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Sinusoidal-amplitude binary phase mask and its application in achieving an ultra-long optical needle

Xu-Zhen Gao, Peng-Cheng Zhao, Jia-Hao Zhao, Xue-Feng Sun, Jin-Jin Liu, Fan Yang, and Yue Pan

Open Access Open Access

Abstract

Optical needle has become a hot research topic in recent years, due to the excellent properties and potential applications. To achieve a sub-diffraction optical needle, there are three common methods including planar diffractive lenses, reflective mirrors or axicons, and high-NA objective lenses with the designed phase or amplitude elements. Here, we propose a new kind of designed phase and amplitude element called the sinusoidal-amplitude binary phase mask (SA-BPM), which modulates the amplitude and phase distributions of the incident vector optical fields (VOFs) simultaneously. Based on Richards-Wolf vector diffraction integral, the corresponding parameters of SA-BPM and the optimal optical needle length are calculated by exhaustive method and genetic algorithm. We further upgrade the SA-BPM by adding a Gaussian function in the amplitude modulation, and design the Gaussian SA-BPM (GSA-BPM). We find that the ultra-long optical needles are achieved with the SA-BPM and GSA-BPM, and the depth of focus of the optical needles are improved by 30%–70% compared with the case of binary phase mask. Such SA-BPM and GSA-BPM we proposed have great potential for manipulation and utilization of the ultra-long optical needles.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

An optical needle is a new type of light field, which concentrates most of its energy continuously on the optical axis and there is no diffraction confinement over several wavelengths along the direction of propagation [1]. In recent years, the optical needle has become a research hotspot due to its excellent characteristics of ultra-long depth of focus (DOF) and important applications, such as nano-lithography [2,3], high-density optical data storage [3,4], super-resolution imaging [5], high-efficient particle capture [6], and multi-plane micromanipulation of nanoparticles [7,8]. Generally, there are three methods [9] to generate the optical needles including planar diffractive lenses [10–16], reflective mirrors or axicons [17–19], and high-NA objective lenses with the designed phase or amplitude elements [1,15,17,18,20–26]. Among these three methods, the planar diffractive lenses are composed of micro or nanostructures, and the sub-diffraction optical needles can be achieved by optimizing the locations and optical responses [11]. Thus, the planar diffractive lenses are very popular in generating optical needles and have been demonstrated experimentally for many applications such as the super-resolution imaging [10,27–29]. The reflective mirrors or axicons are also commonly used in generating the optical needles. Compared with the first two methods, the advantage of the third method using high-NA objective lenses is that fewer belts are needed in the belt-like structures [9].

Along with the development of studying optical needles, the vector optical fields (VOFs) with space-variant polarization distribution have been proved to be very useful in achieving ultra-long optical needles in the focal plane, including the radially polarized VOF (RP-VOF) [1,3–5,8,12,14,17–23,30–42] and vortex azimuthally polarized VOF (VAP-VOF) [15,23,24,41,43,44]. For the method of high-NA objective lenses, a binary phase mask (BPM) is often used to modulate the incident VOF to achieve long optical needles [1,9,20,24,26,30,42,43,45,46]. The BPM is an annular diffraction optical element (DOE) with $\pi$ phase shift between adjacent belts [1]. Using the BPM with simple structure, the length of the optical needle can be significantly improved, proving the importance of the DOE in achieving ultra-long optical needles. Therefore, new and better DOEs are always needed for the method of high-NA objective lenses, in order to provide more approaches in designing longer optical needles.

In this paper, we achieve ultra-long optical needle by applying a new kind of DOE as sinusoidal-amplitude binary phase mask (SA-BPM), which is used to modulate the amplitude and phase of the incident RP-VOF and VAP-VOF. Based on the exhaustive method and genetic algorithm, we calculate the optimal parameters of the SA-BPM to achieve the longest optical needle in the focal plane. Compared with the BPM, the DOF of the optical needle modulated by the SA-BPM is significantly improved. Furthermore, we design a Gaussian SA-BPM (GSA-BPM) to modulate the incident VOFs, which can further improve the DOF of the optical needle. We hope that the proposed SA-BPM and GSA-BPM can open new avenue in achieving ultra-long optical needles, which can provide applications in various areas.

