow waveguide - resonator coupling strength.

## 5. Conclusions

## References and links

1. | M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Transverse lightwave circuits in microstructured optical fibers: waveguides,” Opt. Express |

2. | B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, and A.H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. |

3. | B.H. Lee, J.B. Eom, J. Kim, D.S. Moon, U.-C. Paek, and G.-H. Yang, “Photonic crystal fiber coupler,” Opt. Lett. |

4. | J. Canning, M. A. van Eijkelenborg, T. Ryan, M. Kristensen, and K. Lyytikainen, “Complex mode coupling within air-silica structured optical fibres and applications,” Opt. Commun. |

5. | W.E.P. Padden, M. A. van Eijkelenborg, A. Argyros, and N. A. Issa, “Coupling in a twin-core microstructured polymer optical fiber,” Appl. Phys. Lett. |

6. | H. Kim, J. Kim, U.-C. Paek, B.H. Lee, and K. T. Kim, “Tunable photonic crystal fiber coupler based on a side-polishing technique,” Opt. Lett. |

7. | J. Laegsgaard, O. Bang, and A. Bjarklev, “Photonic crystal fiber design for broadband directional coupling,” Opt. Lett. |

8. | K. Saitoh and M. Koshiba, “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,” Opt. Express |

9. | K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic vrystal fibers,” J. Quantum Electron. |

10. | K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method with perfectly matched layers for anysotropic optical waveguides,” J. Lightwave Technol. |

11. | H.A. Haus, W.P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. |

12. | 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. The fundamental connection between polarization mode dispersion and group-velocity dispersion,” J. Opt. Soc. Am. B |

**OCIS Codes**

(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers

(130.3120) Integrated optics : Integrated optics devices

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 7, 2005

Revised Manuscript: February 1, 2006

Manuscript Accepted: February 2, 2006

Published: February 20, 2006

**Citation**

Maksim Skorobogatiy, Kunimasa Saitoh, and Masanori Koshiba, "Transverse lightwave circuits in microstructured optical fibers: resonator arrays," Opt. Express **14**, 1439-1450 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1439

Sort: Year | Journal | Reset

### References

- M. Skorobogatiy, K. Saitoh, and M. Koshiba, "Transverse lightwave circuits in microstructured optical fibers: waveguides," Opt. Express 13, 7506-7515 (2005)http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-19-7506 [CrossRef] [PubMed]
- B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, and A.H. Greenaway, "Experimental study of dual-core photonic crystal fibre," Electron. Lett. 36, 1358-1359 (2000). [CrossRef]
- B.H. Lee, J.B. Eom, J. Kim, D.S. Moon, U.-C. Paek, and G.-H. Yang, "Photonic crystal fiber coupler," Opt. Lett. 27, 812-814 (2002). [CrossRef]
- J. Canning, M.A. van Eijkelenborg, T. Ryan, M. Kristensen, K. Lyytikainen, "Complex mode coupling within air-silica structured optical fibres and applications," Opt. Commun. 185, 321 (2000). [CrossRef]
- W.E.P. Padden, M.A. van Eijkelenborg, A. Argyros, N. A. Issa, "Coupling in a twin-core microstructured polymer optical fiber," Appl. Phys. Lett. 84, 1689-1691 (2004). [CrossRef]
- H. Kim, J. Kim, U.-C. Paek, B.H. Lee, and K. T. Kim, "Tunable photonic crystal fiber coupler based on a sidepolishing technique," Opt. Lett. 29, 1194-1196 (2004). [CrossRef] [PubMed]
- J. Laegsgaard, O. Bang, and A. Bjarklev, "Photonic crystal fiber design for broadband directional coupling," Opt. Lett. 29, 2473-2475 (2004). [CrossRef] [PubMed]
- K. Saitoh and M. Koshiba, "Leakage loss and group velocity dispersion in air-core photonic bandgap fibers," Opt. Express 11, 3100 (2003). [CrossRef] [PubMed]
- K. Saitoh, M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic vrystal fibers," J. Quantum Electron. 38, 927-933 (2002). [CrossRef]
- K. Saitoh, M. Koshiba, "Full-vectorial imaginary-distance beam propagation method with perfectly matched layers for anysotropic optical waveguides," J. Lightwave Technol. 19, 405-413 (2001). [CrossRef]
- H.A. Haus, W.P. Huang, S. Kawakami, N. A. Whitaker, "Coupled-mode theory of optical waveguides," J. Lightwave Technol. 5, 16 (1987). [CrossRef]
- 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. The fundamental connection between polarization mode dispersion and group-velocity dispersion," J. Opt. Soc. Am. B 19, 2867-2875 (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.