Propagation of Laguerre-Gaussian beam intensities through optically thick turbid media | Scientific Reports

Jun 26, 2025

Scientific Reports volume 15, Article number: 19768 (2025) Cite this article

841 Accesses

Metrics details

Recent research has increasingly focused on Laguerre-Gaussian (LG) beams carrying a topological charge (l) due to their demonstrated ability to enhance signal transmission through turbid media. This promising phenomenon has spurred considerable interest in understanding the complex propagation dynamics of such beams, with the goal of optimizing optical systems for highly scattering environments. In this study, we experimentally investigate the transmission characteristics of LG beams with various topological charges as they propagate through a turbid multiple-scattering medium at different depths. By analyzing the resulting transmission data, we calculate the scattering mean free path (\(\:{{\Lambda\:}}^{*}\)) for each LG mode and show that \(\:{{\Lambda\:}}^{*}\) increases with increasing values of l. Moreover, we show that beams with higher topological charges traverse greater distances within diffuse media while retaining their characteristic donut shape using Monte Carlo simulations. Our findings provide an empirical characterization of the relationship between topological charge and scattering mean free path and also explain the improved transmission with increasing l, at least until optical depths of ~ 40 \(\:{{\Lambda\:}}^{*}\), highlighting potential trends that warrant further investigation regarding the potential of applying Laguerre-Gaussian beams for enhanced optical imaging and spectroscopy through highly scattering media.

Optical spectroscopy and imaging have been central in revealing turbid media’s optical properties and detecting anomalies embedded within them due to their high spatial resolution, molecular specificity, and noninvasive nature1,2,3,4. Due to this, visible and near-infrared (NIR) optical spectroscopy and imaging have expanded their biomedical applications in recent years, from measuring blood oxygen saturation5,6 to three-dimensional diffuse optical imaging7,8. However, its effectiveness is limited regarding imaging depth, which means that anomalies located deep within the turbid media cannot be detected, as the imaging system cannot penetrate the medium to access the object9,10. The primary factor contributing to this limited imaging depth in biomedical optical imaging and sensing is the phenomenon of optical scattering11,12. The scattering of light in biological tissues, caused by photon interactions with various cellular and subcellular structures, leads to the emergence of a fundamental parameter known as the scattering coefficient13.

Defined as the probability of photons undergoing scattering per unit length traveled within a medium. The scattering coefficient represents the inherent scattering properties of the medium and is often denoted by (\(\:{\mu\:}_{s}\))14. It determines whether the propagating photons exhibit a ballistic or diffuse behavior, depending on the magnitude of the scattering coefficient relative to the absorption coefficient (\(\:{\mu\:}_{a}\)), which determines the amount of photon absorption per unit length traveled in the medium15. Photons primarily undergo ballistic propagation when the scattering coefficient is significantly smaller than the absorption coefficient. Conversely, photons exhibit diffuse propagation when the scattering coefficient is comparable to or larger than the absorption coefficient2. The media in which such propagation occurs is called diffuse or turbid media, and the light transmission provides minimal or no direct information for such media. The extent to which light can penetrate a given medium has profound implications for the utility of optical techniques across diverse domains. An increased penetration depth enables the analysis and imaging of deeper layers within samples, thereby enhancing the versatility and applicability of optical methods. As such, recent research efforts have focused on developing innovative technologies and methodologies to extend the penetration depth of optical techniques. These include tailoring the spatial coherence properties of light beams to reduce scattering-induced degradation16, using structured light fields such as vortex or vector beams17,18, adaptive optics for real-time wavefront correction15, and computational imaging methods that exploit scattered light to reconstruct deeper-lying structures19.

One such development is using orbital angular momentum (OAM) of light to image objects through or even behind scattered media. Recently, experiments using beams that contain orbital angular momentum of light have shown a significant increase in the percentage transmission compared to the traditional Gaussian beams20,21. Experiments performed in biological tissues of various thicknesses by Alfano et al.22 and Biton et al.18 clearly showed a reduction in the transition from ballistic to diffuse regime when higher OAM modes (or topological charge) of Laguerre-Gaussian beams (LG) are used compared to Gaussian beams (GB). They argued that the enhanced transmission of higher-order LG beams in scattering media may stem from their unique spatial phase structure, which reduces overlap with typical scattering phase functions and promotes more forward-directed or less-diffused propagation. In particular, the central phase singularity and ring-shaped intensity profile may reduce interactions with scattering centers, allowing a larger fraction of forward-scattered (or snake) photons to remain within the detection aperture. Li et al.23 performed experiments by propagating Ince-gaussian beams (IG) through scattering media and showed that IG beams have greater resilience than scalar Gaussian beams. More recently, Mamani et al.24 studied OAM transmission through mouse brain tissue and showed increased light transmission when higher modes of OAM beams are used.

