## Abstract

We present a new family of diffractive lenses, composite Thue-Morse zone plates (CTMZPs), formed by multiple orders of Thue-Morse zone plates (TMZPs). The typical structure of a CTMZP is a composite of two concentric TMZPs. The focusing properties of the CTMZPs with different parameters have been investigated both theoretically and experimentally. Compared with the TMZPs, the CTMZPs have higher performance in axial intensity and imaging resolution. The CTMZP beams are also found to possess the self-reconstruction property, and would be useful for three-dimensional optical tweezers, laser machining, and optical imaging.

© 2016 Optical Society of America

## 1. Introduction

Diffractive optical elements (DOEs) have been found various new applications in photonics technology. DOEs mainly are phase-only and can be more compact than the geometrical optical elements such as refractive lenses and prisms. DOEs can be applied conveniently and fulfill complex tasks the conventional geometric optical elements are incapable of. For example, diffractive lenses, like zone plates (ZPs) [1], can be used for astronomy [2], x-ray microscopy [3,4], tomographic imaging [5], lithography [6], and optical trapping [7,8].

Conventional Fresnel zone plates are the most common ZPs but with inherent limitations. In recent years, various mathematic aperiodic sequences have been applied to generate new ZPs, such as Fractal Zone Plates (FZPs) [1], Fibonacci Zone Plates [9], and Thue-Morse Zone Plates (TMZPs) [10], which all have special focusing properties and potential applications. FZPs can improve the imaging performance with an extended depth of field and reduce chromatic aberration under white-light illumination [11,12]. Fibonacci Zone Plates and TMZPs both have twin foci along the optic axis. However, foci of the Fibonacci Zone Plates are easily affected under broadband illumination because the twin foci are not self-similar [9]. On the contrary, TMZPs own twin self-similar foci positioned symmetrically axially [10]. Besides, TMZPs combine the advantages of FZPs and Fibonacci Zone Plates, so the chromatic aberration is reduced. However, the relatively low intensity and resolution of axial foci of the TMZPs limits the use in optical fields.

Various ways were proposed to increase the intensity and resolution of axial foci of the ZPs [13]. Photon sieves were developed with high resolution to create sharp images [3,14]. Unlike Fresnel zone plates, the zones of the sieves are broken into a great number of isolated circular holes of different sizes. The structure of pinholes with optimized diameters can improve the transverse resolution. Other modified Fresnel zone plates were found to generate even sharper focal spots than those of the photon sieves [15].

Another modified Fresnel zone plates, namely composite zone plates (CZPs) [16], were constructed with a central zone plate of the first-order focal length surrounded by outer zones of the third-order focal length. CZPs increase the intensity peaks of the axial foci and have an improved resolution. Besides, CZPs can increase the spatial resolution, but the narrowest zone is limited by the fabrication capability [17,18].

In this work a new class of ZPs will be introduced, which is a kind of composite zone plates combined with two TMZPs of different or the same orders. We name this kind of ZPs as Composite Thue-Morse Zone Plates (CTMZPs). The focusing properties and the self-similar property of the CTMZPs will be investigated. We will compare the intensity and resolution of axial foci of a CTMZP with those of a TMZP of the same focal length, as both the ZPs have the similar focusing characteristics. Furthermore, the CTMZP beams will be found to possess the self-reconstruction property, which can find the potential use in three-dimensional optical tweezers.

## 2. Design

A TMZP consists of alternate transparent (*A*) and opaque (*B*) zones. We will construct the Thue–Morse sequence (TM) with elements *A* and *B*, and an initiator seed *D*_{0} = *A* is defined in the first step. During the subsequent steps, the generator of the set is constructed by replacing *A* with *AB* and *B* with *BA*. In this way, the first order is *D*_{1} = *AB* and the following orders are *D*_{2} = *ABBA*, *D*_{3} = *ABBABAAB*, and so on. Figure 1(a) illustrates the one-dimensional (1D) construction of TM sequences from order *S* = 1 to *S* = 4 [10].

