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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 4 — Feb. 21, 2005
  • pp: 1193–1201
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Light transmission in multiple or single subwavelength trefoil channels of microstructured fibers

J.M. Moison, A.M. Apetrei, J.A. Levenson, G. Mélin, Ph. Pédeboscq, A. Fleureau, S. Lempereur, and L. Gasca  »View Author Affiliations


Optics Express, Vol. 13, Issue 4, pp. 1193-1201 (2005)
http://dx.doi.org/10.1364/OPEX.13.001193


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Abstract

We evaluate the trefoil channels present between the holes of microstructured fibers as a potential dense array of small waveguides. In channels with an inner radius of 330nm, calculations indicate possible propagation with a mode waist of ~350nm at λ=670nm, near to the diffraction limit. Actual measurements have been performed on a 1-meter fiber section, with injection by a microlensed fiber and mapping of output by near-field scanning optical microscopy. They show that light can be output in individual channels or in several of them, depending on the injection. The observed waist is ~500nm, possibly due to experimental widening. Estimated propagation losses are <20dB/m. Since each channel occupies only 2µm2, this structure opens a way to dense parallel optical processing.

© 2005 Optical Society of America

1. Introduction

These progresses have not yet been used for building dense waveguide arrays, even though coupling many waveguides for discrete optics shows some fascinating prospects in optical data processing, especially in the non-linear regime [10

10. J. Meier, G.I. Stegeman, Y. Silverberg, R. Morandotti, and J.S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 1–4 (2004). [CrossRef]

]. Starting from conventional fibers, many configurations with multiple cores have been proposed. However, due again to the small Δn, the separation D between neighboring cores is rather large, depending of course on the application. For limited cross-talk between cores, separation is very large, e.g. D~50µm in multi-core fibers for local communication requirements, while in couplers where a short interaction length is required, D is less than 10µm [11

11. A.W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1270 (1972). [CrossRef]

]. Whatever the application, the size reduction initiated by moving to stronger confinement using air cladding – or mostly so – should also allow to reduce this separation. For a given desired effect, D roughly scales with the extension of the evanescent wave in the cladding, which in turn scales with 1/√Δn, so finally the density of waveguides scales with Δn. We then expect from the move to air cladding a density increase of about 100, going with the reduction of the waveguide size. As a guide to the involved sizes, we may note that for the planar air-glass interface, the extension of the evanescent wave in air is classically 0.22 [λ/n], about half the optimal core radius. Here again, this trend will find a limit, with the re-increase of the waist for core sizes less that the optimal value and with the increasing part of the power carried by the evanescent wave. In parallel, building many neighboring waveguides raises another central issue. Distant waveguide structures can still be considered as weakly coupled waveguides retaining their intrinsic specifications, and hence can be treated by coupled mode theory, leading to a continuous power exchange between the waveguides, on a propagation scale given by a coupling length. In this picture, propagation in a single guide still has a meaning. On the opposite, when such structures come close to each other, this approximation is no longer valid, and they must be considered as a whole system, having collective modes or super-modes involving coherently many structures.

In this paper we report a study of the propagation of light in the dense, numerous and nearly ultimate-sized channels present in MOFs by near-field scanning optical microscopy (NSOM). We demonstrate that in spite of an a priori strong coupling between those channels light can be transmitted over significant distances and addressed in a controlled manner to either single or multiple output channels. This behavior is reproduced by simulation results, provided that the actual slight distortions of the structure are taken into account. It also shows that dense, complex, and a priori strongly coupled systems can behave in apparently simple ways, possibly due to deviations from the ideal structure, which opens a way to high-density parallel optical signal processing.

