A general model for different apertures and flat-topped laser beams based on the multi-Gaussian function is developed. The general analytical expression for the propagation of a flat-topped beam through a general double-lens system with apertures is derived using the above model. Then, the propagation characteristics of the flat-topped beam through a spatial filter are investigated by using a simplified analytical expression. Based on the Fluence beam contrast and the Fill factor, the influences of a pinhole size on the propagation of the flat-topped multi-Gaussian beam (FMGB) through the spatial filter are illustrated. An analytical expression for the propagation of the FMGB through the spatial filter with a misaligned pinhole is presented, and the influences of the pinhole offset are evaluated.
© 2009 OSA
Laser beams with flat-topped spatial profiles are widely used in inertial confinement fusion (ICF), optical communications, electron acceleration, and material processing. Hence, the flat-topped laser beam and its propagation have attracted increasing attention in recent years [1–13]. Several models have been proposed to understand this kind of beam, such as the super Gaussian beam model , the superimposed complex Gaussian beam model , and the flatted Gaussian Beam model . Among them, the super Gaussian beam model is the most popular one. However, the propagation of the field with this profile cannot be performed in a closed form. Recently, an analytically solvable beam model, known as the flat-topped multi-Gaussian beam (FMGB), is proposed . It consists of a finite sum of Gaussian beams side by side with the same width, phase curvature, and absolute phase.
Up to now, the propagation of the FMGB has been studied in some articles [6, 8, 17]. But little work has been done on the general analytical expression of the FMGB through a double-lens system with apertures. On the other hand, few studies concerning the propagation property of a square beam through the spatial filter with a square pinhole have appeared in literatures. This paper is organized as follows. First, the multi-Gaussian function is used to establish kinds of flat-topped beam models and aperture models in section 2. Next, the general analytical formulas for the propagation of the FMGB through a double-lens system with apertures are obtained in Section 3. After simplification, some discussions, including the propagation characteristics of the FMGB through the spatial filter with different sizes of pinholes and the propagation characteristics of the FMGB through the spatial filter with different offsets of pinholes, are provided in Section 4. In the end, the conclusions are outlined in section 5.
2. Flat-topped multi-Gaussian beam modules and aperture modules
The multi-Gaussian function consists of a finite sum of Gaussian components side by side with the same width, phase curvature, and absolute phase. The general formula of the one-dimensional multi-Gaussian function is 
where N is the order of the multi-Gaussian function, ω is the width of the individual Gaussian component, and n represents the offset of the corresponding Gaussian component. For the two-dimensional formula, the expression becomes 
When M = N, it represents a square shape. The width of the entire multi-Gaussian profile W is related to ω by 
Based on Eqs. (2) and (3), the different-order multi-Gaussian shapes are plotted, as shown in Fig. 1 . It indicates that the edge becomes sharper when the order N is higher, and the difference between any two successive orders is much smaller. The multi-Gaussian function can also be used to simulate other beam shapes, such as the circle, the ellipse, and the rectangle, as shown in Fig. 2 . The forms of the Eqs. (2) and (3) are invariable for different FMGB shapes, and what we just need do is to establish the relationship between N and M. For instance, we can get the elliptical shape as shown in Fig. 2(b) by inserting into Eq. (2).
Because of numerous advantages, the multi-Gaussian function is also an ideal aperture model. By choosing different ω, N, and M, we can get different kinds of aperture functions. In next section, a general analytical solution for the propagation of the FMGB through a double-lens system with apertures is presented.
3. Propagation of the flat-topped multi-Gaussian beam through a double-lens system with apertures
A double-lens system is comprised of a pair of lenses and an aperture placed between them, as shown in Fig. 3 . , where , represent the optical field amplitude distributions in the input plane, the front surface of L1, the position Sin, the front surface of L2, and the output plane, respectively.
