1. Introduction

Diffractive optical elements (DOE) are widely used in beam shaping applications for their flexibility in creating a desired field distribution with a minimum of optical components [1–3]. Typically, beam-shaping DOEs are designed by iterative Fourier transform algorithm (IFTA), which assumes that the incident beam is fully coherent [4,5]. To achieve good shaping results for these beam shapers, the illumination laser should have a good beam quality. Although coherent DOEs have demonstrated good capabilities in beam shaping, they also induce problems such as speckle noise and unwanted high-order diffractions [6], which limit their applications in high-quality optical imaging and fine processing. An efficient way to suppress the coherent noise induced by fully coherent beams is to reduce the spatial coherence of the beam, which has been proven to be useful in beam homogenization in lithography [7]. This prompted people to study DOEs for shaping spatial partially coherent beams. Another reason is that it is difficult to maintain the TEM₀₀ mode output in high-power lasers. Therefore, it is also important to develop DOEs for shaping high-power multi-mode lasers, which behave as spatial partially coherent beams [8].

Usual representations of partially coherent light introduce four-dimensional second-order correlation functions [8], making it difficult to directly design DOEs for shaping partially coherent beams. One way to handle this task is to use mode expansion methods to reduce the calculation. Recently, an approach has been presented for designing multiple phase DOEs to shape partially coherent light by a set of mutually uncorrelated modes and shows nearly perfect results for two DOEs [9]. Another way is to convert complex DOE designs for partially coherent beam into the coherent beam design. The diffraction pattern of a DOE illuminated with a spatial partially coherent beam can be modeled by the convolution of the coherent diffraction pattern and the inherent degree of coherent function [10,11]. Therefore, the conventional DOE design method for fully coherent beams can also be used to design DOEs for shaping partially coherent beams, as long as the proper target diffraction pattern is determined. The target pattern can be determined by the deconvolution method. However, it is a constrained optimization problem of non-negative variable deconvolution, which usually requires complex deconvolution algorithms to implement [10].

In this work, we propose a new method to design DOEs for shaping partially coherent beams, which is inspired by the optical proximity correction technique in optical lithography. The designed DOE exhibits good performance in beam shaping and noise suppression.

2. Theory

The basic setup of a beam shaper is shown in Fig. 1. The DOE modulates the phase of the incident beam to obtain the desired pattern and energy distribution on the focal plane of the Fourier lens. For spatial partially coherent beams, the setup is the same. However, the output diffraction pattern on the focal plane is no longer a simple Fourier transform of the incident laser and the DOE phase.

 

Fig. 1. The basic set-up of a beam-shaping DOE.

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The incident quasi-monochromatic partially coherent beam can be expressed as the cross-spectral density function in space-frequency as [12]

This four-dimensional integral describes the output of a DOE illuminated with a partially coherent beam, and can be further simplified for quasi-homogeneous, secondary, planar sources such as Schell-model beams. For Gaussian Schell-model (GSM) beams, the cross-spectral density function is [12]

Defining $\textrm{}({\Delta u,\Delta v} )$ and $({\bar{u},\bar{v}} )$ as

Using Eq. (5) and Eq. (6), Eq. (4) can be rewritten as

Equation (9) holds for any quasi-homogeneous, secondary, planar partially coherent sources. For simplicity, in the following sections, we only consider GSM beams. When the DOE is illuminated by a fully coherent Gaussian beam, the convolution kernel ${I_{raw}}$ becomes a Dirac function, and the diffraction pattern is equal to the designed output ${I_{coh}}$. When illuminated by a partially coherent beam, the diffraction pattern will become blurred due to the broadening of the kernel ${I_{raw}}$.

Figure 2 shows the convolution process of obtaining the diffraction patterns of a DOE under lasers with beam quality factors ${M^2}$ of 1.5 and 3, respectively. For GSM beams, ${M^2} = \sqrt {1 + w_0^2/\sigma _0^2} $. As the coherence decreases, the output pattern changes including reduced edges sharpness and pattern deformation. The output intensity distribution will be severely degraded when the target size is comparable to the convolution kernel width. Therefore, DOEs designed for coherent beams are not suitable for low-coherence lasers.

 

Fig. 2. The convolution process of obtaining the diffraction patterns of a DOE under GSM beams with beam quality factors ${M^2}$ of 1.5 and 3, respectively.

