## Abstract

A period-doubling route to chaos with nonstationary transverse pattern is observed in a diode-pumped Nd:YAG laser with a Cr^{4+}:YAG saturable absorber. The nonlinear behavior results from the multitransverse-mode competition with gradually adding the number of the transverse modes. By analyzing the transverse pattern, we find that the bifurcation accompanies with the increase of the higher-order family of transverse modes in sequence. Moreover, a series of period-doubling route to chaos beginning from the low period is attained by varying the pump power at specific cavity configurations.

©2004 Optical Society of America

## 1. Introduction

The technique of Q-switching is extensively employed for the generation of high pulse power in lasers. Among the Q-switched lasers, the passively Q-switched laser is one type that uses a saturable absorbing medium to achieve the pulse output. Recently, all solid-state passively Q-switched lasers have attracted wide attention owing to the advantages of significantly simplified operation, compactness, high efficiency, and low cost. In particular, Cr^{4+}-doped saturable absorbers have been the most popularly used in solid-state lasers because Cr^{4+}-doped crystals have not only large absorption cross section and low saturating intensity but also photochemical and thermal stabilities, moderate excited-state lifetime, and higher damage threshold [1,2]. The pulses with sub-nanosecond durations and several tens of kilowatt peak power have been obtained [1,2].

In such a potential pulsed laser source, the nonlinear instability, which is independent of the technique, is an interesting and intrinsic issue for the researcher. The phenomenon of pulse train bifurcation had been mentioned in a microchip laser system [1], but the characteristic of the nonlinear dynamics was not investigated in details. The deterministic chaos was recently found in a diode-pumped Nd:YAG laser with Cr^{4+}:YAG saturable absorber based on the laser operating at the fundamental mode [3]. The period-doubling route to chaos was observed by choosing the cavity length as a control parameter. By considering the effects of a finite number of doped ions in the host, period doubling could be achieved by increasing the pumping rate in numerical simulations, but they could experimentally resolve bifurcation only to period 2 upon increasing the pump power. Besides the nonlinear dynamics of the fundamental mode, the transverse mode interaction is also important in laser dynamics [4]. The interaction of the transverse mode with cooperative frequency locking could lead to time-independent and complicated, though stationary, pattern [5]. Moreover, the studies about the transverse mode interaction were also extended to include spatial variation of the dynamics, and the first time on the observation of spatiotemporal chaos was accomplished in Na_{2} laser [6]. Recently, pattern formation [7] and optical vortices [8] are interesting and concerning topics. Apparently, the role of transverse mode interaction has attracted a great deal of attention. In diode-pumped solid-state lasers, multitransverse modes are easily excited, because the spot size of laser diode pump source is usually larger than the spot size of the cavity fundamental mode. Thus, the nonlinear dynamics resulted from the transverse modes in these laser systems is worth investigating.

In this paper, we focus on the dynamics of transverse mode competition in a diode pump passively Q-switched Nd:YAG laser with a Cr^{4+}:YAG saturable absorber. When the laser is operated near the transverse mode degeneracy, a completely and clear period-doubling route to chaos occurs upon increasing the pump power. The periodic bifurcation accompanies with the addition of the higher-order families of transverse modes, and nonstationary transverse pattern is formed. The mechanism of the spatiotemporal dynamics is different to the temporal one such as the laser being operated and analyzed at the fundamental mode in Ref. [3]. In addition, a series of period-doubling routes beginning from the low periods, such as period 3 and period 5, is observed by controlling the pump power at the specific cavity length. Transition between the periodic windows can also be obtained by inserting a knife edge.

## 2. Experimental setup

The experimental setup is shown in Fig. 1. A fiber-coupled 16W cw diode laser with 808 nm wavelength and diameter of 800 µm is used as the pump source. An optical imaging accessory focuses the output beam with a ratio 1:1.8 at 13 mm behind the accessory. The Nd:YAG laser crystal has the dimensions of *ϕ*5×10 mm and 1.1 at.% of Nd^{3+} doping. One side of the Nd:YAG crystal which is anti-reflection coated at 808 nm and high-reflection coated at 1064 nm becomes to one cavity mirror of laser. The other side is anti-reflection coated at 1064 nm to reduce the cavity loss. A concave mirror with the radius of curvature ρ of 80 mm with a 90% reflectivity acts as the output mirror. Then the cavity forms a plano-concave cavity configuration. A Cr^{4+}:YAG crystal with a 80% transmission and a length of 5 mm is located in the cavity as a saturable absorber. These two crystals are mounted on water-cooled copper holders to reduce the thermal effect. The laser output is measured by a power meter for recording the average output power, or simultaneously detected by high-speed photodetectors for observing the pulse train and spectrum and pictured by CCD for monitoring the pattern at a screen with the help of beam splitters.

