Tapered Velocity Mode-Selective Couplers - OSA Publishing

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 13, JULY 1, 2013

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Tapered Velocity Mode-Selective Couplers Nicolas Riesen and John D. Love

Abstract—In this paper tapered velocity mode-selective couplers are presented. These tapered couplers are approximately adiabatic devices which unlike standard mode-selective couplers, do not rely on precise phase conditions to be satisfied over an extended length, to achieve their mode-selective functionality. Therefore these mode couplers permit ultra-wideband mode-division multiplexing of few-mode waveguides. Moreover, their behavior is largely independent of parameters such as index contrast and the precise coupler dimensions. The mode-selective functionality is demonstrated in slab geometry, with the principles generalizing to planar or fiber-based devices. Index Terms—Mode-division multiplexing, mode-selective couplers, mode splitters, tapered velocity couplers.

I. INTRODUCTION

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ITH the advent of mode-division multiplexing for potentially increasing the capacity of optical fiber, significant interest has emerged for developing wavelength-independent mode multiplexers/demultiplexers which have high coupling efficiency and low loss. Currently, bulk free-space optics are used to excite and separate the modes in experimental demonstrations of few-mode fiber networks [1]–[3]. However, such approaches are both lossy and cumbersome for commercial implementation. Therefore, significant incentives exist for developing compact waveguide-based mode-multiplexers, which not only share the wavelength-independence of the existing techniques, but are also low loss and low complexity. Previous suggestions for low loss waveguide-based mode multiplexers/demultiplexers include asymmetric Y-junctions [4], [5] and mode-selective couplers [6], [7]. Asymmetric Y-junctions are approximately adiabatic devices whose mode-selective functionality is indeed largely independent of wavelength. However, since they are planar devices, they are limited to the multiplexing/demultiplexing of modes of fixed spatial-orientation such as those of polarization-maintaining or elliptical-core fiber [5]. On the other hand, mode-selective couplers do not have the same restrictions, and it has been shown that three-core variants can in principle demultiplex any mode of a few-mode circularly-symmetric fiber, independent of its incoming spatial-orientation [7]. However, these are essentially interference-based Manuscript received February 23, 2013; revised May 21, 2013; accepted May 21, 2013. Date of publication May 23, 2013; date of current version June 05, 2013. The work of N. Riesen was supported in part by Australian Research Scholarships from both The Australian National University, Canberra and the Commonwealth Scientific and Industrial Research Organisation, Lindfield. The authors are with the Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia (e-mail: nicolas. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2013.2264827

devices and so the mode-selective functionality is dependent on precise phase conditions, such that high wavelength-dependence is inevitable [8]. In this paper it is shown that the benefits of both these waveguide-based multiplexers can be realized in the form of a hybrid structure, referred to as a tapered velocity mode-selective coupler (TVMSC). This device represents an extension of single-mode tapered velocity couplers proposed many years ago, but which found limited use in integrated optics due to practical difficulties and increased interaction lengths outweighing marginal benefits over standard directional couplers [9]–[16]. Whilst there may be little use for wavelength-independent 100% power transfer tapered couplers in single-mode applications, the same is not true if the power transfer occurs between different modes. Therefore these tapered couplers may finally find a niche in the form of the mode-selective variants presented in this paper. Tapered velocity mode-selective couplers could permit the highly-desired, wavelength-independent, mode-division multiplexing of any given spatial-mode in a few-mode fiber network. The mode-selective functionality is also insensitive to parameters such as index contrast and the precise coupler dimensions. In this paper, the mode-selective functionality of these parameter-insensitive devices is demonstrated numerically in slab geometry for simplicity, although the underlying principles apply to planar or all-fiber devices. Realization of these simple couplers as all-fiber devices using novel fabrication techniques such as femtosecond-laser direct write [17], [18], would represent a significant leap forward in the goal of realizing practical wavelength-multiplexed few-mode fiber networks. II. MECHANISM OF OPERATION Tapered velocity couplers [9]–[16] are related to standard directional couplers, with the main difference usually being that an approximately adiabatic taper is introduced into one or both of the waveguides. If the cross-sectional variation of the device is sufficiently slow, power transfers between the waveguides provided that the propagation constants of the modes cross somewhere along the taper. As mentioned, the advantages of these devices compared with standard directional couplers include much lower wavelength-dependence and increased dimensional tolerances, albeit at the expense of increased interaction length. To understand the mechanism of operation of tapered velocity couplers, firstly consider the standard directional coupler. Standard directional couplers typically consist of two uniform waveguides, and support two forward normal modes of propagation. Both of these forward normal modes must be present in order to allow for power to transfer between the two waveguides [9]. Since the two normal modes in general travel at different phase velocities, they can interfere with each other, resulting in

