1. INTRODUCTIONOptical resonators with high-Q whispering gallery modes(WGM’s) or morphology-dependent resonances providethe unique features for technological devices and the fun-damental science for, e.g., microdisk lasers,1,2 channeladd-drop filters,3,4 and cavity QED experiments.5,6 Re-cently, special attention has focused on a new class of mi-croresonators, the so-called asymmetric resonantcavities,7–10 which have two interesting characteristics:(1) they give rise to enhancement of the input–output cou-pling and directional emission,8,9 and (2) they supportchaotic as well as regular ray trajectories.7,10 Unique tothe partially chaotic system, laser-emission patterns withdark and bright regions on the surface of quadrupole-deformed liquid droplets have been explained by the re-fractive escape and dynamical eclipsing of internalrays.8,11
Images of deformed lasing microdroplets, captured atvarious inclination angles, has provided information onboth subsurface ray motions and spatially nonuniformoutput coupling of deformed microdroplets.10–12 How-ever, all the previous experiments10–12 have used externaloptical pumping, which causes spatially varying lasinggain within the droplet volume because of nonuniformdistribution of the internal pump intensity. To isolatethe effect of the spatially varying output coupling on thenonuniform droplet emission pattern, we have to elimi-nate the spatially varying pumping effect.
We achieved uniform optical pumping throughout thedroplet by using chemiluminescence. The nonuniformexcitation associated with external pumping with a laserbeam could thus be avoided. By using chemilumines-
cence as the pumping mechanism and the WGM’s to pro-vide high-Q optical feedback, we demonstrated chemicallypumped lasing in liquid pendant droplets, for the firsttime to our knowledge. The resulting laser-emissionpattern from the upper (lower) half of the pendant drop-let was similar to that of lasing prolate (spherical)microdroplets. Our present results indicate that in theprevious experiments10,11 on the spatial variation of thelaser emission from deformed microcavities, the nonuni-form pumping effect of an external beam is less signifi-cant than the spatial variation of output coupling. Weinvestigated the effect of surface curvature on the outputcoupling by perturbing the droplet surface with the sharptip of a pulled fiber. We obtained a cavity Q of up to3.5 3 108 by measuring decay time of both elastic scatter-ing and stimulated Raman scattering (SRS) from thesame-shape pendant droplet made of glycerol (67%) andwater (33%).
2. CHEMILUMINESCENT MATERIAL ANDPENDANT-DROPLET PRODUCTIONThe chemiluminescent material used in the experimentswas extracted from a commercial light stick (Cyalume byOmniglow), often referred to as cold light.13,14 Theselight sticks are usually used during scuba diving, camp-ing, and amusement-park activities. They consist of anouter sealed-plastic tube containing one solution and aninternal thin, breakable glass ampule containing anothersolution. The first solution contains pheny oxalate esterand a fluorescent dye [9,10-bis(phenyl-ethynyl) anthra-cene for high-intensity light sticks and rubrene for
1999 Optical Society of America
S. Chang et al. Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. B 1225
ultrahigh-intensity light sticks]. The second solutioncontains dilute hydrogen peroxide in a phthalic ester sol-vent. When the external plastic tube is flexed, the innerglass ampoule breaks, causing the two solutions to mixand to initiate a chemiluminescence process that lasts forapproximately 10–30 min, depending on the product.
The chemiluminescence process occurring in the lightstick can be explained by chemically initiated electron-exchange luminescence15 (CIEEL). The CIEEL processwith rubrene as a fluor is shown in Fig. 1. The CIEELprocess is one type of activated chemiluminescence, inwhich the fluor, instead of being a passive acceptor, ac-tively participates in the chemiluminescence reaction it-self. In CIEEL the reaction of peroxide (in our case,dioxetandione, which is produced by the reaction of ox-alate ester and hydrogen peroxide) with the fluor pro-duces a charge-transfer complex. The charge complexbreaks into an excited dye molecule and two CO2 mol-ecules. The excited dye molecules then spontaneouslydecay to the ground state by emitting a photon.
To make chemiluminescent pendant droplets, we ex-tracted and prepared the two solutions from the lightstick in separate containers. After the two solutionswere mixed, the luminous solution was poured into a sy-ringe, which was connected to flexible Tygon tubing thatwas reduced in diameter by the insertion of plastic tub-ing. A glass capillary tube of 700-mm outer diameter wasinserted into the plastic tubing.16 By pushing the pistonof the syringe, the solution reached the open end of the
Fig. 1. Chemically initiated electron-exchange luminescence isa photoemission process associated with chemiluminescent pen-dant droplets formed from ultrahigh-intensity light sticks. Thediagram shows the chemical reactions upon mixing of two solu-tions from the light stick. Dioxetandione is produced from a re-action of oxalate ester and hydrogen peroxide (H2O2). Dioxet-andione (peroxide) and rubrene (fluor) generate a charge-transfer complex. The charge complex breaks into two CO2molecules and an excited dye molecule, which subsequentlyemits a photon.
capillary tube, and a pendant droplet with an equatorialdiameter of ;2 mm was formed. We controlled the finetuning of the droplet size by squeezing flexible Tygon tub-ing between a fixed and a motorized moving aluminumblock.
3. IMAGES OF CHEMILUMINESCENTPENDANT DROPLETSThe color photographs of chemiluminescent pendantdroplets were taken with a photographic film camera (Ni-kon FG-20). To magnify the droplet images, we detachedthe camera lens from the camera body to allow a long im-aging distance. The light emitted from the droplets wascollected by a camera lens adjusted to be ;50 mm awayfrom the droplet. The pendant droplet was imaged ontothe color film, in the camera body, which was located;500 mm from the camera lens. The 103-magnified im-age almost filled the whole area of the color film [Ekta-chrome (ASA 4000)]. The shutter speed was adjusted toacquire adequate exposure for each image.