2. Theory

The schematic of achieving optical needle by tight focusing a VOF with the SA-BPM is shown in Fig. 1. The focusing system to generate optical needle is shown in Fig. 1(a), and the incident VOF is modulated by the SA-BPM with multiple concentric belts. Then the modulated VOF is focused by an objective lens with a high numerical aperture (NA), and the optical needle with a long DOF can be achieved in the focal plane.

 figure: Fig. 1.

Fig. 1. The schematic of generating ultra-long optical needle by tight focusing a VOF with the SA-BPM. (a) Schematic of the focusing system. (b) Polarization and phase distributions of the incident RP-VOF and VAP-VOF. (c) The modulation of phase and intensity distributions of the SA-BPM.

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Here, we choose the RP-VOF and VAP-VOF as the incident VOFs, and the electric field components of the two kinds of VOFs are expressed as

$$\mathbf{E}_{\mathrm{RP}}=\cos \varphi \hat{\mathbf{e}}_{x}+\sin \varphi \hat{\mathbf{e}}_{y}, \quad \mathbf{E}_{\mathrm{VAP}}=e^{j \varphi}\left(-\sin \varphi \hat{\mathbf{e}}_{x}+\cos \varphi \hat{\mathbf{e}}_{y}\right),$$
where $\varphi$ is azimuthal coordinate, $\hat {\mathbf {e}}_{x}$ and $\hat {\mathbf {e}}_{y}$ are the unit vectors in the $x$ and $y$ directions. The polarization and phase distributions of the RP-VOF and VAP-VOF are shown in Fig. 1(b).

To achieve optical needles with longer DOF, we propose the SA-BPM to modulate the amplitude and phase distributions of the incident VOFs. The transmission function of the SA-BPM with multiple concentric belts is written as

$$T_{N}(r)=\sum_{n=1}^{N} S_{n} P_{n},$$
with
$$S_{n}=\sin ^{m}\left(\frac{\pi\left(r-r_{n-1}\right)}{r_{n}-r_{n-1}}\right), \quad P_{n}=2 {\rm mod} (n, 2)-1, \quad r_{n-1}<r \leq r_{n},$$
where $N$ is the number of the concentric belts in the multiple concentric belts structure. $S_{n}$ and $P_{n}$ indicate the amplitude and phase modulations of each belt. $r$ is the radial coordinate, and $r_{n-1}$ and $r_{n}$ are the inner and outer radii of the $n$th concentric belt with $r_{0}$ = 0, respectively. $m$ is the index of the sinusoidal amplitude modulation function. mod ($a$, $b$) is the function giving the remainder on division of $a$ by $b$, which can lead to a $\pi$ phase shift between the adjacent concentric belts. Figure 1(c) shows the modulation of phase and intensity distributions of the SA-BPM. Obviously, the alternating phase of the SA-BPM is 0 or $\pi$, and the intensity changes along radial direction according to sine function.

Based on the Richards-Wolf vector diffraction integral [47,48], the electric field components of the modulated RP-VOF and VAP-VOF in the focal region are described as

