Single-focus x-ray zone plate by stagger arrangement of zones

Xiaohu Chen; Xiaofang Wang

doi:10.1364/OE.21.020005

1. Introduction

Fresnel zone plate (FZP) that consists of alternating opaque and transmitted concentric zones has been widely used in x-ray microscopy system (XMS) for its focusing characteristics in the x-ray region [1, 2]. Its spatial resolution depends on the width of the outmost zone. Lawrence Berkeley National Laboratory has achieved a spatial resolution of 12 nm, which is the highest two-dimension resolution in x-ray microscopy [3]. Nevertheless, the imaging property of a conventional FZP is affected by two defects: low efficiency and multi-order foci. The first- order efficiency of a conventional FZP is around 10% [1], which degrades the energy from a source to the image. The multi foci allocate a large quantity of energy to other unavailable foci, which form background of the image.

Phase shifted FZP (FPZP) may theoretically raise the first-order efficiency to four times of a conventional FZP [4]. FPZP has been developed in recent years. A multilevel zone plate that introduced a four-level phase shift in each zone improved the first-order efficiency to 55% at 7 keV experimentally, and meanwhile decreased the background [5]. But the width of the multistep outmost zone which determines the spatial resolution is technically restricted by electron beam lithography (EBL). A spatial resolution of 0.6 μm at 9.85 keV has been reported by Takeuchi et al. [6], still far away from the sub-15 nm resolution realized by conventional FZP’s in the soft x-ray region [3,7,8]. In the soft x–ray region, a multilayer FZP composed of alternating SiO₂ and Al₂O₃ zones, fabricated with a different technique, may produce a theoretical maximum first-order diffraction efficiency of 13.3% at 1.2 keV [9], but its imaging properties, such as multi-foci formation, are the same as that of a conventional FZP. A different approach is to use a zone plate made of photon sieves [10] or compound photon sieves [11], which could improve the spatial resolution and decrease the higher-order diffractions simultaneously.

Gabor zone plate (GZP) can also improve imaging by eliminating the higher-order diffractions or multi foci. As the transmittance of GZP is a cosinoidal function in the radial direction, GZP has only a pair of conjugated foci. Beynon et al. has proposed a binary Gabor zone plate (BGZP), in which a spatially symmetrical two-dimensional binary pattern is introduced [12]. Choy and Cheng improved the BGZP in both design and performance, and experiments with their newly designed GZP at the wavelength of 633 nm showed that the focusing efficiency is 23% higher than the previous BGZP [13]. But due to fabrication difficulties of the three-dimensional binary structures, no such GZPs have been applied in the x-ray region. More recently Wei et al. designed an annulus-sector-shaped structure, trying to decrease the technical difficulty of manufacturing a binary GZP [14]. Still, there exist micro sector structures which are hard to manufacture by EBL or other techniques.

In this paper a novel single-focus x-ray zone plate is proposed by stagger arrangement of zones, which is called SZP. When integrated in the azimuthal direction, the transmittance of the SZP is a cosinoidal function of radius, which is the same as that of a GZP. In the following, the design and structure, the imaging properties, and the spatial resolution of the SZP will be presented. An advantage of the SZP is its single-order focus. Besides, different from the previous binary GZP’s which have micro-structure sectors [12–14], the SZP only has large staggered zones, and would be technically easier to manufacture.

2. Design of the SZP

Compared to a conventional FZP, the SZP is made up of a number of reshaped half-wave zones. Every two of these reshaped zones are stagger-arranged in a pair and these pairs are randomly orientated in the azimuthal direction. For simplicity, we start the design from the diffraction pattern of a zone plate illuminated normally by a plane wave of wavelength λ and amplitude A₀. The far field diffraction pattern at point P(x, y) can be obtained by the Fresnel-Kirchhoff diffraction formula in Fresnel approximation [15]:

U (P) = \frac{A_{0}}{i λ z} \exp (i k z) \int_{0}^{2 π} \int_{0}^{R} t (ρ, θ) \exp {\frac{i π}{λ z} [{(x - ρ \cos θ)}^{2} + {(y - ρ \sin θ)}^{2}]} ρ d ρ d θ,

where k = 2π/λ, t(ρ, θ) is the transmittance of the plate, R is radius of the zone plate, and ρ and θ are the radial distance and the azimuthal angle of the zone plate in a polar coordinate system, respectively, z is the on-axis distance from the zone plate center to the image plane. When only considering the on-axis imaging property where x = 0 and y = 0, Eq. (1) can be simplified to:

