Optomechanical nonlinearity enhanced optical sensors Jiahua Fan,∗ Chenguang Huang, and Lin Zhu Department of Electrical and Computer Engineering, Center for Optical Material Science and Engineering Technologies, Clemson University, Clemson, South Carolina 29634, USA ∗ [email protected]
Abstract: We propose and investigate an ultra-sensitive optical sensor system based on optomechanically induced nonlinear effects in high-Q optical resonators. In both dispersive and dissipative optomechanical systems, a positive feedback is formed between the optical resonance frequency and the mechanical displacement, which results in nonlinear transmission spectra different from a Lorentzian profile. Given the same resonator design, the optomechanical nonlinearity can increase the overall sensitivity by at least two orders of magnitude. Further improvement is possible by employing the phase sensitive detection. For the stable operation of the proposed sensor, we also analyze the requirement on the input power and the optomechanical coupling rate to overcome the thermal-optically induced frequency shift. © 2015 Optical Society of America OCIS codes: (280.4788) Optical sensing and sensors; (350.4855) Optical tweezers or optical manipulation; (130.3990) Micro-optical devices.
References and links 1. E. Krioukov, D. J. W. Klunder, A. Driessen, J. Greve, and C. Otto, “Sensor based on an integrated optical microcavity,” Opt. Lett. 27, 512–514 (2002). 2. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007). 3. S. Campopiano, R. Bernini, L. Zeni, and P. M. Sarro, “Microfluidic sensor based on integrated optical hollow waveguides,” Opt. Lett. 29, 1894–1896 (2004). 4. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002). 5. K. DeVos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15, 7610–7615 (2007). 6. S. Y. Cho and N. M. Jokerst, “A polymer microdisk photonic sensor integrated onto silicon,” IEEE Photon. Technol. Lett. 18, 2096–2098 (2006). 7. J. J. Lillie, M. A. Thomas, N. M. Jokerst, S. E. Ralph, K. A. Dennis, and C. L. Henderson, “Multimode interferometric sensors on silicon optimized for fully integrated complementary-metal-oxide-semiconductor chemicalbiological sensor systems,” J. Opt. Soc. Am. B 23, 642–651 (2006). 8. D. H Luo, R. A Levy, Y. F Hor, J. F Federici, and R. M Pafchek, “An integrated photonic sensor for in situ monitoring of hazardous organics,” Sensors and Actuators B: Chemical 92, 121–126 (2003). 9. J. Yang and L. J. Guo, “Optical sensors based on active microcavities,” IEEE J. Sel. Top. Quantum Electron. 12, 143–147 (2006). 10. F. Liu, S. Lan, and M. Hossein-Zadeh, “Application of dynamic line narrowing in resonant optical sensing,” Opt. Lett. 36, 4395–4397 (2011). 11. Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q, low index-contrast polymeric photonic crystal nanobeam cavities,” Opt. Express 19, 22191–22197 (2011) 12. Q. Quan, F. Vollmer, I. B. Burgess, P. B. Deotare, I. W. Frank, S. K. Y. Tang, R. Illic, and M. Loncar, “Ultrasensitive on-chip photonic crystal nanobeam sensor using optical bistability,” Conference on Laser and Electro Optics, paper QThH6 (2011)
#225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2973
13. C. Wang and C. P. Search, “Nonlinearly enhanced refractive index sensing in coupled optical microresonators,” Opt. Lett. 39, 26–29 (2014). 14. M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009). 15. M. Wu, A. C. Hryciw, C. Healey, D. P. Lake, H. Jayakumar, M. R. Freeman, J. P. Davis, and P. E. Barclay, “Dissipative and dispersive optomechanics in a nanocavity torque sensor,” Phys. Rev. X 4, 021052 (2014). 16. A. Xuereb, R. Schnabel, and K. Hammerer, “Dissipative optomechanics in a michelson-sagnac interferometer,” Phys. Rev. Lett. 107, 213604 (2011). 17. V. S. Ilchenko and M. L. Gorodetskii, “Thermal nonlinear effects in optical whispering gallery microresonators,” Laser Phys. 2, 1004–1009 (1992). 18. R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448 (1978). 19. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006). 20. C. Huang, J. Fan, and L. Zhu, “Dynamic nonlinear thermal optical effects in coupled ring resonators”, AIP Advances 2, 032131 (2012)
1.
Introduction
Integrated optical sensor has been a subject of interests in the past decade for its compact size, high detection efficiency and chip-scale integration capability with other components [1, 2]. A number of configurations have been demonstrated for optical refractive index (RI) sensors, such as optical waveguides [3], optical micro-cavities [4–6] and Mach-Zehnder interferometers [7, 8]. The RI change of the sample is detected by monitoring the change of the output transmission. In such detection schemes, employing low loss dielectric optical micro-resonators can greatly improve the sensitivity. The long photon lifetime for optical signal traveling inside these resonators provides strong interaction between the optical modes and the surrounding materials, which makes the optical resonance frequency susceptible to environmental perturbations. The change of the RI is then converted to the output signal by detecting the output power or monitoring the frequency shift in the transmission spectrum. Various types of resonators have been adopted for sensing applications, such as the microring resonator and the whispering-gallery mode (WGM) resonator. In optical resonator based RI sensors, high Q-factors are usually desired for a high sensitivity and a low detection limit. However, such high-Q factors may not be readily obtained for chip-scale photonics circuits due to the material absorption and the optical scattering loss. To reduce the reliance on very high Q-factors, several approaches have been proposed to increase the sensor performance by engineering the transmission spectrum. A gain-assisted resonator is adopted to increase the variation range of the transmission [9]. Thermal-optic effect has been used for narrowing the linewidth of the optical resonator [10] and changing the transmission spectrum [11, 12]. Coupled resonators with the nonlinear self-phase modulation have been used to re-shape the transmission function [13]. In this work, we propose to use optomechanical nonlinearity to improve the sensitivity of microresonator based RI sensors. The enhanced optical field inside the high-Q optical microresonator generates an optical force through optomechanical interactions. It can change the mechanical properties, such as the dimension of the resonator or the density of material. This change will in return modify the optical resonance frequency as well as the optical transmission. Because the strength of the optical force depends on the intensity of the intra-cavity light, a positive feedback is formed between the optical resonance frequency and the mechanical displacement, which generates a fast change of the output transmission for a certain frequency range. Utilizing this strong dependence of the output power over the optical resonance shift can improve the sensitivity of the sensor system. In the following discussions, we investigate the feasibility and performance of this concept in different types of optomechanical systems.
#225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2974
2.
