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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 16 — Jul. 30, 2012
  • pp: 18005–18015
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Force-induced optical nonlinearity and Kerr-like coefficient in opto-mechanical ring resonators

Y. F. Yu, M. Ren, J. B. Zhang, T. Bourouina, C. S. Tan, J. M. Tsai, and A. Q. Liu  »View Author Affiliations


Optics Express, Vol. 20, Issue 16, pp. 18005-18015 (2012)
http://dx.doi.org/10.1364/OE.20.018005


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Abstract

This paper demonstrates the optical nonlinearity in opto-mechanical ring resonators that consist of a bus waveguide and two ring resonators, which is induced by the optical gradient force and characterized by the Kerr-like coefficient. Each ring resonator has a free-hanging arc that is perpendicularly deformable by an optical gradient force and subsequently this deformation changes the effective refractive index (ERI) of the ring resonator. The change of the ERI induces optical nonlinearity into the system, which is described by an equivalent Kerr coefficient (Kerr-like coefficient). Based on the experimental results, the Kerr-like coefficient of the ring resonator system falls in the range from 7.64 × 10−12 to 2.01 × 10−10 m2W−1, which is at least 6-order higher than the silicon’s Kerr coefficient. The dramatically improved optical nonlinearity in the opto-mechanical ring resonators promises potential applications in low power optical signal processing, modulation and bio-sensing.

© 2012 OSA

1. Introduction

Optical nonlinearity is a material property, which is defined as the nonlinear response of the polarization P to the electric field E of light when light is propagating in the material. It is used for high speed optical signal processing [1

1. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3(4), 216–219 (2009). [CrossRef]

, 2

2. Y. H. Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen, “Demonstration of wavelength conversion at 40 Gb/s data rate in silicon waveguides,” Opt. Express 14(24), 11721–11726 (2006). [CrossRef] [PubMed]

], broadband electro-optic modulation [3

3. J. M. Brosi, C. Koos, L. C. Andreani, M. Waldow, J. Leuthold, and W. Freude, “High-speed low-voltage electro-optic modulator with a polymer-infiltrated silicon photonic crystal waveguide,” Opt. Express 16(6), 4177–4191 (2008). [CrossRef] [PubMed]

], mass detection [4

4. J. T. Robinson, L. Chen, and M. Lipson, “On-chip gas detection in silicon optical microcavities,” Opt. Express 16(6), 4296–4301 (2008). [CrossRef] [PubMed]

] and Raman lasing [5

5. H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005). [CrossRef] [PubMed]

]. Due to the optical nonlinearity of materials, a variation of the refractive index is induced, which is proportional to the local intensity of light and can be expressed as n=n0+n2I, where n0is the refractive index, and n2is a constant. This phenomenon is well known as the optical Kerr effect and n2is defined as the Kerr coefficient. The optical Kerr effect is one of the important nonlinear optical properties of silicon, which attracts much attention due to the well-developed semiconductor technologies. However, the silicon’s Kerr coefficient is only in the range from 2.8 × 10−18 to 14.5 × 10−18 m2W−1 [6

6. H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23(6), 064007 (2008). [CrossRef]

], which does not have the superior nonlinear characteristics for many potential applications. It is desirable to improve the optical Kerr effect of the silicon material, for example, to decrease the threshold power for lasing or increase the high sensitivity for sensing. There are few methods used to achieve high optical nonlinearity in the silicon material. For example, ion doping is used to change the optical nonlinearity of the silicon [7

7. A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007). [CrossRef] [PubMed]

], but it induces a large free-carrier absorption, which decreases the capability of light intensity enhancement in the silicon–based devices. Therefore, methods to improve the optical nonlinearity effect without inducing a large absorption loss are desired. Optical ring resonator is an efficient method to enhance the optical nonlinearity of the silicon through the thermal-optic effect or the free-carrier absorption effect [8

8. G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005). [CrossRef] [PubMed]

10

10. Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15(3), 924–929 (2007). [CrossRef] [PubMed]