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Fig. 1. (a) Schematic of a single-mode tapered velocity coupler with counter tapered cores A and B, and (b) the propagation constant crossover. The precise taper shape is immaterial, provided it varies sufficiently slowly.

a standing wave or beat pattern [9]. Power transfer occurs due to interference of these two normal modes, and this is why these couplers are often referred to as mode interference directional couplers [9]. In order to obtain 100% power transfer, the device length is made equal to half the beat wavelength. Since these standard directional couplers rely on the interference of normal modes they tend to be highly sensitive to the precise parameters, such as wavelength [13]. As suggested, the parameter sensitivity of standard directional couplers can however be resolved by introducing approximately adiabatic tapers into one or both of the waveguides. The case of a slab geometry single-mode tapered velocity coupler is represented in the schematic of Fig. 1. In this case the waveguide cores are counter tapered. In a tapered velocity coupler the propagation constant difference between the two waveguides and the coupling coefficient, vary along the length of the device [9]. The concept of normal modes is however not strictly valid for tapered velocity couplers due to the non-uniformity of the waveguides. However, if the propagation constant difference and the coupling coefficient vary sufficiently slowly (i.e., approximately adiabatically), we may consider what are known as quasi-normal modes [9]. A quasi-normal mode is a mode of the overall coupler cross-section at a given point along the device. The assumption here is that the parameters vary slowly over a length scale equal to the local beat wavelength. This avoids loss of power to the radiation field or crosstalk between quasi-normal modes. The quasi-normal modes at a particular point along the device differ only slightly from the hypothetical normal modes that would have existed if the waveguides were actually uniform either side of the local point [9]. Now, the quasi-normal modes vary along the taper which means that whilst a given quasi-normal mode may carry all the power in one core at the start of the device, it may turn out to resemble all the power in the other core at the end. This is the requirement for high localization of energy or near 100% power transfer [19], [20]. Therefore the field distribution of the quasi-normal mode slowly changes from that resembling all power in one core to that resembling all power in the adjacent core. Unlike directional couplers, there is no interference between modes. Thus the mechanism for power transfer is quite different to that of standard directional couplers, though not completely unrelated. Moreover, in a tapered velocity coupler it is not necessary to


