Abstract
Optical vortices (OVs) with the phase factor carry orbital angular momentum (OAM) of per photon, where denotes the topological charge (TC), denotes the azimuthal angle, and is the Planck constant. Nowadays, OVs are motivating a plethora of applications[
In 2019, the cross phase (CP), a new kind of phase structure, has been involved in Laguerre–Gauss (LG) beams that open up a new horizon for generation and measurement of OVs[
In this Letter, inspired by the finding above, we propose a new kind of OV called the Hermite–Gaussian-like OV (HGOV), which has an HG-like intensity distribution but still retains the OAM. HGOVs have a novel function of the self-measurement, which reveals the value and sign of TCs. Further, HGOVs effectively decouple the phase of a spherical lens from the phase of a cylindrical lens, which allows us to control the HGOV more flexibly, whether at near-field or far-field. HGOVs also have a good depth of field due to relatively stable distributions at far-field, which has a significant meaning for precise three-dimensional (3D) optical tweezers. In addition, there have been many reports about the generation of multi-singular beams[
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The form of the CP in Cartesian coordinates is where the coefficient controls the conversion rate, and the azimuth factor characterizes the rotation angle of converted beams in one certain plane. It is noteworthy that Eq. (
Namely, a vertically placed cylindrical lens can be equivalent to one spherical lens plus one CP that is rotated by 45°. It is to be noted that the phase of a spherical lens limits the generated beams to shape in the back focal plane (or at the far-field), but utilizing the CP to generate HGOVs effectively decouples the phase of a spherical lens from the phase of a cylindrical lens. It allows us to control the HGOV more flexibly, whether at the near-field or far-field. Decoupling the spherical lens has the following advantages. Firstly, we can use the rotation of the HGOV during the propagation to achieve a curved path for particle 3D manipulation. Secondly, if we need a focused HGOV, we can load the phase of a spherical lens with any focal length by an SLM, which is not limited by the fixed focal length of the cylindrical lens. Further, HGOVs have a good depth of field due to relatively stable distributions at the far-field, which has a significant meaning for precise control of 3D optical tweezers. Besides, due to the split singularities, HGOVs possess a unique advantage that could manipulate multiple particles precisely and dynamically at the same time. It is noteworthy that we define the light field that meets the far-field condition as the far-field hereinafter.
In fact, the coefficient in a CP needs to be adjusted to meet the requirements for different initial conditions, which is similar to the cylindrical lens that has the following requirement for the waist radius of the incident light[
Thus, if is set to , we can derive the expression of as
Without loss of generality, may the form of HGOV with be
According to the Fresnel diffraction integral, when the light field mentioned above propagates a certain distance , the output can be expressed as where denotes the observation plane, and is the Fourier transform.
Firstly, we would like to introduce the propagation properties of an HGOV from the near-field to the far-field. Under the condition of the coefficient , we simulate the propagation of the HGOV from 0 m to 5 m, as shown in Fig.
Figure 1.Simulated propagation of the HGOV from 0 m to 5 m.
Secondly, we would like to generalize the situation of the HGOV from the far-field to infinity to figure out if there is any possibility that the HGOV can reach the 100% mode purity. Figure
When , , which means that we only get a pure HGOV at infinity. Interestingly, due to decoupling the phase of a spherical lens from the phase of a cylindrical lens, we maintain the consistency of OAM growth rather than decreasing even when flipping the sign of OAM[
Nevertheless, HGOVs can still maintain high-mode purity at the far-field. Furthermore, HGOVs keep the OAM of an OV while retaining the function of the self-measurement, as shown in Fig.
Figure 2.Function of the self-measurement of the HGOV at the far-field. (a) Simulated intensity distributions of the HGOV of
We can also use the function of the self-measurement to achieve the self-sign-measurement of the HGOV, as shown in Fig.
