Robust propagation of a steady optical beam through turbulence with extended depth of focus based on spatial light modulator

Finding appropriate strategies to increase the robustness through turbulence with extended depth of focus (DOF) is a common requirement in developing high-resolution imaging through air or water media. However, conventional lenses with a specially designed structure require high manufacturing costs and are limited by a lack of dynamic modulation characteristics. Spatial light modulators (SLMs) are unique flat-panel optical devices which can overcome the distance limitation of beam propagation for the dynamic modulation property. In this work, we address the dynamic generation of a steady optical beam (STOB) based on the mechanism of transverse wave vector elimination. STOBs generated by the SLM have significant advantages over Gaussian beams for the characteristics of peak intensity, robust propagation, extended-DOF beam profile, and dynamic wavefront modulation over a long distance under strong turbulent media. Our versatile, extensible, and flexible method has promising application scenarios for the realization of turbulence-resistant circumstances.


Introduction
Light field variation in a refractive index introduces aberrations as the beam passes through the turbulence. These aberrations greatly reduce the resolution and depth of focus (DOF) of the images [1][2][3]. The prospect of achieving a steady optical beam (STOB) that can robustly propagate in turbulence is a long-term goal [4,5]. Gaussian beams, generated by the laser, are limited by diffraction effects. Bessel beams are generated in terms of anti-diffracting optical wave packets [6,7]. Airy beams receive a significant amount of attention due to their unique representation in the class of non-diffracting optical waves [8][9][10]. Structuring light with a custom light field is highly desirable and of great value for various applications, such as Hermite-Gauss beams [11][12][13], Laguerre-Gaussian beams [14][15][16], annular beams [17][18][19], and optical vortex beams [20][21][22][23]. In this regard, a variety of methods have been proposed to overcome the effect of diffraction and turbulence to maintain robust propagation, such as nonlinear solitons [24,25] and adaptive optics (AO) [26][27][28]. The nonlinear soliton method utilizes the nonlinear self-trapping effect to resist beam broadening, which is not applicable to free-space propagation due to the requirements of the nonlinear medium. The AO method is well-known for the correction of wavefront distortion with feedback. However, the complexity of the AO system, along with its extraordinarily high cost, limits its applications. In 2019, optical pin beams (OPBs) with stable wavefronts against diffraction and ambient turbulence during free-space long-distance propagation were proposed [29,30]. The OPBs exhibited extended diffraction-free propagation in free space, as well as a constant peak intensity along the propagation distance. However, the OPBs were proposed only for the specific case of a radially symmetric Airy phase and lack dynamic modulation characteristics.
The study of structured light [31][32][33], beam shaping techniques [34][35][36][37] and metalenses [38,39] to conquer the limits of optical systems are enthusiastic research regimes. However, conventional lenses with a specially designed structure have high manufacturing costs and are limited by a lack of dynamic modulation characteristics. In engineering, the scope of applications with a single lens is limited, and it is difficult to replace the lens for different cases. Finding solutions to overcome the distance limitation of beam propagation requires significant engineering technology. Spatial light modulators (SLMs), which originated in the 1970s, are indispensable dynamic flat-panel optical devices employed to dynamically manipulate the amplitude, phase, and polarization state of beams. After several decades of device innovation, heavy investment in advanced phase calibration technologies [40,41], as well as extensive material research and development, SLMs have come to the fore as a fundamentally important tool in the field of modern optics [42,43]. In particular, SLM technology has been regarded as a versatile tool for generating arbitrary optical fields and tailoring all degrees of freedom beyond just the phase and amplitude. Facilitated by their ease of use and real-time light manipulation in space and time, SLMs can replace conventional optical elements with a digital equivalent with better dynamic modulation properties.
In this work, we address the dynamic generation of a STOB by superimposing multiple ring structure beams based on the mechanism of transverse wave vector elimination. The ring structure beams are realized by splitting the wavefront of the input Gaussian beam into a collection of ring structure wavefronts. Each concentric ring beam has a focus at a different position along the optical axis. When air or water turbulence introduces a random wave vector in a STOB, the random wave vector can be canceled out continuously at different distances by the proposed multiple concentric ring structure. Moreover, a dynamic flat-panel optical device SLM is employed to realize the dynamic modulation of the STOB. The SLM can overcome the distance limitation of beam propagation in the dynamic modulation property, which provides a new solution for beam propagation in a longer turbulence media and further extends the DOF. In this regard, increasing the flexibility of the beam profile can lead to improved performance in turbulent media. The turbulence effect can be reduced to maintain the intensity distribution in the STOB. We extend the analysis by investigating the properties of STOBs with different numbers of rings and focal length of the innermost ring structure for the first time. We precisely measure the focal length under different parameters and analyze the varying trends of DOF. By refreshing the holograms, long Rayleigh length STOBs with focuses at different distances and variable DOF are achieved. The propagation characteristics of STOBs are systematically studied. We theoretically and experimentally demonstrate that the STOB has superior properties for robust propagation in turbulence with small wavefront distortion and aberration. The proposed STOB has significant advantages over Gaussian beam in terms of peak intensity, robust propagation, extended-DOF beam profile, and dynamic wavefront modulation characteristics over a long distance under strong turbulent conditions. Our analyses provide a more general and complete study of the propagation of STOB in a turbulent media.

