Principle and application of optical information technology (Thirteen) Comparison and calculation of CTF and OTF

A comparison method of coherent and incoherent imaging Some comparisons of coherent and non coherent imaging are made: Cutoff frequency size(截止频率大小): Incoherent system: the maximum frequency of the intensity that can be passed by cosine change .强度呈余弦变化 Coherent system: the maximum frequency of the complex amplitude that can be transmitted .复振幅呈周期变化

Spectral characteristics of image intensity(像强度的频谱特性): In the two case the spectrum of image intensity may vary, but it is still not possible to conclude which is better. Because the imaging results not only depend on the structure of the system and the coherence of the illumination light, but also related to the spatial structure of the object. The ability to distinguish between two points(两点之间的分辨本领) : According to the Rayleigh criterion, incoherent point sources of two equal intensity, the first zero coincidence if a point source of Airy spot center exactly with the second point source of the Airy pattern, is that the two point source was able to distinguish. 瑞利分辨判据仅适用于非相干成像系统

Resolution ability of coherent and incoherent imaging systems The intensity distribution of the two incoherent point sources. Intensity distribution of coherent point source in the two upper coherent point source.

Comparison between incoherent and coherent imaging Through the optical system with a lateral magnification of 1, the exit pupil of the system is a circular aperture with a radius of a, the distance from the exit pupil to the image plane is . λ is the illumination of light wavelength.比较用非相干照明与相干照明成像质量 the complex amplitude transmittance of the object is

Among the formula

The comparison results (1) By using coherent illumination, the cutoff frequency of the circular exit pupil with a radius of a is Condition of the title λ di/b<a<2λdi/b, The object function is expanded into Fourier series.

The fundamental frequency基频 is 2/b>ρ c , 在相干照明下，成像系统只允许零频分量通过，而其它频谱分量均被挡住，所以物不能成像，像面呈均匀强度分布。 In the incoherent illumination condition, the cutoff frequency 2ρc of the system is greater than the frequency of 2/b, so the zero frequency and fundamental frequency can be through participation in imaging system. So there are still images in the image plane. Incoherent illumination is better than coherent illumination. （非相干照明比相干照明好）

The comparison results (2) For coherent illumination, the complex amplitude distribution of the ideal image is ,its frequency is 1/b, according to the problem , the cutoff frequency of the system is moreover 1/b<ρ c . 这个呈余弦分布的复振幅能不受影响地通过此系统成像。 For incoherent illumination, the intensity distribution of the ideal image is the frequency of 2/b, according to the problem of 2/b<2ρc, which is less than the incoherent cutoff frequency. 故此物也能通过系统成像，但幅度要受到衰减。 在这种物结构下，相干照明好于非相干照明。

Examples A cosine type amplitude grating（余弦型振幅光栅）, the complex amplitude transmittance is On the surface of the imaging system, oblique illumination with monochromatic light, the plane wave propagation direction is in the plane, and the angle between the plane wave and the Z axis is in the plane. Lens focal length is , aperture is . The frequency spectrum of the transmission light field of the object; How much is the maximum angle of the stripe in the image plane? Image surface intensity distribution.

Answer (1) monochromatic inclined illumination light can be expressed as The transmission light field of the object can be expressed as

The spectrum of transmitted light field of the object is (2) the image surface intensity distribution can be applied to the convolution formula

Because in this optical magnification for one, the formula like geometrical image can be expressed as Point-spread function is 这里的光瞳函数已经是以空间频率为自变量的光瞳函数。归一化的点扩散函数是以空间频率为自变量的光瞳函数的傅氏变换。

The convolution theorem can be applied to calculate the intensity distribution of the image surface 几何像的傅氏变换可用前面计算的物的频谱

The frequency spectrum of the point-spread function is Fourier transform geometry image is like three functions, the inverse transform is plane wave. 要使像平面出现条纹时，频谱中至少要有两个 函数能够与出瞳函数乘积不为零，变换成为平面波射到物面上相干涉形成条纹。

Exit pupil here is the lens box, so the requirement and the pupil function product is not zero. The position of the other function is farther than the first one, and only needs two functions. 采用近轴光学近似时，角度的正弦等于其弧度值, and the left and right sides of the above two inequalities are required.

The maximum value given in the previous, , as the geometrical Fourier transform in three of only two, the two functions and the pupil function of the product also function, but also because the pupil function in the pupil range as a function of two , before the coefficient is constant. The optical field distribution on the image plane can be obtained by the inverse transform The intensity distribution is the square of the light field distribution

(2)使像平面出现条纹时，物体透射光场的频谱中至少要有两项能够通过透镜的出瞳，射到物面上成像。The next lesson we have to talk about the nature of the pupil's low pass, here we have to answer from the equivalent principle. In the imaging optical path, in the back focal plane of the light bar and the diameter of the lens frame formed by light bar equivalent. 后焦面作为观察平面时，光场复振幅除一个相位因子外，就是物体透射光场的傅里叶变换即其频谱。 so in the back focal plane of optical field for the three use function in the spot the three spot in the spatial frequency

空间频率相对应的后焦面上的坐标是 Considering the near axis optical approximation, the angle of the sine is equal to the value of the arc, which is located in the three points of light.

(2) 使像平面出现条纹时，Clearly required Two spot can be passed in the back focal plane diameter of the equivalent light bar. Moreover 这时只可能有一个光点通过等效光栏射到物面上成像，形成等效的平行光照明的现象。