Optical Trapping by Aperiodic Kinoform Lenses Based on the Baum–Sweet Sequence

Abstract

This work presents a new family of aperiodic diffractive lenses based on the Baum–Sweet sequence. To the best of our knowledge, this is the first report of a diffractive lens derived from this sequence. The study of their focusing properties reveals two focal points with similar intensities along the optical axis. Both the main focal distances and the axial irradiance distribution are correlated with the aperiodic Baum–Sweet sequence. An approximate 60% increase in diffraction efficiency is observed when employing kinoform profiles instead of binary phase lenses. The integration of the Baum–Sweet-based kinoform lens into an optical tweezers system demonstrates its ability to simultaneously trap multiple particles at two distinct focal planes, highlighting its potential for applications in more advanced optical devices.

1. Introduction

In recent decades, diffractive optical elements (DOEs) have emerged as key components in numerous scientific and technological fields because of their ability to manipulate the properties of light, such as phase and intensity. Their versatility, ease of integration, and lightweight nature have promoted their adoption in a wide range of applications, including advanced microscopy, biomedical optics, nanotechnology, materials science, and the design of compact and customized optical systems [1,2,3,4,5,6,7].

Among DOEs, diffractive lenses have proven particularly useful for their ability to generate complex irradiance distributions with multiple focal planes along the optical axis. These lenses are designed from generating sequences—either periodic or aperiodic—that determine the arrangement of concentric zones that are opaque or have varying phase indices. Aperiodic lenses, in particular, have attracted growing interest due to their ability to produce unconventional focusing patterns [8,9,10]. The first achromatic aperiodic Zone Plate was proposed based on the combination of two Zone Plates for different wavelengths in one [11]. In this context, various approaches have been proposed for the design of multifocal lenses based on aperiodic sequences, implemented in binary phase configurations as well as in Kinoform-type lenses, vortex-generating lenses, and binary amplitude lenses [12,13,14]. Examples of such sequences include Fibonacci [15,16,17,18], Thue-Morse [19], m-Bonacci [20], Silver Mean [21], and Walsh functions [22], each contributing unique optical properties linked to their mathematical structure. For instance, a lens based on the Fibonacci sequence produces two focal points whose axial positions approximate the golden ratio. The m-Bonacci lens, an extension of the Fibonacci structure, allows the focal separation to be tuned by adjusting the parameter m. As m increases, the distance between focal planes decreases, providing greater flexibility for multifocal lens designs. The Silver Mean sequence can generate up to four well-defined foci, making it suitable for optical trapping configurations. Diffractive elements based on radial Walsh functions exhibit self-similar structures in both axial and transverse diffraction patterns, allowing for the design of complex and controllable intensity distributions. Finally, Zone Plates designed with the Thue–Morse sequence generate a pair of self-similar, high-intensity foci; under broadband illumination, they provide extended depth of focus and significantly reduced chromatic aberration compared to conventional periodic Zone Plates.

Binary phase lenses, characterized by discrete phase transitions (0 or $\pi$), generate multiple diffraction orders, resulting in several focal spots with intensity distributions that depend on the employed sequence [23]. On the other hand, Kinoform-type lenses, which use a continuous phase modulation between 0 and $2\pi$, allow for the concentration of optical energy into a single diffraction order, producing focal points with higher intensity [24,25]. This feature makes them particularly suitable for applications requiring high energy efficiency, such as optical tweezers.

In the context of optical manipulation, optical tweezers have revolutionized the study of microscopic systems by enabling the trapping and movement of particles via intensity gradients generated by focused beams [26,27]. The integration of DOEs into these systems has significantly expanded their capabilities, enabling more complex configurations such as three-dimensional trapping, simultaneous manipulation of multiple particles, and the design of customized motion trajectories [28]. In particular, multifocal diffractive lenses have proven to be effective tools for enhancing the performance and versatility of optical tweezers, enabling the trapping of particles at different focal planes with precise control [21,29].

This work presents the design of a new family of aperiodic diffractive lenses based on the Baum–Sweet sequence, in both binary phase and Kinoform configurations. The analysis of their focusing properties reveals the generation of two focal planes along the optical axis, whose positions and irradiance distributions are intrinsically linked to the Baum–Sweet sequence. The Kinoform lens outperforms the binary phase version by concentrating energy more efficiently into a single diffraction order, producing focal points of higher intensity. Additionally, the implementation of the Kinoform Baum-Sweet Lens (KBSL) in an experimental optical tweezers setup demonstrates its capability to simultaneously trap particles at two distinct focal planes, highlighting its potential for applications in particle trapping and manipulation.

