1. Introduction

High-intensity foci are applicable for different fields [1]. The foci with high intensity are used for optical lithography [2] and applicable to trap particles stably [3,4], and generate clear images [5] and high-intensity optical vortices [6–9].

Some zone plates produce special foci along the optic axis. For example, fractal zone plates (FraZPs), fractal photon sieves, fractal zone plates with lacunarity and fractional fractal zone plates generate the single main foci with many subsidiary foci [10–13]; Thue-Morse zone plates (TMZPs) and Walsh zone plates have two equal-intensity foci with many subsidiary foci applicable to create the multiple images simultaneously [14,15]; Fibonacci zone plates (FiZPs), generalized Fibonacci photon sieves and Greek ladder zone plates have two equal-intensity foci with the golden mean or other ratios [16–18]. Another interesting way to produce the special foci is to implement m-bonacci zone plates, which are constructed by m-bonacci sequences and generate two main foci related to the m-golden mean [19]. In addition, a square Fresnel zone plate with a spiral phase has a square optical vortex whose topological charge is based on a modulo-4 transmutation rule [20]. Furthermore, an annulus-sector-shaped-element binary Gabor zone plate, which is constructed by annulus-sector-shaped structure apertures, has a cosinusoidal transmittance to generate a single focus along the optic axis [21]. Moreover, a single-focus x-ray zone plate, whose transmittance is a cosinoidal function of a radius, is constructed by stagger arrangement of zones to generate the single first-order focus along the optic axis [22]. A modified photon sieve with multiple regions has two or three equal-intensity foci with high resolution by adjusting the number of pinholes [23]. In addition, petal-like zone plates generate a variety of star-like intensity patterns, whose arms and sizes depend on the two parameters, and the depths and spacings between two foci are easily handled by means of a shifting parameter [24]. However, these zone plates cannot generate the high-intensity foci. There are some zone plates which generate the high-intensity foci, i.e., a conventional composite Fresnel zone plate consists of a central zone plate with first-order focal length and outer zones with third-order focal length, which generate the high-intensity focus with the improved resolution [25]. A composite photon sieve consists of an inner photon sieve using the first-order diffraction surrounded by an outer zone plate using the third-order diffraction, which suppresses sidelobes, and produces a focus with slightly better resolution [26]. Modified photon sieves, which are based on two overlaid binary gratings and a photon sieve through two logical XOR operations, provide better resolution and two-dimensional hard X-ray differential-interference-contrast imaging [27]. Moreover, composite fractal zone plates have the single high-intensity main foci with many subsidiary foci [28]. Composite fractional fractal zone plates generate the tailorable main foci with high intensity and subsidiary foci [29]. In addition, composite Thue-Morse zone plates generate two high-intensity foci with many subsidiary foci, which are applicable to produce two images with the low chromatic aberration [1]. Furthermore, modified Thue-Morse zone plates based on the Thue-Morse sequence of S = 3 provide two high-intensity foci located at the positions satisfying the fixed ratio of 3/5 [30]. Another interesting way to produce the high-intensity foci is to implement devil’s lenses, which are the fractal zone plates with the gradient phases and have the high-intensity main foci with many subsidiary foci [31]. Moreover, kinoform generalized mean lenses generate twin high-intensity main foci with the generalized mean and the suppressed high-order foci along the optic axis [32]. Nevertheless, the above-mentioned methods are not applicable for the other aperiodic zone plates, and for the above composite zone plates, the first-order diffraction areas of the inner and outer parts are not coincident. Some radially modulated zone plates are used to acquire annular beams at the focal planes. For example, Fresnel zone plates with radially shifted phases create annular beams with designed radii at the focal planes [33]; radial phase modulated spiral zone plates generate optical vortices with tailorable radii at the focal plane [34]; annular beams produced by modified spiral zone plates have tailorable radii and segmented phase gradients [35]. However, these zone plates cannot produce focal spots at the designed focal planes and fractal foci. There are some composite zone plates which generate the special foci, i.e., the bi-segment spiral zone plate generates a variety of beam shapes and structures by adjusting the width, topological charge, and radial phase shift of each segment [36]. Composite spiral multi-value zone plates produce equal intensity arrays of petal-like modes as well as dark optical ring lattice structures along the optical axis in multiple focal planes of the diffractive element [37]. Moreover, multi-regioned square zone plates generate square corner-located focal points as well as square top hat-like beams [38]. Furthermore, multi-regioned spiral square zone plates can generate unique features of an array of optical vortices, so that the number of vortices is directly related to the number of regions [39]. Nevertheless, these composite zone plates cannot generate high-intensity foci at the focal planes.