A TMZP can be generated with a transmission function *q*(*ζ*) written as,

*ζ*= (

*r*/a)

^{2},

*ζ*∈[0, 1] is the normalized square radial coordinate,

*r*is the variable of radius,

*a*is the radius of the maximum extent of pupil, and

*S*is the order. There are 2

*sub-intervals in the interval*

^{S}*ζ*∈[0, 1] for a TM sequence of order

*S*, and the length of each of the sub-interval is

*d*= 1/2

_{S}*. The*

^{S}*j*-th sub-interval of the TM sequence of order

*S*is represented with

*D*

_{S}_{,}

*. The transmittance value*

_{j}*t*

_{S}_{,}

*taking at the*

_{j}*j*-th sub-interval is determined by the value of the element

*D*

_{S}_{,}

*. And the transmittance value*

_{j}*t*

_{S}_{,}

*in Eq. (1) is*

_{j}*t*

_{S}_{,}

*= 1 when*

_{j}*D*

_{S}_{,}

*is “*

_{j}*A*”, and

*t*

_{S}_{,}

*= 0 when*

_{j}*D*

_{S}_{,}

*is “*

_{j}*B*”.

When the inner circle with radius *r* = *a*/2 is taken out of a TMZP of order *S*, we find that the inner circle is a TMZP of order (*S-*2). Hence, we can generate the new family of zone plates with multi-orders of *M* = (*S _{in}* ,

*S*), i.e., composite Thue-Morse Zone plates (CTMZPs). The generation steps of a CTMZP are in the following. Firstly, the inner circle of

_{out}*r*=

*a/*2 is taken out of a TMZP with

*r*=

*a*. The remained TMZP with a missing inner circle has an order of

*S*. Then we fill the inner circle with another TMZP with

_{out}*r*=

*a/*2 and order of

*S*. Finally, a CTMZP is formed. Figure 1(b) shows a CTMZP of order

_{in}*M*= (4, 4).

As two TMZPs of different orders are combined as one, the intensity distributions in three-dimensional space will be redistributed because of coaxial interference of the two TMZP beams. Compared with a TMZP, the CTMZP with the same focal length has an axial irradiance increasing more rapidly with *S _{out}* and a higher resolution, when

*S*is unchanged. The outer annular zones of the CTMZP become narrower with an increase of

_{in}*S*. The same goes with

_{out}*S*when

_{in}*S*is unchanged. As realization of the narrowest zone of a zone plate is dependent on the fabrication capability, we will consider the CTMZPs of

_{out}*M*= (

*S*,

_{in}*S*), where

_{out}*S*≤

_{out}*S*+ 1. In this case, the focal length of a CTMZP can be calculated by

_{in}*f*

_{CTMZP}=

*a*

^{2}/(

*λ*2

*).*

^{Sin}## 3. Axial irradiance of CTMZP

As a CTMZP comprises transparent and opaque zones, the transmittance of the CTMZP can be expressed by a matrix of *T _{a}* comprising 0 and 1. Since a phase-only zone plate has much higher diffraction efficiency than the amplitude-only counterpart, we will use phase-only CTMZPs to study the axial irradiances. The matrix

*T*can be replaced with a matrix of phase steps, i.e., 0 and 1 in

_{a}*T*are replaced with phase steps of 0 and

_{a}*π*, respectively. Thus, a binary phase-only CTMZP can be expressed by a matrix of

*T*, which comprises phase steps 0 and

_{p}*π*. Then we use the angular plane-wave spectrum theory to simulate free-space propagation of the electric field expressed in Eq. (2) [8],

*E*and

_{a}*E*are the complex amplitudes of the diffracted patterns,

_{p}*FT*and

*iFT*represent the Fourier transform and the inverse Fourier transform, respectively, and

*H*is the transfer function expressed in Eq. (3),

*z*is the propagation distance,

*λ*is the wavelength of light,

*d*is the size of the zone plate, and

*x*and

*y*are the dimensionless coordinates of the sampling grid. We will use Eq. (2) to evaluate the focusing properties of the CTMZPs. In the simulation, the wavelength of light

*λ*is chosen as 0.532 μm. The ZPs are sampled with a grid of 512 × 512 pixels with a pixel area of 15 × 15 μm

^{2}.