]. However, the intrinsic optical nonlinearity of silicon is not improved. In the contrary, an extremely strong optical nonlinearity is generated due to the opto-mechanical effect in nanoscale mechanical structures, such as two coupled silicon waveguides [11

11. J. Ma and M. L. Povinelli, “Mechanical Kerr nonlinearities due to bipolar optical forces between deformable silicon waveguides,” Opt. Express 19(11), 10102–10110 (2011). [CrossRef] [PubMed]

] or waveguide substrates [12

12. W. H. P. Pernice, M. Li, and H. X. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. 95(12), 123507 (2009). [CrossRef]

]. On the other hand, the optical ring resonator enlarges the opto-mechanical effect via the increase in light intensity, which is widely applied in mesoscale back action [13

13. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008). [CrossRef] [PubMed]

], sideband cooling [14

14. R. Rivière, S. Deléglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83(6), 063835 (2011). [CrossRef]

], thermal nonlinearity [15

15. H. Rokhsari and K. J. Vahala, “Observation of Kerr nonlinearity in microcavities at room temperature,” Opt. Lett. 30(4), 427–429 (2005). [CrossRef] [PubMed]

17

17. I. Grudinin, H. Lee, T. Chen, and K. Vahala, “Compensation of thermal nonlinearity effect in optical resonators,” Opt. Express 19(8), 7365–7372 (2011). [CrossRef] [PubMed]

] and photonic structure controlling [18

18. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical Nonreciprocity in Optomechanical Structures,” Phys. Rev. Lett. 102(21), 213903 (2009). [CrossRef] [PubMed]

].

In this paper, an opto-mechanical ring resonator system is proposed to produce an ultra-high optical nonlinearity via the opto-mechanical effect. The mechanical deflection of the free-hanging arc of the ring resonator caused by the optical gradient force changes the effective refractive index (ERI) [19

19. W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009). [CrossRef] [PubMed]

21

21. J. F. Tao, J. Wu, H. Cai, Q. X. Zhang, J. M. Tsai, J. T. Lin, and A. Q. Liu, “A nanomachined optical logic gate driven by gradient optical force,” Appl. Phys. Lett. 100(11), 113104 (2012). [CrossRef]

], which induces a mechanical optical nonlinearity into the system and a Kerr-like coefficient is derived to describe this phenomenon. The theoretical analysis, the numerical simulations and the experimental results are presented and discussed.

2. Theoretical analysis

A ring resonator system is designed to enhance the optical nonlinearity induced by the optical gradient force as shown in Fig. 1(a)
Fig. 1 (a) Schematic illustration of the opto-mechanical ring resonator system; and (b) demonstration of the deformation of the free-hanging arcs caused by the optical gradient forces.
. It consists of a bus waveguide and two ring resonators, which has a free-hanging arc with radius R in each ring resonator. The propagating light in the bus waveguide is coupled into the two ring resonators and the free-hanging arc is perpendicularly deformable by the optical gradient force. The free-hanging arc has a length of Li = 4R arcsin (Di/X), whereX=2RDi and Di is the height of the arc in ring i (i = 1, 2) as shown in Fig. 1(b).

The input light intensity I0 (measured in Wm−2) is expressed as I0=ε0neffc|E|2/2, and the guiding power Pg in the waveguide equal toPg=I0A, where ε0 is the permittivity of vacuum, neffis the ERI of this guiding mode, c is the speed of light in vacuum and A is the mode area. Therefore, the electric field in the waveguide can be written as |E|=2Pg/Acneffε0. When the optical fields pass through the coupling region, the electric field due to the evanescent coupling can be expressed as [22

22. C. S. Ma, Y. Z. Xu, X. Yan, Z. K. Qin, and X. Y. Wang, “Optimization and analysis of series-coupled microring resonator arrays,” Opt. Commun. 262(1), 41–46 (2006). [CrossRef]

]
[E3E4E5]=[tjr1jr2jr1t10jr20t2][E0E1E2],
(1)
where j is the imaginary unit, ri and ti are the transmission coefficient and the tunnelling constant, respectively [23

23. Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D microresonator lattice optical filters,” IEEE J. Quantum Electron. 41(11), 1410–1418 (2005). [CrossRef]

], and t=1r12r22 is the total tunnelling constant. In the two ring resonators, the optical fields have the relationship of E1=E4a1ejϕ1 and E2=E5a2ejϕ2, where ai are the field attenuation factors in ring i, ϕi=2π[(2πRLi)neff|g=0+Lineff]/λ are the collected phases caused by the light propagation and neff is the ERI. The field buildup factors in the two rings (Bi) and the relative transmitted field (E3/E0) can be expressed as
{B1=|E1E0|=|jr1a1ejϕ11t1a1ejϕ1|B2=|E2E0|=|jr2a2ejϕ21t2a2ejϕ2|,
(2)
and
E3E0=tr12a1ejϕ11t1a1ejϕ1r22a2ejϕ21t2a2ejϕ2.
(3)
Based on the above analysis, the two rings can be treated individually and acted as a reference for each other, which assists in the study and analysis of peak shift. However, to simplify the expression, the discussion of Kerr-like coefficient is limited to one ring. When the field attenuation factor is approximated to one (a01), the circulating light intensity Ic in the ring resonator can be expressed as
Ic=B2I0.
(4)
The Kerr-like coefficient for the free-hanging arc is expressed as
n2=ΔneffΔIc.
(5)
However, it is hard to detect the circulating light intensity in the ring resonator (Ic). Furthermore, the study of the nonlinearity of the free-hanging arc without the ring’s effect is not necessary. Therefore, n2is redefined as the Kerr-like coefficient of the ring resonator, which can be expressed as
n2=ΔneffΔIa,
(6)
where Ia is the effective input intensity for the ring resonator. Ia is equal to the absorbed intensity by the ring resonator and can be expressed as
Ia=(1T(λ))I0(λ)dλ,
(7)
where the transmission T(λ)=|E3(λ)/E0(λ)|2.

In considering the deformation of the free-hanging arcs, the uniform load (Fm) can be expressed as Fm=KeffΔg/L, where Δg is the deflection and Keffis the effective stiffness, which depends on radius of ring, height of the free-hanging arc and the Young’s modulus of the material [24

24. S. Pamidighantam, R. Puers, K. Baert, and H. A. C. Tilmans, “Pull-in voltage analysis of electrostatically actuated beam structures with fixed-fixed and fixed-free end conditions,” J. Micromech. Microeng. 12(4), 458–464 (2002). [CrossRef]

]. Based on the effective index method, the optical gradient force (Fo) can be expressed as [19

19. W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009). [CrossRef] [PubMed]

]
Fo=1neffneffgPcngc,
(8)
where ng=neffλneff/λ is the group velocity [25

25. Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive-index measurement by michelson interferometer,” Opt. Commun. 78(2), 109–112 (1990). [CrossRef]

] and Pc=B2Pg is the circulating power on the ring. The effective input powerPeffis defined as the absorbed power by the ring resonator (Peff=pg(λ)(1T(λ))dλ). Based on Eq. (7), Peff=AIa. The mode area A is assumed to be independent of the wavelength in a small wavelength range. Then, Eq. (6) is written as
n2=AΔneffΔPeff,
(9)
where Δneff/ΔPeff is determined by the experimental results.

In the numerical simulations, the parameters of the ring/waveguide used are as follow: the width b = 450 nm, the height h = 220 nm, the ring radius R = 30 μm, the ring-waveguide gap G = 200 nm, the refractive index of ring/waveguide ns = 3.5 and the initial ring-substrate gap go = 200 nm. The gradient optical force is calculated based on Eqs. (1)- (4) and Eq. (8). First, based on the finite element method (FEM), the ERI (neff) in the gap-wavelength (g-λ) domain is shown in Fig. 2(a)
Fig. 2 Contour plot of (a) the effective refractive index and (b) the optical force in the gap-wavelength domain.
, thus the values of neff/gand neff/λ are determined. Second, according to Eq. (2), ai is approximately 0.99, ri and ti are subsequently simulated by the finite-difference time domain method. Therefore, when the input power Pg is fixed at 1 mW, the normalized optical force Fo is obtained as shown in Fig. 2(b). It can be seen that the maximal values of the optical force is corresponding to the resonant wavelengths (i.e. λ11 = 1545.3 nm and λ21 = 1548.2 nm at g = 180 nm). When the gap g is decreased, the maximal value of the optical force is increased and the corresponding resonant wavelengths are red shifted.

