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Abstract
We analyze the characteristic of amplified spontaneous emission (ASE) in a high power diode pumped Nd:YAG slab gain module. Simultaneously the characteristic of parasitic oscillation (PO) is considered. A high-efficiency three-dimensional coupled ASE calculation model using the geometric optical tracing technique is proposed. The model considers the stimulated emission cross-section correction by thermal effect and diffuse reflection on rough surfaces. From the experiment result of the fluorescence curve and small signal gain coefficient, the impact of ASE and PO on QCW-slab amplification is evident. Quantitative agreement between the ASE numerical model and experiment is achieved. The energy storage as well as the gain limit condition in QCW-slab amplification are investigated with the help of experiment. For the Nd:YAG slab with the size of 150.2 mm*30 mm*2.5 mm, the energy storage limit is 1100 mJ.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High energy solid-state lasers have a pivotal role in worldwide industrial and scientific applications. Traditional rod-based lasers have difficulty scaling to high average power while maintaining a superior beam quality due to thermal effect [1]. For zigzag slab lasers, it’s nearly one-dimensional thermal gradients and zigzag optical path significantly reduce thermal phase distortions, birefringence, and depolarization loss. A key to accumulate energy is suppressing ASE. Spontaneous emission(SE) gets amplified on its way through the pumped medium, which is ASE. Under the condition of high gain medium, closed paths and sufficient feedback, PO generates as the gain overcomes the reflection losses, especially in quasi-continuous waves(QCW)slab amplifiers.
Problems of ASE and PO have been comprehensively discussed since the last century [2–7]. Trenholme [2], McMahon [3], and Contag [4] reported their research on ASE in disk amplifiers. LLNL firstly studied the spherical and circular shape of gain media to observe the influence of geometry on the ASE and PO. For rod amplifiers, a detailed explanation was put forward by McMahon in 1974 [5]. In 1978, the team of Brown presented detailed calculations to introduce their found of the stored energy density distribution in active-mirror and disk amplifiers with parasitic oscillations [6].
Systematic researches have been carried out on the disk laser amplifier with spherical symmetry to find valid methods to suppress ASE and PO [8–18]. Extensive researches have shown the inner back reflection and PO in disk amplifiers can be suppressed with an absorbent cladding located on the crystal edge. Both solid cladding and liquid cladding have been proposed [8–14]. Absorption cladding made of Cr: YAG for Yb: YAG disk media received extensive attention [14]. Kouznetsov’s work verifies the significance of the anti-ASE cap in an increase of the maximal output power [13]. Depending on the report of Yu Qiao in 2017, different thickness anti-ASE caps were calculated to estimate the suppression ability of ASE with theoretical analysis [15].
Research on ASE and PO in slab amplifiers is relatively complicated. A closed-form model was designed by Norman to solve the equations describing the change of population inversion for Nd:YAG rod laser [16]. Qitao Lv proposed a three-dimensional calculation of ASE in high power slab amplifiers. In his model, the calculation is simplified to a two-dimensional plane owing to side pumping [17]. Goren proposed an analytical model for describing ASE in slabs in one and two dimensions. His team also provided the numerical results of small signal gain coefficient in steady-state of three-dimensional systems [18]. At the same time Sridharan developed a series of methods like novel technique, edge bonding slanted undoped crystal and large surface coating to suppress ASE in zig-zag slab amplifier [19]. In 2009, Albach gave a comprehensive explanation of the influence of the gain on the population inversion spatial distribution both theoretically and experimentally [20]. For multi-slabs, a three-dimensional code based on ray-tracing is also presented by Sawicka to describe ASE. The characteristic of this code is not simply expressing ASE on the energy storage of slabs, but also calculating the heat distribution and the amplification of the signal [21]. Sawicka’s team optimized the code in 2013 [22]. In summary, ASE models for slab structure require a large amount of calculation due to the three-dimensional asymmetric structure. The aforementioned models usually reduce dimensionality or drastically simplify the fluorescence’s boundary reflection characteristics. Large computational programs using parallel algorithms require more computational resources and convergence time. For engineering, experiments using large aperture slab are costly, high-timeliness and resource-conserving ASE calculation models are required.
