1. Introduction

Beam smoothing, with a prescribed intensity distribution on the focal plane, is required in many kinds of application, for example, inertial confinement fusion (ICF) [1]. Diffractive optical element (DOE) is a good option for realizing such kind of beam smoothing, due to its high light efficiency, high flexibility in phase design, matured fabricating technology and broad selectivity of materials. The phase design of the DOE is usually considered as an optimization problem. In fact, many kinds of optimization algorithms have been applied for designing DOE, such as Gerchberg-Saxton (GS) algorithm [2,3] and its related modified algorithms [4,5], Yang-Gu (YG) algorithm [6] and Global/local united search algorithm (GLUSA) [7]. The near field phase distribution of the DOE is generally discretized with some same intervals, and the sampling interval on the focal plane is generally chosen as Δ=λf/D, where f, λ and D are the focal length of the Fourier transform lens, the incident wavelength and the aperture size of the DOE respectively. With this sampling interval on the focal plane, good performances of DOE for beam smoothing have been obtained by using any algorithms mentioned above. The intensities of the chosen sampling points on the focal plane are consistent with the required beam smoothing, but the other points are hardly fulfilled. Therefore, the sampling interval on the focal plane should be reduced, i.e. Δ≤λf/D/2. To save the optimization time, let Δ=λf/D/2. Then the intensity of any point on the focal plane will be fully filled with the required beam smoothing, not only the used sampling points in the optimization [8]. However, the performance parameters of the DOE for beam smoothing are generally calculated with a group of discrete sampling points on the focal plane, and the continuous far field intensity distribution of the DOE is not truly reflected in practice. To reflect the continuous far field intensity distribution and actually evaluate the performance of the beam smoothing, based on the spatial-frequency spectrum of the far field intensity distribution of the DOE, a novel evaluation criterion has been developed [9].

However, in practical applications, the designed DOE is very difficult to realize the expected beam smoothing on the focal plane, no matter what the sampling interval on the focal plane is chosen in the optimization. The reason is that the DOE is very sensitive to the incident phase distortion. In a realistic laser system, in particular the high power laser system, the output laser beam is not an ideal plane wave and usually it contains a rather large phase distortion, which exceeds the tolerance of the DOE for beam smoothing. To decrease the influence of the phase distortion and improve the performance of the DOE for beam smoothing, some techniques to realize smoothing have been adopted in time domain, for example, smoothing by spectral dispersion (SSD) in the ICF system [10,11]. For a laser pulse, the effective phase distortion is different at different time, therefore the obtained intensity distribution on the focal plane is a statistic result with numerous phase distortions. And the performance of the DOE for beam smoothing is improved. To explain this phenomenon theoretically, in this paper, based on the spatial-frequency spectrum of the far field intensity distribution of the DOE for beam smoothing, the relationship between the characteristic of the phase distortion and the performance parameters of the DOE for beam smoothing is quantitatively analyzed.

2. Theory of spatial-frequency spectrum analysis

For simplicity, only one-dimensional DOE is considered. In the optimization, the near field phase distribution of the DOE is discretized with same intervals. However, the continuous phase DOE cannot be obtained by interpolation with sinc function from the discrete phase data, as after interpolation, the continuous DOE has not only a phase modulation but also an amplitude modulation. In this case, the designed DOE is a multi-phase level element.

Supposing the size of the DOE is D, the number of the phase cells is K, then the transmittance of the DOE is

where x is the coordinate of the DOE, φ_p is the phase value of the pth cell, and

When an ideal plane wave is incident on the DOE, according to Kirchhoff theory, the far field intensity distribution can be written as

where x′ is the coordinate of the focal plane, f is the focal length of Fourier transform lens, λ is the incident wavelength. In Eq. (3), the sinc function and the phase factor are ignored.

If the phase of the DOE is symmetrical, expanding Eq. (3), then

where $y=-\frac{2\pi D}{K\lambda f}x\prime$,

$A_m=\sqrt{{\left[{\displaystyle \sum _{p=m+1}^K}\cos \left({\phi }_p-{\phi }_{p-m}\right)\right]}^2+{\left[{\displaystyle \sum _{p=m+1}^K}\sin \left({\phi }_p-{\phi }_{p-m}\right)\right]}^2}$ and $B_m=\arctan [{\displaystyle \sum _{p=m+1}^K}\sin \left({\phi }_p-{\phi }_{p-m}\right)/{\displaystyle \sum _{p=m+1}^K}\cos \left({\phi }_p-{\phi }_{p-m}\right)]$ are the amplitude and initial-phase spectra of the intensity distribution respectively.

Based on Eq. (4), the total intensity in the required area of the main lobe of the far field intensity distribution of the DOE for beam smoothing can be calculated. Assuming the required area is d ₁≤x′≤d ₂, the total intensity is

where $y_i=-\frac{2\pi D}{K\lambda f}{d}_i,i=1,2$.

The input intensity can be calculated from Eq. (5) with boundaries y ₁=-π and y ₂=π. The value is 2πK, and the light efficiency η can be defined as

The non-uniformity at the top, rms, can be defined as

where $\overline{I}=\frac{I_{\mathit{total}}}{\left(y_2-y_1\right)}$ is the average intensity in the required area.