3. Experimental results and discussions

The opto-mechanical ring resonator system, as shown in Fig. 4(a)
Fig. 4 (a) SEM image of the opto-mechanical ring resonator system with a released bus waveguide and two partially released ring resonators, and (b) the schematic of experimental setup.
, is fabricated on the silicon-on-insulator wafer by using the nano-fabrication processes. The waveguide-ring gap is 200 nm, the thickness of the silicon structure layer and the buried oxide layer are 220 nm and 2 µm, respectively. Except for the free-hanging arcs, the remaining parts of the system are deposited with a 2-µm thick SiO2 cladding layer. The free-hanging arcs are released by the selective etching process and the gap g between the free-hanging arc and the substrate is controlled to be approximately 200 nm.

The experimental setup is shown in Fig. 4(b). In the experiments, a pair of tapered fiber is used to couple the input light into the nano-waveguide and detect the output light, respectively. The probe light is a broadband light source with a central wavelength of 1550 nm and a bandwidth of 70 nm. With the probe light, each ring has two absorption modes within the spectrum range, i.e. ring resonator 1 with λ11 = 1545.4 nm and λ21 = 1548.2 nm while ring resonator 2 with λ12 = 1545.6 nm and λ22 = 1548.4 nm as shown in Fig. 5
Fig. 5 Transmission spectrum of the opto-mechanical ring resonator system.
. By combining with the numerical results ofneff, ri, and ti, the simulated transmission spectrum and the experimental results are well matched with each other via the optimization of the attenuation factor ai and the height of the free-hanging arcs Di. Subsequently, a control light with an output power of 160 μW and a tunable wavelength ranging from 1500 nm to 1600 nm with a tuning step of 0.01 nm are coupled into the system. When the control light is adjusted to different wavelengths, the transmission spectrum of the ring resonator is affected as shown in Fig. 6
Fig. 6 Shift of absorption peak of ring 1 under different control wavelengths.
. It can be seen that the absorption peaks are not shifted as compared to those peaks in the transmission without the control light when the control wavelength is 1547.97 nm. When the control wavelength increases from 1548.22 nm to 1548.26 nm, the resonant peak λ11 is shifted from 1545.4 nm to 1545.6 nm (melted with λ12). This occurs because the effective input power Peff of the ring 1 is increased when the overlapped range between the control laser peak and the absorption peak (λ21 = 1548.2 nm) increases. The effective input power (Peff, a part of guided power in waveguide, which is absorbed by ring 1) is calculated by the integration of the product of the laser power pL(λ) and the absorption ratio (1- T) in the wavelength domain (Peff=λminλmaxpL(λ)(1T)dλ), in which the laser power pL(λ)can be directly determined by the transmission spectrum. The absorption ratio (1- T) is estimated based on the absorption peak with the central wavelength λ21. The normalized effective input power is associated with the control wavelength as shown in Fig. 7(a)
Fig. 7 (a) Effective input power for ring 1 versus the control wavelength; and (b) Kerr-like coefficient versus the effective input power.
. It is observed that the effective input power is increased from 0 μW to 51.9 μW when the control wavelength increases from 1548.18 nm to 1548.76 nm. Thereafter, the free-hanging arc cannot be kept at the position with the maximum deflection and returns to the initial position by the mechanical force. On the contrary, when the control wavelength decreases from 1548.76 nm to 1548.45 nm, the effective input power is zero in the beginning position, and jumps to 0.98 μW at the wavelength of 1548.5 nm. Thereafter, the effective input power is gradually decreased to zero. It is shown that one control wavelength corresponds to two possible effective input powers in this particular range, which means an opto-mechanical bistability is induced in this system.

