Recently, Bessel-like beams generated by the optical fibers-based method are desirable, which is because they could offer a compact, robust, and alignment-free operation compared with bulky optical systems. So far, there are many fiber-based methods for generating Bessel-like beams, such as fiber microaxicons^[1–3], self-growing micro-tips^[4], or a tapered hollow tube^[5]. Another way is utilizing a long-period grating to excite higher-order cladding mode and then generate Bessel beams with controllable diffraction-resistant distance^[6]. These non-diffracting Bessel-like beams have attracted increasing interest owing to their unique light intensity distributions and self-healing propagation properties, which have been widely used in various domains such as optical trapping^[7] and manipulation^[8], high-order harmonic generation^[9], nonlinear optics^[10], and material processing^[11]. Especially, such properties can also be used in optical wireless communication to overcome the diffraction limits and particle scattering^[12], even if encountering opaque obstacles.

As we all know, underwater optical wireless communication is very useful for many applications such as oceanography, environmental monitoring, and underwater surveillance^[13,14]. However, it suffers from extreme absorption and Mie–Rayleigh scattering in the water or seawater. Blue–green wavelengths as a distinct transmission window of the seawater for underwater optical wireless communication also suffer the Mie–Rayleigh scattering from water molecules, salt ions, phytoplankton, and other transparent biological organisms^[15,16]. Blue–green Bessel-like beams as communication lights underwater could provide a method to solve the problem caused by the Mie–Rayleigh scattering underwater^[17].

In this paper, we aim to design fiber microaxicons or micro-tips and simulate for generating visible Bessel-like beams with long depth of focus (DOF) underwater. It can be utilized in underwater optical wireless communication for long-distance communication, for overcoming such Mie–Rayleigh scattering effects.

An ideal Bessel beam of the $n$th order can be represented by $$E(r,\varphi ,z)=A_0\exp (i{k}_{z}z)J_n(k_{r}r)\exp (\pm in\varphi ),$$(1)where $k_r$ is the radial wave vector, $k_z$ is the longitudinal wave vector, $r$ is the radial coordinate, and $r^2=x^2+y^2$.

For $n=0$, which is the zeroth-order Bessel beam that can be easily written through substituting zero for $n$ in Eq. (1), then, $$E(r,\varphi ,z)=A_0\exp (i{k}_{z}z)J_0(k_{r}r).$$(2)

The power contained in the Bessel beam up to radius $b$ is found by integrating Eq. (2): $$\begin{gathered} P=A_0^2{\int }_0^{2\pi }{\int }_0^b{J}_0(k_{r}r) \\ =A_0^2{B}_w^2\pi [J_0{(k_{r}b)}^2+J_1{(k_{r}b)}^2]. \end{gathered}$$(3)

As for the optical profile of generated Bessel beams over propagation, the beam envelope propagation method (BPM) is adopted to calculate the Bessel-like beam generation and propagation.

Here, the fiber microaxicon generated by the single mode fiber (SMF) is utilized to generate the Bessel-like beams. The BPM is employed to numerically calculate the intensity distribution in the x−z plane for Bessel-like beam generation and propagation. That is because the SMF for the fiber microaxicon is a rotational symmetric structure.

Figure 1(b) shows that the parallel lights from the fiber produce refraction in the microaxicons interface between the fiber and the medium. As we all know, when the refraction angle can reach 90°, the refraction lights into the medium could traverse along the surface of the fiber microaxicon and then disappear, thus being unable to form the Bessel-like beams. The smallest cone angle $\theta$ can be defined by the equation in Fig. 1(b), which comes from the refraction theorem.