Figure 2(a) shows the normalized axial irradiance of the TMZP of order *S* = 4 which belongs to the inner circle of a CTMZP of order *M* = (4, 4), and Fig. 2(b) shows the normalized axial irradiance of the TMZP of order *S* = 4 which belongs to the external circle of the CTMZP of order *M* = (4, 4). Figure 2(c) shows the normalized axial irradiance of the CTMZP with *r* = 256 × 15 μm and order *M* = (4, 4). The foci of the CTMZP and TMZP are located at a reduced axial coordinate *u* = *a*^{2}/(2*λz*), where *z* is the axial distance from the zone plate. The focal lengths of a CTMZP and a TMZP can be calculated by *f*_{CTMZP} = *a*^{2}/(*λ*2* ^{Sin}*) and

*f*

_{TMZP}=

*a*

^{2}/(

*λ*2

*), respectively. Note that the axial irradiances in Fig. 2 are normalized to the intensity peak of the TMZP of order*

^{S}*S*= 4 with

*r*= 128 × 15 μm in Fig. 2(a). The irradiances around

*u*= 2, 6, 10, and 14 correspond to the first, third, fifth, and seventh diffraction orders of the lens, respectively. However, the intensity peaks of the foci in Fig. 2(b) are reduced by about 50%, compared with those in Fig. 2(a). Note that the inner circle of the CTMZP can be regarded as a CTMZP of

*M*= (4, 0), where “0” means that the outer circle of the CTMZP is removed. In the same way, the outer circle of the CTMZP of

*M*= (4, 4) can be regarded as a CTMZP of

*M*= (0, 4). Obviously, the irradiance of

*M*= (4, 4) is the interference of the fields diffracted by the inner CTMZP of

*M*= (4, 0), i.e., a TMZP of

*S*= 4 in Fig. 2(a), and the outer CTMZP of

_{in}*M*= (0, 4), i.e., a TMZP of

*S*= 4 in Fig. 2(b). It can be observed that the intensity peaks of the foci of the CTMZP in Fig. 2(c) are more than two times higher than those of the TMZP in Fig. 2(a), and the CTMZP of

_{out}*M*= (4, 4) generates tighter foci, which will improve the axial resolution. The spot size of the main focus at around

*z*= 320.5 mm for the CTMZP with

*M*= (4, 4) is about two times smaller than that of the TMZP with

*M*= (4, 0), but the total energy in the focal spot of the CTMZP is slightly lower than that of the TMZP.

To further illustrate the focusing properties of the CTMZPs, we present the corresponding axial irradiances, which are shown in Fig. 3(a), of four CTMZPs which are shown in Fig. 3(b) and have the same radius *r* = 256 × 15 μm and the same *S _{in}* = 4 but different

*S*= 2, 3, 4, and 5, respectively. When

_{out}*S*becomes larger, the foci are tighter and the intensities increase rapidly. The intensity peaks of the twin foci of the CTMZP of

_{out}*M*= (4, 5) are more than four times higher than those of the TMZP in Fig. 2(a). Obviously, the peak intensities of the twin foci are not exactly the same, and the left one is a little higher than the right one. The reason is that, in the light propagation, the twin foci of the central TMZP overlaps with the focus of the third-order focal length and the focus of the fifth-order focal length of the outer TMZP. Hence, the composite of the two TMZPs has higher intensity peaks of the axial foci, tighter main foci, and improved axial resolution. Note that the axial irradiances in Fig. 3(a) are normalized to the intensity of axial foci of the TMZP with

*r*= 128 × 15 μm and order

*S*= 4 in Fig. 2(a).