Formalism
Fig. 1. Schematics of a ring resonator based nonlinear optomechanical system. The dispersive optomechanical coupling (gom ) changes the optical resonance frequency of the resonator, and the dissipative optomechanical coupling (γom ) changes the decay rate of the resonator.
One of the simplest optical sensors based on chip scale micro-resonators contains a bus waveguide and a ring resonator. The input light with a fixed wavelength near resonance is launched into the system from the input port and the transmission is detected through the output port. When the effective mode index n, or the optical resonance ω0 , is changed due to environmental perturbations, it can be detected by monitoring the transmitted optical power. To enhance the sensitivity, an additional freedom is introduced to the system to form an optomechanical cavity, as shown in Fig. 1. The light inside the cavity generates an optical force that changes the mechanical properties of the resonator. If this change modifies the optical resonance frequency, the system is dispersive (red); on the other hand, if it modifies the decay rate of the resonator, the system is dissipative (blue). In both types of systems, the amplitudes of the optical field a and mechanical displacement x satisfy the dynamic equations Γ γom x da = [iΔ − igom x − ( + )]a + i Γe + γom xAin (1a) dt 2 2 Fo dx d2x (1b) + Γm + Ω2m x = dt 2 dt m where Ain is the amplitude of the input optical power with |Ain |2 = Pin ; Δ = ω − ω0 is the frequency detuning of the input optical field (ω ) from the resonance (ω0 ); Γ is the total decay rate of the resonator which equals to the sum of the external decay rate Γe and the intrinsic decay rate Γi . The optomechanical coupling strength is characterized by the dispersive coupling coefficient gom = d ω /dx and the dissipative coupling coefficient γom = dΓe /dx. The mechanical motion is modeled as a damped harmonic oscillator driven by optical force Fo . m, Γm , Ωm are the effective mass, decay rate and resonance frequency of the mechanical oscillation, respectively. The optical forces induced by optomechanical coupling have been derived in [14], which are expressed as gom |a|2 ω γom |a|2 Δ Fγ = − ω Γe
Fg = −
#225123 - $15.00 USD (C) 2015 OSA
(2a) (2b)
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2975
The optical field and mechanical displacement can be obtained by solving Eq. (1) and (2). The working scheme of our sensor is set by inputing a continuous wave laser with the fixed wavelength and input power. When the system arrives at the steady state, the change of an environmental parameter, such as temperature, refractive index, geometry, etc., can change the optical resonance frequency, and then modifies the transmitted power. We define the overall sensitivity (S) as the ratio of the optical transmission change to the environmental perturbation, which can be further divided into three parts S=
dT dΔ dn · · dΔ dn d α
(3)
Here T = Pout /Pin is the normalized transmission and d α denotes the change of the environmental parameter. The derivative of the detuning frequency over the mode index (dΔ/dn) depends on the refractive index and geometry of the resonator. For the RI sensor based on high Q microresonators, the transduction of the environmental variation to the change of the mode index (dn/d α ) is usually small and is also determined by the resonator material and geometry. Since the last two parts of S are fixed for a given resonator, further increase of the sensitivity requires the increase of the transduction rate between the resonance shift and the output transmission (dT /dΔ), which is the slope of the transmission spectrum. For the linear resonators, the transmission T follows a Lorentzian function, so that dT /dΔ is fixed given the loss rates of the optical resonator. The idea of the optomechanical nonlinear sensor is to use the optomechanical effect to engineer the transmission spectrum and increase its slope. For simplicity, we analyze purely dispersive and dissipative coupling separately in the following sections. Dispersive coupling Pin [mW]
−8
0.8
0.8
T
0.6 0.4 g
0.2
≠0
om
x 10
0 600
gom≠ 0 g
=0
om
Enhancement
1
|dT/dΔ| [Hz−1]
1
0.6 0.4 0.2
0
Δ [GHz]
5
0 −4
4
6
8
−3
(a)
−2
Δ [GHz]
−1
0
10 600
500
500
400
400
300
300
200
200
100
gom=0 0 −5
2
0 0
Enhancement
3.
100
10
(b)
20
30
40
0 50
gom [GHz/nm]
(c)
Fig. 2. (a) The normalized transmission curves in the dispersive system with (red) and without (blue) the optomechanical coupling for the frequency detuning scanned from negative to positive. (b) The calculated relative sensitivity as a function of the frequency detuning. The cavity resonance is at 1550nm under critical coupling condition (Γe = Γi = 1GHz). Parameters used in the calculation are Pin = 1mW, Ωm = 100MHz, m = 1pg, gom = 28GHz/nm. ω0 = 2π c/1550nm where c is the speed of light. (c) The enhancement factor of the nonlinear sensor system as a function of (black) the optomechanical coupling coefficient with Pin = 1mW and (red) the input power with gom = 28GHz/nm.
Given γom = 0, the steady state mechanical displacement is governed by a cubic function g2om x3 − 2Δgom x2 + (
#225123 - $15.00 USD (C) 2015 OSA
Γ2 gom Γe Pin + Δ2 )x + =0 4 mΩ2m ω
(4)
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2976
The above equation solves for the mechanical displacement given a certain input wavelength and optical√power. The output optical power is then calculated using the output relation PT = |Ain + i Γe a|2 . In Fig. 2(a), the normalized optical transmission spectra are plotted for the proposed optical sensor. If the system has no optomechanical coupling, the transmission spectrum follows a symmetric Lorentzian function, as indicated by the blue curve. When an optomechanically induced resonance shift is considered, the transmission curve becomes asymmetric as indicated by the red curve. For the sensing operation, the input laser is set on the red side (lower frequency side) of the shifted resonance dip. If the optical resonance frequency decreases due to an environmental perturbation, more optical power will enter the resonator. This generates a larger mechanical displacement through the optomechanical effect. Since the increase of the mechanical displacement favors a further decreased optical resonance frequency, a positive feedback is formed, resulting in a fast change of the transmission spectrum for a narrow frequency range. The slope of the transmission curves are plotted in Fig. 2(b). Since the overall sensitivity is proportional to the slope, it is clear that the dispersive optomechanical coupling can enhance the sensitivity of the sensor by at least two orders of magnitude. We define an enhancement factor as the ratio between the maximum slope of the transmission curve in the systems with and without the nonlinear optomechanical effect. As shown in Fig. 2(c), the enhancement factor generally increases with the input power and gom because of the enhanced nonlinear effect. However, when the optomechanical coupling coefficient is further increased beyond a critical value ( 36 GHz/nm in Fig. 2(c)), the system will fall into another bistable state before reaching the maximum sensitivity, which leads to a reduced enhancement factor. This optomechanically induced bistability will be discussed separately in a subsequent section. 4.
