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Optics Communications
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Optomechanical coupling between two optical cavities: Cooling of a micro-mirrorand parametric normal mode splitting
Tarun Kumar a, Aranya B. Bhattacherjee b,⁎, ManMohan a
a Department of Physics and Astrophysics, University of Delhi, Delhi-110007, Indiab Department of Physics, ARSD College, University of Delhi (South Campus), New Delhi-110021, India
We propose a technique aimed at cooling a harmonically oscillating mirror mechanically coupled to anothervibrating mirror to its quantum mechanical ground state. Our method involves optomechanical coupling be-tween two optical cavities. We show that the cooling can be controlled by the mechanical coupling strengthbetween the twomovable mirrors, the phase difference between the mechanical modes of the two oscillatingmirrors and the photon number in each cavity. We also show that both mechanical and optical cooling can beachieved by transferring energy from one cavity to the other. We also analyze the occurrence of normal-mode splitting (NMS). We find that a hybridization of the two oscillating mirrors with the fluctuations ofthe two driving optical fields occurs and leads to a splitting of the mechanical and optical fluctuation spectra.
Recently due to significant advances in technology, optical andmechanical degrees of freedom have become entangled experimen-tally by the mechanism of radiation pressure forces. This fieldknown as cavity optomechanics has played a major role in the explo-ration of the boundaries between classical and quantum mechanicalsystems. The entanglement of mechanical and optical degrees of free-domvia radiation pressuremechanismhas been a topic of early researchin the context of laser cooling [1–3] and gravitational-wave detectors[4]. In recent years, there has been a great interest in the application ofradiation forces to coherently control the center-of-mass motion of me-chanical oscillators covering a huge range of scales from macroscopicmirrors in the Laser Interferometer Gravitational Wave Observatory(LIGO) project [5,6] to nano-mechanicalmirrors [7–12], vibratingmicro-toroids [13,14], micro-membranes [15] and ultra cold atoms [16–19].The primary achievement of the field of cavity optomechanics is thestudy of radiation pressure forces to coherently manipulate the motionalstate of micromechanical oscillators and further use it for designingmicromechanical quantum devices. In particular, it has become possibleexperimentally to significantly cool the thermal excitation of a singleme-chanical mode [20]. With these exciting experimental advancements,micro- and nanomechanical resonators now represent an important
model system with the prospect of demonstrating and controllingquantum effects on a macroscopic scale. Early theoretical work hasalso proposed to use the radiation-pressure coupling for quantumnon-demolition measurements of the light field [21].
It has been demonstrated that ground state cooling of micro/nano-mechanical mirror is experimentally possible only in the resolved sideband regime (RSB)where themechanical resonance frequency is great-er than the bandwidth of the driving resonator [21,22]. The cooling ofmechanical oscillators in the RSB regime at high driving power is ac-companied by the appearance of normal mode splitting (NMS)[23,24]. Recently, it was shown that an optical parametric amplifier in-side an optical cavity considerably improves the cooling of a micro-mechanical mirror by radiation pressure force [25]. The dynamics of amicro mirror was studied in the presence of a nonlinear Kerr mediumplaced inside an optical cavity [26]. It was demonstrated that due tothe photon blockade mechanism, as the Kerr nonlinearity is increased,the NMS progressively decreases. The Kerr medium was found to be anew handle to coherently control the micro-mirror dynamics whichcould lead to a possible quantum device [26].
In thiswork,we propose a technique aimed at cooling a harmonicallyoscillatingmirror (mechanically coupled to another vibratingmirror) toits quantum mechanical ground state. Our method involves optome-chanical coupling between two optical cavities. We show that the cool-ing can be controlled by the mechanical coupling strength between thetwomovable mirrors and the phase difference between themechanicalmodes of the two oscillatingmirrors. We also analyze the occurrence ofnormal-mode splitting (NMS). We find that a hybridization of the twooscillatingmirrors with the fluctuations of the two driving optical fieldsoccurs and leads to a splitting of the mechanical and optical fluctuationspectra. The continuous variable entanglement between twomechanical
301T. Kumar et al. / Optics Communications 285 (2012) 300–306
modes could be used to improve the detection of weak classical forces inoptomechanical devices as atomic force microscopes or gravitationalwave detectors. Optomechanically coupledmirrors has been investigatedearlier [27,28]. A continuous variable entanglement between the twomirrors was maintained by the light bouncing between the mirrorsand was found to be robust against thermal noise [27]. Entanglementbetween twomechanical oscillators coupled to a nonequilibrium en-vironment showed that there is an optimal dissipation strength forwhich the entanglement between two coupled oscillators ismaximized[28]. A new coolingmethodwhich involves the two-sided irradiation ofthe vibratingmirror inside an optical cavity has been proposed recently[29]. This method provides a stiffer trap for cooling the mirror and hasseveral advantages over conventional methods of optomechanicalcooling.