For the FMGB, the electric field amplitude distribution can be written as 
where is a constant, and represent the orders of the FMGB in the x direction and the y direction respectively, and is the width of the individual Gaussian component. The double-lens system includes three apertures, the aperture of L1, the aperture of L2, and the aperture Sin. By using the Huygens-Fresnel diffraction integral and taking the transformation of the thin lens as a phase modulation function, the electric field amplitude distribution in the output plane can be written as
, where j = 1, 2, 3, represent the aperture functions of L1, Sin, and L2, respectively. In Eq. (5) and other equations in this paper, the integral limits are all from minus infinity to plus infinity. Applying the integral formula
Equations (8)-(13), which are the primary results of this section, represent an analytical expression of the output electric field amplitude for the propagation of the FMGB through a double-lens system with apertures. They can be used to study the propagation of various FMGB through the double lens systems with different apertures. The forms of Eqs. (8)-(13) are invariable, and what we just need do is to establish the relationship between Nj and Mj for different kinds of apertures and different shapes of laser beams. In actual systems, only one or two of these apertures are the main influencing factors generally. Hence, Eq. (8) can be simplified greatly by keeping invariant, and letting and , where j denotes the ignored apertures.
4. Results and discussions
In this section, Eqs. (8)-(13) are simplified to investigate the propagation of the FMGB through a spatial filter. We use a square FMGB beam and a square pinhole to simulate the situation of the fusion laser propagation through the spatial filter. When L1 and L2 have the same focal point and the same optic axis, and Sin locates in their common focal plane, the double-lens system shown in Fig. 3 becomes a spatial filter. The spatial filter is the essential component of a high power laser system. It is widely used [19–22]. As a beam expander, it can expand the beam size; as a low-pass filter, it can control the nonlinear growth of spatial component; as an image relaying unit, it can reduce the effective propagation length. However, the limited apertures of the pinholes cause many other problems, such as the diffraction modulation rings and the propagation disturbance caused by the misaligned pinhole. They are very harmful for the laser system.
For an actual spatial filter used in the laser fusion system, the lens apertures are much bigger than the beam aperture. Hence, the influence of the lens apertures can be ignored. For the spatial filter, we also have the relationship and . Therefore, Eqs. (8)-(13) are simplified as
In Eq. (14), there is only one aperture, the pinhole. In order to graphically show the propagation properties of the FMGB, the beam parameters are chosen as, , , , , and in this section. The propagation properties of the FMBG through spatial filter without pinhole are presented first. In order to reduce the influence of the pinhole, the radius of the pinhole is assumed to be 500 times of the diffraction limit of the input beam in Eqs. (14)-(19), and the relationship between and is presented by Eq. (3). Figure 4 shows the propagation properties of the FMGB through the spatial filter at different Fresnel numbers. Here, the Fresnel number is defined as , where is the half-width of the output beam. Figures 4(a) and 4(b) indicate that when the distance is short, the output beam remains flat-topped beam. It is just widened to twice which does accord with the expanding function of the spatial filter when, and its peak intensity becomes a quarter of the input beam intensity, which can be explained by the conservation of optical energy. Figures 4(a) and 4(b) can also be explained by that the widths of individual Gaussian components of the input beam are expanded to twice by the spatial filter, and the distances between any adjacent components are expanded to twice too. Hence, the output beam keeps flat-topped beam and the intensity is reduced to a quarter. Figure 4(c) shows that the Fresnel rings appear gradually with increasing distance . Figure 5(b) is the on-axis intensity distribution of the Fresenel diffraction pattern. It discloses the variations of the Fresnel rings against the Fresnel number to some extent. Figure 4(d) is the intensity distribution of the FMGB near the Fraunhofer area.
4.2 Propagation characteristics with different sizes of pinhole
In this section, the propagation characteristics of the FMGB with different sizes of pinhole are investigated. One of the most important functions of the spatial filter is to remove the high spatial frequency which is harmful. The pinhole cuts off the high spatial Fourier mode spectrum from the beam and allows only the lower spatial modes of the Fourier spectrum to transmit. However, the pinhole induces a low frequency modulation of the lower beam which is also harmful.
Using Eqs. (4) and (7), ignoring the influence of the lens aperture, and just calculating from the input plane to the pinhole plane, the intensity distribution of the FMGB in the focal-plane can be written as
Figure 6 shows the focal-plane intensity distributions of the different order FMGBs when and that of an ideal square beam with the same half width. It indicates that the intensities of the higher frequency components increase with increasing FMGB order, and the multi-Gaussian shape becomes sharper, which accords with the relationship between the shapes and the orders of the multi-Gaussian function as shown in Fig. 1. In high power laser system, in order to illuminate the fusion target uniformly and use the light beam as sufficiently as possible, the beam is flattened. But it causes very strong diffraction rings. Therefore, the soft-edge aperture is used, which makes the edge of the beam not very sharp. Hence, we can use lower order multi-Gaussian function to simulate the flat-topped beam, and use the higher order multi-Gaussian function to simulate the hard edge pinhole. In the following calculations, it is assumed that , , , , and .
