非线性光纤光学中非线性薛定谔方程推导逻辑的探讨

DISCUSSION ON THE LOGIC OF THE DERIVATION OF NONLINER SCHRÖDINGER EQUATION IN NONLINEAR FIBER OPTICS

非线性薛定谔方程是描述光脉冲在光纤中非线性传输的基本方程,广泛应用于光纤通信、光纤激光器等领域。在现有的非线性光纤光学的经典教材中对非线性薛定谔方程的推导,采用的是直接在非线性介电常数中引入非线性折射率效应。本文指出了该方法对于传输方程非线性项处理的局限甚至不妥之处,给出了一种非线性薛定谔方程较严格的推导方法,即从非线性的波动方程出发,讨论不同非线性效应的非线性极化强度驱动源的不同表现形式,并将方程作横向积分处理。本文详细阐述了从非线性源的角度出发推导非线性薛定谔方程的方法,并以四波混频等非线性效应为例,帮助读者理解非线性效应对于传输方程的影响;此外,还探讨了在推导非线性薛定谔方程中的正负频表示等问题。

The nonlinear Schrödinger equation is the basic equation describing the nonlinear transmission of optical pulses in optical fibers,and is widely used in optical fiber communications,fiber lasers and other fields.In the existing classical textbooks on nonlinear fiber optics,the nonlinear Schrödinger equation is derived by directly introducing the nonlinear refractive index effect into the nonlinear dielectric constant.Herein,the limitations and even inappropriateness of this method for dealing with the nonlinear term of the transmission equation are pointed out.A more rigorous derivation of the nonlinear Schrödinger equation is given,i.e.,starting from the nonlinear wave equation,discussing the different manifestations of the nonlinear polarization driving sources for different nonlinear effects,and making the equation a transversal integral.The method of deriving the nonlinear Schrödinger equation from the perspective of nonlinear sources is elaborated in detail,and examples of nonlinear effects,such as four-wave mixing,are given to help readers understand the influence of nonlinear effects on the transmission equation.In addition,issues such as positive and negative frequency repre-sentations in deriving the nonlinear Schrödinger equation is also discussed.