## Abstract

A continuous-phase plate (CPP) is a key element for beam smoothing in a high-power laser system. For the beam-smoothing effect, the surface shape of a CPP is one of the most important facts. In this paper, the change law of the transmission direction of light rays has been analyzed according to the geometrical optical principle. It is discovered that the 2-dimensional histogram of a surface gradient can be used to show the far-field distribution of a CPP. Drawing on the experience of histogram modification technology in digital image processing, a novel method is proposed to design a CPP. The design steps of a 1-dimensional CPP are introduced in detail. The far-field distribution and spatial frequency spectrum of this CPP are calculated. The results show that this method is efficient and can reflect the relationship between the surface figure and the far-field distribution of a CPP directly.

©2013 Optical Society of America

## 1. Introduction

High-power lasers are used in many areas such as new energy sources, industry, and science research. Usually the requirement of energy distribution in a focal plane is uniform. During the operation process, beam quality, especially a laser beam with a large aperture, descends obviously. Poor beam quality is one of the reasons that will cause asymmetrical optical energy distribution in the focus. In order to obtain uniform irradiation in the far field, many kinds of beam-smoothing methods based on diffraction optical theory are established [1–3]. In the meantime, different kinds of beam-smoothing elements are developed [4–6]. A continuous-phase plate (CPP) has attracted widespread attention in recent years [7–10] because of its high diffraction efficiency, high energy utilization, and convenience of shape controlling of the focal spot.

As shown in Fig. 1 , in a high-power laser system the aberrant wavefront will be modulated by a CPP put in front of a lens; then a focal spot with uniform energy distribution will be observed. For beam smoothing, a continuous-phase plate must meet the following conditions: the phase plate must have a uniform far-field distribution, the phase plate should have a random surface figure, and the correlation length of the surface undulation is short enough [11]. So we can say that good design of a CPP is a precondition for efficient beam smoothing.

Nowadays, the modified G–S algorithm (Gerchberg–Saxton algorithm) is widely used to design a CPP [12]. Using a modified G–S algorithm, repeated iteration is unavoidable. It will complicate the designing and optimizing processes. Moreover, this method cannot intuitively explain the relationship between the far-field distribution and the surface pattern of a CPP. In this paper we present a novel design method based on the surface gradient of a CPP.

## 2. Relationship between the surface gradient and the far field distribution of CPP

There are few reports on how to use the surface gradient to describe the focal spot of a CPP, although it is a mature method based on the geometric optical principle. In this section, we will introduce the relationship between the surface gradient and the focal spot.

The surface shape and its gradient of a CPP can be expressed as $\varphi (x,y)$ and $\nabla \varphi $, respectively. $\nabla \varphi $ is a 2-dimensional function. In addition, we can regard surface shape as a 3-dimensional implicit function, which can be expressed as $\varphi (x,y,z)=0$. So $\nabla \varphi $. can be regarded as a 3-dimensional vector, and it is equated with the normal vector of the surface. Using the geometric method the analysis principle of the far field focal spot of a CPP is shown in Fig. 2 .

The incident parallel light from point A propagates through the CPP, where n is the refractive index of the material. Refraction will happen when the light ray spreads to the rear surface. Line AO is the incidence light ray and OP is the refracted ray. OB is the normal line and *θ _{i}* and

*θ*are the incidence and refraction angles, respectively.

_{o}*δ*means the excursion from the original transmission direction. In the incident plane, using small angle approximation the relationship between those angles can be derived as

In Fig. 2, the transmission rule of only one light ray is described. Because the surface undulation is 3-dimensional, it is easy to imagine that when a bundle of light rays propagate through a CPP, different incident planes can be gotten in consideration of the different directions of the refracted rays. The rotation angle of an incident plane is related to the gradient direction of the incident point. If we only pay attention to the direction, a space vector can be expressed by two angles. So in a space rectangular coordinates system the vectors in the incident plane can be transformed as

*α*, which is not marked in Fig. 2, is the angle between the

*x*axis and the intersecting line of the

*x*-

*y*plane and the incident plane.

Substitute Eq. (1) into Eq. (3), and Eq. (3) can be rewritten as

The refracted ray $\overrightarrow{OP}$ can be regarded as a function of the space coordinates. So if we only care about the relative distribution of the far-field of the CPP, the constant term and the scale factor could be omitted. So we can simply say that the propagation direction of the refracted ray is the gradient of the refracted point in the CPP surface. According to the result, the ray-tracing method can be employed to calculate the far-field distribution of a CPP. To prove the effect of the ray-tracing method, we will generate a random surface $\varphi (x,y)$ and calculate the far-field distribution according to the gradient $\nabla \varphi $.

Using the calculation model that is introduced in Reference [13] a CPP can be generated. The surface shape of a CPP can be described as:

where*x*and

*y*are the coordinate variables and

*A*and

*s*are two constants.