The CTMZPs also preserve the self-similarity property. In fact, the four patterns in Fig. 3(a) are self-similar. To support this point, we show the axial irradiances around the respective main foci produced by the CTMZPs of orders *M* = (4, 4) and *M* = (3, 3) in Fig. 3(c). The axial irradiances are normalized to their respective peak intensities of the foci. It can be seen that the lower curves envelope the upper ones. This property is generally applicable for the CTMZPs of *M* = (*S _{in}* ,

*S*) when

_{out}*S*≤

_{out}*S*+ 1. Here, the axial distance of the CTMZP is located at a normalized distance given by

_{in}*u*/

*u*

_{0}, where

*u*

_{0}= 2

*.*

^{Sin}The superposition of interfering Bessel beams has been proven to be self-reconstructive [19]. Each of the narrow rings of a CTMZP will generate a Bessel beam with a radial component of the wave vector during the free-space propagation. Therefore, the CTMZP beam would also possess the self-reconstruction property, which will be verified with simulations in the following. The parameters of the CTMZP in the simulations are *M* = (4, 4) and *a* = 3.84 mm. In the light propagation, an opaque object with a radius of 38.4 μm is placed in the center of the beam at the propagation distance of 262 mm, where a major focus of the CTMZP beam is located. The opaque object blocks light passing through the center of the major focus. For better viewing, in Fig. 4 we present the enlarged central parts of the intensity profiles only. Figures 4(a) and 4(e) show the intensity profiles of the obstructed and unobstructed beams at propagating distances of *z* = 271.1 mm, 4(b) and 4(f) at *z* = 310 mm, 4(c) and 4(g) at z = 321.5 mm, and 4(d) and 4(h) at z = 620 mm, respectively. From Fig. 4 we can see that, although the central part of the major focus has been completely blocked by the opaque object, the obstructed beam still recovers after a propagation distance of several centimeters, which is rather short compared with the focal length. The beams shown in Figs. 4(c) and 4(g), and 4(d) and 4(h) are almost identical, illustrating that the obstructed beam of the CTMZP is self-reconstructive.

To further investigate the light propagation, we sampled cross-sections of the propagating beams with a step of 2.16 mm along the axial direction. Figures 5(a) and 5(b) represent the cross-sections of the intensities of the obstructed and unobstructed beams in the propagation, respectively. For a better view, we only display part of the beams where the opaque object is placed. The horizontal axis in Fig. 5 represents a propagation distance ranging from 217 mm to 346 mm, and the vertical axis is the truncated size of the beam, which is 0.21 mm in diameter. A total of 60 intensity cross-sections are sampled, and the obstacle is placed in the center of the 21st sampled intensity cross-section at *z* = 262 mm and is highlighted with a dashed box in Fig. 5(a). We can observe that both the obstructed and unobstructed beams are almost identical after a short propagation distance. Therefore, the CTMZP beams possess the self-reconstruction property. We have also found that the TMZP beams possess the self-reconstruction property. With the property, the CTMZP beams have the potential in optical trapping multiple particles with the twin foci.

## 4. Experiment

Experiments were implemented to verify the increased axial irradiances and tighter main foci of the CTMZPs. We formed the CTMZP of order *M* = (4,5) and TMZP of order *S* = 4 beams separately with a spatial light modulator (SLM) (Boulder Nonlinear System, model P512-532 nm, 512 × 512 pixels, 15 × 15 μm^{2}/pixel, multilevel phases, reflective type). Separate experiments on the CTMZP and TMZP were implemented. Firstly, an expanded and collimated diode-pumped laser beam (Coherent, Genesis MX532-1000 STM, λ = 532 nm) was modulated and reflected by the SLM, where the pattern of the phase structure of a ZP had been imprinted through a personal computer interface. Secondly, a charge-coupled device (CCD, Newport, LBP-2-USB) was used to measure the intensities of the beam at various distances along the optical axis from the SLM screen. Note that blazed grating phase distributions were added to the transmittance functions of the CTMZP and TMZP, respectively, to deflect the reconstructed beams from the direct component of the SLM. Schematic of the experimental setup can be referred to [20].

