Laser spot position dependent photothermal mode cooling of a micro-cantilever
LLaser spot position dependent photothermal mode cooling of a micro-cantilever
Hao Fu , , Cunding Liu , , Yong Liu , Jiaru Chu , and Gengyu Cao * State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institution of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, P. R. China Graduate University of the Chinese Academy of Sciences, Beijing 100049, P. R. China Department of Precision Machinery and Precision Instrumentation,University of Science and Technology of China, Hefei 230027, P. R. China (Dated: September 17, 2018)We explore the laser spot position (LSP) dependent photothermal mode cooling of a micro-cantilever in aFabry-P´erot (FP) cavity. Depending on the LSP along the lever, photothermal coupling to the first two mechan-ical modes can be either parallel or anti-parallel. This LSP dependent behavior is analyzed theoretically by asimple model, which is in quantitatively agreement with our experimen tal observation. From simulation, theparallel and anti-parallel coupling region is identified along the lever. We conclude that a more efficient modecooling may be achieved in the parallel coupling region.
PACS numbers: 05.40.Jc,42.50.Wk,85.85.+j,42.65.Sf
Micro-resonator has become an ideal candidate for explor-ing quantum phenomena, such as the Heisenberg uncertaintyprinciple and quantum entanglement, at the boundary betweenclassical and quantum realms [1–3]. Preparing the micro-resonator close to its ground mechanical state is a crucial steptowards observation of such quantum effects at macroscale[4–6]. In addition, cooling operation of the micro-resonatoralso attracts enormous interest in diverse areas of applied sci-ence, ranging from ultra-high sensitive measurement [7, 8]to quantum information processing [9]. Recently, coolingschemes for the micro-resonator have been intensely stud-ied [10–13], among which optomechanical schemes are pro-posed as one of the most promising strategies to access themacroscopic quantum regime [14–19]. As demonstrated inthe pioneer works, the brownian vibrational fluctuations ofthe micro-resonator can be quenched significantly by the ac-tively controlled optical forces of radiation pressure [20]. Onthe other hand, implemented through the retarded backac-tion of optical forces, passive optical cooling of the optome-chanical resonator also demonstrates the ability of coolingthe fundamental mechanical mode close to ground mechan-ical state [21–23]. Further investigations on passive opticalcooling scheme show that the direction of photothermal cou-pling is mechanical mode dependent, which could be eitherparallel or anti-parallel for different modes [21, 24]. Sinceanti-parallel coupling may excite one mode while cooling theother, cooling operation could be always benefited from theparallel coupling effect. Thus, it raises an important issue incooling modes simultaneously. Here we study the mechanicalmode dependence of photothermal mode cooling at laser spotpositions (LSP) along the optomechanical resonator of com-pliant micro-cantilever in a low fineness FP cavity. Experi-ments at five different LSPs clearly revealed a LSP dependentbehavior of photothermal mode cooling. With the assistanceof theoretical analysis, two types of coupling regions, the par-allel coupling region (PCR) and anti-parallel coupling region(aPCR), are sketched out along the micro-cantilever accord-ing to the relative direction of bolometric backaction. And we
FIG. 1: (a) Schematics of the experimental setup. (b) The low fine-ness level-based FP micro-cavity formed by fiber end and micro-cantilever surface. As the inset indicated, the reflectivity is changedperiodically with the cavity length. By illuminating the micro-cantilever at position l , vibrational resonance of the optomechanicalresonator is recorded at the working position marked by b and r forblue and red cavity detuning accordingly for each laser power. (c)Shapes of mode 0 and mode 1. The photothermal coupling to thefirst two mechanical modes of micro-lever are investigated at fiveLSPs denoted by P1 to P5. demonstrated that simultaneous cooling of the first two me-chanical modes can be realized at the PCR. It paves the wayfor more efficient cooling of an optomechanical resonator toits classical limit.In our experiment, a single crystal silicon micro-cantileverwith dimension of 480 µ m × µ m × µ m is used [25]. And a r X i v : . [ phy s i c s . op ti c s ] J u l FIG. 2: Mode dependent bolometric backaction at position P3. Ther-mal noise amplitude of mode 0 (a) and mode 1 (b) is detected at P3for the different laser powers. At working point b , cooling of mode0 accompanies with the warming up of mode 1. Insets in graphs (a)and (b) plot the effective damping factors at working points b (dashline) and r (solid