Harmonic mode locking and multiple pulsing in a soft-aperture Kerr-lens mode-locked Ti : sapphire laser

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Harmonic mode locking and multiple pulsing in a soft-aperture

Kerr-lens mode-locked Ti:sapphire laser

Ja-Hon Lin

a

, Wen-Feng Hsieh

a,*

, Hsiao-Hua Wu

b

a_{Institute of Electro-Optical Engineering, National Chiao-Tung University, 1001 Ta-Hsueh Rd., Hsinchu 300, Taiwan} b_{Department of Physics, Tunghai University, 181 Sec.3 Chung Kang Road, Taichung 407, Taiwan}

Received 11 April 2002; received in revised form 11 April 2002; accepted 20 August 2002

Abstract

We report the observation of harmonic mode locking and multiple pulse operations in a soft-aperture Kerr-lens mode-locked Ti:sapphire laser by varying the total intra-cavity dispersion. Second, third, and fourth order harmonic mode locking as well as multiple pulsing with interpulse separation ranging from femtosecond to nanosecond are observed. The laser is characterized in these regimes in terms of wavelength, bandwidth, pulsewidth, average and peak powers as functions of group velocity dispersion. By introducing a loss diﬀerence term into gain dynamic analysis, we conclude that gain depletion and recovery mechanisms are responsible for the underlying physics of current observa-tions.

1. Introduction

Resulting from the soliton-like pulse shaping eﬀect via the interplay between the self phase modulation (SPM) and the group velocity disper-sion (GVD), generation of ultra-short pulses have been achieved in Kerr-lens mode-locked (KLM) Ti:sapphire lasers [1–4]. Because of extremely high nonlinear eﬀects, it is not surprising that the pulse-splitting and multiple pulse operation can be ob-served in these laser systems besides the normal

mode locking (NML). A single pulse per round trip splits into two or three pulses with fs spacing had been observed in a soft-aperture KLM (SAKLM) Ti:sapphire laser [2]. The pulse-splitting is attributed to side-lobe pulse separation by the interplay of the negative GVD and SPM in the laser cavity.

In a hard-aperture KLM (HAKLM) Ti:sap-phire laser, other than the fs-split pulses, multiple pulse operations with asymmetric ns-splitting were also observed [3]. It is explained by the enhanced nonlinear interaction caused by two counter-propagating pulses coinciding within the active medium to stabilize the interpulse spacing equal to the round trip time between the active medium and one of the output couplers. With an ion-implanted semiconductor as the saturable-absorber mirror

Optics Communications 212 (2002) 149–158

www.elsevier.com/locate/optcom

*_{Corresponding author. Tel.: 5712121; fax:}

+886-3-5716631.

E-mail addresses:[email protected](J.-H. Lin), [email protected] (W.-F. Hsieh).

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[4], double to quadruple pulses were observed in a passive mode-locked Ti:sapphire laser. The inter-pulse spacing was obtained ranging from several hundred fs to several ns without reliably repro-ducible spacing [4] when the GVD was decreased below a critical value. The results were directly compared with the simulation from the Ginzburg– Landau equation.

Recently, harmonic mode locking generated in the passive mode-locked fiber ring lasers were re-ported and attributed to soliton interaction by means of acoustic effects [5–7]. Harmonic opera-tion of a passively mode-locked short-cavity Cr4þ_{:YAG laser with one, two, and three pulses in} the cavity has been demonstrated [8] for various values of total cavity GVD. By increasing the in-tracavity power of a passively harmonic mode-locked erbium/ytterbium fiber laser with short cavity, up to 11 pulses in the cavity with nearly equal pulse-to-pulse temporal spacing has also been observed [9]. An analytic model based on the interaction between the pulses through the tran-sient gain depletion and recovery dynamics in the gain medium [10] was proposed to explain these phenomena. Similar gain dynamics had already been successfully applied to explain harmonic mode locking in an external cavity semiconductor laser by adjusting the injection current [11]. It contains two-section multiple quantum well structures of which one providing gain and the other acting as the slow saturable absorber.

We report for the first time to our knowledge the observation of second, third, and fourth order harmonic mode locking in a SAKLM Ti:sapphire laser without saturable absorber. Multiple pulsing with interpulse separation ranging from femto-second to nanofemto-second was also observed. The mode-locked states are distinguished into three regions and characterized in terms of wavelength, time bandwidth product (TBP), pulse width, average and peak powers as function of group velocity dispersion. By introducing a loss differ-ence of the split pulses into theoretical analysis of gain depletion and recovery model [10], we find that multiple pulses with unequal spacing between the pulses can take place in addition to the har-monic mode locking which occurs when the loss difference is neglected.

