Ideality in a Fiber-T aper-Coupled Microresonator System for Application
to Cavity Quantum Electrodynamics
S.M.Spillane,T.J.Kippenberg,O.J.Painter,and K.J.V ahala
Thomas J.W atson Laboratory of Applied Physics,California Institute of T echnology,Pasadena,California91125,USA
(Received13December2002;published22July2003)
The ability to achieve near lossless coupling between a waveguide and a resonator is fundamental to
many quantum-optical studies as well as to practical applications of such structures.The nature of loss
written in the starsat the junction is described by afigure of merit called ideality.It is shown here that under appropriate
conditions ideality in excess of99:97%is possible usingfiber-taper coupling to high-Q silica micro-
spheres.To verify this level of coupling,a technique is introduced that can both measure ideality over a
range of coupling strengths and provide a practical diagnostic of parasitic coupling within thefiber-
taper-waveguide junction.
DOI:10.1103/PhysRevLett.91.043902P ACS numbers:42.81.Qb,42.60.Da
Microresonators have attracted considerable interest for investigation of fundamental process ranging from cav-ity quantum electrodynamics(QED)[1,2]to nonlinear optics[3,4],and in more applied areas such as photonics [5,6]and chemical/biological nsing[7].Resonators that feature modes with ultrahigh quality factors(Q)and small modal volume,such as silica microspheres,can induce strong coupling between an atomic system and the cavity electromagnetic mode[8],as well as drastic reductions of the power necessary to obrve nonlinear effects[9,10].An important requirement for such studies (quantum-optical experiments in particular)is ultralow-lossfield coupling both to and from the microresonator. Parasitic coupling loss in this process is quantified by defining a coupling‘‘ideality’’as the ratio of power coupled to a desired mode divided by power coupled or lost to all modes(including the desired mode).Ideality describes to what extent the coupling process behaves as single mode to single mode.Prism coupling,a popular technique for coupling to ultrahigh-Q microspheres,does not typically posss a high ideality becau of the large number of possible output modes(as evidenced by the obrved spatial‘‘fan’’of output power[11,12]).In con-trast,the small number of modes supported byfiber-taper-bad coupler waveguides suggests that the situ-ati
on could be improved over prism coupling.Tapered fibers have been shown to provide high coupling effi-ciency to microresonators with low nonresonant inrtion loss,and this has prompted our proposal for their u in cavity quantum-optical studies[10,13].Nonetheless, since neither resonator nor taper(in general)are single transver mode devices,and since it is possible to couple to radiation modes by way of the taper,high ideality in this system is not necessarily expected.Here,the ideality of afiber-taper coupler is measured using a novel method. It is shown that for appropriate conditions taper junctions offer near-unity ideality.Becau the conditions for ideality can be generalized,it is expected that the results can apply in a variety of systems.
Typicalfiber tapers are1–4 m diameter air-clad silica cylinders,fabricated byflame heating and pulling standard single-modefiber into a narrow thread[13,14]. Tapers are usually multimode waveguides.For example, six modes are supported in a2:0 m diameter silica taper at an optical wavelength of1550nm.By control of taper adiabaticity[15],it is always possible to launch the fun-damental HE11taper mode.Likewi,excitation of a single resonator mode is possible through a combination of pha-matching and modal frequency lection(the power coupling coefficient to the next resonator mode assuming a mode linewidth of1MHz and paration between modes of10GHz is on the order of10ÿ8). However,a resonator mode,once excited,can trans-fer power back to many t
aper modes(e Fig.1). Additionally,the prence of the taper-waveguide can cau the resonator to couple power into the continuum of radiation modes(and induce scattering at the resona-tor-waveguide junction).
This waveguide/resonator system(Fig.1)can be studied using a simple model[17],bad on the assump-tion of coupling between the resonator and waveguide being weak,and intrinsic resonator loss being low.In such cas,the individual contributions to the cavityfield decay are parable(the assumptions are valid for the fiber tapers and high-Q microresonators ud in the cur-rent work).The internal resonatorfield(a)is determined by considering both excitation by the fundamental wave-guide-mode(input wave amplitude s and coupling am-plitude 0),and decay due to intrinsic resonator loss (round-trip power loss coefficient 20)and output coupling to all available waveguide/radiation modes(coupling am-plitude to a higher-order waveguide-mode denoted by i>0and to radiation modes denoted by rad).On reso-nance the internalfield obeys the equation of motion[18], da
dt
ÿ
1
2
steam什么意思>outfit
X
i
2i 2
rad
2
a i 0s:(1)
The transmission through the waveguide consists of an interference of the partially transmitted input field with the field coupled from the resonator back into the funda-mental HE 11taper mode (this assumes that power coupled into higher-order taper modes is lost upon transition to single-mode fiber),which is given by
T j t 0 i 0a=s j 2:
(2)
In steady state,the transmission can be expresd as
T
1ÿK 1 K 2
;(3)where the coupling parameter K is defined by
K 20
P i Þ0
2i 2rad
20:(4)
K is the ratio of the desired waveguide-mode power
coupling to total system power loss.