For tuning the cavity length, the output coupler is posited on a translation stage with the moving resolution of 10 µm. The cavity length can be tuned from about 43 mm to 68 mm, and the corresponding beam waists of the cold cavity mode are between 116 µm and 98 µm with the wavelength of the laser being 1064 nm. Considering the pump beam focused by an optical imaging accessory with a ratio 1:1.8, we can obtain that the ratio between the radius of the pump beam and the waist of the cavity mode varies from 1.91 to 2.20 in our plano-concave cavity configuration. It appears that the system is easy to excite the higher-order transverse modes. Owing to the lasing threshold is higher in this system, the ratio between the pump power, P_{p}, and the lasing threshold, P_{th}, defined as *w*=P_{p}/P_{th}, is convenient to showing the pumping condition. In this paper, we will concentrate on the nonlinear behavior under the low pump power in which is less than twice the threshold value, i.e., *w*<2.

## 3. Results

Scanning the laser output at various pump powers and cavity lengths, we find that the occurrences of nonlinear dynamics strongly depended on the cavity length under the low pump power. Within the operating range there are three instability regions corresponding to the cavity lengths of: (1) 45.87–45.94 mm, (2) 57.73–57.75 mm, and (3) 64.13–64.16 mm. These regions are close to the configuration having the transverse mode degeneracy. When transverse modes are degenerate, the output power exhibited a dip due to the effects of spatial hole burning in the distribution of population inversion [9,10]. This characteristic can be used to decide the relation between the degeneration points and the geometrical (or practical) cavity length L_{g}. We measure the average output power of the laser as a function of the cavity length at pump power P_{p}=2.78 W, and power dips are found when the cavity lengths become near 46.8, 56.8, and 66.3 mm. By considering the effective cavity length L_{eff}=L_{g}-L_{c}(1-${{\mathrm{n}}_{\mathrm{c}}}^{-1}$) [9] and the indices of Nd:YAG and Cr^{4+}:YAG being both 1.82, the observed power dips correspond to the g-parameter of cavity configuration [11] having g_{1}g_{2}=1/2, 0.3887, and 1/4, respectively. Here L_{c} is the length of the crystal, n_{c} is the index of the crystal, and g_{1}g_{2}=1-L_{eff}/ρ for a plano-concave cavity. Based on the above-mentioned discussions, the instabilities occur at the cavity configurations near g_{1}g_{2}=1/2, 0.3887, and 1/4.

Since similar nonlinear dynamics can be found in each one of the instability regions, we will focus the following discussion on the instability region near g_{1}g_{2}=1/4. Firstly, a complete period-doubling route to chaos is observed by increasing the pump power, as shown in Fig. 2 for a cavity length of 64.13 mm. When the pump power is less than 2.12 W, the laser output showed stable pulse intensity, as shown in Fig. 2(a). After the bifurcation occurring, the pulse train becomes period 2 as shown in Fig. 2(b) with P_{p}=2.4 W. Period-4 and chaotic pulse trains are attained (Figs. 2(c)–(d)) upon increasing the pump power. Figure 3 depicts the bifurcation diagram in which we only indicate the peak frequency of spectrum within the repetition rate and the solid circles correspond to the repetition rate. The resolution of the pump power is restricted by the tuning resolution of the high power laser. Owing to this tuning resolution and the stability of the higher-order period, period 16 is the highest order period to be observed in this scenario. The bifurcating pump power of 2.20 W with *w*=1.05 for period 2 is slightly above the lasing threshold P_{th}=2.10 W. As P_{p}≥3.66 W, the output intensity evolves a chaotic pulse train, in which a broadening spectrum about 230 KHz is obtained. Further calculating the correlation dimension of the chaotic signal, we find that the correlation dimension is 4.583±0.044.