line) against laser power for the two mechanicalmodes accordingly. gold film of 80nm thickness is thermally evaporated ontothe top side of the lever. The intrinsic resonant frequen-cies of mode 0 and mode 1 of the micro-cantilever are f =4443.4Hz and f = 27736.4Hz respectively. In ultra-highvacuum ( × − Torr) and at room temperature, the leverexhibits inherent thermal dissipation with the damping fac-tors Γ =2.9Hz and Γ =15.4Hz for the first two modes. Theexperimental setup is schematically illustrated in Fig.1. Alaser beam of wavelength λ =1310nm is supplied by a semi-conductor diode with power programmable from 10 µ W to5mW. After split by a 10dB directional optical coupler (re-flection = 90%, transmission = 10%), the laser beam is cou-pled into the ultra-high vacuum system by a single mode op-tical fiber. Aligning the polished fiber end and the surface ofmicro-cantilever parallel, a low fineness level-based FP cavityis formed. A fiber stage piezo capable of accurately regulat-ing the fiber position in range of ∼ µ m at 300K is employedto switch the working position from the maximum sensitiveinterference detection point of red detuning r to that of bluedetuning b [26]. Laser reflected off the top surface of micro-cantilever partly couples back to the fiber and interferes withthe laser reflected from the fiber end to produce the micro-cantilever oscillation signal. The oscillation signal is moni-tored by a photodetector and analyzed by an FFT spectrumanalyzer (SR760, Stanford Research System) in the vicinityof the first two resonant frequencies of micro-cantilever.The deformable mirror of FP cavity formed by the compli-ant micro-cantilever is subject to action of optical forces. It inturn modulates the stored optical energy and results in the op-tical forces backaction on the oscillation of micro-cantileveras a consequence. The mutual modulation between the laserfield inside the cavity and the motion of micro-cantileverforms the foundation of passive optical cooling. Under thelow cavity fineness condition, only the retarded backactionof photothermal force (or bolometric backaction) participatesin optomechanical cooling [21]. Other optical forces, such FIG. 3: Photothermal coupling to mode 0 (solid line) and mode 1(dash line) at different LSPs. Graph (a) to (d) show the experimen-tal results of the first two modes at P1, P2, P4 and P5 accordingly.During measurements, the cavity length is about 35 µ m. as radiation pressure, react instantaneously with the oscilla-tion of micro-cantilever, thus modify only the resonant fre-quency. Depending on the detuning condition of micro-cavity,the thermal oscillation of the micro-cantilever can be eitherenhanced or suppressed. However, the direction of bolometricbackaction is mechanical mode dependent. As shown in Fig.2,cooling operation on the fundamental mode (mode 0) alwaysaccompanies the warm-up of mode 1 at position P3 indicatedin Fig.1(c). Its thermal oscillation is enhanced continuouslywith laser power increasing. Once crossing the threshold of Γ eff , =0, mode 1 will be driven into self-sustained oscillation.This result is in consistent with the observation in the work byG. Jourdan et al [24].This mode dependent photothermal coupling is investigatedfurther at other LSPs denoted in Fig.1(c). The photother-mal damping for mode n can be represented by ∆Γ eff ,n =(Γ eff ,n ( b ) + Γ eff ,n ( r ) ) / , where Γ eff ,n ( b ) and Γ eff ,n ( r ) are theeffective damping of micro-cantilever at working position b and r respectively. In the case of ∆Γ eff ,n < , mode n ofthe micro-cantilever is enhanced at blue cavity detuning whilecooled at red cavity detuning, and vice versa for ∆Γ eff ,n > .For mode 0, ∆Γ eff is always a positive value, whereas its valuechanges with illuminating point for mode 1. As showed inFig.3, it indicates a LSP dependent behavior in photothermalmode cooling. At position P1, the bolometric backaction isexerted on the two modes in the same direction. Dependingon the detuning of cavity, it provides a possibility of coolingthe two modes simultaneously. Although the bolometric back-action on mode 1 maintains its direction at those LSPs beyondthe vibration node, its relative strength is varied evidently. Theaccompanied enhancement of mode 1 at position P2 and P5 isso weak that its influences become apparent only in the caseof strong photothermal coupling. However, at position P3 andP4, mode 1 is excited obviously during cooling operation onmode 0.In order to understand such LSP dependent behavior, wedevelop a simple model for the photothermal coupling. Aslong as the displacement of micro-cantilever u ( x, t ) fromits equilibrium position is the summation of all mechanicalmodes, denoted by u ( x, t ) = Σ φ i ( x ) a i ( t ) where φ i ( x ) is theshape of mode i , the motion equation for mode n of