Introduction to Information Optics (6) Mathematical Description of Light wave propagation

what is scalar diffraction theory ？ Diffraction: It is defined as “can not be used to explain any deviation from a straight line on the optical path by the reflection or refraction of light of.” by Sommerfel (索末菲定义) The conditions of scalar diffraction theory： （1） Diffraction aperture is much larger than the wavelength.（衍射孔径比波长大很多） （2）Observation point keep away from diffraction aperture;（观察点离衍射孔不太靠近）

Classical scalar diffraction theory was first proposed by Huygens（惠更斯） in 1678. Fresnel introduced the interference concept in 1818 for complementing Huygens principle. In 1882, Kirchhoff （基尔霍夫）used Green theorem(格林定理) to derive a strict scalar diffraction formula by spherical wave as the wave equation Green function.

Primary contents Scalar wave equation Complex amplitude of the vibration of optical wave Helmholtz equation Spherical wave Plane wave The spatial frequency of plane wave The angle spectrum theorem of plane wave

Scalar wave equation Electric or magnetic field component u as the space and time function meet the passive scalar wave equation on point of any space. （标量波动方程） Where (拉普拉斯算子) Electromagnetic propagation velocity in the medium and is dielectric constant and the magnetic permeability of the medium.（介电系数和磁导率）

A linear combination of solutions to meet the basic solution of the equation is the solution of equations. 满足该方程的基本解的线性组合都是方程的解 Spherical wave and plane wave are the fundamental solution of the equation. Any complex wave can use a linear combination of spherical wave and plane wave satisfied for equation. 任何复杂的波都可以用球面波和平面波的线性组合表示，也都是满足波动方程的解

The definition of the complex amplitude of the vibration of optical wave Take the most simple harmonic vibration as special solution of wave equation, the monochromatic light vibration can be expressed as the optical vibration can be represented by a complex exponential function, the formula becomes It will be simplify

It is called the complex amplitude of the monochromatic light field, which contains the point light vibration amplitude and initial phase, only relating to position coordinates, regardless of the time. The complex amplitude of the light intensity can be used to represent

Helmholtz equation(亥姆霍兹方程) Complex exponential function replace a cosine function of the vibration of light. In the operation real component of complex exponential function is equal to cosine function. 在运算的任何一个阶段对复指数函数取实部，与直接用余弦函数进行运算在同一个阶段得到的结果是相同的 So the complex amplitude wave equation can be simplified to Where k is called wave number, represent a phase change of per unit length, defined as

After the reduction ,wave equation is called the Helmholtz equation (化简后的波动方程称为亥姆霍兹方程),not relate to the time. In the free space ,any complex amplitude of the disturbance of monochromatic light must meet Helmholtz equation. This also means that the complex amplitude distribution not relate to the time describe monochromatic light wave field。 可以用不含时间变量的复振幅分布完善地描述单色光波场

The spherical wave complex amplitude Light waves of a point source propagate in isotropic medium form spherical wave. A complex light source often can be seen as a collection of point light source as well as spherical wave superposition; 一个复杂的光源常常可以看做是许多点光源的集合,它所发出的光波就是球面波的叠加 The incoherent point light source are superposition of intensity , the coherent point light source are superposition of complex amplitude when together. (不相干时是光强的叠加,相干是复振幅的叠加)

When the coordinates of the origin is coincidence with the center of spherical wave , complex amplitude of monochromatic divergent spherical wave can be writted （发散球面波） （会聚球面波） a0 is the amplitude of unit distance from the point light source For a converging spherical wave，

Equal phase lines of spherical wave in the plane

Spherical wave complex amplitude distribution in the plane When a point light source or converge point in any position, there Optical field distribution of x-y plane distance from the point .r can be written as If 泰勒公式：f(x)=f(a) + f'(a)(x-a)/1! + f''(a)(x-a)^2/2! + …… + f(n)(a)(x-a)^n/n!