All the studies above show that beams containing OAM possess a distinct advantage in transmission through multi-scattering media. As a result, these beams have the potential to penetrate deeper into scattering media with higher optical depths, making them promising candidates for various applications requiring efficient light propagation in complex scattering environments3,9,25. However, the utilization of OAM beams for imaging and sensing through multi-scattering media is not without its limitations. The exact reasons behind the enhanced transmission of OAM beams through scattering media remain unconvincingly explained. Additionally, it should be noted that all the transmission experiments conducted thus far have been limited to thin samples, typically up to a few millimeters in thickness.

In this study, we propagate Laguerre-gaussian beams through melamine foam of various thicknesses (1.250 cm, 2.500 cm, and 3.750 cm) to determine the relationship between the topological charge (l) of the LG beam modes and the scattering mean free path (\(\:{{\Lambda\:}}^{*})\) of the LG modes propagating through the sample. In contrast to previous methods, our approach takes into account theoretical considerations to calculate \(\:{{\Lambda\:}}^{*}\)26. Moreover, the optical depth of the sample used in our experiment is significantly thick, extending to at least ~ 40 transport mean free paths. Furthermore, we perform proof-of-concept Monte Carlo simulations to show that LG beams with higher topological charges traverse greater distances within diffuse media while retaining their characteristic donut shape, confirming their robust propagation capabilities even in highly scattering environments. The proposed approach provides empirical observations on the propagation of LG beams through scattering media, revealing trends that may be useful for various applications in optical biomedical imaging, spectroscopy26, and optical communications25,27,28.

This study’s research methodology and progression are detailed as follows: Sect. “Diffuse and ballistic transmission of Laguerre-Gaussian beams through turbid media" provides essential background on LG beams and presents our experimental findings concerning their behavior when transmitted through melamine foam. In Sect. “Results and discussion”, we calculate the scattering mean free path for each LG beam mode, allowing us to establish a clear link between the topological charge of LG beams and the scattering mean free path. We also delve into a comprehensive discussion of our results, highlighting their implications. Finally, Sect. “Conclusions” summarizes our findings and concludes our study.

This section analyzes the diffuse and ballistic transmission of different LG beam modes as they propagate through various thicknesses of melamine foam. To provide a foundation for our experimental work on transmission characteristics, we first delve into the fundamentals of Laguerre-Gaussian beams in Sect. “Laguerre-Gaussian beams”.

Laguerre-Gaussian (LG) beams are optical beams with a unique spatial intensity distribution and carry orbital angular momentum (OAM). These beams are characterized by their azimuthal phase dependence, given by the Laguerre polynomial, and exhibit a helical wavefront with a dark central region. Due to their intriguing properties, LG beams have garnered significant attention in various research fields, including biomedical optics and optical wireless communication. The mathematical expression given by29:

Equation 1 describes the electric field (\(\:{E}_{\left(p,l\right)}^{LG}\)) of a Laguerre–Gaussian (LG) beam in cylindrical coordinates \(\:\left(r,\varphi\:,z\right)\), where the beam is characterized by two mode indices: the radial index \(\:p\), and the topological (or azimuthal) index \(\:l\). Here, \(\:w\left(z\right)\) denotes the beam waist with propagation distance \(\:z\).The term \(\:{exp}\left[i\left(2p+\left|l\right|+1\right)\varphi\:\left(z\right)\right]\) represents the Gouy phase shift, which is an additional phase acquired as the beam propagates through its focus. The factor \(\:{\left(\frac{\sqrt{2}r}{w\left(z\right)}\right)}^{\left|l\right|}\) coupled with the associated Laguerre polynomial \(\:{L}_{p}^{\left|l\right|}\left(\frac{2{r}^{2}}{w{\left(z\right)}^{2}}\right)\) determines the beam’s radial intensity profile and introduces a ring-like structure when \(\:l\ne\:0\). The term ‘\(\:\text{exp}\left(il\varphi\:\right)\)’ imparts the helical phase structure responsible for the beam’s orbital angular momentum.