In this paper, we analyze the impact of the ASE and PO on amplification and energy storage of large-caliber slabs theoretically and experimentally. A three-dimensional coupled calculation model utilizing geometric optical tracing technique is proposed to forecast the slab amplification under ASE. An optical measurement system is built to detect amplification of the slab gain module. Both the small signal gain coefficient and the energy storage limitation are measured. Experimental results have been compared with the calculation data.
2. Theoretical analysis
ASE is intensively dependent on the small signal gain coefficient and path length which fluorescence lights pass through. The latter is dependent on the geometric structure of the material, interface reflection and feedback characteristic of external optical lens.
The internal reflection on the gain medium boundaries can drastically enhance ASE, and positive feedback occurs as long as inner reflections originate from a closed loop. Once the gain exceeds the loss with an intense feedback mechanism, the PO generates. Special structures of slab lead to complex reflection characteristics.
From the rate equation, we get that [1]
For the quasi-continuous pump, we suppose that the pump pulse is ${t_p}$ ignoring the ASE, the small signal gain coefficient can be obtained by solving the formula [1]:
The structure of Nd: YAG slab and the actual products of the slab gain module are shown in Fig. 1. Doped regions bonded to undoped YAG crystal are designed to suppress ASE and PO. The length of the doped region is 120 mm. There are three pairs of parallel planes: large surfaces (150.2 mm×30 mm); side surfaces (150.2 mm×2.5 mm); passing surfaces (30 mm×2.5 mm). The side surfaces are brushed. The large surfaces connected to the heat sink are plated with $\textrm{Si}{O_2}$ film and metallized layer.
Fig. 1. The Nd:YAG slab and quasi-continuous end pumped slab gain module: (a) The structure of Nd:YAG slab;(b) The real products of quasi-continuous end pumped slab gain module
The slab related parameters are shown in the following Table 1.
Table 1. The slab parameters
Relaxation oscillation is a transient process that is difficult for numerical calculation. For PO, we only talk about the possible closed paths where PO can generate. As the slab is small in thickness and width, the reflectivity of the large and side surfaces is low. PO does not generate on these surfaces. There are for two possible closed paths: the axis path and the axial path of zigzag beam. It can be proved by geometrical optics that the incident angle of the positive ray and the incident angle of the reverse ray on the axial path are complementary to each other, which indicates the positive ray and the reverse ray cannot form a total inner reflection(TIR) at the same time. There are four reflective surfaces in the axial loop that satisfies the total reflection condition on the passing surfaces. Once the formula below being satisfied, PO will generate.
${R_1},{R_2}$ is the reflectivity of the passing surfaces, $\alpha $ is the absorption scattering loss coefficient of the gain medium.3. Numerical modeling
3.1 Numerical formulation for slab geometry
Due to the non-circular symmetry of slabs, ASE distribution in slabs is extremely sophisticated. Therefore, the following approximations have been used to minimize computer calculation:
- (1) Considering the low concentration doping of Nd(0.1%) and a large enough size of slab, pump intensity distribution in the doped region is assumed to be homogeneous;
- (2) Within a 4π spatial angle range, spontaneous emission generates stochastically and equiprobably;
- (3) The reflection characteristic of the passing surfaces and undoped YAG crystal have been neglected;
- (4) The reflection characteristic on large surfaces is assumed to follow Fresnel’s law.
Ray tracing is a general method to analyze the ASE effect in the laser gain medium. We are concerned about the integral effect on amplification property rather than the instantaneous evolution of ASE. A simplified model focusing on the amplification path and ASE’s interaction with the medium surfaces is proposed.