micro-cantilever driven by both thermal force F th and bolometricforce F bol can be expressed as m ¨ a n + m Γ n ˙ a n + K n a n = F th − (cid:90) L E ∂ φ n ∂x I l dx (1)where m , E , Γ n and K n are the effective mass, Young’smodulus, damping factor and spring constant of mode n re-spectively [24]. The integration term includes the photother-mal stress generated all along the length L of the micro-cantilever. A detailed understanding of the moment of iner-tia I l = (cid:82) zα ( z )∆ T l ( x, y, z, t ) dydz is extremely difficult,because the dynamic temperature field ∆ T l ( x, y, z, t ) insidethe lever body is intricate, which depends on the opticaland thermal properties of lever materials, structural geom-etry, environment temperature, laser wavelength and so on.For our gold film coated micro-cantilever, the photothermalforce originates primarily from the different thermal expan-sion coefficients between gold film and silicon material of the lever. The small temperature variation through the thick-ness of micro-cantilever is therefore negligible [27]. Pro-viding the small width of the micro-cantilever as compar-ing with the laser spot profile, the temperature distributioncan be reduced to a one dimension field for this compos-ite lever structure. In the limit of small oscillation ampli-tude, the oscillation modulated laser intensity I ( z ) is approx-imated by the Taylor expansion around equilibrium position z : I ( z ) ≈ I ( z ) + ( z − z ) ∇ I ( z ) . Thus the laser induceddynamic temperature field along the micro-cantilever can bedescribed as: ∆ T l ( x, t ) = AD l ( x ) ∇ I ( z ) (cid:90) t h ( t − t (cid:48) ) ˙ u ( l, t (cid:48) ) dt (cid:48) (2) A is the absorptivity of laser power. The retarded backactionof photothermal force is described by the response function h ( t ) = 1 − exp ( − t/τ ) , where τ is referred to as the delaytime constant of bolometric backaction [21]. This expressiongenerally takes the contribution of all mechanical modes intoaccount.Since the photothermal coupling to higher order mechani-cal modes is attenuated by the low pass filter behavior of bolo-metric backaction, in this letter, we consider the contributionof the first two modes only. Combining Eq.1 and Eq.2, theFourier transform yields: m (cid:34) ω − ω + iω Γ + γG ( l ) φ ( l ) m (1+ iωτ ) γG ( l ) φ ( l ) m (1+ iωτ ) γG ( l ) φ ( l ) m (1+ iωτ ) ω − ω + iω Γ + γG ( l ) φ ( l ) m (1+ iωτ ) (cid:35) (cid:20) a ,ω a ,ω (cid:21) = (cid:20) F th,ω F th,ω (cid:21) (3)where γ = AE ∇ I ( z ) (cid:82) zα ( z ) dydz is cavity detuning con-dition dependent. In particular, when working position insidethe micro-cavity changes from blue detuning point b to the ad-jacent red detuning point r , the slope of interference ∇ I ( z ) is reversed and hence γ changes its sign. The mechanicalmode dependent behavior of bolometric backaction is intro-duced by the function G n ( l ) = (cid:82) ( ∂ φ n /∂x ) D l ( x ) dx . Sincethe temperature profile D l ( x ) = D p exp ( − | x − l | /r ) is con-centrated around the LSP [28], it functions to window the con-tribution of mode shape in the vicinity of the laser illumina-tion point only, which decides the LSP dependent nature ofphotothermal mode cooling. As Eq.3 describes, photothermalcoupling involves both self-coupling and inter-modes cou-pling for each mechanical mode, which are represented by thediagonal terms and non-diagonal terms of the matrix, respec-tively. The contribution of inter-modes coupling is typicallyoverwhelmed by the self-coupling. Thus, the responses ofthe micro-cantilever are dominated by the self-coupling terms,with the resonant frequency and damping factor modified bythe real and imaginary part of bolometric backaction accord- ingly. Providing the relatively small frequency shift, the rel-ative photothermal cooling effect of mode n with respect tofundamental mode can be represented by the photothermaldamping ratio β n , which is defined as: β n = d ∆Γ eff ,n d ∆Γ eff , = G n ( l ) φ n ( l )(1 + ω τ ) G ( l ) φ ( l )(1 + ω n τ ) (4)Either the bolometric backaction acting in opposite directionson the two modes simultaneously or not depends completelyon the sign of β n . Simulation shows that three functional re-gions of two types, the PCR and aPCR, are divided for mode1 along the micro-cantilever by two nodes, as illustrated inFig.4. In region I, bolometric backaction applying onto thetwo modes of the optomechanical resonator in the same di-rection results in a parallel coupling. Crossing the vibrationnode of mode 1 such that the sign of φ ( l ) is changed, revers-ing of the vibration phase results in altering the direction ofbolometric backaction. Operating in the aPCR of region II,damping of one mode always accompanies with the enhance-ment of the other at the same time. It is not surprising that FIG. 4: Comparison of