Based on Eq. (9) and the experimental results, the Kerr-like coefficient is calculated. First, the mode area A is calculated based on the optical mode in the nano-waveguide by using the numerical method. Then, neff/Peff for ring 1 is obtained byneffPeff=neffλ11λ11λcλcPeff, where neff/λ11 is determined based on the resonant conditions (i.e.neff/λ11=m/Li), λ11/λc is determined based on the peak shift as shown in Fig. 6, λc/Peffis determined from the curve shown in Fig. 7(a), and the resonant mode order m is determined from the simulation results (i. e., m = 309). Therefore, the Kerr-like coefficient n2 for ring 1 is depending on the effective input power as shown in Fig. 7(b). It can be seen that when the effective input power decreases from 51.9 µW to 0.998 µW, the Kerr-like coefficient is increased from 7.64 × 10−12 to 2.01 × 10−10 m2W−1. In other words, the optical force-induced Kerr-like coefficient in the opto-mechanical ring resonator system is at least 6-order higher than the Kerr coefficient in the silicon material (2.8 × 10−18 to 14.5 × 10−18 m2W−1). When the effective input power is lower than 0.998 µW, the calculated Kerr-like coefficient is further increased. However, the calculation value becomes unreliable because a larger fluctuation is observed at lower effective input power due to the noises in the experiments.

In order to verify the Kerr effect is induced by the optical force but not the thermal optic effect, devices with different gaps between the free hanging arc and the substrate are experimented. The gap is controlled by the etching time in fabrication process, which is ranging from 15 to 30 minutes (etching rate: 10 nm/min). The free-hanging arc in each device is completely released and can be deformed by the optical force, in which the Kerr effect is observed in all cases. On the contrary, when the etching time is shorter than 15 minutes, the free-hanging arc is not fully released from the SiO2 substrate and the maximum observed peak shift is less than 0.1 nm. When the etching time is longer than 30 minutes, the gap between the free-hanging arc and the substrate is larger than 300 nm. The maximum observed peak shift is also less than 0.1 nm because the weak optical force is insufficient to deform the free-hanging arc. Based on these observations, it is verified that the Kerr effect is indeed induced by the optical force. This can also be verified by the measurement of the time-domain response [21

21. J. F. Tao, J. Wu, H. Cai, Q. X. Zhang, J. M. Tsai, J. T. Lin, and A. Q. Liu, “A nanomachined optical logic gate driven by gradient optical force,” Appl. Phys. Lett. 100(11), 113104 (2012). [CrossRef]

, 26

26. H. Rokhsari and K. J. Vahala, “Observation of Kerr nonlinearity in microcavities at room temperature,” Opt. Lett. 30(4), 427–429 (2005). [CrossRef] [PubMed]

].

4. Conclusions

In summary, the theoretical analysis, the numerical simulation and the experimental results of the optical nonlinearity induced by the optical gradient force in the opto-mechanical ring resonator system are presented and discussed. The Kerr-like coefficient is determined, which is in the range from 7.64 × 10−12 to 2.01 × 10−10 m2W−1, and it is at least 6-order higher than the silicon’s Kerr coefficient. The dramatically improved optical nonlinearity in the opto-mechanical ring resonator promises potential applications in low power optical signal processing, modulation and bio-sensing.

Acknowledgments

This work is supported by the Science & Engineering Research Council (SERC) of Singapore through the TSRP research project (Project No. 102 165 0084).

References and links

1.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3(4), 216–219 (2009). [CrossRef]

2.

Y. H. Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen, “Demonstration of wavelength conversion at 40 Gb/s data rate in silicon waveguides,” Opt. Express 14(24), 11721–11726 (2006). [CrossRef] [PubMed]

3.

J. M. Brosi, C. Koos, L. C. Andreani, M. Waldow, J. Leuthold, and W. Freude, “High-speed low-voltage electro-optic modulator with a polymer-infiltrated silicon photonic crystal waveguide,” Opt. Express 16(6), 4177–4191 (2008). [CrossRef] [PubMed]

4.