2. Trefoil channels in microstructured fibers

We consider here a way to create ultimate waveguides which potentially retains the advantages of small mode size, protected interfaces, and massive parallel fabrication. A quasi-neglected side consequence of hole patterning in MOF fibers is the fabrication of numerous small channels in the “cladding” holey region. The behavior of these channels have been considered in contexts fairly different from ours [12

12. M. Hu, C. Wang, Y. Li, Z. Wang, L. Chai, and A.M. Zheltikov, “Multiplex frequency conversion of unamplified 30-fs Ti:sapphire laser pulses by an array of waveguiding wires in a random-hole microstructure fiber,” Opt. Express 12, 6129–6134 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-25-6129 [CrossRef] [PubMed]

,13

13. M.A. van Eijkelenborg, “Imaging with microstructured polymer fibre” Opt. Express 12,342–346 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-2-342 [CrossRef] [PubMed]

]. For the most common hexagonal hole arrangement, the silica between holes takes the form of inverted trefoil channels connected by tiny pillars (Fig. 1).

Fig. 1. sketch of a section of a MOF, detail of a region inside the holey cladding.

These channels are silica regions surrounded mostly by air, just as the center of a conventional MOF – a missing hole which is also a sixfoil channel – but on a much smaller scale, and therefore could act as waveguides. Compared to other extreme confinement devices they can be rather easily fabricated and connected to conventional fiber circuits, and can be built in large numbers in a single process.

The MOF we evaluate here is a 125µm double-coated fiber fabricated from pure silica by a double-stage stack-and-draw process. Its structure is based on a triangular lattice whose average hole diameter and pitch are respectively d=1.86±0.04µm and Λ=2.18±0.08 µm. Clearly, good confinement in channels requires a high diameter/period ratio, which is here 0.85. SEM pictures show a symmetrical hole geometry with elongated first-row holes (Fig. 2).

Fig. 2. scanning electron microscopy image of a section of the microstructured optical fiber.

The diameter of the inner circle of a channel is (2Λ/√3-d), here 0.66µm, the distance between adjacent channels is Λ/√3, here 1.26µm, and the minimum width of a pillar, which expresses the width of the bottleneck between adjacent channels, is (Λ-d), here 0.32µm. The channel density is ~0.5 µm-2, which would correspond to about 6000 channels in a fiber with the standard 125µm diameter. The MOF we consider here displays a rather high morphological stability since the variation of its outer diameter is less than 1.2% over 2km. It is generally admitted - and checked on this fiber by several cleavages - that all sizes scale with this diameter, which would give very stable channel sizes over these 2km, i.e., distances orders of magnitude larger than those envisioned for applications.

3. Simulations: trefoil channel modes

Calculations of the possible guided modes in this fiber have been performed using the finite element and the localized functions methods, which yield similar results. Since the fiber is designed for propagation in the center silica region at λ=1550nm, we have cross-checked the results of both calculations with the experimental NSOM data at this wavelength. The excellent agreement obtained [14

14. J.M. Moison, A. Apetrei, J. Levenson, G. Mélin, S. Lempereur, A. Fleureau, E. Bourov, and L. Gasca, “Evaluation of a highly nonlinear microstructured optical fiber by near-field scanning optical microscopy and simulations: nonlinear coefficient and coupling losses” Appl. Phys. B 80, 73–76 (2005). [CrossRef]

] has confirmed the ability of this structure to deliver high and robust confinements, and validated the use of the simulation methods, even at the low waist/wavelength values involved.

The possibility of propagation in the trefoil channels at shorter wavelengths has been explored by these methods. Two model structures have been considered, (i) a synthetic structure formed by a perfect lattice for all holes except the inner first ring which is represented by an average of the actual elongated circles, and (ii) the actual fiber section obtained from the electron microscopy picture (see Fig. 2). At λ=670nm, apart from a central mode with a waist ~800nm, we obtain various solutions corresponding to propagation in single or multiple trefoil channels. With the synthetic structure, we obtain mostly modes involving propagation in many channels, thus evidencing as expected a super-mode covering spatially the entire holey region (Fig. 3(a)). On the other hand, with the actual structure that takes into account the non-symmetrical and non-periodic character of the fiber section, such a super-mode is not obtained. We obtain much more localized modes involving only a few channels (Fig. 3(b)), including modes involving nearly a single channel (Fig. 3(c)).

