1. Introduction

Thin-slice solid-state lasers with coated dielectric mirrors (LDMs) on both ends, in a laser-diode (LD) end-pumping scheme that uses the thermal lens effect, are the simplest and most promising device configuration for practical use of solid-state lasers if high beam-quality is attained. However, in contrast to LD-pumped solid-state lasers with well-designed external cavity configurations, in which stable transverse eigenmodes are formed, transverse eigenmodes are formed through the pump-dependent thermal-lens effect in thin-slice platelet LDMs, which depends on the pump beam spot size, crystal length, and thermal properties of the laser materials (e.g., thermal conductivity and thermal coefficient of refractive index [1].

On the other hand, such thin-slice Fabry-Perot lasers including surface emitting lasers [2] have revisited for understanding wave formation in microcavities in a different context from microstructure resonators surrounded by hard walls without end mirrors [3,4]. We showed formations of billiard-like laser modes and associated self-induced, high-speed intensity modulation by introducing asymmetric LD end-pumping into thin-slice LDMs which operate in 3-dimensional gradient refractive index (GRIN) optical cavity [5,6]. Wilkinson et al. predicted wave chaos in GRIN optical cavities with tilted planar ends [7].

The purpose of the present paper is two-folds: to show irregular lasing pattern formation in three-dimensional GRIN optical cavities with undulated reflective end surfaces, and to identify the effect of undulated surface reflector ends on lasing pattern formations in thin-slice LDMs operating in a thermally-induced refractive index potential. To study the combined effects of surface roughness and optical confinement, in particular, we employ a laser crystal Nd:GdVO₄ which has a larger thermal conductivity, in which an optical confinement effect can be controlled to a greater extent by the pump power.

We found a transition from an irregular lasing pattern to a Gaussian or billiard-like transverse modes in a thin-slice Nd:GdVO₄ LDM undergoing symmetric or asymmetric LD end-pumping with increasing the pump power. Different irregular lasing patterns appeared in an uncontrolled fashion, critically depending on the pump position, while the surface flatness was as small as λ/10 at wavelength λ = 633 nm [8]. Self-induced instabilities in accordance with the formation of irregular transverse lasing patterns are also demonstrated.

 

Fig. 1. (a) Experimental setup of symmetrically pumped LDM. (b) Example input-output characteristic of Nd:GdVO₄ LDM. PD: photodiode, DO: digital oscilloscope, WFA; wave-front analyzer, SFP: scanning Fabry-Perot interferometer, IRV: PbS infrared viewer. The asymmetric pumping scheme is depicted in the inset.

Download Full Size | PDF

An experimental setup is shown in Fig. 1(a). We used a 0.3-mm-thick, 7-mm-square, a-cut Nd:GdVO₄ crystal with a Nd concentration of 3 at.% (CRYSTECH Inc.) with a mirror coating on each surface (M₁: anti-reflection at 808 nm and 99.9 % reflection at 1063 nm; M₂: 99 % reflection at 1063 nm). The collimated elliptical beam from the LD (wavelength: 808 nm) was shaped into a circular beam using a pair of anamorphic prisms and then focused on the mirror M₁ by a microscope objective lens of N.A. = 0.25, in which pure symmetrical end-pumping was established. The absorption coefficient at 808 nm was 74 cm^-1. The crystal was mounted on a Cu plate with a 5-mm diameter hole across which the sample was pasted using a Si heat-sink compound. An example of input-output characteristics is shown in Fig. 1(b). A linearly-polarized emission along the crystal c-axis was observed. From the measurement of global optical spectra with a multi-wavelength meter, the laser was found to exhibit a single longitudinal-mode operation in the entire pump region in Fig. 1(b).

Let us show lasing far-field patterns. An extremely large Fresnel number of the cavity, i.e., 1.5x10⁵, ensures plane-wave approximation for light propagation along the 300-μm crystal length. In short, lasing field within the resonator can be expressed by the product of transverse eigenmode profile, determined by thermally-induced gradient refractive-index (GRIN) distribution, and a plane-wave which satisfies the standing-wave condition at end mirrors [5,6]. If undulated surfaces on end mirrors are considered, reflected waves at end mirrors suffer wave-front distortions and they are expanded into a series of plane waves having different wave vectors (i.e., plane-wave expansion). In this case, different modes, which satisfy the boundary condition, are expected to be selected from a series of expanded plane waves and contribute to lasing. Consequently, complicated lasing patterns would appear, critically depending on the reflective surface undulation across the lasing beam. A conceptual illustration of oblique lasing paths, which can be expected in a thin-slice circular GRIN lens with undulated reflective end surfaces, is depicted in Fig. 2. Here, surface undulations consist of concave curvatures resulting from the thermal expansion of the crystal and surface roughness.