2. Experiment

The experiment was carried out by a SAKLM Ti:sapphire laser with a near symmetric cavity as shown in Fig. 1. The SAKLM Ti:sapphire laser consists of a 9-mm long Brewster-cut Ti:sapphire rod (Ti:0.1%, FOM>150), two curved mirrors (M1 and M2) of 10-cm radius of curvature tilted by 10° angle, and two flat mirrors – a 99% high reflector M4 and a 95% output coupler M3. Both the dis-tances between M2 and M4 as well as M1and M3 are adjusted around 74 cm to form a near sym-metric configuration. A pair of Brewster-angled SF10 prisms separated by 27 cm is inserted be-tween M1 and M3 for dispersion compensation that allows for femtosecond pulses generation. The prism pair is mounted on the translation stage to finely adjust negative GVD. The negative GVD (b2z) can be calculated by the following expression [12]: b₂z¼ 2k 3 pc2 dn dk 2 Lpþ k3 pc2 d2n dk2 " þ 2n 1 n3 dn dk 2# 4 sinðe=2Þ ½1 n2_sin2_ðe=2Þ1=2 dn dkDkLp þ k 3 pc2 d2n dk2 tanðe=2Þðdp1þ dp2Þ;

where z is single path length of the pulse propa-gation in cavity, n and e are the refractive index and the apex angle of the prisms, c is the speed of light in vacuum, k is the center wavelength, Dk is the optical-spectrum bandwidth, Lp is the prism

Fig. 1. Experimental setup of the Kerr-lens mode-locking Ti:sapphire laser. B1; B2are beam splitter and D1; D2are the

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separation, and dp1 as well as dp2 are the insertion width of prisms P1 and P2, respectively. Thus the total intra-cavity GVD (b00) could be estimated by taking the positive dispersion of the crystal into account. The total cavity length is approximate 160 cm that corresponds to the repetition rate of 93.3 MHz or round trip time of 10.7 ns. An all-line continuous wave (CW) Ar ion laser (Coherent, Innova 300) was employed to pump the Ti:sap-phire crystal via a plano-convex lens with the focal length of 12.7 cm. Two fast photodiodes (Electro-Physics Technology ET-2000 with rise time 100 ps) together with an oscilloscope (LeCroy 94504A, bandwidth 300 MHz) and an rf-spectrum ana-lyzer (HP-8560E) are employed to observe the pulse train and its corresponding power-spectrum. A collinear autocorrelator combined with a GaAsP photodiode (Hamamtsu G117) acting as two-photon absorption detector [13] is con-structed to achieve interferometric autocorrelation measurement of pulse waveforms. Finally, the spectra were measured by using an optical-spectrum analyzer (Ophir wavestar).

3. Results and discussion

3.1. Regimes for normal mode locking, harmonic mode locking, and multiple pulsing

As our previous experimental results [14], the SAKLM operation usually occurs at the fractional degenerate cavity configurations [15] with peculiar mode patterns consisting of superposition of the fundamental mode with low-order degenerate transverse modes. In this experiment, we trans-lated the curved mirror M2along the optic axis of cavity to search for pertinent cavity configuration so that the SAKLM operation was achieved. Throughout the experiment, the pumping power and the insertion width of the first prism P1 into laser beam were kept to the value of 4.5 W and 0.37 mm. Subsequently, we moved the translation stage to change the insertion width of the second prism P2 into laser beam in order to change the negative GVD. Besides the NML, the laser could also be operated with second, third, and fourth order harmonic mode locking as well as stable

multiple pulsing whose interpulse separation ran-ges from femtosecond to nanosecond.