Fig. 3. 9×9µm field maps corresponding to propagation in channels at 670nm obtained by the finite element method. Holes are indicated by black lines. Map (a) (neff=1.368588) is obtained with a synthetic symmetrical structure. Maps (b) (neff=1. 363749) and (c) (neff=1.373892) are obtained with the actual fiber structure.

The slight deviations from symmetry and distortions observed in the actual structure with respect to a regular lattice of holes lead therefore to the existence of localized modes. The choice of the preferential channel(s) also depends on the details of the structure. In addition, the size of the three tails extending towards the neighboring channels via the pillars– and hence the coupling to other channels – depends critically on the width of the pillar, as evidenced by the differences in the three extensions in Fig. 3(c) corresponding to three different pillar widths. On the other hand, the coarse shape of the intensity map inside a channel is about the same for all channels, and corresponds to a waist of 350±40nm. Its nonlinear effective area (square of the sum of intensity over sum of intensity squared) is 0.7±0.1µm2, which means an expected nonlinear coefficient γ~300W-1km-1 at 670nm. Considering that the calculated effective index is in all cases ~1.37, this waist is 0.77 in [λ/n] units, i.e., near to the diffraction limit mentioned above. According to the calculation of reference [5

5. M.A. Foster, K.D. Moll, and A.L. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express 12, 2880–2887 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2880 [CrossRef] [PubMed]

], the silica rod giving the same waist is 0.85±0.1 [λ/n], which compares to the inner radius of the channel, 0.67 [λ/n]. It is then predicted that nearly diffraction-limited modes can propagate in the channels. The actual demonstration of this effect clearly requires sub-wavelength resolution, which is obtained here by NSOM.

4. Experimental procedure for NSOM evaluation of channels

We evaluate by NSOM a 1-meter section of the MOF described above. Light sources are fiber-coupled DFB laser diodes with wavelengths of either 1550 or 670nm. Injection is performed by a microlensed fiber. The tip of this fiber is a cone ended by a spherical section, with a radius of about 4µm. The characteristics of the injection beam are not easily obtained. The good yield of the injection at λ=1550nm in small semiconductor waveguides or centers of MOF fibers suggests that its effective waist lies in ~2µm range, and a similar and possibly better behavior is expected at shorter wavelengths. Considering the small number of channels illuminated and the high convergence of the input beam, no important grating effect is expected. Finally in order to select on a fine scale the point of light launch into the entrance end of the MOF and the focusing distance, the microlensed fiber is fixed to the end of a XYZ piezoelectric drive, with minimum travel steps in the 50nm range.

The NSOM measurements are performed on the cleaved end section of the MOF. The NSOM standard apparatus involves uncoated pulled fiber tips and shear-force distance control. Images of the fiber topography and of the optical power captured by the tip are obtained simultaneously, with a lateral resolution expected to be less than the wavelength.

Large corrugations are usually hard to image correctly by tip microscopy, and NSOM of MOFs, especially on the hole scale, is particularly tricky. A special care has been taken to avoid tip and MOF damage and to increase image quality by reducing the downward travel of the tip while it is located above the holes. However, due to the very steep topographic profile, holes still appear distorted, especially on the side at which the tip emerges during the scan. Since modes mostly lie in a flat region, their maps should not be affected. This is the case for the central region, which is rather wide. For the much smaller channels, care must be exercised as to the regions near the hole edges since distortion due to the transient vertical travel of the tip could occur. Finally, the images can be unambiguously corrected for slight overall distortions after acquisition, using the known hole geometry as a template, thus keeping the uncertainty on the xy scales to less than 5%.

