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Fei Xu, Qin Wang, Jian-Feng Zhou, Wei Hu, and Yan-Qing Lu, Senior Member, IEEE. Abstract—The dispersion characteristics of an optical nanowire microcoil ...
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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 17, NO. 4, JULY/AUGUST 2011

Dispersion Study of Optical Nanowire Microcoil Resonators Fei Xu, Qin Wang, Jian-Feng Zhou, Wei Hu, and Yan-Qing Lu, Senior Member, IEEE

Abstract—The dispersion characteristics of an optical nanowire microcoil resonator (ONMR) are investigated, which show remarkable influence on their performances at a high bitrate. In addition to the waveguide dispersion it was found that the adjacent ring coupling also has notable contribution to the total dispersion. If the nanowire diameter’s range is from 1.5 to 5 μm, the waveguide dispersion and coupling dispersion could almost be canceled with each other so that the total dispersion is well suppressed, which is desired in high-speed (10 Gb/s and beyond) optical systems. On the other hand, both positive and negative dispersion could be obtained by adjusting suitable ONMR parameters. Even a tailored dispersion curve could be designed theoretically. These results are helpful for ONMR-based device design in communication and sensing applications. Index Terms—Fiberoptics components, optical resonators.

I. INTRODUCTION VANESCENT-FIELD-BASED microresonators are versatile wavelength-selective elements that can constitute cascading building blocks for large-scale integrated photonic circuits given their key merits of compact size, wavelength agility, and tunability. Various microresonators have been investigated, including photonic crystal cavities [1], [2], ring resonator [3], microspheres [4], and optical nanowire microcoil resonators (ONMRs) [5]–[10]. ONMR is a new kind of microresonator following the quick development of fabrication technology on subwavelength-diameter optical fiber nanowires (OFN). As the OFN has low cost, low loss, extreme flexibility, and large evanescent filed, it turns out to be an ideal resonatorbuilding material. For example, the 3-D ONMR is difficult to be realized by standard planar light circuit (PLC) technology due to the stereoscopic geometry. However, it could be easily obtained by just wrapping an OFN on a low-index dielectric rod. With recent improvements in fabrication technology of low-loss OFNs,

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the Q-factor of ONMRs could potentially compete with the highest Q-factors currently achieved only in whispering gallery resonators [2]. Up to now, ONMRs have been demonstrated experimentally by a number of researchers [5]–[14]. Various applications, such as optical filters, optical buffers, and delay lines have been proposed [8]. Thanks to the mature fiber slicing and light-coupling technologies, it is convenient to link different ONMR functional units or feed lights in and out. This is a very attractive feature that makes ONMR fully compatible with conventional free space or PLC components in optical networks. Similar to other kinds of microresonators, dispersion is one of its major limitations in high-speed communication. Some earlier reports considered the waveguide dispersion of microresonators. But to the best of our knowledge, the coupling dispersion was ignored in most cases. In an ONMR, the waveguide dispersion is evident and may be very large when the OFN diameter is very small. Besides, the interaction between adjacent turns is another main source of dispersion because of the strong coupling strength and long coupling length. As a consequence, it will be very important and practical to investigate the dispersion characteristics of ONMRs considering both waveguide and coupling contributions. Particularly, the limitation and application by the strong coupling dispersion should be considered in high-speed communication. Because of the similarity between silicon and silica nanowire devices, the theoretical investigation on coupling dispersion should be helpful for future ultralargescale silicon-on-insulator on-chip interconnects based on silicon nanowires. In this paper, both the waveguide and coupling dispersions of a typical two-turn ONMR has been calculated and investigated. The possible influence and limitation in a high-bitrate optical system are discussed, which is helpful for ONMR-based device design and applications.

II. DISPERSION Manuscript received May 5, 2010; revised June 25, 2010 and July 16, 2010; accepted July 19, 2010. Date of publication September 27, 2010; date of current version August 5, 2011. This work was supported in part by the National 973 program of China under Contract 2010CB327803 and Contract 2006CB921805, in part by the National Science Foundation of China program under Contract 60977039, Contract 10874080, and Contract 10775070, and in part by the New Century Excellent Talents program, China Ministry of Education for new century, and Changjiang scholars program. The authors are with the College of Engineering and Applied Sciences and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China (e-mail: [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/JSTQE.2010.2061220

The geometry of a typical two-turn ONMR is shown in Fig. 1(a). If the characteristic transversal dimension of the propagating mode is much smaller than the characteristic bend radius, the adiabatic approximation of parallel transport can be applied. Assuming that the difference between the lower and upper turns’ propagation constants is small, compared to the coupling coefficient, light propagating around the uniform microcoil in the linear regime can be described by the coupled wave equations [8]. The transmission T, which is defined as the transmitted electric field divided by the input electric field, can be described as the following equation for a two-turn ONMR

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XU et al.: DISPERSION STUDY OF OPTICAL NANOWIRE MICROCOIL RESONATORS

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Fig. 2.