U (z) = \frac{A_{0}}{i λ z} \exp (i k z) \int_{0}^{R} T (ρ) \exp (\frac{i π}{λ z} ρ^{2}) ρ d ρ,

where the integrated transmittance T(ρ) is expressed as:

T (ρ) = \int_{0}^{2 π} t (ρ, θ) d θ .

For a GZP, which is azimuthally symmetric, the transmittance can be expressed by:

t_{G} = \frac{1}{2} (1 - \cos \frac{π ρ^{2}}{λ f}) .

Substituted into Eq. (1), the on-axis diffraction pattern of GZP will be written as:

\begin{array}{l} U (z) = \frac{A_{0}}{i λ z} \exp (i k z) \int_{0}^{2 π} \int_{0}^{R} \frac{1}{2} (1 - \cos \frac{π ρ^{2}}{λ f}) \exp (\frac{i π}{λ z} ρ^{2}) ρ d ρ d θ \\ = \frac{2 π A_{0}}{i λ z} \exp (i k z) \int_{0}^{R} \frac{1}{2} (1 - \cos \frac{π ρ^{2}}{λ f}) \exp (\frac{i π}{λ z} ρ^{2}) ρ d ρ \end{array}

Compare Eq. (2) and Eq. (5), when the transmittance T (ρ) of the diffractive plate in Eq. (2) obeys the following relation:

T = π (1 - \cos \frac{π ρ^{2}}{λ f}),

the diffraction pattern of the zone plate would be like that of a GZP. Next we will introduce a design that satisfies the Eq. (6).

Different from the previous binary GZP’s that contained small-scale sector elements [12–14], we design the SZP that satisfies the cosine transmittance Eq. (6) by using large-scale elements of staggered zones, which should be technically easier to manufacture. Figure 1 shows the first pair of staggered zones of the SZP. As in the case of amplitude type, it consists of opaque material (blue-shaded area) and transparent part (red-shaded area). Compared to the 1st and the 2nd zones (boundaries indicated by dash circles in the figure) of the corresponding conventional FZP, the staggered zone pair is composed of two equal parts. The lower part (the region for y≤0) is formed by rotating the upper part (the region for y≥0) by 180 degrees. So, in the following, we only introduce how to generate the upper-part pattern. For the upper part, the integral of the Eq. (3) is simplified to:

T_{u p p e r} (ρ) = \int_{0}^{π} t (ρ, θ) d θ .

For the above integral, when designing the inside boundary of the upper part, we suppose t = 1 for θ in the range from 0 to θ and t = 0 otherwise. In this way, Eq. (7) becomes:

T_{u p p e r} (ρ) = \int_{0}^{θ} d θ = θ .

Taking into account the contribution of the lower part, the total transmittance of this zone pair is

T = 2 θ .

By equaling Eq. (9) to (6), one can get the inside boundary r₁ of the staggered zone pair:

Fig. 1 The first pair of staggered zones. The red-shaded area and blue-shaded area stand for the transparent part and the opaque part, respectively. The dash circles show the outer boundaries of the 1st and the 2nd half-wave zones of a conventional FZP.

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r_{1} : ρ = \sqrt{\frac{λ f}{π} [arc \cos (1 - 2 \frac{θ}{π})]} 0 < θ \leq π .

On the other hand, for the outside boundary of the upper part, we suppose t = 0 for θ in the range from 0 to θ, and t = 1 otherwise. In this way, Eq. (7) becomes

T_{u p p e r} (ρ) = \int_{θ}^{π} d θ = π - θ .

Also, taking into account the contribution of the lower part, the total transmittance of this pair is

T = 2 (π - θ) .