5. Results: single- and multi-channel light transmission

When light is injected at the center of the MOF (best transmission), the main feature of NSOM maps both at λ=670 and 1550nm is a central quasi-gaussian spot [14

14. J.M. Moison, A. Apetrei, J. Levenson, G. Mélin, S. Lempereur, A. Fleureau, E. Bourov, and L. Gasca, “Evaluation of a highly nonlinear microstructured optical fiber by near-field scanning optical microscopy and simulations: nonlinear coefficient and coupling losses” Appl. Phys. B 80, 73–76 (2005). [CrossRef]

]. When injection is offset from the center region, maps depend strongly on the wavelength. At λ=1550nm, the transmission is extremely weak and maps mostly display erratic stray light patterns. At λ=670nm, on the other hand, sharp spots located inside the channels are obtained. Fig. 4 shows the intensity map combined with the topography map around a channel. A pattern of scale circles representing the holes is moved over the complete image until a best fit with the NSOM topographic (distorted) hole patterns is obtained. This shows that the light spot is approximately located at the center of the channel.

Fig. 4. NSOM images around a single channel.

As shown on Fig. 4(b), the intensity map in the channel is about circular, with weak extensions in the regions between adjacent holes. The actual field pattern very near the holes – and especially for the tip upwards travel, on the hole right hand size – is possibly distorted, in view of the topographic distortion, so that the shape of these extensions cannot be fully trusted. On the other hand, the center of the channel mode, which lies in a flat region, should not be distorted in the NSOM intensity map.

This center is correctly approximated by a circular gaussian spot of waist 500±100nm, corresponding to an effective area of 0.9±0.2µm2. These values are somewhat larger than calculated ones. However, at such low scales, one must take into account the resolution provided by the NSOM tip. Besides being very difficult to evaluate since the tip structure is not known on this small scale, the resolution is not unique, and depends on the light pattern considered. If we assume that the mode calculation is correct, then the enlargement of the channel spot by about 30% corresponds to a resolution of about 300nm (~λ/2). This value, which also corresponds to the size of the smallest detail that can be observed on images, lies at the top of the range expected from semi-empiric models of the tip [15

15. J.M. Moison and A.C. Boccara are preparing a manuscript to be called “Semi-quantitative approach to NSOM resolution and sensitivity for the analysis of near-infrared waveguide devices”.

]. Hence the difference between calculation and NSOM data may be due to an experimental enlargement. However, it might also correspond to the simultaneous excitation of higher-order modes: for comparison, in silica wires at similar wavelengths, the critical radius for single-mode operation is about 225nm [7

7. L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1025 [CrossRef] [PubMed]

], which is smaller than the effective radius of the channels.

The power transmitted in a single channel may be evaluated from the integral of the NSOM image over the area of this channel, typically a disk with a radius equal to half the distance between adjacent channels. For a laser power of 1mW at 670nm, this power is at most – i.e., for optimized injection conditions and single channel output – ~0.55µW; for reference, when injection is centered, the power output in the center waveguide is ~20µW at 670nm and ~150µw at 1550nm. This apparent ~33 dB loss in the transmission from the laser to the single-channel output actually encompasses several contributions, namely the transmission loss in the microlensed fiber, the coupling loss to the channel, the coupling to MOF leaky modes, and the transmission loss in the MOF. The first contribution is easily determined by direct power measurements and corresponds to about 3dB. The second one may be evaluated by comparison to measurements at 1550nm. At this wavelength where fibers have only weak transmission losses, ~450mW total power is output, including center mode and ring modes [14

14. J.M. Moison, A. Apetrei, J. Levenson, G. Mélin, S. Lempereur, A. Fleureau, E. Bourov, and L. Gasca, “Evaluation of a highly nonlinear microstructured optical fiber by near-field scanning optical microscopy and simulations: nonlinear coefficient and coupling losses” Appl. Phys. B 80, 73–76 (2005). [CrossRef]

