2.1

Principle of Abbe-Porter imaging and spatial filtering

Based on the principle of Abbe imaging, the object wave is deemed as a group of different spatial frequency information, and the coherent imaging process of a target object is typically achieved in two steps. In the first step, the spatial frequency spectrum is generated on the spectrum plane by Fraunhofer diffraction of the object wave, and in the second step, the image of the target can be constructed on the image plane by coherent superposition of the subwaves emitting from the spectrum plane, which represent different spatial frequency, ^[3]. The two steps means twice operations of Fourier transform. If these operations are ideal completely, the obtained image without any lose is exactly similar as the object^[4].

In Abbe imaging experiment, 4f optical system is one of the most classic configuration in optical information processing^[5]. Figure 1 shows the imaging process of 4f system. The object plane P₁ (x, y) is located on the front focal plane of the Fourier transforming lens L₁, the spectrum plane P₂ (u, v) is located on the back focal plane of the L₁, which is also on the front focal plane of the L₂, the image plane P₃ (x′, y′) is located on the back focal plane of the L₂. Furthermore, the focal length of lens L₁ is equal to that of lens L₂. In this experiment, the specific design of the filter is employed on the spectrum plane to change the spectrum structure of the target to generate the expected imaging.

Fig. 1

The schematic diagram of 4f imaging system. P₁: object plane; P₂: spectrum plane; P₃: image plane; L₁, L₂: lens; A, B, C: object; S_-1, S₀, S₊₁: spatial frequency; A′, B′, C′: image

The transmittance function of a one-dimensional grating, t(x), is expressed as follows

Where d is the grating constant, a is the width of the silt, L is the total width of the grating.

If the filter is a through-hole that allows zero-order spectrum to pass, the wave field distribution on the image plane is given as follows

Where H(f_x) is the transmittance function of the filter.

If the filter is a slit which allow the zero-order, the order +1 and the order -1 spectrum to pass, the optical field distribution on the image plane is given as follows

If the filter is an opaque circular screen which can filter out the zero-order spectrum, the optical field distribution on the image plane is given as follows

2.2

Principle of image addition and subtraction

In this experiment, the spatial filter is a Ronchi grating, and the experiment setup consists of a 4f optical imaging system. Figure 2 shows the experimental configuration of image addition and subtraction with a grating filter. Lens L₂ is a transform lens, lens L₃ is imaging lens, and the focal length of the lens L₂ is equal to that of the lens L₃. The object plane (x₁, y₁) is located on the front focal plane of the lens L₂, the frequency plane (x₂, y₂) is located both on the back focal plane of the L₂ and on the front focal plane of the L₃, the image plane (x₃, y₃) is located on the back focal plane of the L₃.

Fig. 2

The schematic of the image addition and subtraction with a grating filter. A, B: object; (x₁, y₁): object plane; (x₂, y₂): frequency plane; (x₃, y₃): image plane; L₂, L₃: lens; f: focal length

The samples are image A and B, the center coordinates of them are (b, 0) and (-b, 0), respectively. When the plane wave with the wavelength λ illuminates the object vertically, the complex amplitude distribution after the object

The grating of spatial frequency ξ₀=b/λf is placed onto the spectrum plane. The complex amplitude transmittance function is given as

If the initial phase of the grating φ₀=π/2, image subtraction is performed. If the initial phase of the grating φ₀=0, image addition is achieved.

3.1

Abbe-Porter spatial filtering

In the Abbe-Porter spatial filtering experiment, the object is generated by MATLAB program. According to Eq. (1), the parameters both on the x and y direction can be given with pixel number as a=16pixels, d=32pixels, and L=512pixels. Figure 3(a) shows the pattern that is input onto the object plane as a target object, and figure 3(b) shows the spatial frequency of the object. The zero-order spectrum, the order +1 and order -1 spectrum are zoomed in (seen as in red rectangle).

Fig. 3

The simulation result. (a) an input pattern as an object; (b) spatial frequency spectrum

The simulation of the filters and imaging results are shown in figures 4(a) to 4(d) and figures 4(e) to 4(h), respectively. When the filter is a transparent screen shown in figure 4(a), the whole spatial frequency of the object is passed through completely. The imaging result is given in figure 4(e). When the filter is a horizontal slit of figure 4(b), the image on the image plane after spatial filtering is of vertical stripes as shown in figure 4(f). If the filter is a vertical slit in figure 4(c), the spatial filtering result on the image plane is of horizontal stripes as shown in figure 4(g). When the filter made of a slant slit at an angle 45° shown in figure 4(d) is used, the spatial filtering result on the image plane is a pattern of slant strips with angle -45°, shown in figure 4(h).

Fig. 4

The filters and imaging results by filtering. (a) - (d): filters; (e) - (h): the imaging results with respective to through the filters in (a) - (d)

The filter that blocks high frequency components to pass through is referred as low pass filter. It is typically used in noise reduction of digital images. On the other hand, the filter stopping low frequency components is referred as high pass filter, which is used to sharpen the image that is helpful to observe the details. In figure 5(a), the low pass filter allows the zero-order, the order +1 and the order -1 spectrum of the target to pass through the system. The imaging result is some blurry as shown in figure 5(c). With a high pass filter that can filter off the zero-order spectrum, shown in figure 5(b), the image with opposite contrast is formed as in figure 5(d).

Fig. 5

The filters and the imaging results. (a) low pass filter; (b) high pass filter; (c) and (d): the imaging results with respective to filters in (a) and (b).

3.2

Image addition and subtraction

In the virtual experiment of image addition and subtraction with a grating filter, the object is generated by using MATLAB program, shown as in figure 6(a). The window size loading the image is 512×512 pixels, the distance between images A and B is 40 pixels, the width each of two rectangles are 15 pixels.

Fig. 6

(a) The object in the virtual experiment of image addition and subtraction; Simulation of image addition and subtraction, (b)and (c): the grating filter and the result of image addition; (d) and (e): the grating filter and result of image subtraction.

According to formula (7), when the initial phase φ₀=0, the generated grating is shown as in figure 6(b). As a result, image addition is achieved on the coincident position of the diffraction images, shown in figure 6(c). When φ₀=π/2, the generated grating is shown in figure 6(d), and thus, image subtraction is realized on the coincident position of the diffraction images, shown in figure 6(e),.