J. T. Robinson, L. Chen, and M. Lipson, “On-chip gas detection in silicon optical microcavities,” Opt. Express 16(6), 4296–4301 (2008). [CrossRef] [PubMed]

5.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005). [CrossRef] [PubMed]

6.

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23(6), 064007 (2008). [CrossRef]

7.

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007). [CrossRef] [PubMed]

8.

G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005). [CrossRef] [PubMed]

9.

Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006). [CrossRef] [PubMed]

10.

Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15(3), 924–929 (2007). [CrossRef] [PubMed]

11.

J. Ma and M. L. Povinelli, “Mechanical Kerr nonlinearities due to bipolar optical forces between deformable silicon waveguides,” Opt. Express 19(11), 10102–10110 (2011). [CrossRef] [PubMed]

12.

W. H. P. Pernice, M. Li, and H. X. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. 95(12), 123507 (2009). [CrossRef]

13.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008). [CrossRef] [PubMed]

14.

R. Rivière, S. Deléglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83(6), 063835 (2011). [CrossRef]

15.

H. Rokhsari and K. J. Vahala, “Observation of Kerr nonlinearity in microcavities at room temperature,” Opt. Lett. 30(4), 427–429 (2005). [CrossRef] [PubMed]

16.

I. S. Grudinin and K. J. Vahala, “Thermal instability of a compound resonator,” Opt. Express 17(16), 14088–14097 (2009). [CrossRef] [PubMed]

17.

I. Grudinin, H. Lee, T. Chen, and K. Vahala, “Compensation of thermal nonlinearity effect in optical resonators,” Opt. Express 19(8), 7365–7372 (2011). [CrossRef] [PubMed]

18.

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical Nonreciprocity in Optomechanical Structures,” Phys. Rev. Lett. 102(21), 213903 (2009). [CrossRef] [PubMed]

19.

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009). [CrossRef] [PubMed]

20.

H. Cai, K. J. Xu, A. Q. Liu, Q. Fang, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Nano-opto-mechanical actuator driven by gradient optical force,” Appl. Phys. Lett. 100(1), 013108 (2012). [CrossRef]

21.

J. F. Tao, J. Wu, H. Cai, Q. X. Zhang, J. M. Tsai, J. T. Lin, and A. Q. Liu, “A nanomachined optical logic gate driven by gradient optical force,” Appl. Phys. Lett. 100(11), 113104 (2012). [CrossRef]

22.

C. S. Ma, Y. Z. Xu, X. Yan, Z. K. Qin, and X. Y. Wang, “Optimization and analysis of series-coupled microring resonator arrays,” Opt. Commun. 262(1), 41–46 (2006). [CrossRef]

23.

Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D microresonator lattice optical filters,” IEEE J. Quantum Electron. 41(11), 1410–1418 (2005). [CrossRef]

24.

S. Pamidighantam, R. Puers, K. Baert, and H. A. C. Tilmans, “Pull-in voltage analysis of electrostatically actuated beam structures with fixed-fixed and fixed-free end conditions,” J. Micromech. Microeng. 12(4), 458–464 (2002). [CrossRef]

25.

Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive-index measurement by michelson interferometer,” Opt. Commun. 78(2), 109–112 (1990). [CrossRef]

26.

H. Rokhsari and K. J. Vahala, “Observation of Kerr nonlinearity in microcavities at room temperature,” Opt. Lett. 30(4), 427–429 (2005). [CrossRef] [PubMed]

OCIS Codes
(130.0130) Integrated optics : Integrated optics
(190.0190) Nonlinear optics : Nonlinear optics

ToC Category:
Integrated Optics

History
Original Manuscript: April 17, 2012
Revised Manuscript: May 14, 2012
Manuscript Accepted: May 24, 2012
Published: July 23, 2012

Citation
Y. F. Yu, M. Ren, J. B. Zhang, T. Bourouina, C. S. Tan, J. M. Tsai, and A. Q. Liu, "Force-induced optical nonlinearity and Kerr-like coefficient in opto-mechanical ring resonators," Opt. Express 20, 18005-18015 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18005


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References

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