The gain medium is meshed. For a volume element dV, the photons generated by spontaneous emission transmit into 4π stereoscopic space randomly. The spontaneous emission photons emitted to the solid angle dΩ in unit time are:
One of the pivotal mathematical treatment of ASE intensity calculation is computing the amplification path. We assume that the length of the amplification path which the photons generated by formula(4) passed through is L. Distinctly that L is a function L(x, y, z, $\; \varOmega $) of coordinates and direction of spontaneous emission. $\; {g_0}$ is small signal gain coefficient. The decrease of population inversion caused by ASE on the path function L(x, y, z, $\; \varOmega $) is :Ultimately, spontaneous emission, the decrease of population inversion caused by ASE and the increase of population inversion by pumping are levelling off. The stable small signal gain coefficient is called ‘the self-sustained gain’ and the steady-state rate equation of ASE should be amended to:
3.2 Partial parameter correction
3.2.1 Stimulated emission cross-section correction by thermal effect
Part of past ASE models for slab structure are generally simplified to a large extent. This kind of models emphasize the qualitative law and tendency of gain. They are the absence of distinct accuracy in quantitative description and prediction. We amend several factors approximating reality to remedy the model.
Related research by A.Rapaport and B.Chen proved that the stimulated emission cross-section of the gain medium is affected by the thermal effect [23], the relationship between the stimulated emission cross section of Nd: YAG crystal and the temperature is:
According to the above formula, we use the software of finite element analysis (Ansys) to simulate calculate the temperature distribution in the gain medium.From Fig. 3 it’s clear that the cooling performance is well on the account of the large surface’s large enough area. The main thermal effect is concentrated on the incidence plane as well as there is no high temperature gradient. According to the calculation, we modify the internal stimulated cross-section of the gain medium.
Fig. 3. The calculation results for thermal effect in slab(a) Distribution of temperature in the gain medium;(b) Distribution of temperature gradient
3.2.2 Scattering model for side surfaces
For diffuse reflection on the roughen side surfaces, we simulate the situation using a simplified diffuse model. The spatial distribution of diffuse reflection and scattering light is complicated. So the gain medium is deemed to be a Lambertian diffuse reflection structure. The fluorescence light reflection on the roughened side surfaces obeys Lambert’s cosine law (Fig. 4):
3.3 Calculated results
The model which we establish only calculates the ASE process without consideration of parasitic oscillations
3.3.1 Proportion of different surfaces contribution to ASE
Evaluation of each surface’s contribution to ASE is significant in practical engineering, and it provides methods for targeted suppression measures. We calculate the proportion of the population inversion caused by each surface and the weight coefficient of each surface in ASE by the model.
According to Fig. 5, it can be discovered that the reduplicative reflection of spontaneous emission on large surfaces is the uppermost component of the entire ASE, accounting for 90% approximately. The proportion of reflection on side surfaces is approximately 9%. When pump power increases to 8000W, the proportion of population inversion reaches 25%. It is consistent with our prediction because the solid angles formed on large surfaces are much larger than the ones on side and passing surfaces. The gain path of spontaneous emission light generates in the reciprocal reflection between large surfaces. The calculation results also explain that optimizing the side surfaces structure has a positive impact on the suppression of ASE.
Fig. 5. Calculated results :(a) weight coefficient in ASE; (b) Proportion of the population inversion
3.3.2 Small signal gain coefficient with ASE
The calculation result of the small signal gain coefficient is in Fig. 6:
It can be observed from Fig. 6 that the ASE under low pump power has less influence and the small signal gain coefficient increases linearly with pump power. With high pump power condition, the ASE increases rapidly and exponentially with the small signal gain coefficient. The growth of the small signal gain coefficient slows down.
3.3.3 Discussion of the model
Compare to large-scale computational models using parallel algorithms, our model realizes the analysis of ASE in three-dimensional slab structure with reasonable convergence speed (mesh 120*120*2 12h). Due to simplify conditions, the calculation model has the following shortcomings:
- (1) Parasitic oscillation is not integrated into the calculation model due to the uncertainty of the closed path of spontaneous emission light in the three-dimensional slab. The calculation results are without saturation state and the energy storage limit of the slab;
- (2) The small angle reflection on passing surfaces increases the ASE effect. The calculation result of the small signal gain coefficient is larger than the actual result;
- (3) The computational model only performed geometric optical tracing of spontaneous emission rays, ignoring its overlap and diffraction effects.
4. Experimental result
The experiment is divided into two parts. We measure the fluorescence characteristics of the slab and make the slab injected seed to calculate the small signal gain coefficient. On the basis, we explore the energy storage limit of the slab.