Fig. 2. Color photographs of chemiluminescent pendant drop-lets (103 magnification) with and without absorber. The pen-dant droplets are generated at the tip of a capillary tube of700-mm outer diameter. The equatorial diameter of the dropletsis ;2 mm. (a) High-intensity light stick [fluor: 9,10-bis(phenyl-ethynyl) anthracene] without an absorber. Camera ex-posure time (Texp) 5 1/4 s. (b) Ultrahigh-intensity light stick(fluor: rubrene) without an absorber. Texp 5 1/60 s. (c) High-intensity light stick with an absorber (0.4 g/l of Nigrosin). Texp5 16 s. (d) Ultrahigh-intensity light stick with an absorber (1.0g/l of Nigrosin). Texp 5 16 s.
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Color photographs were taken of pendant dropletsmade from light sticks of high intensity [Figs. 2(a) and2(c)] and ultrahigh intensity’’ [Figs. 2(b) and 2(d)], withand without a black absorber (Nigrosin). Without addingany absorber, we observed a highlighted droplet rim aswell as a redshifted color change [see Figs. 2(a) and 2(b)].When enough Nigrosin was added to the droplets, thedroplet rim was no longer highlighted and there was notany color change [see Figs. 2(c) and 2(d)]. The Nigrosinconcentration was 0.4 g/l for the pendant droplet in Fig.2(c) and 1.0 g/l for the pendant droplet in Fig. 2(d). Inboth cases the absorption coefficient aabs was greater than2.8 cm21 in the visible range, and consequently the ab-sorption length (1/aabs 5 3.6 mm) was less than the drop-let circumference (;6 mm), which was the shortest cavitylength required for the WGM’s in the pendant droplet.We note that both the interface curvature and the absorp-tion affect the pendant-droplet emission image.
To further investigate the effect of surface curvature onthe pendant-droplet–emission image, we poked the drop-let surface with a sharp object. A pulled-fiber tip (sizeless than 1 mm in diameter) provided a convenient poker.As the fiber tip touched the droplet surface, the dropletsurface bulged out locally because of the capillary force.The images shown in Fig. 3 were recorded with a CCDcamera, and the false color was chosen to mimic the truecolor. Figure 3(a) shows the image without any poking.The following locations were poked: at the equator [seeFig. 3(b)], at 45° below the equator [see Fig. 3(c)], and at
Fig. 3. False-color presentation of chemiluminescent pendantdroplets with their surfaces perturbed by a pulled-fiber tip: (a)The pendant droplet is formed from the original solution ex-tracted from an ultrahigh-intensity light stick without perturba-tion. The pulled-fiber tip touches (b) the equator, (c) below theequator from the left, and (d) above the equator from the left.All the perturbed locations appear brighter. Note that the drop-let edge diagonally opposite the perturbation location also bright-ens.
45° above the equator [see Fig. 3(d)]. As expected, therewas higher light leakage at the perturbed locations. Sur-prisingly, the droplet edge, diagonally opposite the per-turbed location, also brightened. We conjecture thatrays, launched by the surface perturbation, are guided in-side the droplet along a precessing tilted or a circularequatorial orbit, reaching the diagonally opposite part ofthe droplet with internal incident angle x less than thecritical angle xc 5 sin21(1/n) and hence leak out as re-fractive rays.
4. EMISSION SPECTRA ALONG THE RIMWe took the spectra along the rim of the chemilumines-cent pendant droplets by placing the curved edge of thedroplet image on the slit of an image-preserving spec-trograph (Acton Research Corporation, Spectra Pro-500).At the spectrometer slit the size of the droplet image wasmade to be the same as the real droplet size by selectionof the focal length of the collection lens. The image-preserving spectrograph was able to preserve the integ-rity of the input image along the slit height (z axis), even
Fig. 4. (a) Emission and absorption curves of rubrene (takenfrom Ref. 17). The absorption curve is presented in units ofmolar-extinction coefficient e for the rubrene dissolved in ben-zene with a concentration of 0.1 g/l. The molar-extinction coef-ficient e (mole21 cm21) is defined as e 5 A/c 5 (aabs /c)log10 e,where A (cm21) is the absorbance, c (mole) is the concentration ofthe molecule, and aabs (cm21) is the absorption coefficient. (b)The absorption coefficient of 0.1-g/l nigrosin dissolved in ethanol,measured by a spectrophotometer (Hitachi U-2001).
S. Chang et al. Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. B 1227
after the image was spectrally dispersed by the grating.Thereby, the emission spectra along one side of the drop-let rim were detected simultaneously. The spectrographhad a 0.5-m focal length and a 150-groove/mm gratingwith a liquid-nitrogen-cooled CCD camera (Princeton Ap-plied Research Corporation (PARC, OMA-Vision) at thespectrograph exit plane.
Fig. 5. Emission spectra at selected locations along the rim ofvarious pendant droplets. The inset shows the image of the pen-dant droplet on the slit (along the z axis) of the spectrometer.The emission spectra are taken along the rim at z 5 0 mm (theequator) and from z 5 0.1 mm to z 5 0.4 mm of the upper hemi-sphere. (a) Droplet A is made of the original solution extractedfrom an ultrahigh-intensity light stick. (b) Droplet B is made ofthe original solution plus 0.1-g/l of Nigrosin. (c) Droplet C ismade of the original solution plus 0.4-g/l of Nigrosin.
Emission spectra were taken from three different drop-lets: (1) droplet A, the original solution extracted froman ultrahigh-intensity light stick (containing rubrene); (2)droplet B, the same solution as droplet A with 0.1-g/l ofNigrosin added; and (3) droplet C, the same solution asdroplet A with 0.4-g/l of Nigrosin added. The absorptionand fluorescence curves of rubrene are shown in Fig. 4(a),and the absorption curve of 0.1-g/l of Nigrosin is shown inFig. 4(b).