reduce the coupling coefficient to zero outside of a prescribed coupling length as is the case with standard directional couplers [13]. This is because any interaction stops far from the point of propagation constant matching even if the waveguides may still be close together as in Fig. 1. This essentially explains the insensitivity of tapered velocity couplers to parameters such as wavelength when compared to standard directional couplers. The principles of tapered velocity couplers presented here are shown to extend beyond the realms of single-mode waveguides and also apply to few- or multimode waveguides. Moreover, the power transfer can be between different modes of dissimilar cores. Therefore, provided the adiabatic criterion [21], [22] is satisfied, and a propagation constant crossover point does exist for the relevant modes, tapered velocity couplers can exhibit mode-selective functionality. III. SLAB GEOMETRY TAPERED VELOCITY MODE-SELECTIVE COUPLERS The majority of previous theoretical research on single-mode tapered velocity couplers (TVCs) assumes weak-coupling. For example Louisell [9], Milton and Burns [11], and Smith [12] have long reported analytical work on single-mode weakly-coupled tapered velocity couplers. However, to allow for short device lengths whilst avoiding coupling between the quasi-normal modes or the radiation field strong-coupling is essential [19]. Very limited theoretical work however exists on strongly-coupled devices largely due to the innate theoretical complexity [19]. Various simple numerical techniques do however exist to facilitate analysis of such devices. In this section numerical Beam Propagation Method (BPM) simulations are used to characterize the functionality of tapered velocity mode-selective couplers. For simplicity, the coupling behavior is analyzed in slab geometry, although the principles apply to other waveguide structures such as all-fiber devices. A. Two-Mode Couplers This section discusses two-mode tapered velocity mode-selective couplers (TVMSCs) for the multiplexing/demultiplexing of two slab waveguide modes. The principles of operation are similar to those of a two-arm asymmetric Y-junction [4], [5]. Asymmetric Y-junctions are however generally restricted to slab/planar geometry, whereas TVMSCs are not. This section proceeds to discuss the behavior of slab TVMSCs, with the results being indicative of the performance that can be expected for fiber-based devices. The behavior of two-mode TVMSCs is discussed with reference to the example configuration of Fig. 2(d). The configuration was optimized numerically using various programmed parameter sweeps. The optimization ensured strong coupling, high localization of energy, an acceptable taper ratio and a short taper length. As shown in Fig. 2(d), the interaction region consists of two cores with counter tapers. The centre core initially supports the fundamental T0 and second T1 ( , with cut-off at 2.65 m) slab modes assuming that and nm. The width of the centre core tapers from 5.26 m down to 1.9 m over a length of 5.22 mm, whereas the single-mode outer

RIESEN AND LOVE: TAPERED VELOCITY MODE-SELECTIVE COUPLERS

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Fig. 3. Propagation constant crossover for the two-mode TVMSC, modeled under the assumption of weak-coupling. This approximation of weak-coupling means that the cores can be considered in isolation for the modal analysis.

Fig. 2. Numerical TE simulations of a two-mode TVMSC showing isolation of (a) the fundamental T0 and (b) second T1 slab modes. Simultaneous demultiplexing of both modes is shown in (c). The divergence of the outer core towards the end of the device (d) is crucial to achieving extremely high levels of energy localization.

core tapers from a zero initial width to 1.36 m over a length of 4 mm. The fundamental T0 mode is never cut-off and assuming it propagates either way through the centre core, never couples to the single-mode outer core. This behavior is predicted from Fig. 3, in which the propagation constant of the T0 mode in the centre core never crosses or even comes close to that of the fundamental mode in the outer core. This is true for all locations along the length of the taper. Therefore for mode-selective applications, the fundamental mode of a two-mode waveguide could be excited or detected directly via the centre core as in Fig. 2(a). From Fig. 3, it is also clear that the second T1 mode in the centre core will couple to the fundamental mode of the outer core. This occurs because of the crossover of modal propagation constants along the taper. By reciprocity, the reverse is also true and so the second mode of the centre core can be excited via the fundamental mode of the outer core (see Fig. 2(b)). Thus the second mode of a two-mode core could be excited or separated via the outer core of such a device. Note also that the second T1 mode in the centre core is well cut-off at the end of the taper. This significantly improves the level of mode extinction possible for mode-division multiplexing purposes since residual uncoupled power in the T1 mode is radiated away. This would serve the purpose of avoiding corruption of the fundamental mode data channel. Moreover, the outer core diverges away towards the end of the device, in order to well-separate