Unlike Gaussian beams, LG beams maintain their spatial integrity even when traversing complex and disordered media, such as biological tissues or atmospheric fog. As a result, LG beams have shown promise in applications that involve light propagation through scattering media, such as imaging and sensing within biological tissues2,18,24,25. Another promising application of LG beams lies in optical wireless communication (OWC)20,30,31. The LG beams’ resistance to scattering enables them to maintain their spatial profile over longer distances, making them well-suited for free-space optical communication through turbulent atmospheres. By exploiting LG beams’ OAM modes for multiplexing and demultiplexing data, optical wireless communication can achieve higher data-carrying capacity and enhance the security of the communication link27.

However, It is essential to consider that when OAM beams enter diffuse media, scattering photons interfere with the propagation paths of the photons from various modes. Increasing the width of the media or even the scattering phase and amplitude function of the scatterer reduces the number of ballistic and snake photons that reach the beam profiler, which is essential for a beam to hold its shape. At some point, the output pattern no longer resembles the original mode25.

Our experiment comprises two separate set-ups to measure diffuse and ballistic transmission. The optical set-ups and the transmission characteristics of LG beam modes as they propagate through melamine foam are detailed in the next section.

For any particular sample thickness “d,” the diffuse transmission of any input LG beam with topological charge “l” is given by32:

Here, \(\:{T}_{d}\) is the diffuse transmission, which is computed as the ratio of transmitted optical power to the incident power recorded without the sample in place. g is the scattering anisotropy factor. \(\:l\), \(\:\:d,\) \(\:{\mu\:}_{s}\), and \(\:\:{\mu\:}_{a}\) are the topological charge, sample thickness, scattering coefficient, and absorption coefficient, respectively. Equation (2) is obtained from the diffuse approximation and does not include any dependence on the topological charge. However, Eq. 2 will overestimate the transmission for thick samples as it originates from the classical diffusion approximation, which is known to become less accurate for optically thick or highly anisotropic scattering media. Specifically, it overestimates the transmission by assuming idealized boundary conditions and neglecting higher-order scattering corrections. To address this limitation and better align the model with experimental observations, an empirical correction term \(\left\langle E \right\rangle\) is introduced. This term accounts for systematic deviations between measured and predicted transmission values at larger sample thicknesses and higher topological charges. In the context of our work, \(\left\langle E \right\rangle\) is not derived from first principles but is treated as a phenomenological adjustment. Its inclusion enables a more accurate fit to the measured transmission data across different beam modes and thicknesses. It reflects the accumulated scattering effects that are otherwise unaccounted for in the base diffusion equation. Similar correction approaches have been adopted in prior studies under analogous conditions32. Therefore, for practical reasons, Eq. 2 now takes the form32:

(a) Experimental set-up used to determine the diffuse transmission of the first ten positive and negative modes of LG beams propagating through various thicknesses of melamine foam. (b) Photograph of the experimental set-up from the SLM onwards. The propagation is from left to right. In this Figure, PH represents a pinhole, SLM is the spatial light modulator, L represents a lens, and PD is the photodetector.

The first set-up, depicted in Fig. 1, is employed to measure the diffuse transmission. To serve as the light source, we utilize a continuous wave (CW) Helium-Neon (He-Ne) laser (Melles-Griot 05-LHP-123–496) with a wavelength of 633.2 nm. The beam emanating from the laser possesses an average power of 5 mW, and the spot size of the laser beam is 1.36 mm. After passing through a polarizer, the beam is directed toward the Spatial light modulator’s (Jasper LCoS) center. The Spatial light modulator (SLM) has a resolution of 1920 × 1080, with each pixel measuring 6.45 μm, and operates at a frame rate of 60 Hz. It displays the fork pattern that imparts the desired topological charge to the input beam and converts the Gaussian beam into a Laguerre-Gaussian beam of topological charge “l.”

The reflected LG beam from the SLM, which now contains topological charge, is passed through a 4f set-up consisting of two lenses of focal length = 100 mm and a pinhole iris. The 4f set-up is used to clean the LG beam of any diffraction orders that arise due to the SLM’s response to the input beam and to collimate the beam. The LG beam is then incident normally on melamine foam samples of varying thicknesses. Following propagation through the melamine foam, the transmitted light is collected using a photodetector (PD) (Ophir Optronics PD300-UV SENSOR), which is in turn connected to a power meter (Ophir Optronics Centauri Dual Channel). The PD is placed immediately behind the melamine foam sample to collect the diffuse transmission (as shown in Fig. 1).