Theory
Split the wavefront of the Gaussian beam into a collection of ring structure wavefronts, where each concentric ring beam has a focus at a different position along the optical axis. Suppose that the focal length of each ring beam increases with the radius. The inner beam converges and diverges first, while the outer beam is still in the process of converging. The two beams mix and interfere with each other during propagation. For the cross-section of the beam, the optical field with two ring beams can be expressed as where A 1 and A 2 represent the amplitude, k 1 and k 2 represent the radial wave vector, and r represents the radial length. Assuming that the amplitude of the optical field of the two beams is the same, A 1 = A 2 = A, and the radial wave vector of the optical field of the two beams is the same, k 1 = −k 2 = k, equation (1) can be expressed as Since the radial wave vector of the composite optical field is eliminated, the beam intensity remains stable during propagation. The optical field of a STOB with two ring beams can be expressed as The above derivation only contains two ring beams. When multiple concentric ring beams combine together, the robust propagation length of the STOB can be extended to achieve the spatial behavior of an extended DOF. The produced STOB can be summarized as follows. The focal length of the outer ring beam is a Rayleigh distance r 0 longer than the inner one, which ensures that the light intensity in the central zone of the STOB remains at a high level. In addition, the cross-section area of each ring is equal to ensure that each ring beam contains the same energy. If the number of concentric ring beams is n, the beam cross-section optical field can be expressed as where A 1 , A 2 , . . ., A n represent the amplitude, k 1 , k 2 , . . ., k n represent the radial wave vector, ∆r 2 , ∆r 3 , . . ., ∆r n−1 represents the lateral bias between different ring beams. ∆r n−1 can be expressed as where k n represents the radial wave vector of the nth ring beam. Supposing all beams have the same amplitude A and radial wave vector k, equation (4) can be expressed as Since the phase factor does not have a radial component, its shape can be maintained during propagation. By refreshing the uploaded holograms in the SLM, a dynamic modulation of the STOB that focuses at different distances with variable DOF is realized. The optical field of the STOB with n ring beams can be expressed as The side view of the STOB is shown in figure 1(a). The number of rings is set to 6 with Gaussian intensity distribution. Each ring carries different convergence phases, as shown in figure 1(b). The converging and diverging ring beams are mixed near the optical axis with opposite phases, interfering with each other. The beams form a stable intensity distribution during propagation z 2 − z 5 , as shown in figures 1(c)-(f), which are at 100 times magnification to demonstrate the transverse beam profiles. Almost all of the energy is concentrated at the center of the beams, forming an extended DOF. Figure 1(g) is the output beam after interference, which is a ring shape with uniform intensity distribution. By modulating the parameters of the rings, an extended DOF is realized, as shown in figure 1(h).

Experimental apparatus
The experimental apparatus employed for the generation and verification of a STOB is illustrated in figure 2. BS and PBS refer to beam splitter and polarization beam splitter, respectively. Lens 2 and Lens 3 are a 4-f system. For an experimental demonstration of the proposed STOB, a phase-only SLM (Holoeye GAEA-2, 8-bit gray phase levels, 3840 × 2160 pixels, 3.74 µm × 3.74 µm pixel area) is employed to modulate the phase of a CW fiber laser operating at 532 nm with an output power of about 100 mW. A grating is uploaded with the designed holograms to eliminate the effect of the zeroth-order diffraction. Two CCDs (QHY163M, 4656 × 3522 pixels, 3.8 µm × 3.8 µm pixel area) are exploited to record the beam profile of the STOB and Gaussian beam, respectively. A water tank (35 × 21 × 25 cm 3 ) is adopted to verify the robust propagation characteristics of STOBs with the extended DOF through water turbulence. The position of the water tank can be moved back and forth arbitrarily to simulate the turbulence at different distances.

