We demonstrate a high reflectivity (> 99%), low-loss (< 0.1%) and aberrations-free (2% of λ rms phase fluctuations) concave Bragg mirror (20mm radius of curvature) integrating a photonic crystal with engineered spherical phase and amplitude transfer functions, based on a III-V semiconductors flat photonics technology. This mirror design is of high interest for highly coherent high power stable external cavity semiconductor lasers, exhibiting very low noise. We design the photonic crystal for operation in the pass band. The approach incorporates spatial, spectral (filter bandwidth= 5nm) and polarization filtering capabilities. Thanks to the mirror, a compact single mode TEM(00) 2mm-long air gap high finesse (cold cavity Q-factor 10(6) - 10(7)) stable laser cavity is demonstrated with a GaAs-based quantum-wells 1/2-VCSEL gain structure at 1μm. Excellent laser performances are obtained in single frequency operation: low threshold density of 2kW/cm(2) with high differential efficiency (21%). And high spatial, temporal and polarization coherence: TEM(00) beam close to diffraction limit, linear light polarization (> 60dB), Side Mode Suppression Ratio > 46dB, relative intensity noise at quantum limit (< -150dB) in 1MHz-84GHz radio frequency range, and a theoretical linewidth fundamental limit at 10 Hz (Q-factor ∼ 3.10(13)).