Dissipative coupling
In the dissipative optomechanical system, instead of varying the optical resonance, the change of the mechanical displacement modifies the decay rates of the resonator. Experimental demonstration of such systems can be found in various setups, such as the ring resonator [14] and the Michelson-Sagnac interferometer [16]. Most of the previous works focus on the control and entanglement of the quantum state of the intracavity field. The observation of a nonlinear transmission spectrum has been lacking due to the relative low coupling strength for dissipative systems. Recently the coupled photonic crystal cavities [15] demonstrated a strong dissipative coupling rate that is comparable with the dispersive system, which makes it possible to utilize the dissipative optomechanical nonlinearity. Here we propose a nonlinear optical sensor utilizing the pure optomechanical dissipative coupling. Although a pure dissipative system with a relatively large coupling rate has not been demonstrated so far, the results will be useful for analyzing system that contains a combination of dispersive and dissipative optomechanical coupling. For a pure dissipative optomechanical system, the steady state mechanical displacement is obtained by setting gom = 0 in Eq. (1) and using Eq. (2b). Γ2 1 2 3 1 γ 2 ΔPin γom Pin Δ γom x + γom Γx2 + ( + Δ2 + om2 )x + =0 4 2 4 mΩm ω Γe mΩ2m ω
(5)
The optical transmission is then calculated using Eq. 1 and the output formalism. The idea of the nonlinear sensor is to utilize the dissipative coupling and modify the linewidth of the transmission curve over a narrow frequency range. For an optical resonator sensor without nonlinear effects, the transmission spectrum corresponds to a Lorentzian function with the linewidth equal to Γ. The maximum sensitivity is reached when the input frequency is detuned from the reso#225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2977
√ nance by Δ = Γ/2 3. In the nonlinear sensor, the optical resonator is set in the under-coupled regime (Γe Γ/2). When the dissipative coupling is present, the optical force generated inside the resonator will alter the external coupling rate as well as the trnasmission. In Fig. 3(a), we scan the frequency of the input light and compare its transmission (red curve) with systems without the optomechanical effect. Because the dissipative optical force is proportional to the detuning frequency, the optomechanical effect reduces the external coupling rate on the blue side of the resonance (Δ > 0), and generates narrower linewidth compared with the linear system in the under-coupled regime (blue curve). On the red side of the resonance (Δ < 0), the optomechanical effect increases the external coupling rate. Therefore, the extinction ratio of the transmission curve increases from Δ = 0 to Δ = Δc , where the modified external coupling rate equals to the intrinsic coupling rate and the transmission curve coincide with the linear system under the critically-coupled condition (black curve). Because the power dropped into the resonator increases with the increased extinction ratio, a positive feedback is formed within this frequency range to generate a steep slope. For Δ < Δc , the system falls into the over-coupled regime and its transmission spectrum is very broad. Figure 3(b) shows the slope of the transmission spectra and Fig. 3(c) shows the enhancement factor with respect to the coupling coefficient and input power. Compared with the linear system, the slope of the transmission curve as well as the sensitivity can be improved by more than two orders of magnitude, as shown in Fig. 3(c). Different from the dispersive system discussed before, the dissipative optomechanical coupling tends to broaden the linewidth on the red side of the resonance dip. The sensitivity may be further increased for a hybrid system containing both dispersive and dissipative optomechanical coupling if their induced optical forces have the same sign. However, realizing such hybrid systems can be challenging and requires accurate control of the device parameters. Pin [mW]
−8
−1
|dT/dΔ| [Hz ]
0.8
T
0.6 0.4 γom=0 γom≠ 0
0.2
γom=0, Γe=Γi
0 −4
−2
Δc 0
Δ/Ωm
2
4
(a)
x 10
0.8
0 600
γom≠ 0
0.6 0.4 0.2 0 −1
1
2
3
4
5 600
γom=0
−0.5
0
Δ [GHz]
(b)
0.5
1
500
500
400
400
300
300
200
200
100
100
0 0
10
20
30
Γom [GHz/nm]
40
Enhancement
1
Enhancement
1
0 50
(c)
Fig. 3. (a) Transmission curves for the optical resonator without optomechanical coupling in the under-coupled regime (blue, Γi = 20Γe = 1GHz) and under the criticallycoupled condition (black, Γi = Γe = 1GHz)). The red curve is the transmission for a dissipative optomechanical system. (b) Calculated relative sensitivities for the system with and without optomechanical coupling. The dissipative optomechanical coupling rate is γom = 20GHz/nm. The input power P = 1mW. (c) The enhancement factor of the nonlinear sensor system as a function of (black) the optomechanical coupling coefficient with Pin = 1mW and (red) the input power with γom = 20GHz/nm.
5.
Thermal-optic effect
In the previous sections, we have theoretically analyzed the potential of an enhanced sensitivity for the nonlinear optomechanical sensors. For practical applications, it is also instructive to consider other nonlinear effects that can potentially influence the performance. Here we analyze the impact of the thermal-optic effect for instance. Due to the enhanced light field inside the resonator, the material absorption causes the temperature increase and the red-shifted resonance #225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2978
in materials with positive thermal-optic coefficients. Although the slope of the transmission curve can be further increased by the positive thermal-optic effect because of the positive feedback similar to the optomechanical effects, an additional noise component may be induced on the output transmission due to the susceptibility of the temperature to environmental fluctuations. To avoid the thermal-optical effect in the sensing operation, we need to ensure that the optomechanical coupling strength is large enough to overcome the thermal-optic effect. The resonance shift due to the thermal-optic effect can be expressed as a function of the temperature change [19] dnm Δt · ) ω (Δt) = ω0 (1 − (6) dt nm while nm is the material refractive index. The temperature change Δt = Ph /K depends on the heat power inside the resonator Ph and the thermal conductivity K. To avoid complicated derivation of the coupled equations, here we make a first order estimation to evaluate the minimum optomechanical coupling rate required to overcome the thermal-optic effect. We consider a critically coupled optical resonator and an input laser (Pin = 1mW) blue-detuned by half a linewidth of the resonator. For the resonators fabricated in silicon, dnm /dt = 1.84×10−4 (K −1 ). Since the value of K depends on the geometry of the resonator, here we use a typical value K = 4.5 × 10−4 (J/s · K) for estimation [20]. Assuming all the optical loss contributes to heat power and the thermal dissipation rate of the fabricated device is the same as in bulk material, if the resonance increases by Δω due to the external perturbation, the thermal effect will induce an additional resonance shift of about 2.9Δω . If we require the frequency shift caused by optomechanical effect to be at least an order of magnitude larger to neglect thermal-optic effect, the optomechanical coupling rate needs to be larger than 42.5GHz/nm for the dispersive system and 19.2GHz/nm for the dissipative system. Although these parameters are achievable by use of current fabrication techniques, they can be further reduced by stabilizing the temperature with an additional pump source, or using other materials with a lower thermal-optic coefficient, such as SiO2 or SiN. 6.
