DOI: 10.1007/s00340-002-0999-0 Appl. Phys. B 75, 453–456 (2002)
Lasers and Optics
Applied Physics B
y.f. chen1,✉
y.p. lan2
Spontaneous transverse pattern formation in
a microchip laser excited by a doughnut pump
profile
1Department of Electrophysics, National Chiao Tung University, Hsinchu Taiwan
2Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu, Taiwan
Received: 20 March 2002/Revised version: 20 May 2002 Published online: 25 October 2002 • © Springer-Verlag 2002
ABSTRACTA transition from a pure Laguerre–Gaussian (LG) mode to a pattern of optical vortex lattices in a large-Fresnel-number microchip laser is experimentally demonstrated by con-trolling the cavity Q-factor. The cooperative frequency locking of nearly degenerate modes is found to be a primary process for the generation of the optical vortex lattices in a class-B laser. When the cavity Q-factor is high enough, a LG-like mode and a structure of optical vortex lattices are found to coexist. Competition between coexisting transverse patterns of different symmetry gives rise to chaotic fluctuations.
PACS42.55; 42.65
1 Introduction
Spatio-temporal pattern formations appear spon-taneously in a wide range of systems, including hydrody-namics, granular media, chemical reactions and optics, when they are driven sufficiently far from thermodynamic equilib-rium [1–4]. In an optical system far from equilibequilib-rium, two types of pattern formation have been identified [5]. One is called a pure pattern, which can be successfully described in terms of the empty-cavity eigenmodes, in the absence of non-linearities. Non-linearities imply the interaction of many eigenstates, or pure patterns. The other is essentially non-linear pattern formation that generally requires a large-Fresnel-number resonator.
Depending on the material decay constants and photon de-cay rate, laser media are roughly classified into three type, Class A, B and C. In a Class A laser, both the material polar-ization dephasing and population decay rates are much larger than the photon damping rate, and the material variables can be regarded as being slaved to the latter. A Class B laser dif-fers in that the polarization-dephasing rate greatly exceeds the photon and population decay rates, and hence it is slaved to the other two variables. In a Class C laser, all damping rates are comparable in magnitude. The theoretical investigation of non-linear pattern formation is generally based on solving the order parameter equation (OPE). The OPE for a class-A laser ✉ Fax: +886-35/729134, E-mail: yfchen@cc.nctu.edu.tw
is the complex Swift–Hohenberg equation (CSHE) [6–9], whereas the OPE for a class-B laser is a CSHE coupled to the slow population equation [10–12]. The numerical inte-gration of the CSHE shows that non-zero detuning causes excitation of strip patterns in the one-dimensional (1D) space or of vortex lattices in the 2D case. The vortices are extremely interesting because vortex structure appears so widely in na-ture, in gases, fluids/superfluids, plasmas and even in living things, as in the helix of DNA. It is expected that the laser pat-tern dynamics and dynamics of other distributed non-linear systems have common features. However, it is difficult exper-imentally to observe the non-linearity-controlled patterns in laser systems because the requirements comprise both a large resonator Fresnel number and a high level of degeneracy of transverse mode families.
The recent rapid progress of diode-pumped microchip lasers has driven a renaissance of solid-state laser physics research and has led to novel phenomena [13, 14]. The mi-crochip laser can be easily operated in single longitudinal mode more than ten-times above threshold before the sec-ond longitudinal mode reaches threshold, because the gain medium has a short absorption depth, which reduces the lon-gitudinal spatial-hole-burning effect [15]. Here, we clearly demonstrate the dependence of transverse pattern formation on the cavity quality factor (Q-factor) in a large-Fresnel-number microchip laser excited by a doughnut pump profile. Upon increasing the cavity Q-factor, the transverse pattern shows a transition from a pure Laguerre–Gaussian (LG) mode to a non-linear pattern of square vortex lattices (SVLs). The stability of the SVL is found to depend mainly on the trans-verse mode spacing and the pump power. By further increas-ing the cavity Q-factor, we find the coexistence of transverse patterns of a LG-like mode and a non-linear SVL structure. Competition between coexisting transverse patterns of differ-ent symmetry gives rise to chaotic fluctuations.
2 Experimental setup
The experimental cavity we used is analogous to that described in [16]. The system schematic diagram and the pump profile in the laser system are shown in Fig. 1. The gain medium in the experiment was an a-cut 2.0 at.% 1-mm-long Nd:YVO4microchip crystal. The absorption coefficient of the Nd:YVO4crystal was about 6 mm−1at 809 nm. We set
454 Applied Physics B – Lasers and Optics Pump profile Output coupler Focusing lens Nd:YVO Laser diode 4
FIGURE 1 Schematic of a fiber-coupled diode end-pumped laser; a typical pump profile of a fiber-coupled laser diode away from the focal plane
up the resonator length to be as short as possible for reach-ing sreach-ingle-longitudinal-mode operation. The total length in the present resonator was L= 2.5 mm. The frequency spac-ing between consecutive longitudinal modes∆νLwas about
60 GHz. Since the longitudinal-mode spacing was consider-ably greater than the transverse-mode spacing, the present laser could be easily operated in single longitudinal mode to study pattern formation. The pump source was a 1-W fiber-coupled laser diode (Coherent, F-81-800C-100) with 100µm of core diameter. With a special coupling condition, the output intensity of the fiber-coupled laser diode could be controlled to be like a doughnut distribution. The doughnut pump profile was the key technique in the present investigation of pattern formation and competition. The pump power was focused into the microchip gain medium by using a focusing lens with 0.57 magnification.