In this paper we propose generalized composite aperiodic zone plates (GCAZPs), which are used to produce the clearer images than the corresponding common aperiodic zone plates. The GCAZP is constructed by the radial zone plate (RZP), whose original radius is not zero, and the common aperiodic zone plate, which is used for many aperiodic zone plates. The first-order diffraction areas of the inner and outer parts are coincident, and two parts have the same axial first-order diffraction intensity distribution. Furthermore, the modulation transfer function of the GCAZP basically has bigger value than that of the corresponding common aperiodic zone plate. The GCAZPs have foci with higher intensity than the corresponding common aperiodic zone plates. In addition, the GCAZP with any size can be designed. We will illustrate the construction method of the GCAZP in details. Moreover, it will be proved in the simulations and experiments that the GCAZPs can generate high-contrast images. After a collimator is used in the experiments, the images at the focal planes can be acquired by a target object at infinity.

2. Design

For simplicity, the GCAZP based on the Fibonacci sequence is taken for example to illustrate the construction method of the GCAZP. The GCAZP based on the Fibonacci sequence is composed of the FiZP with (r₀=0, R=αa, S) and the Fibonacci RZP with (r₀=βa, R = a, S). r₀ and R respectively present the original radius and the outermost radius of the FiZP or the Fibonacci RZP, α is a positive number between 0 and $1/\sqrt 2$, β is a positive number between $1/\sqrt 2$ and 1, and S presents the order of the Fibonacci sequence. It should be noted that α and β satisfy the equation α²+β²=1, and these zone plates are designed on the devices with the same size of a×a.

The transmitance function ${q_i}(\zeta )$ of the FiZP with (r₀=0, R=αa, S), i.e., the inner part of the GCAZP based on the Fibonacci sequence, can be calculated by Eq. (1),

The transmitance function ${q_o}(\xi )$ of the Fibonacci RZP with (r₀=βa, R = a, S), i.e., the outer part of the GCAZP based on the Fibonacci sequence, can be calculated by Eq. (2),

It should be noted that the transmitance function of the GCAZP based on the Fibonacci sequence can be expressed as ${q_{GCAZP}}(\varsigma )= {q_i}(\zeta )+ {q_o}(\xi )$, where $\varsigma = {({{r / a}} )^2}$ is the normalized square radial coordinate. The FiZP with (r₀=0, R=αa, S) generates the first-order focus located at the axial position z=(αa)²/(λM). λ and M are the wavelength and the number of the elements of the Fibonacci sequence of order S, respectively. The Fibonacci RZP with (r₀=βa, R = a, S) generates the first-order focus located at the axial position z=(1-β²)a²/(λM). Thus, as the result of the equation α²+β²=1, the FiZP with (r₀=0, R=αa, S) and the Fibonacci RZP with (r₀=βa, R = a, S) generate the first-order focus with the same axial position. That is to say, the inner and outer parts have the coincident first-order diffraction area. In addition, the first-order focal axial position of the GCAZP can be adjustable by designing α and β.

In this paper, for simplicity, a, λ, α and β are set as 3.84 mm, 532 nm, $1/\sqrt 2$ and $1/\sqrt 2$, respectively. Figure 1 illustrates the construction method of the GCAZP based on the Fibonacci sequence. Figure 1(a) presents the binary profile of the FiZP with (r₀=0, R=$a/\sqrt 2$, S = 6) expressed as ${q_i}(\zeta )$. Figure 1(b) presents the binary profile of the Fibonacci RZP with (r₀=$a/\sqrt 2$, R = a, S = 6) expressed as ${q_o}(\xi )$. In fact, the inner and outer zones of the GCAZP based on the Fibonacci sequence are shown in Figs. 1(a) and (b), respectively. The GCAZP based on the Fibonacci sequence in Fig. 1(c) can be constructed by combining the binary profiles in Figs. 1(a) and (b).

 

Fig. 1. Binary profiles of (a) the FiZP with (r₀=0, R=$a/\sqrt 2$, S = 6), (b) the Fibonacci RZP with (r₀=$a/\sqrt 2$, R = a, S = 6) and (c) the corresponding GCAZP.

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3. Focusing properties

The Fresnel approximation expressed as Eq. (3) [16] is used to calculate the axial intensity distribution of the GCAZP.