GVD curve as a function of OFN diameters.

At the resonating wavelengths, the dispersion of an ONMR Fig. 1. (a) Schematic of a two-turn ONMR. (b) Cross section of two adjacent OFNs, r is the radius of OFN and Λ is the distance between the centers of two OFNs.

with uniform pitch and nanowire diameter [8] T = |T | eiϕ = γ = exp(−αs)

[cos βs + i(sin βs − γ −1 sin K)] [cos βs + i(− sin βs + γ sin K)]

is 2πc ∂ 2 ϕ |ω λ2 ∂ω 2 0  2πc β0s (2 − pu) =− 2 λ0 1 + u2 − pu

DR s = −



(1)

where ϕ is the phase delay, s is the length of one turn, β = 2πneff /λ is the propagation constant, neff is the effective index, λ is the wavelength, α is the loss coefficient, K = ks is the coupling parameter, and k is the coupling coefficient due to the overlap of the field modes between neighboring turns as shown in Fig. 1(b) ∞ ∞ ωε0 −∞ −∞ (n2f − n2air )E1∗ E2 dxdy (2) k = ∞ ∞ ∗ ∗ −∞ −∞ uz (E1 × H1 + E1 × H1 )dxdy where ω is the angular frequency and ε0 = 8.854 × 10−7 (F/m), nf and nair are the refractive index of OFN and air, respectively, and E1 and E2 , and H1 and H2 represent the electromagnetic fields at OFN1 and OFN2, respectively. Sumetsky [8, eq. (1)] has the limitation in the radius of OFN and the length of coil. Generally r = s is necessary for small bending loss and satisfying adiabatic approximation of parallel transport, which has been extensively investigated in [6]–[8] and [11]. In this paper, the ONMR with the OFN radius from several hundreds of nanometers to several micrometers has been discussed. The one turn length is set around 1 mm. These typical parameters are easy to achieve with current ONMR fabrication techniques. For an ONMR, the resonance condition is βs = 2mπ + π/2 and sin (Km ) = 1, where m is an integer. The group delay is   ∂ ln T ∂ϕ = Im τ= ∂ω ∂ω    2iβ s − cos βs[ipu + β  squ] + sin βs[qu − iβ  spu] . = Im 1 + u2 + iu[q cos βs + ip sin βs] (3)

=

 β0 su [2(2 − pu)(2u − p) − iq 2 u] (1 + u2 − pu)2

Df s(2 − pu) 1 + u2 − pu −

dβ/dω|β 0 s(du/dλ)[2(2 − pu)(2u − p) − iq 2 u] (1 + u2 − pu)2 (4) −1

−1

where p = γ + γ , q = γ − γ, and u = sin K. For a lossless ONMR, it can be simplified as   8β  su 2πc 2β  s + DR s = − 2 1 − u (1 − u)2 λ0 =

2Df s 2πc 8β  su − 2 = D1 s + D2 s 1−u λ0 (1 − u)2

(5)

where Df is the group velocity dispersion (GVD) of OFN. The dispersive properties of an ONMR are very important. A higher dispersion may broaden the optical pulse then deteriorate the system performance at a high bitrate B. There are two main contributors to the dispersion: the OFN waveguide and material dispersions, and the coupling dispersion, which are shown in the first and second items in [5]. From [5], the coupling coefficient depends on the light frequency, which is easily understandable. Conventional study on microresonators ignores the dispersion of coupling, sometimes even waveguide and material dispersions. This approximation is reasonable in the cases of weakly coupling and low Q-factor. For very strong coupling and ultrahigh Q-factor, we believe that these dispersions have to be all considered. Fig. 2 shows the GVD curve at λ0 = 1.55 μm, as a function of the diameter of an OFN made out of fused silica [15, p. 17]. The Sellmeier polynomial is n2 = 1 +  3 2 2 −3 i=1 ((ai λ )/(λ − bi × 1e )) (λ: μm), with the coefficients