2. Materials and Methods

Baum–Sweet Lens Design

The Baum–Sweet lens is based on the binary sequence of the same name. The k-th element of this sequence, $a_k$, is associated with the number of consecutive zeros in the binary representation of k. If, in the representation of k, all zeros are consecutive and the total number of zeros is even, then $a_k=1$. If the binary representation contains no zeros, $a_k$ is also 1. For other cases, $a_k=0$. This sequence is recursive, so the value of $a_k$ for the k-th term can be determined using the recursion rule $k=m{4}^t$ where m and t are non-negative integers, with the particularity that m is not divisible by 4. Thus, $a_k=1$ if $k=0$; $a_k=0$ if m is even and $a_k=a_{(m-1)/2}$ if m is odd [30,31]. Therefore, the first terms of the sequence are $1,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1,\dots$ Another way to generate this sequence is by applying a particular set of substitution rules. Starting from a seed element $S_1=\{1,1\}$, which corresponds to order $n=1$, the next orders of the Baum–Sweet sequence can be obtained by replacing each pair of consecutive elements with a chain of four elements [32]. In the sequence, specific substitution rules are applied: the pair “11” is replaced by “1101”, “01” becomes “1001”, “10” is changed to “0100”, and “00” is transformed into “0000”. Following this pattern, the sequences of the first five orders are represented as follows:

$S_1=\{1,1\}$

$S_2=\{1,1,0,1\}$

$S_3=\{1,1,0,1,1,0,0,1\}$

$S_4=\{1,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1\}$

$S_5=\{1,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1\}$

Note that the number of elements of a given sequence of order n is $L_n=2^n$ and therefore the ratio between the lengths of two Baum–Sweet sequences of consecutive orders is:

$$\phi ={\displaystyle \frac{L_n}{L_{n-1}}}=2$$

(1)

Based on the aforementioned aperiodic sequence, two new diffractive elements are proposed and studied: a binary phase Zone Plate and a Kinoform lens. In both cases, we define a phase distribution ${\varphi }_n$ along the normalized squared radial coordinate $\zeta ={(r/a)}^2$, where r is the radial coordinate and a is the radius of the lens. The variable $\zeta$ is bounded within the interval $[0;1]$, which is divided into $L_n$ sub-intervals of length $d_n=1/L_n$.

For the Baum–Sweet Zone Plate of order n, the phase at the j-th sub-interval is given by ${\varphi }_{n,j}=\pi S_{n,j}$, where $S_{n,j}$ is the j-th element of the Baum–Sweet sequence of order n, so ${\varphi }_{n,j}=\pi$ when $S_{n,j}=1$ and ${\varphi }_{n,j}=0$ when $S_{n,j}=0$. Mathematically, this function for a given order n can be written as:

$${\varphi }_n\left(\zeta \right)=\pi \sum _{j=1}^{L_n}{S}_{n,j}rect\left({\displaystyle \frac{\zeta -(j-1/2)d_n}{d_n}}\right)$$

(2)

where $rect$ refers to the rectangular function.

Figure 1a shows the phase distribution $\varphi$ of a Baum–Sweet Zone Plate (BSZP) of order 4 as a function of the normalized quadratic radial coordinate $\zeta$, while Figure 1b illustrates its phase distribution along the Cartesian coordinates. In Figure 1b, the white rings correspond to a phase of $\varphi =\pi$, while the black rings correspond to a phase of $\varphi =0$.

On the other hand, the Kinoform Baum–Sweet Lens (KBSL) has a phase function, ${\varphi }_n\left(\zeta \right)$, such that at each occurrence of the pair “10” in the aperiodic sequence, there is a linear variation of the phase between 0 and $2\pi$; otherwise, ${\varphi }_n\left(\zeta \right)=0$ (see Figure 2a). Mathematically, the phase distribution function can be written as:

$${\varphi }_n\left(\zeta \right)={\displaystyle \frac{2\pi }{2d_n}}\sum _{j=1}^{L_n-1}rect\left({\displaystyle \frac{\zeta -j{d}_n}{2d_n}}\right)(\zeta -d_n(j-1)){\delta }_{1,s_{n,j}}{\delta }_{0,s_{n,j+1}}$$

(3)

where ${\delta }_{1,s_{n,j}}$ and ${\delta }_{0,s_{n,j+1}}$ are Kronecker delta functions. In the same way as the BSZP, the KBSL can be generated by applying the revolution symmetry around the optical axis (see Figure 2b).