Based on Eq. (3), the normalized axial irradiances of the GCAZP based on the Fibonacci sequence of S = 6, the inner part of the GCAZP based on the Fibonacci sequence and the outer part of the GCAZP based on the Fibonacci sequence, i.e., the FiZP and the Fibonacci RZP, are compared in Fig. 2, which are shown in Figs. 2(a)–(c), respectively. Moreover, the normalized axial intensity is normalized by the focal maximum intensity of all zone plates. It can be seen in Fig. 2(a) that the GCAZP based on the Fibonacci sequence generates two equal-intensity foci located at u = 9.95 and 16.05. Moreover, the FiZP and the Fibonacci RZP shown in Figs. 2(b) and (c) have twin equal-intensity foci with two same focal positions u = 9.88 and 16.12. Therefore, twin foci of the GCAZP based on the Fibonacci sequence have the approximately same u values as those for the FiZP and the Fibonacci RZP. Moreover, it can be seen in Figs. 2(b) and (c) that the FiZP and the Fibonacci RZP basically have the same axial first-order diffraction intensity distribution in the first-order diffraction area. It should be noted that the GCAZP based on the Fibonacci sequence consists of the inner and outer parts with the same center. Thus, the beams produced by the inner and outer parts propagate along the same optic axis. As is known, two foci generated by the FiZP and the Fibonacci RZP have the same u and first-order diffraction distribution. Thus, the GCAZP based on the Fibonacci sequence composed of the FiZP and the Fibonacci RZP generates twin foci with higher intensity.

 

Fig. 2. Normalized axial intensity distributions of (a) the GCAZP based on the Fibonacci sequence of S = 6, (b) the FiZP and (c) the Fibonacci RZP.

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On the basis of the above study, it can be found that left and right foci of the inner part of the GCAZP based on the Fibonacci sequence respectively have the same u values as those for the corresponding outer part, which have the approximately same u values as those for the GCAZP, respectively. It should be noted that the difference between the u values of the left foci for the GCAZP and the corresponding inner part is the same as that of the right foci. Thus, the difference between the u values of the left foci is equal to that of twin foci. The relative error of the u values of twin foci is defined as $\varepsilon = |{{u_l} - {u_{li}}} |/{u_{li}}$, where ${u_l}$ and ${u_{li}}$ show the u values of the left foci for the GCAZP and the corresponding inner part, respectively. Figure 3 shows the relationship between ɛ for twin foci of the GCAZP and S. It can be found in Fig. 3 that ɛ decreases firstly, then increases, then decreases and remains finally constant with the increasement of S. Specially, ɛ for the GCAZP of S = 8 and 9 are equal to 0. At this time, the u values of twin foci for the GCAZP are the same as those for the corresponding inner or outer part. Thus, the u values of twin foci for the GCAZP based on the Fibonacci sequence of the bigger S are approaching to those for the corresponding inner or outer part.

 

Fig. 3. Graph of ɛ for twin foci of the GCAZP with different S and S.

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The GCAZPs based on the Fibonacci sequences of the different orders can generate two foci with higher intensities. The normalized axial intensity distributions of the GCAZP based on the Fibonacci sequence of S = 7 and the FiZP of S = 7 are shown in Figs. 4(a) and (b), respectively, which are normalized by the focal maximum intensity of the above zone plates. The normalized axial irradiances of the GCAZP based on the Fibonacci sequence of S = 8 and the FiZP of S = 8 are shown in Figs. 4(c) and (d), respectively, which are normalized by the focal maximum intensity of two zone plates. It can be seen in Figs. 4(a) and (b) that the intensities of twin foci generated by the GCAZP based on the Fibonacci sequence of S = 7 are about 3.56 times stronger than those of the FiZP of S = 7. Moreover, it can be also seen in Figs. 4(c) and (d) that the intensities of twin foci generated by the GCAZP based on the Fibonacci sequence of S = 8 are about four times stronger than those of the FiZP of S = 8. Thus, two high-intensity foci can be generated by the GCAZPs based on the Fibonacci sequences of the different orders. As is known, for the GCAZP, the inner and outer parts generate the first-order focus with the same axial position. Therefore, two parts have the coincident first-order diffraction area. Thus, the adding outer area has no effect on the intensity level of the generated focal spots. Moreover, it can be seen in Figs. 2(b) and (c) that the inner and outer parts have the same focal positions. Thus, after adding the outer part to the inner part, the constructed GCAZP has two foci with higher intensity in the first-order diffraction area than the common FiZP.

 

Fig. 4. (a-d) Normalized axial irradiances of the GCAZP based on the Fibonacci sequence of S = 7, the FiZP of S = 7, the GCAZP based on the Fibonacci sequence of S = 8 and the FiZP of S = 8, respectively.