The coupling factor K is compod of an intrinsic
contribution K I 20= 2
0,and a parasitic contribution K P 20= P i Þ0 2i 2rad ,such that K
ÿ1 K ÿ1I K ÿ1P .As noted earlier,ideality is defined as the amount of power coupled into the desired mode (in this ca the fundamental HE 11mode)divided by the amount of power coupled into all modes,and is given by
I 20
P i
2i 2rad ;T
11 K ÿ1
P :(5)
An ideal waveguide coupler (I 1)is characterized by coupling only between the intended resonator a
nd wave-guide modes.The degree of ideality is determined by the parasitic coupling factor K P .K P is,in general,a function of the relative position of the taper and resonator,and,as
such,the deviation of K (K ÿ1 K ÿ1I K ÿ1P
)from ideal behavior (K K I )determines I .
K can be obtained by measuring the dependence of coupling on waveguide-resonator gap and inverting Eq.(3)as follows:
1 T p 1 T p K 20e ÿ 0x 2i e ÿ i x 20
;(6)
where the upper signs are taken for transmission values in
the overcoupled regime,and the lower signs for the under-coupled regime.The cond equality follows from Eq.(4)by noting that the coupling amplitudes 0and i decrea exponentially with resonator/waveguide paration and by assuming that K P is dominated by a single higher-order taper-waveguide mode (as shown below this as-sumption is valid for the data in this work). 0( i )are s
patial decay rates (vs gap x )such that 20;i 20;i exp ÿ 0;i x with x 0corresponding to zero gap.As demonstrated below,upon plotting K vs gap on a logarithmic scale,K I and K P can often be identified,as K P (for higher-order taper mode parasitic coupling)is a line with slope less than that of K I .In particular,ifwe will always love you
2i > 20,then the relation K
ÿ1 K ÿ1I K ÿ1
P results in a roll-off of K for small gap distances due to parasitic
coupling.In situations where 2i < 20,the higher-order
mode coupling is masked and a lower bound on ideality can be established.
梵文翻译器
W e experimentally investigate the ideality of a fiber-taper-microsphere system using the approach given above.The experimental tup consists of a silica microsphere coupled to a tapered optical fiber with the paration distance controlled by a clod-loop stage with a resolu-tion of 20nm.All data were taken for resonances near 1550nm.The transmission data are obtained by normal-izing the on-resonance power transmission with the power transmitted by the taper alone (i.e.,infinite gap).
The dependence of ideality on fiber-taper diameter was investigated by varying the resonator location on the fiber taper.Figure 2shows K vs gap curves for multiple fiber-taper diameters.The taper diameters,measured by a scanning electron microscope,are approximately 1:2 m (circles),1:35 m (stars),and 1:65 m (trian-gles).For the smallest taper diameter measured,three waveguide modes are supported,the HE 11,TE 01,and TM 01modes,although both higher-order modes are near cutoff.There are four modes supported (adding the HE 21mode)for the two larger taper diameters.The data show that for increasing taper diameter there is a devia-tion of K from the single-mode coupling regime (dashed lines)due to higher-order mode coupling.A fit
using
FIG.1.Coupling and loss parameters in a taper-microreso-nator system.The input field is a fundamental taper mode
which couples into the resonator with amplitude 0(trans-mission amplitude t 0).The output field couples into the funda-mental taper mode and higher-order taper modes with coupling constants 0and i ,respectively [16].The prence of the waveguide can also result in a radiated field.The higher-order taper modes are radiated or coupled to cladding modes upon transition of the taper back to single-mode fiber.The round-trip resonator intrinsic power loss is given by 20.
Eq.(6)shows excellent agreement (solid lines),suggest-ing that a single higher-order mode is responsible for the obrved roll-off of K with decreasing gap distance.As the number of modes supported for the two largest taper sizes are identical,the strong variation of the coupling data suggests that pha matching is playing a significant role in determining the coupling behavior,resulting from the change of the taper-waveguide modes’propagation constant as taper diameter is varied.
