## Abstract

Kinoform phase plates (KPPs) are widely used in inertial confinement fusion to improve energy efficiency and to produce an optimum irradiance profile on the target plane. However KPPs are sensitive to beam aberrations and offer little flexibility in temporally tailoring the far-field pattern. To overcome these problems, we developed a multisegmented KPP and demonstrated temporal control of a focusing pattern and protection against phase distortions by numerical simulations.

©2004 Optical Society of America

## 1. Introduction

Improvement of laser irradiation uniformity is an important issue in inertial confinement fusion (ICF) research. Irradiation nonuniformity prevents a fusion target from reaching the density and temperature required for fusion ignition, especially in direct drive laser fusion. To reduce irradiation nonuniformity caused by near-field intensity perturbation and phase aberration, random phase plates (RPPs) [1] that consist of randomly distributed bilevel phase shifts (0 or *π* rad) are widely used. However, the widely spread far-field pattern of a RPP, which follows the Besinc ^{2} function, reduces the amount of energy to the target. Additionally, considering the laser’s absorption distribution caused by the generated plasma, a flat-top irradiance profile on the target plane provides high absorption uniformity of the incident laser. Therefore, it is necessary to introduce a kinoform phase plate (KPP) [2] that is able to produce an optimum focal-plane irradiance profile.

However, in principle the far-field intensity pattern produced with a KPP is fragile against the phase error (wave surface distortion) of an input beam. In designing a KPP, we usually assume that the input beam is perfectly coherent, an assumption that is not correct for practical use. Although the phase reconstruction algorithm can design a desired phase distribution of KPP by considering fixed phase distortions, it is important for future commercial reactors to develop advanced focusing devices, which will be free from arbitrary beam aberrations, because of cost, simplification of the laser system, and freedom from need of beam alignment. To overcome this problem we have developed a multisegmented KPP (MS-KPP). Our MS-KPP consists of many sub-KPPs, each independently creating a far-field pattern, which then reduces the influence of the overall phase distortion because the relative phase error is small in each of the sub-KPPs. Moreover, temporal variation of the diameter of the focusing spot following implosion of a target will not only improve energy efficiency but will also facilitate waveform shaping and improve the power balance accuracy, although a general KPP offers little flexibility in temporally tailoring the far-field pattern.

In this paper we present a design algorithm for a MS-KPP that allows storage of multiple focusing patterns on a piece of the KPP. We demonstrated spatial and temporal control of the focal-plane irradiance profile using our MS-KPP by numerical simulations; thus we show the device’s usefulness and protection against phase distortions.

## 2. Multiple patterning of laser focus spot using a segmented kinoform phase plate

A conventional KPP produces only a single far-field pattern corresponding to the phase distribution and the input beam pattern. We have used a spatial multiplexing technique [3] for the design of KPPs that produce multiple far-field patterns in one phase distribution. This algorithm calculates a phase distribution by varying the input pattern (near-field pattern) and designing a phase distribution to generate a far-field pattern that corresponds to each input pattern. The pattern-designing algorithm is presented below (see Fig. 1). Assume two input beams, A (beam diameter: D_{a}) and B (beam diameter: D_{b}, D_{b} > D_{a}), and far-field patterns F_{A} and F_{B} generated by each, respectively.

(1) First, the phase distribution is determined so that input beam A produces pattern F _{A}. At this time, the phase-retrieval algorithm computes only the region of the diameter D_{a} (region A) on the phase plate surface.

(2) The phase distribution in region A is fixed after the phase distribution for input beam A is determined.

(3) The phase distribution is determined in the region of the diameter D_{b} (region B) on the phase plate surface, wherein the phase distribution is determined so that F_{B} is produced without variation of the phase distribution of region A, canceling the far-field pattern F A.

A general iterative Fourier transformation algorithm is used to determine phase distributions in each region [4, 5, 6, 7, 8]. First, input amplitude and phase distribution are set to be uniform and random, respectively. Second, a far-field pattern is calculated by fast-Fourier transform of the input complex amplitude. The amplitude distribution of the output pattern is replaced with that of the designed pattern; the replaced complex amplitude is inverse transformed to the near-field complex amplitude. The new input amplitude is replaced with the original uniform distribution while the phase distribution remains. When this process is repeated, the error in the calculated far-field patterns is decreased to a small value. In practice it is difficult to fabricate a KPP with a continuous phase distribution. In this algorithm we discretized the phase distribution to binary phase steps (0 or *π*). Figure 1 presents an example in which two far-field patterns are produced with one MS-KPP: Figs. 1(a), 1(b), and 1(c) show the phase distribution and the far-field patterns for input beams A and B, respectively. The algorithm is designed so that input beams A and B generate alphabets A and B in a far-field, respectively. The mesh size is 1024×1024, the intensity ratio of an input beam is of a flat top pattern of 1:1, and the beam diameter ratio is D_{a} : D_{b}=1 : 2. We have successfully tested the algorithm by generating two far-field patterns using only one KPP, as shown in Fig. 1. However, in principle the multiplexed hologram is sensitive to the alignment of the input beam. For example, if the input beam illuminates more than region A on the phase plate, the far-field pattern contains a mixture of the designs A and B in Fig. 1.