On the basis of our observation, we divided mode-locked operation into three regions as a function of the total intra-cavity GVD (b00) that is shown later in Figs. 6 and 7. In region I, b00 is less than )2250 fs2 _{where only the NML is observed.} Further increasing b00 to the region II ()2250 fs2 6_b006_{1950 fs}2_{), the system exhibits two} pos-sible states either in the NML or in second har-monic mode locking (SHML) whose repetition period becomes one-half of the cavity round trip time. Occasionally, with a certain b00in this region, unstable pulse-splitting with ps spacing is found that would slowly turn into the SHML. In region III of which b00>1950 fs2

, multiple pulses with unequal nanosecond interpulse spacing were ob-served and sometimes third harmonic mode lock-ing (THML) or even fourth harmonic mode locking (FHML) pulses could be observed at the far limit of this region. Mostly, multiple pulses with unequal nanosecond interpulse spacing were found immediately as laser operated in this region. However, the NML may also be found initially and maintained in few-seconds to minutes and eventually developed into stable multiple pulse operation. Therefore, fewer data of the NML can be recorded in region III of Figs. 6 and 7 before they developed into pulse-splitting. We can also block the laser to destroy the mode locking at multiple pulsing and initial the mode locking again for observing the NML.

In Fig. 2, we show a typical interference auto-correlation trace (solid line) for the NML mea-sured at b00 2768 fs2 about 60 fs pulsewidth that is fitted with the sech2ð1:76t=t0Þ curve (dot line). The inset is the corresponding optical-spec-trum with center wavelength at 825 nm and bandwidth of 12.4 nm by fitting with a sech2 profile (bold line). The autocorrelator can also be employed to resolve subpicosecond pulse-splitting. In Figs. 3(a) and (b), the autocorrelation trace exhibits pulse-to-pulse separation of 210 and 475 fs, respectively. As mentioned before [16,17], the phase difference between two pulses can be gained from the optical-spectrum with the relation u¼ 2pðx0 x1Þ=ðx2 x1Þ. Here x0is the central frequency of the fitting sech2_{curve and x}

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are the closest frequency to x0. It reveals that the relative phases of the split pulses in Fig. 3(a) and (b) are p and p=2, respectively. Under diﬀerent

operation conditions, the change of relative phase of pulses around p and 0 between pulses was also seen.

We have observed multiple pulses with the in-terpulse separation larger than 10 ps which could not be observed by our autocorrelator but could be displayed by the oscilloscope. The double pulses in Fig. 4(a) shows split spacing of 2.6 ns with its corresponding rf-spectrum in Fig. 4(b). Similarly, we observed stable three asymmetric ns-spacing pulses as shown in Fig. 4(c) whose rf-spectrum is in Fig. 4(d). In addition, the four pulses having pulse spacing 1.8, 1.3, and 2.6 ns followed by a 5-ns blanking are shown in Fig. 4(e) with correspond-ing spectrum in Fig. 4(f). It is worth to mention that the relative spacing and amplitude for multi-ple pulses could be altered at the diﬀerent total intra-cavity GVD while we change the insertion length of the second prism.

The pulse trains together with their corre-sponding rf-spectra for the NML, SHML, THML, and FHML are shown in Fig. 5, respectively. The period of pulse train for the NML shown in Fig. 5(a) is equal to cavity round trip time with four harmonics of longitudinal beating in correspond-ing rf-spectrum [see Fig. 5(b)]. In Fig. 5(c), the SHML state has two pulses within a round trip in which its spectrum shows the first and third har-monics being suppressed below the noise level ()70 dB) in Fig. 5(d) to verify that the pulse train is highly periodic. Similarly, highly periodic and suppressing of the first, second and fourth har-monics below the noise level in Figs. 5(e)–(f) dis-plays the laser is at the THML state. The FHML appears in the system with only the fourth har-monic as shown in Fig. 5(h). However, due to the limited bandwidth of our 300-MHz oscilloscope which is smaller than the 373-MHz repetition rate of the FHML state, large pulse-to-pulse fluctua-tion is recorded in Fig. 5(g).

The variations of the measured average output power (solid symbols), pulsewidth (open symbols), and the intra-cavity peak power versus the total intra-cavity GVD ðb00Þ for the NML, SHML and double pulses are shown in Figs. 6(a) and (b). In Fig. 6(a) the average power (solid squares) and pulsewidth (open squares) of the NML are about 271 mW and 84 fs, respectively, at the largestjb00j

Fig. 2. Typical autocorrelation trace (solid curve) and corre-sponding optical-spectrum (inset) for the NML. The dash curve is ﬁtted with the sech2_function.

Fig. 3. Autocorrelation trace and corresponding optical-spec-trum (inset) for double pulses with 210 separation as well as the relative phase p between two pulses (a), and 475 fs splitting in conjunction with p=2 phase diﬀerence in (b).