]. In the simple picture of the injection seen as coupling mismatched gaussian modes, this coupling efficiency corresponds to a waist of the injector of ~2.35µm, in agreement with the microlensed fiber specifications. This evaluation of the injector waist, which incidentally suggests that injection involves only a few channels, also allows us in turn to calculate the coupling loss to the much smaller channel mode, which is ~11dB. This finally yields a ~20dB/m value for the single-channel propagation loss. This value which corresponds to a propagation length of at least 20cm is pessimistic since it corresponds to the maximum coupling at 1550nm and zero contribution of other modes at 670nm, whereas we measure that the single channel carries only a fraction of the total output power. Though the distribution of power in the rest of the structure would be very useful information, its measurement is difficult and not yet reliable due to its large and seemingly featureless spreading. Therefore these data cannot yet help understanding the details of the propagation process, but more simply give an order of magnitude of the overall transmission for future applications.

Finally Fig. 5 shows examples of the NSOM patterns obtained for various injection conditions. Output in nearly a single channel (Fig. 5(a)) has been clearly observed for several different channels when the injection location is optimized so as to yield the highest NSOM output signal. Output in several neighboring channels (Fig. 5(b)) is obtained when injection is slightly offset from this optimum position.

Fig. 5. NSOM images of the end section of the MOF illuminated at 0.67µm. Image height is 9.2µm. Images are combinations of topographic maps and optical intensity maps. The contrast of each image is normalized to maximum and minimum intensities, so that only relative information is displayed; peak intensity in (a) is about 20 times that in (b).

Our present experimental configuration does not allow us to know where injection actually takes place with respect to the MOF structure, and all we can measure are movements relative to the “optimum” position of the injection fiber. Considering the lateral displacements, the estimation of the injection displacement required to switch from output in a single channel to output in another single channel is in agreement with the actual distance between those channels, up to about 5µm. The axial displacements required to switch form single-channel to multiple channel output is much smaller (<1µm) and cannot be readily understood as mere defocusing due to side effects such as driver hysteresis and parasitic cross displacements, and also to uncertainties in the details of the injection pattern. Establishing clearly the correlation between input and output locations, and between input beam size and output pattern is clearly important but experimentally complex. Nevertheless, our present data demonstrate unambiguously that after propagation over distances in the 1-meter range, selective output in either nearly a single channel or several adjacent channels is possible.

6. Discussion and conclusion

We have evaluated the trefoil channels of a microstructured fiber as a dense array of small waveguide structures. Each channel is indeed near to the ultimate size: in channels with an inner radius of 330nm, calculations indicate possible propagation in a single channel with a mode waist of ~350nm at a wavelength of 670nm, near to the diffraction limit. Each channel occupies only 2µm2, which also opens a way to ultimately dense arrays. Measurements show that light can travel along a 1-meter fiber section and exit in individual channels or in several of them, depending on the injection conditions. Observed waists are in the 500nm range, possibly due to experimental widening, and propagation losses are estimated to be <20dB/m. Beyond this first result, it is clear that important issues need to be explored.

The main issue is clearly the interaction between neighboring channels. Considering the high packing density of the channels and the results of the calculations which indicate that the field of one channel can get deeply into its neighbors, a strong coupling of the channels is predicted. For equivalent channels, this would mean propagation in collective or super-modes involving many channels or in the picture of coupled waveguides a short coupling length L, certainly well smaller than the 1m we considered. Either way, this should lead to an ever-spreading partition of the power among the neighbors of the input channel, i.e. a defocusing effect (discrete diffraction). Nonlinear focusing effects leading to self-organization of light and solitonic propagation which have been demonstrated in 1D [16

16. R. Morandotti, H.S. Eisenberg, Y. Silberberg, M. Sorel, and J.S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001). [CrossRef] [PubMed]

,17

17. D.N. Christodoulides and R.I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed]

] and recently 2D [18

18. E.J. Bochove, P.K. Cheo, and G.G. King, “Self-organization in a multicore fiber laser array” Opt. Lett. 28, 1200–1202 (2003). [CrossRef] [PubMed]

] waveguide arrays are not expected to appear at the small power densities we consider (~1mW/µm2 or 1MW/m2). However, at what is expected to be a late stage of the propagation (many Ls), where we would expect to observe only the guided modes, we obtain power robustly located in a single channel, near to the input one.