For calculations, our designed fiber microaxicon assumed to be axially symmetric with respect to the optical fiber core is shown in Fig. 1, which has large cone angle $\theta$, fiber core diameter of 8 µm, and fiber cladding diameter of 125 µm. Such a fiber microaxicon could be fabricated by a polishing procedure^[2]. A visible light source provides the optical mode excitation regime in this fiber microaxicon. The longitudinal (x−z) power distributions of visible light laser radiation calculated for different cone angles θ are simulated by BPM. Figure 2 shows that visible light Bessel-like beams of 470 nm, 520 nm, and 632 nm are from fiber microaxicons with different cone angles $\theta$ of 80°, 100°, 120°, 140° and propagated in the air. It can be found that the DOF of Bessel-like beams is dependent on the incident wavelength and cone angle $\theta$. With the increase of the cone angles from 100° to 140°, the DOF of the Bessel-like beam at a wavelength of 470 nm is increased from $\sim 3.8\mathrm{\mu m}$ to $\sim 12\mathrm{\mu m}$. With the increase of the incident wavelength from 470 nm to 632 nm, the DOF of the fiber microaxicon with the cone angle of 140° is also increased from $\sim 12\mathrm{\mu m}$ to $\sim 14.21\mathrm{\mu m}$. In a word, the large cone angle incident light with long wavelength could lead to large DOF of the fiber microaxicon. The longitudinal (x−z) power distributions of the Bessel-like beams underwater have been shown in Fig. 3. Comparing Fig. 3 to Fig. 2, it can be found that the underwater Bessel-like beams are easy to generate by using the fiber microaxicon with small cone angles. With the incident light with the wavelength of 470 nm, the fiber microaxicon with the cone angle of 60° could generate the Bessel-like beam underwater, while in the air the fiber microaxicon generating the Bessel-like beam needs the cone angle at about 100°. Thus, the Bessel-like beams in the visible light band have long DOF underwater. When the incident wavelength is changed from 470 to 632 nm at the fiber microaxicon with the cone angle of 140°, its underwater DOF is changed from $\sim 28\mathrm{\mu m}$ to $\sim 50.7\mathrm{\mu m}$.

As we all know, the refractive index of water is about 1.33, which is larger than that of the air. According to the equation in Fig. 1(b), the cone angle $\theta$ underwater could be low to that of smaller than the cone angle in the air. According to the calculation, the smallest cone angle $\theta$ of the fiber microaxicon underwater could be as large as 46°, and the smallest cone angle $\theta$ of the fiber microaxicon in the air could reach 95°. These smallest cone angles are close to the simulation results in Figs. 2 and 3.

Figure 4 demonstrates the results in the air and water of numerical calculations of the normalized full width at half-maximum (FWHM) and DOF values as functions of the fiber microaxicons cone angle $\theta$. As shown in Fig. 4(a), in the air, the smallest focal spot with FWHM about $\sim 100\mathrm{nm}$, $\sim 220\mathrm{nm}$, and $\sim 380\mathrm{nm}$, respectively, for wavelengths of 470 nm, 520 nm, and 632 nm is close to the diffraction limit size and can be achieved using fiber microaxicons with the cone angle of 100° for the wavelengths of 470 and 520 nm and the cone angle of 80° for the 632 nm wavelength. This indicates that the Bessel beam for the short wavelength light has the smallest focal spot with FWHM 100 nm value under the same cone angle. In Fig. 4(b), underwater the smallest focal spot with FWHM $\sim 240\mathrm{nm}$, $\sim 570\mathrm{nm}$, and $\sim 420\mathrm{nm}$, respectively, for wavelengths of 470 nm, 520 nm, and 632 nm, which are all close to their wavelengths and over the diffraction limit size, can be obtained by using the fiber microaxicons with the cone angle of 60° for the wavelengths of 470 and 520 nm and the cone angle of 40° for the 632 nm wavelength. It should be noted that in this case the output beam represents underwater Bessel-like beams with an enlarged DOF, where the DOF value dramatically reaches ∼50.7 µm for the 632 nm wavelength, $\sim 36.12\mathrm{\mu m}$ for the 520 nm wavelength, and $\sim 28\mathrm{\mu m}$ for the 470 nm wavelength at $\theta =140°$. Summing up these results, for Bessel-like beams with large DOF, fiber microaxicons with $\theta >140°$ are needed. Thus, for the Bessel-like beams propagated underwater with long distance, $\theta =140°$ has been chosen.