the experimental measurements with the the-oretical simulation. Photothermal damping ratio of mode 1 obtainedfrom experimental data (dots) at the five LSPs indicated in Fig.1(c)are plotted together with theoretical simulation (solid line). Twotypes of coupling region, the PCR and aPCR, are defined by dashlines. locally laser heating imposes the second node in the vicin-ity of the position ∂ φ /∂x = 0 , which imposes the otherboundary of aPCR. Beyond this node, in region III, the modedependent coupling function G ( l ) changes its sign, whereasmode function φ ( l ) does not. It leads to the second PCRalong the micro-cantilever. By positioning the laser beam intothe PCR, simultaneous cooling of the two modes is achievedat blue cavity detuning condition. The delay time constant τ , which sets the cutoff frequency for the bolometric backac-tion, is calculated to be 2.5ms for our micro-cantilever at roomtemperature [21]. To obtain a more efficient cooling, the di-mension of the micro-cantilever should be designed accordingto experiment temperature to satisfy the optimal cooling con-dition of ωτ = 1 .It is worth noted here that even higher order mechanicalmodes of micro-cantilever may become significant in strongphotothermal coupling condition. The photothermal coolinginvolving higher order mechanical modes can also be ana-lyzed in the same principle. For higher modes, we predict the-oretically that the micro-cantilever could still be divided intothree functional regions, with both boundaries of aPCR ex-panding towards two ends of micro-cantilever continuously asthe order of mechanical modes increases as a result. However,the details inside aPCR become increasingly complicated forhigher order modes, which introduce more nodes within aPCRand emphasize the necessity of a more accurate description ofthe temperature field distribution.In conclusion, by positioning the laser beam onto differ-ent points along the FP optomechanical resonator of the goldfilm coated micro-cantilever, we have investigated the LSPdependent behavior of photothermal mode cooling. Experi-ments show that not only the direction but also the efficiencyof photothermal mode cooling is LSP dependent. Accord-ing to the direction of bolometric backaction, we divide the micro-cantilever into three coupling region of two types, thePCR and aPCR. Simultaneous cooling of the first two modescan be achieved when laser beam is pointed onto the PCR ofmicro-cantilever. After satisfying the optimal cooling condi-tion such that the residual thermal effect is minimized, pho-tothermal cooling limitation may ultimately reach by illumi-nating the micro-cantilever at the cross section of the PCRs ofthe first few mechanical modes.This work was supported by the Grand Project of Instru-mentation and Equipments for Scientific Research of the Chi-nese Academy of Sciences under contract YZ0637. We grate-fully acknowledge Y. Miyatake in Unisoku Scientific Instru-ments for technical supports..It is worth noted here that even higher order mechanicalmodes of micro-cantilever may become significant in strongphotothermal coupling condition. The photothermal coolinginvolving higher order mechanical modes can also be ana-lyzed in the same principle. For higher modes, we predict the-oretically that the micro-cantilever could still be divided intothree functional regions, with both boundaries of aPCR ex-panding towards two ends of micro-cantilever continuously asthe order of mechanical modes increases as a result. However,the details inside aPCR become increasingly complicated forhigher order modes, which introduce more nodes within aPCRand emphasize the necessity of a more accurate description ofthe temperature field distribution.In conclusion, by positioning the laser beam onto differ-ent points along the FP optomechanical resonator of the goldfilm coated micro-cantilever, we have investigated the LSPdependent behavior of photothermal mode cooling. Experi-ments show that not only the direction but also the efficiencyof photothermal mode cooling is LSP dependent. Accord-ing to the direction of bolometric backaction, we divide the micro-cantilever into three coupling region of two types, thePCR and aPCR. Simultaneous cooling of the first two modescan be achieved when laser beam is pointed onto the PCR ofmicro-cantilever. After satisfying the optimal cooling condi-tion such that the residual thermal effect is minimized, pho-tothermal cooling limitation may ultimately reach by illumi-nating the micro-cantilever at the cross section of the PCRs ofthe first few mechanical modes.This work was supported by the Grand Project of Instru-mentation and Equipments for Scientific Research of the Chi-nese Academy of Sciences under contract YZ0637. We grate-fully acknowledge Y. Miyatake in Unisoku Scientific Instru-ments for technical supports.