Analysis of multi-pass pumped thin-disk laser performance with measured disk deformation

Technical Summary: Analysis of Multi-Pass Pumped Thin-Disk Laser Performance with Measured Disk Deformation

Problem Statement
High-power thin-disk (TD) lasers are renowned for their superior thermal management and beam quality; however, their signal power does not scale linearly with pump power at high intensities. While energetic models based on rate equations can predict population inversion and signal extraction, they often fail to account for the spatially varying resonator mode behavior induced by thermally deformed gain media. Specifically, existing models struggle to explain output-power reductions that occur even when the laser maintains a near-diffraction-limited beam quality ($M^2 \approx 1.1$). The central challenge is to determine how disk deformation, beyond simple curvature changes, alters intra-cavity mode formation and contributes to power limitations in the single-mode regime.

Methodology
The authors developed a comprehensive numerical framework that couples a non-lasing multi-pass pump absorption model with two-dimensional (2D) intra-cavity field propagation. The methodology proceeds in two distinct stages:

1. Steady-State Absorption Initialization:

   • The model derives an effective pump absorption coefficient ($\alpha_{eff}$) under non-lasing conditions to account for spatial inhomogeneities in population inversion.
   • Using a quasi-three-level rate equation for Yb:YAG, the model incorporates absorption saturation, gain saturation, and temperature-dependent cross-sections.
   • An implicit steady-state equation is solved to determine the effective absorption coefficient as a function of incident pump intensity and the number of pump passes ($n_{pass}$). This initialization provides a spatially resolved map of population inversion ($\beta$) and single-pass gain prior to lasing, avoiding the computational cost of full iterative coupling during the initial pump phase.

2. 2D Cavity Propagation with Measured Deformation:

   • The initialized gain profile is coupled with a 2D field propagation model. The intra-cavity electric field evolution is simulated by iterating a round-trip operator that includes Rayleigh–Sommerfeld diffraction and the complex phase shift induced by the disk surface.
   • Crucially, the disk-induced phase is not modeled as a simple lens (defocus) but is derived from experimentally measured Optical Path Difference (OPD) maps. These maps capture mechanical, thermal-expansion, and air-convection effects.
   • To isolate the impact of higher-order aberrations, the measured OPD is decomposed using Zernike polynomials. A comparative simulation is performed using only the defocus term (converted to an equivalent radius of curvature, eROC) versus the full measured OPD.

Key Contributions

• Non-Lasing Pump Initialization: The paper introduces a method to initialize the 2D cavity model using a derived steady-state absorption coefficient from a multi-pass pump model. This approach efficiently captures pump saturation and spatial inversion without requiring iterative absorption equilibration during the signal propagation loop.
• Measured OPD Integration: Instead of relying on theoretical thermal deformation models, the study utilizes experimentally measured OPD maps to define the cavity phase. This allows for a direct correlation between physical disk deformation and laser performance.
• Aberration-Resolved Analysis: By comparing simulations using only the defocus term (eROC) against those using the full measured OPD, the study quantitatively isolates the contribution of higher-order aberrations to power loss.

Results
The model was validated against experimental data from a 32-pass Yb:YAG thin-disk laser system:

• Accuracy: The model reproduces the residual pump fraction, signal power, beam diameter, and $M^2$ with errors of 3.0%, 1.7%, and 0.05, respectively.
• Power Reduction Mechanism: Simulations using only the eROC (defocus-only) predict a higher theoretical signal power than observed. When the full measured OPD is applied, the simulated signal power decreases by up to 19.3% as pump intensity increases.
• Role of Higher-Order Aberrations: Zernike decomposition reveals that as pump intensity increases, higher-order spherical aberrations (primary, secondary, tertiary, etc.) grow significantly. These aberrations, distinct from simple defocus, reduce the overlap between the circulating field and the fundamental cavity mode, effectively increasing round-trip loss.
• Mode Deviation: The study shows that the circulating mode size in the presence of measured OPD deviates from the paraxial eigenmode predicted by ray transfer matrix (RTM) analysis using eROC, with mode size reductions of up to 3.6% at high pump intensities.

Significance and Claims
The paper claims that the limitation on power scaling in high-power single-mode thin-disk lasers cannot be assessed solely through pump-energy extraction or equivalent curvature models. The authors identify aberration-induced modal loss as a key limiting factor. Even before significant degradation in beam quality ($M^2$) is observed, higher-order disk aberrations hinder fundamental-mode formation, leading to signal power reduction.

Consequently, the study concludes that accurate prediction and optimization of high-power TD lasers require cavity analysis based on the measured optical state of the disk, specifically resolving aberrations beyond defocus. The proposed framework serves as a practical predictive tool for understanding output characteristics under monochromatic pump and signal assumptions, bridging the gap between energetic efficiency and spatial mode behavior.