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To design the DOE for partially coherent beams, the main task is to determine the appropriate target for conventional IFTA with a fully coherent Gaussian beam as input, to form the desired diffraction pattern through the convolution by the kernel of the degree of coherent function. One way to obtain the target is through the deconvolution method. It is a constrained optimization problem of non-negative variable deconvolution, which usually requires complex deconvolution algorithms to implement. Another way is to directly correct the initial target through certain rules.

3. Design DOEs for partially coherent beams

It is interesting to consider the output of a DOE as an analog of the response of an optical imaging system, where the image of a complex object is the convolution of that object with the point spread function (PSF) [14]. The PSF is especially important in lithography. With the continuous shrinking of critical dimension in advanced technology nodes, the significant optical proximity effects (OPEs) result in imaging quality degradation on the wafer, due to the diffraction-limited property of the lithographic imaging systems [15]. The imaging distortions such as pinching, bridging, line-end shortening, and corner rounding, impact the circuits’ functionality and performance. As an important resolution enhancement technique, optical proximity correction (OPC) successfully compensates for and minimizes the imaging distortions by modifying the original mask patterns, through the adjustment of the distribution of transparent regions of the mask [15–18]. OPC methods can be categorized into rule-based OPC (RBOPC) and model-based OPC (MBOPC). In RBOPC, rules are derived through experiment or simulation to determine specific corrections that should be applied to a given structure in a specific environment. RBOPC are typically fast, but cannot extrapolate to configurations beyond their development set. MBOPC uses mathematical models to characterize the image formation process, and iteratively seeks the global optimum of a merit function to improve the imaging quality. MBOPC techniques produce typically more rigorous correction solutions, but require longer time due to more intensive calculations and simulations required. By adding well-designed serifs to mask pattern boundaries, OPC can effectively improve the quality of diffraction images. The success of OPC techniques in lithography inspired us to use similar proximity correction (PC) methods in DOE design for partially coherent beams.

Once the incident GSM beam is determined, the width ${d_{raw}}$ of convolution kernel ${I_{raw}}$ can be calculated as

Obviously, the target line width cannot be less than kernel width ${d_{raw}}$. The width ${d_{raw}}\; $ can thus be regarded as the coherence limit. When the target size is comparable to ${d_{raw}}$, the output intensity distribution will be severely degraded. Here we use the empirical PC method to design DOEs for partially coherent beams by adding square serifs to the original target. Two basic types of diffraction anomalies are discussed, including line-end shortening and corner rounding.

To determine the appropriate serif size, a merit function is used as shown below:

3.1 Line-end shortening

As the degree of coherence of the incident beam decrease, the output patterns will show inward shrinkage at the line-end (Fig. 3(b) and 3(f)). To overcome this drawback, square serifs are added to the line end of the original target, as shown in Fig. 3(a). Here the center of the square coincides with the corner of the target. Assume the linewidth of the target is d and the serif side length is a. By changing target serifs sizes, the corresponding DOE changes, and the line-end shape and length of the final output pattern can be adjusted. Figure 3(b-i) show zoomed-in views of calculated output line-end shapes of DOEs with different target serif sizes and kernel width. For the original target without PC ($a = 0$), the output of the designed DOE shows obvious line-end shortening and shape rounding for low-coherence incident beams. The wider the convolution kernel width, the shorter the line obtained.

 

Fig. 3. (a) Target correction for line-end shortening, where d is the target line width, and a is the side length of the serifs. (b-i) Enlarged views of diffraction patterns at line-end for DOEs designed according to different target serif sizes under GSM beams with coherent kernel width ${d_{raw}} = d/2$ and ${d_{raw}} = d$, respectively.

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As we can see, the performance of the designed DOE under the partially coherent beam is effectively improved by adding square serifs on the target. By using the merit function, we can determine the appropriate serif size a. For kernel widths ${d_{raw}} = d/2$, the appropriate serif size is $a = 0.3d$; for ${d_{raw}} = d$, the appropriate serif size is $a = 0.5d$.