The spatially resolved (along the z axis) emission spec-tra from the rims of pendant droplets are shown in Figs.5(a)–5(c). From the droplet A the emission spectra at theequatorial edge (z 5 0 mm) are similar to the fluores-cence spectra measured from the chemiluminescent liquidin a cuvette. However, an intensity enhancement forz . 0 mm is observed in the wavelength region ofl . 570 nm, where the self-absorption of the dye mol-ecule is low [see Fig. 4(a)]. For the wavelength region ofl , 570 nm, where the self-absorption is high, there is nointensity enhancement, and, in fact, the intensity forz . 0 is lower than that at the equator. For l,570nm the higher intensity at z 5 0 mm is because the larg-est spatial overlap between the vertical slit and the drop-let rim occurs at the equator (at z 5 0 mm; see the insetimage of Fig. 5). Consequently, even if the emission isuniform along the rim, the amount of fluorescence enter-ing the spectrograph is largest at the equator and is ex-pected to be less for 6z Þ 0. The fact that the intensityat the rim increases for z . 0 mm suggests that amplifi-cation occurs for l . 570 nm and overcomes the slit–rimspatial overlap effect. The redshifted color change ob-served near the rim of droplet A in Fig. 2(b) is due to thiswavelength-dependent fluorescence amplification and ab-sorption of the emitting dye. The WGM peaks for theselarge-sized droplets are not observed because the modedensity is too high to be spectrally resolved by a conven-tional spectrometer (the mode density is proportional tothe droplet radius). For pendant droplets (a 5 1 mm)the mode density is ;303 higher than in the case ofsmaller spherical microdroplets (a 5 35 mm), where las-ing peaks are separated by 1 nm in the wavelength regionnear l 5 600 nm.
For the droplet B [see Fig. 5(b)] in the wavelength re-gion of l . 610 nm, where aabs owing mostly to Nigrosinis less than 2.5 cm21, a smaller intensity enhancement isalso observed. For z . 0 the amplification barely over-comes the effect of the spatial overlap of the slit. The in-tensity enhancement is smaller than the droplet A be-cause of the absorption introduced by Nigrosin. For thedroplet C the spectral shapes at various z’s are identicaland the same as that for the liquid in an optical cell. Thespectrally preserving but decreasing-in-intensity curves(as z is moved away from z 5 0) are totally caused by theslit–rim overlap effect. There is no intensity increasealong z for the whole wavelength region up to 750 nm.The aabs of the droplet C is greater than 2.8 cm21
throughout the whole wavelength region measured.The observed intensity enhancement for z . 0 mm in-
dicates that chemically pumped lasing occurs inside thependant droplet A, when the absorption is small enoughin the wavelength range l . 570 nm. The pendant drop-let A must have extremely high-Q WGM’s and a small ab-
1228 J. Opt. Soc. Am. B/Vol. 16, No. 8 /August 1999 S. Chang et al.
sorption so that the round-trip gain can always be higherthan the round-trip loss. In the droplet B the observedthreshold of lasing appears to be at aabs52.5 cm21, whichcorresponds to a 1/e intensity drop after a 0.7 round tripat the equator.
5. INTENSITY DISTRIBUTION ATDIFFERENT WAVELENGTHSThe spatial profiles of the chemiluminescence intensityalong the rims of pendant droplets A, B, and C are shownin Fig. 6, for several different wavelengths. Each spatialprofile of the intensity at the various wavelengths is ob-tained by integration of photon counts of the CCD cameraover a 2-nm interval, centered at each indicated wave-length. For the droplet A, at l 5 540 nm the intensityprofile follows the amount of fluorescence entering thespectrometer slit at each z, which is the slit–rim overlapeffect. For the wavelength range 570 nm , l , 700 nm
Fig. 6. Intensity profiles (along z) of the chemiluminescent pen-dant droplets at various selective wavelengths. We obtainedeach intensity profile curve by integrating counts from the CCDcamera over 2 nm, centered at the indicated wavelength. Weused the same data as in Fig. 5 for (a) droplet A, (b) droplet B,and (c) droplet C.
the emission intensity increases with z in the upper half(z . 0), up to a point where the pendant droplet touchesthe capillary tube. The maximum intensity increase isobserved at l 5 620 nm and z 5 0.5 mm. No significantchange in the intensity distribution is observed in thelower half (z , 0) of the pendant droplet, but we shouldkeep in mind that away from z 5 0 mm, the spatial over-lap between the vertical spectrograph slit and the curvedrim decreases in both directions. The intensity profile forthe tear-shaped droplet A is similar to that of lasing pro-late droplets in the upper half (z . 0) and to that ofspheres in the lower half (z , 0) of the droplet. For thedroplet B, a small intensity increase with z is alsoobserved in the upper half (z . 0) in the wavelengthrange 610 nm , l , 700 nm. For the nonlasing dropletC, all the intensity profiles at each wavelength decreasefor uzu . 0.
We attempt to explain the cause of the observed non-uniform intensity distribution at different wavelength re-gions in terms of spatially dependent leakage while real-izing that spatially uniform absorption and excitationexist throughout the droplet. First, the Q of a cavitymode is expressed in terms of leakage and absorption.Second, we construct an expression for the transmissionprobability or output coupling for each reflection on acurved interface, where the incident angle x with respectto the surface normal is greater than the critical anglexc 5 sin21(1/n) and n is the index of refraction of thedroplet. Third, the standard laser rate equation is ap-plied to the lasing in WGM orbits in the pendant droplet.The laser output intensity is expressed in terms of theoutput coupling, absorption, and excitation. Finally, in-terpretation of the observed nonuniform intensity distri-butions along the droplet rim is given based on the previ-ous derivations.