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and then suddenly decrease to around 200 mW and 55 fs at b00 2760 fs2in region I. Pulsewidth decrease accompanied with intra-cavity peak power increase [shown as solid squares in Fig. 6(b)] is due to better GVD compensation by the prism pair. In addition, the operation wavelength is continuously red shifted to the maximum ex-cursion at 827 nm where the time bandwidth product (TBP) is near the transform-limit of 0.318 for b00 2760 fs2

as shown in Fig. 7. Then the average power of the NML keeps almost constant until region III with a relatively small deviation of 10 mW or 5% and the pulsewidth is approximately 53 fs.

Since the 5.35 ns-separated pulses in region II can extract more energy from the active medium in a round trip, the SHML pulse (solid circle in Fig. 6(a)) shows higher average output power about 310 mW with 62 fs pulsewidth at b00₂¼ 2220 fs2. However, the fact of two pulses in one round trip

together with the broader pulsewidth for the SHML in comparison with the NML causes the pulses at the SHML (solid circle) in Fig. 6(b) having smaller intra-cavity peak power than that at the NML (solid square). Thus the peak power of single pulse at the SHML declines to 0.54 MW (almost a half of the NML one). Generally, pulses at the NML state have a little higher value of the TBP near 0.39 and recover at the state of the SHML as shown in Fig. 7.

In region III of Figs. 6 and 7, two kinds of double pulses are found in our laser system and labeled with triangle and diamond symbols. One (triangle symbol) shows near equal spacing be-tween pulses on the oscilloscope like that observe in the SHML. However, its odd number of longi-tudinal beatings on rf-spectrum are still above the noise level ()70 dB) revealing a little unequal in-terpulse spacing between pulses. The others with asymmetric nanosecond splitting spacing between

Fig. 4. Pulse trains for double (a), triple (c), and quadruple (e) pulses with asymmetric nanosecond separations and their corresponding rf-spectra (b), (d), and (f).

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pulses are labeled diamond symbol whose time traces are shown in Fig. 4(a). Double pulses in region III of Fig. 6(a) reveals a little shorter pulsewidth than the SHML pulses in region II, but their bandwidth become a little broader to cause the value of the TBP slightly larger than the SHML in Fig. 7. In addition, the average power of double pulses in Fig. 6(a) tends to drop as jb00j declining in region III to cause the peak power fall in Fig. 6(b). The operation wavelength of the pulses gradually decreases throughout the region II and III in Fig. 7 and the lasing wavelength at either the state of the SHML or double pulses is longer than that at the NML state.

3.2. Explanations for harmonic mode locking and multiple pulsing

Pulse-splitting with regular interspacing or harmonic mode locking has been observed in passive mode-locked fiber laser that was explained by the acoustic effect created by the electrostriction in fiber [5–7]. The refractive index change caused by the acoustic effect for a 9-mm Ti:sapphire crystal is too small to cause harmonic mode locking in such a laser. We therefore try to use the transient depletion and recovery of gain dynamics [10] which has been successfully applied to the passive harmonic mode-locked semiconductor

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[11], Cr4þ_{:YAG [8], and Er/Yb ﬁber lasers [9] with} saturable absorber to explain the observed har-monic mode-locked phenomena.

As the pulse goes through a gain medium, the gain saturation induced group velocity drift may drive the pulse toward the higher gain [10]. As-suming, initially, two pulses per round trip with unequal separation, one of pulses with larger de-pleted gain will experience larger group-velocity drift relative to the other pulse. After repeating this process in the cavity, two pulses will become equal spacing in one round trip time if the gain recovery time is longer than the cavity round-trip time and the depleted gain is much smaller than the total net gain [10]. Similar analysis can also be applied for N (N > 2) pulses to result in equal in-terpulse spacing among a cavity round trip period or termed Nth-harmonic mode locking. Since the recovery time of the Ti:sapphire about 2.5 ls is much longer than cavity round trip time of 10.6 ns, the observed harmonic mode locking in our ex-periment can be explained by the aforementioned gain dynamics.