2. Theoretical framework
We consider two Fabry–Perot cavities connected with each otherthrough their movable mirrors as shown in Fig. 1. Here mirror M1 andmirror M4 are fixed and are partially transmitting whereas mirrorsM2 and M3 are movable and totally reflecting. The two mirrors M2and M3 can both oscillate under the effect of the radiation pressure.Themotion of eachmirror is the result of the excitation ofmany oscilla-tionmodeswhich can be either external or internal. The externalmodescorresponds to pendulum modes which leads to global displacementsof the mirror while the internal modes corresponds to deformationsof the mirror surface due to excitation of internal acoustic modes ofthemirror surface. These various degrees of freedom have different reso-nance frequencies and experimentally it is possible to select themechan-ical response of a single mode by using a bandpass filter in the detectioncircuit. Consequently, wewill consider a singlemechanical mode for eachmovable mirror, which will be therefore described as a simple harmonicoscillator. The system under consideration is in contact with thermalbath in equilibrium with thermal bath at temperature T. The movablemirrors are treated as quantum mechanical oscillators with masses m1and m2, frequencies ν1 and ν2, and energy decay rates γm1 and γm2 re-spectively of the mirrors M1 and M4. The system is coherently drivenby two laser fields (a1, in and a2, in) with frequencies ω1 and ω2 asshown in Fig. 1. It is known experimentally that high Q-optical cavitiescan significantly isolate the system from its environment. This stronglyreduces decoherence and ensures that the optical field remains quan-tum mechanical for the duration of the experiment. We also assumethat the induced cavity resonance frequency shift of each cavity is muchsmaller than the longitudinal spacing, so that the model can be limitedto a single longitudinal mode for each cavity. Let ε1 and ε2 be the ampli-tudes of the two laser fields. As we know in a Fabry–Perot cavity, whena photon collides with the surface of the movable mirror, it exerts radia-tion pressure on the mirror and the force that mirror will experience isproportional to the photon number inside the cavity. But in our system
Fig. 1. Schematic description of the system under study. Two Fabry–Perot cavities are connefixed and are partially transmitting whereas mirrors M2 and M3 are movable and totally reflare the output fields.
here, the force experienced by one of the movable mirror, say M2 notonly depends on the number on photons of the corresponding cavitybut also depends on the number of photons of the second cavity. Thisis because the two mirrors are coupled, therefore position of one mirroris influenced by the position of the othermirror.We also assume thatω1,ω2bb πc/L (adiabatic limit); c is the speed of light in vacuum and L is thecavity length in the absence of the cavityfield. (assuming same length forthe two cavities). The Hamiltonian of the system can be written as
H ¼ ℏω1a†1a1 þ ℏω2a
†2a2 þ ℏν1 b†1b1 þ 1=2
� �þ ℏν2 b†2b2 þ 1=2
� �þℏν12 b†1e
−iθ1 þ b1eiθ1
� �b†2e
−iθ2 þ b2eiθ2
� �−ℏg1a
†1a1 b†1e
−iθ1þb1eiθ1
� �
þiℏ�1 a†1−a1� �
−ℏg2a†2a2 b†2e
−iθ2 þ b2eiθ2
� �þ iℏ�2 a†2−a2
� �:
ð1Þ
Here a1 (a1†) and a2 (a2
†) are the annihilation (creation) operators of
the two cavity fields, b1 (b1†) and b2 (b2
†) are the phonon annihilation
(creation) operators of the twomovablemirrorsM2 andM3 respective-ly. The parameters g1 and g2 are the coupling parameters between thecavityfields andfixedmirrorsM1 andM4 respectively,ν12 is the couplingfrequency of the twomovable mirrors and θ1 and θ2 are the phases of thetwo movable mirrors. The phases θ1 and θ2 can be thought of as arisingfrom the complex mirror–photon coupling strengths gi(i=1,2).