$$\mathbf{E}_{\mathrm{RP}}^{f}(\rho, \phi, z)= k f \int_{0}^{\theta_{m}} d \theta T_{N}(r) E_{0} P(\theta) \sin \theta \left(\begin{array}{c} \cos \theta J_{1}(k \rho \sin \theta) \hat{\mathbf{e}}_{\rho} \\ 0 \hat{\mathbf{e}}_{\phi} \\ j \sin \theta J_{0}(k \rho \sin \theta) \hat{\mathbf{e}}_{z} \end{array}\right) \exp(j k z \cos \theta), $$
$$\begin{aligned}\mathbf{E}_{\mathrm{VAP}}^{f}(\rho, \phi, z)=& \frac{j k f}{2} e^{j \phi} \int_{0}^{\theta_{m}} d \theta T_{N}(r) E_{0} P(\theta) \sin \theta \exp (j k z \cos \theta)\\ & \times\left[(\cos \theta-1) J_{2}(k \rho \sin \theta)+(\cos \theta+1) J_{0}(k \rho \sin \theta)\right] \hat{\mathbf{e}}_{\phi}, \end{aligned}$$
where $(\rho, \phi, z)$ are the radial, azimuthal, and longitudinal coordinates in the focal plane, respectively. $E_{0}$ is the constant amplitude of incident optical field, and $k$ is the wavenumber. $f$ is the focal length of the lens, and $J_{i}(\cdot )$ is the $i$th-order Bessel function of the first kind. $P(\theta ) =\sqrt {\cos \theta }$ is the pupil plane apodization function, and $r$ is the radial coordinate with $r = \sin \theta$ in the incident plane. $\theta _{m}$, which is determined by NA = $\sin \theta _{m}$, is the maximum value of convergence angle $\theta$. NA is the numerical aperture of the lens, which is set as NA = 0.95 in the following calculations. The wavelength of the incident light is 532 nm, and the focal length of the objective lens is 45 mm.

Next, we will calculate the optical needles generated by the tightly focused RP-VOF and VAP-VOF with the SA-BPM, based on Eqs. (5) and (6). In the process of numerical calculations, we try to gain optimal parameters of the SA-BPM, in order to achieve the longest optical needles in the focal plane. We first use the method of exhaustion to gain the rough values of these parameters, and further use these values as the initial values in the genetic algorithm. As a result, we can obtain the accurate parameters of the SA-BPM to generate an ultra-long optical needle.

3. Achieving ultra-long optical needles with SA-BPM

Figures 2(a-c) show the total intensity, transverse component, and longitudinal component of the optical needle generated by the tightly focused RP-VOF modulated by the SA-BPM with $m = 3$, $N = 4$ and $(r_{1}, r_{2}, r_{3}) = (0.42R, 0.6R, 0.74R)$, where $R$ is the total radius of incident optical field. For comparison, we also show the intensity of the optical needle originated from a BPM with the same number of the concentric belts and $(r_{1}, r_{2}, r_{3}) = (0.39R, 0.63R, 0.81R)$ in Figs. 2(d-f). It can be seen that the DOF of the tightly focused RP-VOF modulated by SA-BPM can reach 7.4$\lambda$, which is much longer than the DOF of 5.59$\lambda$ for the case of BPM. Besides the data of DOF, the energy of the optical needle generated by the tightly focused RP-VOF modulated by SA-BPM is more concentrated as shown in Figs. 2(a) and 2(d), due to the fact that the optical needle has lower sidelobes along the longitudinal axis. Thus, the length and quality of the optical needle can be significantly improved with SA-BPM we propose. In addition, the optical needle is mainly with longitudinal component and the transverse component is almost zero.

 figure: Fig. 2.

Fig. 2. Optical needles of tightly focused RP-VOF modulated by the SA-BPM and BPM, respectively. The first row shows the case of SA-BPM when $m = 3$, $N = 4$, and $(r_{1}, r_{2}, r_{3}) = (0.42R, 0.6R, 0.74R)$. The second row shows the case of BPM when $N = 4$ and $(r_{1}, r_{2}, r_{3}) = (0.39R, 0.63R, 0.81R)$. The three columns show the total, transverse, and longitudinal intensity patterns in the focal plane.

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Apart from the RP-VOF, we also calculate the optical needles of the tightly focused VAP-VOF modulated by SA-BPM and BPM when $m = 3$ and $N = 4$ in Figs. 3(a) and 3(b), respectively. As the longitudinal component of the tightly focused VAP-VOF is zero, the optical needles shown in Fig. 3 are purely transversely polarized. The red and blue curves of Fig. 3(c) show the normalized intensity profile of the optical needles along longitudinal axis in the above two cases, respectively. The DOF of the tightly focused VAP-VOF modulated by SA-BPM with $m=3$, $N = 4$ and $(r_{1}, r_{2}, r_{3}) = (0.31R, 0.53R, 0.69R)$ can reach 7.6$\lambda$, while the DOF of the tightly focused VAP-VOF modulated by BPM when $N = 4$ and $(r_{1}, r_{2}, r_{3}) = (0.25R, 0.51R, 0.74R)$ is 4.88$\lambda$. These results indicate that for the VAP-VOF, the length of optical needle can also be significantly improved by applying SA-BPM compared with the case of BPM. In addition, the optical needle of the tightly focused VAP-VOF modulated by BPM has strong sidelobes on both sides along the longitudinal axis. When using the SA-BPM, the intensity of the strong sidelobes disappear and the energy is concentrated in the optical needles, which is a great improvement for the quality of the optical needle.

 figure: Fig. 3.