For a general two-mirror resonator, the Fresnel number can be expressed as Fr= a2πω2
o
, where a2is the aperture
area andπω2
ois the area of the lowest-order mode cross. For
an end-pumped microchip laser, the effective aperture is usu-ally determined by the pump cross section, not by the mirror aperture. Namely, the Fresnel number for an end-pumped mi-crochip laser is given by Fr= ω2
p
ω2
o, whereωpis the pump
size of the gain medium. Changing the pump-to-mode size ratioωp/ωo can therefore control the value of Fresnel
num-ber. For the present cavity, the mode size of the microchip is given by ω2
o=
λ√L (R − L)π, where L is the cavity length and R is the radius of curvature of the output cou-pler. Three different output couplers were used in the ex-periment; the radii of curvature were 200 mm, 50 mm and 10 mm. For L= 2.5 mm, the mode size of the microchip was calculated to be 0.087 mm, 0.061 mm and 0.038 mm, respec-tively, for R= 200 mm, R = 50 mm and R = 10 mm. By de-focusing the pump source, the pump size could be adjusted within 0.5–0.8 mm. The maximum pump size depended on the lasing threshold. Usingλ = 1.064 µm and L = 2.5 mm, the Fresnel number could vary from 35 to 85 for R= 200 mm. In contrast, the Fresnel number could vary from 180 to 440 for R= 10 mm. Note that the thermal lensing effect was not significant because the thermal power density on the gain medium was controlled to be less than 0.5 W/mm2.
3 Results
First we used an output coupler of R= 50 mm with reflectivity of 97% in the laser resonator. Near lasing thresh-old, the laser emitted a pure high-order LG TEM0,l-mode with the distribution cos2lφ (or sin2lφ) in azimuthal angle,
having 2l nodes in azimuth. Laser oscillation on a single high-order LG mode resulted from a doughnut-shape pump profile. As shown in Fig. 2, the free-running single-transverse-mode class-B laser displayed relaxation oscillations, which play an important role in the dynamics of multi-transverse-mode class-B lasers. Slightly above lasing threshold, the present laser usually emitted a pair of transverse LG TEM0,l cosine and sine modes with chaotic dynamics. A non-linear form of the Maxwell–Bloch equation [16] has been used to investigate the interaction of two nearly degenerate transverse modes in a class-B laser. It was found that the appearance of dynamic chaos arises from the interaction of the relaxation frequency and the frequency difference between the nearly degenerate modes. The frequency difference between the nearly degen-erate modes generally comes from cross-saturation and other astigmatisms.
The cavity Q-factor is defined as Q= 2π×(energy stored)/(energy lost in one oscillation cycle). To excite a con-tinuum of transverse modes simultaneously, we increased the cavity Q-factor by using an output coupler with a higher reflectivity. With an output coupler of R= 50 mm with a re-flectivity of 99%, we observed a succession of spatially well-organized SVL patterns (Fig. 2), as predicted in [10]. An optical vortex is a point where the field intensity is zero, and the circulation of the phase gradient on any loop which encloses that point is equal to ±2πm where the integer m is the topological charge of the vortex. As found in [17], the square patterns observed are optical vortices of absolute charge one. The complexity of the SVL pattern increases with the Fresnel number and its axes, along with the orien-tation of the optical axes of the microchip gain medium. The experimental result confirms the theoretical prediction that the most natural transverse patterns for large-Fresnel-number
FIGURE 2 Power intensity spectra of laser emission for Laguerre– Gaussian TEM0,23mode near the lasing threshold. The beam profile is shown in the inset
CHENet al. Spontaneous transverse pattern formation in a microchip laser excited by a doughnut pump profile 455 lasers are SVLs if transverse-mode families with a high level
of degeneracy are excited [6, 10]. Nevertheless, it should be noted that all the analytical investigations deal with plane mir-rors, whereas the experimental result presented used curved mirrors. The present experiment also provided the first obser-vation of the transition from linear to essentially non-linear pattern formation in a large-Fresnel-number class-B laser. Measurement of the optical spectrum showed that the SVL pattern was a single-mode emission rather than a combina-tion of multimodes. This result indicates that the formacombina-tion of SVL patterns is a spontaneous process of transverse mode locking of nearly degenerate modes, assisted by the satura-tion process of the laser non-linearity. Accompanying the relaxation frequency, the power spectrum of the SVL pattern displays a self-induced oscillation mode, as shown in Fig. 2. The self-induced oscillation mode is generally found to have the same pump-power dependence as the relaxation oscilla-tion. The numerical analysis of the OPEs [10] show that in general it is difficult for a stable SVL pattern to exist in a large-Fresnel-number class-B laser because inertia of population inversion influences the transverse dynamics. Only when the lasing spectrum range is less than the relaxation oscillation frequency can a stable SVL pattern with self-induced oscilla-tions be found in a class-B laser. This situation is consistent with the experimental result that the cooperative frequency locking of nearly degenerate modes is an essential process for finding the self-induced oscillation accompanied by the relaxation oscillation in the SVL pattern. Although similar transverse mode locking in the generation of optical vortex crystals was demonstrated in broad-area VCSELs [17], to date optical systems have not generated such a large number of vortices.