The Fluence beam contrast (FBC) and Fill factor are employed to evaluate the influence of the Fresnel modulation rings. The FBC is defined as 
where is the average intensity of the beam, is the individual position intensity, and m and n are the sampling numbers. The Fill factor is defined as the ratio of the average intensity to the maximum intensity. Figure 7(a) illustrates the variations of the FBCs and Fill factors against the sizes of pinholes, which are expressed by the times of the diffraction limit (DL). The FBC decreases with increasing pinhole size gradually, and it exhibits obvious periodicity, which can be explained by the variations of the Fresnel rings with the sizes of the pinholes. Figures 8(a) and 8(b) present the Fresnel rings, corresponding to one of the adjacent pairs of minimum and maximum FBCs, when the pinhole sizes are near the 4 × DL. The Fill factors show the opposite trend. Figure 7(b) shows the variations of the FBCs and Fill factors of different order FMGBs against the distance z4 when the pinhole size is 10 × DL. The FBCs tend to be bigger with periodic oscillation when the distance z4 increases. The Fill factors show the opposite trend. From Fig. 7 and Fig. 8, we can find that the maximum Fill factors and minimum FBCs appear at the same size of pinhole or at the same distance z4.
4.3 Transmission characteristics with different offsets of pinhole
In high power laser system, the beam alignment is a great challenge [24–26]. It is very difficult to keep the laser bean pass through the pinhole centre exactly. In order to evaluate the influence of the pinhole offset, the Eq. (14) is further extended to the offset situation.
If the pinhole offset is , the pinhole function can be written as
When , Eq. (23) is simplified as Eq. (14). Equation (22) can also be used to investigate the off-axis beam propagation.
First, the influence of the offset on the beam propagation is presented. Figure 9 shows the output FMGB with different offset at the same position when , and . Here, the offset of the pinhole is just along the y-axis, that is. The times of the pinhole diameter (TPD) is used to describe the offset next. With increasing offset along the y-axis, the number of the Fresnel rings along the x-axis decreases when. It is because that the one side of the focal plane is obstructed gradually, and the higher frequency components on that side, which are farther away from the center, are cut off orderly. Hence, the lower frequency components play a dominant role. When, the lower frequency components are obstructed gradually, and the influence of the other side of higher frequency components displays remarkably. But the intensity decreases rapidly.
Based on the FBC and Fill factor, the influence of the offset on the propagation of the FMGB is investigated. Figure 10(a) shows the variations of the FBCs and Fill factors against the offsets at the same distance . Although the proper offsets can increase the Fill factors, the offsets still have little effect on the FBCs and Fill factors when both the offsets are small () and the distance z4 is short. The increase of the Fill factors can be explained by that one side of the higher frequency components is cut off when the offset increases, while the other side of the much higher frequencies passes through the pinhole. The proper higher frequency components can modulate the top of the beam and make it flat. Figure 11 shows that one side of the output FMGB becomes shaper than the other side when the distance z4 increases, and Fig. 10(b) can also show that the modulation becomes bigger with increasing distance z4. It is because that one side of the high frequency components is shut off and the other side of the higher frequencies passes through the offset pinhole, which appears with increasing distance z4 gradually. Hence, the small offset of the pinhole has little effect on the FBC and Fill factor of the output FMGB when the distance z4 is short, but it can induce much bigger modulation with increasing distance z4. It is very dangerous for the high power laser system [20–23].
A general analytical expression for the propagation of a flat-topped beam through a general double-lens system with apertures is presented based on the multi-Gaussion function first. It can be used to study the propagation of different flat-topped beams through a general double-lens system with various apertures. A simplified analytical expression for the propagation of the FMGB through a spatial filter is obtained. Then, the propagation characteristics of the FMGB through the spatial filter are presented, and the influence of the size of pinhole is studied. Based on the offset multi-Gaussian function, an analytical expression with a misaligned pinhole is derived, and the influence of the offset is evaluated.
This work was supported by the National High Technology Research and Development Program of China (Grant No. 007SQ804) and the Japan-Korea-China Cooperative Project on High Energy Density Science for Laser Fusion Energy.
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