By equaling Eqs. (12) to (6), one can get the outside boundary r₂ of the staggered zone pair:

r_{2} : ρ = \sqrt{\frac{λ f}{π} [arc \cos (1 - 2 \frac{θ}{π}) + π]} 0 < θ \leq π

Similarly, the boundaries of the lower part can be obtained as follows:

\begin{array}{l} r_{3} : ρ = \sqrt{\frac{λ f}{π} [arc \cos (3 - 2 \frac{θ}{π})]} π < θ \leq 2 π \\ r_{4} : ρ = \sqrt{\frac{λ f}{π} [arc \cos (3 - 2 \frac{θ}{π}) + π]} π < θ \leq 2 π \end{array}

As can be found, r₁ starts with ρ(r₁) = 0 when θ = 0 and ends with $ρ (r_{1}) = \sqrt{λ f}$ when θ = π, which is equal to the radius of the first half-wave zone of the corresponding conventional FZP. r₂ starts with $ρ (r_{2}) = \sqrt{λ f}$ when θ = 0 and ends with $ρ (r_{2}) = \sqrt{2 λ f}$ when θ = π, which is equal to the outside boundary radius of the second half-wave zone of the corresponding conventional FZP.

From the boundary curves above, the transmittance of the first pair of staggered zones can be figured out:

t (ρ, θ) = {\begin{cases} 1 0 < θ < \frac{π}{2} - \frac{π}{2} \cos \frac{ρ^{2} π}{λ f}, 0 < ρ \leq \sqrt{λ f} \\ 1 π < θ < \frac{3 π}{2} - \frac{π}{2} \cos \frac{ρ^{2} π}{λ f}, 0 < ρ \leq \sqrt{λ f} \\ 1 \frac{π}{2} + \frac{π}{2} \cos \frac{ρ^{2} π}{λ f} < θ < π, \sqrt{λ f} < ρ \leq \sqrt{2 λ f} \\ 1 \frac{3 π}{2} + \frac{π}{2} \cos \frac{ρ^{2} π}{λ f} < θ < 2 π, \sqrt{λ f} < ρ \leq \sqrt{2 λ f} \\ 0 t h e o p a q u e z o n e \end{cases} .

By integrating the transmittance of Eq. (15), we get the integral transmittance which can be expressed as:

\begin{array}{l} T (ρ) = \int_{0}^{2 π} t (ρ, θ) d θ = {\begin{cases} \int_{0}^{\frac{π}{2} - \frac{π}{2} \cos \frac{ρ^{2} π}{λ f}} t (ρ, θ) d θ + \int_{π}^{\frac{3 π}{2} - \frac{π}{2} \cos \frac{ρ^{2} π}{λ f}} t (ρ, θ) d θ, 0 < ρ \leq \sqrt{λ f} \\ \int_{\frac{π}{2} + \frac{π}{2} \cos \frac{ρ^{2} π}{λ f}}^{π} t (ρ, θ) d θ + \int_{\frac{3 π}{2} + \frac{π}{2} \cos \frac{ρ^{2} π}{λ f}}^{2 π} t (ρ, θ) d θ, \sqrt{λ f} < ρ \leq \sqrt{2 λ f} \end{cases} \\ = π (1 - \cos \frac{π ρ^{2}}{λ f}) . \end{array}

Compare the Eq. (16) to the Eq. (6), this zone pair sector can realize the function of the so-called cosinoidal transmittance. To build a SZP in this way, large quantities of other staggered zone pairs are arranged. In order to generate azimuthal symmetry, these staggered zone pairs are randomly orientated azimuthally, which are shown in Fig. 2.

Fig. 2 (a) Azimuthally random arrangement of the first and the second pair of staggered zones. (b) An SZP of 50 zone pairs. The red-shaded area and blue-shaded area stand for the transparent part and the opaque part, respectively.

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Then the transmittance of such a SZP can be written as:

t (ρ, θ, n) = {\begin{cases} 1 0 < θ - θ_{n}^{'} < \frac{π}{2} - \frac{π}{2} \cos (\frac{ρ^{2} π}{λ f}), \sqrt{2 n λ f} < ρ \leq \sqrt{(2 n + 1) λ f} \\ 1 π < θ - θ_{n}^{'} < \frac{3 π}{2} - \frac{π}{2} \cos \frac{ρ^{2} π}{λ f}, \sqrt{2 n λ f} < ρ \leq \sqrt{(2 n + 1) λ f} \\ 1 \frac{π}{2} + \frac{π}{2} \cos \frac{ρ^{2} π}{λ f} < θ - θ_{n}^{'} < π, \sqrt{(2 n + 1) λ f} < ρ \leq \sqrt{(2 n + 2) λ f} \\ 1 \frac{3 π}{2} + \frac{π}{2} \cos \frac{ρ^{2} π}{λ f} < θ - θ_{n}^{'} < 2 π, \sqrt{(2 n + 1) λ f} < ρ \leq \sqrt{(2 n + 2) λ f} \\ 0 t h e o p a q u e z o n e \end{cases},