Example pump-dependent far-field lasing pattern changes are shown in Fig. 2. In the low pump-power region, the thermal lens was not well formed presumably due to the high thermal conductivity of Nd:GdVO₄ (i.e., K = 11.7 [W/mK]). A weak optical confinement results in a relatively large lasing beam spot size and lasing patterns consisting of multiple eigenmodes can be formed due reflections at undulated end surfaces. Indeed, lasing patterns critically change depending on the pump position of the crystal at a fixed pump power. Irregular patterns always appeared for different 0.3-mm-thick samples. Example far-field patterns observed for different pump positions at the same pump power of 120 mW are shown in Fig. 3, in which the pump position was shifted by 200 μm for each pattern.

 

Fig. 2. Far-field lasing patterns for different pump powers corresponding to Fig. 1(b). Pump power; a: 27 mW, b: 36 mW, c: 87 mW, d: 139 mW, e: 164 mW, f: 241 mW. [Media 1]

Download Full Size | PDF

 

Fig. 3. Far-field patterns observed at different pump positions at a fixed pump power of 121 mW. The crystal was shifted horizontally. Relative positions are indicated.

Download Full Size | PDF

The relative surface roughness, R = flatness/cavity length was on the order of 2x10^-4. Then, a question arises: Does such an extremely small surface roughness result in irregular lasing patterns? To give a physical insight into this problem, we carried out ray path simulations for meridional rays and calculated Poincare sections similar to [7]. Here, we assumed a periodic sinusoidal variation of end surfaces, which is symmetric with respect to the laser axis, although actual surface roughness is not periodic in space. Example results for R = 2x10^-4 are shown in Fig. 4(a)–(c), where different effective focal lengths of thermal lens are assumed. The phase space ray-orbits revealed that KAM islands of stability [7] embedded in a chaotic sea appear even for a small roughness of R = 2x10^-4 while only stable tori appeared in the case of R = 0. (Chaotic waves could be formed in the present optical cavity in the form of a thin GRIN lens with designed corrugated reflective ends with larger R as shown Fig. 4(d).)

 

Fig. 4. Poincare sections indicating ray orbits on [x, sin θ] for different effective focal lengths of GRIN thermal lens. A sinusoidal surface roughness is assumed as shown in the figure. A 7/4 modulation period is assigned for each side (150 μm). (a),(b),(c): R = 2x10^-4, (d): R = 2x10^-3 Effective focal length: (a) 7.5 mm, (b) 4.75 mm, (c) 1.5 mm, (d) 7.5 mm.

Download Full Size | PDF

Moreover, ray orbits in the central part of Poincare sections, which should correspond to actual lasing paths, become simpler as the effective focal length (optical confinement) is shortened (increased) (see (Figs. 3(a)–(c)). This parallels the experimental result shown in Fig. 2.

To further confirm the phenomenological explanation for such combined effects of optical confinement and surface roughness experimentally, we replaced Nd:GdVO₄ laser by a 0.3-mm thick LiNdP₄O₁₂ (LNP) LDM with the same surface flatness and mirror coating. As expected, in the LNP LDM, TEM₀₀ mode operations were observed from the lasing threshold [1] due to the sufficient thermal lens effect, i.e., optical confinement, because of a lower thermal conductivity, K = 3.2 [W/mK]. An effective focal length of a thermal lens, which determines the optical confinement (i.e., lasing beam diameter within the cavity), is proportional to K/[dn/dT + α(n - 1)] , where dn/dT is the coefficient of thermal refractive index change, n is the refractive index, and α is the coefficient of thermal expansion [1]. Here, the denominator value is almost the same for both lasers, i.e., 6.2x10^-6/K (Nd:GdVO₄) [8] and 6.3x10^-6/K (LNP) [9]. (To be specific, thermal coefficients of Nd:GdVO₄ crystals are dn/dT = 4.7x10^-6/K and α = 1.5 x10^-6/K [8]). When the absorbed pump power was 200 mW, a temperature rise at the focused pump beam center of the 0.3-mm-thick Nd:GdVO₄ crystal was measured to be 10°C, while it was 38 °C for the 0.3-mm-thick LNP crystal. The temperature far outside of the pumped region was 18 °C . Therefore, the stronger optical confinement (i.e., smaller lasing beam diameter) is realized in LNP lasers than Nd:GdVO₄ lasers due to the larger temperature rise, yielding TEM₀₀ mode operations. Furthermore, it should be noted that as for 1-mm-thick Nd:GdVO₄ LDMs, pure TEM₀₀ mode operations were obtained from the threshold. Indeed, only stable tori appeared in central part of calculated ray orbits for R = 6.6x10^-5 accordingly.

