Resonance condition of a microfiber knot resonatorimmersed in liquids

Kok Sing Lim,1 Ali A. Jasim,2 Siti S. A. Damanhuri,2 Sulaiman W. Harun,1,2,*B. M. Azizur Rahman,3 and Harith Ahmad1

1Photonics Research Center, Department of Physics, University of Malaya, 50603 Kuala Lumpur, Malaysia2Department of Electrical Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia3Electrical, Electronic and Information Engineering Department, City University, London, UK

*Corresponding author: swharun@um.edu.my

Received 23 May 2011; revised 19 June 2011; accepted 15 July 2011;posted 15 July 2011 (Doc. ID 147966); published 19 October 2011

Effects of immersing a microfiber knot resonator (MKR) in liquid solutions that have refractive indicesclose to that of silica are experimentally demonstrated and theoretically analyzed. Significant improve-ment in resonance extinction ratio within 2 to 10dB was observed. To achieve a better understanding, aqualitative analysis of the coupling ratio and round-trip attenuation of the MKR is performed by using acurve-fitting method. It was observed that the coupling coefficient at the knot region increasedwhen immersed in liquids. However, depending on the initial state of the coupling and the quantityof the increment in the coupling coefficient when immersed in a liquid, it is possible that the MKRmay experience a deficit in the coupling parameter due to the sinusoidal relationship with the couplingcoefficient. © 2011 Optical Society of AmericaOCIS codes: 060.2340, 060.4005, 230.5750.

1. Introduction

Optical microfibers/nanofibers have recently at-tracted considerable interest as promising buildingblocks for awide variety of photonic applications. Thisis due to their unique optical guidance properties,which include a relatively low loss, strong evanescentfields, tight optical confinement, and controllablewaveguide dispersion. Various optical-microfiber-based resonators have also been demonstrated inloop, knot, and coil configurations benefitting fromthe intrinsic advantages of low scattering/absorptionloss, structural simplicity, and direct coupling to in-put/output fibers. One of the important applicationsof these resonators is as high-sensitivity optical sen-sors [1,2], whereby the operating principle normallyrelies on the characteristics of the resonance. The var-iations of the positions of the resonance wavelengthand extinction ratio of the resonators are significantly

dependent on the sensing parameters, such as tem-perature and refractive index [3,4]. The resonancecondition of a resonator also relies on the index con-trast between themicrofiberand its ambientmedium,evanescent field strength, and the distance betweenthe two microfibers in the coupling region.

The large evanescent field that can be found inthinner microfibers is one of the solutions to achiev-ing higher coupling in microfiber resonators. Thelarge fraction of light intensity in the evanescentfield allows stronger mode interaction between twomicrofibers and yields a high coupling coefficient.Caspar and Bachus [5] suggested embedding the mi-crofiber resonator into a medium that has a slightlylower refractive index than that of silica. Because ofthe small index contrast, the microfiber has a largerevanescent field, which yields stronger coupling inthe resonator [5,6]. Besides being used as a postfab-rication remedy for improving the resonance condi-tion of the resonator, embedding also offers goodprotection from the fast aging process and provides

0003-6935/11/305912-05$15.00/0© 2011 Optical Society of America

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portability for the microfiber devices. Vienne et al.reported that, when a microfiber resonator is em-bedded in low-index polymer, the optimal resonancewavelength is downshifted by ∼20% [7]. However,there are very few reports that provide mathematicalanalysis on the effect of embedding in low-indexcontrast medium to the resonance condition of theresonator.

In this paper, we present a mathematical analysisand experimental results on the characteristic of amicrofiber knot resonator (MKR) immersed in liquidsolutions. This study provides a better understand-ing of the effect of replacing surrounding air by aliquid. MKR was used in the experiment due to itsrigid knot structure and strong interfiber coupling[3]. The knot structure and resonance condition couldbe easily maintained during the immersing process.

2. Theory

The amplitude transfer function of a self-coupledmicrofiber resonator can be well defined by the fol-lowing equation [2]:

T ¼ expð−αL=2Þ expðjβLÞ − sinðKÞ1 − expð−αL=2Þ expðjβLÞ sinðKÞ ; ð1Þ

where α is the intensity attenuation constant, β is thepropagation constant along the microfiber, L repre-sents the round-trip length of the resonator, andsinðKÞ represents the intensity coupling ratio. K isdirectly related to the coupling length and it canbe expressed as

K ¼ κl; ð2Þ

where κ is the coupling coefficient and l is the cou-pling length. For every oscillation in an MKR, thecirculating wave undergoes some attenuation in in-tensity attributed to nonuniformity in the microfiberdiameter, material loss, impurity in the ambience ofthe microfiber, and bending loss along the microfiberloop. These losses can be combined and representedby a round-trip attenuation factor expð−αL=2Þ that isa part of Eq. (1). In the condition when the couplingratio is equivalent to the round-trip attenuation, alarge fraction of the wave power at resonance is an-nihilated in the oscillation and thus the transmissionof the resonance wavelength is minimum [8]. Thiscondition is called critical coupling. The resonancecondition is optimal and the resonance extinction ra-tio (RER) is the highest in this condition. Mathema-tically, in this condition the numerator of Eq. (1) isequal to zero and, therefore,

sinðKcÞ ¼ expð−αL=2Þ; ð3Þ

where Kc denotes the coupling parameter at criticalcoupling.