Phase sensitive detection 1
1
1
0.8
0.6
0.6
f =200MHz m
T
T
0.8
Distortion (A.U.)
fm=0
0.4
0.4
f =0
0.2
0.2
m
f =200MHz −3
−2
−1
Δ (GHz)
(a)
0
1
2
0 −3
0.998 0.997 dispersive dissipative 0.996 0
m
0 −4
0.999
−2
−1
0
Δ (GHz)
(b)
1
2
3
50
100
150
200
f (MHz) m
(c)
Fig. 4. The normalized transmission curves with and without phase sensitive detection for (a) dispersive (b) dissipative optomechanical systems. The input laser is modulated at the frequency fm before entering the resonator for the phase sensitive detection . The output signal is then demodulated at fm to obtain the noise-reduced transmission spectrum. The mechanical resonance frequency Ωm =100MHz and the effective mass m=1pg. (c) the distortion factor is calculated as the overlap integral (over a 5GHz frequency span centered at the effective resonance) between the transmission functions of a modulated and a nonmodulated input source.
So far the working scheme of the sensor is to monitor the change of the CW output power. #225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2979
This can be rather difficult for ultrasensitive applications due to various noises embedded in the system, including environmental noises and noises from the electronic components. One approach to improve the signal-to-noise ratio is to use the phase sensitive detection (PSD). To employ this technique, typically the amplitude of the input light is modulated before coupling into the resonator. The PSD only detects the signals whose frequencies are very close to the modulation frequency. Since noises generally contain strong low frequency components, a relatively high modulation frequency allows the system to avoid unwanted noises and detect the signal that would otherwise be overwhelmed. However, in practical applications the modulation frequency can not be very high in order to reduce the cost and to avoid the distortion of the transmission spectrum. In the proposed nonlinear sensor system, the laser frequency is detuned from the resonance to obtain the maximum sensitivity. The amplitude modulation generates two frequency sidebands which interact with the resonator separately. Since the two sidebands have different detuning frequencies, the transmission spectrum of the total output signal extracted from the PSD can be distorted due to the nonlinear nature of the system. In Fig. 4, we calculate the transmission spectra using the PSD technique with a modulation frequency of fm = 200MHz and compare with previous results ( fm = 0). The frequency components (ω , ω ± fm ) interact with the optomechanical sensor separately based on the coupled equations (Eq. (1)). The transmitted signal is then demodulated at the same modulation frequency fm , and the DC component extracted is proportional to ℜ{a(ω ) × [a∗ (ω + fm ) + a∗ (ω − fm )]}. For both dispersive and dissipative systems, the transmission spectra deviate from the non-modulated systems near the regions that correspond to high sensitivities. Since the purpose of utilizing the PSD is to reduce the noise background, such distortion needs to be avoided. We define a distortion factor as the normalized overlap integral between the transmission functions with a modulated and a non-modulated laser source. Therefore, a lower value of this factor corresponds to a larger distortion. As shown in Fig. 5, the distortion becomes larger as the modulation frequency increases in both types of system, which indicates that an optimized range of fm exists to balance the noise reduction and the distortion. Because the frequency components of main system noises are usually below 100KHz, using a modulation frequency between 100KHz and 10MHz can improve the signal-to-noise ratio and still provide an accurate detection of the system response. 7.
Limitation
Generally, nonlinear effects are enhanced with increasing the input power, but high power can also generate limitations which should be considered for practical applications. In the dispersive system, because the optomechanical systems are governed by a cubic function, optomechanically induced bistability exists for a higher input power or a stronger optomechanical coupling rate. Although the bistable effect is useful for various applications, switching between bistable states should be avoided in the proposed sensor where an unambiguous relation between the input and output signal is required. As shown in Fig. 5(a), for an increased input power (Pin = 1.2mW), the transmission spectrum of the dispersive system stays in different states depending on the direction of the frequency scan. When the frequency of the input laser is scanned from the lower frequency side, the induced resonance shift is in the opposite direction of the frequency scan, which generates the positive feedback and the steep transmission curve (red line). When the frequency of the input laser is scanned from the higher frequency side to the resonance, the transmission curve becomes stretched because the effective frequency detuning is increased due to the shifted resonance. The transmission curves are similar to the thermal optically induced bistability. Because the high sensitivity exists only on the lower frequency side of the resonance, this optomechanically induced bistability may jeopardize the stable operation of the sensor. When operating
#225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2980
the proposed sensor with an input power above the threshold of the bistability, the input light should be red detuned from the maximum sensitivity position to avoid crossing to a different state due to perturbations from noise sources. Therefore, the bistability imposes a limitation on the sensitivity depending on the noise levels and the measurement bandwidth. In the dissipative system, the sensitivity is maximized when the input light is near the cavity resonance and decreases for larger detuning frequency. Its maximum value can be further increased by using larger input power or γom , as shown in Fig. 3(c). However, stronger nonlinear effects also generate a faster change of linewidth near the resonance frequency, corresponding to a narrower lineshape of the sensitivity function. In Fig. 5(b), the FWHM of the sensitivity function (red curve in Fig. 3(b)) is calculated for different input powers. It is clear that high powers (strong nonlinear effects) correspond to small bandwidths. For sensing signals containing frequency components over a large freqeuncy range, the sensor system needs to provide enough bandwidth to avoid signal distortion, which thus limits the maximum nonlinear sensitivity enhancement. Additionally, because the measurement bandwidth of the sensor can not be larger than the resonance frequency of the mechanical resonator, the actual bandwidth of the device may be further reduced. 1
40
FWHM (MHz)
0.8
T
0.6 0.4 0.2 0 −5
0
Δ [GHz] (a)
5
35 30 25 20 15 10 1
1.5
2
2.5
3
Pin [mW] (b)
Fig. 5. (a) Optomechanically induced bistability in the dispersive system for Pin = 1.2mW. The system stays in different states depending on the direction of the frequency scan. (b) The calculated FWHM of the sensitivity function in the dissipative system for different input powers.
8.
Conclusion
In conclusion, we propose and analyze a new type of nonlinear optical sensor based on optomechanical effects. With practical parameters, we have shown that by utilizing the optomechanical nonlinearity, very high sensitivities can be obtained in both dispersive and dissipative systems. We have analyzed the required optomechanical coupling rate to overcome thermal opitcal effect and the bandwidth limitation of the proposed sensor. When the optomechanically induced bistability exists, the input light needs to be detuned from the maximum sensitivity to avoid crossing between different bistable states.