The optomechanical system we are considering here is an opensystem since the cavity optical fields are damped due to the leakageof photons through the coupling mirrors. In the absence of the radi-ation pressure coupling, the micro-mirrors would undergo pureBrownian motion, driven by their contact with the thermal environ-ment. The motion of the system can be described by the followingQuantum Langevin equations
†−bi)=pi ; i=1,2.Also D1=g1cos(θ1), D2=g1 sin(θ1), D3=g2cos(θ2), D4=g2 sin (θ2),A=ν12 cos(θ2−θ1), B=ν12 sin (θ2−θ1). Cin
1 and Cin2 are input
cted with each other through their movable mirrors. Here mirror M1 and mirror M4 areecting. The system is coherently driven by two laser fields a1, in and a2, in. a1, out and a2, out
302 T. Kumar et al. / Optics Communications 285 (2012) 300–306
noise operators with zero mean value and obeys following commu-tation relation bδCin
i δCinj †N=δij(t− t′), bδCin
i δCinj N, bδCin
i †δCinj †N=0.
Also pini = i(ξi †−ξi), ξ is the Brownian noise operator, arising due
to the thermal bath. Brownian noise operator has zero mean valueand obeys following correlation at temperature T: bξi(t)ξ j†(t′)N=2γmi
(1+2nT)δij(t−t′) and bξi(t)ξ j(t′)N=bξi†(t)ξ j†(t′)N=bξi†(t)ξ j(t′)N=0, nT ¼ coth ℏω
2kBT
� �, where kB is the Boltzmann constant and T is the
temperature of the thermal bath.
3. Small fluctuations dynamics: normal mode splitting and coolingof a micro mirror
Here we show that the coupling of the two mechanical oscillatorsand the two cavity field fluctuations leads to the splitting of the nor-mal mode into two modes (Normal Mode Splitting (NMS)) for eachcavity depending on the system parameters. The optomechanicalNMS however involves driving four parametrically coupled nonde-generate modes out of equilibrium. The NMS does not appear in thesteady state spectra but rather it appears in the fluctuation spectraof the mirror displacement. In order to study the dynamics of thecoupled mirror, we need to find out the fluctuations in the mirror'sposition. As is clear from the Eqs. (2)–(7) that the problem involvedhere is non-linear. We assume that this non-linearity is small. There-fore we study the dynamics of fluctuations around the steady state ofthe system. We write each canonical operator of the system as a sumof its steady state mean value and a small fluctuation with zero meanvalue, q1→q1s+δq1, q2→q2s+δq2, p1→p1s+δp2, p2→p2s+δp2,a1→a1s+δa1, a2→a2s+δa2. The steady state values are obtainedby putting the left hand side of Eqs. (2)–(7) to zero. In order toachieve ground state cooling, we will always take γmi bb κi, gi b νiand νi N κi (with i=1,2) The last condition is the resolved side bandregime necessary for ground state cooling. Note that these conditionsnecessary for cooling also implies that the system is stable. Lineariz-ing Eqs. (2) to (7) to obtain following Heisenberg–Langevin equationsfor the fluctuation operators:
On Fourier transforming all operators and noise sources of Eqs. (8)to (13) and solving in the frequency domain, the position fluctuationsδq1(ω) of the movable mirror M2 is obtained as
δq1 ωð Þ ¼ 1d ωð ÞfX ωð Þð a�1s
κ1−i ω−Δ1ð Þ δC1in þ
a1sκ1−i ωþ Δ1ð Þ δC
1†in Þ2 ffiffiffiffiffiffiffiffi
2κ1p
þY ωð Þð a�2sκ2−i ω−Δ2ð Þ δC
2in þ
a2sκ2−i ωþ Δ2ð Þ δC
1†in Þ2 ffiffiffiffiffiffiffiffi