Extended DOF
A general and complete analysis of STOBs is performed for the first time by uploading two categories of holograms to realize the dynamic modulation of STOBs, which provides an appropriate strategy to realize focusing at different distances with variable DOF. Each hologram corresponds to one parameter of the rings in the STOB. To provide a general and complete study of STOBs, the number of rings and the focal length of the innermost beam are changed. The beam profiles and intensity distributions of all STOBs are thoroughly measured. In addition, the DOF in all STOBs is measured, and the varying trends of the DOF are analyzed when changing the number of rings and focal length.
On the one hand, the number of rings is modulated while maintaining the focal length of the innermost beam at 0.3 m. The holograms, measured beam profiles, and relative intensity distributions are shown in figure 3. The beam waist is about 50 µm at peak intensity. With the increase in the number of rings, as shown in figures 3(a)-(c), the effect of diffraction is well restrained, especially when the number of rings is greater than five. However, when the number of rings is too large, as shown in figures 3(g)-(l), surrounding side petals appear for the excessively small ring structure designs. The DOF extension trends with the increase in the number of rings are shown in figures 5(b) and (d). The DOF increases rapidly at first, then gradually slows down and tends to be a constant. Thus, when the focal length of the innermost beam is 30 cm, the optimum number of rings is seven to nine, as shown in figures 3(d)-(f). The measured DOF at different numbers of rings is shown in table 1. By changing the number of rings, a dynamic modulation of the STOB that shows variable DOF is realized.
On the other hand, the focal length of the innermost beam is modulated while maintaining the number of rings at nine. The holograms, measured beam profiles, and relative intensity distributions are shown in figure 4. The beam waist is about 50 µm at peak intensity. With the increase in the focal length of the innermost beam, as shown in figures 4(a)-(d), the beam converges well and the effect of diffraction is well restrained, especially when the focal length of the innermost beam is approximately 0.4 m. However, when the focal length of the innermost beam is too long, as shown in figures 4(j)-(l), the convergence has a negative affect on the loss of energy during propagation. The DOF extends slowly and remains almost unchanged with the increase in the focal length of the innermost beam, as shown in figures 5(e) and (i). Thus, when the number of rings is nine, the optimum focal length of the innermost beam is 0.3 m to 0.5 m, as shown in figures 4(j)-(l). The measured DOF at different focal lengths of the innermost beam are presented in table 1 . By changing the focal length of the innermost beam, dynamic modulation of athe STOB with a focus at different distances is achieved.
By combining two dynamic modulation methods and refreshing the uploaded holograms for the first time, dynamic modulation of STOBs can be achieved with the property of peak intensity and focusing at different distances with variable DOF. Compared to a conventional lens, the flexibility of SLMs allow for the dynamic modulation of STOBs under different experimental conditions. This point is crucial in the apparatus, which makes STOB a particular property of displaying extended DOFs with a constant or low-varying beam profile. Compare this with Bessel beams generated by an axicon, whose phase distribution  is linear in the radial direction and whose energy is approximately unchanged during transmission. The phase of an STOB is nonlinear in the radial direction, which is a new type of structured light. By adjusting this nonlinear phase distribution, we can generate STOBs with different focal distances at arbitrary locations.