However, not only the periodic and chaotic pulse trains but also the spatial instabilities with the pattern formation dynamically varying are attained. As P_{p}=2.10 W, the pattern of period-1 pulse train shows a fundamental mode. If we increase the pump power, period 2 occurs and two patterns as shown in Figs. 4(a) and 4(b) come to existence owing to the generation of the higher-order transverse modes. In general, the higher-order transverse modes of the laser resonator can be expressed in terms of Hermite-Gaussian modes, and the Hermite-Gaussian mode has the form [11]

where H_{m} represents a Hermite polynomial of order m, x and y are the Cartesian coordinates, and w_{0} is the spot size of the laser. The experimental pattern can be fitted by using the superposition of the Hermite-Gaussian basis set to be |∑
_{m,n}
*C*_{mn}
×*HG*_{mn}
(*x,y*)|^{2}, where *C*_{mn}
is the complex factor for the corresponding modes and satisfies ∑
_{m,n}
|*C*_{mn}
|^{2}=1 [12]. Besides the pattern in Fig. 4(a) still like a fundamental mode, the other pattern in Fig. 4(b) could be decomposed into |(0.5)^{1/2}×*HG*_{01}
+(0.5)^{1/2}×*HG*_{10}
|^{2} and numerically reconstructed pattern is shown in Fig. 4(c). These two patterns switch each other in time evolution. Continuing to add pump power, we obtained a period-4 pulse train and nonstationary transverse patterns. As period 4, the main transverse pattern can be decomposed as the superposition of the transverse modes with the family of order m+n=2 besides the transverse patterns appeared in period 2. Figure 5 displays some of the transverse patterns observed at the different times in P_{p}=2.68 W. The experimental results are presented in the upper row of Fig. 5 and numerically reconstructed patterns are plotted under the experimental ones. We also list the simulation parameters concluding the amplitude and the relative phase of the participating transverse modes in Fig. 5. When the higher periodic or chaotic pulse train is generated, the transverse patterns are more complicated beyond the resolution of the CCD. Sometimes the pattern corresponding to the superposition of the transverse modes with the family of order m+n=3 can be attained under period 8; however, mostly the recorded pattern not only twinkled but also alternately shrank and enlarged in time evolution. It is an interesting result that each bifurcation accompanies with the participation of the upper-order family of the transverse modes.

If we focus on the sequence of the pattern alternation and the time of the persistence of the pattern, they are irregular from the records of the CCD. A periodically temporal output accompanying with an irregular alternation of the pattern may be paradoxical, we think that it results from the time for the alternation being less than the temporal resolution of CCD with 32 frames per second. For verification, we use two small-area detectors to measure intensities at different transversal positions of the pattern. Figure 6 depicts the temporal signal from these two detectors for a period-4 pulse train. Since the evolutions of these two intensities do not synchronize, it implies that the transverse pattern varies for the neighbor pulse. Since the duration of the pulse train is significantly less than the temporal resolution of the CCD, we cannot observe a regular sequence of alteration if the ratio between the above two time values is an irrational number. In addition, the patterns with the dark dots have been observed in higher-period and chaotic pulse trains. Figures 7(a) and 7(b) display a doughnut pattern and a pattern with two dark dots, respectively. On the basis of similarity, these patterns may be similar to the ones proposed in Ref. [13]. But the patterns in Ref. [13] appear in the stationary configurations and the singularity is located in precisely defined position. Usually, the dark dot of the transverse pattern may represent the existence of the optical vortex or phase singularity. Owing to these pattern are seldom observed, it need further investigate to verify the character.

As the cavity configuration with g_{1}g_{2}=1/4, the longitudinal-mode spacing is triple the transverse-mode spacing, i.e., the transverse modes with m+n=3p families are degenerate, where p is an integer. However, the nondegenerate modes, such as m+n=1 and 2, are attained in our system, because there are two reasons to add the extracting efficiency of the gain for these higher-order modes. One is the size of the pump beam greater than the waist of the cavity beam associated with a ratio of about 2.06 for the effective cavity length of 6 cm. The other is that the profile of the pump beam for the fiber-coupler diode laser can be looked as a top hat, in which offers a better overlap integral between these mode distribution and the pump profile.

Moreover, the different scenario of period-doubling route, which begins from the low period such as period 3 and period 5, is observed within the same instability region. Figure 8 shows the period-doubling route beginning from period 3 by increasing the pump power. The cavity length is 64.17 mm and about 0.03 mm larger than the one in Fig. 2. Figure 8(a) depicts the period-3 pulse train, which is bifurcated from period 1, as the pump power of 2.58 W. Period 6 and period 12 are obtained at the pump power of 2.88 W and 3.30 W, which are shown in Figs. 8(b) and 8(c), respectively. The laser output becomes chaotic at larger pump powers. We also observe that period 5 bifurcated to period 10 upon increasing the pump power, but period 20 is obtained sometimes and the higher-order period was not found. Since the correlation dimension of the chaotic signal close to the ordinary period-doubling route, such as 4.456±0.059 associated with the chaotic signal bifurcated from period 3, we think that these phenomena represent the high-order periodic windows in the period-doubling route to chaos. The periodic window means the period of the attracting orbit before it begins to period double such as period-3 and period-5 windows in the period-doubling orbit diagram of quadratic function [14].