Two, three, and four pulses with asymmetric ns-spacing were observed similar to that reported in the HAKLM Ti:sapphire laser [6] in which two pulses with temporal separation equal to the round trip time from the laser rod to the output coupler and back will collide in the Kerr medium. Because the two linear arms of our laser system are near equal, if aforementioned pulse colliding mechanism does play the role, the SHML should always be observed and ns-split pulse will not be seen. Furthermore, for the THML, three pulses traveling around the cavity are separated by 1/3 of round trip time ðT0Þ. Assume that one of the pulses is located at the Kerr medium at some moment, the other two pulses will be located ahead and behind of 1=3T0, they will never collide one another in the active medium. In most cases the interpulse spacing is arbitrary such as that shown in Fig. 4(a) where interpulse spacing is diﬀerent from the SHML of 5.4 ns or the THML of 3.6 ns. These phenomena observed in our sys-tem cannot be explained in terms of pulse colliding within the active medium, and therefore some other mechanisms should be considered in order to explain our results.

Fig. 6. The average output power (solid symbols) as well as pulsewidth (open symbols) (a) and the intra-cavity peak power; (b) versus the total intra-cavity GVD (or b00) for the NML (square), SHML (circle) and pulse-splitting states (triangle and diamond).

Fig. 7. The wavelength (open symbols) and time bandwidth product (solid symbols) as functions of the total intra-cavity GVD for the NML (square), SHML (circle), and pulse-splitting states (triangle and diamond).

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The wavelength dependent gain profile [18] of Ti:sapphire crystal and reflectance of curved mir-rors are shown in Fig. 8. We can see that the maximum value of Ti:sapphire gain profile [18] is located around 800 nm and the high reflection coating of our mirrors has a bandwidth from 780 to 880 nm. The pulses experience the higher in-stantaneous gain in region II where the operation wavelength is around 802–810 nm. Thus, the dy-namic gain introduces enough repulsive force (group velocity drift) to push the ps-split pulses to equal interpulse spacing and form the SHML in region II. However, for multiple pulsing in region III, the operation wavelength of the laser is less than 802 nm (Fig. 7) so that the loss increases due to abruptly decrease of the reflectance of the mir-rors (Fig. 8). This is consistent with experimental observation as shown in region III of Fig. 6 where the average powers of pulses reduce. Assuming two ps-split pulses with slightly different peak in-tensities, the stronger pulse will experience larger SPM and higher loss than the other.

By introducing an excess loss of the individual pulse into the instantaneous gain to modify the theoretical analysis of gain depletion and recovery model as described in [10], we are able to obtain analytical solution of multiple pulses with unequal spacing occurring in a cavity. The detailed deri-vation is shown in Appendix A. The diﬀerence of interpulse separations in Eq. (A.8) displays an additional term that depends on the loss diﬀerence

between pulses as compared with [10, Eq. (12)], where it always converges to zero without the loss difference between pulses. In our system, the round trip loss without considering the excess loss in-troduced by finite bandwidth of the mirrors is about 3% such that the instantaneous gain gi is 0.321 in order to balance the loss. For the mea-sured interpulse separations 2.6 and 8.1 ns within a round trip, as shown in Fig. 4(a), which corre-sponds to 5.4 ns difference of interpulse separa-tions, we calculated from Eq. (A.8) the loss difference between two pulses is only 1%. This value is quite reasonable for the center wavelength below 795 nm where the mirror reflectance shows strong wavelength dependent (see Fig. 8) to result in ns-pulse-splitting as shown by diamond symbol in Fig. 7. Similar analysis can also be applied for N (N > 2) pulses per round trip with equal interpulse spacing to result in the Nth-harmonic mode lock-ing or multiple pulses with unequal separation.

4. Conclusion

We have observed multiple pulses per round trip occurring in a soft-aperture Kerr-lens mode-locked Ti:sapphire laser. By varying the total in-tra-cavity dispersion, equally spaced harmonic operation and unequally spaced pulses with sepa-ration ranging from femtosecond to nanosecond can be found. The harmonic mode locking is ob-served for the first time to our knowledge without adding saturable absorber that could be reason-ably explained by the gain dynamics. An investi-gation of the laser operated in these regimes in terms of wavelength, bandwidth, pulsewidth, average and peak powers as functions of group velocity dispersion shows that the operating wavelength of unequally spaced pulses is shorter than that of equally spaced pulses. In addition, the information of gain profile for amplifying medium and wavelength-dependent reflectance for mirrors suggest pulses with different wavelength as well as bandwidth may undergo different gain and loss. We then introduce a loss difference of the pulses into gain dynamic analysis and find that the steady state solution of multiple pulses with unequal spacing between the pulses exist in addition to

Fig. 8. The gain proﬁle of Ti:sapphire crystal (solid line) and reﬂectance of curve mirrors (dashed dot line).