Figures 6(a) and 6(b), and 6(e) and 6(f) show the corresponding simulated intensity profiles of the TMZP and CTMZP at distances of 320 mm (*f _{1}* ) and 660 mm (

*f*), respectively.

_{2}*f*and

_{1}*f*represent the distances of the two main foci from the zone plates, respectively. For convenient observation, we display central parts of the beams only. Figures 6(c), 6(d), 6(g), and 6(h) show the corresponding CCD captured intensity profiles of the TMZP and CTMZP around distances of

_{2}*z*= 300 mm, 660 mm, 320 mm, and 670 mm, respectively. Figure 6(i) and 6(j) show the normalized axial irradiances around the main foci and

*z*ranges from

*z*= 217 mm to

*z*= 1516 mm, where the focal length

*f*= 433.1 mm. Note that the axial irradiances are normalized to the intensity of axial foci of the TMZP with

*r*= 128 × 15 μm and order

*S*= 4 in Fig. 2(a).

From Figs. 6 (c), 6(d), 6(g), and 6(h) we can see that the intensities of the foci of the CTMZP are higher than those of the TMZP. The experimental results are in good agreement with the simulation ones.

## 5. Conclusion

A new type of composite zone plates based on the Thue-Morse sequence with high intensity of axial foci and resolution was proposed. The focusing properties of various combinations of the CTMZPs were evaluated with simulations and experiments. Besides, the self-similarity of the axial irradiances of the CTMZPs had been verified. It was shown that the CTMZP presents a significant increase in the intensities of the axial foci and produces two tighter self-similar main foci, compared with a TMZP. The CTMZP beams were also found to possess the self-reconstruction property, which would be useful for optical trapping, especially for constructing three-dimensional optical tweezers. The CTMZPs can be regarded as a composite of two concentric TMZPs. Hence, advantages of the TMZPs and composite diffractive lenses are inherited. The CTMZPs would have the potential applications in such as haze concentration measurement [21], optical trapping [22,23], X-ray microscopy [24,25], ophthalmology [26],spectrum domain OCT [27], and THz imaging [28].

## Acknowledgments

The research was financially supported by the National Natural Science Foundation of China (NSFC) (Grant No. 61178017).

## References and links

**1. **G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. **28**(12), 971–973 (2003). [CrossRef] [PubMed]

**2. **R. A. Hyde, “Eyeglass. 1. Very large aperture diffractive telescopes,” Appl. Opt. **38**(19), 4198–4212 (1999). [CrossRef] [PubMed]

**3. **L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature **414**(6860), 184–188 (2001). [CrossRef] [PubMed]

**4. **E. Di Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, “High-efficiency multilevel zone plates for keV X-rays,” Nature **401**(6756), 895–898 (1999). [CrossRef]

**5. **S. H. Wang, X. C. Zhang, M. P. Maley, M. F. Hundley, L. N. Bulaevskii, A. E. Koshelev, and A. J. Taylor, “Terahertz tomographic imaging with a Fresnel lens,” Opt. Photon. News **13**(12), 58 (2002). [CrossRef]

**6. **R. Menon, D. Gil, G. Barbastathis, and H. I. Smith, “Photon-sieve lithography,” J. Opt. Soc. Am. A **22**(2), 342–345 (2005). [CrossRef] [PubMed]

**7. **S. B. Cheng, S. H. Tao, C. H. Zhou, and L. Wu, “Optical trapping of a dielectric-covered metallic microsphere,” J. Opt. **17**(10), 105613 (2015). [CrossRef]

**8. **S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plate,” Appl. Phys. Lett. **89**(3), 031105 (2006). [CrossRef]

**9. **J. A. Monsoriu Serra, A. Calatayud Calatayud, L. Remón Martín, W. D. Furlan, G. Saavedra, and P. Andrés Bou, “Bifocal Fibonacci diffractive lenses,” IEEE Photonics J. **5**(3), 34001061 (2013).