the light in the cores. This allows for very high levels of energy localization without requiring an impractical device length. Conceptually, the mode-selective functionality observed in Fig. 2 can be described using quasi-normal modes. Since the device is approximately adiabatic a quasi-normal mode excited at the start of the coupler does not couple to any other quasinormal modes. Here, the outer core initially has an approximately zero width to ensure a non-abrupt cross-sectional variation, as is necessary to satisfy the adiabatic criterion. Whilst the quasi-normal mode may resemble the second T1 mode in the centre core at the start of the device, the field distribution changes gradually along the taper until finally it resembles that of all power being in the fundamental T0 mode of the outer core. The diverging outer core then further facilitates isolation of the mode power. 1) Parameter Dependence: In a TVMSC, varying parameters such as wavelength and index contrast, results in the displacement of the propagation constant crossover point along the taper length. However this crossover point need not be precisely defined. This is true provided it occurs somewhere in the middle of the taper. Therefore if the taper is sufficiently long, small relative displacements in the propagation constant crossover point will occur even over a large parameter space. Hence the device’s mode-selective functionality can be made largely independent of parameters such as wavelength, index contrast or the precise dimensions. Thus the precise taper ratios, taper lengths and other dimensions are not critical to a TVMSC’s performance. This is a significant advantage over standard directional mode-selective couplers. The wavelength insensitivity of the two-mode TVMSC is shown in Fig. 4, where in excess of 85% coupling efficiency is possible over a more than 1000 nm bandwidth. Moreover, for a bandwidth spanning 1500 nm to 2300 nm, the coupling efficiency exceeds 98.5%. This is remarkable considering that this bandwidth is approximately an order of magnitude greater than that considered useable in wideband wavelength division multiplexing (WWDM) [23]. In fact, for the extended C+L EDFA bands negligible degradation in the coupling efficiency is observed. The coupling efficiency here is affected only by radiation losses since no measurable mode crosstalk was observed. The radiation losses are a result of the second mode being cut-off

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Fig. 4. Coupling efficiency of the two-mode TVMSC as a function of wavelength. The coupling efficiency exceeds 85% for a 1000 nm bandwidth.

Fig. 6. Movement of the 3 dB point of power transfer from the second T1 mode in the centre core to the fundamental T0 mode of the outer core, as the wavelength is varied from 1.3 m to 2.0 m.

Fig. 5. Mode extinction ratios of the T0 and T1 modes (separately excited in the centre core) of the two-mode TVMSC, as the final taper width of the centre core is varied from 1 m to 4.2 m. The nominal width is 1.9 m, and thus this represents in excess of 100% deviation from the nominal dimension.

Fig. 7. Movement of the 3 dB point of power transfer from the second T1 mode in the centre core to the fundamental T0 mode of the outer core, as the final width of the centre core is varied from 3.0 m to 4.2 m.

towards the end of the device prior to all power being decoupled. If the second mode were not cut-off towards the end of the device, the power otherwise radiated away would corrupt the fundamental mode channel. However since the residual power is very small (i.e., in this case less than 1.5% over a wavelength range of 1500 nm to 2300 nm), only slight degradation of the associated mode extinction ratio would be expected. In either case, ultra-broadband performance can be expected. Furthermore, the high dimensional tolerance of the TVMSC is represented in Fig. 5, by varying the final width of the centre core from 1 m to 4.2 m, with the effect being only marginal changes in the mode extinction ratios. The extinction ratio for each separately excited mode of the centre core, is defined as the power exiting the correct output port relative to the power exiting the wrong output port. Fig. 5 demonstrates in excess of 20 dB ( %) mode extinction for either mode over the entire parameter space. This result demonstrates the high dimensional tolerances attributed to TVMSCs. The parameter insensitivity demonstrated in this section can be explained with reference to Figs. 6, 7 and 8. These figures show the level of power transfer (between T1 of the centre core and T0 of the outer core) along the length of the device, as functions of wavelength, final taper width and absolute index difference, respectively. The plots show that the point of 3 dB power transfer moves along the device length as the respective parame-

ters are varied. Movement of the 3 dB point is synonymous with movement of the propagation constant crossover point in the same direction. However since the mode-selective functionality requires only that the crossover occurs somewhere towards the centre of the taper, and given that the parameter changes have marginal effects on the level of adiabaticity, the performance is essentially unaffected even by significant parameter variations. So, therefore, the point of propagation constant crossover in Figs. 6–8 moves by a relatively small amount compared to the overall length of the taper, the parameter variations have little effect on coupler performance. For this reason the greater the taper length, the greater the parameter insensitivity [9]. Nonetheless, even for the short coupler lengths used here, very high parameter insensitivity (e.g., high bandwidth) can be expected as is shown. For instance, in Fig. 8 the absolute index difference is varied from 0.0125 to 0.04. Despite varying the value by nearly 100% of the nominal, the coupling efficiency still comfortably exceeds 90% over this parameter space. Note also that ripples observed in the outputs of Figs. 6 and 8 can be reduced by careful optimization of the coupling strength whilst simultaneously reducing the taper ratio (or taper angle). B. Few-Mode Couplers In the following section the mode-selective functionality of slab geometry tapered velocity mode-selective couplers is generalized to more than two modes. This generalization is made


Fig. 8. Movement of the 3 dB point of power transfer from the T1 mode in the centre core to the T0 mode of the outer core, as the absolute index difference is varied from 0.0125 to 0.04.