#225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2981
Abstract: We propose and investigate an ultra-sensitive optical sensor system based on optomechanically induced nonlinear effects in high-Q optical resonators. In both dispersive and dissipative optomechanical systems, a positive feedback is formed between the optical resonance frequency and the mechanical displacement, which results in nonlinear transmission spectra different from a Lorentzian profile. Given the same resonator design, the optomechanical nonlinearity can increase the overall sensitivity by at least two orders of magnitude. Further improvement is possible by employing the phase sensitive detection. For the stable operation of the proposed sensor, we also analyze the requirement on the input power and the optomechanical coupling rate to overcome the thermal-optically induced frequency shift. © 2015 Optical Society of America OCIS codes: (280.4788) Optical sensing and sensors; (350.4855) Optical tweezers or optical manipulation; (130.3990) Micro-optical devices.
References and links 1. E. Krioukov, D. J. W. Klunder, A. Driessen, J. Greve, and C. Otto, “Sensor based on an integrated optical microcavity,” Opt. Lett. 27, 512–514 (2002). 2. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007). 3. S. Campopiano, R. Bernini, L. Zeni, and P. M. Sarro, “Microfluidic sensor based on integrated optical hollow waveguides,” Opt. Lett. 29, 1894–1896 (2004). 4. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002). 5. K. DeVos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15, 7610–7615 (2007). 6. S. Y. Cho and N. M. Jokerst, “A polymer microdisk photonic sensor integrated onto silicon,” IEEE Photon. Technol. Lett. 18, 2096–2098 (2006). 7. J. J. Lillie, M. A. Thomas, N. M. Jokerst, S. E. Ralph, K. A. Dennis, and C. L. Henderson, “Multimode interferometric sensors on silicon optimized for fully integrated complementary-metal-oxide-semiconductor chemicalbiological sensor systems,” J. Opt. Soc. Am. B 23, 642–651 (2006). 8. D. H Luo, R. A Levy, Y. F Hor, J. F Federici, and R. M Pafchek, “An integrated photonic sensor for in situ monitoring of hazardous organics,” Sensors and Actuators B: Chemical 92, 121–126 (2003). 9. J. Yang and L. J. Guo, “Optical sensors based on active microcavities,” IEEE J. Sel. Top. Quantum Electron. 12, 143–147 (2006). 10. F. Liu, S. Lan, and M. Hossein-Zadeh, “Application of dynamic line narrowing in resonant optical sensing,” Opt. Lett. 36, 4395–4397 (2011). 11. Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q, low index-contrast polymeric photonic crystal nanobeam cavities,” Opt. Express 19, 22191–22197 (2011) 12. Q. Quan, F. Vollmer, I. B. Burgess, P. B. Deotare, I. W. Frank, S. K. Y. Tang, R. Illic, and M. Loncar, “Ultrasensitive on-chip photonic crystal nanobeam sensor using optical bistability,” Conference on Laser and Electro Optics, paper QThH6 (2011)
#225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2973
13. C. Wang and C. P. Search, “Nonlinearly enhanced refractive index sensing in coupled optical microresonators,” Opt. Lett. 39, 26–29 (2014). 14. M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009). 15. M. Wu, A. C. Hryciw, C. Healey, D. P. Lake, H. Jayakumar, M. R. Freeman, J. P. Davis, and P. E. Barclay, “Dissipative and dispersive optomechanics in a nanocavity torque sensor,” Phys. Rev. X 4, 021052 (2014). 16. A. Xuereb, R. Schnabel, and K. Hammerer, “Dissipative optomechanics in a michelson-sagnac interferometer,” Phys. Rev. Lett. 107, 213604 (2011). 17. V. S. Ilchenko and M. L. Gorodetskii, “Thermal nonlinear effects in optical whispering gallery microresonators,” Laser Phys. 2, 1004–1009 (1992). 18. R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448 (1978). 19. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006). 20. C. Huang, J. Fan, and L. Zhu, “Dynamic nonlinear thermal optical effects in coupled ring resonators”, AIP Advances 2, 032131 (2012)
1.
Introduction
Integrated optical sensor has been a subject of interests in the past decade for its compact size, high detection efficiency and chip-scale integration capability with other components [1, 2]. A number of configurations have been demonstrated for optical refractive index (RI) sensors, such as optical waveguides [3], optical micro-cavities [4–6] and Mach-Zehnder interferometers [7, 8]. The RI change of the sample is detected by monitoring the change of the output transmission. In such detection schemes, employing low loss dielectric optical micro-resonators can greatly improve the sensitivity. The long photon lifetime for optical signal traveling inside these resonators provides strong interaction between the optical modes and the surrounding materials, which makes the optical resonance frequency susceptible to environmental perturbations. The change of the RI is then converted to the output signal by detecting the output power or monitoring the frequency shift in the transmission spectrum. Various types of resonators have been adopted for sensing applications, such as the microring resonator and the whispering-gallery mode (WGM) resonator. In optical resonator based RI sensors, high Q-factors are usually desired for a high sensitivity and a low detection limit. However, such high-Q factors may not be readily obtained for chip-scale photonics circuits due to the material absorption and the optical scattering loss. To reduce the reliance on very high Q-factors, several approaches have been proposed to increase the sensor performance by engineering the transmission spectrum. A gain-assisted resonator is adopted to increase the variation range of the transmission [9]. Thermal-optic effect has been used for narrowing the linewidth of the optical resonator [10] and changing the transmission spectrum [11, 12]. Coupled resonators with the nonlinear self-phase modulation have been used to re-shape the transmission function [13]. In this work, we propose to use optomechanical nonlinearity to improve the sensitivity of microresonator based RI sensors. The enhanced optical field inside the high-Q optical microresonator generates an optical force through optomechanical interactions. It can change the mechanical properties, such as the dimension of the resonator or the density of material. This change will in return modify the optical resonance frequency as well as the optical transmission. Because the strength of the optical force depends on the intensity of the intra-cavity light, a positive feedback is formed between the optical resonance frequency and the mechanical displacement, which generates a fast change of the output transmission for a certain frequency range. Utilizing this strong dependence of the output power over the optical resonance shift can improve the sensitivity of the sensor system. In the following discussions, we investigate the feasibility and performance of this concept in different types of optomechanical systems.
#225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2974
2.
Formalism
Fig. 1. Schematics of a ring resonator based nonlinear optomechanical system. The dispersive optomechanical coupling (gom ) changes the optical resonance frequency of the resonator, and the dissipative optomechanical coupling (γom ) changes the decay rate of the resonator.