A. Q of Cavity ModesThe Q of a cavity mode is defined as (2p 3 storedenergy)/(average energy loss per cycle) and is a measureof energy storage efficiency. In lasing, the Q of a modeand the pumping rate determines the internal intensityand total output intensity of a lasing mode. The Q of amode is not a local property of a cavity but is related tothe round-trip energy loss of a cavity mode. The majorsource of energy loss in the lasing pendant droplet is ab-sorption and leakage; other sources of loss can be scatter-ing and nonlinear optical effects, both of which we will ne-glect. The absorption with and without the addedabsorber (Nigrosin) is uniform throughout the pendantdroplet. Therefore the absorption per unit lengthaabs (cm21) is constant along the round-trip trajectorywith the length L of a WGM orbit, and hence the round-trip absorption is aabsL. However, the leakage per unitlength aleak (cm21) can be nonuniform along the light tra-jectory, as the output coupling is dependent on the sur-face curvature. Therefore the round-trip leakage can beexpressed as
a leakL 5 E0
L
a leakdl, (1)
S. Chang et al. Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. B 1229
where l is the path length along the trajectory and a leak isthe leakage per unit length averaged over one round trip.Then, the overall Q of a mode can be expressed as
1
Q5
1
Qabs1
1
Q leak5
1
v
c
n~aabs 1 a leak!
51
v
c
n S aabs 11
LE
0
L
a leakdl D , (2)
where c is the speed of light and v is the WGM frequency.We assume there are no losses owing to bulk and surfacescattering and to photon depletion associated with othernonlinear optical effects. Unlike linear cavities in whichthe leakage loss occurs only at the output coupler, we notethat leakage from the droplet cavity is distributed andvarying along the trajectories as the curvature is chang-ing for the tilted orbits.
B. Leakage Rate on a Curved SurfaceThe local leakage per unit length a leak is related to thetransmittance T of rays that bounce on the droplet sur-face. When the internal incident angle (relative to thesurface normal) is less than the critical angle for total in-ternal reflection (i.e., x , xc), the transmittance T fromthe curved interface (or refractive leakage) can be ap-proximately obtained by use of Fresnel’s formula for theflat surface. For x > xc (total-internal-reflection condi-tion for the plane wave incident on a planar interface),however, there is no refractive leakage; there is only dif-fractive leakage associated with the curvature of thedroplet surface. To calculate diffractive leakage for asphere, one usually recasts the electromagnetic-fieldequation (Helmholtz equation) of the dielectric sphereinto a quantum-mechanical Schrodinger equation with aneffective potential well/barrier, as shown in Fig. 7.18 Inelectromagnetic notation, WGM’s of a sphere are usuallydesignated by n (radial order) and l (angular modenumber).19,20 The (n, l ) WGM’s correspond to the quasi-bound states (labeled as n1 , n2 , ...) in an l -effective po-tential well/barrier or centrifugal barrier. These boundstates are called quasi-bound states because energy canleak out of the potential well or centrifugal barrier bytunneling21 with an effective barrier thickness deff and aneffective work function Weff . Such a tunneling loss hasbeen referred to as diffraction loss. With WKBapproximation22 and semiclassical approximation23 thetunneling probability (or transmittance T) for x @ xc(quasi-bound states far below the top of the potentialwell) can be calculated to be11
T ' expH 22nka sin x
3 F1 2 S sin xc
sin xD 2G3/2J , (3)
where k 5 2p/l, a is the radius of the sphere, and n isthe index of refraction of the sphere. The quantity ka isthe size parameter, which denotes the circumference ofthe sphere cavity in units of the optical wavelength in air.Note that the T for diffractive or tunneling loss is an ex-ponentially decreasing function of sin x, ka, and n.Therefore the smaller angle of incidence x leads to an ex-ponentially higher leakage rate, provided that x @ xc (seeFig. 7).
The x dependence of T for the diffractive process in asphere can be understood with both ray and wave pic-tures of WGM’s. Using semiclassical approximation, anangle of incidence can be assigned for each (n, l ) WGMby23
sin xn, l 5l
nkn, l a. (4)
The rays having sin xn, l occupy an annular region be-tween an outer shell (surface of the sphere at r 5 a) andan inner shell (the caustic at r 5 a sin xn, l ) to which all
Fig. 7. (a) Effective radial potential well/barrier Veff(r) 5 k2@12 n2(r)# 1 l (l 1 1)/r2 in the quantum-mechanical analogy forWGM’s in a spherical cavity with radius a, where k 5 2p/l,n(r) is the refractive index as a function of r, and l is the angu-lar mode number of an WGM, usually labeled with (n, l ), wheren is the radial order. Then, (n, l ) WGM’s correspond to thequasi-bound states (labeled as n1 ,n2 ,...) in an l -potential well.A WGM with wave number k sees the effective work functionWeff and effective thickness deff of the potential barrier. Thetransmittance T in relation (3) can also be expressed in terms ofWeff and deff , T ' exp(22/3adeff AWeff).
11 (b) Diffractive or tun-neling leakage (in the effective potential picture) occurs evenwhen x . xc in the spherical cavity. The tunneling probabilityis given by relation (3). Using semiclassical approximation,7
each (n, l ) WGM can be related to the angle of incidence x by therelation sin xn, l 5 l /nkn, l a, where a is the radius of the sphere.For a given l potential, lower n WGM’s have higher x, and con-sequently they are more tightly bound to the rim of the sphere,resulting in less tunneling loss and a higher Q value.
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the rays are tangent [see Fig. 7(b)]. This annular region,l /nkn, l , r , a, is exactly the same region in which then state (with a wave number kn, l ) in an l -potential wellis classically allowed, and thus most of theelectromagnetic-wave function is distributed. The lowern is for a given l potential, the larger is the angle of inci-dence x and the closer are the rays confined to the sur-face of the sphere. WGM’s with larger x have exponen-tially smaller diffractive loss or tunneling probabilities@T ' exp(22/3adeffAWeff), where 1.28 , a , 2]11 be-cause the corresponding n state sees thicker effective bar-rier thickness deff and higher effective work function Weff .
