The surface temperature of a single layer
The frequency domain solution for a semi-infinite solid which is heated by a point source with angular frequency  can be expressed by the following equation.
 where  (1)
(Λ: thermal conductivity of the solid, D: thermal diffusivity of the solid, r: radial coordinate)
In a typical time-domain thermoreflectance experiment, the co-aligned laser beams have cylindrical symmetry, therefore the 
Hankel Transform can be used to simplify the computation of the convolution of equation (1) with the distributions of the laser intensities.
(The Hankel transform is an integral transform equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel)
Here g(r) is radially symmetric and by the definition of Hankel transform using Eq. (1),
Since the pump and probe beams used here have Gaussian distribution, the  radius of the pump and probe beam are  and  respectively.
The surface is heated by the pump laser beam with the intensity , i.e.
where  is the amplitude of the heat absorbed by the sample at frequency .
Then the Hankel transform of   is
Then the distributions of temperature oscillations at the surface  is the inverse Hankel transforms of the product  and , i.e.
The surface temperatures are measured due to the change in the reflectivity  with the temperature , i.e. ,
while this change is measured by the changes in the reflected intensity of a probe laser beam.
The probe laser beam measures a weighted average of the temperature , i.e.
 (6a)
This last integral (6a) can be simplified to an integral over :
 (6b)
The surface temperature of a layered structure
In the similar way, frequency domain solution for the surface temperature of a layered structure can be acquired. Instead of Eq. (2), Eq. (7) will be used for a layered structure.
(Λn: thermal conductivity of nth layer, Dn: thermal diffusivity of nth layer, Ln: thickness of nth layer)
Using Eqs. (6) and (7), we can calculate the changes of temperature of a layered structure.
Modeling of data acquired in time-domain thermoreflectance
The acquired data from time-domain thermoreflectance experiments are required to be compared with the model.
(Q: quality factor of the resonant circuit)
This calculated Vf/V0 would be compared with the measured one.