One of the simplest optical sensors based on chip scale micro-resonators contains a bus waveguide and a ring resonator. The input light with a fixed wavelength near resonance is launched into the system from the input port and the transmission is detected through the output port. When the effective mode index n, or the optical resonance ω0 , is changed due to environmental perturbations, it can be detected by monitoring the transmitted optical power. To enhance the sensitivity, an additional freedom is introduced to the system to form an optomechanical cavity, as shown in Fig. 1. The light inside the cavity generates an optical force that changes the mechanical properties of the resonator. If this change modifies the optical resonance frequency, the system is dispersive (red); on the other hand, if it modifies the decay rate of the resonator, the system is dissipative (blue). In both types of systems, the amplitudes of the optical field a and mechanical displacement x satisfy the dynamic equations Γ γom x da = [iΔ − igom x − ( + )]a + i Γe + γom xAin (1a) dt 2 2 Fo dx d2x (1b) + Γm + Ω2m x = dt 2 dt m where Ain is the amplitude of the input optical power with |Ain |2 = Pin ; Δ = ω − ω0 is the frequency detuning of the input optical field (ω ) from the resonance (ω0 ); Γ is the total decay rate of the resonator which equals to the sum of the external decay rate Γe and the intrinsic decay rate Γi . The optomechanical coupling strength is characterized by the dispersive coupling coefficient gom = d ω /dx and the dissipative coupling coefficient γom = dΓe /dx. The mechanical motion is modeled as a damped harmonic oscillator driven by optical force Fo . m, Γm , Ωm are the effective mass, decay rate and resonance frequency of the mechanical oscillation, respectively. The optical forces induced by optomechanical coupling have been derived in [14], which are expressed as gom |a|2 ω γom |a|2 Δ Fγ = − ω Γe
Fg = −
#225123 - $15.00 USD (C) 2015 OSA
(2a) (2b)
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2975
The optical field and mechanical displacement can be obtained by solving Eq. (1) and (2). The working scheme of our sensor is set by inputing a continuous wave laser with the fixed wavelength and input power. When the system arrives at the steady state, the change of an environmental parameter, such as temperature, refractive index, geometry, etc., can change the optical resonance frequency, and then modifies the transmitted power. We define the overall sensitivity (S) as the ratio of the optical transmission change to the environmental perturbation, which can be further divided into three parts S=
dT dΔ dn · · dΔ dn d α
(3)
Here T = Pout /Pin is the normalized transmission and d α denotes the change of the environmental parameter. The derivative of the detuning frequency over the mode index (dΔ/dn) depends on the refractive index and geometry of the resonator. For the RI sensor based on high Q microresonators, the transduction of the environmental variation to the change of the mode index (dn/d α ) is usually small and is also determined by the resonator material and geometry. Since the last two parts of S are fixed for a given resonator, further increase of the sensitivity requires the increase of the transduction rate between the resonance shift and the output transmission (dT /dΔ), which is the slope of the transmission spectrum. For the linear resonators, the transmission T follows a Lorentzian function, so that dT /dΔ is fixed given the loss rates of the optical resonator. The idea of the optomechanical nonlinear sensor is to use the optomechanical effect to engineer the transmission spectrum and increase its slope. For simplicity, we analyze purely dispersive and dissipative coupling separately in the following sections. Dispersive coupling Pin [mW]
−8
0.8
0.8
T
0.6 0.4 g
0.2
≠0
om
x 10
0 600
gom≠ 0 g
=0
om
Enhancement
1
|dT/dΔ| [Hz−1]
1
0.6 0.4 0.2
0
Δ [GHz]
5
0 −4
4
6
8
−3
(a)
−2
Δ [GHz]
−1
0
10 600
500
500
400
400
300
300
200
200
100
gom=0 0 −5
2
0 0
Enhancement
3.
100
10
(b)
20
30
40
0 50
gom [GHz/nm]
(c)
Fig. 2. (a) The normalized transmission curves in the dispersive system with (red) and without (blue) the optomechanical coupling for the frequency detuning scanned from negative to positive. (b) The calculated relative sensitivity as a function of the frequency detuning. The cavity resonance is at 1550nm under critical coupling condition (Γe = Γi = 1GHz). Parameters used in the calculation are Pin = 1mW, Ωm = 100MHz, m = 1pg, gom = 28GHz/nm. ω0 = 2π c/1550nm where c is the speed of light. (c) The enhancement factor of the nonlinear sensor system as a function of (black) the optomechanical coupling coefficient with Pin = 1mW and (red) the input power with gom = 28GHz/nm.
Given γom = 0, the steady state mechanical displacement is governed by a cubic function g2om x3 − 2Δgom x2 + (
#225123 - $15.00 USD (C) 2015 OSA
Γ2 gom Γe Pin + Δ2 )x + =0 4 mΩ2m ω
(4)
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2976
The above equation solves for the mechanical displacement given a certain input wavelength and optical√power. The output optical power is then calculated using the output relation PT = |Ain + i Γe a|2 . In Fig. 2(a), the normalized optical transmission spectra are plotted for the proposed optical sensor. If the system has no optomechanical coupling, the transmission spectrum follows a symmetric Lorentzian function, as indicated by the blue curve. When an optomechanically induced resonance shift is considered, the transmission curve becomes asymmetric as indicated by the red curve. For the sensing operation, the input laser is set on the red side (lower frequency side) of the shifted resonance dip. If the optical resonance frequency decreases due to an environmental perturbation, more optical power will enter the resonator. This generates a larger mechanical displacement through the optomechanical effect. Since the increase of the mechanical displacement favors a further decreased optical resonance frequency, a positive feedback is formed, resulting in a fast change of the transmission spectrum for a narrow frequency range. The slope of the transmission curves are plotted in Fig. 2(b). Since the overall sensitivity is proportional to the slope, it is clear that the dispersive optomechanical coupling can enhance the sensitivity of the sensor by at least two orders of magnitude. We define an enhancement factor as the ratio between the maximum slope of the transmission curve in the systems with and without the nonlinear optomechanical effect. As shown in Fig. 2(c), the enhancement factor generally increases with the input power and gom because of the enhanced nonlinear effect. However, when the optomechanical coupling coefficient is further increased beyond a critical value ( 36 GHz/nm in Fig. 2(c)), the system will fall into another bistable state before reaching the maximum sensitivity, which leads to a reduced enhancement factor. This optomechanically induced bistability will be discussed separately in a subsequent section. 4.
