Deduction and demonstration of Fourier optics

It is a branch formed by introducing communication theory, especially Fourier analysis method into optics. There are many similarities between optical information system and electronic information system. Both of them collect and transmit information, and both have the same mathematical tools-linear system theory and Fourier analysis. From the perspective of information theory, it focuses on the process of information transmission in the system; Similarly, for an optical system, according to the scalar diffraction theory, the relationship between objects and images can also be determined by the propagation of light fields in the system, so the optical system can be regarded as a communication channel. In this way, the mature linear system theory in communication theory can be used to describe most optical systems. When an object is illuminated by incoherent light, the light intensity distribution on the image plane of the system is linear (proportional) to that on the object. The concept of impulse response h(t, τ) is used to describe the response of electrical system to narrow pulse δ(t) (Dirac δ function), and it can also be used to describe the response of optical system to point light source δ(x, y) (image of point light source) H (x, y; ξ, η) to describe the properties of the system. The only difference between them is that the impulse response of electrical system is a one-dimensional function in time, and the impulse function of optical system is a two-dimensional function in space. Both of them are displacement-invariant, the distribution of the former does not change with time displacement, and the distribution of the latter does not change with space displacement (that is, the condition of equal halo). The impulse response of optical system is also called point spread function (see optical transfer function). Once the point spread function of the system is known, the image g(ξ, η) formed by the system for any object f(x, y) can be obtained by the linear superposition of the point spread functions generated by each point source on the object. When the spatial displacement is constant, the superposition integral can be simplified as convolution. Spatial frequency in information theory, the concept of frequency response is also commonly used, that is, input signals of different frequencies and observe the corresponding output of the system, and we can understand the transmission of various frequencies from the frequency response curve. In addition to replacing instantaneous frequency response with spatial frequency response, the concept of frequency response can also be introduced into optical systems. The instantaneous frequency is the reciprocal of the period of the time function acos ω t.

Similarly, the reciprocal v= 1/d (unit: line /mm) of the period d of a spatial function can be defined as the spatial frequency. Taking the simplest object-grating as an example, it can be expressed by the function 1+Acos(2πvx), where v= 1/d and d is the grating constant.

According to Fourier analysis, any complex object f(x, y) can be written as Fourier transform relation.

Where F(vx, vy) is the spatial spectrum of the object. Its physical meaning is to decompose the complex number f(x, y) into many simple elementary functions.

The spatial spectrum F(vx, vy) is just a weight factor, which is added to the corresponding original function. The original function can be more vividly regarded as gratings with different orientations [θ = TG- 1 (vy/VX)] and different spatial periods L= (figure 1), and the proportion of each grating in the objective function is determined by the weight factor-spatial spectrum F(vx, vy). In this way, the response of an optical system to f(x, y) can be decomposed into responses to various elementary functions, and then the total response can be obtained by superimposing each response. Similarly, the inverse transformation can also be written as. For a known object f(x, y), its spatial spectral distribution can be calculated.

The Fourier transform property of lens can be known from scalar diffraction theory. Considering the paraxial approximation condition, in the Fresnel diffraction (near field) region, the relationship between the aperture plane (x, y) and the light field (ξ, η) on the observation plane is as follows.

This is called Fresnel transform. Where f(x, y) is the amplitude of the light field on the diffraction aperture plane, g(ξ, η) is the amplitude of the light field on the observation plane, с is a constant phase factor, u=2πξ/λz, μ = 2π η/λ z is the spatial angular frequency, and z is the distance between planes. The above formula may

Mutual Fourier transform relation

This is a quadratic phase factor. When the observation plane is far away from the diaphragm plane, the above formula becomes Fraunhofer diffraction (far field).

The diffraction image g(ξ, η) at this time is the Fourier transform of the light field distribution f(x, y) on the aperture plane, or the spatial spectrum of f(x, y). What is interesting is the transmittance function of the thin convex lens.

(where f is the focal length of the lens) just cancels the secondary phase factor in Fresnel diffraction, and as a result, the light field distribution g(ξ, η) on the back focal plane of the lens becomes the Fourier transform or spatial spectrum of f(x, y). At this time, the spatial angular frequencies u = 2πψ/λf and ξ = 2π η/λ f, and when the wavelength λ of the incident light wave and the focal length f of the lens are constant, the spatial frequencies vx=ξ/λf and vy=η/λf are directly proportional to the spatial coordinates ξ and η on the back focal plane, respectively. Therefore, the function of convex lens is to draw Fraunhofer diffraction pattern at a distance closer to the back focal plane. It can be proved that when the aperture plane is placed on the front focal plane of the lens, the constant phase factor disappears, and there is an accurate Fourier transform relationship between f(x, y) and g (η, η) (Figure 2).