The periodic window can be also obtained by transversally inserting a knife edge within the cavity under the fixed pump power. The diffraction effect induced by inserting the knife edge redistributes the composition of the transverse modes and modifies the interaction and competition among the transverse modes. Because inserting a knife edge increases the loss, a reverse period-doubling route will be found. By changing the initial axial distance between the knife edge and the output coupler, we can switch between the different periodic windows. For example, period 6 has two reverse routes upon gradually inserting the knife edge; the one follows P_{6}-P_{3}-P_{1} and the other returns to period 4 and follows P_{4}-P_{2}-P_{1}, where the subscripts represent the number of the periods. Owing to the existence of the periodic window and the tunable region for instability being only several tens of µm, it is difficult to sequentially obtain a purely period-doubling route by monotonically increasing or decreasing the cavity length within the instability region.

## 4. Discussions

In experiments, a spatiotemporal instability is found, in which we attain a temporal instability with the periodic or chaotic pulse trains and a spatial instability with the altering patterns simultaneously. Since each temporal bifurcation accompanies with the presence of the upper-order family of the transverse modes, it appears that the temporal bifurcation follows the participation of the new transverse-mode family, i.e., the spatial bifurcation. Therefore, we think that the transverse mode competition is the mainly dynamical mechanism in our system. The competition is controlled by the spatial distribution of the modes with respect to that of the available gain and of the loss profile. Owing to the altering patterns being decomposed by the specific family for a periodic pulse train, the obtained gain is transferred among the different frequencies of transverse-mode families. It implies that the competition by adding a new frequency may change the dynamical behavior, so we observe that the participation of the new family inducing the spatial and temporal bifurcation simultaneously.

Moreover, the instability region exists within several micrometers for tuning cavity length only. We think that the complicated nonlinear behaviors significantly correspond to a cavity-configuration-dependent nonlinear behavior [15]. When the cavity configuration corresponds to the low order resonance, such as g_{1}g_{2}=1/2, 1/4, 3/4, and (2±2^{1/2})/4, the laser will be sensitive to the nonlinear effect, i.e. will be unstable under the persisting nonlinear effect. Thus, a weak interaction or competition among the transverse modes will easily exhibit complicated dynamics under these specific configurations. Owing to the nonlinear resonance is more critical than the transverse mode degeneracy, the instability regions are comparatively less than the region of the dip and extend only several tens µm. Besides, relating to the thermal effect, it is a considerable role in high-power pumped laser. But we think that the thermal effect is not an important role to induce the nonlinear behavior in our results, because the instability region is only several tens of µm for tuning cavity length and the thermal effect cannot dramatically vary within this small region.

In general, quasi-periodic and spatiotemporal behavior is expected in near degenerate configuration based on the several transverse mode competitions [5,16]. Such in Ref. [16], the mixed effect between the beam propagation and gain dynamics make the route to chaos be the interplay of quasi-period and period multiplied bifurcation. But a period-doubling route to chaos is observed in this system. We think that the existence of a saturable absorber adds the role of the gain effect that results period doubling. In fact, the period-doubling bifurcation is a typical route for CO_{2} laser with a saturable absorber in single-mode [17] and two-mode laser systems [18]. The fractal dimension of the strange attractor is 2.02±0.01 in two-mode system [18]. Different to these passively Q-switched systems, the higher correlation dimension appears the participation of multitransverse modes (at least three families from the above-mentioned discussions) and the spatial and temporal instabilities evolve in our system.

Apparently, the nonlinear mechanism with multitransverse modes in our system is also different to the one in Ref. [3], in which the laser is operated and analyzed at the single fundamental mode with considering the effects of a finite number of doped ions in the gain medium. The difference between single-mode and multitransverse-mode mechanisms is also revealed by the characteristics of the threshold instabilities. From the previous study, Lugiato *et al*. claimed that transverse effects could be responsible for low threshold instabilities [19]. If the ratio between the waist of the pump beam and the cavity beam waist at the gain medium became sufficiently large, the threshold for instability decreased dramatically and could be a few percent higher than the ordinary laser threshold. Compared to the pump power of 11.6 W used in Ref. [3], the bifurcating pump powers are 2.20 W (or *w*=1.05) for period 2 and 3.66 W (or *w*=1.74) in our experiment.