3. Results

3.1. Focusing Properties

To determine the focusing properties of the BSZP and a KBSL, we have computed its irradiance along the axial direction using the Fresnel-Kirchhoff diffraction theorem [33]:

$$I\left(u\right)={\left(2\pi u\right)}^2{\left|{\int }_0^1{e}^{i{\varphi }_n\left(\zeta \right)}{e}^{-2i\pi u\zeta }d\zeta \right|}^2$$

(4)

where $u={\displaystyle \frac{a^2}{2\lambda z}}$ is the reduced axial coordinate, $\lambda$ is the wavelength of incident light, z is the axial distance from the lens plane and ${\varphi }_n\left(\zeta \right)$ is the phase function of the lens.

Figure 3 shows the axial intensity distributions generated by the BSZPs of orders $n=4$, $n=5$, and $n=6$. In each case, the irradiance has been normalized with respect to its maximum achieved value. These lenses produce two main foci of similar intensities around the first diffraction order. Higher diffraction orders also appear due the binary nature of the lens, so these two foci are periodically replicated along the coordinate u with period $L_n$. The positions of the two main foci are related to the Baum–Sweet sequences used in the design of these lenses. For order $n=4$, the focal positions are $u\left(f_1\right)=5.64$ and $u\left(f_2\right)=10.47$; for $n=5$, $u\left(f_1\right)=10.43$ and $u\left(f_2\right)=21.67$; and for $n=6$, $u\left(f_1\right)=21.63$, and $u\left(f_2\right)=42.57$. When computing the ratio $u\left(f_2\right)/u\left(f_1\right)$ for each of these orders, the results are $1.86$, $2.08$, and $1.97$, respectively. It is evident that, as the order increases, these values approach $\phi$, which is also involved in the aperiodic Baum–Sweet sequence. It is also evident that, for consecutive orders, the irradiance distribution expands by a factor of $\phi$ along the axial axis

KBSLs retain the same focusing properties as BSZPs (Figure 4). Kinoform lenses of orders $n=4$ and $n=6$ produce two single foci of similar intensity, whose positions match those generated by their ZP counterparts. However, in the irradiance distribution of the KBSL of order $n=5$, two high intensity points are observed, which correspond to the main foci, along with four secondary foci that also exhibit significant intensities, ranging from 40% to 65% of the intensity of the main foci. As a result, a substantial portion of the incident light is concentrated in these secondary planes, considerably reducing the diffraction efficiency of the main foci in this order.

A comparison of the irradiance distributions up to the second diffraction order for both lenses, the BSZP and the KBSL of order $n=4$, reveals that the KBSL directs most of the incoming light to the first diffraction order (Figure 5), whereas the BSZP produces two foci at each diffraction order due to its binary structure. The maximum irradiance value produced by the BSZP was $60\%$ lower than that of the KBSL, demonstrating that the proposed sawtooth profile enhances the diffraction efficiency of these lenses while preserving the focusing properties of the binary phase design.