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In fact, the GCAZP based on the Fibonacci sequence can be designed on the device with an arbitrary size. For simplicity, the GCAZP based on the Fibonacci sequence of S = 6 is taken for example to illustrate it. The binary profiles of the GCAZPs with 1024 × 1024 pixels and 512 × 512 pixels are shown in Figs. 5(a) and (b), respectively, and the corresponding normalized axial irradiances normalized by the maximum of the axial intensities are shown in Figs. 5(c) and (d), respectively. In Figs. 5(c) and (d), the horizontal and vertical coordinates present the axial distance and normalized axial irradiance, respectively. It can be found in Figs. 5(c) and (d) that the GCAZP with 1024 × 1024 pixels generates twin foci located at z = 3.45 and 5.57 m, whose axial positions are about four times than those z = 863.2 and 1393.5 mm of twin foci for the GCAZP with 512 × 512 pixels, respectively. It should be noted that the size of the GCAZP with 1024 × 1024 pixels is about four times than the GCAZP with 512 × 512 pixels. Therefore, the size of the GCAZP have the approximately same scale factor as the axial positions of the generated twin foci. However, twin foci for the GCAZPs with 1024 × 1024 pixels and 512 × 512 pixels have the same intensities, and their intensity distributions along the optic axis in Figs. 5(c) and (d) are similar. Thus, the GCAZP based on the Fibonacci sequence can be designed on the device with an arbitrary size.

 

Fig. 5. Binary profiles of the GCAZPs based on the Fibonacci sequence of S = 6 with (a) 1024 × 1024 pixels and (b) 512 × 512 pixels. (c) and (d) Corresponding normalized axial intensity distributions, respectively.

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In fact, the GCAZP is applicable for the other aperiodic sequences, for example, the Cantor and Thue-Morse sequences. It should be noted that an element B is added after the last element of the Cantor sequence of order S, which represents the annular zone with low transmittance. The main reason is that the FraZP has the single main focus only, and the annular zone with high transmittance, which consist of the outermost annular zone with high transmittance of the inner part and the innermost annular zone with high transmittance of the outer part, splits the single focus into two equal-intensity foci. The Cantor sequence can be constructed by the initial seed A and the substitution rule {A→ABA, B→BBB}, where the elements A and B present the transparent and opaque rings of the constructed zone plate, respectively [14,40]. The zone plate constructed by the Cantor sequence is named as the fractal zone plate, which is firstly proposed by Saavedra et al. [10], not the Cantor zone plate. Since then, the fractal zone plate is used to refer to the zone plate constructed by the Cantor sequence. Based on the Cantor sequence, the fractal zone plate can be constructed by the transmission function equation [10,14]. It should be noted in Figs. 6(a)–(d) that the vertical coordinate presents the normalized axial intensity, which is normalized by the focal maximum intensity of the respective GCAZP. The dashed and solid lines in Fig. 6(a) show the axial intensity distributions of the GCAZP based on the Cantor sequence of S = 2 and the FraZP of S = 2, respectively. The dashed and solid lines in Fig. 6(b) show the axial intensity distributions of the GCAZP based on the Cantor sequence of S = 3 and the FraZP of S = 3, respectively. The dashed and solid lines in Fig. 6(c) show the axial intensity distributions of the GCAZP based on the Thue-Morse sequence of S = 4 and the TMZP of S = 4, respectively. The dashed and solid lines in Fig. 6(d) show the axial intensity distributions of the GCAZP based on the Thue-Morse sequence of S = 5 and the TMZP of S = 5, respectively. It can be seen in Fig. 6(a) that the intensity of the single main focus generated by the GCAZP based on the Cantor sequence of S = 2 is about four times stronger than that of the FraZP of S = 2, respectively. Moreover, it can be also seen in Fig. 6(b) that the intensity of the single main focus generated by the GCAZP based on the Cantor sequence of S = 3 is about four times stronger than that of the FraZP of S = 3, respectively. Interestingly, the main focal intensities of the GCAZPs based on the Cantor sequences of different S have the approximately same multiple of enhancement. Therefore, the single high-intensity main foci can be generated by the GCAZPs based on the Cantor sequences of the different orders. It can be seen in Fig. 6(c) that the intensities of twin foci generated by the GCAZP based on the Thue-Morse sequence of S = 4 are about 3.10 times stronger than those of the TMZP of S = 4. Moreover, it can be also seen in Fig. 6(d) that the intensities of twin foci generated by the GCAZP based on the Thue-Morse sequence of S = 5 are about 3.71 times stronger than those of the TMZP of S = 5. Thus, two high-intensity main foci can be generated by the GCAZPs based on the Thue-Morse sequences of the different orders. Therefore, the GCAZPs based on the different aperiodic sequences generate the high-intensity foci. It should be noted in Fig. 6(c) that the GCAZP based on the Thue-Morse sequence of S = 4 have two foci at u = 10.16 and 21.84, whose intensities are 234.7 and 234.7 a.u., respectively, and the TMZP of S = 4 have two foci at u = 10.56 and 21.44, whose intensities are 75.8 and 75.8 a.u., respectively. Moreover, It should be also noted in Fig. 6(d) that the GCAZP based on the Thue-Morse sequence of S = 5 have two foci at u = 21.92 and 42.08, whose intensities are 753.3 and 753.3 a.u., respectively, and the TMZP of S = 5 have two foci at u = 21.6 and 42.4, whose intensities are 203.1 and 203.1 a.u., respectively. Thus, two foci of the above zone plates based on the Thue-Morse sequences have equal intensity. In fact, the main and side lobes in Fig. 6 are corresponding to the main and subsidiary foci along the optic axis, respectively. In the following, the ratio between the intensities of main and side lobes will be calculated. For simplicity, the side lobe with the highest intensity is taken for an example only. Moreover, if there are two equal-intensity main or side lobes, only one main or side lobe is chosen to calculate the ratio. For the GCAZP based on the Cantor sequence of S = 2 and the FraZP of S = 2 in Fig. 6(a), the ratios between the intensities of main and side lobes are 8.02 and 3.90, respectively. For the GCAZP based on the Cantor sequence of S = 3 and the FraZP of S = 3 in Fig. 6(b), the ratios between the intensities of main and side lobes are 5.80 and 3.55, respectively. For the GCAZP based on the GCAZP based on the Thue-Morse sequence of S = 4 and the TMZP of S = 4 in Fig. 6(c), the ratios between the intensities of main and side lobes are 2.43 and 1.78, respectively. For the GCAZP based on the GCAZP based on the Thue-Morse sequence of S = 5 and the TMZP of S = 5 in Fig. 6(d), the ratios between the intensities of main and side lobes are 1.65 and 1.73, respectively. It can be found that with the increasement of S, the ratios for the zone plates based on the Cantor and Thue-Morse sequences decrease. The main reason is that the intensities of subsidiary foci may increase more quickly with the increasement of S for the above zone plates. Furthermore, for the GCAZP based on the Fibonacci sequence of S = 8 and the FiZP of S = 8 in Fig. 4, the ratios between the intensities of main and side lobes are 11.36 and 9.28, respectively. It can be found that the ratios for the zone plates based on the Fibonacci sequence are bigger than those for the above zone plates based on the Cantor and Thue-Morse sequences. Thus, the intensities of side lobes for the zone plates based on the Cantor and Thue-Morse sequences are higher than those for the zone plates based on the Fibonacci sequence. The main reason is that the zone plate based on the Cantor sequence has the single main lobe with many side lobes, and the zone plate based on the Thue-Morse sequence has two equal-intensity main lobes with many side lobes. However, the zone plates based on the Fibonacci sequence are considered to generate two equal-intensity main lobes only, whose generated side lobes have very low intensity. That is to say, compared with the Fibonacci sequence, the Cantor and Thue-Morse sequences make the constructed zone plates have the side lobes with higher intensity.