Fig. 3. Optical needles of tightly focused VAP-VOF modulated by SA-BPM and BPM. (a) Total intensity pattern of the tightly focused VAP-VOF modulated by SA-BPM when $m = 3$, $N = 4$, and $(r_{1}, r_{2}, r_{3}) = (0.31R, 0.53R, 0.69R)$. (b) Total intensity pattern of the tightly focused VAP-VOF modulated by BPM when $N = 4$ and $(r_{1}, r_{2}, r_{3}) = (0.25R, 0.51R, 0.74R)$. (c) Normalized intensity profile of the optical needles of the tightly focused VAP-VOF modulated by SA-BPM and BPM, respectively.

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In order to further prove the advantage of the SA-BPM, we make a comparison of the DOFs of tightly focused RP-VOF and VAP-VOF modulated by the SA-BPM and BPM when $m = 3$, $N = 3$, 4, 5, 6 and 7, as shown in Fig. 4. From Fig. 4, we can find that the DOF of the optical needle increases with the increasing $N$, and the DOFs of the optical needles of the tightly focused RP-VOF is close to that of the VAP-VOF when applying the same mask with the same $N$. For the cases of same incident VOF, the optical needles with SA-BPM are 32.38$\%$ to 78.76$\%$ longer than those with BPM when $N$ is the same. This also proves the advantage of the SA-BPM in achieving ultra-long optical needles.

 figure: Fig. 4.

Fig. 4. DOFs of the tightly focused RP-VOF and VAP-VOF modulated by SA-BPM and BPM when $m = 3$, $N = 3$, 4, 5, 6 and 7, respectively.

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In the above discussion, we have set the index $m$ of the SA-BPM to be 3. Now we should further discuss the effect of the index $m$ in achieving ultra-long optical needles. Figures 5(a) and 5(b) shows the dependence of the DOFs of the tightly focused RP-VOF and VAP-VOF on the index $m$ for the SA-BPM when $N = 3$, 4, and 5, respectively. As we can see, the DOF of the optical needle increases when $m$ increases from 1 to 6. This is because that the intensity is more concentrated in each belt of the SA-BPM when $m$ increases. Obviously, the index $m$ is an important degree of freedom to increase the DOF of the optical needle.

 figure: Fig. 5.

Fig. 5. Dependence of the DOF of the optical needle on the index $m$ for the SA-BPM when $N = 3$, 4, and 5, respectively. (a) and (b) correspond to the two cases of tightly focused RP-VOF and VAP-VOF, respectively.

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4. Design of GSA-BPM and calculation of optical needles

To further increase the length and quality of the optical needle, we introduce the Gaussian function to modulate the SA-BPM, which is called the Gaussian SA-BPM (GSA-BPM). The transmission function of the GSA-BPM can be expressed as:

$$G_{N}(r)=\sum_{n=1}^{N} G_{n} S_{n} P_{n},$$
with
$$G_{n}=\exp \left[\frac{4 \ln B}{\left(r_{n}-r_{n-1}\right)^{2}}\left(r-\frac{r_{n}+r_{n-1}}{2}\right)^{2}\right],$$
where $G_{n}$ is a Gaussian function. The Gaussian function $G_{n}$, together with the sine function $S_{n}$, modulates the amplitude distribution of the incident field in the $N$th concentric belt of the GSA-BPM. The maximum value of the Gaussian function locates at $r=\left (r_{n}+r_{n-1}\right ) / 2$, which is also the center of each belt. The parameter $B$ indicates the minimum value of the Gaussian function at the boundary of each belt, which also determines the full width at half-maximum (FWHM) of the amplitude of the optical field in each concentric belt. By introducing the Gaussian function, the FWHM of the amplitude in each concentric belt decreases. Figure 6(a) shows the schematic of constructing the GSA-BPM by combining the Gaussian and sine functions, and the superposition of the two functions denotes the amplitude of the optical field behind the GSA-BPM. Figures 6(b) and 6(c) show the normalized intensity profile of the optical field behind the SA-BPM and GSA-BPM, respectively. Obviously, the intensity of the optical field modulated by the GSA-BPM is more concentrated in each belt. In this way, the DOF of the optical needle can be further increased.