Graphs showing the change in the log of the diffuse transmission with respect to the changes in the topological charges and sample thickness. (a–c) show the transmission for 1.25 cm, 2.50 cm, and 3.75 cm thicknesses, respectively.

Figure 2 shows the diffuse transmission of the first ten positive and negative modes of the LG beam for 1.25 cm (Fig. 2a), 2.50 cm (Fig. 2b), and 3.75 cm (Fig. 2c) thickness of melamine foam. It can be seen from the Figure that the diffuse transmission of the optical beam through the melamine foam increases with increasing topological charge (l). As expected for an isotropic and achiral medium such as melamine foam, the sign of the topological charge does not influence the diffuse transmission. The transmission depends solely on the absolute value of l. The thickness of the sample also affects the transmission as the diffuse transmission of any particular beam through a diffuse media of thickness “d” is proportional to \(\:{e}^{-d}\).

Following the measurement of diffuse transmission, we modify the set-up shown in Fig. 1 to separate and estimate only the ballistic transmission. Likewise, the ballistic transmission of the input LG beam through the same sample is given by33:

The set-up to measure the ballistic transmission is shown in Fig. 3. The experimental set-up is similar to the one used to measure diffuse transmission; however, the PD is placed behind another 4f set-up with f = 50 mm to measure the ballistic transmission (Fig. 3). A pinhole iris is placed immediately after the melamine foam sample to block any diffuse or snake photons transmitted through the medium.

(a) Experimental set-up used to determine the ballistic transmission of the first ten positive and negative modes of LG beams propagating through various thicknesses of melamine foam. (b) Photograph of the experimental set-up from the SLM onwards. The propagation is from left to right. In this Figure, PH represents a pinhole, SLM is the spatial light modulator, L is a lens, CL is the 4f collimating lens set-up, and PD is the photodetector. PH3 is placed immediately after the melamine foam to spatially filter out diffuse and snake photons, allowing only ballistic light traveling in the original beam direction to pass through.

Figure 4 displays the ballistic transmission of both positive and negative modes of the LG beam. It illustrates the data for three different melamine foam thicknesses: 1.25 cm (Fig. 4a), 2.50 cm (Fig. 4b), and 3.75 cm (Fig. 4c), respectively. The ballistic transmission of the optical beam through melamine foam increases with a higher topological charge (l), just like the diffuse transmission. Additionally, the thickness of the melamine foam sample also plays a role in the ballistic transmission, following a proportionality of \(\:{e}^{-d}\) as it passes through a diffuse media of thickness “d.” Consistent with the behavior observed in diffuse transmission, the sign of the topological charge does not affect ballistic transmission through the melamine foam. This symmetry is expected, given the medium’s lack of chirality or directional scattering bias.

Graphs showing the change in the log of the ballistic transmission with respect to the changes in the topological charges and sample thickness. (a–c) show the transmission for 1.25 cm, 2.50 cm, and 3.75 cm thicknesses, respectively.

Furthermore, we note that the transmission curves in Figs. 2 and 4 display an exponential increase as the topological charge increases. This trend is reminiscent of a generalized Beer’s Law of the form \(T\sim e^{{{\raise0.7ex\hbox{${ - \alpha \cdot d}$} \!\mathord{\left/ {\vphantom {{ - \alpha \cdot d} {\left| l \right|}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\left| l \right|}$}}}}\), where \(\:\alpha\:\) could reflect the scattering properties of the medium. While such a relation has been suggested in earlier works2,18,22, our present study is focused on determining the scattering mean free path using separate ballistic and diffuse measurements to assess mode-dependent propagation. Therefore, we do not fit this model to the data here, as it falls outside the scope of our analysis.

Following the determination of the ballistic and diffuse transmission of LG beams through various thicknesses of melamine foam, we use the transmission values for each mode of LG beam to calculate their scattering mean free path in the melamine foam sample.