The stabilization mechanism that restores an apparent single-waveguide behavior is not clearly elucidated. The result of the calculations strongly points to the small distortions from the perfect hole lattice, which indeed generate localized modes for which the power is mostly transmitted in a single channel. However, beyond the part played by the local geometry of holes, the potential contribution of effects such as coherent interferences between super-modes for stabilizing localized solutions still has to be evaluated. In reference to the problem of the description of the channels by weakly coupled localized modes or delocalized super-modes, while a description of a perfect array of identical channels in terms of super-modes should be preferred due to the strong coupling, the actual deviations from this scheme seem strong enough to dictate a behavior resembling single channel guidance. Injection conditions certainly also play a part here. The difference between single and multiple channel transmission can be attributed to the selective excitation of the highly localized or more extended modes shown in Fig. 3. Further experimental indications are clearly needed here. Besides more extended calculations, the determination of the correlation between the (single or few) input channel(s) and the single or multiple output channel(s), and of the actual change of injection conditions switching from single to multiple-channel transmission, would be here of prime importance, even though this is by no means a trivial measurement. In parallel, experiments involving small wavelength scans would probably help clearing up the above issues. Such work is under way in our laboratory.

From a more practical point of view, the observation that single or multiple channel transmission can be obtained in a controlled manner opens prospects of applications involving optical parallel data processing (multiplexers, memories, image processing, pattern recognition, neural networks…) through linear and nonlinear effects, with a nearly ultimate processing density. However, any path towards ultimate waveguide size and density involves new problems, and among them propagation losses. The high losses commonly observed in small guides are partly attributed to the large fraction of the power carried travelling in the air cladding and hence experiencing the roughness and contamination of the air/glass interface [19

19. K.K. Lee, D.R. Lim, H.C. Luan, A. Agarwal, J. Foresi, and L.C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: experiments and model” Appl. Phys. Lett. 77, 1617–1619 (2000). [CrossRef]

]. Indeed, losses reported for the optimal-diameter silica rod are in the 100dB/m range [6

6. L. Tong, R.R. Gattass, J.B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003). [CrossRef] [PubMed]

] for λ=633 or 1550nm. Locally tapered MOFs have smaller losses (~20dB/m) and an increased lifetime [8

8. Y.K. Lizé, E.C. Mägi, V.G. Ta’eed, J.A. Bolger, P. Steinvurzel, and B.J. Eggleton “Microstructured optical fiber photonic wires with subwavelength core diameter,” Opt. Express 12, 3209–3217 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3209 [CrossRef] [PubMed]

], possibly due to the protection of the interface from the environment leaving only the roughness problem. Our <20dB/m value compares with the 12dB/m obtained in silica wires for similar values of r/(λ/n). For single-channel nonlinear operation, such values lead to a tradeoff between nonlinear coefficient increase and propagation length decrease that is worse in channels than in a MOF center. However, it is expected that better figures can be obtained in optimized structures, since MOF center sixfoil channels have proven to be much less lossy than silica wires [8

8. Y.K. Lizé, E.C. Mägi, V.G. Ta’eed, J.A. Bolger, P. Steinvurzel, and B.J. Eggleton “Microstructured optical fiber photonic wires with subwavelength core diameter,” Opt. Express 12, 3209–3217 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3209 [CrossRef] [PubMed]

]. Furthermore, in our case part of this loss may not be actual loss but mere spreading part of the power into numerous waveguides. Anyway, considering the coupling length, the present 20cm propagation length is already larger than the expected working length of future devices based on coupling effects.

Acknowledgments

The authors are indebted to A.M. Zheltikov and M. Bensoussan for precious advice and support. Fruitful discussions within the COST P11 action are also acknowledged. This work has been supported by the “Région Ile de France,” by SESAME project n°1377 and by the “Conseil Général de l’Essonne” (contract OPSAVE).