Fig. 9. Mode-selective functionality of a three-mode TVMSC showing isolation of the (a) T0, (b) T2 and, (c) T1 slab modes. The divergence of the outer cores towards the end of the device (d) permits high levels of power localization.

with reference to the carefully optimized configuration shown in Fig. 9(d). This coupler allows for the multiplexing/demultiplexing of three slab modes supported in a centre core. In this case the centre core 2 initially supports the T0 , T1 ( , with cut-off at 2.65 m) and T2 ( , with cut-off at 5.3 m) slab modes, again assuming that and nm. The centre core width tapers from 9.6 m down to 1.43 m over a length of 7.3 mm. In this configuration the third T2 slab mode is demultiplexed using the up-tapered core 1, whereas the second T1 mode is demultiplexed using the up-tapered core 3. The tapered region of core 1 is from a zero initial width to 1.17 m over a length of 3.35 mm, whereas the tapered region of the outer core 3 is from a zero initial width to 0.98 m over 2.2 mm. At the ends of their respective tapers, the outer cores diverge away from the

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Fig. 10. Propagation constant and third modes of the three-mode TVMSC.

crossovers for the second

centre core to maximize the level of energy localization achievable. Moreover, the taper in the centre core 2 is such that the propagation constant crossover points for each mode occur in separate sections of the taper. These crossover points are shown in Fig. 10 where weak-coupling is again assumed. Weak-coupling simplifies the modal analysis since the waveguides can be modeled in pure isolation. However since the coupler is borderline strongly-coupled, marginal (but immaterial) discrepancies in the approximation of Fig. 10 can be expected. When demultiplexing modes it is crucial that the outer cores have a zero or close to zero initial width in order that the adiabatic criterion may be satisfied. Abrupt changes in the cross-section of the coupler would otherwise result in cross-talk between quasi-normal modes resulting in beating, not unlike a standard directional coupler. This could significantly degrade the performance of such a coupler. However, if the outer cores have zero initial width (and the index difference is kept constant) then it is necessary to decouple the highest-order mode first, followed by the second-highest mode and so forth, otherwise power will exit the wrong cores. Therefore it is crucial that the mode of lowest propagation constant is decoupled first, followed by the mode of second-lowest propagation constant and so forth. This avoids unwanted propagation constant crossovers and hence prevents modes exiting more than one output. This explains the coupler configuration used in Fig. 9. Note also that the fundamental mode is not decoupled but is instead accessed directly via the centre core. The taper in the centre core is also such that both of the higher-order modes are cut-off towards the end of the device. This ensures that any residual power from the higher-order modes is radiated away avoiding corruption of the fundamental mode channel. This ensures extremely high levels of mode isolation. Also, the third mode (T2) in the centre core is cut-off prior to the decoupling of the second (T1) mode in order to avoid corruption of the T1 mode channel. 1) Parameter Dependence: The precise positions of the propagation constant crossover points for the T2 T0 and T1 T0 modal transitions shown in Fig. 10 are again unimportant provided they occur somewhere towards the centres of the respective tapers. This again means that the precise dimensions of the structure are not crucial. Therefore high dimensional tolerance is again expected, which could significantly facilitate fabrication. For the same reason the device is also highly wavelength-independent as quantified in Fig. 11.

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Fig. 11. Wavelength-dependence of the three-mode tapered velocity modeselective coupler.

Fig. 12. Wavelength-dependence of total residual power (either lost to radiation or appearing at incorrect outputs), for all three modes separately excited at dB (i.e., %) the start of the centre core. The residual power is less than for all modes over a wavelength range of 300 nm.