 figure: Fig. 6.

Fig. 6. The schematic of constructing the GSA-BPM. (a) The superposition of the Gaussian and sine functions. (b) and (c) Normalized intensity profile of the optical field behind the SA-BPM and GSA-BPM.

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Figure 7(a) shows the optical needle patterns of the tightly focused RP-VOF modulated by GSA-BPM when $N = 4$, $m = 3$, and $B = 0.1$, 0.01 and 0.001, respectively. The corresponding DOFs of the optical needles are 7.66$\lambda$, 7.88$\lambda$ and 8.30$\lambda$, respectively. Compared with the case of the tightly focused RP-VOF modulated by SA-BPM, the DOFs of the three cases are further increased by 3.51$\%$, 6.49$\%$ and 12.16$\%$, respectively. The corresponding results of tightly focused VAP-VOF are described in Fig. 7(b). The DOFs of the optical needles are 7.72$\lambda$, 7.88$\lambda$ and 8.24$\lambda$, which are increased by 1.58$\%$, 3.68$\%$ and 8.42$\%$ compared with the cases of SA-BPM, respectively. It can be seen that the application of GSA-BPM can further lengthen the optical needles, and the optical needle maintains its original advantages at the same time. Figure 7(c) shows the relationship between the DOF and the parameter $B$ of the GSA-BPM for tightly focused RP-VOF and VAP-VOF. Based on Eqs. (7) and (8), and the GSA-BPM turns into SA-BPM when $B$ = 1. We can see that the DOF of the optical needle gradually increases along with decreasing values of $B$, and the increasing trend of RP-VOF and VAP-VOF are similar.

 figure: Fig. 7.

Fig. 7. The optical needle patterns generated by the tightly focused RP-VOF and VAP-VOF modulated by GSA-BPM when $m = 3$, $N = 4$, and $B = 0.1$, 0.01 and 0.001, respectively. (a) and (b) show the cases of RP-VOF and VAP-VOF, respectively. (c) Dependence of the DOFs of the optical needles on the parameter $B$.

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Next, we list the histogram of the DOFs of tightly focused RP-VOF and VAP-VOF modulated by BPM, SA-BPM and GSA-BPM when $N = 4$, as shown in Fig. 8. After applying the SA-BPM with $m = 3$, the DOFs of the optical needles of the tightly focused RP-VOF and VAP-VOF are increased by 32.38$\%$ and 55.74$\%$ compared with the cases of BPM, respectively. After applying the GSA-BPM with $B = 0.001$, the DOFs of the optical needles of the tightly focused RP-VOF and VAP-VOF are further increased by 12.16$\%$ and 8.42$\%$, compared with the cases of SA-BPM, respectively. The above data proves that the SA-BPM and GSA-BPM are both superior to the BPM in generating the ultra-long optical needles, and the GSA-BPM behaves better in increasing the DOF of the optical needle.

 figure: Fig. 8.

Fig. 8. Histogram of the DOFs of tightly focused RP-VOF and VAP-VOF modulated by BPM, SA-BPM and GSA-BPM when $N = 4$, respectively.