2κ2p
þffiffiffiffiffiffiffiffiffiffiffi2γm1
pZ ωð Þδp1in þ
ffiffiffiffiffiffiffiffiffiffiffi2γm2
pT ωð Þδp2in
� �g:ð14Þ
All the variables are defined in the Appendix A. In Eq. (14), the firsttwo terms corresponding to X(ω) and Y(ω) gives rise to the effect of
radiation pressure whereas last two terms corresponding to Z(ω) andT(ω) originate because of the thermal noise. The coupling of the radia-tion field to the mirror shifts the cavity resonance frequency andchanges the optical field inside the cavity in a way to induce a new sta-tionary intensity. The shift in the cavity resonance is seen in the renor-malized detunings Δ1 and Δ2. The change occurs after a transient timedepending on the response of the cavity and the strength of the couplingto themirrors. Now the spectrum of fluctuation of mirror can be definedas
The displacement spectrum of mirror M2 i.e., Sq1(ω) is finallyobtained as
Sq1 ωð Þ ¼ 1d ωð Þd −ωð Þ ½ð 4ja1sj2κ1
κ21 þ ω−Δ1ð Þ2 þ4ja1sj2κ1
κ21 þ ωþ Δ1ð Þ2ÞX ωð ÞX −ωð Þ
þð 4ja2sj2κ2κ22 þ ω−Δ2ð Þ2 þ
4ja2sj2κ2κ22 þ ωþ Δ2ð Þ2ÞY ωð ÞY −ωð Þ
þ 1þ 2nTð Þ γ2m1Z ωð ÞZ −ωð Þ þ γ2
m2T ωð ÞT −ωð Þ� ��:
ð16Þ
Here we have used the commutation relation for ξ. In Eq. (16), thefirst two terms are due to radiation pressure contribution of the opti-cal modes in the two cavities, whereas the last term is due to the ther-mal noise contribution from the two cavities.
In Fig. 2, we show the contour plot of the displacement spectrumSq1 as a function of dimensionless effective detuning Δ1/ν for θ1=0,θ2=π/2,Δ2/ν=0, γ1/ν = γ2/ν = 0.01, g1/ν = g2/ν = 0.6, ν12/ν =0.3, κ1/ν = κ2/ν = 0.1 for different values of the photon numbers inthe two cavities, (a): |a1s|2=0.1 and |a2s|2=0.1, (b): |a1s|2=0.1 and|a2s|2=0.25, (c): |a1s|2=0.25 and |a2s|2=0.1, (d): |a1s|2=0.25 and|a2s|2=0.25. The values of the color scheme are shown in Fig. 6 inthe Appendix A. Clearly, four modes are visible corresponding to thetwo mechanical and two optical modes. The coupling of the cavityfield fluctuations and the mirror fluctuations leads to splitting of thenormal mode of each cavity into two modes (NMS). The NMS is asso-ciated with the mixing between the fluctuation of the cavity fieldaround the steady state and the fluctuations of the mirror modearound the mean field. The origin of the fluctuations of the cavityfield is due to the mixing of the pump photons with the photons scat-tered from the mirrors. We observe from the displacement spectrathat NMS is observed only in plots (a) and (b) where the photonnumber in first cavity is |a1s|2=0.1. Increasing the photon numberin the first cavity destroys the NMS. We also note by comparingplots (b) and (c) that decreasing the photon number in the secondcavity to |a2s|2=0.1 does not restore the NMS. An important pointto note is that in order to observe the NMS experimentally, the energyexchange between the modes (the optical and mirror modes) shouldtake place on a time scale faster than the decoherence of each mode.The parameter regime in which NMS appears implies cooling. Forother values of the system parameters, the observation of NMS is pre-vented by the onset of the parametric instability. Therefore, a pres-ence of NMS cannot be decoupled from the associated coolingwhich we discuss next where we calculate the effective temperature.