Dissipative coupling
In the dissipative optomechanical system, instead of varying the optical resonance, the change of the mechanical displacement modifies the decay rates of the resonator. Experimental demonstration of such systems can be found in various setups, such as the ring resonator [14] and the Michelson-Sagnac interferometer [16]. Most of the previous works focus on the control and entanglement of the quantum state of the intracavity field. The observation of a nonlinear transmission spectrum has been lacking due to the relative low coupling strength for dissipative systems. Recently the coupled photonic crystal cavities [15] demonstrated a strong dissipative coupling rate that is comparable with the dispersive system, which makes it possible to utilize the dissipative optomechanical nonlinearity. Here we propose a nonlinear optical sensor utilizing the pure optomechanical dissipative coupling. Although a pure dissipative system with a relatively large coupling rate has not been demonstrated so far, the results will be useful for analyzing system that contains a combination of dispersive and dissipative optomechanical coupling. For a pure dissipative optomechanical system, the steady state mechanical displacement is obtained by setting gom = 0 in Eq. (1) and using Eq. (2b). Γ2 1 2 3 1 γ 2 ΔPin γom Pin Δ γom x + γom Γx2 + ( + Δ2 + om2 )x + =0 4 2 4 mΩm ω Γe mΩ2m ω
(5)
The optical transmission is then calculated using Eq. 1 and the output formalism. The idea of the nonlinear sensor is to utilize the dissipative coupling and modify the linewidth of the transmission curve over a narrow frequency range. For an optical resonator sensor without nonlinear effects, the transmission spectrum corresponds to a Lorentzian function with the linewidth equal to Γ. The maximum sensitivity is reached when the input frequency is detuned from the reso#225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2977
√ nance by Δ = Γ/2 3. In the nonlinear sensor, the optical resonator is set in the under-coupled regime (Γe Γ/2). When the dissipative coupling is present, the optical force generated inside the resonator will alter the external coupling rate as well as the trnasmission. In Fig. 3(a), we scan the frequency of the input light and compare its transmission (red curve) with systems without the optomechanical effect. Because the dissipative optical force is proportional to the detuning frequency, the optomechanical effect reduces the external coupling rate on the blue side of the resonance (Δ > 0), and generates narrower linewidth compared with the linear system in the under-coupled regime (blue curve). On the red side of the resonance (Δ < 0), the optomechanical effect increases the external coupling rate. Therefore, the extinction ratio of the transmission curve increases from Δ = 0 to Δ = Δc , where the modified external coupling rate equals to the intrinsic coupling rate and the transmission curve coincide with the linear system under the critically-coupled condition (black curve). Because the power dropped into the resonator increases with the increased extinction ratio, a positive feedback is formed within this frequency range to generate a steep slope. For Δ < Δc , the system falls into the over-coupled regime and its transmission spectrum is very broad. Figure 3(b) shows the slope of the transmission spectra and Fig. 3(c) shows the enhancement factor with respect to the coupling coefficient and input power. Compared with the linear system, the slope of the transmission curve as well as the sensitivity can be improved by more than two orders of magnitude, as shown in Fig. 3(c). Different from the dispersive system discussed before, the dissipative optomechanical coupling tends to broaden the linewidth on the red side of the resonance dip. The sensitivity may be further increased for a hybrid system containing both dispersive and dissipative optomechanical coupling if their induced optical forces have the same sign. However, realizing such hybrid systems can be challenging and requires accurate control of the device parameters. Pin [mW]
−8
−1
|dT/dΔ| [Hz ]
0.8
T
0.6 0.4 γom=0 γom≠ 0
0.2
γom=0, Γe=Γi
0 −4
−2
Δc 0
Δ/Ωm
2
4
(a)
x 10
0.8
0 600
γom≠ 0
0.6 0.4 0.2 0 −1
1
2
3
4
5 600
γom=0
−0.5
0
Δ [GHz]
(b)
0.5
1
500
500
400
400
300
300
200
200
100
100
0 0
10
20
30
Γom [GHz/nm]
40
Enhancement
1
Enhancement
1
0 50
(c)
Fig. 3. (a) Transmission curves for the optical resonator without optomechanical coupling in the under-coupled regime (blue, Γi = 20Γe = 1GHz) and under the criticallycoupled condition (black, Γi = Γe = 1GHz)). The red curve is the transmission for a dissipative optomechanical system. (b) Calculated relative sensitivities for the system with and without optomechanical coupling. The dissipative optomechanical coupling rate is γom = 20GHz/nm. The input power P = 1mW. (c) The enhancement factor of the nonlinear sensor system as a function of (black) the optomechanical coupling coefficient with Pin = 1mW and (red) the input power with γom = 20GHz/nm.
5.
Thermal-optic effect
In the previous sections, we have theoretically analyzed the potential of an enhanced sensitivity for the nonlinear optomechanical sensors. For practical applications, it is also instructive to consider other nonlinear effects that can potentially influence the performance. Here we analyze the impact of the thermal-optic effect for instance. Due to the enhanced light field inside the resonator, the material absorption causes the temperature increase and the red-shifted resonance #225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2978
in materials with positive thermal-optic coefficients. Although the slope of the transmission curve can be further increased by the positive thermal-optic effect because of the positive feedback similar to the optomechanical effects, an additional noise component may be induced on the output transmission due to the susceptibility of the temperature to environmental fluctuations. To avoid the thermal-optical effect in the sensing operation, we need to ensure that the optomechanical coupling strength is large enough to overcome the thermal-optic effect. The resonance shift due to the thermal-optic effect can be expressed as a function of the temperature change [19] dnm Δt · ) ω (Δt) = ω0 (1 − (6) dt nm while nm is the material refractive index. The temperature change Δt = Ph /K depends on the heat power inside the resonator Ph and the thermal conductivity K. To avoid complicated derivation of the coupled equations, here we make a first order estimation to evaluate the minimum optomechanical coupling rate required to overcome the thermal-optic effect. We consider a critically coupled optical resonator and an input laser (Pin = 1mW) blue-detuned by half a linewidth of the resonator. For the resonators fabricated in silicon, dnm /dt = 1.84×10−4 (K −1 ). Since the value of K depends on the geometry of the resonator, here we use a typical value K = 4.5 × 10−4 (J/s · K) for estimation [20]. Assuming all the optical loss contributes to heat power and the thermal dissipation rate of the fabricated device is the same as in bulk material, if the resonance increases by Δω due to the external perturbation, the thermal effect will induce an additional resonance shift of about 2.9Δω . If we require the frequency shift caused by optomechanical effect to be at least an order of magnitude larger to neglect thermal-optic effect, the optomechanical coupling rate needs to be larger than 42.5GHz/nm for the dispersive system and 19.2GHz/nm for the dissipative system. Although these parameters are achievable by use of current fabrication techniques, they can be further reduced by stabilizing the temperature with an additional pump source, or using other materials with a lower thermal-optic coefficient, such as SiO2 or SiN. 6.