When the pump power was increased above a critical point, irregular far-field lasing patterns, consisting of coexisting multiple transverse modes as shown later, made an abrupt transition and merged into the TEM₀₀ mode independently of the pump position. Due to the increased lens effect, the lasing beam spot size within the laser cavity becomes smaller and the sufficient optical confinement can dominate the effect of surface flatness. Such a transition occurred when the lasing beam diameter within the cavity, estimated from the beam divergence, became smaller than 60 μm. While, in the LNP laser, the beam diameter at the threshold pump power was 50 μm and decreased further with increasing the pump power.

An example far-field intensity profile and its wave-front (phase) profile of irregular lasing patterns measured by a wave-front analyzer (Imagine Optic Inc.; HASO-32-D) are shown in Fig. 5. It is interesting to point out that the smooth spherical wave front is seen while the intensity distribution is irregular. This implies that the superposition of different modal fields yields a wave-front distribution just like a Gaussian beam in the far-field.

 

Fig. 5. Intensity profiles and wave-front distributions. (a) Symmetric pumping (pump power = 61 mW). (b) Asymmetric pumping (pump power = 481 mW)

Download Full Size | PDF

3. Self-induced pulsations and oscillation spectra

In TEM₀₀ mode operations, the laser exhibited stable oscillations without any dynamic instability. In irregular pattern operations, on the other hand, the laser was modulated at beat frequencies between non-orthogonal transverse eigenmode pairs. Such self-induced modulations resulting from the interference of non-orthogonal transverse mode [5,6] appeared in a complicated fashion, and modulation frequencies changed depending on the pump power.

Measurements of intensity waveforms were performed by focusing the entire beam on the detector to suppress trivial beat notes among usual Hermite-Gaussian modes with perfect mode-orthogonality. In the case of Fig. 1(b), high-frequency modulations appeared in the low pump-power region at first. The beat frequency decreased rapidly as the pump power increased. As the beat frequency entered the domain of a relaxation oscillation frequency or its harmonics, the laser pulsed chaotically through resonances. In Fig. 1, chaotic pulsations occurred in the plateau region where the relaxation oscillation frequency, which is proportional to the output power, was almost fixed relative to the pump power. Example waveforms and their power spectra, which indicate the transition process from chaotic to stable operation, are shown in Figs. 6(a)–(c). In Fig. 6(a), the beat frequency f _B,2 coincided with the relaxation oscillation frequency f _RO, yielding chaotic pulsations, while the pulsations disappeared when f _B,2 was out of resonance with f _RO in Fig. 6(c). Note that another higher frequency beat at f _B,1 appeared in Figs. 6(a)–(b), in which chaotic pulsation waveforms are modulated at higher frequencies, as depicted in the insets of Figs. 6(a)–(b).

 

Fig. 6. Oscillation waveforms and corresponding power spectra indicating resonant excitation of chaotic pulsations in Nd:GdVO₄ LDM. Global views of power spectra indicating high-speed modulations are indicated in insets. Pump power: (a) 240 mW, (b) 260 mW, (c) 271 mW

Download Full Size | PDF

 

Fig. 7. (a) Example scanning Fabry-Perot traces of Nd:GdVO₄ LDM at different pump powers. (b) Chaotic pulsations featuring high-speed modulation at d in (a) where pump power = 155 mW. (c) Power spectrum.

Download Full Size | PDF

To identify such complicated dynamic behaviors, we measured oscillation spectra, intensity waveforms and the corresponding power spectra simultaneously. Example oscillation spectra measured at different pump powers are shown in Fig. 7(a). As expected, irregular lasing patterns were found to consist of multiple non-trivial transverse modes, which are totally different from usual Hermite-Gaussian modes. In the low pump power, a, lasing mode consists of three transverse eigenmodes, in which modes 1 and 2 form a non-orthogonal mode pair. With increasing the pump, mode 1 was found to approach mode 2 as shown by the arrow. The frequency separation became resonant with the relaxation oscillation frequency, f _RO, a chaotic relaxation oscillation occurred up to the pump power, d, although the pair of modes are not distinguished in spectra because f _RO (= 2 MHz) is smaller than the scanning Fabry-Perot frequency resolution of 6.6 MHz. At the pump power of 155 mW at d in Fig. 7(a), another pair of modes 3 and 4 appeared and resulted in a high speed modulation at 416 MHz of chaotic pulsations as shown in Figs. 7(b)–(c). Here, it should be pointed that the high-speed modulation of pulses occur occasionally, with some pulses, indicated by ↓ , being free from modulations. This gives a strong evidence of the transverse field interference of non-orthogonal modes: high-speed modulation occurs only when modal waveforms overlap in time. It is interesting to note that modes 1 and 2 are tend to separate as the pump power increases as depicted by the arrow, implying the avoided crossing of energy levels [6] on the analogy of quantum billiards. Above a critical pump power, an abrupt transition into TEM₀₀ mode occurred, being accompanied by a frequency jump as indicated by → and a spatial shift of the far-field pattern (see, video attached to Fig. 2).