The ideality of the coupler is determined by K P through Eq.(5).Assuming the validity of the two-mod
e model,the dashed and dash-dotted lines in the figure give the K I and K P contributions to K ,respectively.The 1:65 m coupling data (triangles)show significant devi-ations from ideal coupling,with ideality ranging from 88%at microsphere-taper contact to 13%at a 1:5 m gap.The 1:35 m data (stars)exhibit less deviation,with ideality ranging from 99%at contact to 98%at a 2 m gap.Finally,the data corresponding to the 1:2 m taper diameter (circles)reprent apparent ideal behavior over the range of paration gaps measured.The ideality at contact for this taper size is >99:98%and it is not possible to infer a dependence with gap from the data.The influence of nonideality is clearly illustrated at the critical point (dotted lines).For high ideality,the critical point in the data (given by the gap distance where K 1)is identical to the gap paration where K I 1.The data show that this condition holds for the two smaller taper diameters.However,the critical point for the 1:65 m taper diameter data is shifted towards a lower paration than K I (0:1 m shift of data from dashed line),as a result of the lower ideality (the large shift of 0:5 m for K I is mainly a result of pha matching).
The data in Fig.2demonstrate that coupling (and ideality)vs position behavior is very nsitive to the size of the waveguide.In order to further investigate the influence of taper-waveguide diameter on the coupling behavior and ideality of the system,numerical calcula-tions bad on a modified couple
d-mode theory [19]were performed.This model calculated the ideality bad on coupling to the supported waveguide modes (i.e.,it did not include radiation mode coupling).The results were in good agreement with the experimental data in Fig.2(values of K at contact and fundamental mode decay ,slope of K I ,were within 10%of measured values).Finally,the degradation of ideality with increas-ing gap distance is a result of slower evanescent decay for higher-order taper modes.
Finally,by maximizing the value of K for near-contact gaps with a 2- m -diameter taper and using higher Q -factor microspheres (Q >108by measurement of line-width in a 65- m -diameter microsphere),it was possible to obtain even higher values for ideality in near-contact conditions (Fig.3).The int shows the transmission vs paration data for this system,with a maximum over-coupled transmission of 99:95%(determined by using the exponential fit to K at zero gap).Numerical calculations (described above)show that the data slope is consistent with K I .This agreement,combined with a very low radiation mode coupling power loss <0:05%(using the
bindwood
FIG.3(color online).Coupling parameter K vs taper-sphere paration for a 65- m -diameter microsphere.The data show a linear relation between ln K and x ,with a least squares fit (solid line).Ideality inferred at contact is greater than 99:99%.The dotted line marks the critical coupling point (K 1at x 0:91 m ).The int shows transmission vs position data.
FIG.2(color online).K vs position for various taper diam-eters for a 67- m -diameter microsphere.The data reprent taper diameters of approximately 1:2 m (circles),1:35 m (stars),and 1:65 m (triangles).Solid curves are fits using Eq.(6).The ideality at contact in the data (extrapolating fits to zero gap)is >99:98%,99%,and 88%,respectively.Data show that for incread taper diameter higher-order-mode coupling caus a deviation from the ideal ca (dashed line).The dash-dotted line reprents K P ,which is related to ideality through Eq.(5).Dotted lines mark the critical coupling point.
fact that the overcoupled transmission drop from unity is due to intrinsic resonator loss and all other coupling-induced loss),demonstrates that the taper behaves as a nearly ideal coupler(higher-order mode coupling is not obrvable over the range of gaps measured).A lower bound of ideality of99:97%is obtained if the lowest data point at contact is ud.However,using afit to the entire coupling data t(solid line)establishes a lower bound on ideality at contact of99:99%.
The obrvation that afiber taper can obtain high ideality in ultrahigh-Q systems shows that this form of coupling will be uful for the study of process requir-ing very low loss.This includes quantum-optical studies involving cavities in general,with specific examples being the application of strong-coupling cavity QED to quantum information studies[2,20]or of weak-coupling cavity QED to new quantum sources[21,22].In such examples,coupling quantum states to and transport over opticalfiber has been propod[23],making optical tapers an excellent coupling interface.As ideality is dominated by the mode spectrum of the coupler,alternate resonator ,microdisks and recently dem-onstrated ultrahigh-Q microtoroids on a chip[24])should exhibit the same high ideality confirmed in this work when ud with tapers.Furthermore,an additional obr-vation that coupling to radiation modes was negligible for taper diameters near single-mode operation(as given by the magnitude of off-resonance power loss when the resonator is in contact with the taper)suggests that ideal behavior can be obtained,if necessary,by operation at the taper size giving single-mode operation(although it is desirable to operate at the pha-matching point for opti-mal coupling efficiency[13,25]).Finally,the transforma-tion described by Eq.(6)provides a nsitive method of determining the ideality of a coupler and more generally us the resonant system to diagno properties of the coupling.
This work was supported by DARP A,the Caltech Lee Center,and the National Science
Foundation.
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kinda什么意思
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