C. Laser Rate EquationsThe role of the leakage loss (either by the diffractive orthe refractive process) and the absorption loss in affectingthe lasing-emission intensity can be best understoodif the standard laser rate equations are used as an illus-tration. Provided that the dye lasing can be modeled bythe standard four-level system,24,25 the expression ofthe cw-laser output from the pendant droplet may be de-rived from the standard rate equations of the four-levelsystem. We assume that the transition of dye moleculesoccurs among four energy levels of the dye molecules: u0&,u1&, u2&, and u3&. Pumping of molecules occurs from u0& tou3&, and lasing transition occurs between u2& and u1&.From u3& to u2& and from u1& to u0&, we assume that fast(G ' 1012 sec21) nonradiative transitions occur.
Under the usual assumption25 used in the standardrate equations for molecular population of levels 1 and 2and photon flux, we know that
dN2
dt5 Rp 2 s~n!Fn~N2 2 N1! 2 G21N2 ,
dN1
dt5 s~n!Fn~N2 2 N1! 1 G21N2 2 G10N1 ,
dFn
dt5
c
n@ s~n!~N2 2 N1! 2 a leak 2 aabs#Fn , (5)
where Ni (cm23) is the number of molecules per unit vol-ume in level ui&, Rp (sec21 cm23) is the pumping rate pervolume, s(n)@5sa(n) 5 se(n)# is the stimulated absorp-tion and/or emission cross section (cm2), Fn (cm22 sec21)is the photon flux at frequency n @W 5 s(n)Fn (sec21) isthe stimulated transition rate], G21 (sec21) is the decayrate from u2& to u1&, and G10 is the decay rate from u1& to u0&.In the rate equation for the photon flux in the pendantdroplet, the leakage loss is treated as distributed loss,which is quite different from the localized output-couplingmirror loss in a Fabry–Perot laser cavity. In fact, bothgain s(n)(N2 2 N1) and absorption loss aabs are distrib-uted uniformly in the chemically pumped pendant drop-let, and only the leakage loss a leak (or output coupling) iscurvature and hence position dependent. We can obtainthe steady-state (or cw) population inversion DNSS by set-ting dN2 /dt 5 dN1 /dt 5 0 in Eqs. (5):
DNSS 5Rp8
s~n!Fn 1 G21, (6)
where Rp8 is the effective pumping rate @5 Rp(12 G21 /G10)#. The steady-state value of the invertedpopulation DNSS , and thus the gain per unit length@ gSS 5 s(n)DNSS#, are decreasing functions of Fn , lead-ing to the laser saturation behavior. DNSS is maximal atFn 5 0, with a small signal gain g0 5 s(n)DNSS(Fn
5 0) 5 sRp /G21 .The steady-state condition for the photon flux, dFn /dt
5 0, in Eqs. (5) leads to the following condition:
gSS [ s~n!DNSS 5 atot , (7)
where gSS is the optical gain at frequency n and atot5 aabs 1 a leak . By solving Eq. (6) for Fn and using Eq.(7), we can express the internal laser flux Fn in terms ofRp8 and atot . Because counterpropagating laser light inboth directions contributes with equal intensity to the in-ternal laser flux Fn , the one-way circulating laser inten-sity Icirc [hence 1/2 in Eq. (8)] in the cavity is given by
Icirc 5hnFn
25
Isat
2 S g0
atot2 1 D , (8)
where Isat is the saturation intensity given by hnG21 /s.The local output intensity dIout along a length dl is thengiven by
dIout 5 a leakdlIcirc 5 a leakdlS g0
aabs 1 a leak2 1 D Isat
2. (9)
The total output intensity is Iout 5 *0LdIout , where L is
the round-trip length of a WGM in the pendant droplet.While the circulating intensity inside the cavity mono-tonically decreases with increasing output coupling, theoutput intensity has a maximum value at a particularvalue of output coupling, a leak . We reach the key resultthat is pertinent for the case of uniform gain and absorp-tion loss; the laser output intensity is maximal at
a leak-opt 5 2aabs 1 Aaabsg0. (10)
When a leak , a leak-opt , dIout increases with a leak ; whena leak . a leak-opt , dIout decreases with a leak .
D. Interpretation of ResultsInternal circulating laser intensity Icirc in Eq. (8) de-creases as total loss (leakage and absorption) increases.However, local output intensity dIout is proportional to(local leakage rate) 3 Icirc . As long as a leak , a leak-opt ,dIout increases with a leak . However, dIout always de-creases with aabs . In fact, leakage acts as a source ofoutput intensity, and absorption acts as a sink of the out-put intensity. In terms of energy loss from the cavity,the roles of leakage and absorption are similar. Whenthe pump level is reasonably above threshold, the leakageloss (5 dIout) in Eq. (9) can be approximated as26
dIout 'a leak
aabs 1 a leak
Isat
2g0d l, (11)
which can be observed outside the cavity. In the samemanner the absorption loss dIabs can be expressed as
dIabs 'aabs
aabs 1 a leak
Isat
2g0dl, (12)
S. Chang et al. Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. B 1231
which cannot be observed outside the cavity. For a las-ing mode in a Fabry–Perot cavity the laser output is lo-calized at one point (output coupler) along the light tra-jectory. However, for a lasing WGM in the pendantdroplet the laser output is distributed along the orbit tra-jectory, and the local output intensity is proportional tothe local leakage rate a leak that varies along the trajecto-ries.
For the pendant droplet the cavity shape at z 5 0 mmis a circle, where the internal angle of incidence does notchange, nor does transmittance around the equatorial rimchange. The leakage is only through tunneling/diffraction loss and is uniform along the circular equator.Theoretically, extremely high-Q WGM’s in the dropletsare usually limited by intrinsic absorption loss and sur-face fluctuations. Because of the circular symmetry atthe equator, its light leakage rate is slow and is much lessthan the absorption rate, a leak ! aabs . Therefore fromrelations (11) and (12) the laser output is dimmer and theinternal intensity is dominantly absorbed rather thanleaked out.