Using the binomial expansion, and omit the high order item, get it （傍轴近似） Substitute divergent spherical wave equation with the approximate expression, get the plane wave complex amplitude distribution on the plane 思考，公式中的近似条件为何位相里面不考虑成r=z 常量位相因子 二次位相因子

The equipotential lines and phase factor of spherical wave The distribution of complex amplitude phase factor of divergent spherical wave in the plane included :   The constant phase factor is associated with propagation distance. The second item changing with plane coordinates is called secondary phase factor（二次位相因子） of spherical wave, when the expression of complex amplitude distribution include the following factors .

It means in the distance from the plane , there is a spherical wave from point light source through the plane. The trajectory of same phase point , namely the equal phase line equation for concentric circles (等位相线方程为同心圆族)

The plane wave equal phase lines on the x-Y plane

The plane wave complex amplitude At any time, amplitude and phase of plane wave perpendicular to the wave vector is constant. Light wave vector ,the direction cosine a general expression of complex amplitude of plane wave propagating to space point

a is constant amplitude. the direction cosine meet So complex amplitude can be written as Among them

The plane wave phase factor and equipotential lines As similar with the expression of spherical wave ,the plane wave complex amplitude can be divided into two parts: related to coordinate and not related to coordinates; Related to the coordinate plane wave represent linear phase factor(表征平面波特点的线性位相因子). when expression of plane wave complex amplitude included the distribution factor, it’s a plane wave with a direction cosine go through the plane.

Plane wave equal phase line equation Therefore, equal phase lines are parallel lines. 等相位线是一条直线 In front of the picture, the intersection of a wave plane with phase difference 2pi by dotted line and plane x-y is equal phase line ; They are a set of parallel equidistant skew lines 不同C值所对应的等位相线是在xy平面内的一系列平行线，斜线方向的不同对应于不同传播方向的平面波（由cosα和cosβ）决定。

The plane wave spatial frequency Direction x,y,z ,the plane wave space frequency is defined The general expression of plane wave complex amplitude

The reciprocal of spatial frequency is the period of oscillation (X, Y, Z) Spatial frequency represent the numbers of unit complex amplitude cycles . 空间频率表示在 X, Y, Z轴上单位距离内的复振幅周期变化的次数 This is the physical significance of the plane wave spatial frequency. (平面波空间频率的物理意义) Spatial frequencies associated with the direction of propagation of the plane wave. angle between the wave vector and the axis is more greater ,the λ in the projection axis is more larger ,the spatial frequency is more smaller.

The physical meaning of the spatial frequency The spatial frequency of transmission vector k in the plane of the plane wave in the plane x,y.

Two meaning of the spatial frequency Time reciprocal（时间倒数）: frequency; Length reciprocal : spatial frequency, in the unit length periodic function changes the number of cycles. （空间频率，即在单位长度内周期函数变化的周数） (unit: weeks/mm, line to/mm, L/mm, etc.) There are two kinds of spatial frequency in information optics, one is spatial intensity distribution（空间强度分布）, the unit is: cycles/mm, line /mm, L/mm, etc.

The other is a plane wave corresponding spatial frequency, because the wavelengths of the electromagnetic waves in a homogeneous medium is constant, spatial frequency is constant in the direction of propagation . And the three-dimensional coordinates of each direction of spatial frequency (unit: light/mm) express the direction of propagation of electromagnetic waves, or its direction of propagation and the axis Angle, and the size is limited by wavelengths, the largest is the reciprocal of wavelength 表示出的意义实际上是电磁波的传播方向，或其传播方向与坐标轴的夹角，而且大小受到光波长的限制，最大是波长的倒数。

Example of the plane wave angle spectrum(平面波角谱)  A plane wave complex amplitude expression Try to calculate its wavelength and spatial frequency along the direction x y z ,give the complex amplitude distribution of the plane perpendicular to the z axis at z=5mm . ( ) Solution: so

Unit of circular frequency 4,-3,4 is , the corresponding spatial frequency In , complex amplitude distribution of the plane perpendicular to the z axis the unit of x,y is millimeter (mm)