## 3. Temporal control of a focusing pattern with a multisegmented kinoform phase plate

To achieve high target gain in ICF, fuel targets must be compressed to a high density while the lowest possible temperature is maintained (low entropy). A laser pulse suitable for this purpose has a pulse shape whose temporal intensity is suppressed very low at first and increases rapidly with time [9]. However, it is very difficult to generate such a pulse shape and to adjust the power balance between multiple laser beams. Therefore, temporal changes in the the diameter of the focusing spot following implosion of a fuel target will not only improve energy efficiency; it will facilitate control and stabilization of the laser pulse shape. Temporal control can also be achieved with a liquid-crystal device [10]. However, we have used a phase plate to avoid laser damage and fabrication of a large-diameter device.

We have numerically carried out temporal control of a focusing pattern with a MS-KPP designed by the algorithm described in Section 2. The phase distribution of our MS-KPP is designed so that the diameter of the focusing spot shrinks following implosion, corresponding to each of the near-field patterns. The diameter and intensity distribution of an input laser beam are controlled over time with pulse stacking technology. Storage of five patterns was attempted to test practical application of ICF. The assumed design parameters include an incident beam with a circular opening flat-top pattern whose diameter expands gradually to 320 mm, a wavelength of 526.5 nm, and a flat-top intensity distribution in a far-field whose diameter shrinks gradually from 500 *µ*m. These simulations were carried out on a 2048×2048 grid. We estimated the performance of the temporal control of a focusing pattern with the designed MS-KPP utilizing two parameters: energy efficiency and the error from the designed far-field pattern. The energy efficiency is defined as the ratio of the focusing energy on the target surface to the incident energy. The error from the designed far-field pattern was estimated by use of root mean square (RMS) error. Figure 2 shows temporal control of the focal-plane irradiance profile using a MS-KPP. Figure 2(a) shows the near-field intensity distributions at each time step. The area of each ring is equal, and the intensity increases toward the outside; the ratio of the intensity is 1:2:4:6:10. Figure 2(b) shows the far-field intensity distributions in each time step. As shown in Fig. 2, the proper choice of the diameter and relative intensity of an input beam, allows a piece of a KPP to store five patterns; moreover, evaluation of the focusing pattern provided energy efficiencies of 67.3%, 67.7%, 68.0%, 64.3%, and 62.7% and errors of 0.56, 0.62, 0.67, 0.68, and 0.67, respectively. This verifies that the pattern at any time step reproduces a pattern that mostly approximates that designed. To investigate the effect of the misalignment of the input beam, we calculated the far-field pattern in case that the diameter increases with a fixed initial intensity. As the result, we confirmed that the previous patter remains about one tenth in the intensity.

## 4. Multisegmented kinoform phase plate protection against phase distortions

A MS-KPP that has protection against phase distortions was designed to adapt to arbitrary wave surface distortions. This is an application of the algorithm in Section 2, wherein a phase plate is divided into multiple segments (sub-KPPs), and each independently creates a far-field pattern. The aberrational influence on the whole KPP is reduced because the relative phase error is small in each of the sub-KPPs. A conceptual diagram of a MS-KPP with protection against phase distortions is shown in Fig. 3. Figures 3(a), 3(b), and 3(c) represent the segmentation of a KPP, focusing patterns from each segment, and the whole MS-KPP, respectively. One phase plate is divided into several large sets (hereafter referred to as segments; a 4×4 example is shown) as shown in Fig. 3. A general KPP forms one single far-field pattern using a whole phase screen, whereas each segment of a MS-KPP creates only a part of the far-field pattern, and a whole far-field pattern is formed together. For spatial multiplexing, there is a loss of resolution in the diffraction plane. In the case shown in Fig. 3, the resolution is reduced by a factor of 16 (4×4). Therefore, the number of modes that can control in the diffraction plane is reduced by a factor of 16. The method of designing the phase distribution is presented below. The gate function *g _{lm}* (

*x,y*) is defined with

*N*, the total element number in a KPP, and

*n*

^{2}, the number of segment divisions, as

where *l*=1,2,…,*n, m*=1,2, …,*n*, and *a*=*N/n*, (*A _{lm},B_{lm}*) denotes the center coordinates of (

*l,m*) segments, and Π is a rectangle function which satisfies (2).