Fourier optics

Fourier optics

The spatial frequency spectrum of various images can be analyzed, and the images can be identified and classified by using the relationship that the light field distribution on the focal plane before and after the lens is Fourier transform. Using the Fourier transform property and spatial filtering of the lens, the optical system can have mathematical simulation operation ability, which is called "optical computer".

The names of spatial filtering optical information processing, coherent light processing, signal processing, image processing and image (or pattern) recognition are all related to spatial frequency filtering in phase coherent light system.

Based on the interesting fact that the fraunhofer diffraction pattern of the display object on the back focal plane of the convex lens and the relationship between the amplitude of the light field and the Fourier transform of the front focal plane of the lens, the mathematically complex two-dimensional Fourier integral operation can be realized very conveniently by pure optical method. The concept of filtering in information theory is introduced into optics, that is, not only the spatial spectrum of an object can be analyzed, but also the purpose of synthesis can be achieved by filtering. Just as the frequency spectrum of the time function can be changed in some way, the information content of the object can be changed by changing the spatial frequency spectrum of the object function. Examples of Fourier synthesis which has made important progress in modern optics are Zelnik phase contrast microscope, optical matched filter, optical processing of synthetic aperture radar data, various image enhancement technologies, blurred image restoration and so on.

In fact, the concept of spatial filtering is not new, and it has been put forward by E. Abbe in 1873 "Microscope Imaging Theory". 1906a.b. Porter's experiment to verify Abbe's theory is the earliest spatial filtering experiment. In 1950s, French p .-m .· O devoted himself to applying Fourier integral to optics. A. Marechal improved the transfer function of the imaging system through amplitude and phase filtering, and improved the quality of the photos to some extent (Figure 3). His success in this field aroused people's strong interest in optical information processing. In 1960s, due to the appearance of laser, coherent light processing system has an ideal coherent light source, and the research of spatial filtering has developed by leaps and bounds. For example: removing scanning lines and halftone dots, contrast enhancement, edge sharpening, extracting periodic signals from additive noise, aberration balance, data cross-correlation, matched filtering (image recognition), inverse filtering (blurred image restoration) and so on.

The coherent light processing system is shown in Figure 4. The coherent light output by the laser is expanded by the collimation system and irradiated to the object function on the front focal plane of the Fourier lens L 1. The light field on the back focal plane is the Fourier spectrum of the object function. A spatial filter with a change in amplitude (optical density) or phase (optical path) or both is placed on the spectral plane to change the Fourier spectral component of the objective function. The spatially filtered Fourier spectrum is inversely Fourier transformed by the second lens L2 and displayed on the image plane.

Spatial filters can be roughly divided into three categories: amplitude type, phase type and composite type.

The simplest amplitude spatial filters are low-pass, Qualcomm, bandpass and directional filters, as shown in Figure 5. It is binary in optical density, that is, it is only composed of transparent and opaque parts. Low-pass filter can be used to remove the scanning lines of periodic structure in the image, because the image spectrum is generally concentrated near the zero frequency, and the spectrum of periodic structure (scanning lines) is symmetrical with respect to the zero frequency. The low-pass filter is used to let the zero-frequency components in the image pass, and at the same time, the periodic structure spectrum is blocked, and finally the image except the scanning line is displayed on the image plane. Similarly, the directional filter can extract the image information of a certain direction interval, so it is very effective in geological data processing. Fig. 6 shows an example of directional filtering plus low-pass filtering to eliminate scanning lines. Fig. 7 is an example of removing print dots. In addition, the amplitude filter can also use photographic film as needed, and strictly control the optical density to obtain a filter with continuously changing density. This filter is very useful in contrast enhancement and differential operation.

The most famous phase space filter is the phase-shifting plate in Zelnik phase contrast microscope. Generally speaking, the phase filter is made by vacuum evaporation coating or bleaching the photosensitive film.

Composite spatial filter means that the amplitude and phase of the filter need to be changed, and the amplitude and phase filters can be made separately, and then the composite filter can be formed. It can also be done by holography, that is, taking a Fourier hologram of the object function on the spectral plane, which not only records the amplitude of the spectrum, but also records the phase of the spectrum. Making complex spatial filters by holography is a great promotion to optical information processing. Holographic filters can be used for matched filtering, pattern correlation, blurred image processing (see Figure 7), aberration balance, etc.