 

Fig. 6. Normalized axial intensity distributions of (a) the GCAZP based on the Cantor sequence of S = 2 and the FraZP of S = 2, (b) the GCAZP based on the Cantor sequence of S = 3 and the FraZP of S = 3, (c) the GCAZP based on the Thue-Morse sequence of S = 4 and the TMZP of S = 4, and (d) the GCAZP based on the Thue-Morse sequence of S = 5 and the TMZP of S = 5.

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The relation between the focal full width at half maximum (FWHM) and the outermost zone width of GCAZP will be studied in the following. The GCAZP based on the Fibonacci sequence of S = 6 have the outermost zone width of 75 µm and two equal-intensity foci located at u = 9.98 and 16.02, whose full width at half maximum are 109 and 64 µm, respectively. The GCAZP based on the Fibonacci sequence of S = 7 have the outermost zone width of 46 µm and two equal-intensity foci located at u = 16.1 and 25.9, whose FWHM are 64 and 41 µm, respectively. The GCAZP based on the Fibonacci sequence of S = 8 have the outermost zone width of 28 µm and two equal-intensity foci located at u = 26.01 and 41.99, whose FWHM are 41 and 26 µm, respectively. Thus, for the GCAZP based on the Fibonacci sequence, with the increasement of S, two focal FWHM decrease. The GCAZP based on the Cantor sequence of S = 2 have the outermost zone width of 97 µm and the single main focus located at u = 10, whose FWHM is 109 µm. The GCAZP based on the Cantor sequence of S = 3 have the outermost zone width of 34 µm and the single main focus located at u = 28, whose FWHM is 41 µm. Therefore, with the increasement of S of the Cantor sequence, the main focal FWHM decreases. The GCAZP based on the Thue-Morse sequence of S = 4 have the outermost zone width of 60 µm and two equal-intensity foci located at u = 10.16 and 21.84, whose FWHM are 100 and 49 µm, respectively. The GCAZP based on the Thue-Morse sequence of S = 5 have the outermost zone width of 30 µm and two equal-intensity foci located at u = 21.92 and 42.08, whose FWHM are 49 and 26 µm, respectively. Thus, for the GCAZP based on the Thue-Morse sequence, with the increasement of S, two focal FWHM decrease. Therefore, for the GCAZPs based on the different aperiodic sequences, with the increasement of S, the generated focal FWHM decrease.