3. Statistic analysis of influence of phase distortion

If the incident wave is not an ideal plane wave and the phase distortion of the pth cell is Δϕ_p , then the far field intensity distribution is

In the ICF system, the SSD technique is used to improve the performance of the DOE for beam smoothing. The SSD technique does not reduce the phase distortion. It creates an entirely different phase distortion, which changes in time [10]. Supposing the realized number of phase distortions in one laser pulse is L, the intensity distribution on the focal plane can be considered as a sum of L intensity distributions with the corresponding phase distortion. Supposing the pulse shape in time domain is rectangular, and, for convenience, the normalized far field intensity distribution is here concerned,

where Δ${\varphi }_p^l$ is the phase distortion of the pth cell at the lth moment.

Expanding Eq. (9), and

The key problem is how to evaluate $\frac{1}{L}{\displaystyle \sum _{l=1}^L}\left[\exp \left[i\left(\Delta {\varphi }_p^l-\Delta {\varphi }_q^l\right)\right]\right]$. For simplicity, let L be infinity, so the key problem is to obtain the expected value of $\frac{1}{L}{\displaystyle \sum _{l=1}^L}\left[\exp \left[i\left(\Delta {\varphi }_p^l-\Delta {\varphi }_q^l\right)\right]\right]$, i.e.E(exp[i(Δϕ_p -Δϕ_q )]).

In the ICF system, the output laser beam contains high frequency and low frequency phase distortions. The remained high frequency phase distortion after phase correction is approximately a Gaussian distribution [12]. Here, the Gaussian distributed high frequency phase distortion is analyzed, and it can be expressed as

where N[0,σ] takes place the Gaussian distribution with zero mean and σ variance.

The low frequency phase distortion can be simplified as a random phase plate, with a phase distribution written as [13]

where rand (-1,1) and ∗ take place the uniform distribution over (-1,1) and convolution respectively. x_s and a reflect the spatial characteristic size and the magnitude of the phase distortion respectively.

Hence

where Δ${\varphi }_{p\mathit{,}H}^l$ and Δ${\varphi }_{p\mathit{,}L}^l$ are the high and low frequency phase distortions of the pth cell at the lth moment respectively.

Assume, that the high and low frequency phase distortions are mutual independent, then

For the high frequency phase distortion, when q≠p, Δ${\varphi }_{p\mathit{,}H}^l$ and Δ${\varphi }_{q\mathit{,}H}^l$ are mutually independent, then

For the low frequency, q≠p, Δ${\varphi }_p^l$ _,L and Δ${\varphi }_{q\mathit{,}L}^l$ are not mutually independent. According to Eq. (12), for the lth moment,

where $R_k^l$ is the kth random number obeyed with uniform distribution rand(-1,1) at the lth moment, and $F(k,p,q)=a·\left\{\exp \left\{-{\left[\left(k-p\right)\frac{D}{K}/x_s\right]}^2\right\}-\exp \left\{-{[\left(k-q\right)\frac{D}{K}/x_s]}^2\right\}\right\}.$

Supposing $R_k^l$ is mutual independent at different k or different l, then

For each k,

where sin c(x)=sin x/x.

Therefore, Eq. (10) can be rewritten as

where $A_m=\sqrt{C_m^2+D_m^2}$ and B_m =arctan[D_m /C_m ] are the amplitude and initial-phase spectra of the intensity distribution after using the SSD technique respectively.

The forms of Eq. (20) and Eq. (4) are nearly the same, but the amplitude and initial-phase spectra in Eq. (20) are modulated by the characteristic of the phase distortion. According to Eq. (20), the high frequency phase distortion has the same influence on each m(m≥1), while the low frequency phase distortion has different influence on each m(m≥1).

4. Simulated results

For showing an example, the values for f, D, and λ are taken as 600 mm, 100 mm, and 1.053 µm respectively. The size of the main lobe of the far field intensity distribution of the DOE for beam smoothing is 100 µm and K is 256. The sampling interval on the focal plane is chosen as Δ=λf/D/2 to realize a true beam smoothing [8]. The near field phase and the far field intensity distributions of the designed DOE are shown in Fig. 1. Its complex amplitude spatial-frequency spectrum is shown in Fig.2. When Δ=λf/D/2, the GS algorithm or its related modified algorithms and YG algorithm are not suitable to design the DOE for beam smoothing. The adopted optimization algorithm in this paper is a kind of hybrid algorithm which merges hill-climbing (HC) with simulated annealing (SA). The basic structure of this hybrid algorithm is described as follow. This hybrid algorithm has sufficient ability of strong convergence of HC and the global optimization potential of SA.

step1: let phase distribution be a sine function with random period, amplitude and initial phase;

step2: calculate the value cost 1 of the cost function;

step3: choose a sine function with random period and initial phase as the searching direction, and optimize the amplitude of the sine function along this direction(hill-climbing optimization);

step4: calculate the new cost function value cost 2 after every hill-climbing optimization step;

step5: if cost 1>cost 2, accept the optimization step, let cost 1=cost 2, and go to step9 ;

step6: if cost 1<cost 2, calculate the accepted and refused probability p1 and p2,respectively;

step7: if p1>p2, accept the optimization step, and go to step9 ;

step8: if p1<p2, refuse the optimization step, and go to step3;

step9: determine whether the algorithm can stop or not, if can, go to step11, else, go to step3;

step10: stop the algorithm.