3.2. Experimental Results

The focusing properties of the KBSL were investigated using the experimental setup depicted in Figure 6. A He–Ne laser beam ($\lambda =632.8\mathrm{nm}$, power $10\mathrm{mW}$), coupled into an optical fiber, was first collimated by lens $L_a$ (focal length $f_a=160\mathrm{mm}$) and linearly polarized. The collimated beam was then directed to a beam splitter ($BS$), which reflected part of the light onto a liquid-crystal spatial light modulator ($SLM$) (Holoeye PLUTO, $1920\times 1080$ pixels, $8\mathsf{\mu }\mathrm{m}$ pixel pitch, 8-bit gray level), operating in phase-only mode to digitally implement the KBSL. The wavefront modulated by the SLM was transmitted back through the $BS$ and relayed by a telescopic system composed of lenses $L_b$ ($f_b=200\mathrm{mm}$) and $L_c$ ($f_c=100\mathrm{mm}$), which projected the transmittance of the designed lens onto the exit pupil plane. The KBSL included a one-dimensional blazed grating that acted as a linear phase carrier, steering the diffracted light into the first diffraction order in the Fourier plane of lens $L_3$. To isolate this order, a diaphragm ($PH$) was positioned at the focal plane of $L_3$, serving as a spatial filter that transmitted only the first diffraction order. For precise alignment, the SLM was slightly tilted so that the linear phase carrier ensured overlap of the first diffraction order with both the optical axis and the diaphragm aperture. This filtering stage not only isolated the desired signal but also suppressed noise contributions from the zero order and higher diffraction orders produced by the pixelated structure of the SLM. Finally, the focusing profile along the optical axis was recorded using a camera sensor (8-bit gray level, $3.75\mathsf{\mu }\mathrm{m}$ pixel pitch, $1280\times 960$ pixels) mounted on a motorized translation stage (Thorlabs LTS 300; $300\mathrm{mm}$ travel range; $5\mathsf{\mu }\mathrm{m}$ precision), providing an accurate experimental characterization of the lens performance.

The focusing properties of an KBSL of order $n=4$ and radius $a=1.3$ mm were experimentally evaluated using the previous setup. The experimental axial irradiance distribution is shown in Figure 7, along with the one obtained numerically from Equation (4) for comparison purposes. It can be observed that the KBSL generates two focal planes A and B, with a similar intensity. These focal points are located at the axial positions: $z_A=$ 132.8 mm and $z_B=226.3$ mm. The ratio between the positions of these foci, ${\displaystyle \frac{z_B}{z_A}}\approx 1.70$, is in good agreement with the results presented in

Table 1. Comparison between the KBSL and the KFL.

Lenses	u$\left({\mathit{f}}_1\right)$	u$\left({\mathit{f}}_2\right)$	u$\left({\mathit{f}}_2\right)$/u$\left({\mathit{f}}_1\right)$	FWHM$\left(\frac{{\mathit{f}}_1}{{\mathit{f}}_{\mathit{KPL}}}\right)$	FWHM$\left(\frac{{\mathit{f}}_2}{{\mathit{f}}_{\mathit{KPL}}}\right)$
KBSL $n=4$	$5.94$	$10.06$	$1.69$	$1.13$	$1.15$
KFL $\left(16\right)$	$6.14$	$9.96$	$1.62$	$0.99$	$1.01$

.

Once characterized, the KBSL was implemented in an optical tweezers setup, as shown in Figure 8. In this case, a laser light source with a wavelength of $\lambda =1064\mathrm{nm}$ and a power of $P=3\mathrm{W}$ (Laser Quantum, Mod. Opus 1064) is used. A half-wave plate $\lambda /2$ is placed at the output of the laser, followed by a linear polarizer, which changes the propagation direction of the linearly polarized light. Mirrors $M_1$ and $M_2$ direct the light through a lens system consisting of lenses $L_1$$(f_1=50\mathrm{mm})$ and $L_2$$(f_2=150\mathrm{mm})$ with a magnification of three, towards the spatial light modulator $\left(SLM\right)$ (Holoeye PLUTO-2.1-NIR-149, phase-type, pixel size $8\mathsf{\mu }\mathrm{m}$ and resolution $1920\times 1080$ pixels), configured for a phase of $2.1\pi$ at $\lambda =1064\mathrm{nm}$. This modulator projects the KBSL onto a $4f$ lens system using lenses $L_3$$(f_3=150\mathrm{mm})$ and $L_4$$(f_4=150\mathrm{mm})$ to reduce the beam size. A diaphragm $\left(D\right)$ is added at the focal plane of $L_3$, redirecting the light to the first diffraction order and avoiding noise from the specular reflection of higher diffraction orders. The modulator is positioned such that the first diffraction order is aligned with the optical axis of the diaphragm. The lens image then passes through a 40× magnification objective located at the focal plane of $L_4$. To illuminate the sample, a collimated LED (Thorlabs, Mounted High-Power, $1300\mathrm{mA}$, Mod. MCWHL7) light is used, which is then focused using lens $L_5$$(f_5=30\mathrm{mm})$. A beam splitter $\left(BS\right)$ transmits the visible light from the sample through the rear focal plane of the objective, after which the resulting image is focused with lens $L_6$$(f_6=50\mathrm{mm})$ and finally captured by a camera sensor (Edmund Optics, Mod. EO-10012C).