Phase sensitive detection 1
1
1
0.8
0.6
0.6
f =200MHz m
T
T
0.8
Distortion (A.U.)
fm=0
0.4
0.4
f =0
0.2
0.2
m
f =200MHz −3
−2
−1
Δ (GHz)
(a)
0
1
2
0 −3
0.998 0.997 dispersive dissipative 0.996 0
m
0 −4
0.999
−2
−1
0
Δ (GHz)
(b)
1
2
3
50
100
150
200
f (MHz) m
(c)
Fig. 4. The normalized transmission curves with and without phase sensitive detection for (a) dispersive (b) dissipative optomechanical systems. The input laser is modulated at the frequency fm before entering the resonator for the phase sensitive detection . The output signal is then demodulated at fm to obtain the noise-reduced transmission spectrum. The mechanical resonance frequency Ωm =100MHz and the effective mass m=1pg. (c) the distortion factor is calculated as the overlap integral (over a 5GHz frequency span centered at the effective resonance) between the transmission functions of a modulated and a nonmodulated input source.
So far the working scheme of the sensor is to monitor the change of the CW output power. #225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2979
This can be rather difficult for ultrasensitive applications due to various noises embedded in the system, including environmental noises and noises from the electronic components. One approach to improve the signal-to-noise ratio is to use the phase sensitive detection (PSD). To employ this technique, typically the amplitude of the input light is modulated before coupling into the resonator. The PSD only detects the signals whose frequencies are very close to the modulation frequency. Since noises generally contain strong low frequency components, a relatively high modulation frequency allows the system to avoid unwanted noises and detect the signal that would otherwise be overwhelmed. However, in practical applications the modulation frequency can not be very high in order to reduce the cost and to avoid the distortion of the transmission spectrum. In the proposed nonlinear sensor system, the laser frequency is detuned from the resonance to obtain the maximum sensitivity. The amplitude modulation generates two frequency sidebands which interact with the resonator separately. Since the two sidebands have different detuning frequencies, the transmission spectrum of the total output signal extracted from the PSD can be distorted due to the nonlinear nature of the system. In Fig. 4, we calculate the transmission spectra using the PSD technique with a modulation frequency of fm = 200MHz and compare with previous results ( fm = 0). The frequency components (ω , ω ± fm ) interact with the optomechanical sensor separately based on the coupled equations (Eq. (1)). The transmitted signal is then demodulated at the same modulation frequency fm , and the DC component extracted is proportional to ℜ{a(ω ) × [a∗ (ω + fm ) + a∗ (ω − fm )]}. For both dispersive and dissipative systems, the transmission spectra deviate from the non-modulated systems near the regions that correspond to high sensitivities. Since the purpose of utilizing the PSD is to reduce the noise background, such distortion needs to be avoided. We define a distortion factor as the normalized overlap integral between the transmission functions with a modulated and a non-modulated laser source. Therefore, a lower value of this factor corresponds to a larger distortion. As shown in Fig. 5, the distortion becomes larger as the modulation frequency increases in both types of system, which indicates that an optimized range of fm exists to balance the noise reduction and the distortion. Because the frequency components of main system noises are usually below 100KHz, using a modulation frequency between 100KHz and 10MHz can improve the signal-to-noise ratio and still provide an accurate detection of the system response. 7.
Limitation
Generally, nonlinear effects are enhanced with increasing the input power, but high power can also generate limitations which should be considered for practical applications. In the dispersive system, because the optomechanical systems are governed by a cubic function, optomechanically induced bistability exists for a higher input power or a stronger optomechanical coupling rate. Although the bistable effect is useful for various applications, switching between bistable states should be avoided in the proposed sensor where an unambiguous relation between the input and output signal is required. As shown in Fig. 5(a), for an increased input power (Pin = 1.2mW), the transmission spectrum of the dispersive system stays in different states depending on the direction of the frequency scan. When the frequency of the input laser is scanned from the lower frequency side, the induced resonance shift is in the opposite direction of the frequency scan, which generates the positive feedback and the steep transmission curve (red line). When the frequency of the input laser is scanned from the higher frequency side to the resonance, the transmission curve becomes stretched because the effective frequency detuning is increased due to the shifted resonance. The transmission curves are similar to the thermal optically induced bistability. Because the high sensitivity exists only on the lower frequency side of the resonance, this optomechanically induced bistability may jeopardize the stable operation of the sensor. When operating
#225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2980
the proposed sensor with an input power above the threshold of the bistability, the input light should be red detuned from the maximum sensitivity position to avoid crossing to a different state due to perturbations from noise sources. Therefore, the bistability imposes a limitation on the sensitivity depending on the noise levels and the measurement bandwidth. In the dissipative system, the sensitivity is maximized when the input light is near the cavity resonance and decreases for larger detuning frequency. Its maximum value can be further increased by using larger input power or γom , as shown in Fig. 3(c). However, stronger nonlinear effects also generate a faster change of linewidth near the resonance frequency, corresponding to a narrower lineshape of the sensitivity function. In Fig. 5(b), the FWHM of the sensitivity function (red curve in Fig. 3(b)) is calculated for different input powers. It is clear that high powers (strong nonlinear effects) correspond to small bandwidths. For sensing signals containing frequency components over a large freqeuncy range, the sensor system needs to provide enough bandwidth to avoid signal distortion, which thus limits the maximum nonlinear sensitivity enhancement. Additionally, because the measurement bandwidth of the sensor can not be larger than the resonance frequency of the mechanical resonator, the actual bandwidth of the device may be further reduced. 1
40
FWHM (MHz)
0.8
T
0.6 0.4 0.2 0 −5
0
Δ [GHz] (a)
5
35 30 25 20 15 10 1
1.5
2
2.5
3
Pin [mW] (b)
Fig. 5. (a) Optomechanically induced bistability in the dispersive system for Pin = 1.2mW. The system stays in different states depending on the direction of the frequency scan. (b) The calculated FWHM of the sensitivity function in the dissipative system for different input powers.
8.
Conclusion
In conclusion, we propose and analyze a new type of nonlinear optical sensor based on optomechanical effects. With practical parameters, we have shown that by utilizing the optomechanical nonlinearity, very high sensitivities can be obtained in both dispersive and dissipative systems. We have analyzed the required optomechanical coupling rate to overcome thermal opitcal effect and the bandwidth limitation of the proposed sensor. When the optomechanically induced bistability exists, the input light needs to be detuned from the maximum sensitivity to avoid crossing between different bistable states.
#225123 - $15.00 USD (C) 2015 OSA
Received 16 Oct 2014; revised 4 Jan 2015; accepted 19 Jan 2015; published 2 Feb 2015 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002973 | OPTICS EXPRESS 2981