The scattering mean free path (\(\:{{\Lambda\:}}^{\text{*}}\)) of light beams propagating in thick turbid media can be determined using various direct measurement methods33,34,−35. Our approach is to utilize both the ballistic and diffuse transmittance measurements. The motivation for using this approach arises from the need for accuracy and experimental reliability. Independent measurements of the transmitted light’s ballistic and diffuse components help capture the LG beam’s behavior in different propagation regimes and reduce sensitivity to systematic errors or alignment variations. This approach becomes especially important at large sample thicknesses where variations in the scattering mean free path are small and more affected by scattering noise. Measuring both components provides greater resilience to such uncertainties and allows us to investigate how the mean free path depends on the topological charge. For this reason, we do not consider conventional single-path transmittance methods, as they do not offer the same level of precision or insight for the goals of this study32.

The \(\:{{\Lambda\:}}^{*}\) is calculated for each mode of the LG beam used in this section using the diffuse and ballistic transmission measurements (Eqs. 3 and 4). For any particular thickness d, the \(\:{{\Lambda\:}}^{*}\) for any mode l of the LG beam is given by32:

Here, κ is the ratio of the diffuse transmission (\(\:{T}_{d}\)) to the ballistic transmission (\(\:{T}_{b}\)) and is given by:

The \(\:{{\Lambda\:}}^{*}\) values for each mode of the LG beam are shown in Fig. 5. The results indicate that as the topological charge increases, the \(\:{{\Lambda\:}}^{*}\) also increases. To capture this relationship quantitatively, a sigmoidal (logistic) fit was applied using the equation:

Here, \(\:f\left(x\right)\) is the scattering mean free path (\(\:{\varLambda\:}^{*}\left(l,\:d\right)\)), \(\:x\) is the topological charge \(\:\left(l\right)\), the parameter \(\:a(=1.1805)\) represents the upper asymptote, indicating the maximum value that the curve approaches, while the parameter \(\:b(=0.2863)\) governs the steepness of the transition, and \(\:c(=-5.2806)\) marks the inflection point where the rate of increase is greatest.

Calculated \(\:{{\Lambda\:}}^{*}\) for the first ten positive and negative modes of Laguerre-Gaussian beams used in our experiment (green circles). Each point (green circle) in this graph is the average of the \(\:{{\Lambda\:}}^{*}\) calculated for the three propagation distances (1.25 cm, 2.50 cm, and 3.75 cm). The black curve indicates the sigmoidal fit, and the green dotted curves show the prediction bounds (99%) of the sigmoidal fit.

Deductions from the logistic fit in Fig. 5 reveal that the scattering mean free path exhibits a sigmoidal dependence on the topological charge. At lower topological charges, \(\:{\varLambda\:}^{*}\) increases rapidly, indicating that the beam’s resilience to scattering improves significantly in this regime. The change rate gradually diminishes as the topological charge increases and the curve approaches saturation. This behavior suggests a diminishing return in propagation enhancement for very high-order LG modes. While higher-order modes consistently provide improved propagation characteristics, the gains taper off at larger values.

These results provide a quantitative description of how LG beams with topological charges propagate more effectively through turbid media. However, further theoretical studies are needed to explain the mechanisms behind this saturation behavior and better understand structured light’s interaction with strongly scattering media.

To further enhance our understanding of LG beam propagation through turbid media, a Monte Carlo simulation is performed to check the shape-preserving properties of the LG beam. In this simulation, we model the multiple scattering events that photons experience as they travel through a highly scattering medium. By statistically tracking the trajectories of a large number of photons, the simulation replicates the complex interaction dynamics inherent in turbid environments. This approach enables us to assess whether the LG beam’s distinctive spatial intensity profile, a characteristic donut shape, is maintained despite extensive scattering. While this Monte Carlo approach does not account for wave phenomena such as diffraction or interference, it remains computationally efficient and well-suited for modeling photon transport and intensity redistribution in highly scattering environments.

The simulations are performed using a cuboid volume measuring 40 × 40 × 50 \(\:{mm}^{3}\) (height × width × thickness), as depicted in Fig. 6. The volumetric mesh is generated using a MATLAB-based meshing tool called iso2mesh36. Absorption and scattering coefficients of 0.01 \(\:{mm}^{-1}\) and 1.242 \(\:{mm}^{-1}\) are used for this simulation These parameters were derived experimentally by fitting experimental transmission measurements to the methods described in the following references32,33.

(a) Shows the mesh used to simulate the propagation of LG beams through turbid media. (b) Original samples used in the experiment for comparison. The perceived thickness of the samples may appear similar due to the camera angle. However, the actual physical thicknesses are 1.25 cm, 2.50 cm, and 3.75 cm, respectively, as used in the experiments. (c) Examples of LG beam intensities used for the simulation.