We will compare the diffraction efficiencies between a GCAZP and a conventional composite Fresnel zone plate [25] in the following. The diffraction efficiency is defined as the ratio between the total energy of the focal spot and the total input energy of the diffraction element. As the main foci of the aperiodic zone plates have the highest intensity, the comparison between the diffraction efficiencies of the single or two equal-intensity main foci for the GCAZPs and the single foci for the conventional composite Fresnel zone plate are considered only. It should be noted in the simulations that the smallest zone widths of the inner and outer parts for the compared conventional composite Fresnel zone plates are equal to or greater than the specified smallest structure size and approximately the same, and the designed conventional composite Fresnel zone plates have the same first-order focal lengths and radius as the GCAZPs based on the different aperiodic sequences. The GCAZP based on the Fibonacci sequence of S = 8 has the first-order focus located at u = 34, and two equal-intensity foci located at u = 26.01 and 41.99, whose diffraction efficiencies are 8.93% and 3.22%, respectively. The corresponding conventional composite Fresnel zone plate has the first-order focus located at u = 34, whose diffraction efficiency is 3.74%. For the GCAZP based on the Fibonacci sequence, it can be found that the focus located at the smaller u, i.e., the bigger axial position, has higher diffraction efficiency. The main reason is that the focus with the bigger focal length has bigger focal spot. Compared with the focus of the conventional composite Fresnel zone plate, for two foci of the GCAZP based on the Fibonacci sequence, the focus with the bigger focal length has higher diffraction efficiency, and the focus with the smaller focal length has lower diffraction efficiency. The GCAZP based on the Cantor sequence of S = 3 has the first-order main focus located at u = 28, whose diffraction efficiency is 11.38%. The corresponding conventional composite Fresnel zone plate has the first-order focus located at u = 28, whose diffraction efficiency is 4.3%. Compared with the focus of the conventional composite Fresnel zone plate, the single main focus of the GCAZP based on the Cantor sequence has higher diffraction efficiency. The GCAZP based on the Thue-Morse sequence of S = 5 has the first-order main focus located at u = 32, and two equal-intensity foci located at u = 21.92 and 42.08, whose diffraction efficiencies are 8.56% and 2.08%, respectively. The corresponding conventional composite Fresnel zone plate has the first-order focus located at u = 32, whose diffraction efficiency is 3.73%. For the GCAZP based on the Thue-Morse sequence, it can be found that the focus located at the smaller u, i.e., the bigger axial position, has higher diffraction efficiency. Compared with the focus of the conventional composite Fresnel zone plate, for two foci of the GCAZP based on the Thue-Morse sequence, the focus with the bigger focal length has higher diffraction efficiency, and the focus with the smaller focal length has lower diffraction efficiency.

In the following, the modulation transfer function (MTF) of the GCAZP is calculated to study the imaging property of the GCAZP. The MTF can be obtained by the Fourier transform of the corresponding point spread function, which is calculated by the Fresnel diffraction integral formula [41]. For simplicity, the GCAZP based on the Fibonacci sequence of S = 8 and the FiZP of S = 8, the GCAZP based on the Cantor sequence of S = 3 and the FraZP of S = 3, and the GCAZP based on the Thue-Morse sequence of S = 5 and the TMZP of S = 5 are taken for example to illustrate it. It should be noted that the horizontal coordinate shows the spatial frequency, and the vertical coordinate presents the MTF along the central line, which is normalized by the respective maximum MTF. Figure 7(a) shows the MTFs of the GCAZP based on the Fibonacci sequence of S = 8 and the FiZP of S = 8 at two main focal planes. The thick light and black solid lines represent the MTFs at two focal planes located at z = 330.05 and 532.82 mm of the GCAZP based on the Fibonacci sequence of S = 8, respectively. The thin light and black lines present the MTFs at two focal planes located at z = 330.05 and 532.82 mm of the FiZP of S = 8, respectively. It can be found in Fig. 7(a) that the thick light and black lines are above the thin light and black lines, respectively. Thus, the images located at z = 330.05 and 532.82 mm generated by the GCAZP based on the Fibonacci sequence of S = 8 are clearer than those of the FiZP of S = 8, respectively. Figure 7(b) shows the MTFs of the GCAZP based on the Cantor sequence of S = 3 and the FraZP of S = 3 at the main focal plane. The thick solid line represents the MTF at the main focal plane located at z = 495 mm of the GCAZP based on the Cantor sequence of S = 3, respectively. The thin solid line represents the MTF at the main focal plane located at z = 495 mm of the FraZP of S = 3, respectively. It can be found in Fig. 7(b) that the thick solid line is above the thin solid line. Thus, the image located at z = 495 mm generated by the GCAZP based on the Cantor sequence of S = 3 is clearer than that located at z = 495 mm of the FraZP of S = 3. Figure 7(c) shows the MTFs of the GCAZP based on the Thue-Morse sequence of S = 5 and the TMZP of S = 5 at two main focal planes. The thick light and black lines represent the MTFs at two focal planes located at z = 329.34 and 632.24 mm of the GCAZP based on the Thue-Morse sequence of S = 5, respectively. The thin light and black lines present the MTFs at two focal planes located at z = 329.34 and 632.24 mm of the TMZP of S = 5, respectively. It can be found in Fig. 7(c) that the thick light and black lines are above the thin light and black lines, respectively. Thus, the images located at z = 329.34 and 632.24 mm generated by the GCAZP based on the Thue-Morse sequence of S = 5 are clearer than those located at z = 329.34 and 632.24 mm of the TMZP of S = 5, respectively. Thus, the images generated by the GCAZP based on the different aperiodic sequences are clearer than those of the corresponding common aperiodic zone plate, respectively.