The optimized phase distribution is the sum of many sine functions with different period, amplitude and initial phase, which the continuity of the phase distribution can be assured.

The values of η and rms, calculated with Eq. (6) and Eq. (7) are 92.7% and 7.5% respectively. The ideal distribution in Fig. 2 is the corresponding complex amplitude spatial-frequency spectrum of the rectangular function with the same size.

 

Fig. 1. The design results of the DOE for beam smoothing

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Fig. 2. Spatial-frequency spectrum of the designed DOE for beam smoothing

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Now, let us have a discussion. Firstly, in this simulation, only the high frequency phase distortion is considered, i.e. a=0 and sin c(F(k, p, q))=1 for any k, p and q. Then the difference between Eq. (20) and Eq. (4) is the factor exp(-σ ²). According to the definition of the top non-uniformity, the larger the variance σ, the smaller the factor exp(-σ ²) is and the better the top non-uniformity. However, according to the definition of the light efficiency, the larger the variance σ, the smaller the light efficiency is. Calculating with the designed DOE as shown in Fig. 1, the relationship between η, rms and σ is shown in Fig. 3, and the variations of η and rms are both monotone.

Secondly, only the low frequency phase distortion is considered, i.e., σ=0 and exp(-σ ²)=1. From sin c(F(k, p, q)), the qualitative relationship between η, rms and a, x_s is not immediately obvious. When x_s is chosen as 60mm or 80mm, calculating with the designed DOE as shown in Fig. 1, the relationship between η, rms and a is shown in Fig. 4. The variation of η is monotone. When x_s is relatively large, i.e. the spatial frequency of the low frequency phase distortion is relatively low, in this case, the light efficiency decreases relatively slowly. The variation of rms is not monotone but a little bit complicated. rms improves firstly, then goes to bad, and finally improves again.

 

Fig. 3. Relationship between η, rms and σ

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Fig. 4. Relationship between η, rms and a

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Finally, the high and low frequency phase distortions are both considered, when x_s is chosen as 60mm, the relationship between η,rms and a, σ is shown in Fig. 5. σ varies from 0.01λ to 0.10λ step by step.

 

Fig. 5. Relationship between η, rms and a

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5. Conclusion

In this paper, considering the use of the SSD technique in the ICF system, the influence of the high and low frequency phase distortions on DOE for beam smoothing is statistically analyzed based on the spatial frequency spectrum method. The amplitude and initial phase spectra of the far field intensity distribution are modulated by the characteristic of the phase distortion. The relationship between the performance parameters of the beam smoothing and the characteristic of the phase distortion is obtained. For the Gaussian distributed high frequency phase distortion, the light efficiency decreases but the top non-uniformity improves with the increase of the variance of the phase distortion. For the low frequency phase distortion, the light efficiency decreases with the increase of the magnitude of the low frequency phase distortion, but the top non-uniformity improves firstly, then goes to bad, and finally improves again.

With the SSD technique, the rms value is better than the one realized by the DOE with an ideal plane wave. This is very useful and important in a realistic laser system. For high frequency phase distortion, though the value of rms changes slightly when σ varies from 0 to 0.10λ, the rms value will be less than 1% when σ is larger than 0.25λ. With the SSD technique, the top non-uniformity of the far field intensity distribution is correspondingly easy to satisfy the requirement of the beam smoothing. Even if the top non-uniformity of the designed DOE for beam smoothing is not good enough, we can choose or control the characteristic of phase distortion to improve the top non-uniformity according to Eq. (20).

With the SSD technique, the light efficiency is basically determined by the variance σ of the Gaussian distributed high frequency phase distortion and the magnitude a of the low frequency phase distortion. The higher the frequency of the phase distortion, the quicker decrease of the light efficiency is. For high frequency phase distortion, when σ is larger than 0.25λ, the value of η will be less than 30%.

Therefore, the phase distortion should be controlled to be as small as possible, no matter whether the SSD technique is adopted or not. With the SSD technique, the key problem is the light efficiency. Without the SSD technique, the key problem is the top non-uniformity. Eq. (20) affords a theoretical tool to determine whether the characteristic of the phase distortion and the performance of the designed DOE can satisfy the requirement of the light efficiency and the top non-uniformity simultaneously or not.

This method can be easily extended to a two-dimensional DOE for beam smoothing. The relationship between the amplitude and initial phase spectra of the far field intensity distribution of the two-dimensional DOE for beam smoothing and the characteristic of the phase distortion is similar to Eq. (20). In like manner, the Gaussian distributed high frequency phase distortion has the same influence on the amplitude spectrum, except the zero order, of the far field intensity distribution, and the low frequency phase distortion has different influence on the amplitude and initial-phase spectra. The qualitative variation rules of η and rms in two-dimensional case are similar to those in one-dimensional case.