A continuous wave LG light source operating at a wavelength of 639 nm and a spot size of radius 0.5 mm is employed to illuminate the sample, with flux calculated along the propagation direction. The light transport simulations were executed using MCX37,38, and 1 billion photons are used for each simulation. The qualitative and quantitative results of the Monte Carlo simulations are in Figs. 7 and 8, respectively. Figure 7 shows the qualitative representation of the spatial evolution of Laguerre-Gaussian beam intensity profiles as they propagate through a scattering medium. It reveals how the characteristic donut shape is retained at varying depths depending on the topological charge. Concurrently, Fig. 8 presents a quantitative analysis that correlates the topological charge with the depth at which the beam loses its donut shape.

Qualitative results of the Monte Carlo simulation. (a–c) show the simulation results for LG beams with topological charges of 1, 5, and 10, respectively. I) Shows the fluence of the different LG beams as they propagate along the depth of the sample (sliced along half the width). II), IV), and V) show the two-dimensional intensity profiles of the LG beams at varying depths. III) Shows the LG beam intensity profiles at the depth at which the beam loses its characteristic donut shape and resembles a Gaussian distribution.

The results indicate a clear dependence of the propagation characteristics of Laguerre-Gaussian beams on their topological charge in scattering media. Specifically, beams with a lower topological charge exhibit a relatively early breakdown of their ring-shaped intensity profile, whereas beams with a higher topological charge. The observed dependence of transmission on topological charge may be understood through several interrelated physical factors that arise from LG beams’ spatial and phase structure. Firstly, higher-order LG modes exhibit an annular intensity distribution with a larger central dark core and an increasingly extended radial profile as the topological charge increases. This results in a redistribution of optical energy away from the beam center and toward the periphery. This spatial distribution may reduce the probability of interacting with scattering centers near the beam axis in highly scattering media, particularly in media where multiple forward scattering dominates.

Monte Carlo simulation results show the critical depth at which an LG beam carrying a particular topological charge loses its characteristic donut shape.

Secondly, a phase singularity and the helical phase front in LG beams may lead to suppressed backscattering, as the spatial coherence required to reverse the phase structure becomes increasingly difficult to maintain through random scattering events. While our current Monte Carlo simulations are not sensitive to such phase-based effects, the empirical trend of increased mean free path with increasing \(\:l\) supports the notion that the complex structure of the beam may inhibit scattering-induced decoherence22. Finally, from a geometric optics perspective, higher-order LG beams subtend a wider angular spread due to their larger ring radius. As the beam propagates through turbid media, this expanded cross-sectional area may result in a larger fraction of the forward-scattered photons remaining within the collection aperture of the detection system. This effectively enhances the number of “transmitted” photons that are not necessarily ballistic but still preserve some directional information, contributing to the observed transmission.

Together, these factors offer a plausible interpretation of why increasing topological charge correlates with enhanced transmission in diffuse and ballistic regimes. While a complete theoretical treatment involving wave-based scattering models or modal decomposition is required to understand the underlying physics fully, the observed trends highlight the enhanced resilience of beams carrying larger orbital angular momentum, consistent with previous experimental findings on the relationship between topological charge and the scattering mean free path. Such behavior highlights the practical importance of Laguerre-Gaussian modes in optical imaging and communication systems, where preserving beam integrity in turbid environments is crucial for achieving optimal signal fidelity and spatial resolution.

In conclusion, we present a study, using table-top experiments and simulations, into the propagation of Laguerre-Gaussian beams through optically thick turbid media. Our study systematically quantifies both diffuse and ballistic transmission of LG beams with varying topological charges, revealing that the scattering mean free path increases with increasing topological charge, thereby enabling these beams to preserve their characteristic donut shape over extended propagation distances. These findings offer promise for advanced optical imaging, spectroscopy, and free-space communication systems in complex scattering environments where maintaining beam integrity is crucial. Future work will extend these investigations to even higher optical depths, incorporate diverse scattering media, and refine the theoretical models to account for additional scattering phenomena. Moreover, at present, our study does not consider the detailed propagation of the beam’s phase, which may further reveal more information about the evolution of Laguerre-Gaussian fields in scattering environments. However, some insights can be drawn from17, which demonstrates the phase memory of orbital angular momentum (OAM) as twisted light retains its helical phase even in turbid, tissue-like scattering media. Additionally, exploring the polarization properties of Laguerre-Gaussian beams and their wavelength dependence could deepen our understanding of light-tissue interactions, driving advances in biomedical diagnostics and optical communication.