 

Fig. 7. (a) MTFs of the GCAZP based on the Fibonacci sequence of S = 8 and the FiZP of S = 8 at two main focal planes. (b) MTFs of the GCAZP based on the Cantor sequence of S = 3 and the FraZP of S = 3 at the main focal plane. (c) MTFs of the GCAZP based on the Thue-Morse sequence of S = 5 and the TMZP of S = 5 at two main focal planes.

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4. Experiments

In the experiments, the intensity profiles of the GCAZP based on the Fibonacci sequence of S = 8 and the FiZP of S = 8, the GCAZP based on the Cantor sequence of S = 3 and the FraZP of S = 3, and the GCAZP based on the Thue-Morse sequence of S = 5 and the TMZP of S = 5 were captured. It should be noted that these GCAZPs and common aperiodic zone plates have the same R of 3.84 mm and $3.84/\sqrt 2$ mm, respectively. An imaging experimental system is shown in Fig. 8. A laser beam (λ = 532 nm) illuminated a beam expender, a lens and a target object (a letter U) at the focal plane of the lens, and then passed a collimator (a focal length of 550 mm and a diameter of 55 mm), a polarizer, a spatial light modulator (SLM) (CAS MICROSTAR, TSLM07U-A, 1920 × 1080 pixels, 8.5 µm × 8.5 µm /pixel, and transmissive type) with the holograms of the above zone plates, and an analyzer, and was finally captured by a Complementary Metal Oxide Semiconductor (CMOS, Mindvision, MV-UBS300C). It should be noted that a collimator projects a target object at infinity to produce images at the focal planes. Moreover, the added polarizer and analyzer make the SLM work in the best condition.

 

Fig. 8. Experimental setup of the imaging system.

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After the target object is removed from a light path, the focal intensity distribution can be captured. Figures 9(a) and (b) represent the measured intensity distributions of two foci at z≈330 and 530 mm generated by the GCAZP based on the Fibonacci sequence of S = 8, respectively. Figures 9(f) and (g) represent the measured intensity distributions of two foci at z≈330 and 530 mm generated by the FiZP of S = 8, respectively. Figures 9(c) and (h) represent the measured intensity distributions of the foci at the same axial position z≈495 mm generated by the GCAZP based on the Cantor sequence of S = 3 and the FraZP of S = 3, respectively. Figures 9(d) and (e) represent the measured intensity distributions of the foci at z≈330 and 630 mm generated by the GCAZP based on the Thue-Morse sequence of S = 5, respectively. Figures 9(i) and (j) represent the measured intensity distributions of the foci at z≈330 and 630 mm generated by the TMZP of S = 5, respectively. It should be noted that all chosen intensity distributions in Fig. 9 have the same area of 685 µm×685 µm, and a colour bar on the right of Fig. 9 represents the light intensity, which increases from the bottom up. It can be seen in Figs. 9(a), (b), (f) and (g) that the GCAZP based on the Fibonacci sequence produce two main foci with higher intensity than the corresponding FiZP. It can be seen in Figs. 9(c) and (h) that the GCAZP based on the Cantor sequence produce the single main focus with higher intensity than the corresponding FraZP. It can be seen in Figs. 9(d), (e), (i) and (j) that the GCAZP based on the Thue-Morse sequence produce two main foci with higher intensity than the corresponding TMZP. Thus, the GCAZPs based on the different aperiodic sequences can generate the main foci with higher intensity than the corresponding common aperiodic zone plates. It should be noted that as the diffraction of the SLM, the intensity distribution in the shorter diffraction distance has higher intensities than the same intensity distribution in the longer diffraction distance. Thus, it can be found in Fig. 9 that for the bifocal zone plates, the measured focus with the short focal length has higher intensity than that with the long focal length.