Based on Eq. (1), the resonance transmission am-plitude that corresponds to βL ¼ 2mπ is

δres ¼����

expð−αL=2Þ − sinðκlÞ1 − expð−αL=2Þ sinðκlÞ

����; ð4Þ

while the maximum transmission amplitude thatoccurs at βL ¼ ð2mþ 1Þπ is

δmax ¼����

expð−αL=2Þ þ sinðκlÞ1þ expð−αL=2Þ sinðκlÞ

����; ð5Þ

where m is an integer. RER is given by

RER ¼ 20log10ðδmax=δresÞ: ð6Þ

In most scenarios, parameters sinðκlÞ andexpð−αL=2Þ are larger than 0.6 and jδmaxj ∼ 1. There-fore, Eq. (6) can be rewritten as

RER ∼ 20log10ðδresÞ: ð7Þ

A smaller value of δres indicates that the resonancecondition is closer to the critical coupling and ityields a larger value of RER.

3. Experiment and Results

Unlike a microfiber loop resonator (MLR) that ex-ploits van der Waals attractive force to maintainthe structure of the loop, an MKR has a more rigidknot structure, with interfiber twisted coupling be-tween microfibers. Nonetheless, each system consti-tutes a self-coupled loop and they share the sameoptical properties; Eq. (1) can be used to describethe transmission spectra of both structures. The fab-rication of an MKR starts with tapering the fiberusing the flame brushing technique [9]. After a7-cm-long and 3–5-μm-diameter biconical tapered fi-ber was drawn, it was cut into two parts, where thefirst part is twice as long as the second. The longertapered fiber was used for the fabrication of a knot byusing tweezers. The shorter section was used as a col-lector fiber by evanescent coupling with the outputport of the knot resonator. After that, the transmis-sion spectrum of the freestanding MKR in air was re-corded by an optical spectrum analyzer. Next, thespectrum of the MKR immersed in a propan-2-ol so-lution that had a refractive index (RI) of 1.37 wasmeasured. First, the MKR was slowly laid horizon-tally on an earlier prepared flat platform depositedwith a thin layer of propan-2-ol. Using a micropip-ette, a small volume of propan-2-ol solution wasdropped onto the MKR to replace the surroundingmedium of the MKR with the solution. The structureof the microfiber knot was intact and the resonancewas maintained. This is the crucial part that distin-guishes an MKR from an MLR. It is very difficult tomaintain the loop structure and resonance of anMLR when it is immersed in a liquid solution.Figure 1 shows the microscope image of an ∼491 μmknot-diameter MKR immersed in low-index resin.This MKR was assembled from an ∼4:5 μm diametermicrofiber.

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Figure 2 shows the overlaid transmission spectraof the MKR in air (solid) and solution (dashed). Re-ferring to the peak powers of both spectra, it is easyto determine that the MKR had suffered an addi-tional ∼7dB excess loss after it was immersed inthe solution. The drop in the coupling efficiency ofthe output-collector microfiber coupling constitutesa large fraction of this excess loss. This is possiblybecause of the change in index contrast and thedisturbance on the structure of output-collectormicrofiber coupling when the liquid solution was in-troduced. In this work, the transmission spectrum ofthe MKR in the liquid solution was recorded after itwas stable. On the other hand, the RER of the MKRimproves from ∼5 to ∼8dB. It is believed that thestructure of the microfiber knot was more rigid thanthat of output-collector microfiber coupling.

Figures 3(a) and 3(b) show the offset experimentaldata with their best-fit curves for both experimentsin air and solution, respectively, which can be used toanalyze the resonance characteristics. The couplingparameter, sinðκlÞ, and round-trip attenuation factorof MKR, expð−αL=2Þ, can be extracted from fittingthe curve based on the transfer function in Eq. (1).A trial and error approach is employed in the fittingprocess until both the experimental and theoreticalcurves agree with each other. In the air, the best-fit

parameters for the transmission spectrum inFig. 3(a) are sinðκlÞ ¼ 0:6207 and expð−αL=2Þ ¼0:8547. By immersing the MKR in propan-2-ol solu-tion, the best-fit parameters of Fig. 3(b) becomesinðκlÞ ¼ 0:6762 and expð−αL=2Þ ¼ 0:8361. The low-er round-trip attenuation factor (higher round-triploss, α) can be attributed to the higher bending lossof the microfiber knot in a medium with smaller in-dex contrast. It is well known that microfiber is sus-ceptible to scattering due to surface roughness. Theincreased evanescent field amplitude of the microfi-ber in immersion may aggravate the scattering loss.The loss at the output-collector coupling is excludedfrom this analysis as it only affects the total outputpower (position in the vertical axis) and it has al-ready been eliminated in the offset spectrum.