Robust propagation test
So far, the discussion has been restricted to the demonstration of a dynamically extended DOF that performs a constant or low-varying beam profile along the optical path. Another fundamental aspect that requires  investigation is the preservation of the spatial features after interaction with turbulent media. Turbulence can significantly affect the propagation of light beams in various ways. As light travels through a turbulent medium, it encounters random fluctuations in the refractive index of the medium, causing phase distortions and intensity fluctuations. The effect of turbulence on a propagating light beam can lead to a loss of coherence, resulting in beam spreading, distortion, and reduced intensity at the receiver. To perform the spatial analysis, we explore the robust propagation properties of the STOB compared to the Gaussian beam over long distances under strong water turbulent conditions. A silica water tank is placed in the optical path, as shown in figure 2. If we generate the STOB with a conventional lens, the DOF is rather limited, and the water tank is relatively short. Realizing stable beam propagation in a long water tank with a fixed optical element is costly. However, due to the dynamic modulation property of the SLM, by changing the parameters of the uploaded holograms, the position of the water tank can be moved back and forth arbitrarily. Thus, the water tank could be longer than that when using a fixed optical element. The distance limitation of stable beam propagation can be overcome. In engineering, transformability is crucial since it is more convenient than constantly changing lenses for different cases.
The simulation of strong turbulence in the presence of temperature gradients and rapid disturbances in the environment is crucial in understanding the effects of turbulence on the propagation of light beams. To simulate the temperature gradient, hot water at a temperature of 203 degrees Fahrenheit is added, which alters the refractive index. The refractive index is a fundamental optical property that determines how much light is refracted when it passes through a material. By pouring hot water into the water tank, a temperature gradient is created, and the refractive index of the water changes accordingly. Furthermore, to simulate fast disturbance of the environment, the water is stirred quickly, inducing turbulence in the water. This turbulence creates random variations in the refractive index, which can cause light beams to scatter and distort. The combination of a strong temperature gradient and a fast disturbance are intended to mimic the conditions that cause turbulence in the propagation of light beams. By simulating these conditions, we can study and understand the effects of turbulence on the propagation of light beams. We record the variations of the beam profiles, as shown in figure 6(a), in which left indicates the Gaussian beam and the right indicates the STOB. The Gaussian beam and STOB are disturbed by strong turbulence simultaneously. Experiments investigating the properties of robust propagation show that both beam profiles are deformed first. However, the beam structure of the STOB is quickly restored and the central beam profile of the STOB remains at peak intensity compared to that of the Gaussian beam, which shows an evolutionary trend similar to the prediction.
The degree of the intensity variation across the beam profile with time is adopted as a criterion for identifying the robustness since it provides a clear identification of the spatial variances. The degree of intensity variation across the beam profile with time is defined as The results indicate the intensity variation with time of the STOB is obviously smaller compared with the Gaussian beam, as shown in figure 6(b). The blue line indicates the STOB and the red line indicates the Gaussian beam. The magnified beam profile of STOB variation with time is shown above the graph. It can be obtained from the results that in the regime of spatial behavior, STOBs are much more localized in space and have much less fluctuation in time compared with that of the Gaussian beams.
The beam profiles and the degree of intensity variation with time provide an indication that the proposed theory in equation (7) is a stable light field. The interference between ring beams generates a stable light field and is maintained along the optical path of the wave field during propagation, which makes the STOB an excellent candidate for improving transmission in turbulent media. This cannot happen with a Gaussian beam since there is no spatial compensation to begin with.

Conclusion
To conclude, with a flexible apparatus, a thorough analysis of how to generate STOBs with the characteristics of robust propagation and extended DOF by the SLM is performed for the first time. The SLM can overcome the distance limitation of stable beam propagation, which conquers the limits of a conventional lens. Increasing the flexibility of the beam profile through SLM can lead to improved performance in turbulent media. More specifically, the STOB is generated by superimposing multiple ring structure beams based on the mechanism of transverse wave vector elimination. When air or water turbulence introduces a random wave vector, the influence can be canceled out continuously to maintain the intensity distribution of the STOB. Notably, the DOF increases with the number of rings and focal length of the innermost beam. By combining two dynamic modulation methods of changing the number of rings and focal length of the innermost beam with the SLM, the dynamic modulation property of STOB is realized, which guarantees the extended DOF with a constant or low-varying beam profile. The robust propagation tests indicate that the intensity variation with time of STOBs is small and the central beam profile of STOBs remain at peak intensity, which makes STOBs an excellent candidate for improving the transmission in turbulent media. The STOB has significant advantages with the features of peak intensity, robust propagation, extended-DOF beam profile, and dynamic wavefront modulation characteristics over a long distance under strong turbulent conditions. This versatile, extensible, and flexible method has promising application scenarios for the realization of turbulence-resistant circumstances.
Furthermore, the results of the study provide valuable insights into the behavior of STOBs undergoing turbulence processes and the transmission of information through STOBs, including investigating the influence of different turbulent media and further expanding the region of turbulence. The results can also inform the development of new strategies and technologies to mitigate the effects of turbulence in efficient and reliable optical communication systems.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).