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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 4 — Feb. 20, 2006
  • pp: 1439–1450
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Transverse lightwave circuits in microstructured optical fibers: resonator arrays

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


Optics Express, Vol. 14, Issue 4, pp. 1439-1450 (2006)
http://dx.doi.org/10.1364/OE.14.001439


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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

In this paper we present another type of transverse lightguides - weakly coupled resonator arrays which are considerably less polarization sensitive. We believe that such arrays is an excellent starting point for design of polarization insensitive resonant couples in the multicore MOFs.

2. Structure and definitions

Fig. 1. Schematic of a two hollow core MOF coupler. Cores are separated by N lattice periods. Transverse waveguide is formed by a periodic array of holes of smaller diameter (resonators) separated by Nrr periods; Nwr holes closest to the cores are not modified.

3. Weakly coupled fiber arrays

In what follows, dispersion relations of the supermodes of a complete coupler (comprised of the two hollow cores and resonators) were calculated with a Finite Element Method (FEM) mode solver [9

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]

]. Predictions of the supermode analysis for the coupling length and coupler losses were then verified by a FEM Beam Propagation Method (BPM)[10

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]

].

Ft(rt,λ)=i=1Nr+2Pi(λ)Fti0(rt,λ0),
(1)

β(λ)NP¯(λ)=ND(λ)P¯(λ)+ΔHP¯(λ).
(2)

Wavelength independent mode overlap N and coupling ΔH matrixes, and diagonal matrix D(λ) are defined as:

Nj,i=14ẑdrt(Eti0×Htj0*+Etj0*×Hti0)
ΔHj,i=ω4drt(εεi)(Etj0*·Eti0*+εjεEzj0*Ezi0).
Di,i(λ)=βi0(λ)
(3)

In the analysis of weakly coupled resonator arrays, matrices N and ΔH 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:

N=1NwrLNwrL*1NrrNrr*1Nrr...Nrr*1NrrNrr*1NwrR*NwrR1;ΔH=0CwrLCrwL0CrrLRCrrRL0CrrLR...CrrRL0CrrLRCrrRL0CrwRCwrR0,
(4)

In the case of coupling through an array of weakly interacting resonators, supermode dispersion relations and coupling characteristics will be only moderately polarization dependent. Particularly, fundamental modes of the stand alone hollow core waveguides and resonators considered in this paper are degenerate doublets labelled as 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.

Finally, we would like to note that not all the modes of a resonator are degenerate doublets, some of the higher order resonator modes can be true singlets. When operating at the wavelengths corresponding to the excitation of such singlet resonator modes, coupler will exhibit strong polarization dependence. Particularly, even if the mode of operation of a hollow waveguide is a degenerate doublet supporting 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.

In what follows, we disregard the so-called “self-slowing” terms Ni,j ; jj 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 CwrCrr 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

We start with a case of two hollow cores coupled via one resonator. In Fig. 2(a) dispersion relations β (λ) of the supermodes relative to a dispersion relation of a core mode of a stand alone hollow core fiber βw0(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.

Fig. 2. Schematic of the supermode dispersion relations in weakly coupled a) 1, b) 2, c) 3 resonator arrays. Dispersion relations are plotted with respect to the one of a stand alone hollow core waveguide. Solid red curves - even supermodes, dotted blue curves - odd su-permodes. Black dotted curve - dispersion relation of a fundamental mode of a stand alone resonator. Inserts - schematics of the field distributions at a phase matching point λ 0.

Out of the resonance λλ 0 we adopt the following definition of a coupling length:

Lc=λ(2Δneff)
Δneff=min(Re(neff1neff2),Re(neff2neff3)).
(5)

Ld=λ(4πmax(Im(neff1),Im(neff2),Im(neff3)).
P(Lc)P(0)=exp(LcLd)
(6)

3.2. Coupling via two resonators

In Fig. 2(b), dispersion relation of the supermodes relative to a dispersion relation of a core mode of a stand alone hollow core fiber are shown for the case of two hollow waveguides coupled via an array of two resonators. As in the case of one resonator, at a phase matching point all the degenerate modes (two of the hollow waveguides and two of the resonators) will split by forming four properly symmetrized supermodes. Modes 1 and 4 will be split the most with a difference in their propagation constants (β 4 - β 1) ~ Crr 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) ~ Cwr2/CrrCwr . Field distribution in these supermodes will be dominated by the fields of the hollow waveguide core modes.