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In Table 1, we list the specific parameters of the above cases for achieving ultra-long optical needles, including ring radii of the concentric belts of the masks, DOFs of the optical needles, and lateral FWHMs of the optical needles, respectively. Based on the exhaustive method and genetic algorithm, we calculate the ring radii of the concentric belts of the BPM, SA-BPM, and GSA-BPM, to achieve the longest needles in the focal plane, as shown in Table 1. The lateral FWHM in Table 1 represents the lateral size of the optical needle, and the lateral FWHM here is the average value because the lateral FWHMs at different positions are slightly different for the same optical needle. We can find that for the BPM, SA-BPM and GSA-BPM, the lateral FWHM of the optical needle for the incident RP-VOF is slightly larger than that of the incident VAP-VOF. Meanwhile, the DOFs of the optical needles generated with the help of the SA-BPM and GSA-BPM are much larger than the cases of the BPM, proving the advantage of the new DOEs we propose. Due to the importance of the lateral FWHM, we further discuss the lateral FWHMs of all the optical needles in this paper. The lateral FWHMs of the optical needles for the case of incident RP-VOF and VAP-VOF modulated by BPM with 3-7 belts are in range of 0.4$\lambda$-0.45$\lambda$. The lateral FWHMs of the optical needles for the case of incident RP-VOF modulated by SA-BPM are in range of 0.45$\lambda$-0.59$\lambda$, and the range is 0.44$\lambda$-0.51$\lambda$ for the case of VAP-VOF modulated by SA-BPM. The lateral FWHMs of the optical needles for the case of incident RP-VOF modulated by GSA-BPM are in range of 0.5$\lambda$-0.53$\lambda$, and the range is 0.46$\lambda$-0.48$\lambda$ for the case of VAP-VOF modulated by GSA-BPM. Compared with the case of BPM, the lateral FWHMs are increased when applying the SA-BPM and GSA-BPM, and the lateral FWHMs are smaller for the case of incident VAP-VOF. This also agrees with the above discussion.

Tables Icon

Table 1. Ring radius, DOF, and lateral FWHM for RP-VOF and VAP-VOF modulated by BPM, SA-BPM and GSA-BPM when $\textbf {N = 4}$.

View Table

For comparison, we refer to the former references and the DOFs of the optical needles are 4$\lambda$ -5.35$\lambda$ when the input RP-VOF or VAP-VOF are modulated by the simple BPM with 5-7 belts [1,24]. The DOFs of these optical needles are slightly different from the data we provide, due to the different simulation conditions and the differences of the intensity of the input field. The DOFs of the optical needles are 8.82$\lambda$ -10.52$\lambda$ when the input VOF are modulated by the SA-BPM with 5-7 belts, and the DOFs of the optical needles are 9.28$\lambda$ -10.7$\lambda$ for the case of the GSA-BPM. This further proves the advantage of the SA-BPM and GSA-BPM when generating longer optical needles, compared with the simple BPM.

5. Conclusion

In summary, we propose the SA-BPM and GSA-BPM to modulate the incident VOFs, assisting in generating ultra-long optical needles. Based on Richards-Wolf vector diffraction integral, the optimal parameters of the masks and the longest optical needle are calculated by exhaustive method and genetic algorithm. Compared with the BPM, the DOFs of the optical needles with SA-BPM and GSA-BPM are significantly increased. The SA-BPM and GSA-BPM add new degree of freedom to modulate the incident VOF, which open new avenue in designing and generating ultra-long optical needles. The optical needles we generate can be applied in various areas including nano lithography, high-density optical data storage, super-resolution imaging and efficient particle capture.

Funding

National Natural Science Foundation of China (11904199, 11804187); China Postdoctoral Science Foundation (2020M682142).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are publicly available and may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are publicly available and may be obtained from the authors upon reasonable request.

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Figures (8)
Fig. 1.
Fig. 1. The schematic of generating ultra-long optical needle by tight focusing a VOF with the SA-BPM. (a) Schematic of the focusing system. (b) Polarization and phase distributions of the incident RP-VOF and VAP-VOF. (c) The modulation of phase and intensity distributions of the SA-BPM.

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Fig. 2.
Fig. 2. Optical needles of tightly focused RP-VOF modulated by the SA-BPM and BPM, respectively. The first row shows the case of SA-BPM when $m = 3$, $N = 4$, and $(r_{1}, r_{2}, r_{3}) = (0.42R, 0.6R, 0.74R)$. The second row shows the case of BPM when $N = 4$ and $(r_{1}, r_{2}, r_{3}) = (0.39R, 0.63R, 0.81R)$. The three columns show the total, transverse, and longitudinal intensity patterns in the focal plane.