*Rand(x,y)*means a white noise with uniform distribution and * is the convolution symbol.

Figure 3(a) shows a typical surface that is generated by Eq. (5). The CPP is foursquare and the side length is 512 mm. It is very intuitive that the surface undulation of this element is random and continuous. The gradient of any point on the surface can be described as $[qx,qy]=\nabla \varphi $. In order to keep consistent with previous 3-dimensional equations in the form, the gradient can be expressed as $[qx,qy,-1]=\nabla \varphi $. Using the ray-tracing method the far-field distribution of the CPP can be obtained as shown in Fig. 3(b).

As we know, the transmission direction of light rays determines the size of the focal spot. From Eq. (4) we can see that the transmission direction $\overrightarrow{OP}$ will be influenced by the gradient function $\nabla \varphi $. So if we multiply the gradient functions by an appropriate coefficient, we will adjust the spot size on the premise that the relative intensity distribution of the focal spot remains unchanged. This characteristic brings us convenience on the design of the CPP.

In Fig. 3(b) the density of the tracing points means the amount of light rays that arrived at a certain position. Since a light ray represents the energy propagation direction, the density of the tracing points can be seen as the intensity distribution in the focus. In other words, the focal spot of a CPP can be expressed as the 2-dimensional histogram of the surface gradient $hist(qx,qy)$. The unit of the independent variables in function $hist(qx,qy)$ is radian. In order to be clearer, we can transform the angle unit into a length unit in the focal plane by multiplying a constant (i.e., the focal length of a lens) during the calculation process. Furthermore, to calculate $hist(qx,qy)$, the form of $qx$ and $qy$must be discrete.

Using this idea, we calculate the 2-dimensional histogram of the CPP that is shown in Fig. 3(a). For comparison, we calculate the far-field distribution of the same phase plate by the traditional Fourier transform method. Both of the calculation results are shown in Fig. 4 . In this paper the focal length of the lens is 4 mm and the wavelength of the incident light is 351 nm.

As shown in Fig. 4, the diameter of the focal spot is about 0.2 mm. Compared with the results shown in Fig. 4(b), the speckles with high spatial frequency do not appear in the focal spot that is expressed by the histogram. If we ignore the influence of the speckles or filtering the Fourier transform result by a low-pass filter, the two results are consistent. This phenomenon is caused by the approximate treatment that employed in the histogram calculation. Using the geometric optical method, the influence of the coherence characteristics of the laser is not considered. According to the Huygens–Fresnel principle, after passing through a CPP, a laser beam can be divided into a number of wavelets. In the far field the coherence superposition of the wavelets will cause the high spatial frequency speckles as shown in Fig. 4(b). As for a CPP, we do not care about the characteristics that are relevant to the speckles in the far field. Our main concern is the characteristics of the focal spot with low spatial frequency. Using the geometrical method, or in other words with the incoherent approximation, the results will meet the request for the application well. So we can expect that the geometrical method will be an efficient way to design and analyze a CPP.

## 3. Design method based on the histogram modification

Generally, a good design of a CPP means the far-field distribution is uniform and the shape of the focal spot should meet the application requirements. Since the focal spot of a CPP is equal to the histogram of the surface gradient $hist(qx,qy)$, the distribution of the focal spot will be changed with the modulation of the histogram.

It is very interesting that in image processing technology there is an analogous histogram modification method that can be used for reference [14]. Because there are two coordinate vectors in the 2-dimensional condition, some complex deduction processes will lead into the transformation. To explain the design principle and process more clearly, it is worthwhile to avoid complex formula derivation and transformation skills. So we will introduce the design idea and procedures by a 1-dimensional CPP design process.

During the histogram modification process, a transform $p(x)=T\left[q(x)\right]$ is employed. Since there is no direct relationship between the transform process and the coordinate variable, we can express the transform as $p=T\left[q\right]$. Here $q(x)$ or $q$means the gradient of the original surface. And $p(x)$ or $p$ means the gradient of the transformed surface. Here the problem that needs to be solved is to find out the specific expression of $T\left[\right]$, while the histograms $q(x)$ and $p(x)$ are both known. To discuss this problem, we must notice the properties of the histogram. More specifically, when $T\left[\right]$ is a monotone function, the integral of the histograms has the relationship

In Eq. (6), the upper limits of the integrals are variables. Since the histogram of $p(x)$ is known, the specific form of $hist(p)$ can be written as

Equation (7) indicates that in the interval [a,b] the distribution of the focal spot is uniform. Substitute Eq. (7) into Eq. (6) we can get

Because the histogram of $q(x)$ is also known, Eq. (8) is just the specific expression of$T\left[\right]$.

Now we need to have further discussion as to the mathematical and physical interpretation of Eq. (6), which is the starting point of the entire transformation.