where n = 0,1…N-1 stands for the nth zone pair, N is the number of zone pairs, θ’_n is a random number from 0 to π.

When apply such a SZP to x-rays, the material of the opaque zone becomes partially transparent. Then the amplitude-type SZP in Eq. (17) changes into a phase-type one and the transmittance is changed into:

t (ρ, θ, n) = {\begin{cases} 1 0 < θ - θ_{n}^{'} < \frac{π}{2} - \frac{π}{2} \cos (\frac{ρ^{2} π}{λ f}), \sqrt{2 n λ f} < ρ \leq \sqrt{(2 n + 1) λ f} \\ 1 π < θ - θ_{n}^{'} < \frac{3 π}{2} - \frac{π}{2} \cos \frac{ρ^{2} π}{λ f}, \sqrt{2 n λ f} < ρ \leq \sqrt{(2 n + 1) λ f} \\ 1 \frac{π}{2} + \frac{π}{2} \cos \frac{ρ^{2} π}{λ f} < θ - θ_{n}^{'} < π, \sqrt{(2 n + 1) λ f} < ρ \leq \sqrt{(2 n + 2) λ f} \\ 1 \frac{3 π}{2} + \frac{π}{2} \cos \frac{ρ^{2} π}{λ f} < θ - θ_{n}^{'} < 2 π, \sqrt{(2 n + 1) λ f} < ρ \leq \sqrt{(2 n + 2) λ f} \\ \exp [- k d (β + i δ)] t h e o p a q u e z o n e \end{cases},

where β and δ are related to the complex refraction index of the zone material by

\tilde{n} = 1 - δ + i β

, which can be obtained from [16].

From the SZP structures, as shown in Fig. 2 and given by Eq. (18), the narrowest zone width of the outmost zone pair is the same as the outmost half-wave zone width of the corresponding conventional FZP. In fabricating such an SZP by EBL, the technical requirement for the e-beam writer would be the same as for the FZP. It will be shown in the following simulations that the resolution limit is also the same for the two kinds of zone plates.

3. Simulations and discussions

The parameters of the SZP used in the following simulations are shown in Table 1, if otherwise indicated. Referred to Fig. 2(b), it consists of 50 staggered zone pairs. The diameter of the SZP is 140 μm. The x-ray wavelength is chosen to be 0.275 nm, i.e. the Ti-K_α line at 4.51 keV. The zone thickness is 0.9 μm in order to optimize the diffraction efficiency at this wavelength. For the outmost zone pair, the outmost half-wave zone width of the corresponding conventional FZP is Δr_out = 0.35 μm. For the wavelength of 0.275 nm, the focal length is 177.8 mm [17].

Table 1. Parameters of the SZP Used in the Simulation

View Table

To validate the single-order imaging property of the SZP, the diffraction pattern is numerically simulated. For a plane wave normally incident on the SZP, the diffraction is shown schematically in Fig. 3.

Fig. 3 The schematic diagram of diffractions by the SZP. The z-axis is the optical axis that goes through the SZP center.

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Figure 4(a) shows the intensity distribution along the optical axis (z-axis). The data were obtained by using Eq. (1). There is a clear peak at the focal length of f, contributed by the 1st-order diffraction, but no other peaks at the 3rd-order foci at f/3 or at the other focal positions. Note a conventional FZP would produce the same intensity at f/3, f/5, f/7, etc.

Fig. 4 (a) The normalized intensity distribution along the optical axis of an SZP. (b) Profile of the Airy pattern along the x-direction on the focal plane. The inset shows the profile in a wider range. (c) The fraction of energy contained within a circle of radius r_c. The dash line shows 62.6%, the total transmission of the SZP.