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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 17, NO. 4, JULY/AUGUST 2011

Fig. 3. (a) Coupling parameter K and (b) its dispersion dK/dλ versus OFN diameter d and pitch Λ/d.

a1 = 0.6965325, a2 = 0.4083099, and a3 = 0.8969766, and b1 = 4.863809, b2 = 13.94999, and b3 = 97933.99 [15, p. 17]. From the Fig. 2, the GVD is sensitive to the OFN diameter. It could be positive, negative, or even around zero with a proper diameter. When the diameter is less than 1 μm, the GVD value changes remarkably while the flat GVD region corresponds to larger diameters. These interesting features reflect the competition of dispersions from different sources, which may have some applications. For example, a zero-dispersion is favorable for high-speed optical systems, while a negative dispersion is useful for dispersion compensating. In this paper, we only consider d ∈ [0.5 μm, 5 μm] and Λ/d ∈ [2], [6], which are the reasonable parameters for practical ONMRs with current technique. The coupling parameters and its dispersion as functions of OFN diameter d and pitch Λ are shown in Fig. 3, assuming s = 1 mm. For a given diameter d, K decreases with the pitch, it is as large as 300 at d = 750 nm and Λ = 2d, and as small as nearly zero when the pitch is large enough. The dispersion of K (dK/dλ) has a wide range, from 0 to 400 μm−1 . Fig. 4 shows the total dispersion D1 , D2 , and DR as a function of d and Λ/d. Here, we only consider high Q-factor resonator assuming K is nearly at resonating: sin (K) is about sin(1.1π/2) and the Q-factor ∼βs/(K − Km )2 is about 105 . DR can achieve as high as −1 × 104 and 2 × 104 ps/nm/mm, and as low as

Fig. 4. Total dispersion D1 , D2 , and DR as a function of d and Λ/d at nearly resonating.

0. The figure of DR is very similar to D2 , which means D2 is the dominator in most cases, especially when D2 is very large. Because both β  and β  depend on d only, while k depend on d and Λ, we may compare more clearly and simply the competition between waveguide and coupling dispersions by comparing the maximum of DR (MaxDR ) and minimum of DR (MinDR ) among different Λ at the same OFN diameter. Fig. 5 shows the curves of MaxDR , D1 (independent of the coupling dispersion, only dependent on d), and MinDR at different d, the coupling dispersion D2 at strong coupling is dominant because D1  MaxDR . The advantage of strong coupling dispersion is that the coupling dispersion is possible to cancel the waveguide dispersion: MinDR = 0, at d ∈ [∼0.6 μm, ∼0.8 μm] or [∼1.5 μm, 5 μm]. When d ∈ [∼0.8 μm, ∼1.5 μm], D1  MinDR , the coupling dispersion is totally dominant.

XU et al.: DISPERSION STUDY OF OPTICAL NANOWIRE MICROCOIL RESONATORS

Fig. 5.

Comparing among MaxDR , D1 , and MinDR as functions of d.

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Fig. 7. Absolute of total dispersion DR as a function of loss for Λ = 2d and d = 1 μm (solid line), d = 2 μm (dashed line), and d = 3 μm (dotted line) at nearly resonating.

Then, the dispersion of OFN decreases slowly in the loss range of 0.001–0.01 dB/mm, matching the level of current fabrication technologies. III. BITRATE The strong dispersion causes serious effect on ONMR performance, especially bitrate B (when DR is zero, the third-order has to be considered) [16, pp. xvii and 546]. Bm ax

Fig. 6. Total dispersion DR as a function of wavelength for Λ = 2d and d = 1 μm (solid line), d = 2 μm (dashed line), and d = 3 μm (dotted line) at nearly resonating.

From Figs. 2–5, the total dispersion of an ONMR is dominant by its physical parameters, such as OFN diameter and ring pitch. Both positive and negative dispersion could be achieved by selecting suitable ONMR parameters. There could be a possible way to engineer-tailored dispersion curves by means of cascading ONMRs. As the coupling strength between ONMR turns is sensitive to the environmental index, even tunable dispersion might be obtained. The dispersion versus the wavelength was calculated as shown in Fig. 6 for Λ = 2d and d = 1 μm (solid line), d = 2 μm (dashed line), and d = 3 μm (dotted line), respectively. It strongly depends on the wavelength when d is small but is nearly independent on wavelength when d is large (>3 μm). Loss is unavoidable in practical devices. Fig. 7 shows the dispersion versus the loss for Λ = 2d and d = 1 μm (solid line), d = 2 μm (dashed line), and d = 3 μm (dotted line), respectively. It reaches the maximum at a proper loss (∼5 × 10−4 dB/mm), then decreases with an increasing loss. The dispersion at 0.001 dB/mm is close to the dispersion at zero loss.