3.2 Corner rounding

Another diffraction anomaly occurs at the corner formed by two lines, where the proximity effect forces the corner to round. The outer corner will contract inward, while the inner corner will extend outward (Fig. 4(b) and 4(f)). To correct this anomaly, square serifs are added to the outer corners and subtracted from the inner corners of the original target, as shown in Fig. 4(a). The side length of inner corner serifs is b, the side length of outer corner serif is c and the linewidth of the target is d. Similar to line-end shortening, changing the sizes of serifs can correct the shape of the corner in diffraction patterns. It is worth noting that the outer corner case is same to the line-end shortening under this condition, and thus the appropriate serif size c is equal to a in the previous section. Figure 4(b-i) show enlarged views of calculated outputs of DOEs with different serif sizes and different GSM beams illumination. The lower coherence led to more serious corner rounding in the diffraction pattern and thus need bigger serifs.

 

Fig. 4. (a) Target correction for corner rounding, where d is the target line width, b is the side length of the inner corner serifs and c is the side length of the outer corner serifs. (b-i) Enlarged views of diffraction patterns at corner for DOEs designed according to different target serif sizes under GSM beams with coherent kernel width ${d_{raw}} = d/2$ and ${d_{raw}} = d$, respectively.

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The results show that the performance of the designed DOE is effectively improved by using serifs on the target. By using the merit function, the appropriate serif size for GSM kernel widths ${d_{raw}} = d/2$ is $b = 0.25d$; and for ${d_{raw}} = d$, the appropriate serif size is $b = 0.6d$.

Using the PC method mentioned before, we can design DOEs for a certain partially coherent beam. The design steps are shown in the Fig. 5.

 

Fig. 5. Flow chart of DOEs design for partially coherent beams by proximity correction.

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4. Simulation results

Here we take the case in Fig. 1 as an example. The incident 532 nm GSM beam has a waist radius of 2 mm, the beam quality factor ${M^2}$ is 3, and the f of the focal lens is 100 mm. The coherence kernel width ${d_{raw}}$ (24 µm) is less than the target line width (40 µm), so the diffraction pattern could be achieved under this beam. Results of the DOE designed with the original target are shown in Fig. 6(a-d). The DOE works well under a coherent beam but shows obvious line-end shortening and corner rounding under the given GSM beam. Figure 6(e-f) shows the results of the DOE designed with PC. After adding serifs, the target with PC looks quite different from the original target. As expected, the DOE works well for the given GSM beam. Diffraction anomalies such as line-end shortening and corner rounding are successfully compensated. The results show that the PC method can effectively improve the performance of DOE under a certain partially coherent beam.

 

Fig. 6. Simulation results of DOEs designed according to the original target and the target with PC. (a) Target shape, (b) phase of the DOE, (c) coherent diffraction and (d) GSM diffraction of the DOE designed from the original target. (e) Target shape, (f) phase of the DOE, (g) coherent diffraction and (h) GSM diffraction of the DOE designed from the target modified by PC.

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In Fig. 6(c) and Fig. 6(h), we have shown that DOEs can be designed for both coherent beams and GSM beams with the same shape of the diffraction pattern. However, the difference in coherence makes them have different characteristics. Figure 7 shows the mesh plot of Fig. 6(c) and Fig. 6(h). As we can see, the coherent DOE has sharper edges and a flatter top area than the GSM DOE. Although the diffraction pattern of the coherent DOE is more concentrated than that of the GSM DOE, it has many diffraction noise peaks around the target. It may cause unwanted damage in fine laser processing. Inaccurate transformation of the input beam will lead to diffraction noise. In general, IFTAs cannot obtain exact DOE solutions for arbitrary target shapes, so it is difficult to remove the diffraction noise of coherent DOEs. But diffraction noise can be easily suppressed by low-coherence beams of GSM DOE as shown in Fig. 6(h) and Fig. 7(b). Therefore, coherent DOEs and GSM DOEs are suitable for different applications.

 

Fig. 7. Mesh plots of diffraction patterns for (a) the coherent DOE and (b) the GSM DOE.

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5. Conclusion

In summary, we propose a new method to design DOEs for shaping partially coherent beams by proximity correction. The diffraction pattern of a DOE illuminated with a spatial partially coherent beam can be modeled by the convolution of the coherent diffraction pattern and the inherent degree of coherent function. Two basic types of diffraction anomalies of DOEs under partially coherent beams are discussed, including line-end shortening and corner rounding. A proximity correction method is used to compensate for these anomalies. The designed DOE for the partially coherent beam exhibits good performance in beam shaping and noise suppression. This design method is expected to have important application value in fine laser processing.