First, we should explain the meaning of the above functions. In physics, if $q(x)$is discrete, every discrete element reflects a light ray. The histogram of $q(x)$ reflects the focal spot of CPP. In mathematics we can regard *q* as a random variable. So the histogram of $q(x)$ is the probability density of variable *q*. The integral of the histogram is just the probability distribution function of *q*. The discussions on $p(x)$ and *p* are the same.

We can rewrite Eq. (6) as another form,

where $\Delta p$ and $\Delta q$ are the neighborhoods of*p*and

*q*, respectively. In mathematics Eq. (9) just describes that the probability is constant. And in physics, this formula states a fact that the amount of the light rays is constant too.

Using this method, $hist(p)$ is not limited to uniform distribution. It can be expanded to design any distribution form in the focus. So in this paper we name the method as histogram modification rather than histogram equalization [14].

Now we introduce the design procedures as follows:

where in the discrete 1-dimensional situation, functions $p(x)$ and $q(x)$ are both expressed as subscripted arrays. $i\in (1,N]$, where*N*is the amount of the discrete points.

- IV. Integrate the surface gradient function $p(x)$ to obtain the final surface function of a 1-dimensional CPP.

Through the above steps, we transform a random surface into a surface in which far-field distribution can meet the design requirement. So we can also regard the design process as an optimizing process of a random surface.

## 4. Design results and discussions

According to the design steps that were presented in the previous section, we achieve a 1-dimensional CPP and the results are shown in Figs. 5 ,6 ,7 ,8 ,9 . In the numerical calculation process, 10,000 discrete points are used to reflect the statistical law. The correlation length of the CPP is 9 mm.

Figure 5 shows the original surface and the optimized surface. From Fig. 5 we find out that in sharp contrast to the design results by G-S algorithm, the variation law of the surface shape shows little change. The correlation length of the surface shows little change too. This is very helpful in order to control the final surface figure of a CPP to meet the fabrication requirements. During the design process, the continuity of the surface will not be destroyed due to the monotonicity of the transformation.

We calculate the spatial frequency spectrum distributions of the two surface figures, and the results are shown in Fig. 6.

As shown in Fig. 6, after the histogram transformation the spectrum width of the surface changes a little. In fact, the spectrum width is reduced slightly and it can reduce the difficulty of the fabrication of a CPP.

To evaluate the availability of our design results, we calculate the far-field distribution of the two surface figures that are shown in Fig. 5. Figure 7 shows the intensity distributions of the focal spots that are calculated by the Fourier transform, and those calculated by histogram modification are shown in Fig. 8.

Compare the distribution of the focal spot before and after the transformation and it is clear that the uniformity of the focal spot is improved significantly. At the same time, the beam smoothing process does not change the size of the focal spot. The spot size of the 1-dimensional CPP is about 0.02 mm. So we can say that the designed result of the 1-dimensional CPP achieves the design goal of getting a focal spot with uniform energy distribution.

This method can be extended to a 2-dimensional condition. In a 2-dimension design, the histogram transform can be written as

On the left side of this equation, ${p}_{x}$ and ${p}_{y}$are two components of a gradient function in two perpendicular directions. To construct the surface figure of a 2-dimesional CPP the extra relationship of these components must be taken into account.

Solve Eqs. (11) and (12) simultaneously, and the expressions of ${p}_{x}$ and ${p}_{y}$can be achieved. Then we can get the final surface figure of a 2-dimensional CPP by using the following formula:

Figure 9 shows a foursquare 2-dimensional design result with aperture 512 mm × 512 mm.

Using the Fourier transform method the far-field distribution of the 2-dimensional CPP is calculated, and the results are shown in Fig. 10 .

Figure 10(a) shows the direct numerical calculation result of the focal spot. Figure 10(b) shows the focal spot after the low-passing filtering. The results prove that we have achieved a continuous-phase plate with uniform far-field distribution. Furthermore, the shape of the focal spot can be designed to meet actual requirements, such as a circular or rectangular focal spot. Though the physical principle and the design steps of the 2-dimensional CPP are consistent with the 1-dimensional CPP, we need to overcome some more difficulties of mathematical aspects because there are two coordinate components in the surface function. However, this is not the research emphasis of this paper.

## 5. Conclusion

A continuous-phase plate has a complex surface figure. After passing through a CPP, the transmission direction of light rays will be changed. Based on a geometrical analysis method, we study the effect of the surface gradient of a CPP to the transmission direction of light rays. We find out that the far-field distribution of a CPP is equal to the 2-dimensional histogram of its surface gradient. This result gives us the idea that we can change the intensity distribution in the focus by changing the transmission direction of the light rays. So we propose a scheme to change the transmission direction of the light rays by additional phase changes on a phase plate. According to this ideal, we present a novel method to design a CPP. In this paper, we introduce the design principle and the steps in detail. The design results of a 1-dimensional CPP are discussed. As shown in the calculation results, this design method is efficient and can reflect the relationship between the surface figure and the energy distribution in the far field more directly.

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