References and links

1.

J. Knight, J. Broeng, T. Birks, and P. Russell, “Photonic band guidance in optical fibers,” Science 282, 1476–1478 (1998). [CrossRef] [PubMed]

2.

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

3.

J. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

4.

A.M. Zheltikov, “The physical limit for the waveguide enhancement on nonlinear optical processes,” Optics & Spectroscopy 95, 410–415 (2003). [CrossRef]

5.

M.A. Foster, K.D. Moll, and A.L. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express 12, 2880–2887 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2880 [CrossRef] [PubMed]

6.

L. Tong, R.R. Gattass, J.B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003). [CrossRef] [PubMed]

7.

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1025 [CrossRef] [PubMed]

8.

Y.K. Lizé, E.C. Mägi, V.G. Ta’eed, J.A. Bolger, P. Steinvurzel, and B.J. Eggleton “Microstructured optical fiber photonic wires with subwavelength core diameter,” Opt. Express 12, 3209–3217 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3209 [CrossRef] [PubMed]

9.

M.A. Foster and A.L. Gaeta, “Ultra-low threshold supercontinuum generation in sub-wavelength waveguides,” Opt. Express 12, 3137–3143 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3137 [CrossRef] [PubMed]

10.

J. Meier, G.I. Stegeman, Y. Silverberg, R. Morandotti, and J.S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 1–4 (2004). [CrossRef]

11.

A.W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1270 (1972). [CrossRef]

12.

M. Hu, C. Wang, Y. Li, Z. Wang, L. Chai, and A.M. Zheltikov, “Multiplex frequency conversion of unamplified 30-fs Ti:sapphire laser pulses by an array of waveguiding wires in a random-hole microstructure fiber,” Opt. Express 12, 6129–6134 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-25-6129 [CrossRef] [PubMed]

13.

M.A. van Eijkelenborg, “Imaging with microstructured polymer fibre” Opt. Express 12,342–346 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-2-342 [CrossRef] [PubMed]

14.

J.M. Moison, A. Apetrei, J. Levenson, G. Mélin, S. Lempereur, A. Fleureau, E. Bourov, and L. Gasca, “Evaluation of a highly nonlinear microstructured optical fiber by near-field scanning optical microscopy and simulations: nonlinear coefficient and coupling losses” Appl. Phys. B 80, 73–76 (2005). [CrossRef]

15.

J.M. Moison and A.C. Boccara are preparing a manuscript to be called “Semi-quantitative approach to NSOM resolution and sensitivity for the analysis of near-infrared waveguide devices”.

16.

R. Morandotti, H.S. Eisenberg, Y. Silberberg, M. Sorel, and J.S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001). [CrossRef] [PubMed]

17.

D.N. Christodoulides and R.I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed]

18.

E.J. Bochove, P.K. Cheo, and G.G. King, “Self-organization in a multicore fiber laser array” Opt. Lett. 28, 1200–1202 (2003). [CrossRef] [PubMed]

19.

K.K. Lee, D.R. Lim, H.C. Luan, A. Agarwal, J. Foresi, and L.C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: experiments and model” Appl. Phys. Lett. 77, 1617–1619 (2000). [CrossRef]

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(130.2790) Integrated optics : Guided waves
(250.5300) Optoelectronics : Photonic integrated circuits
(350.3950) Other areas of optics : Micro-optics

ToC Category:
Research Papers

History
Original Manuscript: January 6, 2005
Revised Manuscript: January 4, 2005
Published: February 21, 2005

Citation
J. M. Moison, A. M. Apetrei, J. A. Levenson, G. Melin, Ph. Pedeboseq, A. Fleureau, S. Lempereur, and L. Gasca, "Light transmission in multiple or single subwavelength trefoil channels of microstructured fibers," Opt. Express 13, 1193-1201 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-4-1193


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References

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