Visible lights indicated into water are attenuated by absorption and scattering. Absorption in water is mainly from an irreversible thermal process whereby the light interacts with water molecules and other organic or non-organic particulates. Scattering in water is described by the process of a photons path changed due to the interaction with particulates in the water. The suspended particles in water such as phytoplankton or other transparent biological organisms are larger than the wavelength of visible lights, which cause Mie scattering. The water molecules and salt ions in water are much smaller than the wavelength of light, which can form Rayleigh scattering. Thus, compared to the effect of scattering, the absorption of the Bessel-like beams underwater for visible lights has a slight effect in this paper. The model of Mie scattering and Rayleigh scattering for Bessel-like beams propagated in water can be established by the Henyey–Greenstein–Rayleigh phase function^[18,19], which can be written as $$P_{\mathrm{HG}}({\theta }^{\prime },g)=\frac{3}{2}\frac{1-g^2}{2+g^2}\frac{1+\cos {({\theta }^{\prime })}^2}{[1+g^2-2g\cos {({\theta }^{\prime })}^{3/2}]},$$(4)where $g$ is the asymmetry factor, which can be expressed as^[18]$$g=\langle \cos ({\theta }^{\prime })\rangle .$$(5)

The Henyey–Greenstein–Rayleigh phase function model can be simulated by using Monte Carlo methods. A pioneering bio-optical model for the spectral absorption coefficient at visible lights was developed by Prieur and Sathyendranath, which can be expressed as^[20]$$\alpha (\lambda )=[{\alpha }_W(\lambda )+0.06{\alpha }_c^{*}(\lambda )C^{0.65}]\{1+0.2\exp [-0.014(\lambda -440)]\}.$$(6)

Here, ${\alpha }_W(\lambda )$ is the absorption coefficient of pure water, and ${\alpha }_c^{*}(\lambda )$ is a nondimensional, statistically derived chlorophyll-specific absorption coefficient. The $C$ is the chlorophyll concentration. The value of ${\alpha }_W(\lambda )$ and ${\alpha }_c^{*}(\lambda )$ can be obtained from Ref. [20]. From Fig. 3, it is known that the Bessel-like beams generated by the fiber microaxicon have a long DOF underwater, and thus it is suitable for propagation underwater. Lights propagated underwater could suffer from spectral absorption, Mie scattering, and Rayleigh scattering. In this paper, the underwater propagation proceeding of Bessel-like beams generated by the fiber microaxicons has been simulated by using the Henyey–Greenstein–Rayleigh phase function and spectral absorption by a bio-optical model due to Monte Carlo methods. The transverse power distributions of Bessel-like beams generated by the fiber microaxicon with $\theta =140°$ for three propagation distances of 500 m, 2000 m, and 4000 m are shown in Fig. 5 for visible lights of 470, 520, and 632 nm. It can be shown that the propagation distance is enlarged to 4000 m underwater by Bessel-like beams. It is found in this case that the Bessel-like beams with blue light have a better transmission characteristic than other Bessel-like beams with the green and red lights. This is because the absorption and divergence angle of the blue light are smaller than those of other visible lights. The smallest divergence angle of the central focal spot for the blue light came from the smallest FWHM of the blue Bessel-like mode light generated by the fiber microaxicon with $\theta =140°$. Underwater, the Bessel-like beams could transmit at least 4000 m. With the increase of the propagation distance, the central focal spot of Bessel-like beams for red light was broadened to be larger than that of blue and green lights. Especially, the Bessel-like beams with the red light transmitted underwater have larger absorption and scattering, and thus its transmission length is no longer than 4000 m. In a word, the blue–green Bessel-like beams are more suitable for underwater communication than the Bessel-like beams with the red light. Figure 6 shows the transverse power distributions of Gaussian beams for the propagation distance of 500 m. Comparing Figs. 5 and 6, under the same propagation distance of 500 m, the light spots expansion of Bessel-like beams caused by the Mie–Rayleigh scattering is smaller than that of Gaussian beams. The transverse power distributions of Bessel-like beams are clearer and smaller than that of Gaussian beams. Thus, it means that Bessel-like beams are more suitable for underwater optical wireless communication.

Bessel-like beams have been generated by fiber microaxicons with large cone angles in the air and small cone angles in water. Underwater, the cone angle of the fiber microaxicons could be as small as 60° for generating Bessel-like beams for blue light. The Bessel-like beams propagated underwater from the fiber microaxicons with 140° large cone angle have enhanced DOF about 28 µm, 36.12 µm, and 50.7 µm for 470 nm, 520 nm, and 632 nm lights. The enhanced DOF of visible Bessel-like beams generated by fiber microaxicons with cone angle $\theta$ of 140° could improve the propagation distance underwater, which has been simulated to solve the problem of the propagation distance of 1000 m for underwater optical communication. Especially, the Bessel-like beams with blue–green lights could have better focal spots than that of the red light, due to their Bessel-like beams with lower optical absorption and scattering and small divergence angle.