1 ≈ 4



λ2 DR s 0 2πc

−1 (6)

where c = 3 × 108 m/s. The maximum bitrate is very poor (∼1.6 Gb/s) when DR ∼2 × 104 ps/nm/mm. When designing and manufacturing ONMRs for very highspeed communication system, it is ideal to select d and Λ, where DR is nearly zero, of course, we know it is not suitable to select d ∈[∼0.8 μm, ∼1.5 μm]. The dark zones in Fig. 8 show the allowable d and Λ with Bm ax > 10 Gb/s at nearly resonating and Q of ∼105 [see Fig. 8(a)] and ∼107 [see Fig. 8(a)], respectively. It can be seen from Fig. 8(a) that d > 1.5 μm is good at high-speed communication, but larger diameter gives rise to weaker coupling and more bending loss. It is a good choice to use OFNs with diameters of 1.5–5 μm for practical fabrication. Very thin OFN with the diameter of hundreds of nanometers is not recommended. Similar trend exhibits in Fig. 6(b), where 10 Gb/s bitrate at nearly resonating and Q-factors of (a) ∼105 and (b) ∼107 .

In this study, we only consider the ideal two-turn ONMR with uniform pitch and OFN diameter. Variations in the diameter, separation between adjacent sections has to be very slow and small in practical application, otherwise there is possibly coupling of polarization states and change the total dispersion. For ONMR with OFNs above 1 μm, this effect can be ignored with current mature fabrication technology. IV. CONCLUSION In summary, we studied the characteristics in ONMR with strong coupling and waveguide dispersions. The dispersioninduced limitation on performance of ONMR was discussed. At high-speed optical commutation (bitrate above 10 Gb/s), OFNs with diameters larger than 1 μm were recommended for high-speed ONMRs. These results are very helpful for future device design and system applications. REFERENCES [1] Y. Mehmet Fatih, F. Shanhui, and S. Marin, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett., vol. 83, pp. 2739–2741, Oct. 2003. [2] K. J. Vahala, “Optical microcavities,” Nature, vol. 424, pp. 839–846, Aug. 2003. [3] B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightw. Technol., vol. 15, no. 6, pp. 998–1005, Jun. 1997.

Fei Xu received the Ph.D. degree from Optoelectronics Research Center, University of Southampton, U.K., in 2008. He is currently an Associate Professor at the College of Engineering and Applied Sciences and the National Laboratory of Solid State Microstructures, Nanjing University, Nanjing, China. He is the author or coauthor of more than 20 peer-reviewed papers on optoelectronic materials and devices. His research interests include nanophotonics and fiberoptics. Qin Wang received the Ph.D. degree in physics from Nanjing University, Nanjing, China, in 2010. She is currently a Research Associate at the College of Engineering and Applied Sciences and the National Laboratory of Solid State Microstructures, Nanjing University. Her research interests include nonlinear optics and fiberoptics. Jian-Feng Zhou received the Ph.D. degree from Nanjing University, Nanjing, China, in 2007. He is currently an Associate Professor at the College of Engineering and Applied Sciences and the National Laboratory of Solid State Microstructures, Nanjing University. He is the author or coauthor of more than 10 peer-reviewed papers on optoelectronic materials and devices. His research interests include nanomaterials. Wei Hu received the Ph.D. degree from Jilin University, Changchun, China, in 2009. He is currently a Lecturer at the College of Engineering and Applied Sciences and the National Laboratory of Solid State Microstructures, Nanjing University, Nanjing, China. He is the author or coauthor of more than 10 peer-reviewed papers on optoelectronic materials and devices. His research interests include liquid crystal optoelectronic materials. Yan-Qing Lu (SM’03) received the Ph.D. degree from Nanjing University, Nanjing, China, in 1996. He is currently a Professor at the College of Engineering and Applied Sciences and the National Laboratory of Solid State Microstructures, Nanjing University. He is the author or coauthor of more than 60 peer-reviewed papers on optoelectronic materials and devices and holds 15 U.S. and China patents. His research interests include fiber optics, nanophotonics, and liquid crystal devices.