Fig. 11 demonstrates that more than 99% of the power appears at the correct output ports over a wavelength span of 300 nm from 1400 nm to 1700 nm. This wavelength range far exceeds the combined C L EDFA bands spanning approximately 1530–1610 nm [23]. Therefore ultra-wideband performance is again confirmed. Moreover, considering that most of the residual 1% power (see also Fig. 12) is radiation loss due to modal cut-offs in the centre core, the mode extinction ratios of such a device are especially impressive as seen in Fig. 13. Fig. 13 shows that extremely high mode extinction ratios are achievable (i.e., to 60 dB) for all three modes, in part due to the centre core being tapered in such a way that residual uncoupled power in higher-order modes is lost to radiation towards the end of the device. Similar parameter-insensitivity is also observed for precise coupler dimensions and index contrast. The parameter insensitivity is again due to relatively small displacements in the propagation constant crossover points, even for large variations in the parameters. IV. FABRICATION CHALLENGES The advantages of tapered velocity mode-selective couplers over standard directional mode couplers are clear. Wavelengthindependent behavior can be expected well beyond the extended C+L EDFA bands, and this behavior is also largely independent

Fig. 13. The mode extinction ratios achievable for the three-mode TVMSC as a function of wavelength. The extinction ratio of each individual mode of the centre core, describes the power exiting the correct output port relative to that exciting the wrong port(s).

of parameters such as index contrast, and the precise coupler dimensions. Moreover, using suitable diverging output ports, short interaction lengths can be realized. The coupling performance shown in this paper is also representative of that to be expected for all-fiber based devices, although in this case the coupling behavior would also depend on the modal spatial-orientations as described in [7]. Whilst fabrication of TVCs using thin film, diffused channel or rib waveguides [19], [20], [24] is relatively straight-forward, the same is however unlikely to be true for all-fiber devices. The advent of 3D waveguide fabrication techniques such as femtosecond-laser direct write [17], [18], as well as advances in microstructured fiber technology, however significantly improves the likelihood for successful fabrication. Moreover, fabrication is likely to be facilitated by the increased dimensional tolerances compared with standard mode-selective couplers. This means that manufacturing variations could result in negligible degradation of performance. V. CONCLUSION In this paper, tapered velocity mode-selective couplers were proposed and analyzed. These approximately adiabatic devices could be used for wideband mode-division multiplexing of few-mode waveguides. Their mode-selective functionality was demonstrated for slab geometry, although the performance is representative of planar or all-fiber devices. These tapered mode couplers were also shown to be independent of the precise parameters such as index contrast, and the precise coupler dimensions. This is because, unlike standard interference-based mode couplers, the mode-selective functionality does not depend on satisfying precise phase conditions over extended lengths. Therefore provided that the adiabatic condition is satisfied, and that perfect phase-matching still occurs somewhere in the centre of the taper, the performance can be made insensitive to parameter variations. The successful fabrication of such couplers as fiber-based devices, would signify the realization of practical, low loss, and compact wavelength-independent mode multiplexers/demultiplexers. This could significantly progress the development of practical few-mode fiber networks.


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Nicolas Riesen was born in Canberra, Australia, in 1987. He received the B.Sci. degree majoring in physics and the B.Eng. degree in systems engineering with first class honors and The University Medal, from The Australian National University, Canberra, in 2011. He is currently working in collaboration with the Commonwealth Scientific and Industrial Research Organisation toward the Ph.D. degree at the Research School of Physics and Engineering, The Australian National University, Canberra. His main research interests include mode-division multiplexing (MDM), and distributed fiber sensing.

John D. Love was born in the U.K. in 1942. He received the M.A. and M.Maths. degrees in mathematics from the University of Cambridge, U.K., and the M.A., D.Phil., and D.Sc. degrees in mathematics from the University of Oxford, U.K. Since 1973 he has worked at The Australian National University in Canberra, researching theoretical aspects of fiber optics, planar waveguides and associated light processing devices. He co-authored Optical Waveguide Theory in 1983 and Silica-based Buried Channel Waveguides and Devices in 1996. His research activities encompass both academic and industrial problems. He is an honorary Life Member of the Australian Optical Society.