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Figure 4(b) shows the intensity distributions along the x-axis of the diffraction pattern on the focal plane. Here the data were obtained by using Eq. (19), without the Fresnel approximation, which will be discussed later, and the intensity is relative to that of the incident plane wave. Like that of a conventional FZP, the pattern of the SZP is quite centrosymmetric, due to the azimuthally random orientations of the staggered zone pairs. In Fig. 4(b), the width between the intensity peak and the first minimum is 0.427 μm. According to the Rayleigh criterion, this width corresponds to an angular resolution limit of 2.40 × 10⁻⁶ rad. This resolution limit agrees with that of the corresponding conventional FZP [18], which is given by 1.22Δr_out/f = 2.40 × 10⁻⁶ rad, where Δr_out is the outmost zone width of the FZP.

From the schematic diagram of Fig. 3, the region of the 0th- (undiffracted) and −1st-order diffraction on the focal plane can be figured out, which occupies a circular area of radius 70 μm and 140 μm, separately. In combination with the numerical data of Fig. 4(b), the 1st- and 0th-order efficiency relative to the incoming light onto the SZP is evaluated to be 11.5% and 23.3%, respectively. Here, for the numerical evaluation of the 1st-order efficiency, only the energy inside the Airy pattern is taken into account, not including the contributions of the sidelobes of the 1st-order diffraction. For the 0th-order efficiency, the numerical evaluation is somewhat rough due to its superposition on the −1st-order diffraction and an additional diffraction background. It is more difficulty to evaluate the efficiency of the −1st-order diffraction, which is distributed more widely and mixed with the background. But it’s reasonable to assume that the −1st-order diffraction efficiency is equal to that of the 1st-order diffraction [4]. For the SZP, its theoretical total transmission is 62.6%, this means that there is about 16.3%, by deducting the 1st-, 0th-, and −1st-order, of the incoming light as the background, which is distributed more widely as will be discussed next in Fig. 4(c).

Spreading out from the Airy pattern center, Fig. 4(c) shows the energy contained within the circle of radius r_c in fraction of the energy of the incoming light. For the region r_c <140 μm, which contains the geometrical region including the 1st-, −1st-, 0th-order diffractions, and some background as well, it has a fraction of 54.3% of the energy of the incoming light. Outside of this region, for r_c >140 μm, there remained 8.3% of the energy. For this remained fraction, in addition to the possible contributions by the sidelobes of the 1st- and −1st-order diffractions, there is a diffraction background contributed by the SZP zone-pair structures, the latter is distributed in a much wider range and causes the fraction to increase gradually versus the radius r_c. Up to r_c = 2000 μm, the total fraction of energy is 62.5%, which is very close to the total transmission of the SZP.

Because of this background, the SZP may not be a good candidate in scanning transmission x-ray microscopy (STXM). In STXM, usually an order-sorting-aperture (OSA) of a diameter on the order of one third of the FZP size is positioned at f/3 in front of the 1st-order focus to block the 0th- and other higher-order diffractions [1]. By setting the OSA diameter to be 46 μm for the present SZP, it is estimated that the ratio of the energy of the Airy pattern to the total energy through the OSA is about 55%, while for the corresponding conventional FZP, this ratio is about 80%. The smaller ratio for the SZP is due to more 0th-order diffraction and the diffraction background as indicated above.

Nevertheless, the SZP is of single-order focusing, and its 1st-order efficiency is higher than that of a GZP, given by 6.25% [4], owing to the partial transmission and phase-shift of the SZP material in the x-ray region. The phase-shift effect is confirmed by comparing the 1st-order efficiency to that of an amplitude-type (or black-white type) SZP given by Eq. (17). The latter gives 5.2%, which is close to that of GZP.

By changing the x-ray photon energy of the incoming light, it is found that the SZP still keeps the property of single-order focusing like Fig. 4(a). The 1st-order efficiency changes, correspondingly, due to the change of index of refraction of the SZP material versus the photon energy. Figure 5 shows the 1st-order diffraction efficiency versus the photon energy. As a comparison, the 1st-order efficiency of the corresponding conventional FZP is also shown. Here the data were obtained based on Eq. (1). The two kinds of zone plates have the same material (Au), the same thickness (0.9 μm), the same total number of zones or zone pairs (50). The only difference is the structure of the zones (refer to Fig. 1), which causes different diffraction orders, and the different fractions of energy in each diffraction order, as well as the background by the SZP.