Energy exchange between themodes of the two cavities depends onthe two phases θ1 and θ2. In Fig. 3 we show the contour plot of the dis-placement spectrum Sq1(ω) as a function of dimensionless effectivedetuning for θ1=0, θ2=π/4 (left plot) and θ1=0, θ2=3π/4 (rightplot). Clearly we observe energy exchange between the modes as wego from left to the right plot. Such energy exchange also implies thatwe can selectively cool onemirror at the expense of the other. In generalit is known from basic physics that energy exchange between two
0.0a b
c d
-0.5
-1.0
-1.5
-2.00.90 0.95 1.00 1.05 1.10
vΔ
/1
ω v/
0.0
-0.5
-1.0
-1.5
-2.00.90 0.95 1.00 1.05 1.10
vΔ
/1
ω v/
0.0
-0.5
-1.0
-1.5
-2.00.90 0.95 1.00 1.05 1.10
vΔ
/1
ω v/
0.0
-0.5
-1.0
-1.5
-2.00.90 0.95 1.00 1.05 1.10
vΔ
/1
ω v/
Fig. 2. Contour plot of the displacement spectrum Sq1 as a function of normalized effective detuningΔ1 for the following parameters: θ1=0, θ2=π/2,Δ2/ν=0, γ1/ν= γ2/ν=0.01, g1/ν=g2/ν=0.2,ν12/ν=0.03, κ1/ν= κ2/ν=0.1. (a): |a1s|2=0.1 and |a2s|2=0.1, (b): |a1s|2=0.1 and |a2s|2=0.25, (c): |a1s|2=0.25 and |a2s|2=0.1, (d): |a1s|2=0.25 and |a2s|2=0.25. The valuesof the color scheme are shown in Fig. 5 in the Appendix.
303T. Kumar et al. / Optics Communications 285 (2012) 300–306
mechanical oscillators takes place only for the anti-symmetric mode i.e.when each mechanical oscillator is initially displaced from its positionin opposite direction. In our case, such energy exchange between thetwo cavities can be achieved by tuning the phases θ1 and θ2.
We now calculate the effective temperature of the mirror M2. Inorder to calculate the effective temperature, we need the spectrumof the momentum of the mirrorM2 in Fourier space. In a similar man-ner as above, we can calculate the momentum spectrum of the mirrorM2, which is found as:
Sp1 ωð Þ ¼ 1t15 ωð Þt15 −ωð Þ ½ 4ja1sj2κ1
κ21 þ ω−Δ1ð Þ2 þ4ja1sj2κ1
κ21 þ ωþ Δ1ð Þ2 !
t16 ωð Þt16 −ωð Þ
þ 4ja2sj2κ2κ22 þ ω−Δ2ð Þ2 þ
4ja2sj2κ2κ22 þ ωþ Δ2ð Þ2
!t17 ωð Þt17 −ωð Þ
þ 1þ 2nTð Þt18 ωð Þt18 −ωð Þ γ2m1 þ γ2
m2
� ��:ð17Þ
For a driven system, effective temperature can be defined as [25]:
Teff ¼bδp2Nþ bδq2N
2; ð18Þ
where the variances are calculated as,
bδq2N ¼ 12π
∫∞−∞
Sq ωð Þdω ð19Þ
bδp2N ¼ 12π
∫∞−∞
Sp ωð Þdω: ð20Þ
The equation for the effective temperature is one of our key resultswhich tells how the temperature of one mirror depends on the varioussystem parameters. Note that in general bδq2N ≠ bδp2N. This impliesthat one does not have energy equipartition. This means that the steadystate of the system is not, strictly speaking, a thermal equilibrium state.However, in order to get to the quantum ground state, both varianceshave to tend to 1/2 and therefore energy equipartition has to be satis-fied in the optimal regime close to the ground state.
A plot of the effective temperature Teff of the mirror M2 as a func-tion of dimensionless mirror–mirror coupling strength ν12/ν for fourdifferent combinations of |a1s|2 and |a2s|2. (a): |a1s|2=0.1 and|a2s|2=0.1, (b):|a1s|2=0.1 and |a2s|2=0.25, (c): |a1s|2=0.25 and|a2s|2=0.1, (d): |a1s|2=0.25 and |a2s|2=0.25 is shown in Fig. 4. Weclearly observe that minimum temperature is attained when boththe cavities have low photon numbers. Increasing the photon numberin any one of the cavity increases the temperature and the influenceof |a1s|2 is more profound than |a2s|2. This is consistent with our ear-lier result on the NMS where we mentioned that the presence of theNMS also indicates cooling. Plots (a) and (b) in Fig. 4 which showmin-imum temperature corresponds to plots (a) and (b) in Fig. 2 whichshow NMS. Mechanical cooling of the mirror mode by the radiationpressure can be understood from elementary thermodynamics. The
Fig. 3. Contour plot of the displacement spectrum Sq1 as a function of dimensionless effective detuning Δ1/ν for the following parameters: Δ2/ν=0, γ1/ν = γ2/ν = 0.01, g1/ν =g2/ν = 0.2, ν12/ν = 0.03, κ1/ν = κ2/ν = 0.1, |a1s|2=0.1, |a2s|2=0.1, θ1=0, θ2=π/4 (left plot), θ1=0, θ2=3π/4 (right plot).