The laser output intensity at z Þ 0 is enhanced be-cause the leakage is greater from the tilted precessingorbit27 (see Fig. 8). When the cavity is a perfect circle,there is no effective torque exerted on the rays at each re-flection (riFeff , where Feff is the effective force exerted onthe rays), and hence the effective angular momentum Leffof the ray orbit is conserved. This effective angular-momentum conservation manifests itself as the existenceof angular mode number l in the electromagnetic field of aperfect sphere. When the cavity is not a perfect circle,however, the effective torque exerted on the rays at eachreflection is nonzero (Feff , which is parallel to the surfacenormal, is not parallel to r anymore), and hence Leff is notconserved. As a consequence, a ray of light circumnavi-gating along a tilted orbit does not go back to its original
Fig. 8. Precessing tilted orbits in the pendant droplet. Thependant droplet is axisymmetric (axis of symmetry: z axis), andhence Lz is conserved. As a result, the angular-momentum vec-tor L, which is the normal of the plane of motion, rotates aroundthe z axis. The V denotes the precession frequency. For eachprecessing tilted orbit the highest curvature occurs at the high-est latitude of the orbit because of the shape in the upper half ofthe pendant droplet. The bottom half is more like a hemisphere.
position after completing one round trip along a greatcircle. However, if the cavity is axisymmetric (z axis asthe axis of symmetry), the effective torque does not havethe z component, and therefore (Leff)z is conserved. Foraxisymmetric cavities the vector Leff rotates around the zaxis with constant (Leff)z as shown in Fig. 8. As a result,the rays circumnavigate the cavity with their plane of mo-tion rotating around the z axis, the so-called precessingorbits.27,28
The pendant droplet is axisymmetric, and in particularbecause of the shape in the upper half of the pendantdroplet, the curvature along a tilted precessing orbit ismaximum at the highest latitude of the orbit, as shown inFig. 8. At this highest curvature point the curvature islargest and the angle of incidence (x) is smallest eventhough x . xc . Consequently, the rays encounter themaximum diffractive leakage [see Eq. (3)] at the highestlatitude of the tilted precessing orbit. The largest lightleakage along the path length Dl in the region of highestcurvature, a leakDl, is
a leakDl . aabsDl, (13)
where aabs ' 1.5 3 1023 cm21 or Qabs ' 108 for l. 570 nm. Therefore from relations (11) and (12) the la-ser output at z Þ 0 is expected to be brighter and the in-ternal intensity leaks out more than it is absorbed. Forthe other parts (particularly the lower half of the pendantdroplet) of the precessing orbit, the curvature along thetrajectory is lower and does not deviate much from that ofthe sphere. As z increases away from the equator, laseroutput intensity increases because the leakage is frommore tilted-precessing orbits and x decreases as the cur-vature increases at the highest latitude of the orbit.Note that the integrated quantity Q leak [see Eq. (2)] for atilted orbit need not deviate much from that for the equa-torial orbit because a leakDl near the localized (Dl ! l)highest curvature has only a small contribution to the in-tegrated leakage loss.
6. LOCAL CURVATURE PERTURBATIONThe emission spectra from the rim of the pendant dropletA, poked at the equator, are shown in Fig. 9(a). Atz Þ 0 mm the spectra are the same as those from the un-perturbed droplet A. At z 5 0 mm, however, the spectrahave a higher intensity in the wavelength regionl . 570 nm, where lasing occurs. Whereas the spectraat z 5 0 for unperturbed droplet A did not exhibit muchdetectable lasing emission [see Fig. 5(a)] because of lowa leak at the equator, the increased laser output of the pre-turbed droplet, poked at z 5 0 [see Fig. 9(a)], is support-ive of the hypothesis that the enhanced intensity iscaused by an increased leakage rate at the localized per-turbed location. The intensity spatial profiles along z forvarious wavelengths are shown in Fig. 9(b). The inten-sity is highest at the equator, where curvature at thepoked location is highest, and at l 5 620 nm, where thenet gain is the largest. The effect of curvature becomesclear when the intensity spatial profile in Fig. 9(b) is com-pared with the emission-intensity spatial profile of unper-turbed droplet A [see Fig. 6(a)], which has its maximumat the highest curvature (z 5 0.5 mm), near the neck of
1232 J. Opt. Soc. Am. B/Vol. 16, No. 8 /August 1999 S. Chang et al.
the pendant droplet (the region where the pendant drop-let touches the capillary tube).
7. CAVITY-LIFETIME MEASUREMENTSWe obtained Q’s of the modes in the pendant droplet bymeasuring the radiation decay time t0 (where Q 5 vt0)of both elastic scattering and stimulated Raman scatter-ing (SRS) from the pendant droplet. To avoid the com-plication of self-absorption (aabs Þ 0) inherent with thelaser dye, we chose not to use the radiation decay of laseremission.
As a droplet material, glycerol (n 5 1.47) diluted withwater was used in order to approach the refractive indexof the chemiluminescence material (n 5 1.48). Water(n 5 1.33) had to be introduced to reduce the viscosity ofthe glycerol for easier droplet generation. The volume
Fig. 9. (a) Emission spectra at selected locations along the rimof pendant droplet A, made of the original solution extractedfrom an ultrahigh-intensity light stick, with surface perturbationapplied at the equator (z 5 0) by a sharp pulled-fiber tip. Theenhanced light emission at z 5 0 is due to increased output cou-pling of lasing light at the perturbed equatorial location. (b)Spatial intensity profile along z at selected wavelength regions(each intensity curve is obtained by integration of the photoncounts over 2 nm) of the same droplet. The largest enhance-ment occurs at l 5 620 nm, where the net unsaturated gain oflasing is the highest.
ratio of glycerol to water, 2:1 (hereafter referred to asdroplet D), was used in the experiment and had a final re-fractive index n 5 1.431.