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

El-Sharkawy, Y. H., Elbasuney, S. & Radwan, S. M. Non-invasive diffused reflected/transmitted signature accompanied with hyperspectral imaging for breast cancer early diagnosis. Opt. Laser Technol. 169, 110151 (2024).

Article CAS Google Scholar

Gianani, I. et al. Transmission of vector vortex beams in dispersive media. Adv. Photonics. 2, 1 (2020).

Article Google Scholar

Lyons, A. et al. Computational time-of-flight diffuse optical tomography. Nat. Photonics. 13, 575–579 (2019).

Article ADS CAS Google Scholar

Yoon, S. et al. Deep optical imaging within complex scattering media. Nat. Rev. Phys. 2, 141–158 (2020).

Article ADS Google Scholar

Fredriksson, I., Larsson, M. & Strömberg, T. Machine learning for direct oxygen saturation and hemoglobin concentration assessment using diffuse reflectance spectroscopy. J Biomed. Opt 25, (2020).

Wang, X., Xie, X., Ku, G., Wang, L. V. & Stoica, G. Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic tomography. J. Biomed. Opt. 11, 024015 (2006).

Article ADS PubMed Google Scholar

Yoo, J. et al. Deep learning diffuse optical tomography. IEEE Trans. Med. Imaging. 39, 877–887 (2020).

Article PubMed Google Scholar

Balasubramaniam, G. M. & Arnon, S. Regression-based neural network for improving image reconstruction in diffuse optical tomography. Biomed. Opt. Express. 13, 2006 (2022).

Article PubMed PubMed Central Google Scholar

Gigan, S. Imaging and computing with disorder. Nat. Phys. 18, 980–985 (2022).

Article CAS Google Scholar

Bertolotti, J. & Katz, O. Imaging in complex media. Nat. Phys. 18, 1008–1017 (2022).

Article CAS Google Scholar

Wang, L. V. & Hsin, I. Biomedical optics: principles and imaging. https://doi.org/10.1002/9780470177013 (2012).

Article Google Scholar

Vo-Dinh, T. Biomedical photonics: handbook. https://doi.org/10.1117/1.1776177 (2003).

Article Google Scholar

Lindell, D. B. & Wetzstein, G. Three-dimensional imaging through scattering media based on confocal diffuse tomography. Nat. Commun. 11 (2020).

Balasubramaniam, G. M. et al. Tutorial on the use of deep learning in diffuse optical tomography. Electron 11, 305 (2022).

Article CAS Google Scholar

Katz, O., Heidmann, P., Fink, M. & Gigan, S. Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations. Nat. Photonics. 8, 784–790 (2014).

Article ADS CAS Google Scholar

Gbur, G. & Wolf, E. Spreading of partially coherent beams in random media. J. Opt. Soc. Am. A. 19, 1592 (2002).

Article ADS Google Scholar

Meglinski, I., Lopushenko, I., Sdobnov, A. & Bykov, A. Phase preservation of orbital angular momentum of light in multiple scattering environment. Light Sci. Appl. 13, 214 (2024).

Article CAS PubMed PubMed Central Google Scholar

Biton, N., Kupferman, J. & Arnon, S. OAM light propagation through tissue. Sci Rep 11 (2021).

Rosen, J. et al. Roadmap on computational methods in optical imaging and holography [invited]. Appl. Phys. B Lasers Opt. 130, 166 (2024).

Article ADS CAS Google Scholar

Suprano, A. et al. Propagation of structured light through tissue-mimicking phantoms. Opt. Express. 28, 35427 (2020).

Article ADS CAS PubMed Google Scholar

Perez, N., Preece, D., Wilson, R. & Bezryadina, A. Conservation of orbital angular momentum and polarization through biological waveguides. Sci. Rep. 12, 14144 (2022).

Article ADS CAS PubMed PubMed Central Google Scholar

Wang, W. B., Gozali, R., Shi, L., Lindwasser, L. & Alfano, R. R. Deep transmission of Laguerre–Gaussian vortex beams through turbid scattering media. Opt. Lett. 41, 2069 (2016).

