Research Background
In the field of Wave Optics, formulas of diffraction is so complicated and sometimes may not even find analytical solutions. To simplify them, paraxial approximation is widely used to solve actual problems. For instance, Fraunhofer diffraction describes well for far-field diffraction as an approximation of Fresnel diffraction. In Young’s double-slit interference, the result matches well with the theory when the paraxial approximation is applied. These show some clues of analyzing far-field diffraction behavior. In my paper, the plane wave incidents on a strip zone plate is analyzed numerically with paraxial approximation. By using this new method, the one to one correspondence between the shape of the aperture plane and the observation plane can also be visualized in this work, which contribute to understanding the diffraction behavior of the light in a simpler way instead of undersranding it by abstract integrations. The following picture shows the physical situation of the simulation.
Narrow Strip Element Model Analysis
The following picture shows the differential element in a strip zone plate, which we call “narrow strip element”. A simple idea is that we can build a strip zone plate by superposing numbers of narrow strip elements in the vertical direction. We can firstly calculate the light field distribution of a narrow strip on the observation plane and superpose the distribution of each narrow strips arrayed along the vertical direction to get the diffraction image of a strip zone plate. This method is correct only when the system is linear. When the system is linear, suppose an input x1 generates an output f(x1), and an input x2 generates an output f(x2), then the input a*x1+b*x2 should generate the output a*f(x1)+b*f(x2). Luckily the system is linear as the electric field on the observation plane is based on the Fraunhofer diffraction and the Fourier transform .
As the strip element can be built by a number of narrow strips, an easy way of calculating the light distribution of it on the observation plane should be proposed. One way is to regard a narrow strip element as a bunch of rectangular holes arrayed along the horizontal direction. For each hole, the distribution of electric field can be quickly found by referring to the formulas of Fraunhofer diffraction. The horizontal distribution of a narrow strip element f can be superposed by these holes (am represents the width of the mth wave zone, bm represents the distance between the mth wave zone’s edge and the central point in the horizontal direction). The following three formulas provide the horizontal electric field distribution of a narrow strip. These can be easily dealt by MATLAB.
In the following simulation, distance between the aperture plane and the observation plane is set as 2.5m (to satisfy far-field diffraction) and the wavelength is set as 400nm. The result shows below:
As we can see, when the number of wave zones increases (”波带“ means wave zone, which is each black part of the aperture plane shown in the above picture), the focusing property becomes noticeable. When there is only one wave zone, it is just a rectangular hole and the distribution shows a sinc function. For 39 wave-zone case, the light intensity on the observation plane is focused at the center, showing the focusing characteristic of the zone plate. The left picture clearly shows the horizontal characteristics.
By using the paraxial approximation, the phase on the observation plane along the vertical direction for each edge of the wave zone is calculated as a function of the height of that point and the number index of the wave zone. After finishing this part, the contribution of each wave zone will add up to get the contribution of a narrow strip.
The picture on the left side shows geometric relationships and the formulas below show steps of approximation. (typo correction: Rj-Rj-1 is nearly equal to 2z should be Rj+Rj-1 is nearly equal to 2z).
As the amplitude difference between adjacent wave zones is very small, the final formula can be easily acquired. This method also appears in some textbooks about Wave Optics. However, error analyses are also given in the next part to verify the correctness of all approximations shown in my work. Considering distributions of two orthogonal directions, we can get the two-dimensional distribution of a narrow strip element with 0.1mm height and 39 wave bands.
Superpose in The Vertical Direction
We have known that a strip zone plate is composed of numerous narrow strip elements, the next question is to determine the exact way of superposing light field in the observation plane. An obvious fact is that every narrow strip element contributes to the light field of a given point in the observation plane. For two adjacent narrow strip, the contribution of the upper element to a given point is the same with the contribution of the lower element to a point that is also lower than the given point. Instead of calculating the total contribution from the first element to the last element for a given point, we are able to calculate the whole vertical light field distribution of the first element. The following picture shows relevant formulas.
In order to prevent vague meanings, parameters in formulas should be explicated. hm refers to the height of a strip zone plate; 1 refers to the number index of the first narrow strip element; N refers to the number of wave bands; C is a constant; h refers to a vertical variable; b1 refers to the width of the first wave band. Considering the final formula, when a point locates in the height of y, then the light intensity contribution from the vertical direction should be the square of ER. As we have acquired the horizontal distribution formula, the two-dimensional light intensity distribution can be thus calculated.
We choose four combinations of parameters to illustrate behaviors of diffraction imaging (all are simulated when the wavelength is 400nm, and the distance between the aperture plane and the observation plane is 2.5m). The first is for 1 wave zone, low height strip zone plate, it’s actually a rectangular hole; the second is for 39 wave zones, low height strip zone plate, it’s actually a narrow strip element; the third is for 3 wave zones, high height strip zone plate and the last one is for 39 wave zones, high height (0.02m) strip zone plate, and this one can be considered as a typical strip zone plate.
Error Analyses
The last step is to analyze whether approximation will largely affect the value of optical phase. The difference between exact phase expression and approximated expression is shown as follows. By referring to the picture, we find out that even the maxium error is around 0.1π (rad) which is far smaller than π (rad). This result proves that our new method to analyze the diffraction simulation under the situation that monochromatic plane wave passes through a strip zone plate is reasonable.
Paper Current Status
Unfortunately, the paper is not received successfully because the review has expired, and the review comments have not been replied (no other reviews are received) in the Chinese journal EXPERIMENT SCIENCE AND TECHNOLOGY. Considering it’s not received and I no longer focus on this project anymore, I upload this work as a first writer in the following link to memorize past contributions.