The droplet D was pumped by a 130-mJ pulse of thesecond harmonic (lp 5 532 nm) of a Q-switched, diode-pumped Nd:YAG laser (Spectra-Physics X30-532Q) at 100
Fig. 10. Cavity decay time t0 measurements from the drop-let D, pumped by a 30-ns laser pulse at 532 nm. (a) The longestdecay times measured from SRS: t0 5 60 ns (Q 5 1.8 3 108)when pumped at the equator and detected at the equator(equatorial orbit); t0 5 86 ns (Q 5 2.6 3 108) when pumpedat u 5 145° and detected at u 5 135° (precessing orbit). (b)The longest decay time measured at the green wavelength (532nm) when pumped at the equator by a green beam and detectedat the equator (equatorial orbit): t0 5 31 ns (Q 5 1.1 3 108).(c) Various decay times measured at the green wavelength (532nm) when pumped by a green beam at u 5 145° and detected atu 5 135° (precessing tilted orbit): (i) t0 5 37 ns (Q 5 1.33 108), (ii) t0 5 68 ns (Q 5 2.4 3 108), and (iii) t0 5 40 ns (Q5 1.4 3 108) and t0 5 98 ns (Q 5 3.5 3 108).
S. Chang et al. Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. B 1233
Hz. The FWHM of the temporal profile for the pump-laser pulse was tp ' 30 ns (see Fig. 10 for the pump-pulsetemporal profile). The pump laser was focused at a par-ticular point of the droplet rim (P1 and P2), as shown inthe insets of Fig. 10. The droplet was imaged (with 63magnification) onto a photomultiplier tube (PMT,Hamamatsu H5783), which has a response time ,0.5 ns,after passing through appropriate (color or interference)filters. The relatively large area (;1 cm2) of the PMTphotocathode was covered by a spatial filter with a pin-hole (4 mm2). Only a small part of the droplet image (D1or D2 in Fig. 10) projected on the spatial filter passedthrough the pinhole. The PMT signal was then moni-tored by a digital oscilloscope (HP 5450A, 2-GSa/s sam-pling rate). Labview (National Instrument) was used totake and store data by a computer.
For SRS from a pendant droplet a long-wavelength-pass, sharp-cut color filter at 550 nm (Corning) was usedto block the pump wavelength. The red SRS wavelengthwas at 630 nm. To measure t0 of the WGM on an equa-torial orbit, the equator ( u 5 90°) was pumped (P1) by atightly focused beam and detected at the equator (D1)through the pinhole of the spatial filter [see Fig. 10(a)].To measure t0 of tilted precessing modes, u 5 45° waspumped (P2) and detected at u 5 135° (D2). Occasion-ally, high-intensity SRS signals were observed with longdecay times ( t . 10 ns). The longest lifetimes mea-sured were t0 5 60 ns (Q 5 1.8 3 108) from the equato-rial mode and t0 5 86 ns (Q 5 2.6 3 108) from the 45°tilted precessing mode [see Fig. 10(a)]. SRS from a pen-dant droplet of liquid H2 was observed to exhibit a longdecay time for the tilted orbits, as long as 600 ns.29
The SRS was always accompanied by an intensity en-hancement of the green intensity, at or near the pumpwavelength (532 nm). The generation of SRS was some-times accompanied by a burst of sound, which lasted for afraction of a second. This could be caused by stimulatedBrillouin scattering30 (SBS) produced by the pump laser.For the decay-time measurement at ;532 nm, a bandpassinterference filter centered at 532 nm was used. There-fore we cannot distinguish the elastic scattering of thepump radiation from the SBS. The pump radiation mayor may not be on a WGM (i.e., on an input resonance), be-cause the laser linewidth is less than 0.3 cm21. The SBSmust be on a WGM to provide the necessary feedback (i.e.,on an output resonance). From the equatorial orbit, thelongest decay time measured for the green elastic scatter-ing was t0 5 31 ns (Q 5 1.1 3 108), as shown in Fig.10(b). For the tilted precessing orbit the following threedifferent decay behaviors were observed [see Fig. 10(c)]:(i) short decay time, (ii) long decay time, and (iii) a com-bination of short and long decay time. The short decaytime (t0 5 37 ns, Q 5 1.3 3 108) behavior may be causedby pump depletion that results from the generationof SRS at 630 nm. The long decay time (t0 5 68 ns,Q 5 2.4 3 108) behavior may be possible because thepump or SBS intensity was not high enough to achieveSRS. The combination of short and long decay-timebehavior may result from excitation of two differentWGM’s with different Q’s. By separating the short andlong temporal profiles into an earlier part (60–100 ns)and a later part (150–350 ns), we obtained the two decay
times: t0 5 40 ns (Q 5 1.4 3 108) and t0 5 98 ns (Q5 3.5 3 108). Q 5 3.5 3 108 corresponds to a 5 53 1024 cm21, which corresponds to the 1/e intensity droppath length of 20 m (3000 round trips at the equator).
Longer t0 (higher Q) was measured from the tilted pre-cessing mode rather than from the equatorial mode atboth 532 nm (pump or SBS) and 630 nm (SRS). Thistemporal measurement result appears at first to be theopposite of the lasing image results, i.e., the tilted pre-cessing orbits are brighter than the equatorial orbit, andhence the tilted precessing orbits with more diffractiveleakage than that of the equatorial orbit should have alower Q. We propose the following model that can recon-cile the apparent contradiction.
During the external laser excitation process, WGM’swith the tilted precessing orbits are excited with a higheraverage angle of incidence x than are WGM’s with theequatorial orbits. The focused input laser localized atthe edge of the pendant droplet can input couple withmany different angles of incidence x ’s. Because the ex-ternal angle of most incidence rays from the air into thedroplet satisfies xext ' 90°, internal rays with x > xc canbe excited through the refractive coupling (x ' xc) anddiffractive or tunneling coupling (x . xc), where x is theusual internal angle with respect to the surface normal.The external beam thus sets up an initial internal anglex0 distribution for the internal rays at the input locationwith an intensity I(x0), which is determined according tothe tunneling probability in Eq. (3). Because the dis-tance between the center of the input beam and the drop-let surface was kept the same, the functional dependenceof I(x0) on x0 at u 5 90° and u 5 45° should be the sameexcept that the overall value of I(x0) at u 5 45° is largerthan I(x0) at u 5 90° because the tunneling probabilityat a higher curvature region is larger, as shown in Eq. (3).
The time evolution of x is, however, different dependingon the launch locations. For the internal rays launchedat u 5 90°, x does not change because for a circular cav-ity, the x remains fixed at each reflection. Therefore thetime average of x for an equatorial WGM satisfies
x 5 x0 , (14)
where x0 is the initial value of x in the excited WGM.For the internal rays launched at u 5 45°, x does changeas a function of t and has the lowest value at the high-est latitude of the orbit, i.e., at the input location ofu 5 45°. Therefore the time average of x for a tilted pre-cessing WGM satisfies
x . x0 . (15)
So that the initial distribution of x0 is the same for bothexcitation locations, WGM’s excited at u 5 45° have ahigher probability of a larger x than those excited atu 5 90°. In terms of the Q value, the same externalbeam is able to launch WGM’s with a higher Q at u5 45° than WGM’s at u 5 90°. Therefore at 532 nm(pump or SBS) the excited WGM’s with tilted-precessingorbits have a lower average tunneling probability andthus a higher Q than the excited WGM’s with equatorialorbits. It is a well-established fact that higher-Q WGM’spump SRS with higher-Q WGM’s because of better spatialoverlap between the pump and the nonlinearly generated
1234 J. Opt. Soc. Am. B/Vol. 16, No. 8 /August 1999 S. Chang et al.
waves.31 Consequently, 532 nm (SBS or pump) pumpsSRS at higher-Q WGM’s for the tilted precessing orbitsthan for the equatorial orbits.
Recall that the observed higher laser output from thetilted precessing WGM’s in the chemiluminescent pen-dant droplet is caused by the higher local leakage rate atthe highest latitude of the orbit, not by the lower Q. Infact, Q, which is dependent on the averaged leakage rateover the particular trajectory [see Eq. (2)], can be higherfor the WGM’s with tilted precessing orbits because themajor part of their trajectory can have a lower leakagerate than the entire trajectory of the WGM’s with anequatorial orbit. Hence we believe that there exists aconsistency between the lasing-image measurement andthe cavity-Q determination for the tilted precessing orbitsand the equatorial orbits.
8. SUMMARYTear-shape pendant droplets containing chemilumines-cent material that chemically pumps dye molecules uni-formly over the droplet volume show color-shifted inten-sity enhancement along the droplet rim. The intensityenhancement is due to lasing, which occurs in the wave-length region where the absorption is small. The thresh-old of the intensity enhancement is determined by the ad-dition of an absorbing dye until aabs 5 2.5 cm21, whichcorresponds to a 1/e intensity decrease after a 0.7 roundtrip around the equator. Chemiluminescence emissionintensity along the rim is measured at a different wave-length. In the lasing-wavelength region the emission in-tensity increases as the latitude increases in the upperhalf of the pendant droplet. Because the molecule exci-tation rate is uniform throughout the volume of the pen-dant droplet, we conclude that the observed nonuniformlaser emission is caused by variations of the leakage rateassociated with the droplet surface curvature. The ob-served light emission is caused by diffractive leakage thatoccurs on a curved surface with the internal incidentangle x . xc . The diffractive leakage rate from a spherecan be estimated by WKB approximation. In this modelthe tunneling leakage rate is determined to be an expo-nentially decreasing function of sin x, index of refractionn, and the size parameter ka.
For an axisymmetric pendant droplet, the WGM’s withequatorial orbits have x fixed at each internal reflectionand hence a uniform leakage rate, a leak . The lasing ac-tion around the equator of the pendant droplet can bemodeled by the conventional one-dimensional four-levellaser rate equations, with distributed optical gain, cavityleakage, and dye absorption. This model indicates thatthe Iout of a lasing droplet increases with a leak providedthat a leak , aabs . The spatially uniform excitationachieved by chemical pumping enables us to isolate theeffect of surface curvature on the laser-emission distribu-tion and hence supports ongoing research on the laser-emission pattern from deformed microdroplets.10–12 Tofurther investigate the effect of the curvature, weachieved localized surface perturbations of lasing pen-dant droplets by poking the droplet with a sharp pulled-fiber tip. At the location of the poking as well as at thediagonally opposite edge, their intensities were greatly
enhanced. This result supports the exponentially in-creasing diffractive tunneling leakage rate as x → xc ,which occurs when the surface curvature increases fortilted precessing orbits. The observed spectra at the pok-ing and diagonally opposite locations show that the inten-sity enhancement results from increased leakage of lasinglight and not from the chemiluminescence.
The cavity Q of a pendant droplet made of glycerol dis-solved in water (n 5 1.431) was determined from the di-rect cavity-lifetime measurements. The lifetime wasmeasured both in the green (532 nm, elastic scattering,and SBS) and in the red (630 nm, SRS). Cavity Q valuesas high as 2.6 3 108 from the SRS and 3.5 3 108 from theelastic scattering or SBS were measured from the tiltedprecessing orbits. The measured cavity lifetime waslonger for the tilted precessing modes than for the equa-torial mode. This can be explained by the average angleof incidence x in the tilted precessing modes being largerthan that of the equatorial mode for the same internal-ray-launching condition by the external pump beam atu545° and u 5 90°. Consequently, the round-trip dif-fractive tunneling leakage loss is smaller for the tiltedprecessing orbits than for the equatorial mode.
ACKNOWLEDGMENTSWe gratefully acknowledge support of this research byNational Science Foundation grant PHY 9612200.
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