Figure 3.Self-sign-measurement of the HGOV. (a) Simulated intensity distributions of the HGOV of
We would like to discuss the influence of the coefficient on the generation of HGOVs. The coefficient controls the conversion rate of an HGOV to the intensity distribution of the HG beam. However, we find that the mode purity decreases from 99.25% to 92.71% as the coefficient increases from 0.5 to 2.0 at the far-field, as shown in Fig.
Figure 4.Simulated distributions of HGOVs with different
However, it is not enough to change the relative positions of singularities. We also need to accurately control the direction of an HGOV if we want to achieve multi-particle manipulation at any position. We know that the azimuth factor characterizes the rotation angle of an HGOV in one certain plane in Eq. (
Figure 5.Simulated results of HGOVs of
In summary, we propose a new kind of OV called the HGOV. Firstly, we show how the CP is decoupled from the phase of a cylindrical lens, which allows us to control the HGOV more flexibly, whether at the near-field or far-field. Secondly, we investigate the propagation characteristics of HGOVs, which have an HG-like intensity distribution but still retains the OAM at the far-field. HGOVs have a good depth of field due to relatively stable distributions at the far-field, which has a significant meaning for precise control of 3D optical tweezers. Theoretically, we derived the diffraction integral formula of HGOVs, which confirms that pure HGOVs only exist at infinity. We also confirm this conclusion from the perspective of energy flow. Thirdly, we introduce a novel function of HGOVs that the function of the self-measurement, which reveals the value and sign of TCs, no longer needs interferometry to measure it. Further, we discuss the influence of the coefficient on the generation of HGOVs and find that the mode purity decreases as the coefficient increases at the far-field. In addition, we show that we can change the relative positions of singularities and the direction of HGOVs precisely, which is of great value in the field of multi-particle manipulation.
References
[1] P. Chen, L. L. Ma, W. Duan, J. Chen, S. J. Ge, Z. H. Zhu, M. J. Tang, R. Xu, W. Gao, T. Li, W. Hu, Y. Q. Lu. Adv. Mater., 30, 1705865(2018).
[3] Y. Q. Zhang, X. Y. Zeng, L. Ma, R. R. Zhang, Z. J. Zhan, C. Chen, X. R. Ren, C. W. He, C. X. Liu, C. F. Cheng. Adv. Opt. Mater., 7, 9(2019).
[4] J. Wang. Chin. Opt. Lett., 16, 050006(2018).
[5] G. Cossu. Chin. Opt. Lett., 17, 100009(2019).
[6] X. Yang, S. Wei, S. Kou, F. Yuan, E. Cheng. Chin. Opt. Lett., 17(2019).
[8] S. Qiu, T. Liu, Z. Li, C. Wang, Y. Ren, Q. Shao, C. Xing. Appl. Opt., 58, 2650(2019).
[9] W. Zhang, D. Zhang, X. Qiu, L. Chen. Phys. Rev. A, 100, 043832(2019).
[10] G. Liang, Q. Wang. Opt. Express, 27, 10684(2019).
[11] D. Shen, D. Zhao. Opt. Lett., 44, 2334(2019).
[12] C. Wang, Y. Ren, T. Liu, C. Luo, S. Qiu, Z. Li, H. Wu. Appl. Opt., 59, 4040(2020).
[13] Y. Shen, Y. Meng, X. Fu, M. Gong. J. Opt. Soc. Am. A Opt. Image Sci. Vis., 36, 578(2019).
[14] Y. Wang, Y. Chen, Y. Zhang, H. Chen, S. Yu. J. Opt., 18, 055001(2016).
[15] J. Zhou, W. Zhang, L. Chen. Appl. Phys. Lett., 108, 8185(2016).
[16] H. I. Sztul, R. R. Alfano. Opt. Express, 16, 9411(2008).
[17] L. Torner, J. Torres, S. Carrasco. Opt. Express, 13, 873(2005).
[18] A. Y. Bekshaev, M. S. Soskin, M. V. Vasnetsov. Opt. Commun., 241, 237(2004).
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