The transverse mode spacing∆νTgoverns the coupling
strength between the transverse modes and thus rules the in-fluence of the non-linearity on the dynamical behavior. The-oretical analysis of the OPEs indicates that when the re-duced pump parameterε exceeds the critical value C∆νT/κ,
a chaotic regime can appear and the vortices in the square pat-tern annihilate and nucleate. Here,ε = (Io/Ith) − 1, Io is the
FIGURE 3 Power intensity spectra of laser emission at Fr≈ 200, ε = 1.0 and∆νT= 4.3 GHz. The beam profile of laser emission is shown in the inset
incident intensity, Ithis the intensity at the threshold, C is a
co-efficient of order of unity, andκ is the decay rate of the optical field. To investigate the influence of transverse mode spac-ing∆νT, we replaced the output coupler with a R= 200 mm
concave mirror. The∆νTchanged from 4.3 GHz to 2.1 GHz
at the same cavity length. In this case, the critical pump pa-rameter for the onset of chaos was aboutεc= 1.7. Near the
lasing threshold, the dependence of pattern formation on the Fresnel number was almost identical to the previous result, except that a smaller square pattern was emitted due to the larger mode size. When the pump power was increased to ε > 2.0, chaotically moving lattice defects suddenly appear in the SVL pattern and the vortices annihilate and nucle-ate continuously, as shown in Fig. 4a. A time-series of pic-tures of the chaotically moving lattice defects is depicted in Fig. 4b. Measurement of the optical spectrum showed that several different transverse-mode frequencies were simul-taneously emitted around 1064 nm. The observation of the spatio-temporal instability is in agreement with the theoretical prediction.
By further increasing the cavity Q-factor by using an out-put coupler of R= 50 mm with a reflectivity of 99.9%, a simi-lar SVL pattern was emitted near the pump threshold. For
FIGURE 4 a Power intensity spectra of laser emission for the square pat-tern at Fr≈ 45, ε = 2.1 and ∆νT= 2.1 GHz. The beam profile is shown in
456 Applied Physics B – Lasers and Optics
FIGURE 5 Power intensity spectra of laser emission for the square pattern at∆νT= 4.3 GHz and ε = 1.2: a Fr ≈ 70 and b Fr ≈ 140. The beam profiles
are shown in the insets
moderate values of the pump parameter power (1< ε < εc),
the transverse mode displayed coexisting transverse patterns that consisted of a LG-like mode on the pump region and a well-organized SVL mode in the center of the boundary, as shown in Fig. 5. The coexistence is understood from the fact that the most natural pattern for minimizing the free energy at the reflecting lateral boundaries is the formation of square vortex lattices; in contrast, the super-high cavity Q-factor with the doughnut-shape pump profile leads to the excitation of a LG-like mode. The first evidence of coexist-ing patterns of different symmetries was provided in an ex-periment on parametrically excited surface waves [18]. The different symmetries could be due to either to the selection of different wave vectors corresponding to the same wave-length or to the selection of different wavewave-lengths. The present result belongs to the former case. Recently, the coexistence of domains of different wavelengths has been observed in parametrically excited surface waves [19] and in passive non-linear optics [20]. In laser systems, this is the first evidence of coexisting transverse patterns of different symmetry. The broadening of the power spectra shown in Fig. 4 indicates that the interaction of two patterns with different symme-tries gives rise to time-chaotic fluctuations. The appearance
of chaotic oscillations in a coexisting transverse pattern with-out external periodic perturbations is of considerable inter-est. The two-mode route to chaotic relaxation oscillations has been observed in a microchip laser with the TEM00 mode
output in a two-longitudinal-mode oscillation regime [21]. The weak cross-gain coupling among two longitudinal modes has been proposed to explain the relaxation oscillation in-stabilities. The chaotic oscillation of the coexisting trans-verse pattern can be explained by the same non-linear gain mechanism.
4 Summary
In summary, we have demonstrated a transition from a pure LG-mode to a pattern of optical vortex lattices in a large-Fresnel-number microchip laser by controlling the cavity Q-factor. Experimental results reveal that the coop-erative frequency locking of nearly degenerate modes is an essential process for the formation of stable optical vortex lattices in a class-B laser. The dependence of the SVL dy-namics on the transverse-mode spacing agrees very well with the numerical analysis of the order parameter equations. The most striking observation of this investigation is that a super-high cavity Q-factor leads to the coexistence of two transverse patterns with different symmetries and pattern-competition-induced chaotic oscillations.
ACKNOWLEDGEMENTS The authors thank the National Sci-ence Council of the Republic of China for financially supporting this research under Contract No. NSC-90-2112-M-009-034.
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