Compact modulation transfer spectroscopy module for laser frequency stabilization in transportable rubidium fountain clocks

We design and demonstrate a compact modulation transfer spectroscopy (MTS) laser frequency stabilization module for transportable rubidium atomic fountain clocks. The module measures only 140 mm × 120 mm × 60 mm and employs a hybrid fiber–free–space optical path that eliminates polarizing beamsplitters and mirrors. It simultaneously meets the frequency–locking requirements for both the cooling light ( 87 Rb D 2 line, F = 2 → F′ = 3) and the repumping light (D 2 line, F = 1 → F′ = 0,1 crossover peak). For the first time, we systematically characterize the distinct dependence of the error signal on probe power under parallel versus orthogonal linear polarizations, and based on this we optimize the polarization configuration. Together with our intensity–noise–equivalent laser power (INELP) method, we optimize the optical power and vapor cell temperature. As a result, the cooling transition achieves a frequency stability (Allan deviation) of 2.5 × 10⁻ 13 at 1 s integration time, and the repumping transition reaches 6.51 × 10⁻ 13 . For integration times up to 10 4 s, the stability of both lines remains below 5 × 10⁻ 12 . To the best of our knowledge, this work represents the first demonstration of frequency stabilization at the 10⁻ 13 level for transitions associated with the F = 1 → F′ = 0,1,2 manifold. The module provides a high–performance frequency reference suitable for various dual–wavelength architectures in rubidium fountain clocks, including independent dual lasers, master–slave optical phase–locked loops, and single–laser high–frequency modulation sideband schemes.