## 5. Conclusions

In conclusion, we observe the spatial and temporal instabilities in a diode-pumped Nd:YAG laser with a Cr^{4+}:YAG saturable absorber. A period-doubling route to chaos with nonstationary transverse pattern is found, and the bifurcating powers are close to the lasing threshold. By numerically reconstructing the pattern based on a Hermite-Gaussian basis set, the bifurcation accompanies with the increase of the higher-order families of transverse modes in sequence. The nonlinear behavior results from the multitransverse-mode competition with gradually adding the number of the transverse modes. Although the period-doubling route to chaos is observed, the mechanism is different to the one resulted from the fundamental mode in the same laser and the other one resulted from two mode interaction in a CO_{2} laser with a saturable absorber. A series of period-doubling routes to chaos beginning from the low periods including period 3 and period 5 is observed by varying the pump power at different cavity lengths. They correspond to the periodic windows in period-doubling bifurcation. The transition between the periodic windows can be achieved by inserting the knife edge to redistribute the composition of the transverse modes.

## Acknowledgments

The authors would like to thank Professor W.-F. Hsieh for fruitful discussions. This work was supported by the Nation Science Council (NSC) of the Republic of China under Grant NSC 92-2112-035-006.

## References

**1. **J. J. Zayhowski and C. Dill III, “Diode-pumped passively Q-switched picosecond microchip lasers,” Opt. Lett. **19**, 1427 (1994). [CrossRef] [PubMed]

**2. **I. Freitag, A Tunnermann, and H. Welling, “Passively Q-switched Nd:YAG ring lasers with high average output power in single-frequency operation,” Opt. Lett. **22**, 706–708 (1997). [CrossRef] [PubMed]

**3. **D. Y. Tang, S. P. Ng, L. J. Qin, and X. L. Meng, “Deterministic chaos in a diode-pumped Nd:YAG laser passively Q switched by a Cr^{4+}:YAG crystal,” Opt. Lett. **28**, 325 (2003). [CrossRef] [PubMed]

**4. **N. B. Abraham and W. J. Firth, “Overview of transverse effects in nonlinear-optical systems,” J. Opt. Soc. Am. B **7**, 951 (1990) [CrossRef]

**5. **L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, and M. A. Pernigo, “Instabilities and spatial complexity in a laser,” J. Opt. Soc. Am. B **7**, 1019–1033 (1990). [CrossRef]

**6. **K. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continue Na_{2} laser,” Phys. Rev. A **39**, 919–922 (1989). [CrossRef] [PubMed]

**7. **F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. **318**, 1–83 (1999). [CrossRef]

**8. **M. Vasnetsov and K. Staliunas, *Optical Vortices* (Nova Science, New York, 1999).

**9. **Q. Zhang, B. Ozygus, and H. Weber, “Degeneration effects in laser cavities,” Eur. Phys. J. AP **6**, 293–298 (1999). [CrossRef]

**10. **H.-H. Wu, C.-C. Sheu, T.-W. Chen, M.-D. Wei, and W.-F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an an pumped Nd:YVO_{4} laser,” Opt. Commun. **165**, 225–229 (1999). [CrossRef]

**11. **A. E. Siegman, *Lasers* (University Science, Mill Valley, CA., 1986).

**12. **J.-H. Lin, M.-D. Wei, W.-F. Hsieh, and H.-H. Wu, “Cavity configurations for soft-aperturing Kerr-lens mode-locking and multiple-period bifurcation in Ti▫sapphire laser,” J. Opt. Soc. Am. B **18**, 1069–1075 (2001). [CrossRef]

**13. **M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phy. Rev. A **43**, 5090–5113 (1991). [CrossRef]

**14. **R. L. Devaney, *A First Course in Chaotic Dynamical Systems — Theory and Experiment* (Addison-Wesley, Sydney, 1992), *Chap. 8*.

**15. **M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. **146**, 201–207 (1998). [CrossRef]

**16. **C.-H. Chen, M.-D. Wei, and W.-F. Hsieh, “Beam-propagation-dominant instability in an axially pumped solid-state laser near degenerate resonator configuration,” J. Opt. Soc. Am. B **18**, 1076–1083 (2001). [CrossRef]

**17. **M. Tachikawa, F.-L. Hong, K. Tanii, and T. Shimizu, “Deterministic chaos in passive Q-switching pulsation of a CO_{2} laser with saturable absorber,” Phy. Rev. Lett. **60**, 2266–2268 (1988). [CrossRef]

**18. **K. Tanii, T. Tohei, T. Sugawara, M. Tachikawa, and T. Shimizu, “Two different routes to chaos in a two-mode CO_{2} laser with a saturable absorber,” Phy. Rev. E **59**, 1600–1604 (1999). [CrossRef]

**19. **L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, “Role of transverse effects in laser instabilities,” Phy. Rev. A **37**, 3847–3866 (1988). [CrossRef]