From the comparison between the two spectraof Fig. 3, the transmission spectrum with propan-2-ol has higher coupling and a smaller round-trip at-tenuation factor, which results in a smaller value ofδres ¼ 0:3679 compared with δres ¼ 0:5060 obtainedfrom the spectrum for air. This explains the largerRER of the transmission spectrum in Fig. 3(b).

Fig. 1. Optical microscope image of an MKR immersed in low-index resin.

Fig. 2. The transmission spectra of the MKR in the (solid curve)and propan-2-ol solution (dashed curve).

Fig. 3. Offset experimental data (circles) with its best-fit curve(solid curve): (a) air, RI ∼ 1:00; (b) propan-2-ol, RI ∼ 1:37.

Fig. 4. Transmission spectra of theMKR in different surroundingmedia: (a) air, RI ∼ 1:00; (b) low-index resin, RI ∼ 1:36.

5914 APPLIED OPTICS / Vol. 50, No. 30 / 20 October 2011

However, it is also possible that the large incrementin the coupling coefficient κ may have forwarded thephase of κl to the next quarter-cycle of the sinusoidalfunction of sinðκlÞ) and, thus, produces a lower cou-pling value. The next experimental data providean example for such a self-defeating scenario.Figure 4 compares the transmission spectra of anMKR in air and low-index UV-curable resin (UV-Opti-clad 1.36RCM from OPTEM Inc) with an RIof ∼1:36. The coupling parameter, sinðκlÞ for bothspectra is then calculated from the fitting curves.It drops from 0.7132 to 0.6247 when the MKR wasimmersed in water. On the other hand, the round-trip attenuation factor is also calculated from the fit-ting and it suffers a greater fall from 0.9432 to0.7538. In spite of that, the RER had increased from∼2 to ∼10dB. This is in agreement with the decreas-ing value of δres from 0.7027 to 0.2440 and the state ofresonance is closer to the critical coupling condition.

Immersing MKR in a near-index medium does notalways promise an improvement in the resonancecondition or RER. There is a possibility that changesin the round-trip attenuation factor and the couplingparameter yield a larger value of δres and decreasesthe RER. Figure 5 gives an example for this scenario.The best-fit parameters for sinðκlÞ and expð−αL=2Þare calculated to be 0.6235 and 0.8145, respectively,for immersing the MKR in air, as indicated by thesolid curve in Fig. 5. After the MKR was immersedin water (dashed curve), the values of sinðκlÞ andexpð−αL=2Þ changed to 0.7833 and 0.9339, respec-tively. Although there was no sign of dust/particle de-posited on the microfiber or change in the microfiberknot structure before and after immersion in thewater in the observation with an optical microscope,it is believed that the low value of the round-tripattenuation factor can be attributed to the largeamount of unseen dust deposited on the microfibersurface when it came into contact with the tweezersduring the fabrication of the microfiber knot. After itwas immersed in the water, some portion of the de-posited dust or particles that were responsible for thescattering loss might have been “washed” away andthat increases the round-trip attenuation factor. Thedecrease in the value of δres from 0.3881 to 0.5609 isan indication that the resonance condition deviatesfrom critical coupling.

Table 1 tabulates all the fitting parameters of theMKR before and after its immersion in different li-quid solutions. It is understood that the process ofimmersion in liquid solution may possibly disturbthe structure of the microfiber knot and the out-put-collector microfiber coupling. A disturbance tothe structure inmicroscale may induce significant in-fluence to the resonance condition of the MKR. Dif-ferent microfiber waist diameter and orientation ofthe microfibers in the coupling region have an impor-tant relationship with the resonance of the MKR.More investigations pertaining to those parametersare needed. The main focus of this work is to demon-strate the characterization of the MKR resonancecondition using the curve-fitting technique.

4. Conclusion

An investigation on the resonance condition of MKRimmersed in a solution has been conducted. To assistthe analysis, the coupling parameter and the round-trip attenuation factor are obtained by curving fit-ting the experimental data with a transmission func-tion. The extracted parameters from the best-fitcurves provide more quantitative information aboutthe varied resonance condition of an MKR immersedin liquid solutions. In the observation, the couplingcoefficient was increased when immersed in liquids.However, depending in the initial state of the cou-pling and the size of the increment in coupling coef-ficient when immersed in a liquid, it is possible thatthe MKR may experience a deficit in the couplingparameter due to the sinusoidal relationship withthe coupling coefficient.

This project was funded by the Ministry of Science,Technology and Innovation (MOSTI) under theBrain-Gain Malaysia program.

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Fig. 5. Example of an MKR with decreased RER after it isimmersed in water (RI ∼ 1:33).

Table 1. Fitting Parameters of the MKR Beforeand After Immersion in Liquids

Parameters

Propan-2-ol Low-Index Resin Water

Before After Before After Before After

sinðκlÞ 0.6207 0.6762 0.7132 0.6247 0.6235 0.7833expð−αL=2Þ 0.8547 0.8361 0.9432 0.7538 0.8145 0.9339δres 0.5060 0.3679 0.7027 0.2440 0.3881 0.5609α ðm−1Þ 170 200 72 347 180 60L ð10−3 mÞ 1.80 1.63 2.28

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