Next, we present the experimental results demonstrating the trapping and manipulation of microparticles in the two focal planes generated by the KBSL. Figure 9 shows the stable confinement of polystyrene particles with a diameter of ∼2 $\mathsf{\mu }\mathrm{m}$, achieved using a KBSL of order $n=4$. The sequence of images illustrates the simultaneous trapping of two particles, one in each focal plane along the axial axis. In the central region, the trapped particles are visible; meanwhile, at the bottom, a reference particle highlights the displacement of the scene, confirming that the confined particles remain fixed in the traps. The slight tilt of the objective enables the simultaneous observation of both focal planes and their respective trapped particles. As predicted theoretically, the bifocal property of the KBSL allows the formation of two independent optical traps distributed axially, enabling multiple particle capture (see Video S1). These results demonstrate that the KBSL, through its kinoform design and bifocal focusing properties, provides a novel approach for the realization of volumetric optical traps. The ability to simultaneously confine microparticles at different axial positions extends the functionality of optical tweezers and opens new possibilities for applications in three-dimensional optical manipulation.

4. Discussion

To contextualize the proposed diffractive element within the framework of aperiodic diffractive lenses, the irradiance distribution produced by the KBSL with $n=4$ has been compared with that of the equivalent kinoform periodic lens (KPL) and the kinoform Fibonacci lens (KFL), another intrinsically bifocal aperiodic element. In all cases the first 16 elements of the corresponding sequences were considered. The results are shown in Figure 10. The irradiances were normalized with respect to the maximum value of the KPL.

Table 1 summarizes the parameters corresponding to the intensity peaks for each lens. The results indicate that the Fibonacci lens exhibits slightly more intense and better-defined peaks compared to the KBSL. It is noteworthy that the position ratio between the focal points $(u\left(f_2\right)/u\left(f_1\right))$ of both the KBSL and the FKL exhibits similar values. However, as the sequence order increases, this ratio tends to stabilize at characteristic values specific to each type of lens. In the case of the KBSL, it approaches 2, with $u\left(f_2\right)/u\left(f_1\right)=2.02$ for $n=5$ and $u\left(f_2\right)/u\left(f_1\right)=1.94$ for $n=6$, whereas for the KFL, the ratio tends toward the golden ratio. This suggests that lenses based on the Baum–Sweet sequence achieve a greater separation between focal planes than Fibonacci lenses with the same number of zones, thereby offering more flexibility in designing diffractive elements with specific focusing properties.

The experimental results presented above demonstrate the effective trapping and manipulation of microparticles at the two focal planes generated by the KBSL. The simultaneous trapping of two particles, one in each focal plane along the axial axis, highlights the KBSL’s ability to create independent optical traps distributed axially.

Importantly, the bifocal focusing capability of the KBSL enables multiple particle capture along the optical axis, extending the conventional functionality of optical tweezers. The kinoform design of the KBSL ensures precise light modulation, allowing volumetric optical trapping and manipulation. This characteristic opens new opportunities for three-dimensional optical manipulation, including applications in microfluidics, biological systems, and complex particle assembly.

Overall, these findings illustrate that the KBSL represents a versatile and novel approach for realizing volumetric optical traps, providing a foundation for future studies aimed at exploring multiparticle interactions and more complex trapping configurations. Further investigations could focus on optimizing trap stability, evaluating particle dynamics, and expanding the system to different particle types and sizes.

5. Conclusions

A new diffractive optical element based on the aperiodic Baum–Sweet sequence has been presented and studied. Two different designs for this lens were proposed: a binary phase Zone Plate and a kinoform lens. It was demonstrated that both designs exhibit a bifocal behavior and that the positions of their main foci were correlated with the properties of the sequence upon which this structure is based. Moreover, the kinoform design improves the diffraction efficiency of this DOE extending its suitability to different optical applications. Finally, the KBSL was implemented in an optical tweezers setup enabled the simultaneous trapping of particles in two distinct focal planes along the optical axis. The distinctive properties of this aperiodic lens suggest its potential for applications in a variety of research areas, including microscopy and quantum computing. By exploiting its unique focusing and structural characteristics, this lens could provide new opportunities for advanced optical manipulation and imaging techniques, opening pathways for further developments in both fundamental and applied science.