 

Fig. 9. Intensity distributions at z≈330 and 530 mm of (a) and (b) the GCAZP based on the Fibonacci sequence of S = 8. Intensity distributions at z≈330 and 530 mm of (f) and (g) the FiZP of S = 8. Intensity distributions at z≈495 mm of (c) the GCAZP based on the Cantor sequence of S = 3 and (h) the FraZP of S = 3. Intensity distributions at z≈330 and 630 mm of (d) and (e) the GCAZP based on the Thue-Morse sequence of S = 5. Intensity distributions at z≈330 and 630 mm of (i) and (j) the TMZP of S = 5.

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After the target object is added to the light path, the generated images of these zone plates can be captured. In Fig. 10, the captured images have the same area about 1160 µm×1160 µm, and a colour bar represents the light intensity, which increases from the bottom up. Figures 10(a) and (b) represent two images at z≈330 and 530 mm generated by the GCAZP based on the Fibonacci sequence of S = 8, respectively. Figures 10(f) and (g) represent two captured images at z≈330 and 530 mm generated by the FiZP of S = 8, respectively. It can be seen that the letters in Figs. 10(a) and (b) have higher contrast than those in Figs. 10(f) and (g), respectively. The main reason is that the GCAZP based on the Fibonacci sequence has twin foci with higher intensity than the corresponding FiZP. Thus, intensity differences between the letters and background in Figs. 10(a) and (b) are bigger than those in Figs. 10(f) and (g), respectively. Therefore, the letters in Figs. 10(a) and (b) are clearer. Figures 10(c) and (h) represent the measured images at the same axial position z≈495 mm generated by the GCAZP based on the Cantor sequence of S = 3 and the FraZP of S = 3, respectively. It can be seen that the image in Fig. 10(c) have higher contrast than that in Fig. 10(h). The main reason is that the GCAZP based on the Cantor sequence has the single main focus with higher intensity than the corresponding FraZP. Thus, it can be seen that intensity difference between the letter and background in Fig. 10(c) is bigger than that in Fig. 10(h), so that the letter in Fig. 10(c) is clearer than that in Fig. 10(h). Figures 10(d) and (e) represent two images at z≈330 and 630 mm generated by the GCAZP based on the Thue-Morse sequence of S = 5, respectively. Figures 9(i) and (j) represent two images at z≈330 and 630 mm generated by the TMZP of S = 5, respectively. It can be seen that the images in Figs. 10(d) and (e) are clearer than those in Figs. 10(i) and (j), respectively. The main reason is that the GCAZP based on the Thue-Morse sequence has twin foci with higher intensity than the corresponding TMZP. Thus, intensity differences between the letters and background in Figs. 10(d) and (e) are bigger than those in Figs. 10(i) and (j), respectively. Therefore, the letters in Figs. 10(d) and (e) are clearer. Thus, the GCAZPs have clearer images than the corresponding common aperiodic zone plates.

 

Fig. 10. Captured images at z≈330 and 530 mm of (a) and (b) the GCAZP based on the Fibonacci sequence of S = 8. Captured images at z≈330 and 530 mm of (f) and (g) the FiZP of S = 8. Captured images at z≈495 mm of (c) the GCAZP based on the Cantor sequence of S = 3 and (h) the FraZP of S = 3. Captured images at z≈330 and 630 mm of (d) and (e) the GCAZP based on the Thue-Morse sequence of S = 5. Captured images at z≈330 and 630 mm of (i) and (j) the TMZP of S = 5.

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5. Conclusions

We proposed the GCAZPs to generate the clearer images in the focal planes. We illustrated the construction method of the GCAZP. We also proved that the proposed radial zone plates and the common aperiodic zone plates have the coincident first-order diffraction area and the same axial first-order diffraction intensity distribution so that the constructed GCAZPs have the main foci with higher intensity. It was proved numerically that the GCAZPs of the different orders had the high-intensity foci, and the GCAZP can be designed on the device with the arbitrary size. We also studied the focusing properties of the GCAZP in the simulations and experiments. In addition, it was found that modulation transfer functions at the focal planes of the GCAZPs have higher values than those of the corresponding common aperiodic zone plates. The results showed that the GCAZP can generate the fractal foci with higher intensity and images with higher contrast than the corresponding common aperiodic zone plates. The proposed zone plate had the potential applications in the fields of the optical imaging, X-ray microscopy and optical trapping.