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Fig. 3.
Fig. 3. Optical needles of tightly focused VAP-VOF modulated by SA-BPM and BPM. (a) Total intensity pattern of the tightly focused VAP-VOF modulated by SA-BPM when $m = 3$, $N = 4$, and $(r_{1}, r_{2}, r_{3}) = (0.31R, 0.53R, 0.69R)$. (b) Total intensity pattern of the tightly focused VAP-VOF modulated by BPM when $N = 4$ and $(r_{1}, r_{2}, r_{3}) = (0.25R, 0.51R, 0.74R)$. (c) Normalized intensity profile of the optical needles of the tightly focused VAP-VOF modulated by SA-BPM and BPM, respectively.

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Fig. 4.
Fig. 4. DOFs of the tightly focused RP-VOF and VAP-VOF modulated by SA-BPM and BPM when $m = 3$, $N = 3$, 4, 5, 6 and 7, respectively.

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Fig. 5.
Fig. 5. Dependence of the DOF of the optical needle on the index $m$ for the SA-BPM when $N = 3$, 4, and 5, respectively. (a) and (b) correspond to the two cases of tightly focused RP-VOF and VAP-VOF, respectively.

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Fig. 6.
Fig. 6. The schematic of constructing the GSA-BPM. (a) The superposition of the Gaussian and sine functions. (b) and (c) Normalized intensity profile of the optical field behind the SA-BPM and GSA-BPM.

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Fig. 7.
Fig. 7. The optical needle patterns generated by the tightly focused RP-VOF and VAP-VOF modulated by GSA-BPM when $m = 3$, $N = 4$, and $B = 0.1$, 0.01 and 0.001, respectively. (a) and (b) show the cases of RP-VOF and VAP-VOF, respectively. (c) Dependence of the DOFs of the optical needles on the parameter $B$.

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Fig. 8.
Fig. 8. Histogram of the DOFs of tightly focused RP-VOF and VAP-VOF modulated by BPM, SA-BPM and GSA-BPM when $N = 4$, respectively.

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Tables (1)

Tables Icon

Table 1. Ring radius, DOF, and lateral FWHM for RP-VOF and VAP-VOF modulated by BPM, SA-BPM and GSA-BPM when N = 4 .

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Equations (7)

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E R P = cos ⁡ φ e ^ x + sin ⁡ φ e ^ y , E V A P = e j φ ( − sin ⁡ φ e ^ x + cos ⁡ φ e ^ y ) ,
T N ( r ) = ∑ n = 1 N S n P n ,
S n = sin m ⁡ ( π ( r − r n − 1 ) r n − r n − 1 ) , P n = 2 m o d ( n , 2 ) − 1 , r n − 1 < r ≤ r n ,
E R P f ( ρ , ϕ , z ) = k f ∫ 0 θ m d θ T N ( r ) E 0 P ( θ ) sin ⁡ θ ( cos ⁡ θ J 1 ( k ρ sin ⁡ θ ) e ^ ρ 0 e ^ ϕ j sin ⁡ θ J 0 ( k ρ sin ⁡ θ ) e ^ z ) exp ⁡ ( j k z cos ⁡ θ ) ,
E V A P f ( ρ , ϕ , z ) = j k f 2 e j ϕ ∫ 0 θ m d θ T N ( r ) E 0 P ( θ ) sin ⁡ θ exp ⁡ ( j k z cos ⁡ θ ) × [ ( cos ⁡ θ − 1 ) J 2 ( k ρ sin ⁡ θ ) + ( cos ⁡ θ + 1 ) J 0 ( k ρ sin ⁡ θ ) ] e ^ ϕ ,
G N ( r ) = ∑ n = 1 N G n S n P n ,
G n = exp ⁡ [ 4 ln ⁡ B ( r n − r n − 1 ) 2 ( r − r n + r n − 1 2 ) 2 ] ,
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