**10. **V. Ferrando, F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Bifractal focusing and imaging properties of Thue-Morse Zone Plates,” Opt. Express **23**(15), 19846–19853 (2015). [CrossRef] [PubMed]

**11. **J. A. Monsoriu, W. D. Furlan, G. Saavedra, and F. Giménez, “Devil’s lenses,” Opt. Express **15**(21), 13858–13864 (2007). [CrossRef] [PubMed]

**12. **W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “White-light imaging with fractal zone plates,” Opt. Lett. **32**(15), 2109–2111 (2007). [CrossRef] [PubMed]

**13. **F. Giménez, W. D. Furlan, A. Calatayud, and J. A. Monsoriu, “Multifractal zone plates,” J. Opt. Soc. Am. A **27**(8), 1851–1855 (2010). [CrossRef] [PubMed]

**14. **G. Andersen, “Large optical photon sieve,” Opt. Lett. **30**(22), 2976–2978 (2005). [CrossRef] [PubMed]

**15. **Q. Cao and J. Jahns, “Modified Fresnel zone plates that produce sharp Gaussian focal spots,” J. Opt. Soc. Am. A **20**(8), 1576–1581 (2003). [CrossRef] [PubMed]

**16. **M. J. Simpson and A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta (Lond.) **31**(4), 403–413 (1984). [CrossRef]

**17. **E. H. Anderson, V. Boegli, and L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B **13**(6), 2529–2534 (1995). [CrossRef]

**18. **E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B **18**(6), 2970–2975 (2000). [CrossRef]

**19. **D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. **28**(8), 657–659 (2003). [CrossRef] [PubMed]

**20. **S. Tao and W. Yu, “Beam shaping of complex amplitude with separate constraints on the output beam,” Opt. Express **23**(2), 1052–1062 (2015). [CrossRef] [PubMed]

**21. **S. H. Tao, B. C. Yang, H. Xia, and W. X. Yu, “Tailorable three-dimensional distribution of laser foci based on customized fractal zone plates,” Laser Phys. Lett. **10**(3), 035003 (2013). [CrossRef]

**22. **S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A **21**(7), 1192–1197 (2004). [CrossRef] [PubMed]

**23. **J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. **197**(4–6), 239–245 (2001). [CrossRef]

**24. **X. Ge, Z. L. Wang, K. Gao, D. J. Wang, Z. Wu, J. Chen, K. Zhang, Y. L. Hong, P. P. Zhu, and Z. Y. Wu, “Effects of the condenser fractal zone plate in a transmission X-ray microscope,” Radiat. Phys. Chem. **95**, 424–427 (2014). [CrossRef]

**25. **X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission X-ray microscopy,” Anal. Bioanal. Chem. **404**(5), 1303–1309 (2012). [CrossRef] [PubMed]

**26. **J. A. Davison and M. J. Simpson, “History and development of the apodized diffractive intraocular lens,” J. Cataract Refract. Surg. **32**(5), 849–858 (2006). [CrossRef] [PubMed]

**27. **Q. Q. Zhang, J. G. Wang, M. W. Wang, J. Bu, S. W. Zhu, R. Wang, B. Z. Gao, and X. C. Yuan, “A modified fractal zone plate with extended depth of focus in spectral domain optical coherence tomography,” J. Opt. **13**(5), 055301 (2011). [CrossRef] [PubMed]

**28. **A. Siemion, A. Siemion, M. Makowski, J. Suszek, J. Bomba, A. Czerwiński, F. Garet, J.-L. Coutaz, and M. Sypek, “Diffractive paper lens for terahertz optics,” Opt. Lett. **37**(20), 4320–4322 (2012). [CrossRef] [PubMed]