The key feature of this system is appearance of two new resonances at the wavelengths 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 vgCrr , where vg 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 Lc ~ Cwr1 and independent of the inter-resonator coupling strength.

3.3. Coupling via three resonators

In Fig. 2(c), dispersion relation of the supermodes relative to a dispersion relation of a core mode of a stand alone hollow core fiber are shown for the case of two hollow waveguides coupled via an array of three resonators. As in the case of two resonators, there are several resonant wavelenghts 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.

Fig. 3. Supermode analysis predictions. Coupling length, loss per one coupling length, and intensities of electric fields at resonances for the weakly coupled a) 1, b) 2, c) 3 resonator arrays. Dotted blue curves - x polarization, solid red curves - y polarization.

3.4. Coupling via more than three resonators

In a case of more than three resonators Nr > 3 analytical expressions for the propagation constants at a phase matching point become cumbersome. However, in the limit when CwrCrr /Nr coupling via an array of Nr resonators can still be readily understood (Fig. 4). First of all, at a phase matching point λ 0 degenerate modes will split into Nr + 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 - βw0 ≃ 2Crrsin (πi/(Nr +1)), where i = -[Nr /2], [Nr /2]. Field distributions in the two or three modes with the propagation constants closest to the propagation constant βw0 of a mode of a stand alone hollow core waveguide will exhibit strong mixing of the resonator modes with the modes of the hollow cores. In the case of even number of resonators, the split in the propagation constants of the two supermodes closest to βw0 will be proportional to Cwr2/CrrCwr . In the case of odd number of resonators (Fig. 4), the split in the propagation constants of the three supermodes closest to βw0 will be proportional to Cwr .

In spectral domain one observes appearance of Nr resonances. Spacing between such resonances will be proportional to the inter-resonator coupling strength divided by the total number of resonators, while |λi , - λ 0|/λ 0 ~ λ 0 vg 2Crrsin(πi/(Nr + 1)), i = - [Nr /2], [Nr /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 Nr 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 CwrCrr /Nr . Split in the propagation constants of interacting supermodes will be proportional to the hollow waveguide - resonator coupling strength Cwr . At such resonant wavelengths complete power transfer from one hollow core into the other is possible after propagation over a coupling length Lc ~ Cwr1 inversely proportional to the hollow waveguide - resonator coupling strength, independent of the inter-resonator coupling strength.

4. Comparison of a supermode analysis and beam propagation method

Directly at resonance complete power transfer is expected so that after propagation over one coupling length Lc 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 1.311μm < λ < 1314μm 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%.

Fig. 4. Schematic of supermode dispersion relations in a weakly coupled Nr = 11 resonator array. Dispersion relations are plotted with respect to the one of a stand alone hollow core waveguide. Solid red curves - even supermodes, dotted blue curves - odd supermodes, black dotted curve - resonator mode in the absence of coupling. At each of the Nr coupling resonances λi,i = [-5,5] energy transfer from one hollow core into the other is possible via excitation of one of the collective resonance of an 11 resonator array.

Necessary condition for polarization insensitive operation of a coupler is the equality of the coupling lengths for the two polarizations. In Fig. 5 thin vertical dotted lines signify two wave-lenghts (which we further call wavelengths of polarization insensitive operation) at which this condition is satisfied. Such wavelengths do not coincide with coupling resonances neither for 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 ≃ 1.3118μm. 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 Lc = 10.5cm 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 0.25nm for x and 0.5nm for y polarizations. In principle, to increase coupler bandwidth one has to augment hollow waveguide - resonator coupling strength.

Fig. 5. Comparison between modal analysis predictions and BPM simulations for the a) coupling length, b) loss per one coupling length. Solid curves - mode analysis, dotted curves with circles - BPM. Blue curves - x polarization, red curves - y polarization; c) BPM simulations of power transfer from the left core into the right core. Blue dotted curves (x polarization) and red solid curves (y polarization) - power in the left and right cores after one coupling length; black curves - total power in both cores after one coupling length. Cyan dotted curves with circles (x polarization) and magenta solid curves with circles (x polarization) - power in the cores after propagation over a fixed distance Lc = 10.5cm corresponding to a coupling length at the resonance λ ≃ 1.3118μm.

5. Conclusions

References and links

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]

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]

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]

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


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References

  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, 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, 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 sidepolishing 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]
  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]
  9. 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]
  10. 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]
  11. H.A. Haus, W.P. Huang, S. Kawakami, 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]

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