Article ADS CAS PubMed Google Scholar

Li, Z. et al. Robust transmission of Ince-Gaussian vector beams through scattering medium. Optik (Stuttg). 257, 168766 (2022).

Article Google Scholar

Mamani, S. et al. OAM transmission of polarized multipole laser beams in rat cerebellum tissue. Opt. Commun. 532, 129241 (2023).

Article CAS Google Scholar

Balasubramaniam, G. M., Biton, N. & Arnon, S. Imaging through diffuse media using multi-mode vortex beams and deep learning. Sci. Rep. 12, 1561 (2022).

Article ADS CAS PubMed PubMed Central Google Scholar

Angelo, J. P. et al. Review of structured light in diffuse optical imaging. J. Biomed. Opt. 24, 1 (2018).

Article PubMed Google Scholar

Fu, S. & Gao, C. Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams. Photonics Res. 4, B1 (2016).

Article Google Scholar

Cheng, W., Haus, J. W. & Zhan, Q. Propagation of vector vortex beams through a turbulent atmosphere. Opt. Express. 17, 17829 (2009).

Article ADS CAS PubMed Google Scholar

Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A. 45, 8185–8189 (1992).

Article ADS CAS PubMed Google Scholar

Wang, J. et al. Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat. Photonics. 6, 488–496 (2012).

Article ADS CAS Google Scholar

Willner, A. E. et al. Optical communications using orbital angular momentum beams. Adv. Opt. Photonics. 7, 66 (2015).

Article ADS Google Scholar

Hohmann, M. et al. Direct measurement of the scattering coefficient. Biomed. Opt. Express. 12, 320 (2021).

Article PubMed Google Scholar

Michels, R., Foschum, F. & Kienle, A. Optical properties of fat emulsions. Opt. Express. 16, 5907 (2008).

Article ADS CAS PubMed Google Scholar

Wilson, B. C. & Jacques, S. L. Optical reflectance and transmittance of tissues: principles and applications. IEEE J. Quantum Electron. 26, 2186–2199 (1990).

Article ADS Google Scholar

Hohmann, M. et al. Measurement of optical properties of pig esophagus by using a modified spectrometer set-up. J. Biophotonics 11 (2018).

Fang, Q. & Boas, D. A. Tetrahedral mesh generation from volumetric binary and grayscale images. In Proceedings – 2009 IEEE International Symposium on Biomedical Imaging: From Nano to Macro, ISBI 2009 1142–1145 https://doi.org/10.1109/ISBI.2009.5193259 IEEE (2009).

Fang, Q. & Boas, D. A. Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units. Opt. Express. 17, 20178 (2009).

Article ADS CAS PubMed Google Scholar

Yuan, Y., Yan, S. & Fang, Q. Light transport modeling in highly complex tissues using the implicit mesh-based Monte Carlo algorithm. Biomed. Opt. Express. 12, 147 (2021).

Article PubMed Google Scholar

Download references

The authors thank the Israel Science Foundation (ISF) for funding the “Deep learning for multi-dimensional classification of complex structured light beams in optical wireless communication applications” project. The authors also thank the Kreitman School of Advanced Graduate Studies and the Ben-Gurion University of the Negev for providing fellowships to continue the research.

Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Be’er Sheva, 8441405, Israel

Ganesh M. Balasubramaniam, Gokul Manavalan & Shlomi Arnon

Search author on:PubMed Google Scholar

Search author on:PubMed Google Scholar

Search author on:PubMed Google Scholar

GMB developed the idea, performed the experiments and simulations, analyzed the data, and contributed to the preparation of the manuscript (initial draft, writing, review, and english corrections). GM assisted GMB during the experiments, analyzed the measurements, and contributed to the preparation of the manuscript (writing, review, and english corrections). SA led the project, advised how to design and perform the experiments, and contributed to the analysis of the results and manuscript preparation (review and corrections).

Correspondence to Ganesh M. Balasubramaniam.

The authors declare no competing interests.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.

Reprints and permissions

Balasubramaniam, G.M., Manavalan, G. & Arnon, S. Propagation of Laguerre-Gaussian beam intensities through optically thick turbid media. Sci Rep 15, 19768 (2025). https://doi.org/10.1038/s41598-025-03445-2

Download citation

Received: 08 March 2025

Accepted: 20 May 2025

Published: 05 June 2025

DOI: https://doi.org/10.1038/s41598-025-03445-2

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative