Suppression and hopping of whispering gallery modesin multiple-ring-coupled microcavity lasers
Xingwang Zhang,1 Hao Li,1 Xin Tu,1 Xiang Wu,1 Liying Liu,1 and Lei Xu1,2,3,*1Key Lab for Micro and Nanophotonic Structures (Ministry of Education), Department of Optical Science and
Engineering, School of Information Science and Engineering, Fudan University, Shanghai 200433, China2Department of Physics, Fudan University, Shanghai 200433, China
Received September 9, 2010; revised December 11, 2010; accepted December 13, 2010;posted December 15, 2010 (Doc. ID 134869); published February 24, 2011
1. INTRODUCTIONSingle-frequency microcavity lasers have received a lot ofattention due to their applications in optical communication,biochemical sensing, and spectroscopy [1]. Generally, a single-frequency operation can be realized by reducing the length of aresonator so that the free spectral range of the cavity exceedsthe spectral width of the gain medium. Other methods includethe use of distributed Bragg reflectors (DBRs) [2] and distrib-uted feedback (DFB) gratings [3]. In a coupledmicroring filter,where coupled rings with different sizes have different reso-nant frequencies, the common resonant frequency has the lar-gest transmission, and other frequencies are relatively filtered[4]. In our previous work, we also used a coupled-cavity struc-ture as an approach to obtain single-frequency whispering gal-lery mode (WGM) lasing [5]. For the single-frequency laser,side-mode suppression ability is one of the important param-eters in applications such as optical communication systems[6]. The improvement of the side-mode-suppression ratio(SMSR) has long been intensively investigated [7–10]. In gen-eral, the SMSR can be enhanced by enlarging the loss differ-ence between the main and the side modes. For instance, alarge SMSR can be achieved by modifying the wavelength de-pendence of the gain profile, as in a DFB laser or by using awavelength dependent reflector, as in a DBR laser [11]. How-ever, so far, no report has been published in improving the side-mode suppression ability of single-frequency WGM lasers.
On the other hand, larger SMSR does not necessarily meanbetter single-mode stability. Mode hopping occurs when theenvironment changes. We have reported that mode hoppingof a single-frequency coupled microcavity laser can be usedin ultrasensitive biochemical sensing [12]. But up to now,the mechanism of mode hopping in the single-mode-coupledWGM microcavity laser is still unclear.
In this paper, two issues are addressed. First, an easy meth-od is employed to enhance the side-mode suppression abilityof the coupled microcavity laser. We evaluate the side-
mode suppression ability by two parameters, the SMSR(¼ 10log10ðPm=PsÞ, where Pm is the lasing power of the mainmode and Ps is the laser power of the side modes) and theratio between the threshold of the side and the main modes(¼ Isth=Ith). We will show that by coupling multiple rings withdifferent diameters to form a coupled cavity, Isth=Ith can beimproved to 2.5 and the SMSR can be enhanced to 28dB. Sec-ond, mechanisms of single-mode lasing hopping were investi-gated. We found that in a two-ring-coupled cavity, when thetemperature changes gradually, both the envelope of themodulated gain spectrum and the mode resonant wavelengthshift, but with a different shifting speed; as a result, the singlelaser mode hops in steps of the mode spacing one by oneregularly.
2. THEORETICAL FRAMEWORKWe developed a cold cavity model to qualitatively describe theperformance of a ring-coupled microcavity. Figure 1 is theschematic diagram of a two-ring-coupled cavity. Take cavity1 as the master cavity, and cavity 2 as the slave cavity; Einc andEref denote the incident and reflected field amplitudes incavity 1 before and after the coupling. We have
Eref ¼ rEinc þ κ�g2Ecirc; ð1Þ
where Ecirc denotes the field amplitude inside cavity 2 aftercoupling, and it satisfies
Ecirc ¼ κEinc þ r�g2Ecirc; ð2Þin which
κ ¼ κ0 expðjψÞ; ð3Þ
r ¼ffiffiffiffiffiffiffiffiffiffiffiffi1 − κ20
qexpðjφÞ ð4Þ
Zhang et al. Vol. 28, No. 3 / March 2011 / J. Opt. Soc. Am. B 483
are the coupling coefficient and reflectivity, respectively, and
g2 ¼ expð−γL2 − jnkL2Þ ð5Þ
is the amplitude and phase change of Ecirc after going one cir-cle around cavity 2. Here γ is the loss (or gain) coefficient, n isthe refractive index of the cavity, and L2 is the round-triplength of the cavity.
The slave cavity (cavity 2) can be replaced by an effectivereflectivity Reff2 [13–15]:
Reff2 ¼Eref
Einc¼ r − 2jrj2g2 þ g2
1 − r�g2: ð6Þ
Assuming that the initial field intensity in the cavity is 1, theself-consistent field in the master cavity is modulated by Reff2
and can be written as
E ¼ E1 ¼1
1 − Reff2g1: ð7Þ
The same process above can be used to deduce the field incavity 2 by taking cavity 2 as the master cavity and cavity1 as the slave cavity; we have
E ¼ E2 ¼1
1 − Reff1g2; ð8Þ
where
Reff1 ¼r� − 2jrj2g1 þ g1
1 − rg1ð9Þ
is the effective reflectivity of cavity 1.Thus, the total observable field in the coupled cavity is
E ¼ E1 þ E2 ¼1
1 − Reff2g1þ 11 − Reff1g2
: ð10Þ
In the process of derivation, the phase ψ in Eq. (3) is canceledout and the phase φ in Eq. (4) can be phenomenologically trea-ted as
φ ¼ Cnkl; ð11Þwhere C is a factor for phenomenological modeling of the ad-ditional phase shift, k ¼ 2π=λ0, n is the refractive index of thecavity, and l is the coupling length. Both n and l are tempera-ture dependent and can be written as
n ¼ n0 þ α ×ΔT; ð12Þ
l ¼ l0ð1þ β ×ΔTÞ; ð13Þ
where α is the thermo-optic coefficient and β is the linear ther-mal-expansion coefficient.
Similarly, the field intensity of three- and four-ring-coupledcavities can be obtained following the same procedure. For
simplification, we neglect the difference of the coupling coef-ficients in the coupling areas.
For a three-ring-coupled cavity, the three cavities can firstbe transformed to an effective two-ring cavity and then trans-formed again to one cavity. The self-consistent field intensityE in the three-ring-coupled cavity is
E ¼ E1 þ E2 þ E3
¼ 11 − Reff23g1
þ 11 − Reff1Reff3g2
þ 11 − Reff21g3
; ð14Þ
in which
Reff23 ¼r − 2jrj2Reff3g2 þ Reff3g2
1 − r�Reff3g2; ð15Þ
Reff3 ¼r − 2jrj2g3 þ g3
1 − r�g3; ð16Þ
Reff21 ¼r� − 2jrj2Reff1g2 þ Reff1g2
1 − rReff1g2: ð17Þ
And the self-consistent field intensity in the four-ring-coupledcavity is
E ¼ E1 þ E2 þ E3 þ E4
¼ 11 − Reff234g1
þ 11 − Reff1Reff34g2
þ 11 − Reff21Reff4g3
þ 11 − Reff321g4
; ð18Þ
where
Reff234 ¼r − 2jrj2Reff34g2 þ Reff34g2
1 − r�Reff34g2; ð19Þ
Reff34 ¼r − 2jrj2Reff4g3 þ Reff4g3
1 − r�Reff4g3; ð20Þ
Reff4 ¼r − 2jrj2g4 þ g4
1 − r�g4; ð21Þ
Reff321 ¼r� − 2jrj2Reff21g3 þ Reff21g3
1 − rReff21g3: ð22Þ
3. EXPERIMENTThe fabrication process of coupled microcavity lasers is basi-cally the same as in our earlier works [5], and it is schemati-cally shown in Fig. 2(a). First, commercial glass fibers werewet-etched to different sizes and then coated with a thin film(refractive index n ≈ 1:52, thickness d ≈ 2 μm). The coatingmaterials were Rhodamine-B-doped organic/inorganic hybridglass materials. Thus, in a plane perpendicular to the fiber, thethin film on the fiber forms a circular ring cavity. As WGMspropagate along the inner surface of the film, the diameter
Fig. 1. Schematic diagram for theoretical treatment of a two-ring-coupled cavity.
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of the cavity equals roughly the external diameter of the ring.To fabricate the coupled microcavity laser, fibers that variedin diameter with a thin coating film were placed in parallel andmoved toward each other until they stacked together. Thestacked fibers were then put into an oven to consolidate thecoatings. In this experiment, we fabricated four- (125=105=95=85 μm, sample A), three- (125=115=95 μm, sample B), and two-ring-coupled (125=115 μm, sample C) microcavity lasers. Theside views of the coupled microcavity lasers are shown inFig. 2(b).
The coupled microcavity lasers were pumped by a 532 nmmode-locked laser (30ps pulse width, 10Hz repetition rate).The emitted laser light was collected by a multimode fiber(1mm in diameter) and was transmitted to a spectrometer.In the mode-hopping experiment, the 125=115 μm two-ring-coupled microcavity laser was immersed in a beaker thatwas full of water. The temperature of the water was controlledby a hot plate with feedback from a thermal coupler immersedin the water. The control accuracy is about �0:2 °C. As thecoating materials have a negative thermo-optical coefficientand a positive thermal-expansion coefficient, both the refrac-tive index of the microcavity and the coupling length changewhen the temperature of water varies.
4. RESULTS & DISCUSSIONA. Side-Mode Suppression Ability EnhancementThe resonant condition for a WGM is neffL ¼ qλ, where neff isthe effective refractive index of the resonant mode, L is theround-trip length of the cavity, and q is an integer which in-dicates the order of the resonant mode. In a two-ring-coupledcavity, as the size of each cavity is different, their resonantwavelengths are different. The common resonant mode (both
of the rings are on resonant) has a higher gain, and the sidemodes are relatively suppressed. In a multiple-ring-coupledcavity, all the cavities coupled together have different modespacing. The common mode has a much lower lasing thresh-old compared with the two-ring cavity because the total gainlength at the common mode is much longer. Meanwhile theside-mode resonance still depends on the individual ring reso-nant condition but with a higher threshold, because each cav-ity couples to other cavities and energy dissipates at a higherrate. Therefore the side-mode suppression ability is enhanced.
Figure 3 shows the relation between the SMSR and relativepump energy density (Ipump=Ith) of the two-, three-, and four-ring-coupled microcavity lasers. The corresponding emissionspectra under different pump energy densities are shown inFig. 4. When the pump energy density reaches the lasingthreshold (i.e., Ith), a single-frequency lasing emerges. Asthe pump energy density increases, the SMSR becomes larger.Once the pump energy density exceeds the threshold of theside-mode lasing (i.e., Isth), side modes appear (see the insetsof Fig. 4), and, subsequently, the SMSR drops immediately.Obviously, compared to the two-ring-coupled cavity, three-and four-ring-coupled cavities have a larger Isth=Ith becausemore ring numbers lead to less Ith and higher Isth. In Fig. 3,the ratio Isth=Ith of the two-ring-coupled cavity is about 1.3,the three-ring-coupled cavity is 1.77, and the four-ring-coupledcavity is up to 2.53. Therefore, adding more rings to thecoupled cavity can better suppress the side-mode resonances.
The SMSR reaches the maximum when the pump energydensity is just below Isth. For the three-ring-coupled cavity,the maximum SMSR is 25:4 dB, and for the four-ring-coupledcavity, the maximal SMSR is 28:6dB; they are much higherthan that of the two-ring-coupled cavity (18:9 dB). Further-more, for the four-ring-coupled cavity, the SMSR is above20dB in a much larger pump energy range.
The larger SMSR comes from better suppression of the sidemodes lasing in multiple-ring cavities. We can use the theoryabove to give a semiquantitative description.
IfΔT ¼ 0, then φ is a constant (recall that φ is the couplingphase and φ ¼ Cn0kl0). Take l0 ¼ 30 μm, n0 ¼ 1:52, λ0 ¼0:6 μm, and assume C ¼ 1 for simplification. Figure 5 showsthe simulated spectra with γ ¼ 1:8 × 10−3 μm−1 and κ ¼ 0:6.Apparently, the simulated intensity spectra have clear wave-length modulations. Although the simulation is on cold cav-ities, it still provides useful information when laser cavities
Fig. 2. (Color online) (a) Fabrication process of coupled microcavitylasers. (b) Microscopic photos (side view) of four- (sample A), three-(sample B), and two-ring (sample C) cavities.
Fig. 3. (Color online) Plots of the SMSR versus relative pump energydensity (Ipump=Ith).
Zhang et al. Vol. 28, No. 3 / March 2011 / J. Opt. Soc. Am. B 485
are concerned: the wavelength modulation in a cold cavity in-tensity spectrum represents threshold modulation in a lasercavity. In other words, gain is now spectrally modulated. Ad-justing the pump level, we can select from the modulated gaineither one mode or multiple modes to lase. When the pumppower reaches Ith, single-mode lasing occurs at the mode thatis closest to the modulation envelope center where the gain isthe highest. As Fig. 5 shows, three- and four-ring-coupled cav-ities have a larger modulation depth (i.e., a larger difference in
intensity between the common and side modes)—as a result,Isth=Ith is higher. The simulated results qualitatively supportthe experimental findings.
B. Mode HoppingWhen a laser operates in single-mode resonance, instability ofthe gain spectral modulation will lead to mode hopping. Forexample, in an external Fabry–Perot cavity laser, mode hop-ping occurs as a result of the cavity length variations [16,17].In a single-mode WGM coupled microcavity laser, the singlelasing mode will also hop if the coupling coefficient or cou-pling phase changes [12]. Mode hopping must be suppressedto achieve better single-mode stability. On the other hand,mode hopping provides a scheme for sensitive sensing [12,18].Thus mode hopping has both positive and negative effects.For applications based on either side of the effect, a full under-standing of the mechanism of mode hopping in a WGMcoupled microcavity laser is important.
In a coupled WGM microcavity, the resonant spectrum ismodulated. Each resonant mode wavelength is determinedby neffL ¼ qλ, and the modulation envelope center relies onthe coupling phase φ. Because both neff and φ depend on therefractive index of the cavity material or the coupling length,an environmentally (for example, temperature) induced re-fractive index change and the coupling length change will leadto a φ and neff change. As a consequence, the resonant modeposition and modulation envelope center will shift. Obviously,if the envelope center moves with a different speed than thespeed at which the resonant modes move, the resonant modethat originally locates closest to the envelope center will moveaway, and the adjacent resonant mode will become the onethat is closest to the center [19]. In single-mode operation, thismeans that a single mode hops to the adjacent mode.
Experimentally, we changed neff and φ by varying the tem-perature.When cavitieswere pumped at intensitiesmuch high-er than Isth, clear spectrum modulation can be observed [seeFig. 6(a)]. When we raise the temperature gradually, the mod-ulation envelope center moves to the blue side. Meanwhile,each resonant peak is also blueshifting. Figure 6(b) plots theblueshifts of the envelope center and the resonant peak as afunction of temperature. Note that the two lines have differentslopes. The slope of the modulation envelope shift is around
Fig. 4. (Color online) Emission spectra from (a) two-, (b) three-, and(c) four-ring-coupled microcavity lasers. The three spectra in eachplot represent the emission when the pump light is around the laserthreshold (bottom), just below Isth (middle), and above Isth (top). Theinsets in the upper graphs are the enlarged spectra that show theemergence of side modes. The highest SMSRs are 18.9, 25.4, and28:6 dB for the two-, three-, and four-ring cavities.
Fig. 5. (Color online) Calculated spectra of (a) single-ring laser and(b) two-, (c) three-, and (d) four-ring-coupled microcavity lasers.When more rings are added, the single-mode performance becomesclearer.
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0:254nm=°C, which is much higher than that of the laser modeshift (¼ 0:032 nm=°C). This means that the envelope centermoves much faster than the resonant peak does. As a result,in the single-mode lasing scenario, we can see clearly in Fig. 7that the single mode hops to a shorter wavelength. As the tem-perature increases continuously, the above process repeats.Thus, the laser mode hops in the steps of the mode spacingone by one to a shorter wavelength.
We also simulated themovement of spectrummodulation byusing the theory thatwe described above. In the simulation, thethermo-optic coefficient α ¼ −2:0 × 10−4 °C−1 [20] and the line-ar thermal-expansion coefficient β ¼ 3:0 × 10−4 °C−1 [20] of thecavitymaterial were used; all the other parameters in the simu-lation were the same as in Fig. 5. The simulated spectrum isshown in Fig. 8(a). When the temperature increases, boththemode resonantwavelength and the spectralmodulation en-velope center shift to a shorter wavelength. Figure 8(b) plotsthe change of the modulation envelope center versus tempera-ture; the slope of the shift is v2 ¼ 0:275nm=°C, which agreeswith our experimental observations. On the other hand, thelaser mode shifts at a rate of v1 ∼ λ0α=n0 ∼ 0:078nm=°C. InFig. 8(b),wealsoplot the calculated shift of the resonancepeakas a function of temperature. Obviously v2 is much larger thanv1, and the singlemode hops accordingly. Note that v2 is almost1 order of magnitude higher than v1, so it is possible to use theenvelope shift as a sensitive sensing scheme.
5. CONCLUSIONIn conclusion, we demonstrated that the side-mode suppres-sion ability of WGM coupled microcavity lasers can be im-proved by adding more coupling size mismatched multiplerings to form the cavity. We also investigated the mechanismof mode hopping of a two-ring-coupled cavity both in experi-ment and theory, and we concluded that hopping comes fromthe mismatch between the shifting speed of the laser mode
Fig. 6. (Color online) (a) The modulation envelope shifts as the tem-perature changes. (b) Plot of modulation envelope and mode resonantposition shifts versus temperature. Dots are experimental data, andthe lines are linear fittings.
Fig. 7. (Color online) Laser mode hops in steps of the mode spacingas the temperature changes.
Fig. 8. (Color online) (a) Simulated modulation spectra as the tem-perature changes. (b) Plot of modulation envelope and mode resonantposition shifts versus temperature.
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and that of the threshold envelope, which results from the var-iation of the additional phase shift in the coupling area. Thehopping may give a new sensing scheme that can reach amuch higher sensitivity.
ACKNOWLEDGMENTSThis work is supported in part by the National Natural ScienceFoundation of China (NSFC) (grants 60638010, 10874033,60907011, 61078052, 11074051), the Shanghai Commissionof Science and Technology (SCST) (grant 08XD14006), andthe National Basic Research Program of China (973 Program)under grant 2011CB921802.
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