Principle and Applications Of Optical Information Technology (七) The angular spectrum theory of scalar diffraction

The propagation of the plane wave complex amplitude Three spatial frequency is not separated, Because So The plane wave complex amplitude can be written to

Among The equation expressed distance z plane complex amplitude distribution, in the plane of z=0,complex amplitude is given by a complex exponential function related to propagation distance and direction. It shows that the influence of complex amplitude distribution in propagation, has solved the fundamental problem of plane wave diffraction.

The angular spectrum distribution of complex amplitude Two-dimensional Fourier transform of the complex amplitude distribution is the frequency spectrum distribution . A monochromatic light(单色光) wave project onto x , y plane along the z direction, the optical field distribution at z distance is the frequency spectrum distribution（频谱分布） by a two-dimensional Fourier transform The different spatial frequency space Fourier component is considered to be a plane wave propagation along different direction

A(x , y , z)称为复振幅分布U(x , y , z)的空间频谱,即角谱。 so spatial spectrum is called angular spectrum of complex amplitude distribution .空间频谱为平面波谱即复振幅分布的角谱 Its inverse transformation Monochromatic light wave field distribution in a plane respectively, can be regarded as a superposition of different propagation direction plane wave ,their amplitude and phase is magnitude and angle of angular spectrum. (单色光波在某一平面上的光场分别可以看作是不同传播方向的平面波的叠加，在叠加时各平面波有自己的振幅和位相，它们的值分别为角谱的模和幅角) A(x , y , z)称为复振幅分布U(x , y , z)的空间频谱,即角谱。

The spread of the Angle of the plane wave spectrum Space spectrum of complex amplitude distribution express by angle of plane wave propagation direction The optical field distribution of plane z=0 and of z=z can be respectively 孔径平面场与角谱 观察平面 场与角谱

The spread of the complex amplitude distribution and angular spectrum

Propagation law from the Helmholtz equation According to the above , study the spread of the angular spectrum is to find the relation between z=0 and z=z angular spectrum. The expression in Helmholtz equation , the equation can be deduced   because 二维线性微分方程  

The basic solution of the second order ordinary differential equation The z=0 plane Angle spectrum Finally get the angle spectrum propagation law between z=0and z=z

The meaning of two parallel plane angle spectrum propagation law The angular spectrum according to the optical field distribution can be deduced by the two parallel plane angle spectrum propagation law . Through Fourier inverse transform（傅里叶逆变换） to calculate the optical field distribution . Angle spectrum analysis method is mathematical model of the diffraction of light propagation in the free space. 自由空间衍射的数理模型，即光传播的角谱分析方法

Propagation and evanescent wave (传导波和倏逝波) When the propagation direction（一般为z方向）cosine meet （传导波分析） After distance z , just the relative phase of each angle spectral component change. Define a phase delay factor（位相延迟因子） 这是由于每个平面波分量在不同方向上传播，它们到达给定的点所经过的距离不同。

when , square root of angular spectrum spread equation is imaginary, get it Among them If it is a positive number , it indicated that meet the component of the volatility will decrease by the exponential decay with the increase z.（将随Z的增大而按指数 衰减） In several wavelength range ,the wave soon decay to zero called evanescent wave.

The transfer function of free space propagation Under the condition of the scalar diffraction approximation theory, evanescent wave is always ignored, because so the transfer function can be expressed as 输出频谱 输入频谱 传递函数

Therefore, the light wave propagation can be a spatial filter(可以把光波的传播现象看作一个空间滤波器). It has a limited bandwidth(具有有限带宽) (see chart).In the circular area of frequency space, the magnitude of the transfer function is 1 and has no effect on the amplitude of each frequency component. Smaller than the wavelength, the fine structure can’t spread in the original direction under monochromatic light illumination. Light travels in free space, the ability to carry information is limited.

Limited spatial bandwidth of free space 在频率平面上半径为1/的圆形区域内，传递函数的模为1，对各频率分量的振幅没有影响，但引入了与频率有关的相移； 在圆形区域之外，传递函数为零。

Angle spectrum of different diffraction aperture In the figure , in a hole in a infinite opaque screen(不透明屏), the transmission function of the hole is:

Along the propagation direction, the complex amplitude of light incident on the aperture is ，the complex amplitude of emergence light field close to the aperture plane is so Through Fourier transform, angular spectrum can be represented as 时域的乘积对应频域的卷积，卷积使得频谱展宽。

The convolution operation can broaden bandwidth（带宽展宽） The convolution operation can broaden bandwidth（带宽展宽）. Therefore, the space limited diffraction effect is broadening the angle spectrum of the light . 受限制的衍射孔径的效应就是展宽了光波的角谱 Angular spectrum broadening is containing in the emergent wave of the same direction in addition to different direction light wave as well as high spatial frequency wave called the diffraction wave. 由于衍射孔径使得频谱展宽，增加了一些高空间频率的波，即衍射波，也是级次波的由来。

Diffraction Huygens –Fresnel-Kirchhoff The development of Scalar diffraction theory :the classic scalar diffraction theory was proposed by Huygens originally in 1678, put forward by the Huygens ,Fresnel introducing the concept of intervention in 1818.kirchhoff used green's theorem in 1882, using spherical wave as green's function for solving wave equation, to deduce strictly scalar diffraction formula. 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其它各点的复振幅表示出来。显然，这是一个根据边界条件求解波动方程的问题。

Huygens - Fresnel proposed wavelet interference principle are almost consistent with Kirchhoff solution of wave equation as similar 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，都可以表示成类似的衍射公式

Light diffraction of point source through the screen The diffraction formula Inclination factor Complex constant

Fresnel diffraction formula Light diffraction formula can be applied to any monochromatic illumination. Any complex waves can be decomposed into linear combination of the simple spherical wave. According to the Kirchhoff ‘s boundary conditions(基尔霍夫边界条件) , so the integral limit can be extended to the infinite

Under the paraxial approximation, by the binomial approximation Fresnel diffraction formula by the above approximation 近似得到菲涅耳衍射计算公式

The plane wave angular spectrum diffraction theory Angular spectrum can be obtained by Fourier transform of the known plane optical field distribution the angular spectrum of plane z=z Through inverse Fourier transform diffraction light field distribution can be gained. 本章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 basic formula of plane wave angular spectrum diffraction theory Fourier inverse transform Angular spectrum expression of the diffraction plane

Different diffraction area according to the propagation distance

Fresnel formula by the theory of angular spectrum diffraction Assume that the distance between the aperture and the observation plane is greater than line width of the aperture and only a small observation area, 假定孔径和观察平面之间的距离远远大于孔径的线度 there So

Using binomial expansion, keep first harmonic, ignore high harmonic, then so

Fresnel formula by the theory of angular spectrum diffraction Through the gauss function Fourier transform and Fourier transform similarity (利用高斯函数的傅里叶变换和傅里叶变换的相似性定理)     so

The equation is exactly same with Huygens - Fresnel - Kirchhoff scalar theory of Fresnel diffraction formula, the more common Fresnel diffraction formula : 注意：在菲涅尔衍射区观察到的光场，除了与（x0，y0）无关的一些振幅和位相因子外，可以看成 的傅里叶变换

Fresnel diffraction conditions 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