Transient four-wave mixing (TFWM) is a nonlinear optical process that involves the interaction of multiple laser beams within a material with a third-order nonlinearity, such as an optical fiber. The underlying physical mechanism of TFWM can be described mathematically using the principles of nonlinear optics.

The nonlinear response of the material can be described by the nonlinear susceptibility tensor, which relates the induced polarization of the material to the applied electric field. In the case of third-order nonlinearity, the nonlinear susceptibility tensor is a third-rank tensor that describes the material’s response to the third power of the electric field.

Consider three input fields with frequencies ω1, ω2, and ω3, which are co-propagating through the material. The electric fields of these input fields can be represented as E1(t), E2(t), and E3(t), respectively. The electric field of the resulting output field, known as the signal field, can be represented as Es(t).

The interaction of the input fields can be modeled using the third-order nonlinear susceptibility tensor. The polarization P(t) induced by the electric field can be written as:

P(t) = ε0χ(3)E1(t)E2(t)E3(t)

where ε0 is the electric permittivity of free space, and χ(3) is the third-order nonlinear susceptibility tensor.

The induced polarization P(t) generates a new electric field, which can be expressed using the wave equation as:

∇^2Es – (1/vg^2)∂^2Es/∂t^2 = -μ0∂^2P(t)/∂t^2

where vg is the group velocity of the signal field, and μ0 is the magnetic permeability of free space.

The solution to this equation can be written as:

Es(t) = ∫∫∫ g(ω1, ω2, ω3) E1(t-τ1) E2(t-τ2) E3(t-τ3) dω1 dω2 dω3 dτ1 dτ2 dτ3

where g(ω1, ω2, ω3) is the complex coupling coefficient that describes the strength of the nonlinear interaction between the input fields, and τ1, τ2, and τ3 are the time delays between the input fields.

This equation represents the output signal field Es(t) as a convolution of the input fields E1(t), E2(t), and E3(t), with a nonlinear coupling coefficient g(ω1, ω2, ω3).

The frequency of the output signal field Es(t) is determined by the energy and frequency difference between the input fields. This frequency conversion property of TFWM makes it useful for applications such as wavelength conversion and optical signal processing.

In summary, the mathematical description of TFWM involves the use of the third-order nonlinear susceptibility tensor to model the interaction between multiple input laser fields within a material. The resulting output signal field is determined by the nonlinear coupling coefficient that depends on the frequencies and time delays of the input fields. This mathematical framework provides a theoretical basis for understanding and exploiting the properties of TFWM in a variety of applications in nonlinear optics and photonics.