The complex amplitude in the segment (*l,m*) is given by Eq. (3), with the complex amplitude of the whole beam *f* (*x,y*), as

The complex amplitude in a near field is set to Eq. (3), and the phase distribution of the segment is computed in the phase-retrieval algorithm. The overall phase distribution is determined by repeating this process for *l*=1,2,…,*n, m*=1,2,…,*n*. If the input beam is a flat top, it is not necessary to compute the pattern for each segment. Any segment can be computed and then repeated spatially.

We examined numerically whether MS-KPP is less sensitive to phase distortion and determined the optimum number of divisions. A one-dimensional MS-KPP was designed with the following parameters. Both the laser beam incident to the MS-KPP and the designed pattern in the far-field have flat-top profiles that consist of a 1024 grid. Here an assumed phase distortion was a sine function with an amplitude of *I* sin (*α*
*x*), where *I* is the amplitude of the distortion, *α* is the spatial period, and *x* denotes the coordinates on the KPP. After the overall phase distribution was designed, we added this phase distortion and calculated the far-field pattern and error from designed pattern. Figure 4 shows the dependencies of the number of divisions on the influence of phase distortion for cases of two, four, and eight divisions compared with that for a monolithic KPP. Figures 4(a), 4(b), and 4(c) plot RMS errors against the amplitude of the distortion *I*, for a half-period (*α*=0.5), a single period (*α*=1), and a double period (*α*=2), respectively, where the red, green, blue, and orange lines represent the KPP without divisions, with two, four, and eight divisions, respectively. A tendency for a MS-KPP to keep the fidelity of the far-field pattern even with addition of distortion as the number of the division increases can clearly be seen in Fig. 4. A MS-KPP reduces the influence of the overall phase distortion because higher spatial frequency noise on the input beam is lost due to the loss of resolution. In addition, the MS-KPP must be divided into greater than 2*α* segments to prevent a phase distortion that has the spatial period *α*. We observe a similar tendency in the case of binary phase distributions. Continuous phase profiles are sensitive to phase errors as compared with binary phase in that the difference between the error with and without phase distortion is large.

In a practical ICF laser system, laser irradiation uniformity has been improved with a two-dimensional scheme of smoothing by spectral dispersion (SSD) [11, 12, 13]. A laser beam is frequency modulated, expanded, and reflected by a diffraction grating. The focused interfering beams from the phase plates have a smooth time-averaged intensity to drive a target by use of SSD. We designed the MS-KPP for practical ICF application and investigated numerically the focal properties of the MS-KPP with SSD. The adopted simulation parameters of SSD and the MS-KPP are as follows. The rate of angular spectral dispersion is 300 mrad/nm at a wavelength of 351 nm. The modulation frequencies are 9 and 13 GHz. The input beam to the MS-KPP has a flat-top profile of 320-mm diameter. The focal length is 5000 mm. The element (pixel) size of the MS-KPP is 500 *µ*m. The designed pattern has a flat-top profile diameter of 600 *µ*m. The defocus length is 5 mm to reduce a coherent spike caused by the coating thickness error of this binary KPP. A linear combination of three Zernike coefficients—defocus, coma, and spherical aberration—was adopted as an added phase distortion in consideration of the properties of the present laser. We assumed that the irradiation amplitude in the near field is constant, although a MS-KPP is sensitive to irradiation nonuniformity. Figure 5(a) shows the designed focal pattern and its line profile without phase distortion. This far-field pattern was strongly distorted by addition of the aberration as shown in Fig. 5(b). Improvements in this distorted pattern are evident in Figs. 5(c) and 5(d), which were calculated for segmentations of 2×2 and 4×4, respectively. As seen in Fig. 5, the degradation of the focal pattern decreases with increasing number of divisions. Especially the MS-KPP with a 4×4 segmentation still maintains the originally designed flat-top profile in spite of the added phase distortion. We examined the focal property of the MS-KPP from the view point of its energy efficiencies, which were defined as the ratio of energy within a focal spot of 500 µm to the incident energy, and are inserted in Fig. 5. By use of the designed MS-KPP, energy efficiency can be kept as high as 70%, which is comparable with that obtained without phase distortion (72%).

## 5. Conclusion

We presented a design algorithm for a MS-KPP that allows storage of multiple focusing patterns on a piece of the KPP. Using our MS-KPP by numerical simulations, we demonstrated temporal control of a focusing pattern, which shrinks following the implosion of a target. A MS-KPP that can protect against phase distortions was designed so that it could adapt to arbitrary wave surface distortion. We examined the optimum number of divisions and demonstrated the useful in terms of beam aberrations and energy efficiency.

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