4.1 Fluorescence characteristics of the slab gain module
We use a double-ended pumped slab crystal and a high-speed photoelectric probe to observe the scattered light on the passing surface of the slab. Since there is no seed light injection, the scattered light detected by the probe is fluorescence light of spontaneous emission and their intensity are proportional to the intensity of ASE and PO within the solid angle covered by the detector.
Obviously, when the current is high enough or the pulse width is wide enough, PO occurs in the slab. However, as soon as the population inversion density is less than the threshold of PO, the oscillation ceases and the upper level continues to accumulate and reverse the number of population inversion. The cycle reciprocates to form a relaxation oscillation until the pump ends. The presence of the PO limits the maximum energy storage of the slab. Continuing to increase the current or pump pulse width, the population inversion density of the upper level is maintained near the threshold of PO. Under the quasi-continuous pumping condition, the pump peak power and the pump pulse width determine the pump energy. We experimentally investigate the waveform of PO and the threshold current under the pump pulse width of 150 µs, 250µs, 300µs, 350µs respectively. The pump current for parasitic oscillations are 160A, 140A and 130A. The fitting oscillograms using Matlab are shown in Fig. 7
Fig. 7. Parasitic oscillation and the threshold current under the different pump pulse width: (a) 250 µs 160 A;(b) 300 µs 140 A;(c) 350 µs 130 A
Limited by the accuracy of the oscilloscope, parasitic oscillation images are not obvious enough, but the flat top can be observed. Figure 7 shows an overview that the larger the pump pulse width, the lower the threshold. It is explained that the larger the pulse width, the longer the accumulation time of the upper-level particle number, and the corresponding decrease of the threshold current required. The required threshold current exceeds the power supply range when the pump pulse is 150 µs.
With a comparison of the experimental result of the slab with the size of 150.2 mm×10 mm×2.5 mm, we find that pump power density of the 10 slab corresponding to the parasitic oscillation generates is in line with the experimental value of 30 slab. As the slab’s width is larger, the path which ASE light gets amplified becomes longer, meanwhile only the length of the axial direction is invariable. It indicates that parasitic oscillation occurs in the axial direction and ultimately limits the energy storage of the slabs.
4.2 Small signal gain coefficient of the slab gain module
The injected seed light is set with a particular angle to measure the small signal gain. The experimental configuration of the small signal gain coefficient is shown in Fig. 8.
The seed light is emitted by a fiber laser. The seed light output power is 0.81 W, the beam quality factor isM^2 = 1.5. The isolator is utilized to prevent light from returning. The power meter is far from the slab module and a light bean which is just enough to amplify the light is placed in front of the power meter to eliminate background fluorescence at different pump currents. The experimental results are shown in Fig. 9.
It can be observed in Fig. 9 that the small signal gain coefficient increases linearly with the current approximately when the pump power is below 10 kW. The calculation results are consistent with the experimental results and the relative error is small. When the pump power is high enough to reach saturation, the slab energy storage cannot be further increased for the existence of the PO. The output power remains unchanged. Through experimentation, the energy storage limit of the slab under quasi-continuous pulse pumping is 1100 mJ by calculating formula.
5. Conclusion
In conclusion, we theoretically analyze the ASE characteristics of the high power diode pumped slab gain module, indicating the contribution of the ASE is dominant and parasitic oscillation limits the energy storage. A theoretical model to depict ASE is proposed. The model rectifies the influence of thermal effect on stimulated emission cross-section, and considers the diffuse reflection on the rough surfaces. Compared with the past three-dimensional ASE calculation model, we make a more accurate approximation of the fluorescence boundary reflection conditions. Rationalization simplifications of the algorithm make the model maintain reasonable convergence time. The experimental results also verify the accuracy of the model calculation. The calculation model will provide favorable guidance for end-pumped slab amplification experiments in the engineering field. Finally, we obtain that the energy storage limit of the existing slab is 1100mJ.
Funding
Innovation Funding of China Academy of Engineering Physics and Key Laboratory of Science and Technology of High Power Laser.
Acknowledgements
This work was supported by the Key Laboratory of Science and Technology on High Energy Laser, CAEP.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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