Rapid method for the detection of microlens filling factor

An effective and fast detecting, method for microlens array filling factors is introduced in this paper. The relationship between the diffraction field of microlens array and lens diameter, sag height is analyzed, and theoretical model is derived with the scalar diffraction theory, The formulation of diffraction field of microlens array is established computational simulation is carried out for arrays with different filling factors using this formulation, and experiment systems are constructed to obtain actual diffraction field. The simulation intensity distribution calculated definitely approaches the actual diffraction field through contrast. Based on the above analysis, a semi-quantificationally relationship between diffraction field of microlens array and filling factor is given, and a new effective and fast detecting method for microlens array is presented.

Theoretical analysis of far-field diffraction spots of microlens array:

This article mainly utilizes the distortion of the diffraction spot in the far field of the microlens array to achieve the detection of the microlens fill factor. According to the scalar diffraction theory of light, the relationship between the diffraction spot of an object in the far field and the transmittance function of the diffracted object is given by Equation (1). In Equation (1), U₀ represents the distribution of the far-field diffraction spot of the object, U1 represents the transmittance function of the object, z represents the distance between the exit plane and the diffraction plane, and r represents the distance from a point on the object to a point on the diffraction plane. For a parabolic microlens, the transmittance function can be expressed as Equation (2). In Equation (2), U₁(x₁, y₁) represents the transmittance function of a single parabolic microlens, R is a constant related to the lens sag, and a represents the half-width of the microlens aperture. Substituting Equation (2) into Equation (1), the far-field diffraction spot after a plane wave passes through the microlens array can be expressed as Equation (3).

In equation (3), U1 represents the exit surface, r represents the distance from the object point to the diffraction point, and z represents the distance between the exit surface and the diffraction surface. Considering that in the diffraction process, z>> (x₁-x₀)² and (y₁-y₀)², r can be simplified to equation (4). Substituting equation (4) into equation (3) and rearranging, we obtain equations (5) and (6). In equation (6), c is a constant related to relief depth and diffraction distance. When the condition z >>f is satisfied, the light intensity distribution on the output plane can be approximately expressed as equation (7). In equation (7), it represents the Fourier transform. The area where the microlens sag is not zero is defined as the microlens effective aperture Ω. Equation (7) describes the relationship between the microlens surface shape and the far-field diffraction spot. From the above relationship, it is not difficult to see that the far-field diffraction spot of the microlens is exactly the Fourier transform of the microlens surface shape, and it can also be seen as the Fresnel diffraction spot formed by the free transmission of the lens effective aperture Ω in space for a distance of f; based on this Fresnel diffraction spot, rapid detection of the microlens fill factor can be achieved.

Experimental research:

We first designed an experimental setup as shown in Figure 1 to sample the far-field diffraction pattern of the microlens. The focal length of the microlens is f, the distance between the observation plane and the lens is Z, and the size of the diffraction pattern is L. The experiment uses a laser beam to illuminate the microlens array perpendicularly.

When the distance Z between the observation plane and the lens is much greater than the focal length of the microlens, the far-field diffraction pattern of the plane wave passing through the microlens is obtained at a distance Z from the lens. Figure 2 shows a microlens array with a fill factor of 100% used for testing, and the far-field diffraction pattern of this microlens array is shown in Figure 3.

To compare the degree of approximation between the actual light field diffraction behavior and the above-derived results, we used equation (7) as the theoretical model to develop a far-field diffraction simulation program for the microlens. In Figure 2, the microlens has a diameter of 200μm, a sag height of 30μm, a rotationally parabolic surface, and a distance of 1×10⁶μm from the receiving screen. Figure 4 shows the diffraction pattern of this microlens simulated by a computer based on equation (7). By comparing the experimental results in Figure 3 with the simulation results in Figure 4, it can be seen that equation (7) accurately represents the diffraction behavior of the microlens in the far field.

The far-field diffraction pattern of the microlens array is the same as the Fresnel diffraction pattern at the focal plane after light passes through the effective aperture of the microlens. In the Fresnel diffraction region, the diffraction fringes of the light spot are the same as the effective aperture of the microlens. For example, in Figure 2, the microlens array is arranged in a square pattern, so the undistorted microlens aperture with a fill factor of 100% is also square, and its diffraction pattern in the Fresnel diffraction region is necessarily square as well. Therefore, the shape of the microlens aperture in the experiment can be determined based on the shape of the far-field diffraction pattern of the microlens array, and the fill factor of the lens can be further determined. The light spot diffraction fringes in Figure 3 are distributed in a square pattern, indicating that the microlens aperture is square and the fill factor is approximately 100%.

To further analyze the relationship between the microlens fill factor and the far-field diffraction pattern, we detected the far-field diffraction patterns of microlens arrays with fill factors of 90% and 60% (as shown in Figures 5 and 8, respectively). The experimental results are shown in Figures 6 and 9, and Figures 7 and 10 show the far-field diffraction patterns of 90% and 60% fill factor microlenses calculated and simulated based on equation (8), respectively.

By comparing the experimental results with the simulation results, we found that the distribution of the far-field diffraction pattern of the microlens array basically matches the shape of the effective aperture of the microlens. By comparing the closeness of the shape of the lens diffraction pattern to a square, the fill factor can be calculated to be approximately 90% and 60%.

Conclusion:

By analyzing the far-field diffraction spot of the microlens array, we have established a relationship between the microlens fill factor and the far-field diffraction spot, and derived a theoretical model for the far-field diffraction spot of the microlens array. This model indicates that the far-field diffraction spot of the microlens array is the same as the Fresnel diffraction spot of the microlens effective aperture. Comparing the actual diffraction spot with the simulation results, we can see that the theoretical and experimental simulations basically reflect the far-field diffraction behavior of the light field after parallel light passes through the microlens. Therefore, without expensive specialized micro-nano structure detection instruments, the fill factor of the microlens can be quickly and easily obtained through simple testing and analysis of the far-field diffraction spot of the microlens array. Similarly, based on the far-field diffraction spot of the microlens, we can further understand parameters such as the microlens surface shape, geometric structure, and focal length. This method provides a way for us to achieve inexpensive and rapid detection of microstructure parameters.