  Laser cooling of rare-earth doped solids has been demonstrated across a wide
range of material platforms, inspiring the development of simple
phenomenological models such as the four-level model to elucidate the universal
properties of laser cooling under various operating conditions. However, these
models usually require the input of full absorption spectra that must be
provided experimentally or by additional complicated atomic modeling. In this
letter, we propose that a four-level model, when extended to admit effective
energy levels adaptive to the pumping photon energy, can accurately predict the
cooling efficiency as a function of temperature and pumping frequency using
only few inputs such as the absorption coefficient measured at a single
frequency and temperature. Our model exploits the quasi-equilibrium properties
of the excitation of rare-earth ions for the determination of the effective
four energy levels. The model is validated against published experimental
results for a span of materials including ytterbium/thulium-doped glass and
crystals. With the verified model, we derive explicit expressions for the
optimal frequency and the operating bandwidth of pumping laser. Our model
significantly simplifies the modeling process of laser cooling, and is expected
to stimulate further development of optical refrigeration.