Fig. 5 The 1st-order diffraction efficiency versus x-ray photon energy. Solid curve: SZP. Dash dot curve: FZP.

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Finally we discuss the effect of the SZP zone-pair structure on the image’s skewness. Suppose a point source, still, emitting at 0.275 nm. The numerical simulation of its image is carried out for a typical situation: the image-to-source magnification of 10, i.e., the object distance is 1.1f and the image distance is 11f. When applying the SZP to image the source away from the optical axis, the off-axis imaging property of the SZP should be considered. In this case, the Fresnel approximation adopted in Eq. (1) may not be appropriate, instead the Fresnel-Kirchhoff formula should be used [17]:

U (P) = - \frac{i A_{0}}{λ} \iint_{Σ} \frac{t (ρ, θ) e^{i k (r + s)}}{r s} ρ d ρ d θ,

where r is the distance between the point source and a point on the SZP, s is the distance between the point on the SZP and the position P on the imaging plane, and Σ is the integration area of the zone plate.

Figure 6(a) presents, for the point source on the z-axis, the image’s intensity distributions along the x- and y-direction, separately. In the plot, position 0 is the Airy pattern center. It shows that, in either direction, the Airy pattern is quite symmetric, and the left-right asymmetry (the left-right intensity difference relative to the central peak) is less than 2 × 10⁻⁵%. Compare the x-direction intensity distribution to the y-direction one, there is a slight difference between them, which is, relative to the central peak, less than 0.7%. Moreover, the full width at half maximum is the same in the two directions. Therefore, the image’s asymmetry is negligible. The intensity difference between the x- and y-direction is caused by the limited number of zone pairs (50 here), albeit of the azimuthally random arrangement of the staggered zones (refer to Fig. 2). When increasing the zone-pair number, the difference becomes smaller, e.g. for 100 zone pairs, the simulation indicating that the intensity difference between the x- and y-direction is less than 0.2%.

Fig. 6 The image’s intensity distributions along the x- and the y-direction, separately. The intensity is relative to that of the point source. Position 0 is the Airy pattern center. The inset shows the sidelobes. Solid curve: x-direction. Dot curve: y-direction. (a) The point source is on the optical axis. (b) The point source is, along the x-direction, 3 mm off the optical axis.

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Figure 6(b) presents the image’s intensity distributions when the point source is, along the x-direction, 3 mm away from the optical axis. For this off-axis case, the left-right asymmetry of the Airy pattern is less than 5 × 10⁻³% in the y-direction and 0.1% in the x-direction, separately. The larger asymmetry in the x-direction is because the point source is away from the optical axis along the x-direction. The intensity difference between the x- and y-direction is still small, less than 0.7%. Thus, for the SZP’s imaging, the image’s asymmetry or skewness should be negligible.

4. Conclusion

We have designed a novel Gabor-type x-ray zone plate, SZP, by stagger arrangement of zones. Simulations of the SZP imaging have shown a single-order focus, and the diffraction efficiency is 11.5% at 0.275 nm. The SZP can also work for single-order focusing at other x-ray wavelengths. In addition, the SZP can produce diffraction-limited imaging. As the SZP has only large-scale zone structures, it would be technically easier to manufacture. Such a single-focus zone plate may become an alternative diffraction device in x-ray and ultraviolet applications.

Acknowledgments

We thank Yayun Yuan and Weiwei Zhang for help on the work. This work was supported partially by the China National Science Foundation (No.11075160), China National Science and Technology Major Project and the Chinese Academy of Sciences.

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Material	gold
Diameter D (μm)	140
Number of zone pairs	50
Zone thickness d (μm)	0.9
Δr_out (μm)	0.35
Wavelength λ (nm)	0.275
Focal length f (mm)	177.8

Single-focus x-ray zone plate by stagger arrangement of zones

Abstract

1. Introduction

2. Design of the SZP

3. Simulations and discussions

4. Conclusion

Acknowledgments

References and links

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Optics Express