Principle and application of optical information technology (Eight) Fresnel and Fraunhofer diffraction

Huygens-Fresnel-Kirchhoff scalar diffraction theory 基尔霍夫利用格林定理，采用球面波作为求解波动方程的格林函数，导出了严格的标量衍射公式 The problem to be solved by the diffraction theory is that the complex amplitude of any point in the light field can be expressed by the complex amplitude of the other points in the light field. 衍射理论要解决的问题是：光场中任意一点为的复振幅能否用光场中其它各点的复振幅表示出来

Obviously, this is a problem of solving wave equation according to the boundary condition. 这是一个根据边界条件求解波动方程的问题 Wavelet interference principle proposed by Huygens and Fresnel is very is very consistent with Kirchhoff's result of solving wave equation, which can be expressed as similar to the diffraction formula. 惠更斯—菲涅尔提出的子波干涉原理与基尔霍夫求解波动方程所得的结果十分一致，可表示成类似的衍射公式.

Point-source illumination of a plane screen Diffraction formula Tilt factor（倾斜因子） Complex constant

Fresnel Diffraction Calculations According to the boundary condition of Kirchhoff's assumption on the plane screen, the integral limit in the shadow area beyond the aperture can be extended to infinity. In the paraxial approximation(傍轴近似), using the binomial approximation（二项式近似） The above approximated calculations were plugged to obtain the Fresnel Diffraction Calculations.

Diffraction Theory of Angular Spectrum of Plane Waves 本书的重点是从频域的角度即用平面波角谱方法来讨论衍射问题. The propagation of the angular spectrum in the frequency domain has been discussed . The angular spectrum can be obtained by distribution of light field and Fourier transform

Therefore , the angular spectrum propagating to can be calculated Finally, the diffracted light field distribution is obtained by Fourier inverse transform, which can be used to obtain the diffraction formula in the airspace.

Diffraction Theory of Angular Spectrum of Plane Waves Using Fourier inverse transform We plug it into the expression for the angular spectrum of the diffraction plane 上式的四重积分是类似基尔霍夫公式的一个精确的表达式。下面还是要按照菲涅耳的办法进行化简，首先对不同传播距离衍射的情况做个直观的说明。

Dividing diffraction area by propagation distance Far field region 投影区 Near field region

Fresnel Equation Deduced by Diffraction Theory of Angular Spectrum It is assumed that the distance between the aperture and the observation plane is much larger than dimension of the aperture, and it is observed only in a small area near the axis. Therefore, Thus By using the binomial expansion, only one item, omitting the higher order term

Using Fourier transform of Gauss's function and similarity theorem of Fourier transform Therefore, the four integration can be expressed as

Therefore The equation is the same as Fresnel diffraction equation deduced by the Huygens-Fresnel-Kirchhoff scalar diffraction theory. The more commonly used Fresnel diffraction formula is as follows

Conditions for the establishment of Fresnel diffraction The condition of Fresnel diffraction so So observation distance need to meet

The maximum size of the aperture and observation area, is respectively This approximation is called Fresnel approximation or paraxial approximation(菲涅耳近似或近轴近似), the transfer function can be expressed as

Fraunhofer diffraction and Fourier transform

Fraunhofer diffraction（夫朗禾费衍射）: In the Fresnel diffraction formula, taking stronger restrictions on the diffraction hole, namely （夫朗禾费近似） The square phase factor is approximately 1 on the whole aperture, and （夫朗禾费衍射公式）

In the Fraunhofer approximation, the field distribution on the observation plane is equal to the product of Fourier transform of field distribution on diffraction aperture and quadratic phase term. 观察面上的场分布等于衍射孔径上场分布的傅里叶变换和一个二次位相因子的乘积。 For the general light detector which only responds to the light intensity and does not respond to the phase, there is no difference between Fraunhofer diffraction and Fourier transform of the light.

Fraunhofer diffraction pattern of a rectangular hole, single slit and hole

Example of Fraunhofer diffraction 1. Diffraction of a rectangular hole and single slit 2. Diffraction of double slit 3. Diffraction of circular