4. Asymmetric pumping of Nd:GdVO₄ laser

To confirm further the combined effects of surface undulation and thermal lens on lasing pattern formation discussed above, we carried out experiments of the Nd:GdVO₄ laser with LD asymmetric end-pumping.

For asymmetric pumping, we removed the anamorphic prism pair in Fig. 1(a) as depicted in the inset similar to ref. [5]. Example far-field patterns are shown in Fig. 8 for increasing pump power (also, see a video). In the low pump-power regime, the laser exhibited irregular lasing patterns depending on the pump position as expected. As the pump power was increased, the irregular patterns made a transition into complicated patterns possessing fourfold symmetries and the system exhibited successive pattern changes similar to the asymmetrically-pumped LNP lasers [5,6]. While in the LNP lasers with a sufficient lens effect, i.e., asymmetric refractive-index optical confinement, irregular far-field lasing patterns were not observed even in the low pump-power region. Dynamical instabilities similar to the symmetrically-pumped Nd:GdVO₄ LDM like Figs. 6–7 were also observed.

 

Fig. 8. Lasing pattern changes with increasing the pump power of asymmetrically LD-pumped Nd:GdVO₄ LDM. Pump power; a: 165 mW, b: 241 mW, c: 291 mW, d: 366 mW, e: 394 mW, f: 434 mW. [Media 2]

Download Full Size | PDF

5. Numerical simulation and summary

The simulation was carried out by using the following model equations including the modal interference effect [5], assuming three modes:

Here, N_i, is the normalized excess population inversion, E_i is the normalized field amplitude, g is the interference parameter, β is the cross-saturation parameter, w = P/P_th is the relative pump power normalized by the threshold, K = κτ is the fluorescence lifetime normalized by the damping rate of the optical cavity, κ=(2τ _p)^-1 (τ _p: photon lifetime), ϕ _i,i±1 (t) is the phase difference between the i-th lasing mode and its adjacent mode, ΔΩ_i,i±1 = δω_i,i±1 /κ is the normalized frequency difference between the i-th lasing mode and its adjacent mode, and the time is scaled to the damping rate of the optical cavity. The last term of the phase equation expresses the frequency fluctuation, where ξ_i(t) is the Gaussian white noise with <ξl(t) > = 0 and <ξ_i(t) ξ_i(t’)>= δ(t - t’) that is δ-correlated in time.

Example intensity waveform and the corresponding power spectra are shown in Fig. 9. Experimental results shown in Figs. 6(b)–(c) are reproduced remarkably well. In particular, occasional high-speed modulations of pulses shown in Figs. 7(b) are clearly demonstrated.

 

Fig. 9. Numerical result. Adopted parameters are w=1.05, K=2000, ΔΩ_1,2=0.008, ΔΩ_2,3=1.670, β=0.667, g ₁=0.02, g ₂=0.250, D ₁=D ₂=10^-5. In notations in ref. [5], the definition of K should read K = τκ, where τ is the fluorescence lifetime and κ = 1/2τ _p is the cavity damping rate.

Download Full Size | PDF

In summary, spatially irregular stationary lasing patterns and associated self-pulsations have been observed in a laser-diode-pumped thin-slice solid-state platelet laser under free-running condition. The observed irregular lasing pattern would provide insights into transverse effects in three-dimensional optical cavities formed in a thin gradient refractive index lens with undulated reflective end surfaces which possess a conventional flatness. Our findings suggest that LD-pumped thin-slice solid-state lasers could be a promising candidate for forming a variety of transverse lasing patterns by fabricating well-designed corrugations on end reflective surfaces. From a practical point of view, it is shown that the surface flatness with respect to the cavity thickness (i.e., relative surface roughness) is a key factor for realizing stable and high beam-quality operations of laser-diode-pumped thin-slice solid-state lasers with coated dielectric mirrors.