304 T. Kumar et al. / Optics Communications 285 (2012) 300–306
micro-mirror is coupled to the optical cavity mode via the radiationpressure and the optical cavity mode behaves as an effective additionalreservoir for the mechanical oscillator. As a consequence, the mirrormode attains an effective temperature which is intermediate betweenthat of the initial thermal reservoir temperature and that of the opticalreservoir, which is practically zero due to the condition that the meannumber of photons is extremely small. Therefore one can approachthemechanical ground state of themirror when the number of photonsis small. In our case, themechanical mode ofmirrorM2 not only couplesto optical mode of first cavity but also to the optical mode of the secondcavity via themechanical mode of the mirrorM3. This explains why sig-nificant mechanical cooling of the mechanical mode is obtained whennumber of photons in both the cavities is low. Fig. 5 shows the Teff ofthe mirror M2 as a function of detuning Δ1/ν corresponding to the casesshown in Fig. 2. Interestinglyweobserve that the peaks in Teff correspondsexactly to the points in Fig. 2where themirror displacement ismaximum.These plots also illustrate energy exchange between the variousmodes aswe vary mean photon numbers in the two cavities. In this system, thepresence of the additionalmodes of the second cavity allows one to trans-fer energy from the mechanical mode of the mirror M2 and the opticalmode a1 to the mechanical mode of the mirror M3 and the optical modea2. This shows that both mechanical and optical cooling can be achievedby transferring energy from one cavity to the other. From the experimen-tal point of view, the mirror's position can be measured by means of aphase-sensitive detection of the cavity output, which is then fed back to
d cb a
0.00 0.01 0.02 0.03 0.040.0
0.5
1.0
1.5
2.0
2.5
Tef
f
12 /ν ν
Fig. 4. A plot of the effective temperature Teff of the mirror M2 as a function of mirror–mirror coupling strength ν12 for four different values of |a1s|2 and |a2s|2. (a): |a1s|2=0.1and |a2s|2=0.1, (b): |a1s|2=0.1 and |a2s|2=0.25, (c): |a1s|2=0.25 and |a2s|2=0.1,(d): |a1s|2=0.25 and |a2s|2=0.25. Other parameters are same as in Fig. 2.
the mirror by applying a force whose intensity is proportional to thetime derivative of the output signal, and therefore to themirror's velocity.
4. Conclusion
In thiswork,we have proposed a new technique to cool a harmonical-ly oscillating mirror of an optical cavity mechanically coupled to anothermovable mirror of a second optical cavity. The system behaves as fourcoupled oscillators exchanging energy. Energy exchange canbe coherent-ly controlled by the phases of the opto-mechanical coupling strength.Wefind that a hybridization of the two oscillating mirrors with the fluctua-tions of the two driving optical fields occurs and leads to a splitting ofthemechanical and optical fluctuation spectra.We also showed that nor-mal mode splitting (NMS) leads to mechanical cooling. Significant me-chanical cooling can be achieved by controlling the photon number inthe two cavities. In addition, we demonstrate for the first time that bycoupling two cavities, we can cool one cavity (both in the mechanicaland optical sense) by transferring energy to the other. A continuous var-iable entanglement between the twomechanical modes could be used toimprove the detection of weak classical forces in optomechanical devicesas atomic force microscopes or gravitational wave detectors.
Acknowledgements
One of the authors Tarun Kumar thanks the University GrantsCommission, New Delhi for the Research Fellowship.