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harmonic mode locking. The solution expects that only 1% loss diﬀerence is suﬃciently to cause 2.6 ns of pulse-splitting.

Acknowledgements

This work is partially supported by the Na-tional Science Council (NSC) and the Ministry of Education of the Republic of China (R.O.C.) un-der Grants NSC90-2112-009-050, NSC90-2215-E-029-001 and 90-E-6-FA-1-4. Mr. Ja-Hon Lin would like to thank NSC for providing fellowship and Mr. Zhi-Cheng Zhan for experimental help.

Appendix A. Gain dynamics including the loss diﬀerence

We consider two pulses in a laser cavity with a fixed round-trip time of T0. The interpulse sepa-ration of the first pulse to the second pulse is T1 and that of the second to the first is T2 (note that T1þ T2¼ T0). The strength of the differential gain across the first pulse is G1 and across the second pulse is G2. By following the derivative in [10], an analytic description of the interaction dynamics between two pulses is replicated below

T1ðZ þ DZÞ ¼ T1ðZÞ þ G½ 1ðZÞ G2ðZÞ DZ 2 ; ðA:1aÞ T2ðZ þ DZÞ ¼ T2ðZÞ ½G1ðZÞ G2ðZÞ DZ 2 : ðA:1bÞ These equations describe the relative locations of the pulses as they propagate from a distance Z to Zþ DZ with DZ being the round trip distance. Assume the drift of the pulses per round trip time are small and DZ 1, we can convert the ence equations (A.1a) and (A.1b) into the diﬀer-ential equations as dT1 dz ¼ 1 2ðG1ðZÞ G2ðZÞÞ; ðA:2aÞ dT2 dZ ¼ 1 2ðG1ðZÞ G2ðZÞÞ: ðA:2bÞ

In addition, the instantaneous gain of the two pulses is given by g1 and g2 with the relation g2ðZÞ ¼ g1ðZÞ G1ðZÞ þ cT1ðZÞ; ðA:3aÞ g1ðZ þ DZÞ ¼ g2ðZÞ G2ðZÞ þ cT2ðZÞ; ðA:3bÞ where c¼ T0=s 1 and s is the normalized re-covery time of the gain medium. Now, we assume that the two pulses in the cavity with different wavelengths and linewidths may undergo different losses lidue to wavelength dependent reflectance of laser mirrors. Thus we modify the relation of the time dependent differential gain Giin [10] by adding the extra losses lito the instantaneous gain gias Gi¼

1

aðgi liÞ; ðA:4Þ

where a is a constant of proportionality which depends upon the pulse energy and magnitude of the total gain. Incorporating this into (A.3a) and (A.3b) yields the iterative equations

aðG2ðZÞ G1ðZÞÞ þ G1ðZÞ þ ðl2 l1Þ ¼ cT1; ðA:5aÞ aðG1ðZ þ DZÞ G2ðZÞÞ þ G2ðZÞ þ ðl1 l2Þ ¼ cT2:

ðA:5bÞ By adding and subtracting Eqs. (A.5a) and (A.5b), the approximation equation could be ob-tained by G1þ G2¼ cðT1þ T2Þ ¼ cT0; ðA:6aÞ ðG1 G2Þ ¼ c ð2a 1ÞðT1 T2Þ þ 2 ð2a 1Þðl2 l1Þ; ðA:6bÞ where, Taylor expansion G1ðZ þ DZÞ ¼ G1ðZÞ þ DZdG1ðZÞ=dZ þ is used and the OðDZ) is dropped since DZ 1. Insert Eqs. (A.6b) into Eqs. (A.2a),(A.2b) yields

d dZðT1 T2Þ ¼ c 2a 1ðT1 T2Þ þ 2 2a 1ðl2 l1Þ; ðA:7Þ whose solution is T1 T2¼ 2Dl c þ exp c 2a 1Z : ðA:8Þ

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This relation describes that the difference of in-terpulse separations in the cavity will eventually converge to a constant value only determined by the loss difference Dl¼ l2 l1. We can see that, if Dl¼ 0, the only steady-state solution is equally spaced pulses with T1¼ T2 which reduces to the result of [10]. On the contrary, if Dl6¼ 0, the in-terpulse separation difference T1 T2¼ 2Dl=c is no longer zero that will result in unequal spaced pulses.

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