Appendix A. Constants used in this paper are defined as follows.
where, d wð Þ ¼ M′− RPK ′ ;
X ωð Þ ¼ RK ′
AC′
− BLF ′C
� �þ N
C′− AB
F ′C
� �� D1 þ D2;
Y ωð Þ ¼ RK ′
ABC′F
þ LF ′
� �þ A
F ′þ NBC′F
� �� D3 þ D4;
Z ωð Þ ¼ RK ′
AC′
− LBF ′C
� �þ NC′
− ABF ′C
;
T ωð Þ ¼ RK ′
ABC′F
þ LF ′
� �þ AF ′
þ BNC′F
;
K ′ ωð Þ ¼ K− AC′
BGF
−A� �
− LF ′
Gþ ABC
� �;
a
2.0 1.5 1.0 0.5 0.00
20
40
60
80
100
120
140T
eff
Tef
f
Tef
f
Tef
f
b
2.0 1.5 1.0 0.5 0.00
20
40
60
80
100
120
140
c
2.0 1.5 1.0 0.5 0.00
20
40
60
80
100
120
140d
2.0 1.5 1.0 0.5 0.00
20
40
60
80
100
120
140
1/Δ ν 1/Δ ν
1/Δ ν 1/Δ ν
Fig. 5. Plot of Teff of the mirror M2 as a function of detuning Δ1/ν corresponding to the cases shown in Fig. 2. Other parameters are same as in Fig. 2.
305T. Kumar et al. / Optics Communications 285 (2012) 300–306
C′ ωð Þ ¼ C þ B2
F; F ′ ¼ F þ B2
C;
P ωð Þ ¼ −B− LF ′
EBF ′
þ A� �
þ AC′
E−ABF
� �;
M′ ωð Þ ¼ M þ AF ′
EBC
þ A� �
− NC′
E−ABF
� �;
R ωð Þ ¼ Bþ AF ′
ABC
þ G� �
þ NC′
−Aþ BGF
� �;
-1.1
-1.2
-1.3
-1.4
-1.50960 0965 0970 0975 0980
Maximum
Minimum
0985 0990
vΔ
/1
ω v/
Fig. 6. Color scheme for the contour plots.
C ωð Þ ¼ γm1−iωþ 4D1D2ja1sj2Δ1
κ1−iωð Þ2 þ Δ21
; E ωð Þ ¼ −ν1 þ4D2
1ja1sj2Δ1
κ1−iωð Þ2 þ Δ21
;
F ωð Þ ¼ γm2−iωþ 4D3D4ja2sj2Δ2
κ2−iωð Þ2 þ Δ22
; G ωð Þ ¼ −ν2 þ4D2
3ja2sj2Δ2
κ2−iωð Þ2 þ Δ22
;
M ωð Þ ¼ −iωþ 4D1D2ja1sj2Δ1
κ1−iωð Þ2 þ Δ21
; N ωð Þ ¼ ν1 þ4D2
2ja1sj2Δ1
κ1−iωð Þ2 þ Δ21
;
K ωð Þ ¼ −iωþ 4D3D4ja2sj2Δ2
κ2−iωð Þ2 þ Δ22
; L ωð Þ ¼ ν2 þ4D2
4ja2sj2Δ2
κ2−iωð Þ2 þ Δ22
;
Δ1 ¼ ω1−q1sD1 þ p1sD2; Δ2 ¼ ω2−q2sD3 þ p2sD4;
t1 ωð Þ ¼ M þ B2
K;
t2 ωð Þ ¼ N þ ABK
;
t3 ωð Þ ¼ Aþ LBK
;
t4 ωð Þ ¼ K þ B2
N;
t5 ωð Þ ¼ A−BNM
;
t6 ωð Þ ¼ L−ABM
:
306 T. Kumar et al. / Optics Communications 285 (2012) 300–306
t7 ωð Þ ¼ C− Et2t1
þ At5t4
;
t8 ωð Þ ¼ Bþ Et2t1
−At6t4
;
t9 ωð Þ ¼ D1 þD2Ft1
þ D2ABMt4
;
t10 ωð Þ ¼ D4FBt1K
−AD4
t4;
t11 ωð Þ ¼ F−Gt6t4
þ At3t1
;
t12 ωð Þ ¼ Gt5t4
−At2t1
−B;
t13 ωð Þ ¼ −D2BGMt4
−AD2
t1;
t14 ωð Þ ¼ D3 þD4Gt4
−AD2
t1;
t15 ωð Þ ¼ t7−t8t12t11
;
t16 ωð Þ ¼ t9 þt8t13t11
;
t17 ωð Þ ¼ t10 þt8t14t11
;
t18 ωð Þ ¼ t8t11
;
References
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